Thesis Arithmetic Concepts for Daily Life
The research carried out in this thesis tries to understand how children learn
some aspects of graphic representations in elementary mathematics and how
they face the challenges that these present. This will be a pending task to be
carried out, but for now, I prefer to consider this work as a continuity of said
research, now venturing into the writing and conceptualization of elementary
mathematics. with whom I have undertaken the task of venturing into issues
and theories that have been quite a challenge but at the same time a great
satisfaction. The influence of the work carried out by the members of this line
of research and mainly by its founder, Dr. Jorge Vaca, has been so important
that it is very difficult to determine which has been the purely original part
from which this work is carried out. research and also to identify the main
differences and continuities with which the consolidation of this branch in the
line of research begins as a natural part. It is important to express that this
thesis is the result of a collective work in which the members of the Written
Language and Basic Mathematics Research Seminar participated, as well as the
members of the tutoring committee and all the people who somehow intervened
so that this thesis could perform and be defended. For this reason I will use the
plural in the writing of this thesis, although that does not mean also sharing
the responsibility of what is written in this way. To all the members and
collaborators of the Written Language Research Seminar and In particular my
most sincere thanks: THANKS Machine Translated by Google To Dr. Gérard
Vergnaud, who received me at the Paris 8 University for an academic stay with
him. For me it was a great experience for three reasons: the first is that I
had the opportunity to present my thesis to him and get his opinion, he made
suggestions and he positively assessed my work; This is very significant to
me because he is the author of the theory on which I base this research. The
second reason is because he chose from among his facsimiles of his publications
those that he considered could help me on the subject raised and he gave them
to me, a detail that I value very much. The third reason is because, being an
academic who has contributed to the development of cognitive sciences and
specific didactics, particularly that of mathematics didactics in France and
whose reference is practically obligatory, he had the attention to listen to me,
guide me and make efforts. for establishing a pleasant and clear communication
and with a respectful and friendly treatment, his simplicity and humility are
for me an example that great theorists and scientists are also simple people
who love their work and like to share what they know with people who They
show interest in their work. 6 comments and suggestions to improve it. In a
special way I thank Verónica Aguilar, for her support in the revision of the
text in Spanish and translations from French and English; Francisco Martínez
for his support in field work and data preparation; Aracely Hernández for her
contributions to this work; Amanda Cano and Denise Hernández for their
suggestions; to Noel Xilot for his support in revising the text, to Edgardo
Domitilo and Eréndira Espinoza for their comments and Francia Gutiérrez and
Luis Gadea who at the time were members of the research line and the seminar.
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To the directors and teachers of the schools who must remain anonymous
but who allowed me access to their workspaces and mainly to the students
who kindly gave me the opportunity to learn how they solve a mathematical
problem with the additional effort to feel watched, and possibly evaluated,
while doing it. Thanks to them this thesis was possible. To the colleagues
who, at the time as officials of the Universidad Veracruzana, supported me
from their trenches to carry out the activities of the doctoral program: to Dr.
Ragueb Chain, who has supported not only my career in this postgraduate
program but has also been pending the good development Machine Translated
by Google Thank you all. • Dr. Mario Miguel Ojeda 7 To Lic. Alan Pérez
for his support in computing matters. • Dr. Alejandro Gómez To Dr. Alicia
Ávila as editor of the Educación Matemática magazine for her support and
understanding, mainly regarding the bureaucratic aspects of this program. •
Dr. Gustavo Martínez Sierra To the Mtra. Martha Romero for her support
in managing access to the schools where the field work was carried out and
for her comments on the thesis. of this thesis and of my academic life; to
Dr. Miguel Casillas who, as director of the Humanities Area, supported me
to carry out a stay in France; to Dr. Mario Miguel Ojeda who, as general
director of the Postgraduate Studies Unit, contributed so that he could reach
the end of this project and to Mtro. Juan Carlos Ortega, PhD fellow and
colleague who gave me his support in difficult moments. To the Master Héctor
Merino and Lic. Cynthia Palomino for their support from the CPUe magazine
editorial. • Dr. Abraham Cuesta • Dr. Rosa del Carmen Flores To Lic.
Guadalupe Zárate for her support in the administrative procedures. To the
Master Francisco Sánchez for his support to access one of the secondary schools.
To the readers of this thesis and members of the jury whose contributions
and criticism allowed us to improve this thesis, in alphabetical order: Machine
Translated by Google 8 Chapter IV reports the results of the first phase of
empirical exploration, which constituted the inputs for the publication of a
research article (curricular requirement of the doctoral program, Bustamante
and Vaca, 2014)1 and in Chapter V we perform the microgenetic analysis
. of a case that was the basis for the preparation of the second required
article (Bustamante and Flores, in press). The didactics of mathematics, in
the context of Mexican basic education, has been oriented mainly towards
teaching itself, through the design and application of increasingly refined
didactic situations on the different mathematical contents. We believe that
this didactics could be complemented with the type of work that we propose,
more focused on the psychology of learning. Our results show the need to
make explicit the ideas that students build about mathematical content, about
their symbolic representations and about their procedures that could be useful
to improve the design and development of situations. Chapter II presents the
theoretical references that frame the research, namely: the theory of conceptual
fields, the operational theory of representation and the approach to situated
microgenesis. Although we recognize that there are multiple research paradigms
in the field of educational mathematics, we limit ourselves to this conceptual
framework. Chapter VI analyzes 30 clinical interviews conducted with a
2
selected group of students who participated in the previous phase. In Chapter
I the research work is introduced, the problem statement, the justification and
its background are exposed. Of these, the works of the Written Language and
Basic Mathematics line stand out : acquisitions, practices and uses, a research
group from the Universidad Veracruzana that in recent years has explored
the psychological, sociological and anthropological aspects of the acquisition
and use of both written language as well as mathematics, in basic education
students. This thesis seeks to understand in greater depth the role played
by the various representations mobilized by students to solve a mathematical
problem. Finally, in this chapter we review the curriculum by competencies in
basic education and the didactic approach that locates problem solving as the
axis for the teaching of mathematics and as a tool for evaluation and research
in educational mathematics. In Chapter III we present the methodological
tools with which the empirical work was carried out: the experimental design,
the clinical interview and the microgenetic analysis. We describe the empirical
referent and the phases and characteristics of data collection. SUMMARY 1
For the format of the references we have adhered, as far as possible, to the
suggestions of The APA Publications Manual, 2010, 3rd edition in Spanish.
Machine Translated by Google The work also made it possible to identify
the need for students to differentiate representation systems and reconstruct
their own rules and conventions in order to be able to use them properly.
Only in this way can they assume the function of tools capable of supporting
reasoning, promoting reflection and helping to maintain control of the activity
during the resolution process, at the same time that they support working
memory and, finally, enable adequate communication. of the procedures used
and the results obtained. Without this clear differentiation and subsequent
correspondence, instead of working as useful tools for solutions, the diversity
of representations becomes a source of confusion that hinders or hinders said
solutions. The results obtained in this research indicate that language and
writing allow students to make explicit theorems and concepts-in-act, put
them to the test and eventually be able to direct their modification towards
more conventional forms. 9 In the last section we present the conclusions and
final reflections. Furthermore, the translation, made by the thesis student,
of the last publication that we know of on the theory of conceptual fields
proposed by Gérard Vergnaud, is also included as an annex, to contribute to
its dissemination. Machine Translated by Google In Chapter VI we analyze
30 clinical interviews with a selected group of students who participated in
the previous phase. Teaching mathematics, in the context of the Mexican
basic education, has been primarily oriented toward teaching itself, through
the design and implementation of increasingly refined teaching situations on
different mathematical contents. We sustain that this approach could be
enriched by the kind of work we propose here, more focused on the psychology
of learning. Our results show the need for specialists to become aware of
the ideas that students build on the mathematical content, their symbolic
representations and procedures, because that could be useful to design and
improve mathematical situations. In Chapter II we expose the theoretical
3
framework of the research: the theory of conceptual fields, the operational
representation theory (La théorie opératoire de la répresentation) and the
situated microgenesis approach. Although we recognize that there are multiple
research paradigms in the field of mathematical education, we limit ourselves
to that conceptual frame. 10 In Chapter I the research is introduced and the
problem statement is expressed as well as the justification and background.
It highlights the work of the program research Written Language and Basic
Mathematics: acquisitions, practices and usages, from which this work derives.
In recent years, this research group of the Universidad Veracruzana has explored
psychological, sociological and anthropological features of the acquisition and
use of written language and mathematics in the basic education. This thesis
seeks to understand in depth the role of the representations mobilized by
students to solve a problem. Finally, in this chapter the competency-based
curriculum in basic education is reviewed, as well as problem solving as a hub
for the teaching of mathematics and as a tool for evaluation and research. In
Chapter III we present the research design and methodological tools, which the
empirical work was performed with: the experimental design, clinical interview
and microgenetic analysis. We describe the empirical referent and the stages
and characteristics of the data collection. Chapter IV reports the results of
the first phase of empirical exploration. These results were the inputs to the
publication of a research article (curricular requirement of the doctoral program,
Bustamante and Vaca, 2014). In Chapter V we make the microgenetic analysis
of a case, which was also the basis for the development of the second article
required (Bustamante and Flores, in press). ABSTRACT Machine Translated
by Google In the last section we present the conclusions and final thoughts.
Finally, we annex the PHD student’s translation of the last publication that we
know about the Theory of Conceptual Fields written by Gérard Vergnaud, in
order to divulgate it. Results obtained in this research indicate that language
and writing help make explicit the concepts and theorems-in-action of the
students, to test them and eventually orient their transformation toward
more conventional forms. 11 The work also identified the need for students
to differentiate the representation systems and rebuild their own rules and
conventions, in order to use them properly. Only in this way those systems
can assume the role of tools that support the reasoning, encourage reflection
and help maintain control of the activity during the resolution process, while
supporting working memory and finally make the proper communication of
the procedures used and the results obtained. Without such clear differentiation and subsequent correspondences, instead of working as useful tools
for solutions, the diversity of representations becomes a source of confusion
that complicates and blocks those solutions. Machine Translated by Google
CONTENT I.1.1 Readers and their contexts............................................
....................................... twenty I.3.2.2 Troubleshooting, a means of evaluating competencies..................................
39 I. CHAPTER I. Introduction............................................ ......................................... 19 58 III.2.1 Procedure ................................................ ................................................................
....72 I.3.2 The competency-focused model ................................................
4
............................ 35 II.3.3 Theorems and concepts-in-act .......................................................
.................................... 54 III.1.5 Application of the instrument............................................
.................................... 69 II.3.1 The conceptual field of multiplicative structures.................................................. 49 I.3 Rationale...............................................
................................................................ ................. 29 III.1.3 Tools...................................
................................................
.......
67 II.2 Mathematical notation
.................................................. .................................................. 46 III.1.1
Problematic situation............................................ ...................................... 58
I.1.2 Main aspects observed in the students’ procedures.......... 25 CHAPTER III.
Methodology .................................................. .......................................... I.1
Background ............................................................. ..................................................................
................ 20 CHAPTER II. Theoretical framework..................................................
...................................... 42 III.3 The clinical method ................................................
................................................................ ........ 73 I.3.2.1 Life skills..................................................
.................................... 36 II.3.4 Linguistic signifiers..................................................
........................................... 56 III.2 Clinical interviews...................................................
................................................................ .....71 III.1.4 participants...................................................
................................................
.......
68 I.3.1 Competences in the latest educational reforms.................................................
..33 II.3.2 Scheme
.....................................................
..................................................................
.............
53 III.1.2 Analysis of the problem ............................................
...................................... 60 I.2 Research questions .....................................................
......................................... 27 II.3 Theory of conceptual fields..................................................
....................................
48 12 II.1 Operational theory of representation …………………………………….
.............................
44 III.1 Methodological
design ...................................................
..................................................
.58 Machine Translated by Google IV.2.1 Results by response categories.................................................. ...................... 89 IV.1 General results................................................... ................................................................
..82 V.1.2.3 Partial closing............................................ ................................................................
139 VI.1.2.1 Writing with directionality of the alphabetic system..................................
170 VI.1.4.2 Empirical feasibility ............................................ ....................................
178 V.1.2.1 First phase............................................ ................................................................
... 110 SAW. 1.1.3 Autonomous verification..................................................
............................ 169 VI.1.4 Result ........................................ ................................................................
.............
176 V.1.1 Description of the case................................................
................................................ 108 VI.1.1.1 Figure of the problem: proportional relationship .................................. ...... 155 VI.1 Results according
to the identified themes................................................ ...................... 155
VI.1.2.4 Dividend greater than the divisor .................................. .......................
172 VI.1.3.1 Autonomous assignment of units ................................................
......................
174 IV.3 Conclusions from the experimental phase results................................................. ......... 102 CHAPTER V. Results of the
microgenetic analysis................................... ................... 106 IV.2 Difficulty
analysis ................................................... ............................................ 89 V.2
Conclusions of the analysis................................................ ............................................
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150 VI 1.2.2 Calculator, orality/writing ........................................ .........................
171 13 V.1.2.2 Second part of the interview............................................
............................ 129 III.4 Situated microgenesis.................................................
.................................................... 75 CHAPTER IV. Results of the experimental phase................................................... ............ 82 VI.1.2 Division
algorithm..................................................
.......................................
169
VI.1.4.1 Relationship between the symbolic plane and the concrete plane
.............................
176 V.1.2 Microanalysis...............................................
................................................... ..... 109 VI.1.1.2 Correct and autonomous
resolution...........................................
......................
168 VI.1.3 Units
............................................................. ..................................................................
............. 174 VI.1.3.2 Carrying the decimal fraction to the unit .....................................
......... 174 V.1 Andrea’s case ................................................ ................................................................
.......
107 CHAPTER VI. Results of the analysis of the clinical interviews.................................. 153 VI.1.1 Problematic ..............................................
................................................ .......... 155 VI.1.2.3 Mechanics of the algorithm ..........................................
....................................
172 Machine
Translated by Google CONCLUSIONS ................................................
................................................................
.......183 14 VI.2 Conclusions of
the chapter............................................ ............................................ 181 References................................................. ................................................................
.................... 190 Annex 1. Translation of Gérard Vergnaud’s theory of
conceptual fields. .... 201 Machine Translated by Google Table 8. Distribution
based on the type of magnitudes................................................. ..88 15 INDEX
OF TABLES Table 7. Distribution according to the versions with or without
distractor ..................................... 87 Table 9. Distribution of results by categories..................................................... .......... 89 Table 16. Issues related to
the category Result ................................................ ............. 176 Table 4. Distribution of the students clinically interviewed..................................... 154 Table 2. Results by sex................................................. .........................................
82 Table 11. Classification of data from clinical interviews..................................................
153 Table 1. Distribution of the ages of the students..................................................
..........69 Table 10.
Distribution of the Sub-categories of Category
6................................................... 98 Table 4. Distribution of the results
by type of locality.................................................. 84 TABLE INDEX Table 13. Subclasses of Figure of the problem..................................................
...................... 156 Table 3. Results by educational level..................................................
............................. 83 Table 12. Topics related to the Problem category
........................................... ........ 155 Table 2. Distribution of students
by gender and school.................................................. ..68 Table 6. Results
according to the version of the problem............................................ ..............
86 Table 15. Issues related to the Units heading..................................................
.............. 174 Table 3. Distribution of groups and calculator availability ........................................... 70 Table 1. Secondary school curricular
map............................................ .................................. 32 Table 5. Results
depending on the calculator. .................................................. ....... 85 Table
6
14. Issues related to the Algorithm category of the division .............................
169 Machine Translated by Google Image 30. Inverts only the numerals of the
division but not the units. ................. 132 Image 3. Small steps (we had to
provide him with an additional sheet, which is tied here, so that he could finish
his drawing). .................................................. .. 25 Image 1. Elizabeth’s procedure. ............................................................................ 23 Image 10. Example
of resolution of category 5. ....................................... ................. 97 Image
19. Division that Andrea considers to be wrong................................................
.......118 Image 28. Say 70 divided by 35 and write 35/70....................................
............................ 127 Image 37. The first item of information stated is
not always the dividend.................................. 143 Image 8. Example of
resolution of category 4. ........................................... ..................94 Image
17. Andrea’s productions in the experimental phase. .......................................
108 Image 26. 3500/70= 50 cm. ................................................................
......................................... 124 Image 33. Awareness of the organization
of data in the division..................... 134 Image 35. Division of 5 children
between ten bars........................................... .................... 136 Image 6. Example of resolution classified in category 2.................................................
.92 Image 15.
Another example of resolution of Sub-category 6 d.
....................................
100 Image 13.
Example of resolution of Subcategory 6 c................................................ .........99 Image 22. Sum of the
components of a correspondence table. ...................... 120 Image 24. Obtains
50 or 500 as a quotient.................................. ......................... 122 Image
31. Difficulty in assigning units to the quotient. .........................................
133 Image 4. Steps. Item released from the PISA 2003 test. Taken from
INEE, 2013... 59 Image 2. Estimated result. ..................................................
....................................... 24 Image 11. Prototypical resolution of Sub-category
6 a..................................... ........ 98 Image 20. Complement the quotient
with units of measurement. .................................... 119 Image 29. Conventional writing of the division. ................................................................
.......130 16 Image 9. Modification of the text of the problem due to an
incorrect result..................... 96 INDEX OF IMAGES Image 18. Correspondence table................................................. ................................. 113 Image
27. Check 50*70= 3500..................................... ................................. 125
Image 36. Division of 20 marbles among 40 children..................................
..................136 Image 7.
Example of resolution located in category
3................................................. ......94 Image 16. Example of resolution of
Sub-category 6 e.................................. ....... 101 Image 23. Division 3500 cm by
70................................................ ................................. 121 Image 25. Division
3500/70................................................ .................................................. 123
Image 32. Conventional reading of the division..................................................
...................... 133 Image 34. Division of 3 chocolate bars between 2 children.
......................................... 135 Image 5. Problem A, Brandon aged 11 years
8 months (11.8) primary school A (urban), no calculator ..........................
................................................................ ......................................... 90 Image
12. Example of resolution of Sub-category 6 b.................................. .........99
7
Image 14. Example of resolution of Sub-category 6 d..................................
....... 100 Image 21. Relationship between the algorithm and the correspondence table............................ 120 Machine Translated by Google Image 49.
What is meant by a step........................................... ............................ 165
Image 43. Division carried out under the theorem-in-act, the meters are
located in the dividend................................. ..................................................
......................................... 149 Image 48. Successive multiplications to find
the quotient. ................................. 164 Image 50. The word average refers
to measurement and size to an intensive quantity. Image 42. Iconic representation of 20 pesos among 10 children. .................................... 148 Image 45.
From iconic representation to the rule of three..................................................
.......160 Image 51.
Mathematical writing with alphabetic directionality..................................
171 Image 44.
Division carried out under
the theorem-in-act:
the first item of information is located in the
dividend..................................
................................................................
..................................... 150 .................................................................................................................................
167 Image 47. Representation of the relationships of the problem through
a number line. Image 38. 20 minutes divided by 60 seconds is equal
to 0.3 hour and this is equal to 3 min... 143 Image 53. Shows the approximate distance of one meter. ............................................ 179 Image
46. Yéssica’s response obtained from a process of deduction and mental
calculation.....................................
................................................................
............................................ 161 Image 52. Gesture to indicate the size of
a step. ................................................................ ..177 Image 40. Graphic
representation of three tenths of an hour. ......................................... 145
.........................................................................................................................................
17 Image 41. If a tenth of an hour is 6 minutes then 3 tenths are 18.....................
146 Image 39. Adjust the ratio as required. ................................................................
......... 143 163 Image 54. Shows the approximate distance of two meters.
......................................... 180 Machine Translated by Google 18 Figure 6.
Diagram of relationships of the linear function............................................
............... 65 Figure 1. Representation plans (taken from Vergnaud, in press).
............................ Four. Five Figure 3. Scheme for defining a step. Source: self
made. ............... 62 Figure 7. Diagram of relationships operating on the coefficients of proportionality. . 66 Figure 2. Diagram of relationships corresponding
to the problem of steps. ........................ 60 Figure 4. Diagram of vertical
relationships, with scalar operators. ............................ 63 Figure 8. Scheme
of relationships by scalars.................................................. ...................... 112
Figure 5. Scheme of horizontal relationships, with proportional function
operators..... 64 INDEX OF FIGURES Machine Translated by Google 2 I.
CHAPTER I. Introduction It is important to clarify that the term capacity is
used here and in the rest of the document as the possibility that the subject
has to build schemes based on the situations they face. There will be situations
in which the subject may not have schemes to face them successfully, but that
does not mean that they cannot develop or enrich them. Our interest is focused
on the conceptualization processes that imply mathematical knowledge in the
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process of construction, and that are indirectly observable through verbal
expressions and graphic representations made during the process of solving a
mathematical problem. We recognize the importance of research of a cognitive
nature to understand where the obstacles are found during the resolution of
problems and deepen the understanding of the conceptualization processes
of the problems. For this reason, this research focuses its attention on the
learning process, on the student more than on the teacher and their teaching,
highlights the mechanisms by which students face a problematic situation, the
way to mobilize their mathematical tools to solve the problem. problem, the
difficulties they experience and the means by which these difficulties might be
overcome. To do this, we proposed to carry out a study that would allow us
to deepen our understanding of the resolution procedures of students in the
sixth grade of primary school and third grade of secondary school. Through
the clinical interview, we seek to infer the statements and concepts evoked
during the resolution process that could interfere with reaching an expected
result. We believe that the presence of mistaken ideas or incorrect procedures
does not mean that students do not have the capacity2 to solve problems,
but rather that in the resolution process when they are not mastered at a
level necessary to function as tools to favor the resolution process, they could
even complicate it. The solid construction of such knowledge requires various
situations and in the long term, which will allow students the possibility of
using them appropriately. mathematical knowledge. 19 Machine Translated by
Google I.1.1 READERS AND THEIR CONTEXTS I.1 BACKGROUND In
mathematics, some problematic situations were explored in order to identify
variations in the resolution procedures, differences in the graphic representations used and in the application of mathematical knowledge depending on the
educational levels and sociocultural contexts in which the students were. The
purpose of the exploration was to know how students face one of the items
applied in one of the PISA tests. If the results reflect a low score, then we were
interested in knowing specifically where the obstacles that prevented students
from adequately solving an item were located. 20 The object of study was
built from a first approximation to the contexts in which students in the last
grades of primary, secondary and high school from 4 locations in the state
of Veracruz develop, through the research project Readers and their contexts
. (Vaca et al., 2010) whose objective was to investigate the specific weight
that the school, the family, the locality, the classes they receive and their own
cognitive processes have in the learning of reading and mathematics. In the
procedures for solving the problems, interesting aspects were observed that
require a more in-depth study for their understanding: differences between
the graphic representations of the same concept, variation in the levels of
conceptualization of mathematical knowledge necessary for the resolution of
the tasks. proposals and various positions shown by the students before the
applied problems. Since these items are designed for students who are 15
years old on average, we required a problem that could be solved by students
in sixth grade and third grade in secondary school. For this reason, we set
ourselves the task of building one with the same theme as PISA but without
9
the algebra requirements, a content that is not worked on by primary school
students. Machine Translated by Google Version A was discarded because we
wanted to observe what students were doing with data that required filtering
(”by the minute”) and operate on the relevant data. your steps? Hector walks
35 meters taking 70 steps per minute. What is the size of each of his steps?
We wondered who would take it into account, who would not, what obstacles
Hector takes 70 steps per minute and advances 35 meters. What is the size
of each of his steps? The second version (B) was also discarded because we
consider that it presents two pieces of information (the speed in steps per
minute and the distance traveled) that can be interpreted as independent of
the unknown: the size of each one of the steps. Hector walks 35 steps taking 70
steps per minute. What is the size of would represent. c) The research had a
phase where the instruments were tested through a pilot study. Three problem
proposals with some variations in wording were made and tested: 21 b) At
the time, it was considered that the chosen version allowed the ”additional
data” to be maintained and by including the word ”giving”, the decoupling
of the data with the unknown was resolved. However, after collecting the
data and obtaining and Hector walks 35 meters taking 70 steps, what is the
size of each one? Based on the data collected from the pilot study, version
”C” of the problem was chosen as one of the instruments to apply it to all
the research subjects (110): of his steps? a) Synthesis of the research results
Readers and their contexts concerning the resolution of the mathematical
problem Machine Translated by Google Notwithstanding the foregoing, the
results obtained with this ”defective” problem and with interviews carried
out by several participants, some without experience in clinical interviews
(students who mainly did social service), important veins were found to deepen,
now more clearly and clearly. controlled by a single interviewer, the thesis
student. The step problem mentioned above was applied to the entire sample
(110 students). Although an important part of the interest of the authors was
to know the procedures, analyzes were also carried out based on the results
they arrived at. Only a quarter (24.5%) solved it correctly and autonomously
(without the interviewer’s support), including those who argued that it could
not be solved; 46.4% managed to reach a satisfactory answer, but with some
intervention from the interviewer, while 29.1% could not solve it yet and with
the support of the interviewer. Let us remember that this problem was applied
to the last grades of the three educational levels: primary, secondary and high
school. For example, Brenda, who studies the third grade of telesecundaria
in a rural town, says orally that she divided 35 by 70, but in the graphical
representation The approaches made were classified for analysis into 4 types:
adequate (which include operations that can lead to the expected result as
a rule of three and division with the correct data), division (but with data
problems), other operations, and no written operations. Most of the students
of the three educational levels opted for the division algorithm as the main
resolution tool, but a large part of them also inverted the data in the written
representation. Some interesting examples are described below (Vaca et al.,
2010) where the aforementioned problematic situation was applied individually
10
and at the same time clinical interviews were conducted with each of the
students. To share some results, we received comments from mathematics
didactics specialists in which they expressed that the problem was not well
written, that in those terms it was unsolvable because it required making at
least two assumptions: that the distance is covered in one minute and that the
average step size was sought. 22 Machine Translated by Google Would your
step point five centimeters? [Five centimeter point? How much would five
centimeter point be?] Five point... oh no, in millimeters that says [millimeters?
and that would be how much, outlining it] for me they would be small, right?
have been one foot on the other foot and so, right? [Aha, do you think that’s
the case?] I say yes, I can’t find any other way. Another similar example (See
Image 1) is given by Elizabeth, who was a telebaccalaureate student. The result
you get is 0.5 and you doubt which unit of measurement you should use. which
you do from left to right, first write 35 then draw the little house and then write
70 (as a dividend); For this reason, the data is inverted: 70/35 and therefore
obtains a result of 2. It will be necessary to delve into this aspect during the
clinical interviews to find out the characteristics of these interactions and what
causes this type of response in both primary school students like middle school
and high school. Here we can appreciate the convergence of the aspects already
mentioned and that are interesting to explore in greater depth in this thesis:
the variations between the symbolic systems, that is, between what is expressed
verbally and what is effectively written, the reflections on their own written
representations and its relationship with mathematical conceptualizations and
graphic conventions. It first interprets the result as 0.5 centimeters and then
as 0.5 millimeters, which also explains the feasibility of the result if you walk
with one foot on top of the other foot. The following snippet shows how a
result is misinterpreted by a problem reading its own writing, specifically the
decimal point. 23 Image 1. Elizabeth’s procedure. Machine Translated by
Google Unlike María José, Marcos does consider the data of the problem and
also right of the 35, think for a moment and write in the quotient 5 then in
another space multiply 70*5= 350, go back to the division and multiply the 5
by 70 and write in the remainder 00, add a point to the left side of the quotient
( .5) and asks:) Point five? Could it be .5 millimeters? On the other hand,
we find students who use solution strategies that obey schemes activated by
having represented the problem in a way that is divergent from the expected
or non-canonical in terms of Flores (2003). makes a correct interpretation of it,
expressed iconically (See Image 3). We have also found it with other students,
who, when obtaining a decimal point in the quotient, consider that if the unit
is fractional, the referent used must also be. If we are talking about meters,
then it cannot be 0.5 meters but 0.5 centimeters or in this case 0.5 millimeters.
24 María José, for example (See Image 2), thinks about the size of her own step
and makes an estimate of its measurement. It does not take into account the
data of the problem and therefore does not carry out any arithmetic approach.
It has been considered that when a problem goes beyond the students’ schemes,
they can give up the problem or, as in this case, move away from conventional
resolution procedures. [I tell you something? if what you want is to divide
11
the 35 meters by 70 steps, divide that distance by 70 then, 35 goes in and
70 goes outside if that’s what you want to do] (He laughs and writes 35/70,
whispers and adds a point and a zero Image 2. Estimated result. Machine
Translated by Google STUDENTS I.1.2 MAIN ASPECTS OBSERVED IN
THE PROCEDURES OF THE Image 3. Small steps (we had to provide him
with an additional sheet, which is tied here, so that he could finish his drawing).
• Contradictions between representation systems (oral and written) 25 Marcos
manages to graphically represent the problematic situation but fails to propose
a mathematical calculation procedure to deal with the situation. He represents
the action of walking through the drawings, taking care of the one-to-one
correspondence between the jumps drawn and the number of steps mentioned
in the text of the problem. Where he did not take care of that correspondence
was in the line that represents the distance traveled. After completing his
procedure, he concludes that Hector’s step size is “small”. Here we observe
contrasts between the initial representation of the problem (figure in terms
of Gréco and understanding scheme according to Flores, 2003), the drawing,
and the applied resolution scheme. Choose a procedure that leads to intensive
quantification. employed by the student. With the above, a series of resolution
procedures have been shown that obey various schemes, examples of how the
problem is assimilated to the students’ schemes and also represents a certain
familiarity that motivates them to solve it. • Problems with the unit of mean
in quotients less than one. The previous examples show some aspects that are
addressed in this research: • Reflection on the feasibility of the result obtained.
Machine Translated by Google relation to other writing systems Those who did,
were able to solve the problem, even mentally. Due to the breadth of topics to
be explored in the research ”Readers and their contexts” and the methodology
used for these purposes, in addition to the participation of students who
were doing their bachelor’s social service or who were studying their master’s
degree, with little experience and lack training in clinical interviews, in said
investigation we were not able to fully elucidate the reasons for the failures in
resolution; In addition, at that time we did not have the necessary theoretical
tools to be able to build the observables necessary to interpret the data
collected. The realization of this thesis constituted the framework to deepen
the understanding of the difficulties of solving this problem and to know in
depth Vergnaud’s theory of conceptual fields, which seemed pertinent and
coherent with our purpose. • Evidence of lack of practice of algorithms and
multiplication tables, mainly in secondary and high school, perhaps due to
the use of the calculator. • Mathematical tools activated during the solution
of the problem in • Arbitrary use of the decimal point. • Partial knowledge
of the division algorithm. function of the initial scheme of the problem
(figure). 26 • Difficulties with the graphical representation of the division
algorithm and its • Difficulties in the analysis of the relationship between
the quantities 35 and 70. • The non-canonical representation of the problem
and iconic solution procedure with a result with intensive magnitudes such as
”grades” or ”small” despite being in a school context. Machine Translated by
Google I.2 RESEARCH QUESTIONS students during the resolution of the
12
problematic situations raised? we explore others with less intensity. We mainly
concentrate more on treating graphic representation and conceptualization
of the structures involved in the • What are the knowledge underlying the
symbolic representations of • What relationships do students establish between
alphabetic writing, students during clinical interviews, we delve into some
topics and To do this, we aim to explore the characteristics of the relationship
between oral language, multiplicative by primary and secondary students from
rural contexts and • How does the use of the calculator influence the resolution
of written algorithms, 27 resolution of a problem (with four variants) of the
conceptual field of structures of the division algorithm and oral language?
• How are the relationships between symbolization and conceptualization
characterized? • What are the conceptual implications of the distinction
between 35/70 and 70/35 per urban. In addition to some specific questions
derived from this formulation: particularly with the division this small group of
problems? of mathematical problems that cause low scores in the results of the
concept of multiplicative structures? Notwithstanding the foregoing and as the
investigation progressed, when working with the evaluations, what specifically
these difficulties consist of. Initially we ask ourselves if the students experience
difficulties in solving in the procedures for solving a problem belonging to
the field part of the students? Machine Translated by Google outsourcing,
but we agree with the statement that mathematical knowledge 28 own rules.
concepts, to their reorganization, to reflect on them, to the awareness and
It is built through action and not through language. Piaget would say: it
is necessary but In addition to the previous questions, the availability of a
calculator will be included as a variable in this investigation. As already
documented, in the PISA test applications even officials give free calculators
to students in the hope that they will improve their results. Likewise, in the
Plan and study programs of basic education it is suggested the incorporation
of the new The latter has not been differentiated from other writing systems
and its We conceive that the domain of the mathematical representation
system is It is important to underline, even if it is obvious, that oral language
is a system Therefore, the objective of this study is to try to understand
in greater detail the difficulties faced by basic education students when
facing a mathematical problem: the relationship of these difficulties with the
representation systems and with the level of conceptualization of the notions.
involved in the problem. not enough. so to master this representation system
it is also important to have to understand the relationship between systems of
representation and conceptualization When the student says ”I am going to
divide 35 by 70” he is speaking and is expressing a closely linked to the level of
construction of the mathematical concepts represented of representation and
that its adjustment or correspondence with other systems is substantial. The
hypothesis under which this work is carried out is that the characteristics of
the system has writing in the process of conceptualization; helps to identify
the correspondence between systems. writing graph interfere with those of
the mathematical notation system when it mathematics. consolidated the
corresponding meanings. However, we recognize the role that relationship;
13
when he writes it, he sometimes reverses the relationship, which shows the
lack of Machine Translated by Google We consider that some difficulties
that students experience are related to the way in which they interact the
different representation systems involved in the resolution process, specifically
alphabetic writing, oral language, mathematical notation, in addition to those
linked to the level of conceptual development, which they have achieved, of
the mathematical notions involved. 29 technologies in the learning processes,
including calculators, but in the application of tests such as ENLACE they
are prohibited (Vaca et al., 2010, pp. 93 and 946). Due to such actions, we
wanted to explore more systematically if the use of the calculator is truly a
support for more students to give a correct answer to the proposed problem.
Low scores have been reported in the results of standardized tests that purport
to assess the competencies of Mexican students in mathematics, even in the
national press (Avilés, September 8, 2009). For example, the Ministry of
Public Education (2013) reports that the national average in the 2009 PISA
test is 419 points compared to the Organization for Economic Development
Cooperation (OECD) average of 501 points. Due to this, the interest arose,
on the part of the applicant, to carry out a doctoral thesis that would allow
knowing in detail what happens when students, between 11 and 15 years old,
face a problem in the conceptual field of multiplicative structures. If they
fail in their responses, be able to specifically observe and describe what they
are failing at, and to the extent possible be able to understand and explain
those failures. This study has as its main background the research Readers
and their contexts (Vaca, Bustamante, Gutiérrez, Tiburcio, 2010) in which the
sociocultural conditions of basic education students in some locations in the
state of Veracruz are analyzed to search for relationships between their family,
school and cultural contexts, and the way they deal with situations of written
language and mathematics. In this last case, the authors identified that some
of the difficulties experienced by students could be related to the linking of the
different symbolic systems that intervene in the resolution processes, as well
as with the I.3 JUSTIFICATION Machine Translated by Google (PRONAP),
now the General Directorate of Continuous Training of Teachers in Service.
Therefore, we set out to explore the relationships between the different forms
of symbolization that students use while solving an arithmetic problem and the
mathematical concepts that they mobilize during its resolution. We decided to
consider the solution to a single problem, since our interest is to deepen the
deductions and inferences that students make, as well as their calculations, and
in particular relationships between different forms of symbolization without
prioritizing the generality of knowledge in different mathematical situations.
and in a complete conceptual field. Due to the fact that the incorporation
of the so-called ”modern mathematics” in the educational systems in several
countries did not have the expected results (Kline, 1988), in Mexico the 1993
reform was proposed, which incorporates the results of the constructivist
investigations carried out in the which at first was called the DIECINVESTAV
Psychomathematics Laboratory, in whose works I. Fuenlabrada and D. Block
take up the contributions of the French school, mainly from the currents 30 If
14
we start from the idea that mathematics is a body of knowledge that can help
people better deal with the various activities and contexts in which they operate,
we would also have to recognize that most of us find it difficult to do a practical
application of the mathematical knowledge learned during our school training
(terminologies, concepts, formulas, operations, definitions, equivalences, etc.)
in non-school problems. It seems that this difficulty is now minimized thanks
to the technology with which we access computer and information programs
that allow us to resolve the issues of daily life. constructivists of knowledge,
mainly from Brousseau (Fuenlabrada, 2007; Fuenlabrada and Block 2009).
The processes for incorporating new proposals are generally long-term. It
has been mentioned that even after 10 years of said reform there were still
indications that many teachers continued to work in a traditional way without
accepting or fully learning about the new proposal, despite the efforts of the
SEP and specifically the Undersecretariat. of Basic Education that launched
the National Program for Permanent Updating of Basic Education Teachers in
Service weak construction of some mathematical concepts. Machine Translated
by Google Gérard Vergnaud (Brousseau, 1994) Brousseau has expressed the
retos that the didactic of This approach has been developed in research in
Mathematics Education as well as in curricular development projects, for
more than two decades, in various parts of the world, these studies have been
carried out from a constructivist conception of learning (p. 2) mathematics
had to overcome, among which stand out: the training of teachers, necessary
conditions for the student to interact with the learning objects and mobilize
strategies for solving problems with their own resources, while the teacher’s job
consisted, in principle, in guiding the student so that, based on spontaneous
strategies and resources individuals, could build more conventional notions
and knowledge through the prepared didactic situations. important efforts to
improve teaching and learning trying to incorporate the results of research on
learning and teaching mathematics, for example in Mexico from institutions
such as the Department of Educational Research and the Department of Educational Mathematics, belonging to However, currently there is a new curricular
reform, new textbooks and a curriculum now based on the development of
competencies with the justification that previous reforms have not resolved
the problems: school failure, lack of application of mathematical knowledge
in real problems of daily life and currently the low results in the different
evaluations both nationally and internationally. Of the latest curricular
reforms, at least in 1993, to the Center for Research and Advanced Studies of
the National Polytechnic Institute (CINVESTAV-IPN). Fuenlabrada and Block
(2009) express it as follows: excessive criticism of the results of didactics, false
ideas about the This is not exclusive to Mexico. In the Tribute made to Guy
Brousseau and 31 The methodological approach proposed by the new Plan
and programs for Basic Education published in 1993 locates, […], problems
as the core of learning around which teaching is organized. The educational
reform of 1993 guides teaching efforts to create the Machine Translated by
Google Table 1. Secondary school curricular map. Note: Source: (SEP, 2006,
p. 31). specialized both for the training of mathematicians and of didactic In
15
the basic education curriculum in Mexico (Secretaría de Educación Pública,
2009), mathematics has an important space along with Spanish and science.
For example, for fourth, fifth and sixth grades of primary school, 200 hours are
granted annually (although in practice this does not necessarily happen), which
implies a considerable scheduled time for working with the selected contents.
didactics He considered that they were on the right track to overcome them,
creating institutions mathematics and teachers In secondary school, the time
allocated to mathematics is the same as Spanish and science takes on greater
importance, as can be seen in the number of hours assigned weekly in Table 1.
32 Machine Translated by Google I.3.1 COMPETENCES IN THE LATEST
EDUCATIONAL REFORMS The reform initiated in preschool, both in terms
of school management and the development of skills, is expressed as follows:
The central purpose of the curricular reform of this educational level has been
the transformation and improvement of pedagogical practices, guiding them
to favor the development of competences in children; This implies changes in
the conceptions that have predominated for a long time about young children,
their development and learning processes and, consequently, The efforts made
in terms of school work to promote the learning of mathematical tools have
also been guided by the cultural requirements of each country. In Mexico,
traditionally, both the learning of the language and that of mathematics have
been valued, considering these as knowledge that the new generations must
master in order to function adequately in the different areas in which they
interact. The global changes in economic and political matters in the world
oriented the curricular design of education, in Mexico and in many countries,
particularly in basic education, to the development of ”competencies” that
we consider is a notion that is not clearly defined and that therefore, it
provokes divergent interpretations both theoretically and practically (Vaca,
Aguilar, Gutiérrez, Cano, and Bustamante, in press). In this way, the Mexican
educational system begins its transition to this new objective, which is initially
specified in preschool. The curricular reform of preschool education aims to
contribute to the transformation and improvement of pedagogical practices
and the conceptions that support them (beliefs about how young children
are and learn and what is the function of preschool education), so that so
that the little ones develop the cognitive and socio-affective skills that are the
basis for lifelong learning. (Secretary of Public Education, 2009, p. 27) 33
Machine Translated by Google This reform of basic education puts the term
of competences as a central axis: At the end of the period of generalization
of the reform in secondary school, the The main strategy for achieving this
objective in basic education proposes carrying out a comprehensive reform of
basic education, in which the notion of competencies is resumed, which allows
addressing the challenges facing the country in the face of the new century, and
helped to achieve greater articulation and better efficiency between preschool,
primary and secondary. (Education secretary (Secretary of Public Education,
2009, p. 27) Integral Reform of Secondary Education (RIES) and which had
several phases as a training process, of which the last one was carried out in
the 2008-2008 school year. reform in primary education that has its first phase
16
in the year 2009. In 2006 it was the turn of the secondary school, through what
was called 2009: 34 The primary challenges are focused on raising quality and
incorporating into the curriculum and daily activities: the renewal of learning
content and new teaching strategies, the intercultural approach, the use of
information and communication technologies as support. for teaching and
learning; learning a mother tongue, be it an indigenous language or Spanish,
and an additional language (indigenous, Spanish or English) as a subject of
state order; the renewal of the Civic Education subject for Civic Education and
Ethics, and the innovation of educational management. (Secretary of Public
Education, 2009, p. 9) [...] in the 2008-2009 school year the generalization
of the third grade was concluded, however its consolidation is still a pending
task; Without this consolidation, the articulation of all basic education will
not be possible. Difficulties arose in generalization due to the heterogeneity
of the secondary level, not only because of its modalities (general, technical,
telesecundaria), but also because of the operating conditions in the states (and
within them, in the different regions) [Secretariat of Public Education, 2006,
p. 29]. about what corresponds to the school to do to favor learning. Machine
Translated by Google I.3.2 THE COMPETENCE-CENTRED MODEL We
know that the term competition has its history and its own evolution (Vaca, J.
et al., in press) and we agree with the observations made by Vergnaud in the
interview given to Baudelot (2005) about this concept, who considers that it
is not it is a scientific concept and it is not a self-sufficient concept; for him it
is necessary to take into account the situations in which the subjects develop.
He considers that competence is a value judgement, one is competent or one
is not competent or one is more competent today than yesterday, or else, X
is more competent than Y. The term competence has been used in the study
programs at the higher levels and has now been adopted in basic education. It
is necessary to make a brief analysis of the official documents that take up this
concept. This model, which serves as the central axis in the reform of basic
education, that is, from preschool to high school, can be analyzed from different
perspectives. The document of the SEP, in the Plan and study programs
for primary education expresses his: He mentions that for there to be value
judgments like these, there must be criteria Educational research has sought to
specify the term competencies, agreeing that these are closely linked to solid
knowledge; since its realization implies the incorporation and mobilization of
specific knowledge, so there are no skills without knowledge. 35 A competence
implies knowing how to do (skills) with knowing (knowledge), as well as the
assessment of the consequences of that doing (values and attitudes). In other
words, the manifestation of a competence reveals the putting into practice
of knowledge, skills, attitudes and values for the achievement of purposes in
diverse contexts and situations, for this reason competence is conceived as
the mobilization of knowledge (Perrenoud, 1999 ). Making basic education
contribute to the formation of citizens with these characteristics implies
considering the development of competencies as a central educational purpose.
(2009, p. 40) Public, 2009, p. 9) Machine Translated by Google I.3.2.1 LIFE
SKILLS 3. technological evolution (Vaca et al., 2010, p. 62). discharge: The
17
performance criterion: I am more competent today than yesterday because
I can do things that I did not know how to do yesterday. The result of the
activity is enough to say that X is more competent today than yesterday or
that it is more competent than Y. Performance: X is more competent if it
performs better, faster, more reliably, more compatible with the work of others
in a group. However, you have to analyze the activity. The idea that there
are ways of proceeding that are stronger, simpler than the others, cheaper,
faster, more reliable is very important in the definition of competence. If we
consider the criteria that Vergnaud proposes to make a value judgment that
would correspond, therefore, to the assessment of competence, the complexity
of the concept is observed when it is linked to the educational field, mainly
in teaching based on this curricular organization and on the evaluation of
the competences developed under the following profile. 36 1. Resources and
their choice: X is more competent if he has resources that he can adjust to
different problem situations and finds a way to cope with the task. It is seen
there that it is not a scheme but a set of schemes and the competition would
consist of choosing the best ones to mobilize in each problematic situation that
arises. 2. The competencies that the SEP proposes are intended to contribute
to the achievement of the profile of Protection in new situations: The person
who is less helpless in a new situation is more competent. This criterion is
very important today in the business world and is closely linked to the rapid
evolution of society and the of which he distinguishes four: • Competences
for permanent learning. They imply the possibility of learning, 4. Machine
Translated by Google • Competences for information management. They are
related to the search, • Skills for handling situations. They are those linked to
the possibility of organizing and designing life projects, considering various aspects, such as historical, social, political, cultural, geographical, environmental,
economic, academic and affective aspects, and having the initiative to carry
them out, managing time , promote changes and face those that arise; make
decisions and assume their consequences, face risk and uncertainty, propose
and carry out procedures or alternatives for problem solving, and manage
failure and disappointment. • Skills for life in society. They refer to the
ability to decide and • Competencies for coexistence. They imply relating
harmoniously with others and with nature; communicate effectively; teamwork;
make agreements and negotiate with others; grow with others; harmoniously
manage personal and emotional relationships; develop personal and social
identity; Recognize and value the elements of ethnic, cultural and linguistic
diversity that characterize our country, raising awareness and feeling part of it
from recognizing the traditions of their community, their personal changes and
the world. identification, evaluation, selection and systematization of information; thinking, reflecting, arguing and expressing critical judgments; analyze,
synthesize, use and share information; knowledge and management of different
logics of knowledge construction in various disciplines and in different cultural
areas. assume and direct their own learning throughout life, to integrate into
the written culture, as well as to mobilize the various cultural, linguistic, social,
scientific and technological knowledge to understand reality. act critically in
18
the face of social and cultural values and norms; proceed in favor of democracy,
freedom, peace, respect for legality and human rights; participate taking into
account the social implications of the use of technology; participate, manage
and develop activities that promote the development of localities, regions, the
country and the world; act with respect towards sociocultural diversity; combat
discrimination and racism, and manifest an awareness of belonging to their
culture, their country and the world. (Secretary of Public Education, 2009, p.
43) 37 Machine Translated by Google h) Promotes and assumes care for health
and the environment, as conditions that favor an active and healthy lifestyle.
e) Knows and exercises human rights and values that favor democratic life, acts
and fights for social responsibility and adherence to the law. i) Take advantage
of the technological resources at your fingertips, as means to communicate,
obtain information and build knowledge. basic manages to display the following
features: b) Argue and reason when analyzing situations, identify problems,
formulate questions, make judgments, propose solutions and make decisions.
Values reasoning and evidence provided by others and may modify own views
accordingly. f) Assume and practice interculturality as a wealth and form
of coexistence in social, ethnic, cultural and linguistic diversity. a) Use oral
and written language to communicate clearly and fluently and interact in
different social and cultural contexts. In addition, they have the basic tools to
communicate in an additional language. c) Search, select, analyze, evaluate and
use information from various sources. 38 g) Know and value your characteristics
and potential as a human being; knows how to work in a team; recognizes,
respects and appreciates the diversity of abilities in others, and undertakes
and strives to achieve personal or collective projects. d) Interpret and explain
social, economic, financial, cultural and natural processes to make individual or
collective decisions, based on the common good. During the formative process
it is intended that the student at the end of the education Machine Translated
by Google • Program for International Student Assessment (PISA). • Trends
in International Mathematics and Science Study (TIMSS) These features, once
concretized, closely resemble the purposes of the 1993 reform. 2º Acquire the
fundamental knowledge to understand natural phenomena, particularly those
related to the preservation of health, environmental protection and the rational
use of natural resources, as well as those that provide an organized vision of
history and the geography of Mexico. 1º Acquire and develop intellectual skills
(reading and writing, oral expression, information search and selection, the
application of mathematics to reality) that allow them to learn permanently
and independently, as well as act with efficiency and initiative in the practical
matters of everyday life. 3rd They are educated ethically through knowledge
of their rights and duties and the practice of values in their personal lives, in
their relationships with others and as members of the national community. •
Tests for Educational Quality and Achievement (Excale). In the end, whether
they are objectives, purposes, traits or competencies, the measurement of their
achievement is carried out through ”standardized” tests whose main tool is the
formulation of written problems. In Mexico, 4 tests have been applied mainly:
39 4th Develop attitudes conducive to the appreciation and enjoyment of the
19
arts and physical exercise and sports (Secretary of Public Education, 1993). j)
Recognizes various manifestations of art, appreciates the aesthetic dimension
and is capable of artistic expression. (Secretary of Public Education, 2009,
p. 43) I.3.2.2 PROBLEM RESOLUTION, MEANS TO EVALUATE THE
SKILLS Machine Translated by Google They confirm the heterogeneous nature
of the ways of working and the universes of reference mobilized by the students
evaluated by PISA, regarding the impossibility of reducing them to simple
reading and written comprehension skills. These reinforce the hypothesis
according to which the PISA tests, which their creators present as evaluators of
this type of competence, evaluate more complex and heterogeneous processes
than the former (p. 115). Bautier et al. (2010) carried out a study in
France on the PISA tests and consider that the conclusions they reached: •
National Assessment of Academic Achievement in School Centers (Link 3 ). 40
According to the INEE 2005 Pisa document for teachers, in general Mexico
ranked 38 out of 43 in most of the mathematics reagents released by the OECD;
in others, last place. If we take into account what mathematical competences
imply from Vergnaud’s perspective, we wonder which of the criteria If we
take into account the competency-based model and the assessment style of
national and international tests, we note that problem solving continues to
be the means to assess mathematical learning, although based on the results
of the research Readers and their contexts (Vaca et al., 2010) and other
reflections (Bautier, Crinon, Rayou & Rochex, 2010; Vaca, 2005) it is clear
that the standardized assessment of students’ knowledge presents considerable
methodological problems and therefore the results of the evaluations must be
interpreted taking into account the sociocultural contexts of the students who
participate; however, educational programs and policies were created based
on those results and the results did not change substantially. The apparent
elimination of the ENLACE test could mean that the rejection of all the
actors directly involved in it has finally been recognized: teachers, students,
parents and also researchers, as reported by the latest research exposed in
national forums ( Perez, 2013; Ray, 2013). However, there is also the possibility
that another form of evaluation with the same characteristics or with greater
problems is promoted instead. 3 Proof that the current secretary of public
education had announced that for this 2014 it will no longer be applied to
evaluate the performance of students and teachers in the Mexican educational
system. Machine Translated by Google For all these lines of discussion, we are
interested in knowing in detail how What he proposes to make this value judgment coincides with those of the PISA test to place students in that position.
It is clear that one thing is to give the answer to a problem and another is how
the resolution procedure was. Two students can arrive at the same result but by
different procedures: one more powerful and general and the other with detours,
hesitating and spending more time on it. The above taking into account the
second criterion proposed by Vergnaud, that of development. Our students face
a math problem. 41 Machine Translated by Google CHAPTER II. Theoretical
framework The current knowledge of the subject comes from the interaction
between his It results mainly from the internalization process. The theoretical
20
referent that we have chosen to approach this problem is the operational
theory of representation and the theory of conceptual fields by Vergnaud
(1990, 2009), which we consider a contemporary constructivism that takes up
the contributions of Piagetian theory, of the ”group of strategies” by Bärber
Inhelder (Inhelder and Caprona, 2007) and the reflections of Pierre Gréco
(1979-1980; Gilis, in press) on the subject’s cognitive functioning. This theory
of Vergnaud also has important links with the theory of didactic situations
by Brousseau (1997) and finally the studies of those of situated microgenesis,
a psychological current that we consider to be the continuity of Piagetian
constructivism proposed by Saada-Robert and Balslev (in press). . formulation
of his theories: previous knowledge and experience. The main theoretical and
epistemological influences that guide this work are from Gérard Vergnaud, who
has taken up the main theses of Jean Piaget in the Knowledge is a process
of adaptation, it derives from the adaptation of the However, because Piaget
sought the explanation and description of knowledge and intelligence through
logical structures and because these structures are not applied systematically
to the same types of tasks or problems if, for example, the content is changed,
what was called décalage (lag), is that a current of studies emerged that delve
into the application of cognitive structures to specific situations, to this new
current that we call contemporary constructivism. Instead of characterizing
Piaget’s ”epistemic subject”, these investigations were oriented to investigate
how the ”psychological subject” acts in a situation, that is, how an individual
updates such logical structures or meaning structures in specific situations and
in particular contexts. Are Knowledge comes fundamentally from action on
the world. 42 individual to his environment. Symbolic activity is the internal
counterpart of overt activity and Machine Translated by Google Rouchier
(Artigue, Gras, Laborde y Tavignot, 1994) considers that both Vergnaud
and He is part of a small pioneering group in the didactics of the disciplines
and in 43 Vergnaud takes up Piaget’s interactionist thesis and delves into
the interaction relationship between subject and object, he proposes that, if
knowledge is adaptation, what adapts are the subject’s schemes to situations.
Therefore, the Situation Schema pair is theoretically more productive than
the subject-object pair. It is not just a terminological change, but it implies
on the one hand not seeing isolated objects but situations and how they are
faced not by the average, abstract subject, but by a flesh and blood subject
in a determined and situated context (the pragmatics of the subject by Pierre
Gréco). Therefore, the analysis of situations acquires a central role in the
understanding of the cognitive development of the subject, a theory developed
in depth by Guy Brousseau for the didactics of mathematics. He is director
emeritus of research at the National Center for Scientific Research in France
(CNRS for its acronym in French), has directed more than 80 doctoral theses,
was a student of Piaget whose thesis on aesthetics he defended at the Sorbonne
in 1968 before the Genevan teacher. the didactics of mathematics in France
and Switzerland. But who is Vergnaud? Merri (2007), in the presentation
of the work in Tribute to Vergnaud “Activité Humaine et conceptualisation.
Questions à Gérard Vergnaud” mentions that his influence is not only related
21
to the best-known topics, namely his theory of the schema and of conceptual
fields, but also to his lesser-known works such as the one on the notion of
calculable representation published in 1975 whose Translation has been made
in the anthology ”What the hell are the skills” (Vaca et al., in press). En
la obra “Twenty years of mathematics education in France”, André Annick
Weil-Barais in the preface to the same work (Merri, 2007) describes him as an
incessant activist for the development of a psychology capable of responding
to society’s problems, which include both access for all children to knowledge
values of French culture (mathematics and science) as well as the development
and transmission of professional experience. Research was carried out by
Piaget’s collaborators and students: Bärber Inhelder, Pierre Gréco, Jaqueline
Bideaud, Saada-Roberts and also by Gérard Vergnaud. Machine Translated by
Google II.1 OPERATORY THEORY OF REPRESENTATION Given that he
conceives that the representation is not unequivocal to reality, but that there
is These computations are carried out on different representation planes, hence
the second concept, ”calculable representation”. According to this author ”The
notion of representation has returned to the forefront of psychologists’ concerns,
after having been deliberately ignored by a large number of experimentalists
for many years” (Vergnaud, in press, p. 179) and ” The notion of relational
calculus helps to clarify and make explicit the too vague notion of reasoning.
Brousseau are the ones who have contributed the most to the development
of the didactics of mathematics. But Guy Brousseau has concentrated more
on the definition, classification and description of situations (even more than
is necessary according to Vergnaud) than on the structures of meanings or
schemes of the subject. Two of the concepts that we will use to account
for the observed processes and analyze the students’ responses is relational
calculus and calculable representation, both proposed by Vergnaud (in press).
The author states that there are two large categories of relational calculus:
Vergnaud (in press) defines representations as hypothetical constructions that
must be inferred from the subject’s observable behaviors. A great variety of
hypotheses can be formulated about these representations, so it is only possible
to operationalize this notion if the hypothetical representations are calculable,
that is, they lend themselves to relational computation. 44 (Vergnaud, 2004, p.
28). 179). verified or accepted. • Deduce new relationships from established
or accepted relationships (p. Deduce a behavior or a rule of conduct from
relationships Machine Translated by Google Figure 1. Representation plans
(taken from Vergnaud, in press). several representations and are of different
levels, then he also considers the correspondences between the different planes
of representation, which he calls homomorphisms (same structure). It means
that in the representation of reality there must be a homomorphous structure
that allows operations (of thought) to be carried out that will eventually lead
to the solution of the problem, but on the level of reality. different planes
of representation. However, a system cannot calculate on another system if
there is no homomorphism of the represented system in the representative
system. Who says homomorphism does not say isomorphism. In the realityrepresentation application, they are classes of aspects, relations and processes
22
that are represented to enter into the calculations, not singular phenomena
(Vergnaud, 1991, p 15. The translation is ours). 45 The notion of homomorphism is crucial to understanding the relationships between Figure 1 outlines
the relationships between different planes of representation. In mathematics
didactics, Vergnaud considers that it is necessary to seek to understand what
are the prior representations that students must have (or master) in order to
better understand canonical representation systems. Machine Translated by
Google II.2 MATHEMATICAL NOTATION precisions regarding mathematical
notation. From the above, it is necessary to define some concepts that are
important different. Therefore, it is important to make a small detour to
make some 4. The representation is applied to states, events that are state
diachronies, relationships or transformations that are classes of synchronies
or diachronies, to classes of relationships or transformations that are already
invariant (p. 189) The canonical representation that this author defines is
socially consensual and Mathematical writing has played a very important
role in the to understand the resolution process: conceptual field and scheme.
In relation to writing, Harris (1999) proposes an integrational semiological
theory, in which he also takes into account mathematical and musical notation
that are practically not contemplated in the classifications that tried to form
grammatology. history, evolution, which has rules and principles. As well as
writing systems Finally Vergnaud (in press) reaches 4 conclusions: agreed. It
is a cultural product and in mathematics it obeys a system with its own 46
2. This allows explaining how a subject solves the problems of reality but
also how certain high-level representations are preceded by other weaker ones
that it is important to analyze. specific representation that by its nature and
function obey rules and principles 3. Morphisms between representations are
at the center of cognitive activity. Undoubtedly, you have to look for the
best representations, but it is also useful to put into play the correspondences
between different representations. 1. The notion of representation is not
univocal, in the sense that one can never speak of a single possible and useful
representation. have their own characteristics, for mathematics systems have
been developed Machine Translated by Google Harris (1999) mentions that
the most decisive invention of antiquity was the According to Harris (1999,
p. 187) ”the scriptural structure of mathematics is designed with the aim
of integrating various types of calculations” and does not obey an attempt
to match this notation with the language or with phonetic units, so He calls
it non-glottal writing. In fact, Ferreiro clearly established that writing has
never been intended to represent speech, but rather language. Hence its
important distinction between transcription code and representation system
(Ferreiro, 2006). It can be thought that a constituted system of representation
is acquired by new users as if it were a code. Such is the underlying conviction
in most of the pedagogical proposals related to the teaching of reading: the
child is prepared to recognize the letter units by exercises to recognize the
phonetic units and the units are related to each other. them (avoiding as
much as possible the so-called ”exceptions” to the principle of bi-univocity
between letters and phonemes). Epistemological questions are thus ignored
23
and, nevertheless, the greatest difficulties of children are of an epistemological
nature, as the studies reported in this chapter show (Ferreiro 2006, para.
236). 47 In the case of writing mathematics, it is not a transcription code
but a representation system. Therefore, the processes of acquisition of this
system by students can face the same difficulties as with the alphabetic system,
which have already been widely studied. What we have observed is that when
students begin to understand the mathematical representation system, they
mix the glottic of the alphabetic writing system with the writing of numbers,
for example, since numerals are a logographic representation system, children
use it morphographically. In the same way that they graphically represent
a division respecting the directionality of alphabetic writing (from left to
right) based on the linguistic expression of the division; as if each graphic
mathematical element corresponded to a segment of the expression. Detailed
analyzes of this will be made later. The construction of mathematics is a
system of representation, which, like any system, is socially constructed and
has evolved throughout history. Machine Translated by Google This theory,
according to its author, has the purpose of describing and analyzing the
progressive complexity of the mathematical competences that students develop
inside and outside the school, and establishing the relationships between
the operative form and the predicative form of knowledge. Consider that a
conceptual field: [...] from a semiological perspective, this invention represents
not an evolution, but a complete break with the notion that written signs
must agree with or reflect the structures of oral language. In other words,
that notation could not have been invented at all if integration with speech
had been the inventor’s primary concern (p. 190) 48 It is at the same time
a set of situations and a set of closely linked concepts. By this I mean that
the meaning of a concept does not come from a single situation but from a
variety of situations and that, reciprocally, a situation cannot be analyzed with
a single concept, but rather with several, forming systems (Vergnaud, 2009). ,
p.86). Sumerian notation based on position and that marks a difference with
glottal writing systems. In the case of division writing -which concerns us in
the problem to be investigated-, its layout is related to the syntagmatic of
mathematical writing that it calls tabular, referring to the tables of squares and
square roots of Babylonian origin ”where two sets of factors jointly determine
the significance of an interrelated series of discrete shapes” (Harris, 1999, p.
195). It is then a writing system with its own characteristics and logic, which
favor calculations and support memory. It is a representation system that
must be reconstructed by students in order to be understood and must be
differentiated from other writing systems, especially alphabetic writing, which
is the one that most frequently interacts both inside and outside the school.
II.3 THEORY OF CONCEPTUAL FIELDS Machine Translated by Google
In general terms, the main categories that Vergnaud (1994) personal, October
20, 2010) whose construction implies interaction with multiple situations with
each one to find their invariances that strengthen their singularities but at the
same time differentiate them from each other. In this way, any other concept
necessarily implies the relationship with other concepts forming systems; that of
24
automobile, requires the link with other concepts, for example: transportation,
engine, driver, speed, tires, etc. proposes for multiplicative structures: For
elementary mathematics, Vergnaud (1990) proposes two conceptual fields from
which mathematical problems can be classified: that of additive structures and
that of multiplicative structures. The first defines it as the set of situations that
require an addition, a subtraction or a combination of said operations. The
conceptual field of multiplicative structures defines it as the set of situations
that require a multiplication or a division or a combination of such operations.
For the purposes of this work, the four main types of problems of multiplicative
structures will be described (Vergnaud, 1990). Vergnaud (1990, p.197) defines it
as the set of situations whose treatment involves one or several multiplications
or divisions, and the set of concepts and theorems that allow analyzing these
situations: simple proportion and multiple proportion, linear and non- linear,
direct and inverse scalar ratio, quotient and product of dimensions, linear
combination and linear application, fraction ratio, rational number, multiple
and divisor, etc. 49 From the situational point of view, Vergnaud (1994, p. 46),
the conceptual field of multiplicative structures comprises a large number of
situations that need to be carefully classified and analyzed, in such a way that
the competencies developed by the students can be hierarchically described.
students inside and outside of school. For example, the concept of ”son” cannot
be fully understood without the concepts of mother, father, family, etc. (Dr.
Rosa del Carmen Flores, communication II.3.1 THE CONCEPTUAL FIELD
OF MULTIPLICATIVE STRUCTURES Machine Translated by Google • •
• • • He proposes four classes of elementary problems: For this, he mentions
that the key point is to consider the action of the subject in the situation and
the organization of his conduct. It states that there are two types of situations:
the first is when the subject has the cognitive tools to face the situation or
problem and in these cases automatically applies a scheme. The second type is
when the subject does not have the ”skills” and needs to reflect, search and try,
try various schemes, thus favoring discovery. He also mentions that the analysis
of multiplicative structures is completely different from the analysis of additive
structures, since the base relations are not ternary but quaternary. He states
that the simplest multiplication and division problems involve the simple ratio
of two or more variables, one in relation to the other (Vergnaud, 1994, p.153). •
The division-partition. Simple proportion. He proposes (didactically) a graphic
representation system to show the relationships between the magnitudes of
the data of the problems that favors its analysis (Vergnaud, in press). Use
rectangles for states and roundels for transformations or relationships. The
multiplication. Vergnaud mentions that in multiplicative structures, unlike
additive ones, ”the simplest base relations are not ternary but quaternary,
because the simplest multiplication and division problems involve the simple
ratio of two variables in relation to each other.” ” (1990, p. 170 and 2001, p.
214) Comparison between ratios and proportions. The fourth proportional. •
Concatenation of simple proportions. The division-quotation (or grouping). 50
Machine Translated by Google 51 Without this scheme, someone could say,
first let’s determine how many marbles he won or lost in total, if he first lost
25
13 and then he won 7 then he lost 6 in total. proposes to use the scheme
of Figure 2: (Vergnaud, in press, p. 181). But what is the nature of these
numbers? The representation shown above helps to understand that 13 and 7
are transformations and we can also do operations with them and not only with
the states. For example, from the composition of both transformations, we
obtain -6, which represents the total number of marbles that Pedro lost. If it
is known that applying this transformation to the initial state (unknown of the
problem) the final state is obtained, then if the reciprocal of said transformation
is applied to the final state (45) it is possible to calculate the initial state: that
is, 51 marbles. In Figure 3 “Pedro has just played two games of marbles. He
lost 13 in the first game, won 7 in the second, and now he has 45. How many
did he have before you started?” the simplest base relations are represented
in multiplicative structures. In this way it is easier for the student to identify,
or in his case, for the teacher to explain it, that if two transformations give
rise to a final state, then applying the reciprocal of such transformations to
that final state would allow knowing the initial state. . For example, to better
understand the relationships in the following problem Figure 2. Example of a
sagittal diagram to represent states and relationships of a problem. Adapted
from Vergnaud, in press, p. 181 Machine Translated by Google Figure 3.
Relationship diagrams of the conceptual field of multiplicative structures
(Vergnaud, 1990). Finally, in the fourth scheme, the relationships of what
we know about the rule of three are represented. It is about the relationship
between two proportional variables that, knowing this relationship in one type
of measurement, can be applied reciprocally to the other to find These schemes
express the relationship between two proportional variables. In the multiplication scheme (upper part of the figure) of the four measurements, the unit value
is known and the unknown data is the value of c. Which can be deduced from
the scalar operator that makes the value of bo go from unity to the proportional
function operator that makes it go from one type of measurement to another.
To explain the transformation in mathematical knowledge from interaction
with someone more expert, Vergnaud takes up the notion of development zone.
the missing value. In division-partition, the unit value that can be obtained
by applying the scalar operator that passes from unit to b (multiplying) but
now in the opposite direction (dividing) is unknown. In the division-quotation
(or grouping) the unit value is known and the value of c is unknown, which
can be obtained by means of the proportionality coefficients between the
two variables or with the scalar operator. 52 Machine Translated by Google
• Its generative aspect involves rules to generate the activity, specifically
the sequence of actions, information gathering and controls. aspects: • Its
epistemic aspect involves operative invariants, specifically concepts-in-act and
theorems-in-act. Its main function is to collect and select relevant information
and infer goals and rules from it. For example, substituting the theorem-in-act
”the largest number goes inside the little house” by ”the dividend can be less
than the divisor” is a progress in knowledge, generally the product of facing
various situations, directed or not to such end, by the teacher. Vergnaud
(2009) proposes that if the theory that knowledge is an adaptation process
26
is assumed, then the question must be answered: what is it that adapts and
to what? • Its intentional aspect involves one or more goals that can be 53
He considers that the most reasonable answer is that what adapts are the
forms of organization of the activity, that is, the schemes, and they adapt to
the situations. developed into subgoals and anticipations. He takes up the
notion of schema from Piaget, complements it and enriches it. For example,
he departs from the fundamental distinction between epistemic subject-object
and focuses more on the observation of the subject ”in situation”, that flesh
and blood subject of constructivism contemporary functionalist. According to
him, the schemes comprise several next proposal by Vygotsky (1996), defined
as one in which the apprentice knows how to do something with the help
of another, but cannot do it completely alone. It is situated between the
current level of development and the level of potential development; however,
it tells us that this zone of development can be better understood if what
it consists of is specified. In this regard, Vergnaud (2006) proposes that the
current development would be the repertoire of schemes that a subject has
to face certain situations and the potential development would be made up
of those that the subject could strengthen or the new ones to generate when
facing a set of specific situations. (of a conceptual field) that, with the help
of the teacher or some other person with this role, he could achieve. II.3.2
SCHEME Machine Translated by Google 54 Knowledge is derived from activity
and is organized in schemas or, more specifically, in its epistemic components.
Vergnaud distinguishes between theorem-in The theorem defines it as follows:
• Its computational aspect involves inference possibilities. They are essential to
understand that thought is made up of intense computational activity, even in
seemingly simple situations, and even more so in novel situations. When facing
”situations”, ”tasks” or ”problems” we need to generate goals, subgoals and
rules, as well as properties and relationships that are not observable (Vergnaud,
2009, p.88). By definition, a theorem-in-act is a proposition held to be true
in the activity. In fact, the study of the development of competences in the
course of learning or in the course of experience shows that the same concept
can, depending on its state of elaboration, be associated with more or less
numerous and more or less rich theorems. The range of theorems-in-act that
can be associated with the same concept is generally very large, particularly in
scientific and technical disciplines, in such a way that, frequently, there is no
act and concept-in-act. Gérard Vergnaud proposes a classification of situations
based on the mathematical relationships and the conceptualizations involved in
each of them. It gives special emphasis to the notion of representation and its
different planes that are homomorphic with each other and with reality; For our
analysis of the procedures, its notion of schema is very relevant to understand
the answers of the students and that is why the theory of conceptual fields
seems to us to be a very useful tool in order to explain the resolution processes
based on the conceptualizations mobilized and the representation systems used
by students in the proposed experimental situation. It is for this reason that
this problematic situation and the resolution processes by the students and the
underlying conceptualizations are analyzed in the light of Vergnaud’s theory of
27
conceptual fields (1990; 2009). II.3.3 THEOREMS AND CONCEPTS-IN-ACT
Machine Translated by Google 55 According to what is mentioned in this
fragment, a theorem can be true or false because it is a proposition (or a statement). A concept is not a statement, it is a category, it cannot be considered
true or false, but only relevant or not relevant to the situation. It clarifies that
theorems-in-act can be false but be considered by the student (or the person)
as true (Vergnaud, 2009, p. 88). We had the opportunity to personally ask
Vergnaud if it was necessary to express the theorems-in-act mathematically
and he said no, that they are affirmations, or implicit beliefs and can be
recognized in different fields, not only in the field of mathematics. He gave
us the following example: Regarding marriage, the concept of husband and
wife entails a series of affirmations, beliefs that are assumed to be true for the
subject and from there he has certain expectations, forms of behavior, rules of
action, etc. meaning to declare that such a subject understood such a concept.
It would be necessary to be able to specify in each case which theorems-in-act
he is capable of using in the situation. Let us not forget that theorems are
propositions and not concepts, even though they are evidently constituted
by concepts. Inferences are relations between propositions and are chained
by metatheorems (or higher order theorems), such as Aristotelian syllogisms
or the transitivity of order relations: a>b and b>c => a>c. (Vergnaud and
Recopé, in press, p. 307) Some examples of theorems and concepts-in-act can
help to understand these notions. In relation to algorithms, some students
actually believe that all multiplication results in a number greater than those
multiplied (which is true only for integers) or that the result of subtraction is
always less (which is true when not negative numbers are involved). Another
theorem-in-act is to consider that in division the largest number will always
be the dividend and the smallest will always be the divisor. These statements
are considered true by students and they are not always aware of them; they
are theorems not externalized and built in action, they are theorems-in-act.
Propositions are composed of concepts with different levels of appropriation,
some properties of concepts may not be able to be expressed through language
or, like theorems-in-act, not be quotients of them: they are concepts-in-act.
Machine Translated by Google II.3.4 LINGUISTIC MEANINGS theorems.
This statement, considered by the authors as an implicit primitive model, from
Vergnaud’s perspective would correspond to a theorem-in-act, which also has
the characteristic that it generally cannot be made explicit by the subject. It
has a behavioral basis because it is built in action and for this reason he assigns
it the name of theorem-in-act or in action. Thus, although both notions can
refer to the same phenomenon, Vergnaud’s perspective is chosen, who integrates
this notion as part of the operative invariants that in turn are components of
the scheme. However, during the analysis of the results, some coincidences
with those obtained under the notion of tacit or implicit models by Fischbein
et al. (1985). So, any concept has tied to it a certain number of theorems in
action that are more or less explicit and derived from experience, the cultural
environment, etc. of which one does not become aware until facing a particular
situation that ”falsifies” said statements. This notion of theorem-in-act is
28
analogous to that of implicit primitive models proposed by Fischbein, Deri, and
Sciolis (1985). They maintain that some primitive behavioral models come into
play in arithmetic operations that make students, even if they know how to
divide conventionally, to cite an example, be influenced by one of these models
and make mistakes. One of these implicit models is that in division interpreted
as distribution (partition) and in division interpreted as grouping (quotation),
both the divisor and the quotient must be smaller numbers than the dividend
(p. 14). On the other hand, Vergnaud (1994) raises the need to analyze and
classify the variety of linguistic and symbolic signifiers that we can use when we
communicate and think about the conceptual field of multiplicative structures.
He considers that an essential empirical and theoretical task for researchers is
to understand why a particular symbolic representation can be useful, under
what conditions, when and why it can be adequately replaced by a more
general and abstract one. He proposes a scheme to exemplify the different
representation systems based on the planes of representation in which they
are located (See Figure 2). It also expounds the need to pay close attention
to the comparative difficulty of different kinds of 56 Machine Translated by
Google Figure 4. Representation systems. Taken from Vergnaud (in press, p.
183). The psychological current that allows us to account for these processes
arising when solving a task in a short period of time is Situated Microgenesis
proposed by SaadaRobert and Balslev (in press). Relevant aspects of it and the
clinical/critical method proposed by Piaget will then be reviewed to support
the methodology used in the present study. problems and procedures, as
well as the different verbal and written expressions produced by the students
(Vergnaud 1994, p. 43). In Figure 4 four representation planes are shown; the
plane of the iconic representation of groups of objects, the plane of the sets,
the cardinals and the written numbers. Each of these planes involves different
forms of symbolization although they are homomorphic to each other. It means
that each representation system has the same structure as the others but they
are not equal to each other (they are not isomorphic) some properties are
preserved but others change. 57 Machine Translated by Google CHAPTER III.
Methodology III.1 METHODOLOGICAL DESIGN III.1.1 PROBLEMATIC
SITUATION The second phase consisted of conducting clinical interviews
with 30 students who gave interesting answers to understand in depth the
relationship between the different representation systems used and the justifications they give for using them. The investigation was divided into two
phases: the first was the group application of a problem from the conceptual
field of multiplicative structures from which 4 versions were generated that
are described later. The basic problem is the following: It is known that the
sociocultural contexts of the students favor the type of situations they face on
a daily basis, and therefore these contexts influence the development of the
skills needed to solve them satisfactorily (Vaca et al., 2010). We also know that
there are a wide variety of conditions in schools (Rockwell, 2005), and different
levels of teacher performance that could affect the achievement that students
can achieve in learning mathematics. In addition to the fact that Mexico
moved towards a curricular change based on competencies and that transition
29
has caused confusion in the actors of the educational system at its different
levels, but mainly it has caused confusion in teachers, who are attributed the
responsibility of the low scores obtained on standardized tests. However, for
this work we will only focus on the resolution process to identify the main
difficulties that students face when faced with these problems in the conceptual
field of Hector walks 35 meters and takes 70 steps. On average, how big are
your steps? This problem belongs to the conceptual field of multiplicative
structures (Vergnaud 1990, 2009) because the base relation is quaternary (four
quantities are related) and because of the isomorphism of measures that relates
two measures of a different nature, in this case number of steps and number of
meters. 58 Machine Translated by Google the multiplicative structures derived
from one of the items of the PISA test. retain a certain degree of difficulty and
that could promote different strategies of This problem and its versions are an
adaptation of the step problem, reactive resolution and forms of representation.
used in the PISA 2003 test (National Institute for Educational Evaluation,
2013): Since the original problem requires mastering algebra content that is
not 59 are part of the curricular structure of the primary level, an adaptation
was made that Image 4. Steps. Item released from the PISA 2003 test. Taken
from INEE, 2013. Machine Translated by Google III.1.2 ANALYSIS OF THE
PROBLEM dimension to try to define what a step is. However, defining it is
not trivial and requires certain considerations. Since it is about establishing
the relationship between the number of steps and the distance traveled in
meters to calculate the average size of each step, this problem is located in
the conceptual field of multiplicative structures in the types of isomorphism
problems of measures (Vergnaud 2004, p. 218). The Dictionary of the Spanish
Language (Royal Spanish Academy, 2013) contains What is a step? Figure 2
shows the scheme that represents these relationships. 60 This is a problem that
looks for the value of a part or an object, in this case the average size of the
steps. A step can be considered to be something common when talking about
walking, without Before continuing with the analysis of the problem, it is
necessary to make a small Figure 2. Diagram of relationships corresponding to
the problem of steps. Machine Translated by Google 6. m. Regular movement
with which an animal walks with legs, raising its limbs one by one and without
giving rise to any jump or suspension. 5. m. Roman step. To take advantage
of the benefits of graphical representations, which we believe help us to carry
out this small analysis, we made the following scheme (See Figure 3). 4. m.
Mode or way of walking. If it is a question of measuring the advance with
a step, the head of the person who walks can be taken as a reference point
and draw a line perpendicular to the ground that will be the starting point
and when moving one foot forward, it would have to be drawn another line
from the center of the head to the ground to determine the point of arrival
and with it the actual distance advanced. 61 3. m. Continuous movement
with which an animated being walks. 1. m. Successive movement of both feet
when walking. As an instrument for measuring through displacement, some
clarifications must be made: 2. m. Distance traveled in each movement while
walking. Passed. We can consider the step as a movement when walking or as
30
a unit of measurement. As for the first, it is worth considering whether a step
is the movement of only one foot (forward or backward) or the movement of
both feet should also be considered. Regarding the second consideration, it is
essential that, as a measurement instrument, it remain invariant (that is why
in the problem we use the expression ”on average”) and it must be determined
from where to where said measurement will be made. meanings to the term, of
which 8 can be used in the context of the problem: 11 m. Each of the advances
made by a counting device. 9. m. Print or footprint that remains printed when
walking. Machine Translated by Google Figure 3. Scheme for defining a step.
Source: self made. of a single foot, the distance traveled is from line A to line
B. If we consider that the upper drawing indicates the point of origin of the
displacement of one foot that culminates in the drawing below, the following
cases can be defined: Case 3. If we take the person’s feet as a reference, we
must choose a point of reference, for example the heel of the feet. When moving
a single foot, the distance traveled would be from line A to C, or if we chose the
toe as reference, the distance traveled would be from line B to line D. Both are
the same size and the distance would be the same as if both feet are moved and
the head is taken as reference (Case 2). Case 2. If we again take the head as a
reference and the other foot is moved to reach the first one already displaced,
then the distance traveled would be from line A to C , which would be twice the
size of the first one but the displacement of both is required feet. Case 1. If we
take as reference the head of the person and the displacement If we arbitrarily
consider that the average size of a person’s steps is half a meter (Historically
it has been calculated that it is approximately .7 m as an anthropomorphic
measure), this distance is achieved in cases 2 and 3 because in case 1 the
distance traveled would be half, 25 cm. 62 Machine Translated by Google After
this note, we continue with the analysis of the problem. The first requires
applying a scalar operator. If in the steps column (See Figure 4) the scalar4
operator (x70) changes from 1 step to 70 steps, then in the meters column
the operator (:70) is applied, which is the inverse operator and will go from
35 meters to the unit value that acts as unknown. The second procedure (See
Figure 5) is to apply the function operator (:2 meters/step), which converts 70
steps to its measurement in meters, that is, 35 meters, which implies dividing
1 by 2 to obtain the unit value. There are at least two procedures to solve
this type of problem (Vergnaud 2004, p. 202): In the text of the problem it
is mentioned that Hector walks a distance of 35 meters and takes 70 steps. It
means that for this condition to be fulfilled, we must assume any of the cases
2 or 3 as a step because with case 1, with 70 steps the distance of 35 meters
would not be reached. 63 Figure 4. Diagram of vertical relationships, with
scalar operators. 4 Vergnaud represents the operators with circles in the body
of the text as they appear in the diagram, however for typographical reasons
we will represent them with parentheses. Machine Translated by Google Figure
5. Diagram of horizontal relationships, with proportional function operators.
If we substitute the values of the problem in the previous approach we would
obtain: This procedure is more complex than the first because the same
relational calculations are not involved and because it is a function operator
31
that passes from one measurement category to another. 70 steps = 35 meters
ÿ 1 step = 35 meters / 70 ƒ(nx1) = nƒ (1) ÿ ƒ(1)= ƒ (nx1)/n In order to make a
demonstration of these solutions, the analysis of the mathematical properties
that Vergnaud (2001, p. 215) makes in this type of problems is taken up and
adapted, to find the solution of this one, where he considers that it is necessary
to recognize a property of the linear function ƒ that joins the measure of the
distance covered with the corresponding number of steps. This means that the
size of n times a step (70 steps) is equal to n times the size of a step (35 meters),
therefore the size of a step is equal to the size of n 64 times a step (35 meters)
divided by n times (70). Machine Translated by Google Another possibility in
relational calculus that Vergnaud analyzes is to operate on the proportionality
coefficients between the two variables (horizontal relationships in the scheme.
See Figure 7). 65 These relationships operate under scalar quantities, that is,
dimensionless. If the ratio of a step to 70 steps is 70 times more (the scalar
operator x70 is applied), then the unit value of a step is obtained by applying
the reciprocal of that ratio. The value of a step is 35 meters “70 times less”
or to be more precise with the language: 70 times less (the inverse scalar
operator (:70) is applied). The solution procedure of scalar type [vertical in
the schematic] ) establishes the relationship between magnitudes of the same
type. Steps with steps and meters with meters. Of course, when students
face this situation, if they were able to recognize these relationships and make
these deductions, it does not necessarily mean that they can express them in
natural language; It is then a question of knowledge contained in the schemes
but at an operative and non-predicative level, that is to say, they are theorems
and concepts-in-act. The graphical scheme that shows this relationship of the
linear function would be that of Figure 6. Figure 6. Diagram of relationships
of the linear function. Machine Translated by Google Figure 7. Diagram of
relationships operating on the coefficients of proportionality. ƒ(n) = 2n(1) ÿ
(1) = ƒ(n)/2n If the reasoning operates on the proportionality quotient of two
different magnitudes, it is identified that one is twice the other; then, the
relationship to go from 35 meters to 70 steps is the operation of multiplication
x2, but according to Vergnaud, this multiplication does not mean two times
more but rather a coefficient of proportionality that joins two variables, one
with the other. another (2 steps per meter), that is, a composition of measures.
It means that the other relationship to go from a step to meters must be applied
the inverse operation that would be (:2), which does not mean two times less,
but rather divide by two meters per step. The result would be 0.5 meters. The
composition of measures could be equivalent to what Schwartz (1988) proposes
as ”intensive measures” (derived from proportional relationships) as a quotient
of two extensive quantities (subject to additive relationships). However, since
Vergnaud’s proposal is used as a theoretical reference, it was decided to use
his The functional operator is constituted, according to Vergnaud (2001), at a
more elaborate level, since it implies not only the notion of numerical relation
but also that of quotient of dimensions. Until now there is no data that any
student has opted for this procedure. The theorem on which this reasoning
rests is stated by Vergnaud (2001) in these terms: 66 Machine Translated
32
by Google magnitudes. terminology. In addition, it is considered that the
notion of composition of measures is didactically more productive than the
one proposed by Schwartz of intensive measures. Therefore, the problem thus
posed to the students has the following characteristics: These are four versions
of the problem already analyzed. Each version presents some modification that
allows us to observe very specific aspects and we initially consider that each
one has a different level of difficulty: due to the numerical values used and the
transparency or opacity of the double/half relationship and the presence in two
versions of unnecessary information or useless that we call a ”distractor” (the
mention of time, which is irrelevant for the solution of the problem), although
the structure of the four versions is the same and has already been analyzed. In
addition to the above, we included the variable ”availability of the calculator”
considering that if some of the difficulties experienced by the students were
related to the lack of consolidation of the algorithms, then the group of those
who had it at their disposal would reflect significant differences in relation to
the group that did not have it. • The number of steps is twice the number
of meters, a relationship that children identify but that in many cases is not
enough for them to make the appropriate calculation and reach the expected
result. • It is located in the conceptual field of multiplicative structures. •
Relationships are established between four pieces of information: two explicit
(35 meters and 70 steps), one implicit (the unit value of the step) and the
unknown (the measure of the step). 67 • It is a problem of isomorphism of
measures of the division-partition type. • The corresponding division is with
a decimal quotient less than one. • The level of difficulty is high because the
unit value of one of the III.1.3 INSTRUMENTS Machine Translated by Google
problem versions In one minute, Hector walks 20 meters and takes 40 steps. On
average, how big are your steps? Version A. With distractor (”in one minute”)
and with a 35/70 ratio. The number of students who faced these versions of
the problem are 329: 161 are women and 168 are men. These students belong
to the last grade of primary and secondary school in five public schools in
the state of Veracruz, three primary schools (one rural and two urban) and
two secondary schools (one from a rural context and the other from an urban
context). The distribution of students by school and by gender is presented in
the Version D. Without distractor and with a 20/40 ratio. (S/D, 20/40) Version
B. Without distractor and with a 35/70 ratio. In one minute, Héctor walks 35
meters and takes 70 steps. On average, how big are your steps? Table 2. 68
Hector walks 35 meters and takes 70 steps. On average, how big are your steps?
Hector walks 20 meters and takes 40 steps. On average, how big are your steps?
Version C. With distractor (in one minute) and with the 20/40 ratio. Table 2.
Distribution of students by gender and school III.1.4 PARTICIPANTS 26 18
Elementary A Urban 31 70 Secondary B Rural 17 Total general 25 Elementary
C Rural School Women Men 21 58 122 35 329 Elementary B Urban 64 168
69 Secondary A Urban 161 42 57 Total general 27 Machine Translated by
Google On average, primary school students are 11.9 years old and secondary
school students are 14.4 years old. The ages vary between 10 and 16 years as
shown in Table 1. In The application of the instruments was carried out in
33
their classrooms at a time kindly granted by the teachers. In total, we worked
with 13 groups from the five schools. Half of the groups were distributed a
calculator that they had at their disposal so that they could freely use it if they
so decided. The other half of the groups did not have access to the calculator.
69 (in white) 172 Total general 98 12.7 11 4 98 13 6 Secondary 55 1.8 5 N 16.7 6
N 3 14 5.1 329 55 3 31.0 Age in years 42 10 0.9 10 16 7 1 6 12 1 41 29.7 Primary
N Total general 17 98 % 102 1.8 15 157 100 III.1.5 APPLICATION OF THE
INSTRUMENT Table 1. Distribution of student ages Machine Translated by
Google Table 3. Distribution of the groups and the availability of a calculator.
0 Secondary B 32 Total general 0 25 0 B 0 0 Elementary A schools G 0 0
H 22 17 J B 9 Elementary B 20 With calc. B B 0 0 29 Sin calc. 24 A 0 18
High School A 0 19 173 C 34 groups A I 0 A A 9 156 36 Elementary C 35
Students were given a sheet with only one of the four versions of the problem.
Said sheet was placed inverted so that when it was finished distributing to the
entire group, the instruction was given to turn the sheet over and everyone
would start at the same time. They were asked to solve the problem with a
pen to avoid erasing and if they made a mistake they could cross out what
they wanted and continue elsewhere. These graphic traces could be relevant
to us. The groups were organized in such a way that men were located at
one end and women at the other, in order to distribute the versions of the
problem among the students and have relatively equal subgroups in number.
70 The distribution of the groups according to the use of the calculator appears
in Table 3. Note: cal.: calculator In the case of elementary school C, which
has three groups of sixth graders, group a was allowed to use the calculator,
group b was not, and in group c only half of the students were allowed to use
it. Machine Translated by Google III.2 CLINICAL INTERVIEWS 5 Thanks
to Mr. Francisco Javier Martínez Ortega, who collaborated as an assistant in
the field work for the application of the instruments, as well as in the video
recording of the interviews. This support was required due to the difficulty
of recording both the written productions of the students and the verbal and
gestural expressions that could give indications of the meanings mobilized
during the resolution of the problem. The instruction was given for the students
to fill in the identification data The criteria for choosing these students were
the following: firstly, it was sought that there were students in the 6 categories
into which the responses of the students were classified in the experimental
phase (See Table 9, page 63): on the sheet and subsequently answer the
question. 3. Correct numerical result but no reference units 1. Correct answer
It is important to highlight that the situation of application of the instruments
occurs in the school context, in the presence of the teachers and sometimes
with the directors of the schools present, who present the applicators to the
students5 . The instructions given to the students, the arrangement of the seats
and the care of the applicators to deliver and collect the instruments are typical
characteristics of a school evaluation context. Therefore, it is assumed that
in this situation, the student’s responses are mediated by a didactic contract
(Brousseau, 1997). When a student finished, they were asked to turn their
paper over and quietly wait for all their classmates to finish. Only until the
34
last student finished, were the sheets collected in the same order in which they
were handed out. 4. Inversion of data in division (with reference units) 71 2.
Difficulties only with units of measurement Subsequently, 30 interviews were
conducted with students who participated in the group application described
above. 13 students from the third grade of secondary school and 17 from the
sixth grade of primary school were selected. Machine Translated by Google
III.2.1 PROCEDURE First of all, the student was thanked for his participation,
he was informed that the intention of the interview was to find out how he
solved a mathematical problem, rather than whether he gave a correct answer
or not. It was emphasized to them that it was not an evaluation and that their
result was not going to be reflected in their grades. What we were interested
in was knowing if the problem was difficult or not and why. Even with these
clarifications, some students expressed nervousness, which decreased as the
activity developed. One of the questions that they were usually asked when we
noticed any nervousness or little interest in the activity was about the class
their group was in at that moment and if they preferred to return to their
class. Only one sixth grade student preferred to return to his class and not
participate in the interview. 5. Inversion of data in division (without reference
units) 6. Lack of understanding of relationships between data Subsequently, he
was given the sheet with the version of the problem that he solved in Another
criterion used was to select the answers that were considered interesting from
the point of view of the graphic representations used or the answers given,
either because of their originality or because they were not understood by the
interviewer. Since in the analysis of the responses, inferences were made that
needed to be corroborated with a different methodological strategy, in this case,
through clinical interviews. The interviews were conducted in spaces provided
by the school principals. They were generally spaces for the activities of the
teachers or computer centers in which there were generally teachers carrying
out some activities, so there is often interference in the audios, in addition to
the sounds of the activities carried out by the students outside their classroom.
of classes. As already documented (Vaca et al., 2010), one of the characteristics
of public schools is that they are noisy. 72 Machine Translated by Google
III.3 THE CLINICAL METHOD The interviews were videotaped and for this,
as already mentioned, there was the support of an assistant, who operated
the video camera with the intention of recording the detail of the written
productions made by the students, the movements, indications, gestures that
could give indications of the representations made during the process of solving
the tasks during the interview. The reasoning routes were highly variable
due to the justifications or specific difficulties experienced by the students
during the resolution of the problem and the development of the interview. In
some interviews, topics were explored in greater detail than in others; such
as fractions, decimal numbers, proportional relationships, writing the division
algorithm and units of measurement, according to the relevance given by the
interviewer-teacher at that time. the experimental phase. They were asked
to read the problem again and to solve it again. When they concluded the
resolution, they proceeded to explore the meanings attributed to their graphic
35
productions or to the result obtained. Each student represents a different
case, with reasoning paths that may coincide with other students on some
points and not on other points. On the one hand, its autonomous resolution
was sought without the intervention of the interviewer, later we sought to
know their conceptions regarding the general themes and finally we gave
ourselves the freedom to follow the reasoning and procedures that the student
manifested, to try to understand their resolution paths. and, to the extent
possible, the level of consolidation of the concepts evoked. The clinical/critical
method proposed by Piaget has been one of the main instruments for collecting
data in psychogenetic research of the Genevan school. Despite some criticism
and its long history, it is still valid. As demonstrated by Bond and Bunting
(1995) who carried out a study in which they affirm that the results obtained
are sufficient to convince the skeptical empiricists of the validity of the 73
Machine Translated by Google He considered that the form and function
of thought is expressed when the subject interacts with other children or
with the adult and that, although the interaction can be observed from the
outside, the content could not be discovered depending on the subjects and
the objects of representation (Piaget , 1978). But, with Vergnaud’s proposal
(in press) of relational calculus and calculable representation, the analysis of
the situations in which these interactions take place makes it possible to more
clearly infer said contents. Piagetian theory when it was criticized for using a
very small sample of subjects, unlike the statistical treatment of populations
in Anglo-Saxon studies. Piaget (1978) considers that this method of data
collection retains the advantages of both observation and tests, but without
their disadvantages, such as missing essential problems and spontaneous
interests. 74 The art of the clinician consists, not in making people answer,
but in making people speak freely and in discovering spontaneous tendencies,
instead of channeling them and putting them on the dam. It consists of putting
every symptom in a mental context, instead of abstracting from that context.
(Piaget, 1978, p. 258) But unlike the other methods, this one is not easy
to carry out; requires training and certain characteristics on the part of the
clinician: One of the objectives of the clinical interview as we use it will be to
propose problematic situations that can trigger resolution procedures in the
student through which it is possible to infer the theorems- and concepts-in-act
contained in their activated schemes to face it. , and make them explicit
as much as possible. Piaget expressed it in other words: The content is a
system of intimate beliefs, and a special technique is needed to discover them.
It is above all a system of tendencies, of spiritual orientations, of which the
child himself has never been aware and has never spoken (Piaget, 1978, p.
258). Machine Translated by Google III.4 SITUATED MICROGENESES
Microgenetic studies, for us, represent a state of the evolution of Piagetian
constructivism, it is a research perspective with at least 5 currents of study,
all of them have as their object of study to understand how the subjects
perform in the activity in a certain situation. , that is, how their schemes are
mobilized, updated, combined and built in action. These studies are related to
activity theory and socioconstructivist theories. They also give a central role to
36
language as a mediator since the interactions occur mostly through linguistic
signifiers, as well as the interaction between subjects and with the object or
situation. Indeed, the object of study is both the construction of knowledge
and the way in which formal and informal knowledge is presented by the most
expert participant, and the way in which it is transformed into internalized
knowledge by the apprentice. Situated microgenesis is thus characterized by
a double construction process: an asymmetric co-construction between the
participants, regarding knowledge, and the asymmetric co-construction of a
zone of understanding, ensured by the exchange of meanings between them
(SaadaRoberts, M. and K. Balslev, in press, p 305). 75 The current of these
studies on which we base ourselves to carry out this research and which is the
last one developed up to now is that of situated microgenesis that seeks, in
addition to the aforementioned object of study, to understand how meanings
move through analysis. of the microprocesses and the interactions of the
student with the teacher or whoever assumes that role in actual teaching
situations in a school context. The zone of understanding is understood as
the period of interaction time during which the teacher’s didactic intention is
expressed (verbally or not) and is understood by the student. There are, of
course, important differences between the mastery of knowledge by the teacher
and the student, but the task of mediation on the part of the teacher is to carry
out the transposition so that it can be apprehended by the student, in case it
is successful and is at the student level, then you can talk Machine Translated
by Google They are in a zone of understanding. Sometimes it happens that
the teacher’s interventions are not fully understood by the student or that
he interprets them in a different way from the teacher’s intentions. In these
cases, they are not considered to be in a zone of understanding. 76 Machine
Translated by Google III.4.1 General aspects of microgenesis For the present
investigation, we consider as an object of study the one that is The evolution
of genetic psychology has studied the processes of construction of knowledge
by the ”epistemic” subject, through the great periods of development (stages)
and now also by the study of the microprocesses of knowledge acquisition by
a subject. “psychological” through the study of microgenesis (Inhelder and
Caprona, 2007). 77 outlines the situated microgenesis (Saada-Roberts, M. and
K. Balslev, in press): The object of study of microgenesis is different depending
on the current in which it is located: These studies, although originally seeking
to find analogies between the knowledge construction processes of macrogenesis
but on a smaller scale, have differed in that they are now not about analyzing
what a group of subjects do according to their level of development ( epistemic
subject) but to analyze how he updates his knowledge in the face of a specific
task in a particular situation and conditions. The object of study no longer
directly addresses development processes, but rather the construction of
cognitive representations, as instances of organization of previously acquired
knowledge (development pole) with the properties potentially contained in
problem solving situations ( pole of situated learning) (Saada-Roberts, M. and
K. The reciprocal adjustment between the adult and the child. The analyzes
present results in terms of the diversity of actions produced by the children,
37
the more or less complex linkages between their actions to achieve a goal and
the effects of these actions on adult interventions, as well as the reciprocal
effects of these actions. latest in children’s learning. The question of reciprocal
adjustment between the participants in the construction of knowledge is what
constitutes, from our point of view, the originality of these works (P. 300).
Balslev, a prensa, p. 298). Machine Translated by Google 78 Only under this
condition can such a study be linked to the explanation of the microprocesses
of knowledge acquisition. Regarding the situated character of microgenesis,
it refers to the didactic dimension of this acquisition, in other words, to
the triadic study of the construction of teaching and learning knowledge as
they function in situ . More precisely, situated microgenesis deals with the
didactic dimension through the analysis of the progression of knowledge linked
to the exchange of meanings between participants, in real time and place
(SaadaRobert and Balslev, in press, p. 293 ). This flexible modality of the
clinical interview reminded us of a work by Blanche Benveniste and Ferreiro
(1998), who, when investigating the expression of denial in Spanish-speaking
children to reflect on writing, proposed to the children the game of ”saying
things backwards”. ”. For example, one of the questions asked to the students
was: A first definition of microgenesis is ”the study of knowledge acquisition
processes in a short time and in a particular situation among the possible
acquisition situations, solving problems by instruction, by free exploration,
etc.” (Nguyan-Xuan, 1990, p. 197 cited in Saada-Robert and Balslev, in press,
p. 291). To this definition, the authors add the didactic dimension to conform
situated microgenesis: In this research, the interviewer also adopts the role of
teacher, who during the interaction phase, in addition to clinically exploring
the processes of solving a mathematical problem by the students, seeks to
present situations that lead to the confrontation of their ideas, take them to
a certain limit so that, as far as possible, they are made explicit, tested and
modified to generate new and more general ones. That is, work in the zone
of proximal development through interventions that require the student to
update or modify their schemes for the specific content of the task performed.
During the interview, the interviewer proposes new problems from different
fields and even some that could be judged as ”absurd”. Some of them achieve
the purpose of making the student reflect on the mobilized concepts. On other
occasions, these problems do not achieve this objective, given the nature of
improvisation based on the reasoning routes shown by each student. Machine
Translated by Google The units of analysis for this study are determined by
the resolution schemes that the students mobilize, respecting their sequence.
Since these schemata are the invariant organization of behavior in a class
of situations and the students only faced a problematic situation, we cannot
accurately infer them. However, we assume that in solving the problem one
or more schemes are applied, whose components include goals and sub-goals,
inference, operational invariants and action rules, as mentioned above. What
is the reverse grape? The most frequent response was “uvo”, which led the
authors to discover the immense potential for reflection in very young children
regarding the morphological structure of their language. In the example,
38
the morphological opposition of feminine/masculine, without any semantic
correlate (”uvo” does not exist in the language). What could be considered a
very general slogan, even interpreted as absurd, allowed the authors to inquire
about the knowledge that students put into play, for example: deciding on
a semantic or a formal answer, mastery of the morphology of the language
and contributed to building indices on the representation of the word unit
outside the written sphere. Therefore, to be consistent with the microgenetic
analysis and attend to the temporal dimension of the resolution process,
it was decided to determine the units of analysis through the spontaneous
segmentations that arose during the process. Data analysis We have proceeded
in a similar way when formulating problems for the children that seem absurd
(dividing children between palettes) but that allowed them to reflect on the
elements of the graphical algorithm of division and its distribution. Based on
these antecedents, a clinical interview and a microgenetic analysis are carried
out with the purpose of making an investigation of the level of construction
of the mobilized mathematical concepts, inferred from the exploration of
their resolution procedures and from the linguistic interactions between the
participants. students, the interviewer-teacher, the problem and the specific
situation. 79 Units of analysis Machine Translated by Google • Those of
knowledge in operation. problem solving by the student. These segmentations
are generally related to sub-goals, small detours or sub-procedures that occur
while the subject seeks to achieve the main goal in the problematic situation,
that is, to find the solution to the problem. Some of these sub-procedures
may consume a large part of the interview or be very brief; In all cases, the
corresponding segmentation is marked to determine the beginning and end of
each unit of analysis. • Those of the progression of knowledge in the form
of microgenetic “sequences”. • Those corresponding to the direction in the
process of construction through the Situated microgenesis is not supported by
an artificial simulation device, but is methodologically based on the analysis
of interactions, mainly but not exclusively verbal, interactions whose hidden
meanings one tries to reconstruct, depending on the context in which the
participants evolve. ” (P.301) According to Saada-Roberts and Balslev (in
press): • The relevant indices to explain an area of common understanding
among the participants, the dynamics of which vary from one sequence to
another. Regarding the indices of knowledge in operation, we return to Gérard
Vergnaud’s theory of conceptual fields to identify the mobilized mathematical
concepts, their relationship with other concepts and try to infer their theorems
and concepts-in-act during the sequences. Regarding the indices that guide
the meaning in the construction process, we pay special attention to the verbal
and gestural expressions and graphic productions, both of the student and of
the interviewer, that is to say, to operational indices 80 specific enunciative
modalities. We have taken into account the following operational indices
proposed by Saada-Robert and Balslev (in press, p. 307) to account for the
construction of knowledge through the meanings between the student and the
interviewer in a zone of understanding: Machine Translated by Google the
predicative part of knowledge. Those related to the progression of knowledge
39
were carried out by signaling the segments of the microgenetic sequences, as
already mentioned in the previous section. The progressions are expressed
in the knowledge mobilized when they occurred. 81 Machine Translated by
Google parts of the total) these problems present difficulties for them to solve.
Are analyzed students responded adequately, which means that for the rest
(two thirds 82 reference units (half meter or its equivalent). One third of the
statistically significant (See Table 2). secondary school obtain better results
than those in primary school. investigation, that is to say, that they indicate
the measure of a step by means of the magnitude and the How many students
solved the problem in the expected way? Of the total of 329 indeed there are
more men who arrive at the correct answer and the difference is By educational
level, the results of Table 3 indicate that students of students, 113 (34%) gave
the answer that is considered correct for the purposes of the Regarding the
results obtained by men and women, it is observed that Note: Test Chi2 =
4.663, sig. 0.031 location, calculator availability, and problem versions. Below
are the results based on some variables such as sex, educational level, Female
Sex 46 Masculine 113 329 67 113 115 Total 216 Incorrect Correct Grand Total
101 216 CHAPTER IV. Results of the experimental phase 47% 51% 71% 53%
49% 49% 100% 41% 100% 100% 100% 100% 40% 34% 29% 66% 100% 60%
Table 2. Results by sex IV.1 GENERAL RESULTS Machine Translated by
Google more students in rural locations considered the real feasibility of their
answers; better deal with problems of this type. However, it is necessary to
highlight the In a research report, Chain, et al. (2009) analyzed some items
of the were more likely to have constructed or enriched schemas for It seemed
strange to them and they also did not take into consideration its relationship
with reality. that others in high school did not make it. required the calculation
of the weight of a cow in grams or the one that requested the calculation of the
In primary school, the majority had an incorrect result (79%). On the other
hand, in the fact that there were primary school students who gave a correct
answer, while ENLACE test and reported that some items were absurd, such
as the one 83 As expected, more high school students achieved a correct result,
some difference because, in a previous study (Vaca et al., 2010) it was reported
that The results obtained in the rural and urban localities are practically the
same as those capacity of a water tank in milliliters. Surely these reagents
were created by the secondary the percentages are similar; almost half solved it
correctly. same (See Table 4) and there are no significant differences. however
it was expected opportunity to face a greater number of problematic situations
and therefore unlike those who, when obtaining steps of 2 meters or 2 cm
as a result, do not compared to those in primary school. To some extent
it is obvious since they have had the 100% 100% 48% 100% 79% 49% 52%
68% 72% 51% 32% 37% 21% 63% 28% 100% 100% 100% Table 3. Results
by educational level 136 Total 216 Total general 329 Level 77 36 Wrong right
Primary 113 157 172 Secondary 80 Machine Translated by Google 34% 100%
100% 100% 67% 100% 66% 35% 100% 65% 27% 100% 72% 28% 34% 28% 72%
72% 156 81 237 Rural 60 Urbana 32 Total general 216 92 Location Incorrect
Correct Total 113 329 designers under the idea that the student would show
40
the ability to operate with empirical feasibility. large numbers and also in
situations of ”everyday life”, although in it it is not Note: Chi2 test =0.011 sig.
= 0.917 These data in Table 4 indicate that there are no significant differences
between the The students are then familiar with problematic situations the
cows are not weighed in grams nor is the capacity of the water tanks specified
in milliliters. localities, at least in terms of the final result obtained. 84
whose results should not necessarily be anchored in reality and therefore this
Regarding the availability of the calculator, these results (See Table 5) show
that there is no significant difference between those who had a could lead them
to not establish this common sense reference to determine their calculator and
those who did not have one. Table 4. Distribution of the results by type of
locality Machine Translated by Google strengthen the four basic operations of
elementary arithmetic. However, in 31% got a correct answer. Of those who
had a calculator, a total of 173, the become a disadvantage when high school
students did not have their horizontal we see that of the 156 students who
solved the problem without a calculator, use is more permitted by teachers.
On the other hand, this apparent advantage may calculator does not guarantee
better results if you do not properly understand the resolution difficulties. Such
difficulties with the algorithm appear in the roundups Note: Chi2 =1.684 sig.
194 38% get a correct answer. These data may indicate that the use of the
availability of a calculator and possible lack of practice with algorithms could
cause decimal, in the quotient position, for example. Of the 216 students who
did not give the correct answer, half had at their disposal prevent students
from using the calculator because they feel it does not allow them to problem
or it is not known how to interpret the result obtained. that some do to find
the quotient of the division or in the handling of the point 85 It is possible to
analyze this table in two senses: in the vertical sense we see that from Some
teachers with whom we have spoken mention that in primary school correctly,
58% were able to benefit from the technological device. In the sense Secondary
school most of the students use the calculator on a daily basis and their a
calculator and the other half not. Of the 113 that solved the problem 100%
100% 53% 31% 38% 58% 66% 47% 62% 42% 50% 100% 50% 100% 69% 34%
100% 100% Table 5. Results depending on the calculator. 48 Total 216 329
Total general 108 Wrong right 173 Availability no calculator 113 65 156 with
calculator 108 Machine Translated by Google 24% 26% 58% 21% 26% 20%
23% 20% 66% 100% 100% 100% 71% 100% 24% 28% 100% 74% 100% 26%
31% 26% 100% 30% 34% 100% 41% 29% 42% 59% 57 80 incorrect correct
B (s/d, 35/70) C (c/d, 20/40) 23 Version 64 87 35 84 A 23 (c/d, 35/70) 46
32 78 329 49 216 113 D (s/d, 20/40) Total Total ”transparent” (20 and 40)
more students correctly solve version D than version By joining the versions
with the distracting data A and C and contrasting it with the union of the
Note: c/d: with distractor, s/d: without distractor problem. Remember that
the difference between version A and B is that the first second does not have
it (s/d), so the latter is considered to have a lower degree of version C (41%
against 29%). The data shows that indeed more students correctly solve the
contains irrelevant data (one minute) that works as a distractor (c/d) and the
41
difficulty. The same is true for versions C and D. versions B and D, which do
not have it, the difference between the results obtained (See table 7). version B
than A (42% against 26%) and, in the same way, in the versions with quantities
86 Table 6 shows the results obtained according to the version of the Table 6.
Results according to the version of the problem Machine Translated by Google
Table 7. Distribution according to the versions with or without distractor 162
(100%) Sin distractor 167 (100%) versions Total (B y D) incorrect correct Con
distractor 95 (58%) Total 67 (42%) (A and C) 216 (66%) 113 (44%) 329 (100%)
121 (72%) 46 (28%) numbers in the text of the problem change, from being
35 and 70 in the first ones, they change to 20 can face a problematic situation
by applying guided resolution schemes The difference between versions A and
B and versions C and D is that the values demonstrated that distractors
significantly affect performance, in terms The difference is significant (with
0.008 of sig.) and this suggests that the students generals. are necessary. On
the other hand, in the literature on problem solving, it is 87 then concentrate
on finding the relationships between the data and discard the ones that do not
there are no major differences. Using some data, students could modify this
idea and allow them to Once again, if the versions are grouped according to
the type of magnitudes, it can be seen that face various problematic situations,
with variations in the relevance or otherwise of Numerical numbers could make
it easier or harder to find the proportional relationship between them. appear
in the text of a problem should be used in the solution. It is through and 40
in the second. This difference was made under the hypothesis that the values
by the belief (theorem-in-act) that could be expressed thus: “all the data that
Machine Translated by Google Table 8. Distribution according to the type of
magnitudes (C y D) 216 (66%) 113 (44%) 329 (100%) versions 35/70 Total
Incorrect Correct Total (A and B) 113 (66%) 58 (34%) 171 (100%) 20/ 40
103 (65%) 55 (35%) 158 (100%) of the problem and its resolution procedures.
the conditions of the problem can change, if very large quantities are used or
students wrote, which in some way indicate the representation that was made
the reasons why practically those two-thirds did not resolve function of the
magnitudes used. However, it is also known that the magnitudes students.
The difficulties caused give us the opportunity to analyze which were properly.
To do this, the responses that the participants received were classified into six
categories. represent the relationships between the data in order to discard
those that are not relevant for calculus is a more complex activity than calculus
itself in With these data it can be inferred that, for this particular problem,
expected because it is a ”problem” and must offer some resistance to the Note:
Chi2 = 0.029 with sig.=0.865 the problem was solved correctly and therefore
two thirds did not. It is The previous results indicate that a part of the students
who faced as reported by Vergnaud (1994). 88 very small, decimal or integer
and also depending on the domain of experience, Machine Translated by Google
Table 9. Distribution of results by categories IV.2.1 RESULTS BY RESPONSE
CATEGORIES IV.2 ANALYSIS OF THE DIFFICULTIES 82 23 3. Correct
numerical result but no reference units reference) Total 113 7.3% 34.4% 7.0%
6. Lack of understanding of relationships between data Frequency Percentage
42
7.0% 23 2. Difficulties only with units of measurement 19.5% 100% 1. Correct
answer 64 24.9% Category 5. Inversion of the data in the division (without
units of 329 24 4. Inversion of data in division (with reference units) through
clinical interviews in the next phase of the investigation. For this analysis For
example, who answers ½ meter; the student could not do written operations
and only the graphic productions of the students were available and therefore
the Category 1. Correct answer classified the 329 responses of students, both
primary and secondary. arrive at that result by mental calculation (without
ruling out that you can do checks). information is limited. Table 9 shows the
distribution of the results in six categories. HE In this category were located the
answers that we considered correct (N= 113, 34.4%) It is important to clarify
that the following interpretations will be sought to be confirmed by 89 and can
be expressed in various ways depending on the procedure performed to obtain
it, Machine Translated by Google 90 In the example (See Image 5), Brandon
arrived at the correct answer, which It implies having coordinated several
processes without neglecting any of them. If we resume analysis of the problem
we can then make the following inferences: Picture 5. Problem A, Brandon age
11 years 8 months (11.8) primary school A (urban), no calculator The schema
activated to face a situation in which it is involved class of problems where
proportionality comes into play, that is, theorem-in-act of proportionality, the
concepts-in-act of the problem statement (problem reading). He understood
that what allows to recognize the concepts and theorems-in-act, to generate a
proportionality, coefficient of proportionality, measurements, the notion of • In
the first place, it has a scheme that allows it to identify at the level of resolution
procedure and monitor its performance. In the following function among others.
operational invariants, the relationships in the problem to locate it in the the
proportionality of two variables allows you to properly interpret points these
invariants are described but they are related to the Machine Translated by
Google ƒ(nx1)=n ƒ(1) ÿƒ(1)= ƒ(nx1)/n That in Spanish it would be expressed
(returning to the analysis that Vergnaud meters) is equal to 70 times the size
of a step, therefore, the size times. Hence the 35/70 split.” [2001] makes this
property) “the size of 70 times a step (35 of a step is equal to the size of 70 times
a step (35 meters) divided by 70 91 found in students who have solved this
problem. • Correctly identified that the time data is irrelevant and does not
take it The division that it performs is based on the theorem analyzed above:
have been 1/2, a procedure that is also correct and has not been done until now.
• He operated with the scalars, that is, vertically in the analysis of the (50 cm).
taken into account in your calculations. allows you to locate the problem within
the proportionality situations. Figure 3. In case of operating horizontally, the
division would have to operate now with another unit, centimeters, and write
it conventionally • Established the relevant relationships between the data.
Brandon’s scheme steps of different sizes when walking and cover the same
distance result to whole numbers by multiplying the dividend by 100 to with
the same number of steps. quantity and not a possible range of measurements
that would indicate that there may be of the decimal numbers in the division.
He then chose to convert the pide is the average of the size of each step of
43
Héctor, since he gives only one numerical, although it is appreciated that it
had difficulties in proper handling • Was able to determine the appropriate
reference unit for the result Machine Translated by Google The prototypical
example is that of Miguel Ángel (11.6) in the sixth grade of primary school in
decimal metric, which combined produce responses such as “5 cm”, “0.5 cm”,
“5 mm”. correct numeric but assigned the wrong unit of measure. problem
in the conceptual clarity of two notions: decimal numbers and the system
In this category are located the results of the students who reach the result
write 5 cm. as a result. In most of the answers classified in this category, a
Category 2. Difficulties only with units of measurement (N23, 6.9%). they did
not correctly solve the problem, because they faced some difficulty in one or
quotient and properly handles the decimal point in division. However, in the
end several points above. In the rest of the categories, the answers of all the
students who correctly the algorithm with the support of other multiplications
to calculate the recognition of the proportional relationships involved in the
problem, apply a rural locality (See Image 6). The division you choose rests
on the 92 • Does correct numerical calculations, although with some detours.
Image 6. Example of resolution classified in category 2 Machine Translated by
Google 93 decimal numbers, knows that they represent a fraction of the unit;
so as a continuous quantity, in addition to conceptualizing the characteristics
and logic of both the account in solving a problem is the numerical result. As
Jeovany (14.0 units to the final result (See Image 7). unknown; So, only meters
left. But from the experience they have had with the Accepting that it is
possible to write 0.5 meters requires conceptualizing the meter decimal number.
Then they deduce that the unit must be centimeters, which are importance
of specifying the type of magnitudes as an integral part of the result: what
years) who after some attempts reaches the numerical result 0.5 although he
does not add units “meters and steps”, he could have thought that not steps,
because it does not correspond to the meter and that is how they are called:
”meter”) and therefore cannot be accompanied by a fractions of a meter, and
that explains this type of response. They didn’t consider it important. This
may mean that they have not understood the dilemma of which unit of measure
to assign. As you have surely been working with the In the example shown,
Michelangelo arrives at the numerical result 0.5 by half material,” such as the
wooden rulers teachers use on the blackboard, which measure a possibly they
did not want to face that dilemma and decided not to write anything, or else
that of the algorithm (but the same happens with those who use the calculator)
and faces the under construction and therefore cannot be properly coordinated.
integers”, as discrete quantities (possibly representing a meter, ”concrete or
In category 3 (N=24, 7.29% of the total), those students who During the
interviews, some students comment that ”meters are Although it requires a
more detailed analysis, it can be anticipated that these are notions Category 3.
Correct numerical result but without reference units. You may have considered
that it is a contradiction to write “0.5 meters”. decimal metric system, like the
numeral system. Machine Translated by Google Image 8. Example of category
4 resolution. Image 7. Example of resolution located in category 3 attempt
44
in which it does write ”meters”, which indicates that it could have been a
simple forgetfulness division 70 steps/35 meters (N=64, 19.45%); understood
the problem statement In this example it can be seen that he identified the
problem data, discarded result two meters, which also ruled out. Possibly
he considered that it was not possible which did not specify the units. time
correctly and initially operated with the 70/35 split, obtaining as take steps
of that size and by means of another calculation, not represented graphically,
arrived correctly chose the data and the appropriate algorithm, they obtain
as a result 2 and the Category 4. Inversion of the data in the division (with
reference units). The vast majority assign a unit of length (such as meters). to
the result 0.5, this time without defining the reference units, unlike the first 94
In this category are grouped the results of the students who choose the Machine
Translated by Google described in the analysis of the relationships depicted
in Figure 3. What is the reason why by the study by Fischbein et al. (1985)
show that indeed students The two possibilities are not mutually exclusive
and it is necessary to deepen 35/70 as it would correspond according to the
relational calculus by means of scalar operators through clinical interviews to
confirm them. However, the results obtained Those who answer two meters
is because they divided 70/35 instead of 2 steps per meter (uncommon) and
get the correct result. that a person take steps of 2 meters; then possibly they
thought of another alternative: reverse the divisor and dividend or change the
interpretation of the result by this procedure, as in the example of Image 8,
but they did not consider it feasible It is considered that there are at least two
possibilities: 35/70? Some students who gave a correct answer (Category 1)
earlier passed which one fifth of all students divide 70/35 instead of dividing
95 about the divider”. produce a graphical representation in the following
order: write first left to right: the 35; then, to the right of it, they write the
galley and at the end, inside it, the 70. The directionality of the alphabetic
writing system ”interferes” with that of the • Another possible explanation is
that they operate under the theorem-in-act “on division, the With the above
it is represented graphically (the arrow indicates the division algorithm, so
students write the division in the The larger number always takes the role of
dividend while the smaller number, the directionality of writing) the statement
”thirty-five out of seventy” of and they write it in the order in which it is
expressed orally and from left to right, order and directionality of alphabetic
writing. That is, if they decide to divide 35/70 Machine Translated by Google
didactic. It surely intervenes based on the relationships established between
the conflict by adapting the text of the problem to its result. For example John
Paul (12, 3), It is true that these answers can also be interpreted in terms of the
contract conflict. However, there are those who do it and still maintain their
result and solve the problem. Whoever accepts this result as correct generally
does not evaluate the answers, but they could not be considered independent
and exclusive. For example, Vergnaud’s perspective are called theorems and
concepts-in-act. feasibility of taking two meter steps in the real world and
therefore does not cause you any Other types of considerations would also be
relevant to explain these of the following category, which together make up a
45
little more than a quarter of the They face new situations that put them to
the test. These invariants are the ones that since 96 answers (26.35%). define
its role in the graphical algorithm and therefore explain this set of responses
and the students to develop invariants that continue to direct their activity
until they operate under the tacit model of the characteristics of numbers
in the operation to use types of problems with these characteristics that are
the first approach to (See Image 9). teacher, students and the division, since
during the teaching of the algorithm from primary school C (rural), who says
”He doesn’t walk, he hops and jumps two meters” Image 9. Modification of
the text of the problem due to an incorrect result. Machine Translated by
Google we could say that they did not understand the problem or that in that
specific situation they did not 97 result 2 and that way you write it without
needing a reference unit. It is necessary to highlight that only a quarter of the
students are from those who perform one or more operations with the data of
the problem, adding or not units procedures for its resolution. of measurement.
Others decide to make some intuitive approximation of the size of the steps.
proportional relationship and therefore its resolution procedures are oriented to
answer. For example, Jaqueline (13, 11) from the urban high school, obtains as
involved in this type of problem and therefore cannot generate they obtained
and prefer not to write units or do not consider them as part of the difference
is that they only give the value of the magnitude but without reference units
(N=82, 25%) or simply did not understand. They could not establish that have
faced various problematic situations that help them build the notions (N=23,
6.9%). They can mean that the students were not clear on the result The
results of this category have the same characteristics as the previous one. The
understood the relationship between the quantities expressed in the statement
of the problem Students whose response was classified in this category may
not have Category 5. Inversion of data in division (without reference units) In
this category were grouped the results that make us infer that the students did
not half are primary school students (65.8%). Category 6. Misunderstanding
of the relationships between the data. They had the necessary schemes to
deal with it adequately. Of that 25%, more Image 10. Example of category
5 resolution. Machine Translated by Google This category was divided into
5 subcategories, due to the characteristics of the What is expected of them
when faced with a mathematical problem is that they perform an operation
and give a results (See Table 10). answer, but since they are not clear when
establishing the relationships between the data of the 6 a.m. Here the results
were grouped that allow us to infer that the students 98 problem, then some
choose to multiply them, others to add or subtract them and give Ernesto
(11.9) from primary school A, gives us a typical example of this kind of a result
to which some unit of measure is assigned (N=20, 24%). they assume that they
must comply with the expectation of the situation and consider that what is
resolution (See Image 11). Sub-categories 20 6 d 54 100 16 22 6 6 26 6 c Total
5 5 18 Primary % 6 6 b 28 4 17 Total 1 7 6 a 11 6 and 24 21 21 Secondary
82 16 12 Image 11. Prototypical resolution of Sub-category 6 a. Table 10.
Distribution of the Sub-categories of Category 6 Machine Translated by Google
46
This is the case of Jaqueline (14.4), from an urban high school, who as a
process of 6c. In this sub-category are grouped the results of the students who
make an intuitive approximation of the size of the steps without apparently
taking into account the data of the problem; or if they had been considered,
they did not use them to operate arithmetically and they give answers based
on an intuitive approximation using length units of the decimal metric system
to express the size of the steps (N=17, 21%). units (N=21, 26%). units,
allow us to infer that the allocation of units is one of the main solve applies
division and multiplication to the data in the unassigned problem These data,
together with those corresponding to the categories in which they do not add
difficulties that students experience when solving a problem and therefore a
Viviana (14, 6) from the urban high school, gives two result options without
units of reference to the numerical result (See Image 12). mediates some
written operation, apparently by approximation (See Image 13). important
issue for teachers to consider. 99 6 b. Here the students do what the students
in the previous category do, but do not add Image 12. Example of resolution
of Sub-category 6 b. Image 13. Example of resolution of Sub-category 6 c.
Machine Translated by Google students are very ”elementary”. 6 e. In this
group were classified those answers that cannot be included in much interest
because they show that the mathematical conceptualizations of these be small
and then correct and decide to be large (See Image 14). Image 15). students,
who represent 7% of this category and less than 1% of the population Laura
(15, 4) from the Rural secondary school decides that the steps are small (See
the previous ones and that show a frank lack of understanding of the problem.
There are only 6 It is noteworthy that the responses of sub-categories 6 c and
6 d contradict in a way 6 d. Here were located the results of those who make
the same approximation as 100 total. as a reference for their answers and they
give answers such as ”small”, ”medium”, numbers given to obtain a result. In
any case, they are responses from “normal” (N=18, 22%). These results are
obtained in both primary and secondary. For example, Ángel (14.9), from
the urban high school, first thinks that they should in the previous case but
without specifying units of length of the metric system decimal flagrant the
didactic contract, since at least they would be expected to use the values Image
15. Another example of resolution of Sub-category 6 d. Image 14. Example of
resolution of Sub-category 6 d. Machine Translated by Google rejection of the
activity. approximation of the number of steps and not of their measurements
(See Image 16). It can also be answers given without reflection or as a sample
of 101 Like José Alfredo (15.4), from the urban high school, who responds with
a Image 16. Example of resolution of Sub-category 6 e. Machine Translated
by Google IV.3 CONCLUSIONS OF THE PHASE RESULTS although they
probably have constructed the notions involved, they still require the theory
of conceptual fields, the construction of concepts does not occur from 102
recognized a proportional relationship but, due to problems with the units or
with the reference, then the students who gave an answer from the second class
(41%), consolidate and coordinate knowledge about the numbering system and
about the construct the implied notions (for example, proportionality) and that,
47
as correct answers, which integrates the answers in which it can be inferred that
indicate that the students did not understand the proportional relationships
in the require mastery of the multiplication tables, whose memorization is
essential, so a single situation, but through many and varied situations and
in the long term; to the concepts and not isolated concepts as conceived by
Vergnaud. In our study it is algorithm, they did not give an adequate answer
and the third that includes the answers that algorithms. The first is not
enough to give a correct answer; besides, At the same time, a situation always
involves several concepts forming systems of If it is assumed that those who
adequately solved the problem (34%) have complete results (with units), since
its specification is not only they are precisely developing those notions. problem
or were not sufficiently involved in its resolution. decimal numbers and the
metric decimal system. Furthermore, they must take care to express their
of a quarter of the population and the majority are primary school students,
who Based on the analyzed results, three large classes can be identified in
which succeeded in coordinating both the numerical calculation with the
determination of the units of The responses of the third class may suggest that
the students still lack which we can group the answers given by the students:
the one that integrates the managed to construct the notions involved in the
problematic situation and have also conventional but is an integral part of the
solution. EXPERIMENTAL Machine Translated by Google 103 variation in
the magnitudes of the problem, which did not generate significant differences
in the irrelevant as a distractor is an element that causes greater difficulties
in support in some aspects of the problem solving process, for example in
characterization of the resolution procedures with the help of the calculator
and without they could be explored further to find finer distinctions; without
know the scope and limitations during problem solving. we coincided Although
only the results of a first approximation are reported, but it could that they
expressed to operate with the mobilized concepts. significant in the results
obtained, at least statistically. However favored better results, if the criteria
we used to operators, with the use and interpretation of the reference units
and with the interactions between the student and the interviewer in testing
the theorems and based on reasoning routes, procedures that students results.
students to get an adequate representation of the problem, unlike the arithmetic
calculations and verification of numerical results. But he. This was due to the
fact that during the clinical interviews, topics arose that were However, only
the statistical tests were applied to the data collected in the first with some
positions that are in favor of its use, that is, that it can be a Regarding the
hypotheses initially raised, the presence of a piece of data having carried out a
methodological design that would allow reporting in greater detail the On the
other hand, the differences between the results of the rural and urban localities
We believe that it is a topic that should be further explored in greater depth
to determine a correct result (correct numerical result and reference units).
concepts-in-act identified. Exploring the domain of different content followed
during problem solving and graphical and non-graphical representations On the
other hand, the availability of the calculator did not generate differences We
48
differ in that it is a support in relation to the identification of the relationships
between the results analysis. That is why we consider that it was not a resource
that gave them priority, such as changes in meanings due to the Machine
Translated by Google Mastery of mathematical tools. Regarding this issue and
despite the between rural and urban, however it is possible that with another
strategy rural and urban populations. different evaluations, but that does not
mean that they do not reason as they have sometimes correct answer. to be
confirmed or refuted in subsequent investigations to know if indeed these tools,
mainly, is that difficulties arise and they fail to resolve Nor do they allow one
to explore the existence of other theorems-in-act that may be mathematician
than students from urban locations. That is, it is easier algorithms well
mastered, mathematical notation well learned, both in the skills that would
allow them to adequately solve the problem. Difficulties Finally, it becomes
evident that there are differences in the systems of the problem; however, given
the methodology used for this phase, the data does not results we maintain the
hypothesis that students from rural locations can methodological differences
can be found between the resolution procedures and the With all of the
above we can conclude that most of our students do meant to suggest when
interpreting the results of standardized tests, the 104 there is a way to deal
with clearly different mathematical situations between phase of the field work,
from which it is concluded that there were no differences the expected way the
problems and can be the cause of the low scores in the determining resolution
procedures that prevent students from reaching a resolution could assess their
results based on their empirical feasibility, but this will take production and
interpretation (writing and reading). Lacking mastery of representation and
degree of consolidation of the mathematical concepts involved in allow us to
confirm the nature of these relationships between representation systems. more
easily use common sense when solving a problem they reason, that is, they do
perform the pertinent relational calculations, they activate the schemas they
arise when they lack the solid mathematical tools to operate. That is to say the
mainly international. Machine Translated by Google resolution of the problem,
as well as clinical interviews and analysis of the procedures 105 To deepen the
above, direct observation is required during the that students do, which we
expose in the following chapters. Machine Translated by Google symbolic representations and mathematical conceptualization in solving problems Finally,
from this microgenetic analysis, axes of secondary solved the problem of steps.
The records obtained allowed us depth in clinical interviews according to our
object of study, namely, the relationship of problems. Andrea was willing
to work in depth on each of them. In the first place, the way in which 329
primary school students and graphic representations used, interpretations of
the problem and knowledge microgenetically the case of Andrea and document
it in this chapter to account for analysis to be explored in the rest of the
clinical interviews. We also highlight 106 classify their answers and identify the
diversity of resolution procedures, of Subsequently, of the 30 clinical interviews
carried out, it was chosen to analyze particular topics that were of didactic
interest. difficulties encountered by students. mathematics as well as the
49
co-construction of knowledge in a specific situation. mathematicians mobilized
to face the situation as well as to identify the main of the complexity of the
procedures, both of the resolution of a problem sixth grade student from an
urban school. We consider it pertinent ”experimental phase”, we were able
to identify themes that should be explored more themes that appeared in an
isolated way in the rest of the students, in addition to the fact that describe
the methodological path followed to reach this point. This chapter reports the
microgenetic analysis of the clinical interview with Andrea, a With the results
obtained in this stage of the investigation, which we call This case was chosen
because in the preliminary review we noted that it covered many CHAPTER V.
Results of the microgenetic analysis Machine Translated by Google V.1 THE
CASE OF ANDREA interviewer not only collected information for research
purposes, but also Situated microgenesis approach as an interview analysis tool.
The interview is analyzed in fragments that correspond to the application of
one or so that Andrea could adequately solve the problems raised. He assumed
the role of ”teacher”, seeking to propose situations that favored the For this
reason, the methodological and analytical principles of the as well as resolution
procedures. interviewer is guided by the schemes that are activated by the
situation faced only a data collection instrument, the interview became a space
107 various procedures. We assume that the activity of both the student and
the modification of those theorems by other more productive ones. Therefore,
instead of being procedure, but will be used to describe something that we
consider pertinent new reflections that would allow, firstly, to recognize them
and, secondly, (both one and the other). The components of the scheme will
not be described in each of didactic work in which the errors detected were
used to promote student actions during the resolution activity. The notion
of schema that During the interview, theorems-in-act were identified that
generated obstacles written production (Dolz, Gagnon, and Vuillet, 2011).
Vergnaud proposes in his theory enables us to explain both the concepts
mobilized The theory of conceptual fields allows us to analyze in sufficient
detail the highlight. overcome them, in a similar way to what Joaquim Dolz
proposes for the teaching of Machine Translated by Google Image 17. Andrea’s
productions in the experimental phase. Andrea canonically represented the
written division algorithm and they had at their disposal a calculator that they
were free to use. Despite this, the multiplication. solved correctly. Furthermore,
it is interesting that he has repeated the same problem group gave an answer
considered correct. In your group, everyone two divisions have the operators
interchanged (70/35 and 35/70) and then the Image 17 shows the result and
the graphical operations it performed. operation with the same result, because
generally when students perform He obtained it, apparently, through the V.1.1
DESCRIPTION OF THE CASE 108 The answer it gives is .5 meters6 . a
public school in the capital of the State of Veracruz. In addition, it can be seen
that he carried out a verification by means of a In the first phase of the study,
she was one of the students who in the application Andrea (11.3), at the time of
the interview, was in the last bimester of sixth grade at division algorithm, an
operation he performed twice and with the same result. 6 Transcription code:
50
between square brackets the expressions and productions of the interviewer,
except those of the student. Written productions are underlined. Some
clarifications or comments are in parentheses and some key expressions in the
analysis are highlighted in bold. When a phrase or words were not heard or
understood, we put the number and F or P respectively in parentheses for
a phrase, for example (1F) or 3 words (3P). We emphasize with bold what
we consider important to highlight. Machine Translated by Google V.1.2
MICROANALYSIS of his graphic productions. It lasted 59 minutes. Andrea
showed up with the This second phase, in turn, was segmented according to
sub-goals or sub-objectives due to During the interview, Andrea is placed in
an evaluation situation in a and gestures during problem solving and identify
order and directionality restlessness of not being left with doubts and, realizing
that in reality he did not master to test those hypotheses and, as far as possible,
try to modify them. She solves problems and is convinced of her results. as a
teacher, so she assumed the role of student. The interview took place in a The
interview was prolonged, but she was constantly monitored for fatigue. the
difficulties encountered, the procedures chosen or the interventions of the 109
school context with the figure of an interlocutor to whom she surely attributed
the role some basic questions, he wanted to understand and learn them. It is
for this reason that the interviewer. administrative staff and talked to each
other, which at times made it difficult to listen clearly. The student chooses the
result that he considers most appropriate. Despite having his For the analysis
of the interview, it is divided into two phases. The first consists of the small
teachers’ room where two teachers filled out some forms Andrea, throughout the
interview, appears very confident and willing to The interview was videotaped
with the intention of recording their verbal expressions. Andrea’s hypothesis
and the approach of new problems by the interviewer respond to questions
and problems raised. Verbalize what you think while available a calculator,
performed written operations and developed them. Andrea’s expressions were
clear. resolution of the problem and its verification and the second in the
exploration of the Machine Translated by Google how long is each step [aha]
(divide 70/35= 2) and gives two [and gives, two. how fast, we consider the
assumption of a didactic contract (Brousseau, 1997). meanings. So , should I
divide 35 meters by 70?... so that it gives me the result of you are very fast...
And two, what will it be? Write your answer] two meters [two could commit
to adequately solving the problem raised, which resolution, if necessary, and to
the extent possible, build new distance of 35 meters. He expresses orally, with
some doubt, that he can obtain the Steps? [As from where to where] (Indicates
an approximate distance of 2 meters) To achieve the goal of finding the average
size of each step, she 110 Andrea reads the problem and understands that
Hector walks 70 steps and covers a meters?] Yes [Ok, what do you think of
your result?] Well, how big are the expresses orally that he plans to divide
35 meters between 70 steps, with some doubt, inferred different from the
one stated: 70/35. It can be assumed that Andrea recognizes a problematic
situation that must be result “dividing 35 by 70”; however, it does graphically
represent a division [So they are very big, but that gave] aha. The interview
51
is aimed at exploring the knowledge that Andrea is capable of Steps? [What
did you understand?] I say that he walks 35 meters in 70 steps, Therefore,
she as a student could be considered in an evaluation exercise. In doing so,
mobilize to identify these problems, reformulate its procedure of Reading the
problem and initial resolution Hector walks 35 meters and takes 70 steps.
On average, how big are your resolve by finding yourself in a school context,
and as already said, with a teacher and V.1.2.1 FIRST PHASE Machine
Translated by Google express graphically and orally as two meters; In addition,
he recognizes that they are very large Division other than the one that he
enunciated orally. Carry out the algorithm and get determine the result. It is
considered that it is not only about the passage from the oral to the written,
ability to switch from one register to another between representation systems.
But, operator function proportional (:2) to the unit value of a step to find its
or not, and neither does it autonomously recognize the difference between its
verbal expression and the division, graphically represented the oral expression
of the operation using the rules theoretical), it is possible to find the value of a
step through two operators, applying the may be used to in the classroom, she
concludes the process of the result by means of the scalar operator (70 times
less) without dimension. In the oral expression ”should I divide 35 meters by
70?... for him to give me the answer that she considers as the solution to the
problem: 2 meters. The change between the important. the steps. quotient 2
without specifying reference units, which at the request of the interviewer she
but that the graphical representation has conceptual implications both of the
algorithm of What specifically would such an ability consist of? 111 written
in the division, nor that the order of the “data” in the division is important
to of interrogative intonation. During the activity graph 70/35; is a of the
alphabetic writing system. In terms of Duval (2006), this would imply a lack
scalar operator (70 times less) to the known measure, in this case the 35 meters
or the resolution. She does not consider herself the possibility of checking if
the result is correct However, instead of graphically representing the operation
by means of a Oral expression and written representation of the division have
conceptual implications. As can be seen in the schema of relational calculations
(See Chapter Possibly, due to the situation of interaction and the level of
demand to which division as the proportional relationships at stake. result
of how long each step is” adequately expresses the operation to find Andrea
autonomously complies with the objective (of the contract) of giving a Machine
Translated by Google Figure 8. Scheme of relationships by scalars. we would
need a place that was 35 meters and... we would measure step by step obtained
with the real feasibility of taking two-meter steps, but this is not enough [How
could we do if we wanted to check that your result is correct] step... it could
be [It could be] or we add it up [do we do it?] yes, 6 meters the same Andrea
correctly solves the algorithm and manages to relate the result Check Request
112 to deduce that it is wrong. 3 steps, right? And so... (He rotates his hand as
a gesture of iteration, of repetition) with correct. She proposes two alternatives,
both valid. For the first, comment that corresponding measurement in meters.
The interviewer asks if there is any way to check if your result is correct. find
52
the value of the ratio of the proportional function (2) and not the measure of
a step. the second is to “add it up”. Andrea divides 70 steps by 35 meters,
which in the scheme would imply ”We would need a place that was 35 meters
and... we would go measuring step by step”; Machine Translated by Google
Image 18. Correspondence table. (See Image 18). The first alternative refers
to a specific plane of representation and the second of the activity by means of
the graphic records in a correspondence table board). practices and is asked
to do the second (although it must be admitted that it would not have been
12 m meters - 6 steps refers to an arithmetic plane; the first check is discarded
for reasons 6 m meters - 3 steps so then I multiply it From this moment on,
Andrea’s actions are directed to another the table of two [Let’s see, do it. It
is a way to check it and it is very good and 48 m meters - equal to 24 steps
bad idea to go to the schoolyard and put it into practice). 24m meters - equal
to 12 steps completely convinced and it is already verified] yes [if we do it
it has to come out of 35 meters. To achieve this, she structures a strategy
that allows her to take control the same] uh huh [you would be too lazy to
do it] no [do we do it?] (Complete the idea] and 35 is double 70 and since
they are two steps [then you are objective: to determine if taking 70 steps of
two meters is enough to cover a distance 113 Machine Translated by Google
steps down the line; in the same way it does it with the 48 meters and the
this is almost done (he laughs as he points to the twelve meters and then the
oh…no, or yes? No, because here you have to take up to 70 steps (points to the
column of between the two columns: in the first it represents the distances in
meters, with a series 114 I haven’t already passed it. Oh no, I haven’t gone
too far. Here until it hits 70 (points to the column and I already spent even
more [aha]. allows you to identify the next number in each of the columns,
but at the same time (56)] because it is double of... (Realizes that there is a
mistake) then they would be 48) should be here (it makes two small marks,
one in the column of 384-192 steps... (Looks at his correspondence table) I’ve
already passed [have you already that the number of steps corresponding to 6
meters is 3. table, relying on the characteristics of the graphic representation
that somehow 48 steps) [(I point to the meter column) and here you add 6, 12,
then...] twelve steps from the line below, then point to 24 meters and then 24
the steps). [Follow him]. that starts with 6 meters and that is doubling at each
level; in the other column represents of the steps). [Let’s see, follow him]... oh
no, I already passed, yes I already passed [why?] 56 m meters - equal to 48
steps As seen in Image 18, Andrea identifies and reproduces a relationship time
loses control over the objective of the procedure and the goal to be achieved.
here 96. [96?] Yeah and see... [How far do you have to go?] Until... the...
ah meters between 24 and 48, and another in the column of steps, between
12 and 24) Andrea finds regularities during the construction procedure of the
They confirm the appropriateness of your procedure. Generates a calculation
rule that 24 because it is twice 12 [and here (48)] 48 because it is twice 24
[and here? 192 - 96 steps did you pass?] yes because it is 35 meters (indicate
the number in the meter column the steps and, since he considers the steps
to be 2 meters, calculates correctly Machine Translated by Google 115 mark
53
graphically. purpose of your procedure. However, it is from the written record
that she can add 12 + 5 = 17) [17 what is it?] 17 steps and one meter left
over. [But tell me 17 steps of 2 meters and you say that there is 1 meter left
over] uh-huh [and this is the result of measures. The result he arrives at is very
original because 35 meters correspond to 24 meters out of 35, so we still have...
11 meters to go. But? [11 meters for 35] uh-huh eleven [ah, 11, I’m missing
11 for 35, equals...] to 5 steps and one meter left over [and In this sequence of
actions, he systematically varies a quantity in it would be 15 (write 15 below
the 12) yes, right? [I did not understand you] well we are here he advances 24
meters it’s because he took 12 steps, we’re going well up to there] uh-huh [then
you said, why did you choose 6?] times two (refers to 3 (steps) times 2 equals
6) (nods) [and is this a way to check?] yes [are you convinced?] yes Concludes
the series with a relationship of 35 meters is equal to ”17 steps and Let’s see,
then, well, we already have 12 steps (12) and we are missing 6 (he means that,
redirect the activity to the initial goal. Locate an arrival point in each column
and (point to the sum) what are you doing? 17 steps, what does it correspond
to?] It would be the check if you can take steps of 2 meters] uh-huh [you say
that in order not to have 17.5 steps of two meters each, but to avoid decimal
numbers, which makes [uh-huh] we are 11 meters short, we already have 12
steps, we are 6 plus 5 more (write the The interviewer is required to intervene
for her to regain control and the this 12 what is] 12 plus 5 steps is equal to
17 [it means that 35 meters is equal to proportion to the other, which means
that it identifies a proportional relationship between both (make a mark in the
meter column below the number 24); we have (24) is missing 15 for 35, here
are the 15, I point to the second result] no, [100%?] mmm yeah. “There is one
meter left.” as 12 steps and 24 meters correspond, there are 6 meters left to
reach 30) and that... [Have you already screwed up?] yes [don’t worry, we’ll
understand him (I point to 6 m) [ok, if he takes 3 steps he walks 6 meters, if he
walks 12 meters he walks 6 steps] (nods) [yes to go to a distance of 35 meters
and walk it we can do this (table)] Machine Translated by Google ... [do we?]
yes, 6 meters equals 3 steps right? And so... (turns his hand as a gesture of
iteration, of repetition7 ) with the table of two. if he advances 12 meters he
takes 6 steps] (nods). 116 wanted (35 or 70). However, he applied the rule
systematically in both columns, relationship. He expresses it recurrently and
through different signifiers: • two (refers to the fact that 3 steps times 2 equals
6) [ok, if he takes 3 steps he covers 6 meters, and applying “multiplication by
two”, it would not arrive at either of the two numbers the table and cleverly
and ingeniously solved the series finale. She considers completely convinced
and it is already proven] yes. The conclusion you reach seems acceptable to you,
but it does not correspond to the • identified certain arithmetic regularities
based on the distribution of numbers in ... and 35 is double 70 and since there
are two steps [then you are data of the problem, since Héctor walks 35 meters
taking 70 steps and not 17, which meters per step is correct. is to keep the 17
steps and the half step, which must measure one meter, he prefers to leave it
• Correspondence table. wrong way, which was an effective way of checking
that your result of 2 • • The difficulties of this procedure are related to the
54
magnitudes that it identified from the beginning and that the result obtained
in the division coincides with this ... [Don’t worry, we’ll understand (I point to
6 m) why did you choose 6? ] initials you chose to build the correspondence
table; He did not anticipate that with them, as residue. It can be inferred that
Andrea is focused on the relationship between the two magnitudes ... [and
here (48)] 48 because it is twice 24. 7 Note that the gesture of turning the
hand is also a signifier, with which we emphasize our main thesis, referring to
the relationship between mathematical conceptualizations and representation
systems. It is not useless to remember that Piaget identified the origin of the
first semiotic systems with the internalization of action schemes. Vygotsky
also emphasized the internalization of gestures as the origin of representation
systems. Machine Translated by Google interviewer wants you to realize the
mistake and try a new strategy to is observable for her (given the sequence of
reasoning, the starting data ÿ2 is the [What does the problem say?] Hector
walks 35 meters and takes 70 steps on average, because it allows you to observe
the relationships and therefore attribute the lack of I think it gives 50... no, yes
it can... because it would be centimeters (add cm to the with the problem data.
She correctly reads the data and identifies that it is different from her Andrea
finds the error to be in the split after a review, which 350 centimeters (divide
by 70 you get 50 and cross out the division) but it is countersuggestion (70)
so since they are 35 meters I thought they were two meters each meters] 17?
No... yes [but here in the problem does it say how much?] seventy [and here
very convinced] uh-huh [later we wanted to check but you’re already doubting]
uh-huh [for [let’s see] (looks at the sheet for a moment) you couldn’t because
there are 70 steps (points to the find the correct result. Then begins another
sequence of actions aimed at resultÿ and the difficulties encountered due to the
random start ÿthe relationship 3-6ÿ). He how big are your steps? [He says
that he walks 35 meters and how many steps does he take?] correspondence
to the result of the division and not to the verification procedure. 117 result
but keep thinking about the two-to-one relationship between 70 and 35. it
means a loss of control of the activity (”he balls up”); it is something that
is not it may mean that the verification procedure seems more “transparent”
to you wrong... [have you had enough?, why do you say that this is not (I
point to the division)] why not She is asked some questions so that she can
identify that her result does not match and I already divided it (he points to
the 70/35 division) or maybe the division is wrong. where would you like to
start] by checking it [which] the two (division quotient) data in the text of
the problem) between 35 meters (points to the data) and each meter are… to
achieve these new partial objectives. It would be that instead of 17 steps it
would be 17... [But how many steps does he take to advance 35 how many did
he give?] 17 but here it is easy because this (35) is twice as much as this [Well,
we already have several problems here, one, you did the division and you were
Machine Translated by Google performance during the activity, is generally
circumstantial and therefore temporary and but for a time it causes him some
difficulties. problem text). (2011, p. 33). The first, we will use it to describe
a fault product of the local. On the other hand, the error is a systematic
55
failure indicating the process of loses 3500 (three thousand five hundred). Later
he realizes the error and corrects it, because 350 cm is fine because they are
35 meters (point to the data of 35 meters in the it was not possible; Then,
solve the difficulty of making a division with a dividend less than the This
wrong calculation does not mean that she does not know how to multiply by
multiples of At first he considered the possibility of dividing 35/70 but thought
that construction of mobilized knowledge. 118 350 cm and not 3500 cm. This
strategy is very appropriate, relevant and simple. Without quotient) and would
be fifty point meters (add a decimal point to the left in which it could have
influenced that, in its conversion logic, it considers that each meter divisor
through the erroneous conversion of meters to centimeters, since it affirms
that they are 10 or that does not dominate the conversion; It is perhaps a
mistake when mentally calculating meters?] uh huh , but I don’t think that...
there I have a question [what] I don’t know why, but we consider a mistake to
differentiate it from the error as proposed by Dolz et al. centimeters, keeping
in the result the lexical base of one hundred (three hundred), which is I feel
that it is wrong [yes?, what do you feel, let’s see?] that the division is wrong
of the quotient and write one m to the right of cm) [¿50 centimeters is equal
to 0.50 However, he does an incorrect mental calculation by multiplying 35 by
100 to get 350. it is 100 centimeters; and if it is 35 meters, the result would be
three hundred and fifty Image 19. Division that Andrea considers to be wrong.
Machine Translated by Google Once again, the interviewer seeks to explore
whether she manages to sustain that result. ball up, 50 cm times four (write
4) [why times 4?] 4 times 50 is the same result may be wrong by looking at
the division quotient. After doubting, think to the problem, the procedure by
which he arrives at it is not clear to Andrea. convinced that this is the correct
answer. In this segment we can see [How could we do to verify it] I am going
to write it here so that it does not He performs relational calculations based
on writing, that is, he deduces that his a decimal point and the abbreviation
for centimeters. However, when seeking to specify that they are conversion
and manipulation of the decimal point. 119 that it is possible that the result
is 50 because it is about centimeters and then add clearly the interconnection
between the empirical feasibility of the result, the millimeters. When rewriting,
correct and only leave 50 cm. If he had done the 350/70 division correctly, he
should have obtained as a result The following sequence of actions is intended
to check if 50 cm corresponds to 50 centimeters, actually represents graphically
0.50 centimeters, that is, 5 new check partial and gets 50. However she initially
says she ”feels” wrong, but Although the final result is correct, since it responds
as expected the first check. Start the series with “4 times 50 equals 200.”
more for the result than for the algorithm’s resolution procedure. 5. But not
fully mastering the algorithm, you arbitrarily add a zero to the quotient the
right answer. Andrea again makes a table of correspondences as in Image 20.
Complement the quotient with units of measurement. Machine Translated by
Google meters (mark a decimal point to get 35.0). [Let’s see, explain this to me,
200 cm is equal to 4 steps] because 50 plus 50 (count 2 Well yes, but that’s how
I go… [Okay, 4 what would it be] four… steps? (add a pa (draw a horizontal
56
line under the quantities and add) and give 350 and with your fingers) there
are two steps, there are 100, then another two are 200 and here there are 3,
4) then it would only be another 3 steps equal to 150 cm so it is added here
it is 7 p.m. Steps. (Write 7p in the left column.) 120 are 150 and here (the
result) would be 350 centimeters (add cm) which would be 35 to 200 (write 4=
200 cm) [I didn’t understand you, why 4 times? Could it have been 6?] Image
21. Relationship between the algorithm and the correspondence table. Image
22. Sum of the components of a correspondence table. Machine Translated by
Google Initial error identification convert it to 3500) [kind of works itself out,
right?] yeah. She identifies with only the interlocutor’s question and doubt
that 350 centimeters that the result is correct even though Andrea thinks it
is. initial: that 35 meters is equivalent to 3500 centimeters and not 350 cm,
as she considered. calculation. You are asked to check again, the interviewer
considers that now The following intervention by the interviewer seeks Andrea
to reflect on the error equals 3.5 meters and easily corrects, confirming that
it was a mistake 350 centimeters [a little? How do you know that?] Because
meters have 100 Its verification has two characteristics: it starts with an error,
since [Tell me why you put 350 (in the division)] because it is 35 meters and
it is equivalent to everything will become clearer for her. of correspondences
with the data of the problem: 7 steps and 350 centimeters is similar to meters
to be 350, then it would be another zero (add a zero to the dividend to 121 70
steps and 35 meters. Again the procedure does not allow a clear demonstration
Consider that 35 percent is equal to 350; establishes a relation of the result
of your table centimeters... [Aha] oh no... then it would be three point fifty,
three point five Image 23. Division 3500 cm by 70. [Look, I would like you
not to worry, we are testing the options] uh-huh [do you want try it like this?]
yes [try it, do it here if you want] (does a division 3500/70 =50 does not write
Machine Translated by Google residues, only the quotient) gives me 50 [does it
give you 50?] yes, no... to 500 (doubt) [What did you write here? (3500/70)]
Seventy out of 3500. Image 24. Obtains 50 or 500 as a quotient. of the division
is done “by columns”, which means that the dividend is of one of the other
three operations of elementary arithmetic. In the second part of Paraphrasing
Martínez (1997, P. 71) the procedure for the usual algorithm breaks down, in
this case, into units of thousands, hundreds, tens and units. HE usual division
algorithm. The same does not happen if you use the calculator or if you try
depth this relationship and seeks to problematize their distinction. that the
relative value of each one be considered. this interview, once the resolution of
the problem is finished, the requires dividing each of these elements separately
or joining two or more figures without Once Andrea corrects the conversion
from meters to centimeters and writes the division 122 About the division
algorithm In the division Andrea wrote: problem posed, is now faced with the
resolution of the algorithm. This particular relationship between oral language
and writing only occurs with the corresponding to the operation that you
plan to carry out, and that it is adequate to resolve the problem. Machine
Translated by Google Image 25. Division 3500/70. steps of the algorithm,
it is absurd to seek to distribute the zero and find that it touches zero Let’s
57
analyze Andrea’s procedure: forming 350 tens. 350 tens can be distributed
among 70. Touch 5; this 5 This last step is counterintuitive, since if you want
to be systematic in the times. This step caused difficulties for some students
who faced this the logic of the numbering system can deduce this. Hundreds
cannot be divided between 70 either, so they are ”joined” with the tens, Since
there is nothing left over, zero is written as the remainder; the following figure
is “lowered”, which in remain and may be related to the mistake Andrea made
when dividing 350/70 and means 5 tens and must be written above the dividing
point in the tens place. problem because it is at this point where they do not
know what to do with the zeros that 123 distribute the ”zero” among the 70,
which touches 0, which is written in the dividend in the An alternative way
would be to reason it like this: “We divide the 350 tens among 70, this case
is a zero. If we wanted to continue with the procedure we would have to look
for get the result 50 instead of 5. 35 zero. units, the quotient is complemented
by writing zero units”. but only dominating 70; Since it cannot be done, it has
to be ”joined” with the hundreds, forming 35 hundreds8 . First, you want to
divide 3, which actually means 3 thousand units, by units position. Finally, the
remainder is written, which, in this case, is also what touches 5 tens; there is no
more to distribute, the result is 5 tens, that is, 50 8 In the terminology used in
a system to teach algorithms called ”The Arithmetic Table” (The Arithmetic
Flea in its digital version, 2003) there was talk of ”changes” from thousands to
hundreds, etc. See: http://www.uv.mx/cpue/coleccion/n_2526/publjor3.htm
Machine Translated by Google (as a discrete quantity) cannot be divided by 70
because it contravenes a rule 124 it would be 50 (write a zero in the quotient)
[good] and now 50 [what is that 50, Oral level: ”I don’t think 3 fits in 70”,
which may mean that 3 for her fundamental of the division that is that each
one receives the same; if he distributes the three As already mentioned, it
is counterintuitive to continue with the same allocation procedure; quotient)
because it goes 5 times and zero is left over and we lower the zero and it does
not give anything, (add a c before the m). would be 35” which arithmetically
means that 3 thousand units are converted to what does it mean?] 50 meters
per step (add one m), no, 50 centimeters per step There will be 67 who don’t
get anything. Then he expresses ”so we’ll see if it’s for the second, Evidently,
Andrea dominates, at the level of operative invariants, a series of [How do you
solve a division like that?] I don’t think the 3 will fit in the 3 who cannot
divide 35 by 70 and takes the following figure, which would mean converting
the hundreds and add to the existing 5 hundreds resulting in 35 hundreds.
He realizes it fits, but it can’t be added... then we’ll see if it works for the
second, it would be 35 the procedure mentioned above, but again express it in
an inverted way to zero left over The next step is to lower the next number,
which in this case is a zero. As but it does not work either because it is half
of 70 so it would be 350 and it would be 5 (in the [that the 3 does not fit in
the 3?] no, in the 70 [that the 3 does not fit in the 70?] well, yes knowledge
of both division and the number system; for example, use 35 hundreds to tens
forming 350 tens. He does accept that, and deduces that it is 5 and Image 26.
3500/70= 50 cm. Machine Translated by Google centimeters (add cm to the
58
multiplication result) and nothing is left over. In terms of Saada-Robert and
Balslev (in press), the interviewer seeks that space of if the result is correct. It
makes him see that the result obtained is the same as that obtained what?]
let’s see... (Multiply 50*70=3500) no, yes it gives [yes it gives?] yes, because
they are Define the procedure that most convinces you Then it self-corrects
and concludes that it is centimeters. You are also required to check You can
check the result by doing a multiplication. division. Before performing the
multiplication, anticipate what the result should be, but 125 in a different
division. She is wanted to anticipate what she will do and she mentions that
To achieve the goal of checking, recognize the inverse operation to the [well
now let’s check it] aha [how can we know what that is] she solves it apparently
without difficulty, expresses orally ”we lower the zero and it doesn’t give
surprised when she realizes that 50*70 does indeed equal 3500. Finally, [Very
good; Well, he already gave us the same thing that he had given you here,
right? 350/70= 50] uh-huh through some kind of relational calculus, you infer
that you won’t get it. It shows It is necessary that the interviewer asks you to
specify the meaning of the quotient 35 [let’s see?] no, 3500 because they are
centimeters. But it doesn’t give, I don’t think it will give [why have residues
and are part of the result; expresses that nothing is left over. found. First
think in meters, perhaps because of the original problem statement. nothing,
it would be 50”. multiplying? [What would you multiply?] 50 by 70 [how
much should it give?] should give Andrea knows that in the division, unlike in
the other three operations, she can Image 27. Check 50*70= 3500. Machine
Translated by Google seems more appropriate. the interviewer’s meanings with
Andrea’s. problem) [Point five what?] Meters (add an m next to the 5) [meters,
why [Do you know a tape measure?] Yes [where would the answer be?] look in
the 126 division and the multiplication that proves it) [Yes?] Yes [What would
be the numerical representation and the concrete plane is asked to mark on a
tape measure its Mark correctly on the tape measure the point that represents
the measurement of the two?] this one that went out to two (points to the
first division (70/35=2 m) and this verification procedure by means of the
correspondence table. Although it was expected that she would directly choose
the latter procedure, Andrea where point five meters would be (I point to the
result he wrote)] here? (points to I saw… I just remembered that I did it last
time and it also gave me this [So which of everything you did convinces you the
most] I am most convinced by these The interviewer asks you to review your
results and choose the one that best suits you. if you were using 50 and now…]
because they are centimeters (point to the last few far left (as other children
have done, and select 5 centimeters) answer to the question?] point five (write
0.5 next to the question of the interaction to determine if in the work done so
far there are coincidences between result. passed. Furthermore, the notion of
equivalence operates correctly by mentioning the term and (3500/70=50 cm)
but I think this one convinces me more (he makes a mark next to To determine
that Andrea establishes an adequate relationship between the plane of the same
place) [same?] the same, because it is the equivalence [how did you see it?]
result (points to 0.5m) I just remembered. two (point to the correspondence
59
table and the last division (3500/70) [what operations) so I changed it to
meters. considers that the first one also convinces him, possibly because of the
clarity of the ah, it’s centimeters right? So it would be here (correctly points
to 50 cm) [and in Machine Translated by Google [What division would you
have to do on the calculator...if you had to do a 127 that helped you because:
what you “write” or type into the calculator, as you do with the usual written
division. operation, which one would you do?] It would be 70 out of 35 because
I made a mistake here (points to The application of this problem was solved
with the help of a calculator and considers correspondences) and if I had it
wrong, then I quickly saw the result and nothing more do you know which one
goes in and which one goes out?] because we would have… the…. what… if it
would be With her I was not making balls with the answers (points to the table
of 70/35=2) 70 goes outside and 35 goes inside and that’s why it went wrong
[how The interviewer regains control, suggests solving the problem with the
pointing to the same place for different signifiers 50 cm and 0.5m. counting 35
steps, it would be the other way around. So it would be 70 steps on centimeters
posted the results... thus (points to 70/35=2) it would be the 35 meters (points
to the dividend)… we would be From your comment on the use of the calculator,
we explored the way to intention to confirm if it maintains the correspondence
between what is expressed verbally and 28). “write” the same algorithm on
that device. She mentions that in the group phase of Solve using calculator
calculator, but first he asks him to verbally express what he will do on it,
this with the (points to 70 and 35 while enunciating it from left to right. See
Image Image 28. Say 70 divided by 35 and write 35/70. Machine Translated
by Google conventional written division, says that on the calculator he will
divide 70/35 and points to the the data in the division, you know that changing
them will get a different result. She says theorem-in-act: the distance always
goes inside the house. From this invariant [okay, write it down (I pass you the
calculator)] (type 35 / 70 = ) equals point five that must be problematized
and therefore analyzed. Because of the difficulties they experience false even
though for her they are true (Vergnaud, 2009). would you dictate] 35 divided
by 70 [ok, here what division did you make (I point 3500/70] seventy presents
and can get a correct result. identification of invariants in the interaction
process, also builds theorems between seventy [What is the division you would
make] this (mark with a circle 35/70) [I’m going to lend you the calculator and
you’re going to do the division, okay?] Uh huh [what are you going to In this
fragment it can be seen that Andrea actually “writes” in the of children who,
like Andrea, do not establish the distinction between oral expression and 70/35
division, also explicitly mentions ”70 goes outside and 35 goes inside”. That’s
why it ”went wrong”. Also for the first time he expresses verbally and in a
surgery is that you will face the following situations, causing you that, in some
[aha, so it’s very good, right?] yes [because you work very well] yes (laughs, he
writes 128 In the following sequences of activities we can identify the genesis
of a In this part of the interview, Andrea already establishes a difference in
the order of between three thousand five hundred [and here what division did
you make? (70/35)] 70 out of 35 Inversion in the graphical notation of the
60
division algorithm is a component en-acto that allow you to interact with
the situations faced, but that are also [if you were to tell a child write this
division (I point to the same 35/70) how calculator in the order in which it is
expressed verbally. This data confirms that many written representation of
the division, with the use of the calculator this problem is not Despite these
significant transformations of their knowledge, from the Sometimes I can’t find
the right answers. write in the division, what are you going to write here] here
would be thirty-five your name on the sheet) Machine Translated by Google
V.1.2.2 SECOND PART OF THE INTERVIEW First theorem-in-act. In the
division the distance goes inside (of the house) and problem you solved and
the interactions with the interviewer allowed you to arrive at this the best
way the work undertaken and his desire to learn, he was raised a about these
distinctions. others outside... This is the construction he made during the
first part of the interview. He Due to the interest that Andrea showed in
the activity, the effort made to make with the interviewer, will construct new
explanations related to the issue of the order of I don’t know where to put the
data in the slice, but it has happened to me in conclusion because initially he
divided the steps between the meters. It is important to highlight voiced by
Andrea; otherwise they might have remained implicit and therefore series of
problems based on the ideas that he was formulating so that, in interaction
[From the division, what is the...? What is the hardest for you?] I have a hard
time sometimes that it is through the interviewer’s specific questions that these
ideas are Andrea expresses some ideas that are problematized by the researcher
and goes students, it is a content that probably has not been reflected enough
in write outside?] because it would be the distance… because if it is distance the
the data in the division. rarely [if that is difficult for you, how do you decide
what to write inside and what 129 division and the significance of the graphical
representations in the algorithm. The student is given a series of problems
that allow them to reflect distance that you already have complete would be
inside. the school. It requires a detailed analysis of the situations where a
reformulating until he manages to recognize that he must reflect on some issues.
problem would be the distance inside and the steps or whatever outside, the
Machine Translated by Google Image 29. Conventional writing of the division.
What can you think of?] An ordinary child can be one hundred and ninety
three meters conventional and general to various situations. [Let’s see, let’s do
some exercises] An example? [For example of distance in a race 193 m but they
are divided between fifty-eight stations 58 would remain for a long period until
they could be replaced by other more From knowing this idea, the interviewer
tries to invent some situations At the end write down the divisor. See Image
39) [here is the calculator] (does this Countersuggestions to your hypotheses
so it would be 193/58 (write the dividend first, then draw the galley and
determine the order of the data in the division, that, when changing the order,
the result Therefore, they would have generated unproductive rules of action in
the situations that most each station would be] three point thirty-two meters.
problems where Andrea, on her own, recognizes that it is important operation
on the calculator and write the result it gives you) 3.327 [then determined by
61
the characteristics of the problem situations in which therefore it will require
a new, more general construction. 130 I could eventually put them to the
test. Surely some of the following ideas ahead could face. The modification,
sooner or later, would be it is also different; to realize that the distance will
not always be the dividend and Machine Translated by Google meaning of the
result. punto algo y cacho [with the calculator] (carry out the operation and
write down the result division starting with the dividend, then makes the galley
and at the end the divisor, that is, from point thirty-two meters. On previous
occasions it was necessary to ask him the In the following intervention, we try
to show you that by changing the order of the divisor, then the galley and
finally the 193 representing 58/193) would give me It is understood that you
must find the distance between each station. Second, write the division: it no
longer establishes that relationship with oral language; has been disassociated
for this case [What would happen if you had done it the other way around?]
It would give me another result that was 0.30 m.) is point thirty [point thirty
meters, it is very different from this one (points to the 131 right to left. This
represents an important change in the way of writing the data in the algorithm
also changes the meaning of the result. operation 193/58 and 58/193. See
Image 30)]. As mentioned, this is not a problem for students but, given the In
this problem invented by Andrea, several points can be observed backwards,
don’t you get the size of each station?] No (shakes her head) [What specific.
Third, type the data into the calculator in the proper way. wrong [would it
be wrong?, how could we know? So if you had done it initially mentioned,
but from a different scope than Hector and the size of interpret the result as
a distance magnitude according to the problem: three between fifty and eight
meters [let’s see, try what comes out?] write 58 as your steps. It is now a
race in which there are seasons. Although he doesn’t mention it, interesting to
analyze: first of all, it creates a problem in the terms that previous change, for
the writing in the calculator the order was not reversed. Finally, comes out?]
comes out (thinks for a moment) if it were… one hundred and ninety-three
seasons Machine Translated by Google Image 30. Inverts only the numerals
of the division but not the units. what, but you know It also anticipates very
well a decimal number as a quotient. 132 division, Andrea keeps the units
in their original position: the meters inside the inverting the data the result
will be “wrong”. There are no elements to explain why There is a difference
between the meanings of the interviewer and Andrea regarding the is correct)
or would be 30 centimeters (so next to the quotient add = 30 cm. stations
instead of subways, it’s out of reach. By changing the data in the will divide
among 193 stations. He constructed this rule to be consistent with his theorem
that identifies that if you divide meters you will get meters and if you divide
stations you will get dividend and seasons outside. Now instead of 193 meters
it has 58 meters that content that at that time is trying to problematize. While
the interviewer searches distance as dividend. This idea will remain until the
end of the interview, where in this case meters or centimeters. The following
intervention seeks to focus your attention en-acto, ”the distance goes inside
the little house”; change the magnitudes but leave the seasons, Andrea only
62
considers as a possibility a quotient of units of length, interviewer and Andrea),
because while the interviewer wanted her to realize Andrea interprets the result
as 0.30 meters; however, it is clear that [How do you know it’s meters?] In
meters because I divided it into meters (which realize that reversing the order
of the data in the division would result in In this case, we can mention that
there is no comprehension zone (between the it tries to show you that you can
change not only the numerals but also the units. in the unit assigned to the
result. Machine Translated by Google fifty-eight stations between one hundred
and ninety-three meters?] (Write first the divisor 193, secondly draw the galley
and finally the stations next to number 58. See Image 31)] Yes. [And these
are...? (points to number 193, Image 31)] meters [the first division [And what
was this? (points to operation 58/193, See Image 32)] fifty-seven dividend 58,
without resolving the operation) 193/58, which would be the same as here how
is? (points to division 193/58, Image 32)] one hundred ninety-three between
eight meters between one hundred and ninety-three stations [and can they
be divided 133 fifty-eight meters [and that gives you three point thirty-two
meters]. See Image 29) [These are stations (interviewer writes the word Image
32. Conventional reading of the division Image 31. Difficulty in assigning
units to the quotient. Machine Translated by Google result when reversing the
data in the division. Even absurd divisions are proposed 134 This reasoning is
understood when the logic with which it operates is known. quantities in order
to identify the differences in the meanings of the like dividing children between
cakes or between bars of chocolate. five 1.5 (also add the remainder 10) [what
does this mean?] a bar and 58 seasons] it’s the same [is it the same?] yeah.
difficulty of the operating rules of the division algorithm, some conclusion
that two out of three [How would you do it?] It would be… (He thinks for a
moment and does the galley Andrea: for her it is clear that distances should
always be the dividend. Before the [But now if I tell you divide… uh… three
bars of chocolate between two children] proof. (points to the first division,
193/58 stations) because the meters go here may be clear to the student she
stands her ground until something makes her of the division) (See Image 32)
[that is, if I tell you ”divide 58 stations by 193 meters” what The interviewer
uses simpler problems, from different fields and with different types divisor)
3/2 [Um… the three bars, can you finish it?] yes, I would give one point you
write like this (points to division 193/58] uh-huh [but if I tell you divide 193
meters between (points to the dividend of the last division 193/58, bottom of
the image) Differences in results when inverting the data in the algorithm ...
the three… the three bars (add the dividend) 3 between the two children (add
the Image 33. Awareness of the organization of data in the division. Machine
Translated by Google dividend, then the galley and finally the divisor. See
Image 32) 10/5 [and what solve the difficulty expressed by Andrea: knowing
how to accommodate the data. When typing in [Now if I tell you…divide…five
children between ten bars] (Write first the how much would I give you?] I
would give two 2 [Two what?] Mmm… two bars. They find themselves in a
zone of understanding, where they both share the same goal of the arrangement
of the data. In this case, the interviewer asks you to make the division It’s
63
about dividing the children between the bars. The logical and usual thing is
what Andrea answered: first the galley, expresses the conflict that must be
resolved and shows that he thinks about In this domain, experience dominates
completely and does not allow you to realize that the data inverted. media.
135 because that way they will be able to reflect on the result of the next
operation with two bars to each child, although the interviewer’s intention was
different. First speak ”two out of three”, then think for a moment and draw
first the out. galley. This action can be interpreted as both the interviewer
and Andrea You are asked to divide children between bars, which should result
in children per Image 34. Division of 3 chocolate bars between 2 children.
Machine Translated by Google Image 35. Division of 5 children between ten
bars. Image 36. Division of 20 marbles among 40 children. similar division,
but now with paddles with the dividend less than the divisor. Normally there
are twenty marbles among forty children, I would not get one for each one,
Since it is unrealistic for children to distribute “half marbles”, a [How do you
divide...ten lollipops among twenty children?] Well, the lollipops inside and
the they play?] they play…five 5… [Is it possible?] No, but I wouldn’t give it
either because that way five [um...um... are you bored already?] No [No?].
they would have to be forty and up. children outside (write down the dividend
and then the divisor) 10/20 so it would be a meaning of what divides. Still
mastered the logical relationship of the marbles between the In this division
we can see that Andrea begins to understand the 136 Twenty marbles in and
forty goes out 20/40 (write the number first). number of marbles to be able
to distribute them. dividend and then the divisor) twenty marbles divided by
forty [How many do you [How do you divide uh...let’s put...twenty marbles
among forty children?] children, and not the other way around, but he realizes
that there should at least be the same Machine Translated by Google as a
dividend the first data stated verbally by the interviewer. Now I know The
interviewer focuses Andrea’s attention on the operation of dividing children
operation without being prompted) the result is five 5. It does not represent
any difficulty for her and until now she has been systematic in writing change
the meaning of the relationship. What will be divided are children between
meters of ribbon. the theorem-in-act ”the distance in meters corresponds to
the divisor” prevails. [Five meters of ribbon between ten children] (Write 5/10
and this time solve the real possibility of dividing meters of ribbon between
children. It must be remembered that Andrea meters] (corrects and writes
about the previous operation and solves it) 10/2 would be between meters.
Andrea orally repeats the proposed division, correctly expresses how he realizes
that he operates with another relationship, he distributes the ribbon meters
among the children. Now a division is proposed to you with the divisor greater
than the dividend and with the [And if you divide ten children between two
meters of ribbon?] 2/2 [ten children between two place the data in the division
and marks an operation with the same data. But not [Now twenty meters of
ribbon between two children] twenty meters of ribbon between two It raises
the division correctly, although it does not solve it correctly; besides, Andrea
changes the meaning of the relationship: instead of dividing the children among
64
the 137 Consider that the meters go inside the house. five 5 [five what?] m five
meters each. [Er... can this division be possible?] Yes [ten children between 2
meters?] Yes (touches his invariance that will become a theorem-in-act: “the
first object stated will be the without writing the residue) 10 to ten meters each.
that what is first stated is what goes inside the house. on this occasion dividend
and the second, the divisor. here it can be seen that it is through repetition that
he begins to identify a children 20/2 (write the dividend, then add the divisor
and at the end the quotient meters of ribbon distribute the meters of ribbon
among the children. Furthermore, it contravenes the idea Machine Translated
by Google Given this response, a similar division is proposed, but now with a
divisor data order. It is until the next division that questions the possibility
of dividing the children and not the slats. Andrea assimilates the relationship
and therefore on this occasion theorem of the first stated fact. less than the
dividend and then another division with the same data but inverted. Andrea’s
answers are based on logical reasoning; she does not attend to Another division
is dictated to him to emphasize that he is now being asked to divide well in
the logical relationship. his fingers between the sheets that are inside a folder)
touch two [and children. children? [See, that’s what I’ve been saying for a
while now (laughs)]. It takes the first stated data as criteria to use it as a
dividend. It is based more [Ok and a child between two meters of ribbon] (This
time he does not write, he introduces [Um... let’s see, how do you divide...
five children between...?] How can I divide the ribbon] 10/1 (write the divisor
first and then the dividend) touch ten 10 hair) [tell me what are you going to
divide?] is divided… uh… the two children (points to the well at half a meter
[very good, well the truth is, look, I’m making you 138 [Very good, let’s see
if we can do another more difficult one, a child between ten meters high One
meter between two children?] (Moves the pen and does not write the operation
either) It is a detail that takes time to process, it is a construction of inside
(points to the dividend) and the two children outside (points to the divisor
and adds Again, the theorem of length in meters prevails as a divisor over the
good] (touches her hair and shakes her head). next to the dividend the letter
m). divisor) between... (Points to the galley) [Aha] the ten meters of ribbon
that would be because ten divided by one equals ten. trick questions to see how
you answer, but the truth is that you work very Machine Translated by Google
that the divisor represents a length in meters. It is the first time in which that
meters… fifty centimeters [uh, how else?] half… half a meter divisions again to
confirm understanding. possibility of dividing a child between two slats write
the division correctly despite criterion is no longer a priority and also verbally
expresses the result in a creative way would you say?] children [how else would
you say point five meters?] point five Once both share the idea of being able
to divide children between strips, they consider [then we are going to suppose
that a child can be divided between two meters of to see if it occurs to you
to apply the terminology of other fields different to this one. [and in kilos?]
they were kilos... half a kilo, fifty grams [and if they were children?] [If I told
you a child between two meters of ribbon] it would be two meters between a
child and correct. It does not occur to him to mention half a child. So other
65
questions arise. half a child (laughs) it would be... I just don’t know how
to say... well it could be... child who meanings between the interviewer and
Andrea in this zone of understanding that little by little half child: 139 ribbon]
1/2 [how much does it give you there?] (Solve the operation) 0.5 would give
me about five The following questions explore whether Andrea can express
0.5 of child as Once this route has been completed, the difference is expressed
between dividing meters of ribbon problematize the distinction between the
order of the data and the result obtained. Andrea expresses that it should be
two meters between a child, but in the hypothetical Meters [and if they were
kilos? point five, what?] kilos [and... if they were children, how would they
Accepts the possibility of dividing the children between the slats and solves
it correctly little were coinciding up to this point. This detour was necessary
in order to be point five of child. [If they were meters, how would you say
it?] If they were meters? [Yes, how would you say them?] V.1.2.3 PARTIAL
CLOSURE Machine Translated by Google begins to carry out the operation,
writing down the dividend first and then the dividend and then the divisor) 2/4
=0.5 car equals point five car I understand. [Now divide… uh… ten popsicles
among five children.] (On a new piece of paper divisor) 10/5 = 2 that’s it [It’s
easy, isn’t it?] yes, I already realized that the first [Divide hmm… two cars by
four people] (Again add first the we divided two meters of ribbon between a
child, did you get this part yet?] Yes, ok, ok above reasoning holds. At the
same time, Andrea explains the Andrea expresses that she has noticed that the
first word spoken is the [car? Is that what it says?] Yes. of which it knows that
the graphical representation will be adequate. Doesn’t mention half a car To
consolidate this achievement, new problems are posed in order to observe if the
word is inside. It is observed that when writing the dividend, it is focused on
controlling this data, from Second hypothesis: The first word is the one that
goes inside between children and vice versa. interviewer: the first word acquires
the role of dividend. Andrea was able to 140 generalizations you made during
the interview. divisor, which is correct in the divisions stated orally by the as
the interviewer would have expected. and you say that it touches two meters,
but no, because what is being divided is the child dividend greater than the
divisor. with previous relations, changing the scopes and Andrea solves them
correctly. and we divided that between the two meters of ribbon, it is very
different than if [Notice that a while ago I asked you ”how much was a child
between two meters of ribbon” It begins with a simple or intuitive division:
distribute popsicles among children, and also with the recognize this invariant
and also make it explicit. More divisions are proposed Machine Translated by
Google Robert and Balslev (in press) mention regarding the meaning that is
discussed and correctly the result but it is the interviewer who considers that
it is incorrect. then it would touch a quarter of a pie 0.25, point twenty-five
[let’s see, in an inverted way; does not carry out the operation but by mental
calculation obtains 0.5 and four divided by two] (write the dividend and then
the divisor) 4/2 [is that easy? Because there are four children between a cake,
or else it would be the other way around (write the aptly as a quarter. hand
the division 4/2= 0.5 children, to solve the operation again)] four sheet) 4/1
66
aahhh… 4 to four [four?] four children (write next to the quotient) It is notable
how Andrea inverts the direction of the division and in case of dividing a 25)
[are you going to divide four children between a cake?] yes, the four children
are divided giving the same result: because… the children are the ones that are
divided by four [mmm… no] yes [no] built by the different participants: Andrea
does not accept that she is wrong and defends her position. It shows what
Saada explain it to me again, it’s very interesting] (silence) [to see four children
between one mentions that 2 children between 4 cakes would be 0.5 children,
which is correct even though the 141 divisor, then the galley and at the end
the dividend) 1/4 [I don’t know, let’s see] yes and In the next division it is the
interviewer who makes a mistake. andrea express In the following fragment it
is observed that a simple division of 4/2 is interpreted by pies between two kids
oh no I mean four kids between two pies [divide there kids [let’s see, there’s
a mistake there] no, it’s okay because it’s one times four, four. pie between
four children anticipates very well that the result would be 0.25, which he
interprets [Let’s see, and four children between two cakes?] 4/2= 0.5 children
[why does this give you?] No? [let’s see, there (takes the pen and gives it to the
interviewee, then covers it with the [Now four children between a cake] (Write
the operation again on the pie] would be to point twenty-five (point to the
operation you just wrote 1/4= . between one [and four children between one
gives us four] yes. Request was to split 4 children between two cakes. You are
prompted to repeat the operation and continue Machine Translated by Google
write down the remainder 0) [so you said they were boys] oh, yes, two boys,
that’s quickly) 2/4= 2 children [but finish it all, how do you divide?] two after
the divider) 40/80 [doesn’t they become difficult for you anymore?] (He makes
a movement with Once you have managed to build this action rule, you are
presented with a problem 142 [Um... let’s see... how do you divide twenty by
fifty?] Twenty by fifty (points to the 20/40 split)] yes, but this is very different
because…uh…it [thirty minutes?] (Nods) no, maybe and I’m wrong because
Andrea can now graphically represent the division that is expressed to her in a
(Write in the following order: the dividend, the galley, and the divisor) 20/40
[ago [Er... five pesos between twenty meters] (First write the dividend and the
final This... How do we do to... know if... a person in an hour manages to
do how long does it take to make a drawing?] Hmm… How many drawings?
[Twenty] that I got balls here (points to the previous division 4/2= 0.5) [okay,
are we going by two, four and zero left over (this time he does the complete
procedure and the head of denial). where the first stated data does not have the
dividend function for the purpose of 20/50 [you won’t do it and… forty between
eighty] (add the dividend and division?] yes [do it] (gets 5 again) [let’s see,
again] there is…is it… two (writes The first thing you tell me is what goes inside
[aha]. would be…sixty-twenty in (write in another order, the divisor in the first
orally, as shown below: for a while you told me that the meters go inside, that
the distances go inside twenty drawings?] But... like... in an hour you have
sixty minutes... [How much (Write the dividend and then the divisor. See
Image 34) 60/20= 30 min understanding?] yes. The following intervention tries
to compare the first theorem-in-act with the second the divisor) 5/20 [wow]
67
(laughs) [Ehh… twenty kilos between forty people] challenge your hypothesis.
Machine Translated by Google were thirty minutes, it couldn’t be because... if
not, it would only be minutes [the what?] hour (he thinks and writes next to
the quotient) = 3 min minutes, it can’t be, and this one? (points to 0.3 hour)]
well, it’s period (to the quotient 60/20= 30 min)] would be twenty minutes
between… (He thinks for a moment and stays [Because? Let’s see, what are
you going to divide? (hand covers anterior division quiet) would be… [Let’s see]
(Silence, write 20/60) [Yes, you can do these add a zero before the three) 0.03
which means 0.3 hours (one tenth is equal to 6 Are you doubting? It’s okay
because it’s not a common question] but yes minutes, 0.3 tenths is equal to
18 minutes). divisions?] Yes 0.3 is to point three, (drops pen on desk) [point
three 143 place 60 and after the dividend 20, See Image 37) 20/60 Image 37.
The first item of information stated is not always the dividend. Image 38. 20
minutes between 60 seconds is equal to 0.3 hour and this is equal to 3 min.
Image 39. Adjust the ratio as required. Machine Translated by Google 144
conversion and applies his rule: the first item of data is the dividend and he
obtains as data of the hour that appears first is not the dividend; on the other
hand you have to minute because what he divided were minutes and not hours.
The interesting thing is that at The foregoing shows that, in solving a problem,
not only results in a [what does point three mean? (take the pen and write .3
hr = 3 min, See properly, the operation and also returns a correct numerical
result, but the Determine how many minutes “0.3 hour” equals. The interview
then takes another basic and whose difficulty is widely reported (Charles,
2011). cover the division so that she orally expresses what she will divide and
then the rule procedure is only applicable to a decimal system and the hours
and minutes are based on it’s 3 minutes. Andrea generalizes the procedure
to convert the units of quantities and their relationships. very unlikely to
be used in school or everyday life, but involves the result 30 minutes. When
the interviewer repeats the result back to her, she considers that convert to
minutes and the quotient is a decimal less than one. she does the ask him what
that result means, he says ”point three... hour” and makes the equivalence
to obstacle the mechanics of the algorithm, but also the interpretation of the
units of the Image 37)] this point three... then... point three would be from
before... it would be... interpretation of the unit is not because instead of 0.3
of an hour it would have to be 0.3 of The impromptu problem causes Andrea
many difficulties. On one hand, the direction: clarify how much 0.3 hours is
equivalent to in minutes. [Well, this brings us to trying to understand what
point three o’clock means] yes above can be effectively applied in this case.
She represents in writing, a sexagesimal system. Realizes something isn’t right,
but it’s complicated She tries to convert what she considers to be 0.3 hours to
minutes. It is a content with use of rational numbers that must be understood
by education students there may be an error and then it hesitates, inverts
the data in the division. The interviewer minutes as if it were a conversion
from the metric system and expresses that length from meters to centimeters
to hours and minutes. However, that quotient and also plays an important
role the scope of the problem, the magnitudes, Machine Translated by Google
68
Image 40. Graphic representation of three tenths of an hour. and that the set
of ten tenths form the unit. the bar that divided. See Image 40) [why do you
divide it by three?] Because it is rational numbers, knows that 0.3 is equal
to 3 tenths and that each tenth is less than one It coordinates very well the
different forms of representation of the decimal fraction: 145 in three the bar
drawn on the sheet) would be here (write .3 in one of the sections of because
then we mark point four 0.4 after 0.3). the 0.3; on the other, it establishes the
equality in writing between the decimal number and in the form of equal to a
third of an hour 0.33 =ÿ [ah, what if it were point four?] it would be around
here On the one hand, in the bar that represents the integer, it marks the
approximate point where it would be. Andrea, despite having spent almost an
hour in the interview, is very interested There are ten tenths and of those ten
tenths it will not reach one... the tenths third of an hour. This is an example
where the different forms of representation, fraction 0.33= 1/3; In addition, he
verbally expresses 0.33 correctly: ”is equal to a tenths [if we wanted to convert
it? imagine this is an integer (draw a notes that he has a good command of
the subject of decimal numbers and some aspects of conceptualization of the
mobilized mathematical content, in this case the use of numbers horizontal
bar)] yes [where would three tenths be?] (Takes pen) divide are equal to one,
then (points to the 0.3) [so what is the point three?] three in understanding
and learning, she is still very active and committed to the task. HE Oral
language, writing and the iconic representation of the bar coincide with the
Machine Translated by Google Image 41. If a tenth of an hour is 6 minutes
then 3 tenths are 18. six because there are six in every tenth of an hour [mmm…
that is a bit difficult and time operates on that result to convert the decimal
fraction to minutes, you get 18 and [Well, just so that we can move forward
and clarify things, if this is your because it is divided... no, it is by six! Or
point six? [Point six or six?] No, it’s five point of an hour?] would be… mmm
thirty 30 [What’s that called?] thirty divide it by 10; first it approximates
saying that it is seven, then it corrects and at the same hours is equivalent
to multiplying 6 minutes by 3: divide by ten [by ten, well, they are already
divided, so we count three] you understood him, right?] yes [they are things...]
I understood him better [have we gotten tired yet?] 146 whole and you have
tenths, it means that you make ten little pieces] then you can minutes (add
the word) minutes point five equals half an hour [already pen and think for a
moment) [How much would be an integer? How much would it be?] rational.
one, two, three [okay, so what would three tenths of an hour be?] (Take the
hmm… not yet. necessary to give an explanation about the meaning of tenths.
From there Andrea three (write in the following order) 0.6x3=18 min [why for
point six?] is integer is divided by 10 and the integer is an hour, you need to
convert it to minutes and then very quickly deduces that a tenth of an hour
is equal to 6 minutes and therefore 0.3 In order to continue with the theme
of the dividend and the divisor, it is done One hour [and point three?] seven
[eh…] No!... then it would be point six times Andrea does mental calculations
with some precision. For example, deduce that if the Machine Translated by
Google 147 who has understood the relationship between the decimal fraction
69
of an hour and its conversion to He mentions that it is six because there are
six in every tenth of an hour. Andrea is asked if hour and she answers that
20 drawings out of sixty. If you correctly solve this drawings?] twenty, so to
verify multiply twenty by eighteen x 20 how many drawings does a gentleman
make?… he makes twenty drawings in an hour, how much is going to divide by
what to determine the order they will have in the division. When commenting
on this gives eighteen (indicates the corresponding operation 6 x 3 =18 min,
See Image in a problem the data and you have to accommodate them; so how
do you now she is able to spell it correctly; However, when faced with a [Well,
just to confirm, I had told you that... do you remember the see another very
interesting theorem-in-act: consider that if the values are changed correct?] I
could get something less than twenty... (Silence again) [Let’s see... different
from solving a problem, we were already making divisions where we already
minutes, at least at the level of tenths. knows how much 0.5 of an hour is and
she answers correctly. It is a way to check division, what you would get would
be the drawings you make per minute. To suggest that you [How many do you
think? More or less?] Eh… (He thinks about the answer for a few minutes)
How long does it take you to draw a picture?] Well… [What do you have to
divide by what?] He is asked what he would have to divide to find out how
many drawings the man makes in one Write the unit in an abbreviated way: 18
min. Lets see the calculation made when 38) [and if it takes eighteen minutes
for each drawing, then how many do you know? What are you going to divide
and between what?] Mmm… uh… [Look carefully, word problem and she has to
decide how to arrange the data, she must think about what questions?] What
was it… was it point three? Then multiply by six and twenty minutes between
an hour, why did I ask you this question? because it is very do you know how
to write the data if I dictate it to you...] yes [but it is very different if you see
In the following fragment, Andrea is explained that if a division is dictated to
her, correct is reverse division you are asked what would happen if the data is
changed. she leaves numerical in the division the units of each one remain fixed:
uh... I don’t know [How much do you expect it to come out?, How much would
you add to it to come out Machine Translated by Google draw a boy next to
the number 10) yes, here are the ten boys (ten years ago (write the dividend
and then the divisor) 60/20 would be… [And if you change it?] [how do you
divide?] there are twenty pesos between ten children (points to the divisor and
dashes to symbolize children) and here are the twenty pesos (draw 148 you
say I’m interested] (laughs) they are… if I put the sixty inside and the twenty
outside previously) 20/60. each dash that equals children. See Image 39) then
it would be two So it goes twenty between sixty (points to the operation that he
had already written twenty dashes and begins to group two lines that represent
the pesos per [Um... let’s try what you say... if you have twenty pesos between
ten children, Twenty out of sixty [twenty drawings out of sixty minutes? And
would it matter The interviewer explores this idea and tries to prevent Andrea
from staying with it, through the following questions: [to two…]. sixty drawings
between twenty minutes [ah… but then the reverse changes] yes division (covers
the quotient with his finger, See Image 39)] (keeps silent) (Write the dividend
70
and then the divisor. (See Image 42) 5 children [you change [let’s see, let’s
see that is very cool, what you are saying to me, everything that if you divide
sixty minutes by twenty drawings?] No, because then it would be how much
does it give?] twenty pesos 10/20= 5 gives five [five?, why?, let’s see, do well
[Let’s see, what would happen if we divided ten children among twenty pesos?]
10/20 Image 42. Iconic representation of 20 pesos among 10 children. Machine
Translated by Google complete are two pesos] yes. they went inside and the
rest outside?]. Yes, later I said that the first thing you said was what went
inside and the rest went Does it touch point five?] Yes, for each peso it touches
point five children [and for the child It allowed him to express and modify his
hypotheses about the division. Recognizes that 149 To summarize, Andrea
manages to reconstruct, together with the interviewer, the process that outside
(points to the operation in Fig. 11) there it is fine [yes, it is fine there]. divisor,
but he also recognizes that the idea that changing the data in you said that
dividing twenty pesos between ten children gives you two pesos] yes [but if you
some of them were correct, such as: the first thing that is stated is the ten
pesos between twenty children (he affirms) [so, you can change… divide twenty
[Well then let’s take a tour, first did you think that the meters pesos divided
by ten children or ten children divided by twenty pesos, is it possible?] yes
[then change means that here I am dividing ten pesos between twenty children]
the division units remain fixed: Image 43. Division carried out under the
theorem-in-act, the meters are located in the dividend. Machine Translated
by Google Image 44. Division carried out under the theorem-in-act: the first
item of information is located in the dividend. V.2 CONCLUSIONS OF THE
ANALYSIS . reformulation. meters and depending on how you want to divide
it is how you are going to accommodate it. his theorems-in-act, put them to the
test and, as far as possible, promote their Well, what are you going to divide
by what... you can divide meters by steps, steps by this… we took a long time
but it was worth it] yes [thank you]. Note: The interview Andrea, we can now
infer new explanations for why students choose to Are you going to forget?]
no [well...] now for the other one [you worked very well, thank you very much,
Based on the microgenetic analysis carried out in the resolution process that
makes From this tour, it is possible to notice the advances in terms of difficult
to assign units of measurement to the numerical results obtained. lasted 59
minutes. the 70/35 split instead of 35/70 (not just because it’s easier) and why
you find it but what’s wrong is that I said that if things change it’s still what
a personalized approach to Andrea’s resolution processes to identify same, but
no, things change [everything changes, so you should always think changes in
the meanings of both the interviewer and the student. It required 150 Machine
Translated by Google 151 mathematics, leading them to relate biunivocally
each element of the expression consolidated, students use theorems-in-act from
other areas that already Thought, in Vergnaud’s (2004) terms, consists of
operations both requires then a long-term didactic work that allows students
to It is important to highlight that it was through the scaffolding provided
by the interviewer, identification of invariants, construction of action rules
and their subsequent The difficulties that Andrea experiences are due to the
71
flimsy construction of a part of all the relationships and meanings that must
be contemplated in the resolution graphically the inverted relationship. give
meaning to what he does when, on his own or through the intervention of
the teacher or in that have links between them and with the meaning, then
Andrea shows us how confrontations to different situations in which they have
to use various systems The use of the calculator could only be a support as a
complement to these verbal with a graphic signifier in a linear manner and from
left to right, which dominate, such as those of the alphabetic writing system
to write the notation conceptual as pre-conceptual about the meanings and at
the same time about operations realize the limitations of their conceptions and,
therefore, delve into the through specific questions regarding the difficulties
observed or errors committed generalization, and if necessary, its reformulation.
If we consider that the Faced with situations that involve concepts, notions
and procedures not yet the concepts involved during the resolution and not
to ”their ability to reason”. of a math problem. It is important to highlight
Andrea’s intense surgical activity through the In this case, the interviewer
realizes an error. symbols to find the regularities and at the same time
identify their differences. reflections and on par with the domain of algorithms.
However, it only refers to causes that before a correct verbal expression of a
division is represented symbolic on the signifiers, which form several different
symbolic systems, employs the different symbolic systems of representation,
operates with them and tries to detail of the mathematical contents and
written algorithms, through Machine Translated by Google a very difficult
task due to the working conditions in which he works: the number of concepts
and theorems-in-act that lead to the understanding of the problem, as well as
the the student to confront their ideas and manage to overcome them. This
would mean for the teacher students per group, the large amount of content to
be addressed in reduced times, because the interviewer has a referent of how
relationships evolve between different level of complexity (Flores 2005). labor.
However, you can find some strategies such as working in pairs, influence of the
conceptual aspects that give rise to the problems having a extra-class activities
and, finally, what Elsie Rockwell calls the double day as observed with Andrea,
show cognitive activity to make sense of the by Andrea, that her theoremsand concepts-in-act are made explicit and put to the test 152 The errors that
students make when facing a mathematical problem, group reviews, etc. having
generated rules of action that were not very productive in resolving situations
underlying reasoning and from that they would propose new situations to help
belonging to the same conceptual field. This scaffolding process is possible
thanks to for her. Otherwise, they could have remained for a long time
and, therefore, faced situation. It would be very important for teachers to
reveal the Machine Translated by Google understanding the relationships
between mathematical conceptualization and systems of as well as in Andrea’s
microgenetic analysis (See Table 11). The results gave us an overview of the
responses of the 329 students and the representation, which in the experimental
phase remained at the level of inferences, we chose experimentally, we build
some observables to explore in the analysis of the 30 The clinical interviews
72
are part of the second phase of the field work and are This time allowed us to
do statistical analyzes to find similarities and differences of Based on Andrea’s
microgenetic analysis and with the results of the phase clinical interviews. The
data was classified into 12 themes grouped into 4 aspects of conceived as a
methodological resource that would allow us to delve into the according to
the variables that we included in the methodological design. To delve into
153 the resolution of the problem that was highlighted as important in the
experimental phase interpretations that we made from the results of the first
phase, whose Problem Relationship between the symbolic-concrete planes Algorithm reading with alphabetic directionality Units Mathematical writing with
alphabetic directionality Algorithm mechanics Empirical feasibility Calculator,
orality/writing Result Problem figure Autonomous drive allocation Thematic
dividend greater than divisor Autonomous check division algorithm Correct
and autonomous resolution of the problem Carrying the decimal fraction to
unity clinical interviews CHAPTER VI. Results of the analysis of Table 11.
Classification of data from clinical interviews Machine Translated by Google
Table 4. Distribution of clinically interviewed students experimental there was
a process of preliminary analysis of the results to prepare the They are in
the biggest or the smallest, so he says that it is best to start with (the group
application and the interview) was varied, because after the phase mention
that they are the simplest facts. But he wonders if the simplest facts explored.
In the case of high school it was a period of one month and in the case of
primary schools, In this way, in the categories that were elaborated in the
experimental phase, one to two weeks apart. clinical interviews, choosing the
students who would be interviewed and the topics to be the regular cases. In
total, 30 students were interviewed as shown in Table 4. In that It is necessary
to remember that the students interviewed had already faced this 30 students
whose answers were located in the categories defined in the analysis We review
the cases and choose those prototypes of their category. 154 For the selection
of the students, we follow the recommendations made by Poincaré (1963) for
the selection of the facts to be investigated. He suggests selecting the previous
chapter, so it is also accounted for in subsequent analyses. asking them if they
remembered their result. The time elapsed between the two moments the most
interesting facts and they are the ones that can be used several times. Besides
statistical. Andrea is also included in the total, although a description of her
case has been made in same problem in group app. Interviews are usually
initiated Primary Level 8 17 Total 8 Total 17 13 9 13 30 Men women secondary
5 Machine Translated by Google Problem Correct and autonomous resolution
of the problem 15 50.0 Thematic 6 20.0 Frequency Percentage problem figure
24 autonomous check 80.0 VI.1.1.1 FIGURE OF THE PROBLEM: PROPORTIONAL RELATIONSHIP VI.1 RESULTS ACCORDING TO THE ISSUES
IDENTIFIED VI.1.1 PROBLEMATIC Table 12. Issues related to the Problem
category student to deal with the problem. It is taken up from Pierre Gréco
(Gilis, in press), who quantities, which we infer from the resource of division,
however it is interpreted. The Figure of the problem is considered to be the
first resolution scheme executed by the developed a figure of the problem in
73
terms of proportional relationships between both relations. This item groups
the data related to the topics on the interpretation approach generates a
provisional mental model of it that activates one or more In his model of the
cognitive functioning of the subject, he proposes this term to express will pose;
it is considered that they were 24 of the 30, 80%. The rest established another
type of 155 the subject’s first approach to the situation; according to him, from
said related to the resolution process and the autonomy to explore a Starts
making adjustments based on new data initial problem (which we will call
problem figure), also the data schemes that correspond to that first model of
the situation. In this way the subject The table above shows the frequency
with which students who verification of the obtained result. incorporated as
the activity progresses. Machine Translated by Google Can you solve it again
and tell me how you did it? Pray?] Héctor walks... In from right to left)
Twenty in... (1F). subcategory Split 20/40, although at the time of writing it
represented [... this problem, right now I’ll show you how you solved it. Do
you remember?] Yes [Do you that its result will be point five. Therefore your
answer was classified in the first [Umjú] No, just... divide forty [Uh-huh] Well,
divide twenty by problem, the interviewer asks you what you understood about
the problem and then how you would go about one minute Hector walks twenty
meters and takes forty steps, the size (1F) the steps he took and it’s going to
be point five (2P) [Umjú. How do you remember? 156 In the initial part of the
interview, after the student has read the To make a better description of this
initial ”configuration” of the problem, it is whether verbal or written, was the
one considered to integrate these 8 types. 40/20, first the dividend 40, then the
galley and at the end the divisor 20, that is resolve it, or immediately start your
resolution process. The first expression, Let’s see, how did you do it? Do that
division, you divided what by what?] (Write solve the problem (See Box 12).
then it does: He mentions that to solve the problem he will divide 20 between
the steps and anticipates sub-divided into 8 types based on the first scheme
activated by the students to For example, Gustavo remembers how he solved
the problem, explains his process, and 5 c. 35/70 20/40 split without filtering
time d. Rule of three e. Deduction to. Division 35/70 20/40 1 1 3.3 problem
figure Frequency Percentage 29.9 F. multiplication g. another interpretation b.
Split 70/35 40/20 1 40 3.3 12 1 3.3 3.3 9 16.6 Table 13. Subclasses of Figure of
the problem Machine Translated by Google What he did to solve the problem
was divide 20 by 40: resolution route verbally. This does not mean that this
is actually the case. during the resolution process when new relationships are
identified, new Example: Irene, a high school student from an urban area,
mentions that she Hector walks twenty meters and takes forty steps, on average
what It means that you have reconfigured a representation of the problem and
posed a specific schemes to deal with it. This first approach often changes to.
Division 35/70 20/40 (12 students) well, but I would like to know how you did
it] I don’t remember [Umju, let’s see, well represented in writing and solved
the algorithm. From verbal approach to resolution as illustrated by Andrea’s
microanalysis. data or when difficulties are encountered along the way. size
are its steps [Umhoo, that’s the problem you solved and you solved it In the
74
end, there are several possible paths to follow, depending on the difficulties
encountered, first approach to the problem, they expressed that to solve it
they would divide 35 by 70 or 40/20. This variation could be caused by other
aspects that will be analyzed further. twenty out of forty 157 The results of
the clinical interviews show that 40% of the students, in a try to figure it out]
(thinks for a moment) I can’t [You can’t] I think I divided b. 70/35 40/20
split (9 students) In this way, by figure of the problem we understand the first
configuration of the Students proposed from the beginning the appropriate
procedure to solve it. reflect on him and mentions that he can’t. Finally he
says that he divided 20 by 40. problem in which some operative invariants are
identified that allow to activate forward. 20 out of 40, as appropriate to the
version of the problem. which means that 12 Irene says she doesn’t remember
how she solved the problem. try to go back to Machine Translated by Google
Antonio, a telesecundaria student, says that he divided 40 by 20 and got They
initially proposed dividing 70 by 35 or 40 by 20. There are 8 students who
he... this... walked and took forty steps [Umjú] It’s the same thing I did
[He’s verbally that he divided 40 by 20 and when he writes it he does it in a
conventional way possibility of a dividend less than the divisor as in the case of
Luis Antonio. (2P) half a meter [Umjú] That was what I did and then to verify
what it was, to understand…]. raising this division. It means that they possibly
identified among the data of the and he takes forty steps [Umjú] On average,
he says, what size are his steps? I happened?] Five times two... ten (1P) forty,
twenty, (1P) point five... [Umjú] did] I divided forty by twenty (points to 40
and 20 as he could be oriented by the theorem-in-act ”the dividend is the
largest number in a step that is correct (half a meter) and the result through
the written algorithm. as 0.5, which he interprets as half a meter: established
this relationship and represent a quarter of the total. Example: Louis very
easy, right?] Let’s see, do it (Perform the 40/20 division. Start from right
to There is no variation between both representation systems. The problem
that 158 I multiplied those points five by forty [Umjú] (1P) which were twenty
meters that In this subcategory were classified the data that expresses that the
students As can be seen, Luis Antonio, unlike Gustavo, expresses problem a
proportional relationship but when proposing a division they do not consider
the Um... I divided forty by twenty and it gave me uh... zero point five, right?
That (Pause) [Okay, what’s the problem? Don’t worry huh, we’re trying
division”. For this reason there was no correspondence between his deduction of
the size of a The results show that 8 students started solving the problem [How
did you go about solving it? Try to remember] Héctor walks twenty meters
left by the dividend, then the galley and finally the divider) [Let’s see, how
mentions) [Umjú] He is forty out of twenty [Umjú] (Pause for a moment) [What
you are facing is that you are not getting the result you expected. Your initial
resolution scheme Machine Translated by Google again the 70 that appears in
the problem), uh... what came out, what d. Rule of three (one student) by 60,
which correspond to the sixty seconds that make up the minute mentioned in
the trein... No, the 35 (point to the 35 that appears in the problem) among the
70 (point to I multiplied by 60. problem without this data. that he divided 70
75
by 35, although he corrects it immediately, and that the result was multiplied
by [I would like to see how you solved it] How did I…? [Uh-huh] It’s your
initial resolution process. In the experimental phase of this study, it was shown
that María de los Ángeles was the only one (of the students interviewed) who
raised analyzing the situation through an iconic representation of the problem,
putting problem text: Ruth was the only student of all those interviewed who
considered the time in initially a proportional relationship through the rule of
three. Starts saying? Seem to you? Or do you want to see what you wrote
and explain to me what you wrote, c. Split 35/70 20/40 unfiltered time (one
student) problem with distractor and those who solved the version without
distractor. the time data, 159 I don’t remember... [You don’t remember, and
if you solve it again and you give it to me? There were significant differences
between the results of those who solved the version of the in correspondence
the distance in meters with the number of steps. Then try to do resolution
taking into account the ”distracting” data of time. the meters [Aha] and I
divided the 70 (point to the 70 that appears in the problem) by fewer students
solved it correctly, compared to those who faced the This is Ruth, a high school
student from the urban town. she mentions This subcategory is similar to the
first; However, the student began the process of yes?] Yes [Let’s see, what did
you do? Let’s see] It’s that, well, I made an approximation of that should be
discarded because it does not enter into the pertinent relationships, was the
reason that Machine Translated by Google 1 under the seventy, it is a rule of
three, then write the division 35/70, put multiplication 60x35) More or less
like that (laughs), because I don’t remember [Aha, [Well yes, if you want, how
many are there? In centimeters] Ah, well... (Write 70, 35 and a zero in the
dividend and a point in the quotient, write seven next to it.) Seven (Draw
a line and a doll, write 70 steps and 35 meters, then write the the numbers,
so I always put little drawings [Ah] And that’s how it’s represented write the
answer “0.5 meters”). So? [Umhoo] And now [Yes] Wow. ok, what did you do
here? First... talk to me] It’s that good, I... I get bored with times seven... no,
seven times three, seven times four (write 5 in the quotient, to the side distance
of thirty-five meters [Yes] And this is his race tape [Ah... good. a conversion
from meters to centimeters to consider in which units the Until the approach
of the rule of three, there are three aspects in the resolution process of María
de los Ángeles that are worth highlighting. The first is the change in planes
of easier [Umjú] That’s why I put Héctor who takes seventy steps [Umjú] And
this is a Although we could consider the figure of the problem from the drawing
made correct result: 100x35) [What?] Isn’t it? A... because they are, do they
have to be removed in centimeters? 160 result. Finally, correctly state a rule
of three, do the division and arrive at the And then, here you did....] Well, you
have to take it out... Oh no (write the multiplication representation; goes from
an iconic representation of the problem to an approach Image 45. From iconic
representation to the rule of three. Machine Translated by Google some written
operation. 161 algorithm correctly and makes an appropriate interpretation of
the result. that he found among the quantities and concludes that he takes
two steps per meter without making It’s supposed to be thirty-five meters. In
76
about thirty-five meters walk same relations defined in the rule of three. The
third refers to the fact that it resolves the pass the data from the rule of three
to the division algorithm correctly. They are thirty-five, so he takes two steps
for every meter. [Very good] (Laughs) [It’s In the corresponding section, an
analysis of its procedure will be carried out to seventy steps. The seventy steps
uh... which is supposed to be half of... of seventy They analyzed the problem
and were able to estimate the size of the step, practically by correct and
conventional rule of three, one of the most sophisticated tools that [Umhoo, or
what you said, so I don’t forget. Ah, yes, yes] (Writes) [Then and. Deduction
(6 students) easy, right?] Aha [Now, could you answer it?] I’ll just post the
answer or... correctly the proportional relations and in the same way, the
written algorithm Example: Yéssica, from the telesecundaria school, verbally
expresses the relationship half of 70 is 35 and takes 2 steps for every meter
of the division with the data placed appropriately, that is, maintaining the
used by students in this study. The second aspect is that it represents Mental
calculation, establishing the relationships between the corresponding quantities.
for you it was very easy] Hmm, not that much, but yes. Image 46. Yéssica’s
response obtained from a process of deduction and mental calculation. Machine
Translated by Google manages to give a correct answer without having to
resort to other planes of representation in correctly, but when they pass to the
level of written representation they fail to maintain urban town. walk twenty
meters and give forty, it would go too far [If it were one meter, it would go
too far] 162 half a meter. Well, that was exactly how... how you wrote it,
right? before you thought the students did not. Even so, they made a first
interpretation of the in the following way: if you advance 35 meters and take
70 steps, it means that if you took steps forty steps, I put that his... that his...
that his steps are... must be of Don’t worry, that’s why we also wanted to work
with you, so this... nothing quickly, very well, that is the correct answer, really
very few succeeded These 5 sub-categories group the responses of the students
who achieved the initial phase of the interview. homomorphism between the
planes and face several difficulties. In this case, Jessica [... Do you remember
the problem?] Umhoo [Yes, sure?] Yes, yes [Let’s see, could you Umjú [Then
how did you know they could be fifty] Because... half... As in the case of
Carlos, a primary school student at one of the schools in the We have noticed
that some students manage to do this mental calculation that one meter and
you crossed it out, why? Do you remember?] Yes, because if he says yes
problem that allowed them to explore some paths in resolution. of one meter
would advance 70 meters, as it traveled half, then the size of the step is fifty
centimeters each [Aha, come on] (Write “50 centimeters”) [Very solve like you
did and you didn’t need a calculator?] I turned it off. establish a proportional
relationship between the quantities of the problem. In the other three, There
are students who solve the problem by deduction, they frequently operate solve
it again?] (Takes up the pen and reads silently) [You solved it very well, but
I want to see how you did it] It is that here it says, if you walk twenty meters
and give because half, if you take forty steps, it will take you twenty [Umjú]
Well, you did it very Machine Translated by Google Image 47. Representation
77
of the relationships of the problem through a number line. From 1 to 5).
Hector walks thirty-four meters and takes seventy steps, on average what write
“35” at the bottom, divide the line into five segments, number them [Can you
read out loud, please...?] Name... [No, here, here...] In a minute No... [Let’s
see, you can read it again] In a minute Héctor walks thirty steps would be
five meters [Mmm... let’s see, write it down, your answer] (Write how big are
your steps? [Yes, right] Umju [And do you remember how you solved it?] I
think I did it like this [Umho, how?] because seven times five thirty-five, their
[Could you solve it again? (hands him a pen)] You solved the other one well,
F. Multiplication (a student) 163 and five meters and takes seventy steps on
average, how big are his steps? “R=5m.”). technique. She writes the data and
then draws a number line that represents the 35 matter, you can do it again]
(Write 1=35m=70, draw a line, to the center and meters and initially divides
it into 5 segments: In this subcategory, the interpretation made by Irasema,
from the secondary school, was integrated. that’s why we want to see how
you did it, maybe you already forgot, but no Machine Translated by Google
Image 48. Successive multiplications to find the quotient. urban locality,
performs the same strategy, but unlike Irasema she operates with As it finally
ended up operating with 7 steps and 35 meters, it was considered reverse of
division. In the experimental phase, Frida, a primary school student from the
actual amounts and through written calculations. This example also shows the
interprets them as 5 meters. problem. However, the strategy is adequate. puts
two in correspondence memory. a new type of answer for this data, because
it includes a different quantity than the one in the utility of the graphical
representation, helps to maintain control of the activity and the we can infer
that he is doing a division mentally and therefore loses the From his description,
it can be inferred that the relational calculation he made could be the 164
quantities and look for a number that multiplied by one of them gives the other
quantity, 70 to reduce the difficulty of the calculation, but then forgot that it
had only been a form problem should have found a number that multiplied by
70 would give 35, which is the operation to abbreviate, which were not seven
but 70. That is why he concludes that the numerical result is 5 and following:
he looked for a number that multiplied by 7 would give 35. He decided to use
7 instead of control of its activity by using other values. If you have operated
with the amounts of the Machine Translated by Google Image 49. What is
meant by a step. that the size of a step is from the toe of one foot to the heel of
the other, instead of asks to solve the problem and only answers that the steps
are ”medium”. The interviewer asks him that if the question of the problem
had not been about what is. However, most of the students thought of it as a
no-brainer. Gabriel believes assumes that the displacement of a reference point
such as the person’s head I would divide 35 by 70. consider it from heel to heel
or from toe to toe. This is adequate reasoning if one size are the steps but
how long each step is, what would be your answer. Say what travel is reduced
by half compared to other reference points, for g. Another interpretation (a
student) (it would correspond to Case 1 of the step definition, page 63) but as
the distance 165 First, he wonders what a step is. As we have already seen in
78
the section on the so defined. After it is explained to you what counts as a
step, you will be analysis of the problem, of a very pertinent clarification and
although it seems trivial, it is not Gabriel, from an urban elementary school,
makes a very peculiar interpretation of the problem. In example the heels or
the tips of the feet, it would not be possible to advance 35 meters with 70 steps
Machine Translated by Google to determine that the steps are medium. But if
precision of the measurement of intensive quantification of step sizes in which,
somehow, it must have arithmetic. It seemed strange to us that in a school
context, with a mathematical problem they probably did understand it and
that they were also able to make an interpretation classified in subcategory 2.
However, once you perform the necessary calculations, of the present work we
have detected this type of responses, which at first we had type of response and
represent 22% of those who considered that they did not A similar case is that
of Laura, a telesecundaria student. she input explorations of the answers of the
students, both in the investigation “Los lectores amounts. Once the question is
specified (as in the case of Gabriel), the relationship between the Otherwise the
question would have been the size of the steps, it is likely that they would have
mathematical tools to do it, but they did not consider it necessary, as each
step, then it proposes the appropriate division for it. consider the data of the
problem, establish some kind of relationship between the quantities This type
of response was given in students in the sixth grade of primary school and third
grade of secondary school. different from the text of the problem, considering
that the size of the steps can 166 considered as an ”intensive quantification” in
which calculation was avoided Therefore, the first interpretation, his figure of
the problem, orients him to a they understood the problem. However, we can
now understand that very proposes to divide 70/35, so his first interpretation
of the problem is and their contexts” (Vaca et al., 2010), in the pilot study, as
well as in the experimental phase The results of the experimental phase show
that 18 students gave this required to make a more precise calculation. It is not
then that they lacked It happened with Gabriel. This data is very important
for this research, because in the different Now we know that it may be due
to the interpretation that is made of the problem because, a quantities and
arithmetic calculation is used proposing the appropriate division of the express
themselves with language terms such as ”small”, ”medium” or ”big”. yes for
him Machine Translated by Google thirty-five meters (1P) five, are seventy
[Umjú] Which is equivalent to seventy therefore they do not represent the
entire sample. Your selection was with the criteria above I don’t remember, I
think I divided seventy by two and then... here number and that already gives
me (1F), so he walks thirty-five meters that... Steps. Eh... and your steps
are (1P) on average, that’s why I put that already, Remember that these 30
students were not randomly selected and you did to solve it, it’s not that it’s
right or wrong and so] I don’t know [Umju] It’s just that who walks thirty-five
meters, then he takes seventy steps, then... I don’t know, meters each step.
mentioned and the intention was to delve into the different moments of the
process of asked how big his steps were] Umju, so well... I mean, he says I put
it medium because... more or less there it gives you that they were about two
79
167 go...compare with what you did...as you can think of] (Starts to concludes
that the steps are 2 meters and adds that on average they are For Laura, in
the text of the problem there are two questions: on the one hand, the average
and that I divided it and... [Don’t you remember? Why don’t you try to figure
it out? and then [...yeah, I don’t think you remember how you did it to solve
it, uh... put your that... that is, I put about two meters each, because... it
gives me a 2 meters, adds that they are medium. name, please] (Write your
name) [Our idea is this... to know how medium: solve it in silence, he writes 3,
but crosses it out) His steps were... this... of I mean on the other, the size of
the steps. That’s why once you get to the extent of the step of Image 50. The
word average refers to measurement and size to an intensive quantity. Machine
Translated by Google most frequently related to the domain of the algorithm
and its representation conditions of the situation are different. In the group
phase, students could concepts-in-act related to situations of proportionality
that allows them to But the difficulties that prevent them from having a correct
resolution are graph. When this is mastered, the problem is adequately solved.
experimental, in which only 34% succeeded (113 of 329). It must be considered
that the students, in what Vergnaud considers the epistemic part, there are
theorems- and to face it. In this category are grouped the cases in which the
students give an answer commit or not to its resolution more freely. Instead,
in the interview observed and evaluated, in addition to the fact that the
resolution processes and the interactions with the interviewer were videotaped.
This and other elements such as non-random selection identify the problem in
this way and choose some of the available tools clinic, due to their individual
character, could feel more committed to feeling of the problem that have been
made, that is why it is important the initial configuration that is resolution
that stood out in the results of the analysis of the experimental phase of 50
cm or ½ meter) and that also specify the unit of measure. solve the problem
There are a wide variety of ways to represent it to yourself according to the
figure correct, that is to say, that they arrive at the result of .5 meters or its
equivalents (half a meter, 168 Most of the students interpret the problem as
a situation in which built and how well consolidated and structured they are.
solve the problem autonomously, unlike the results in the phase proportionality
is involved. Which means that in the schematics of these chapter IV. do and
this depends on the situation you are facing and the schemes you have without
the intervention of the interviewer. Half of the students interviewed achieved
VI.1.1.2 CORRECT AND AUTONOMOUS RESOLUTION Machine Translated by Google Table 14. Issues related to the Division Algorithm category
169 of the students did it, it means that although it is a fifth, it indicates that
the related to the division algorithm. verification of their result without being
requested by the interviewer. only 20% could use in their resolution process.
This depends on the criteria An important part of the problem solving process
is related to the verification of the result is a resource that students have
available and that teachers. to find the answer to the problem question. Only
6 of the 24 did it for of the students to carry out the clinical interviews were
able to generate these rigor assumed, learned by the students and the criteria
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demanded by the Division algorithm. Most of the students used it as a tool
categories and will show some examples of student productions. divided ”in
the mind.” For this reason, a section is dedicated to the analysis of the data.
This category highlights those students who carry out some type of differences
in results. Each of these will be described in detail in the corresponding
sections. deduction and did not require the written algorithm, but it does
not mean that they have not Percentage 63.3 Thematic 19 46.7 division
Frequency Calculator, orality/writing 36.7 Algorithm of the Mathematical
writing with alphabetic directionality Algorithm reading with alphabetic
directionality 15 dividend greater than divisor 11 Algorithm mechanics 14 7
23.3 50.0 SAW. 1.1.3 AUTONOMOUS TEST VI.1.2 DIVISION ALGORITHM
Machine Translated by Google VI.1.2.1 WRITING WITH DIRECTIONALITY
OF THE ALPHABETICAL SYSTEM The way to graphically represent the
division algorithm was observed and the will operate and the referent of the
result. found that 36.7% of the students (11 cases) did so following the rules
of the representing 35 between 70 represents 70/35 in the conventional system.
more than a third must be considered, because it determines the relationships
between the magnitudes with which students by representing the algorithm
(including with the calculator) and its interpretation the glottic writing that
they dominate, in terms of Harris (1999). they put in 170 That is why in this
section we analyze the graphic productions of the alphabetic writing system,
that is, they wrote a script strongly influenced by with the rules of the division;
and explore the ideas that students This is a very broad subject because it
is not only related to the ”steps” To represent in writing what is verbally
(writting and reading); the way to solve it (mechanics of the algorithm),
which has to do correspondence between orality and writing and recover the
directionality from left to right of alphabetic writing9 . interpretation because
it is a graphic resource. In addition, each of its components division. while
they are speaking it, 35 the galley of the division for “between” and 70. So
instead of is related to an arithmetic meaning and its organization in graphic
space to divide, but their written production and their build on the way to
organize the data within the written algorithm of the express: ”35 out of 70”,
for example, they write from left to right, even some 9 A similar case and that
serves to better explain what we mean by the expression ”strong influence of
glottal writing” we have verified in children who we ask them to write large
amounts that they do not know how to write conventionally. Some children
clearly match numbers or groups of numbers with fragments of the name of
the number to be written. For example, the number three hundred forty-five
represents it, even with self-dictation, ”three hundred” 300 ”forty” 40 and
”five” 5. The result is 300405. This procedure has been confirmed by other
researchers (Sinclair, A. 1988; Pontecorvo, C. 1985). Children match elements
of the graphic number system (which can be considered ”ideographic” or, more
precisely, logographic, with words or word segments that form the name of the
number). Machine Translated by Google Image 51. Mathematical writing with
alphabetic directionality link with the alphabetic writing system that should be
made independent. The questions 171 usable to solve proportionality problems.
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algorithm written at some point in the interview, which again reflects a that
arise are: when will they become independent?, what is required for that to
happen? There is elements (35, ”between”, 70) and therefore the writing is
conventional. of the 7 use the algorithm properly even when they recognize it
as a tool In the same way, when they are asked to read the division they made,
they do so Reading with directionality of the alphabetic system Remember
that they are sixth graders and third graders of secondary school. of the
alphabetical system, “35 between 70”. part of the students interviewed follows
this algorithm writing procedure, calculator to solve the problem. If we take
into account that typing the data in the left to right. For example, in the
following image it would be read, with the directionality Within this group,
the cases were also considered in which a of mathematical notation, each with
different properties and purposes. It is a fact Half of the students interviewed
made this type of interpretation of the alphabetic and mathematical writing
disappears. They type in the order in which the words are listed. very
important to consider, since it constitutes an obstacle that prevents students
from which means that they have not succeeded in making alphabetic writing
systems independent from the calculator is somewhat similar to writing them,
so this difference between VI 1.2.2 CALCULATOR, SPEAKING/ WRITING
Machine Translated by Google VI.1.2.3 MECHANICS OF THE ALGORITHM
VI.1.2.4 DIVIDEND GREATER THAN THE DIVIDER division. technique as
a sequence of steps, sometimes meaningless, these can be forgotten or unusually
frequent its application to the graphical algorithm of the division. difficulties
to reconstruct the steps or even recognize that they do not remember how to
do the A part of these difficulties is linked to the fact of not knowing by heart
the accommodate the data in graphic space. Since many students learned
the influencing the tendency towards arithmetic writing with alphabetical
direction, since it is In relation to the Mechanics of the usual algorithm of the
division, by columns, it is analyzed calculate ”how much does 25 between 4”
have to use ”the table of 4” but, if you don’t have changed. To make sense
of both this arithmetic operation and the rest of the 172 multiplication tables,
which makes the numerical estimation required by the algorithm difficult: for
operations, they create theorems-in-act that guide the rules of action of their
lead to calculation errors. students who used the calculator in the interview,
all expressed a achievement of submissions, etc. the mastery of the students
of the steps to follow and when they modify them giving from it, the estimate
is costly and tends to distract the flow of reasoning, the negatively on the
result. 46.75) adequately apply the usual algorithm technique. the other half
have One of the difficulties encountered with respect to the usual division
algorithm is how The use of the calculator to solve mathematical problems
could be term-to-term correspondence between speaking and typing, without
this having any repercussions The results show that slightly less than half
of the students (N=14, Machine Translated by Google 173 than the divisor”,
and for those who abide by this rule, the accommodation of the data of the
rule at the time of writing down the division at least once. activity to solve
them. One such theorem is “that the dividend is always greater division is not
82
a problem. The results indicate that 63% of the interviewed students applied
this Machine Translated by Google Percentage Units Frequency 36.7 Thematic
11 13.3 Autonomous drive allocation Carrying the decimal fraction to unity 4
Table 15. Issues related to the Units item VI.1.3 UNITS product of arithmetic
calculation. A recurring theme in clinical interviews is the assignment of units
of measurement to the numerical results obtained by the students through
some operation. result. The rest, almost two thirds, were satisfied with the
numerical result, autonomous allocation of units and what we call “the carry
of the fraction divide 35 by 70, either in writing or with the calculator, you
obtain the quotient Although it is a very broad subject, for this section only
two aspects will be analyzed: the Regarding the carry of the decimal fraction,
it can be mentioned that it occurs when at They located 15 cases in which they
added ”cm” as a unit of measurement. in the phase of decimal number, they
interpret that being a fraction, the unit of measurement must also decimal to
the unit”, that is, that some students when finding as a result a 0.5 and must
assign the corresponding unit of measure. In the experimental phase, 36.7%
of the students assigned a unit of measurement as part of the 174 change.
clinical interviews also found 4 cases. One of them is Emiliano: VI.1.3.1
AUTONOMOUS UNIT ASSIGNMENT VI.1.3.2 CARRYING THE DECIMAL
FRACTION TO THE UNIT Machine Translated by Google of the students
with a similar answer, is the theorem-in-act “as the result is not It will be
millimeters here, cent.... I just got confused. Well, now (2P) millimeters now
The explanation that Emiliano gives, and that may be the same idea that the
rest have an integer, then neither can the unit of measurement be”; therefore
175 (Write ml next to your answer, “0.5”) [Why did you hesitate? You were
going to say [Umhoo] [You were going to put centimeters but you said no.
Because... since it’s not integer, so It may be that later they will reflect on the
real impossibility of taking half a step [Millimeters?] (1P, 9:31) millimeters
[Can’t it be centimeters?] Pus maybe yes They are looking for a fraction of a
meter: centimeters or millimeters. millimeters?] Half a centimeter [Like where
to where is it from?] Over here [Mmm... so point five what is it?, how long is
this?, what did you say?] means 0.5 centimeters or 0.5 millimeters, difficulty
is also related to the you said?] Uh huh [So it’s millimeters] (Nods) [What
is point five millimeter (what we call empirical feasibility) or even that they
do not identify what Just spelled). Here, zero point five, what would be the
measurement] Well, centimeters... how much? Point five millimeters?] No, I
don’t know... I wouldn’t measure the step like that. situations are described
below. no, millimeters [Millimeters? Let’s see, then put millimeters] To this
one [Yes... there] Fifty centimeters [This... let’s see, write it down, if not, I
forget (3P)] (Write 50 cm) [Come on, here you are putting a number and the
measurement (indicating what you (points) [Would that be... a centimeter?]
One centimeter [Then your step measures, level of construction of the concept
of rational number (Charles, 2011). Both Machine Translated by Google
Percentage Result Frequency 70.0 Thematic 21 66.7 Relationship between the
symbolic-concrete planes Empirical 20 feasibility Table 16. Issues related to the
category Result VI.1.4 RESULT VI.1.4.1 RELATIONSHIP BETWEEN THE
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SYMBOLIC PLANE AND THE CONCRETE PLANE students (N=21) were
able to correctly identify the measurement on a specific plane concrete and
symbolic representation, and the empirical feasibility of the result. In it First,
the type of relationship that students conceptualize between the Regarding
the result, two elements stand out: the relationship between the planes of a
ruler or tape measure the segment that represents the result obtained. 70%
of the second, if they consider that their result is possible in the real world,
for example that [This... from where to where is half a meter?] From... [If
you could point to me] Maybe represented on the symbolic level. For example,
Gustavo is clear about the distance graphical representation of the result and
its correspondence with the concrete plane. In it approximately half a meter
and you can also accurately locate it on a measuring tape. both hands on one
end of the table, then measure it and the difference is someone can take steps
of 2 meters. This piece is half meter (approximate size to half meter showing
with centimeters (or two meters, 2 cm, depending on their answers)”, or they
are asked to point to 176 Regardless of their result, they are asked “from where
to where are 50 minimum, between 5 and 10 centimeters.) Machine Translated
by Google Image 52. Gesture to indicate the size of a step. (pointing vaguely)
[From the table to here, how much is it?] Like a meter and 177 In other cases,
there is no clarity about the specific distance that the result represents. like
how far, can you tell me with your hands?] Like from there to here medium
[But thirty centimeters?] From to... from this table to the other table [From
here from wonder why. If we take into account that the youngest students
This... if you take two steps of half a meter, how far do you go?] One meter.
centimeters is equal to the distance between two poles (tubes that support a
roof on the (pointing approx. 2 meters), yes? Eh... you, how big do you take
your steps?] I don’t know obtained. For example, Carlos, a sixth grader in an
urban primary school, mentions that 30 my finger or from here to there] From
here to there [Is this to there a foot? Alright, can you imagine this... well,
about how much is that, a foot, bigger than from here to there] (Nods) [Oh,
little by little, really? (laughs). schoolyard with an approximate distance of
3 meters between them): [How much do you imagine?] Fifty [Fifty? So your
steps are gives a tape measure)] (Measures a segment of the table, points to
the fifty post [A light post?] No, the ones that are out there [Ah... I didn’t see,
representation. However, a third do not do so, which is why it is important
centimeters) Here [There. Can you take half-meter steps?] Yes [Yes, right?
[Yes? How do you know?] (1F) [Eh?] Calculating it [Calculating it, let’s see,
with this one (he From where to where will it be?] From... from... (Looks
around the room). From one post there to another Most of the students
establish the homomorphism between both planes of Machine Translated by
Google that is not bad, nor is it good] Umju [Isn’t it bad?] No, it’s good, I
don’t know why inch?] Mmm... three... mmm... about three... centimeters,
something like that [Three?] Three Regarding empirical feasibility, most
students also take into account 2 cm) [What do you think of your answer?]
Hmm... regular? [regular, means you guys [Hmm... do you know the tape
measure?] Umhoo [Here on the tape measure? the tape) [Are you going to
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show me two inches?] Umhoo [Like how long is a (66.7%, N=20). Only in 10
cases do they not do so, despite obtaining steps of time, point it to me here]
(Point to 20 cm) Here [There, is this... here would be two? point... [Point eight]
Eight [Hmm... I think these are inches right?] conflict that the step size is that
size maybe because you have not lent consideration the real possibility of the
size of the steps that they obtain as a result where would you place the result
it gave you?] Mmm... (Points) [Here?] Umhoo [Another Javier considers that
Hector’s step is 2 centimeters, it does not cause him any For example, Javier,
sixth grader at an urban elementary school: They are in sixth grade and that
for this educational level the use of measuring instruments, That would be two
inches, right? [Two inches?] Two inches [Mmm...] (Strike out cm in 2 meters,
2 centimeters or 0.5 cm. centimeters?] No... two centimeters (apparently he
corrects pointing) [Ah...] Those 178 It is important to point this out and
keep an eye on the progression of mathematical knowledge steps would be two
centimeters? [Yeah. What you believe] (Write as answer inches [Two inches]
Well, it gives me like uh... here (points to between 12 and 13 cm in elementals
mainly for magnitudes of length it must be a content already mastered, that is
[Yeah. And what would be the answer?] Two? [Two what] Two... ah... the
size of the your answer and write inches). Can I write with a pencil? Well
give inches, two VI.1.4.2 EMPIRICAL FEASIBILITY Machine Translated by
Google inches. It is once again a ”diffuse” or lax conceptualization of the results,
We believe that the first interpretation is more likely because it is something
that distance cannot measure two centimeters, so it resorts to another unit of
measurement; the or of mathematics in general, conceiving that a result may
be right for (points). real possibility, or that he does not know how to indicate
the amount of 2 cm on the measuring tape. obtained, for example with the
arbitrary use of the decimal point by means of rules for you”. [Umjú] And the
other would be here (with the arms in the same position, he places them to the
we have been observing other students when they make “adjustments” to the
results some and bad for others, as expressed through his statement: “no, it’s
good. I don’t know 179 close to what is empirically possible. Another fact that
allows us to affirm this inference is pay attention to its meaning. When asked
to show on a tape measure, you do not point two meters and is clear about
the distance it represents. external to the numbering system or measurement
system so that its result is more On the other hand, Irene, from the urban
secondary school, considers that it is possible to take steps centimeters. It can
mean either that by becoming aware of the size that they represent Because
you made the adjustment to make the length longer and you know that a lot?]
I think so [How far is two meters?] Well, I don’t know, one meter would be
like this 2 centimeters he wanted to arbitrarily adjust the length to correspond
to a properly the point where it is located, but rather indicates the position of
the 20 the change that Javier makes to the unit of measurement, going from
centimeters to inches. [Yes, it can be like that?... So what will it be?] Well,
two meters is a lot [What, is it Image 53. Shows the approximate distance
of one meter. Machine Translated by Google take steps so small or so big.
However, what these results show is This case is similar to the previous one in
85
that it considers that it is possible to take steps according to different plans
and there are also few who do not consider the real feasibility of that a more
systematic didactic work is required that considers various planes of considers
that it establishes the appropriate correspondence between the symbolic plane
and the corresponding, without neglecting the idea that mathematics should
serve to result obtained, in this case 2 meters. But the difference is that Irene
representation in which students can build relationships depending on the size
of the person. The fit does not do so in mathematical terms, left of the marked
segment to indicate another step) [can you take a step some references to reflect
on their results. concrete and still considers it empirically feasible to take steps
of that size real issues, at least in basic education, so that students have [And
which one convinces you more, two centimeters or two meters?] Two meters
[Yes? Well There are few students who fail to establish the homomorphism
between the write it] (Write “2m”) [Very good]. like this?] Well... yes, it
depends on the person, if he is very big, well he can give them but what adjusts
to the result is reality. 180 Image 54. Shows the approximate distance of
two meters. Machine Translated by Google VI.2 CONCLUSIONS OF THE
CHAPTER interprets it in the same way, that is, it reads it from left to right.
students (almost two thirds) do not consider it necessary to specify the we can
rescue is that there are those who spontaneously make a verification (20%)
students write the algorithm with an alphabetical directionality and half of
them write it Based on the above, we consider that beyond the old discussions
about Regarding the units, these results show that the majority of the problem,
in addition to the fact that the application conditions are very different. what
yes with the rigor of learned mathematics. ourselves, we must overcome that
stage in the didactics of mathematics and give it the value reference units
to your numerical result. Again this data makes us students as well as their
teachers. Is the matter of the units an issue that should be but most do not; We
consider that it is a resource that is related teach algorithms or not because of
the mechanical and thoughtless application of the reflect on the epistemological
conception they have of mathematics, both as a tool to find the result, but
only half use it both in its conceptual and procedural part, it allows students
to reflect 181 On the other hand, most of the students resorted to the division
algorithm They have support for problem solving. The domain of algorithms,
teach as such? or when taking into account the rigor of the approaches problem
and autonomously, unlike the results of the experimental phase, Another
important fact to highlight is that more than a third of the and allows them to
outsource their concepts. We must remember that this is the second time these
students have solved the same problem. Although these data show that half of
the students correctly solve the properly; the other half have difficulty using
it or even remembering it. about the mathematical concepts involved, the
relationships between the data in the problem Machine Translated by Google
20% of students fail to adequately establish this relationship. Therefore it is
The carry of the fraction from the numerical unit to the unit of measurement is
a sample of rendering plans. In this problem the relationships seem somewhat
trivial, but the It is advisable not to lose sight in the didactic work of these
86
correspondences concepts, requires building over time and through multiple
situations. as was seen in the topic of measurements in the theoretical chapter,
it has taken centuries in the major, with the study of algebra for example.
the cognitive activity of the students to try to understand these notions that,
between the different planes, since in later educational levels the complexity
will be Another point that is important to highlight is that most of the
students Mathematicians should not consider the possibility of giving only
the numerical result without study of mathematics understand them. 182
The above regarding the conception of mathematics, but with respect to the
results obtained. This relationship, Vergnaud suggests, should be monitored
between the different construction of knowledge, the definition of measures
is an issue that, like all specify the unit of measure. establishes an adequate
relationship between the symbolic plane and the concrete plane in the Machine
Translated by Google CONCLUSIONS that they have the ability to learn and
apply mathematical knowledge and that if Therefore, representation systems
play a very important role in Based on the results obtained, it is possible to
show that the students of while solving a mathematical task in interaction
with a mediator. Means experience difficulties, these are mainly related to the
lack of representation. multiplicative structures. problems that arise in school
contexts, build knowledge through representation systems. conceptualization
processes, but we do not mean by this that knowledge language. As we were
able to show, most students are proficient in mathematics. basic education
use their mathematical knowledge to face situations consolidation of said
knowledge at a predicative level, that is, externalized to is built only from
the mastery of these systems, for example only through the concepts and
theorems-in-act to make sense of a new situation. representation systems and
the consolidation of mathematical concepts. When there is 183 during the
interaction with mathematical objects through the formulation of The data
also show that there is a close relationship between mastery of in its operative
form, but they face difficulties in the predicative form. writing, language, and
mathematical conceptualization through the study of capable of reasoning, of
mobilizing their mathematical knowledge and of constructing new very lax or
“liquid” (as Zygmunt Bauman would say) the rules of the systems of process
of solving a mathematical problem in the conceptual field of With this work
we deepen the understanding of the relationship between the The students
who were part of this study also showed that they are a loosely constructed,
fuzzy conceptualization, students apply Machine Translated by Google results
obtained. necessary. In this regard, we consider that the understanding of
algorithms, their domain and interaction (in this case with the interviewer and
with knowledge), language and writing In other words, most of the difficulties
that students their correspondences between them. This does not mean that
mastering the systems of problematic and from this they were able to identify
its invariances, very important role in the basic education curriculum; have gone
from being understanding of several interrelated concepts and various levels of
complexity: the some of its properties and relationships with other objects. But
this language is protagonists in the didactics of mathematics to practically play
87
the antagonistic role. 184 iceberg whose base, generally invisible, consists of
the conceptualization must be re-constructed by students to use them properly
and take the tool that most students, both elementary and secondary, do not
time. allows to maintain control of the activity during the resolution process,
which supports consequence of the insistence of educators and specialists not
to teach working memory and that allows to adequately communicate the
procedures and algorithms and has been taken to the other extreme, not giving
them the importance The results obtained in this investigation also indicate
that, during the the possibility of using their respective graphic representations
also implies the they faced are related to the lack of mastery of the systems
of representation and helped students recognize the objects involved in the
situation On the other hand, with respect to the algorithms, we consider that
they have been given a representation students will consequently master mathematical concepts. Vergnaud has been very clear in describing symbolization
as the tip of the great organized into systems and each system has its own rules
and conventions that From this research we can conclude, about algorithms,
that they are a constructed through interaction with multiple situations over
long periods of time function of a tool that supports reasoning, that favors
reflection, that dominate according to what is expected for this educational
level. Possibly this is a Machine Translated by Google decontextualized
teaching. The same happens with multiplication tables, which to know in
advance the difficulties that students face in order to be able to more abstract
mathematical content such as algebra. Instead, the flimsy numbering system,
measures and composition of measures, units of measure, etc Therefore, not
only is mastery of the ”mechanics” of the algorithm required, but also find
themselves helpless before problems that we could even consider trivial. The as
if that meant something anti-pedagogical, they can guide students to activities
or their interventions. solving the algorithm can become, as we have already
seen, a more complicated task consume time and processing to deduce that
information, from which they should be able to 185 The domain of written
algorithms of elementary arithmetic includes both its The 2009 primary study
plan and programs recognize the importance of In relation to the learning
process, the modification of meanings, powerful tools to confront a wide variety
of problems, both school However, in practice (in the classroom) it seems that
something else happens. It’s important to give them interviewer and student.
The problematic situation demands the application of the the importance
they have, without this meaning a thoughtless return to the past with their
corresponding schemes; the teacher must master the problematized content
and as non-schoolers, and it is also of great importance for the understanding
of differentiate conceptual errors from local misunderstandings and in this way
guide the decimal numbers, rational numbers, proportionality, ratio, fraction,
comprehension and consequently its failed application make students some
teachers in order to prevent students from learning them ”by heart”, also its
teaching is an opportunity to reflect on the concepts involved in the operation
carried out. than the problem it is supposed to solve. have it ”automatically”,
to be able to use it in the resolution procedures. graphic representation as well
88
as its interpretation, makes these become teaching algorithms in the context
of problem solving. Without revealed by microgenetic analysis, occurred in
interaction with both knowledge, Machine Translated by Google the students
generated and that at the same time prevented them from being able to
solve training in mathematics and didactics of mathematics would favor the
identification The teacher therefore has the responsibility to train himself to
master with Vergnaud considers that one of Piaget’s fundamental inheritances
to that students build knowledge through activity about objects and face
the didactics of the different disciplines but in particular the didactics of
the situation, that is, by identifying the characteristics of the problem, the
resources to guide teaching activities and thus promote their improvement.
math. It is required that they know the mathematical contents that that
requires for its solution, the possible difficulties that could represent for the
186 proving his theorems-in-act through multiple situations, where contents,
that is, the most efficient way in which they can guide the activity research like
this and with individual clinical interviews with a few students. to improve or
replace said theorems, reorganize their schemes, combine them or knowledge
expected from educational institutions. daily mathematical content but also
the content of other disciplines In the cases we have analyzed, we were able to
unveil some theorems-in-act that many students (on average 40 students per
group). However, a good generate new ones. of students’ difficulties, would
allow teachers to take advantage of their mistakes education was to consider
activity as the origin of knowledge. Means sufficient depth the contents that
it intends to teach. This has been a challenge appropriately for that specific
situation. However, this was achieved thanks to the analysis the situations.
Therefore, if we want to take advantage of this heritage, it is important that
the teachers encourage students to act on reality, experimenting and will be
taught to the students and also that they know the didactics of said students
and ways to get around them. Of course this happens in a job They will
invariably make mistakes that at the same time become opportunities. of
the students to face didactic situations to ensure that they build the The
reality of Mexican teachers is very different, because they must not only teach
Machine Translated by Google notion of the decimal point and the different
uses and meanings of its representation Although the data obtained do not
show significant differences with respect to the this second aspect of work has
been ignored in the didactics developed in Mexico The didactics of mathematics
in basic education in Mexico has been oriented curricular contents (Secretary
of Public Education, 2012). It has focused more on From this work, veins
of research are opened on the relationship between the didactic interest and
should be explored in depth considering both the construction of a broader
exploration of the variety of problems where one can highlight symbolic systems
of representation and mathematical concepts. required now of the notion as the
evolution of representation systems. What was not achieved 187 Knowledge
is the product of the subject’s adaptation to the situations faced. In different
planes of representation through the analysis of the classes of situations that
concepts involved, particularly that of rational numbers. This topic is located
89
by the teachers, both in the design of the didactic situations and in the analysis
problems belonging to the conceptual fields involved. tangentially in the
clinical interviews and in the analyzes and interpretations of the Some of the
issues that we highlight in this research are: the evolution of same. of the
schemes that students evoke and use to face them. We consider that use of
the calculator, we consider that it is necessary to deepen the matter through
to the development, revision and reconstruction of didactic situations for the
teaching of the and it is important to coordinate both perspectives. symbolic
in different situations; The notion of measurement is another topic that is of
great the object of teaching than in the cognitive processes of the subject. from
the perspective theoretical and epistemological on which we base this research,
it is conceived that the carry out research on the construction of mathematical
concepts and their deepen, among other topics, was in the investigation of the
domain of the different In this sense, the construction of knowledge should
be favored by the consideration, give meaning to these concepts. For this it
is necessary to explore a variety of as central to the difficulties faced by the
students and explores Machine Translated by Google highly industrialized;
the same happens in family contexts. According to but that came out” or
”it’s fine for me, I don’t know about you” or that ”they appear from the
methodological design that is focused on exploring this aspect in depth, will be
the advantages and disadvantages of this technological resource in the learning
process of the Finally, we support the hypothesis that empirical feasibility
is a referent and those of appropriation and use of basic mathematical tools
to live in the century both its use, development, use and application will be
different. Therefore, for their needs, shows that they conceive of mathematics
as a school content XXI. that mathematics is valued and used as one would
expect it to be, it is also 188 allow us to deal more regularly and freely
with problems on a specific level It is important to take into account that
mathematics is a cultural tool that is in which they live. contextual, they do
so by emphasizing other planes of representation. Even though in assessment
depends on the context of each student, family, school, locality, state or some
of the conceptions that students have of mathematics regarding the country.
They will not attribute the same value to it in a rural location as in a locality
rigor and precision. Expressions like ”you can’t take steps that size our
results the differences were not significant, we consider that with a nothing a
decimal point” that they strategically place in the result to adjust it, according
to solving mathematical problems. possible to know better and better the
processes of acquisition of mathematical knowledge the configuration of the
contexts is that a value will be given to this tool and therefore to which students
from rural localities resort more, unlike what happens with students from
urban localities, because the conditions in which they live On the other hand,
we share the perspective of Gérard Vergnaud when considering that would
require improving the living conditions of all people and improving localities
first while the students of urban schools, due to their same conditions conforms
of objects, procedures, demonstrations and that as a cultural object its Related
to the above, this research also allowed us to identify Machine Translated by
90
Google contextual, idiosyncratic and cultural characteristics of those who use
them. It is important to continue the regulations, red tape (there is always
the possibility of ”arranging” to investigations to delve into these conceptions
of both students and can be arbitrarily adjusted by the subject depending on
their intuitions, Seems like a cultural trait: stats can be used with those too
skip some steps). needs and expectations. motives, mainly those that try to
show the achievements in the different types of teachers and be able to make
didactic suggestions in this regard. Therefore, the conception of mathematics
does not escape the aspects 189 government reports. The laws could have
this same treatment, the flexible, partial and imprecise, that is, whose procedures, components and results Machine Translated by Google REFERENCES
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Editions. Machine Translated by Google 10 CONCEPTUALS BY GERARD
VERGNAUD. ANNEX 1. TRANSLATION OF FIELD THEORY Science is
reduction. But not all reductions are fruitful. Today it is more or less accepted
that Piaget provided a magnificent contribution to developmental psychology,
when the behaviorists had not been able to do so. Despite this, he declined his
work in the analysis of mathematical content due to his fascination with logic
and his hope of being able to reduce the increasing complexity obtained by
children to logical structures: for example, his analysis of the ”formal stage ”
allowed him to identify the group of INRC transformations as the characteristic
that could help to understand proportionality in children. In doing this, he did
not pay enough attention to content that is specific to mathematics, namely,
the properties of functions. Paris 8 University, Saint-Denis, France The theory
of conceptual fields is a theory of development. It has two purposes. Human
Development 2009; 52:83–94 Keywords THE THEORY OF CONCEPTUAL
FIELDS10 DOI: 10.1159/000202727 (1) Describe and analyze the progressive
complexity, in the medium and long term, of the mathematical skills that students develop inside and outside of school, and (2) establish better connections
between the operational form of knowledge, which consists of acting in the
physical and social world, and the predicative form of knowledge, which consists
of the linguistic and symbolic expressions of this knowledge. As it deals with
the progressive complexity of knowledge, the framework of the conceptual field
is also useful to help teachers organize didactic situations and interventions,
according to the epistemology of mathematics and a better understanding of
the conceptualization process of students. Conceptual fields, Theory of development, Progressive complexity of knowledge Introduction Gérard Vergnaud 201
94
Summary Translation made by Alfonso Javier Bustamante Santos, November
2011, revised by Jorge Vaca and Veronica Aguilar Machine Translated by
Google From this starting point, several questions arise: First I will emphasize
the importance of activity, schemas, and situations for psychology, and then I
present a definition of a conceptual field as the set of situations and concepts.
The concept of schema also requires some attention, because it plays a crucial
role in the analysis of the operational form of knowledge, as distinct from the
predicative form. Finally, I will try to discuss different and complementary
aspects of the concept of representation. 1. Is it possible to theorize about
reasoning with the concept of schema, and specifically about mathematical
reasoning? ÿ the isomorphic properties of linear functions: the constant
coefficient f(x) = kx The concept of schema f(x + y) = f(x) + f(y) y f(ax) =
af(x) A double proportion is when one variable is proportional to two other
independent variables. So the properties of bilinear functions are relevant. 2.
What is the role of schemas in the functioning of representation? Why and
how are they components of the representation? The concept of schema was
not introduced by Piaget: several 19th century philosophers mentioned it after
Kant introduced it and it was also used by various psychologists during the first
decades of the 20th century, notably Revault d’Allonnes (eg 1915, 1920) and
Janet (eg 1928) in France. However, Piaget was the first to provide concrete and
compelling examples of its meaning with his descriptions of early development
in infants and young children. His book La naissannce de l’intelligence chez
l’enfant (”The birth of intelligence in the child”, Piaget, 1968a) is not only
the ”invention” of the cognitive development of the child as a new field of
research, but also the demonstration that gestures and perceptual acts are
the empirical basis for their analyses. Therefore, the sequential organization
of activity for a certain situation is the primitive and prototypical reference
for the concept of schema. 202 These properties could have better described
the different emerging skills in a few years: the recognition and analysis of
products and ratios of dimensions as is the case for relationships between
length, area and volume, or between measurements in physics (for more details,
see Vergnaud , 1983). However, in the theory of conceptual fields, I borrow
from Piaget other important aspects of his work: first, the concept of schema,
to which I give a broader interpretation than his; the thesis that knowledge is
adaptation (accommodation and assimilation); as well as the general Piagetian
conception that action and representation play the main role in development.
The simple proportion is a function of one variable; and two types of properties
are essential: Machine Translated by Google 203 How is this theory related to
the development of mathematical knowledge? Activity, schemes and situations
Therefore, the schema/situation pair is conceptually more interesting and
powerful than the response/stimulus pair, and it is also more feasible to
describe and analyze behavior and representation using the schema/situation
pair than the subject/object pair. Do we have some examples of schematics in
mathematics? The theory that knowledge is an adaptive process is essential,
but what is adapting and to what? To date, the most reasonable answer is that
what adapts are the forms of organization of the activity, the schemes, and
95
they adapt to the situations. If the first reference for schemata is what Piaget
(and most psychologists in the early 20th century) called ”sensory-motor”
activity, the first theoretical question to ask is how gestural and perceptual
actions performed in the world real are or become internal resources. It is
not enough to say that schematics are found in neurons and genes, because it
is impossible to try to describe the organization of a single schematic as an
organized sequence of active neurons, or as a configuration of genes, due to
the trillions of elements. involved. The first example I will give is the scheme
of counting objects. When children are able to count a small set of objects,
they use three different repertoires of gestures: arm and finger movements,
eye movements, and words. The effectiveness of the scheme depends on the
one-to-one correspondence between these three activities and with the set of
objects in the physical world. It also depends on the ability to conclude the
episode by naming the cardinal of the set, which is more than the last element
of the set: the cardinals can be added while the last elements cannot. The
concept of number is characterized by the additive property of cardinals, a
property that equivalence and order relations do not have. The concept of
cardinal is implicit in the child’s activity: it is a concept-in-act. Furthermore,
this biological description ignores the critical point of relating the external
and internal parts of the activity, which is an essential point in promoting an
integrated psychological framework. The most fruitful idea I can find is that of
internalization (or internalization) of activity, both Piagetian and Vygotskian.
This idea is well developed in Piaget’s (1968b) book La formation du symbole
chez l’enfant and in the first chapter of Vygotsky’s Thought and Language
(1962). The paradox is that, in his radical critique of the Piagetian ”egocentric
feature” of infant language, Vygotsky develops the idea that egocentrism is
rather ”a step in the process of internalizing” dialogues, and offers the very idea
of ” internalized imitation” that Piaget understands as one of the first processes
of representation. 3. What is its relationship with other components such as
concepts, linguistic entities and symbols? Machine Translated by Google 204
In the first case, the binary combination of two parts into a whole, only two
kinds of problems can be generated: knowing the two parts, find the whole,
and knowing the whole and one of the parts, find the other part. What is a
conceptual field? Because schemata and situations are the roots of cognitive
development, and because concepts-in-act are essential parts of schemata (see
definition below), the development of a conceptual domain requires children
to know about and deal with contrasting situations. Researchers also need to
carefully analyze the different ways in which children deal with them. In this
paper, I will give only one example, the conceptual field of additive structures.
However, there are other good examples, such as multiplicative structures, the
geometry of figures, positions and transformations, and elementary algebra. In
the second case, six classes of problems can be created: knowing the initial state
and the transformation, find the final state (by increasing or decreasing the
initial quantity); Knowing the initial and final state, find the transformation,
when the final state is greater or less than the initial state; Knowing the final
state and the transformation, find the initial state by increasing or decreasing
96
the final state. Of these six kinds of situations, four require subtraction and
only two require addition. Addition and subtraction are not just inverses of
each other. It is at the same time a set of situations and a set of concepts
linked together. By this I mean that the meaning of a concept does not come
from a single situation but from a variety of situations and that, reciprocally,
a situation cannot be analyzed with a single concept, but rather with several,
forming systems. There are two prototypical situations for addition: the binary
combination of two parts into a whole (”4 boys and 5 girls are on Kath’s
birthday, how many in all?”) and the increment of an initial state (”Richard
had 4 marbles, you won 5; how many marbles do you have now?”) which can
best be modeled by a unit operation, a function from the set of possible initial
states to the set of final states. There are wide and significant differences in
the success or failure achieved by children when dealing with the different
kinds of problems that can be generated from The distinction between these
two prototypes becomes clear when one considers the variety of problems that
can be generated. Another early example of a schema in mathematics is the
perceptual activity used to recognize a construction or figure as symmetrical.
Examining symmetry may be more sophisticated than what 10-year-olds are
capable of (for example, they may not check for equality of angles or even
equality of distances with the axis of symmetry). But even when control is lost,
some invariant properties of symmetry are considered: these are also concepts
in action. Machine Translated by Google -1 Fig. 1. Sagittal diagram. where
I is the initial state, F the final state, T the forward transformation, and T
the inverse transformation. This theorem can also be represented by a sagittal
diagram (fig. 1). The simplest addition and subtraction situations can be
solved by some 4-year-olds, yet some situations requiring only one addition are
not solved by most 13- or 14-year-olds: “Robert played two games of marbles;
he remembers that he lost 7 marbles in the second game, but he doesn’t
remember what happened in the first. When he counts his marbles at the end,
he realizes that he won a total of 5 marbles. There are not only contrasts
between situations, but also between schemes, ie, between ways of dealing with
situations. There are of course wrong ways, but one can also observe different
useful schemes for the same kind of situations, depending for example on the
numerical values of the variables. Some children can of course subtract 7 from
11, others can count backwards from 11 to 7 and then count the number of
digits, others can count from 7 forward to 11, and still others can even make
a hypothesis about from the initial state (5, for example), apply the increment
of 7 marbles, find 12, which is very large, and then correct your hypothesis.
This last scheme is mainly due to the conceptual difficulty of reversing the
increment of 7, by applying a subtraction of 7 to the final state. This operation
of thought requires a theorem-in-act: What happened in the first game? Let’s
take the following situation: “John just won 7 marbles playing with Meredith.
These two symbolic representations (the algebraic and the diagram) show that
there are also contrasts between the ways of symbolizing objects and their
relationships. undoubtedly the If T(I) = F then I = T-1 (F) 205 Now he
has 11 marbles. How many marbles did he have before playing?” these two
97
prototypes and the other cases such as the quantified comparison of quantities
(“Who has more and how much more? Find the compared or the referred
quantity”), or also the combination and decomposition of transformations.
Machine Translated by Google be developed into subgoals and anticipations.
The operational form of knowledge - The generative aspect of the schemes
involves rules to generate the activity, specifically the sequence of actions, the
collection of information and the controls. - You lose what you have won; or
win what you have lost. - You return to someone the amount they gave you
or you recover from someone the amount of money that you lent them. A
debt is therefore the inverse of a balance in favor. Researchers dealing with
the development of mathematical skills cannot be satisfied with the idea that
mathematical words and sentences, as they appear in textbooks or in teachers’
comments and explanations, can be a sufficient criterion for assessing student
competencies. Testing their activity in situation is essential, particularly in
new situations, when they have to adapt their cognitive resources and face
a problem never before known. The function of the schemas, in the present
theory, is both to describe the ordinary ways of doing for already mastered
situations and to give clues on how to deal with new situations. Schemas
are adaptable resources: they assimilate new situations by adjusting to them.
Therefore, the schema definition must contain pre-made rules, tricks, and
procedures that have been shaped by situations already mastered; but these
components should also offer the possibility of adapting to new situations. On
the one hand, a schema is the invariant organization of activity for a certain
class of situations; on the other hand, this analytical definition must contain
open concepts and possibilities of inference. From these considerations it is
clear that schemes comprise several aspects, defined as follows: - You go back
as many steps as you have advanced and vice versa. Not only are increase
and decrease, or movements back and forth, the empirical roots of positive
and negative numbers, but also the relationships between two people (lender
and debtor) are examples of positive and negative numbers. This is important
for teaching algebra and accounting. 206 - The intentional aspect of the
schemes involves one or several goals that can be A tentative conclusion is
that the development of a conceptual field involves situations, schemata, and
symbolic tools of representation. A comprehensive definition of representation
is required, but I will get to it only in the conclusion of this paper. Algebraic
representation may not be useful for children at the elementary level while
the sagittal diagram may at least carry the meaning of going back and forth.
This is essential, but it doesn’t solve the problem of understanding that +7
and -7 are inverses of each other. Children need several examples of the
inverse character of the addition and subtraction operation. Several types of
awareness are needed: Machine Translated by Google The following example,
in the domain of multiplicative structures, shows the difference very clearly:
“suppose a student needs to find the amount of flour that he can make with
the corn production of a large farm: 182 t. He knows that it takes 1.2 kg of
corn to make 1 kg of flour.” The scheme that comes to mind after some time
(meaning it’s not a simple idea) is to try to find the ratio between 182 t and
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1.2 kg. This ratio is a scalar, a number that does not refer to a dimension,
since it is the quotient of two magnitudes of the same type (amounts of corn).
But the choice to calculate that ratio comes from the idea that it can be used
to find the corresponding amount of flour: it is the same - The computational
aspect involves possibilities of inference. They are essential to understand
that thought is made up of intense computational activity, even in seemingly
simple situations, and even more so in novel situations. We need to generate
goals, subgoals, and rules, as well as properties and relationships that are not
observable. The dialectical relationship between situations and schemas is so
intricate that sometimes an expression relative to situations is used to refer
to a schema, for example high jump or solving equations with two unknowns,
as well as expressions relative to schemas to refer to situations, for example,
rule of three situations (the rule of three is a scheme, not a situation). 207
The main points that I need to highlight in this definition are the generative
property of schemas, and the fact that they contain conceptual components,
without which they would be unable to adapt the activity to the variety of
cases usually faced by the subject. I also feel the need to add several comments
on the following. Another clarification concerns the relationship between
concepts and theorems: the link is so intricate that many researchers tend to
confuse them. The difference is that a theorem can be true or false, because it
is a statement (or a proposition). A concept is not a statement and therefore
cannot be true or false, but only relevant or irrelevant. Another important
point is that a statement can be thought to be true when in fact it is false, it
is still a theorem-in-act. There is little difference, from the point of view of
the activity, between a true proposition and a false one that is considered true.
The relationship between theorems and concepts is dialectical, in the sense
that there is no theorem without concepts, and there is no concept without
theorems. But the distinction is important for conceptual field theory, because
it is a developmental theory. For example, the analysis of additive structures
shows that the concepts of addition and subtraction are developed over a long
period of time, through situations that evoke theorems of very different levels.
- The epistemic aspect of schemes involves operational invariants, specifically
concepts-in-act and theorems-in-act. Its main function is to collect and select
relevant information and infer goals and rules from it. Machine Translated by
Google 208 The operative form and the predicative form of knowledge F (rate
X 1.2 kg) = rate XF (1.2 kg) The calculation also requires a change of units,
from tons to kilograms. The problem could be simplified if the farm production
were given in kilograms, but it is not usual to do that for large productions.
Complexity not only comes from doing, but also from putting something into
words and saying it. This theorem is completely implicit and the process
requires, also implicitly, that F (182 tons) be identified as F (ratio X 1.2 kg),
and that F (1.2 kg) be identified with 1 kg of flour. The scalar ratio between
182 tons and 1.2 kilograms is a concept-in-act, not a theorem-in-act, but its
use is invoked by the theorem. Enunciation plays an essential part in the
conceptualization process. One of the difficulties that students encounter when
learning mathematics is that some mathematical statements and symbolic
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expressions are as complex as the situations and thought operations required to
deal with them. Some researchers even consider the difficulty of mathematics
to be primarily a linguistic difficulty. This idea is wrong because mathematics
is not a language, but knowledge. The above scheme is not an algorithm,
but it could be formalized into the following algorithm: ”in a four-term
proportion, find the ratio between the two magnitudes referring to the same
type of quantity, and then apply it to the other quantity”. It is one of the
practical responsibilities of mathematicians to discover or invent algorithms
and the job of students is to learn them. Algorithms are schemes, but not
all schemes are algorithms. The reason for this is that schemes do not have
all the characteristics of algorithms: they lack ”effectiveness”, namely the
property of reaching a solution, if any, in a finite number of steps. However, the
organization of the activity is very similar in terms of schemes and algorithms.
This similarity includes the fact that the algorithms taught to students are
often appropriated by them under a simplified organization; they may even
switch, after some time, to wrong schemata. ratio between the two quantities
of corn (182 t and 1.2 kg) and the corresponding two quantities of flour.
Therefore, when you know the reason, all you have to do is multiply it by
the amount of flour corresponding to 1.2 kg of corn: Machine Translated by
Google However, understanding and verbalizing mathematical sentences plays
a significant role in the difficulties encountered by students. To illustrate this
point, let’s take two situations in which students have to draw the symmetrical
figure of a given figure. These situations contrast with one another, both from
the point of view of the schemata that are necessary for construction, and from
the point of view of the statements that one has to understand or produce
on these occasions (fig. 2). In the first case, there are some coordination
difficulties because the child needs to draw a straight line just above the dotted
line, not too high and not too low, and everyone knows that this is not easy
with a ruler; It is the same type of difficulty for the starting point and the
arrival point. There are also conditional rules. For example, “one square
to the left in the part already drawn, one square to the right in the part to
be drawn”, or also “two squares down in the figure on the left, two squares
down in the one on the left”. right”, or “one square to the right in the left
figure, one square to the left in the right figure”, starting from a reference
point homologous to the starting point on the left. Fig. 2. Two situations
for symmetry The first figure corresponds to the appropriate situation to be
presented to students aged 8 to 10, in which they must complete the drawing
of the fortress symmetrically to the vertical axis. The second situation might
typically be presented to 12- to 14-yearold students in France: construct a
triangle symmetrical to triangle ABC in relation to d (“d” here refers to
the dotted line). These rules are not very complex. However, they depend
on various concepts-in-act and theorems-in-act concerning symmetry and the
conservation of lengths and angles. Since all the angles are right and the lengths
are expressed in discrete units (squares), the difficulty is minimal. In the second
case, drawing triangle A’B’C’, symmetrical to triangle ABC in relation to line
d, is much more complex with the usual instruments in the classroom (ruler,
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compass and set of squares). Even the reduction of the triangle to its vertices
209 Machine Translated by Google S( A´B´C´, ABC,d) Triangle A´B´C´ is
symmetrical to triangle ABC in relation to the Cl (s) y Ca (s) The fortress is
symmetrical. Symmetry is an isometry. 210 Symmetry preserves lengths and
angles. Between statement 1 and statement 2, there is already a qualitative
leap: the symmetric adjective moves from the status of a one-element predicate
to the status of a three-element predicate (A is symmetric to B relative to C).
as sufficient elements to complete the task is an abstraction that some students
do not readily accept because they see the figure as a non-decomposable whole.
One step further, using the line d as the axis of symmetry for the segments
AA´, BB´, CC´, is far from trivial. Why draw a circle with its center at A,
and why should we be interested in the intersections of that circle with line
d? line d. The epistemological leap from the first to the second situation is
obvious. But there is also S(f) great leaps between the different statements that
it is possible to articulate on these occasions.11 One can also use a set square
and draw a perpendicular line from A to d, measure the distance from A to d,
traverse the line d to construct A’ at the same distance from A to d. But how
can I think that the distance is the same when there is no line yet? Between
statement 2 and statement 3, the symmetrical predicate is transformed into
an object of thought, symmetry, which has its specific properties: it preserves
lengths and angles. Nominalization (ie, forming a name from another class
of words or group of words) is the most common linguistic process used to
transform predicates into objects. In statements 1 and 2, the idea of symmetry
is a predicate (a propositional function); in statement 3, it was converted to
an object (an argument). The lowercase “s” is the type of symbol used by
logicians for arguments, while the uppercase “S” is used for predicates. The two
new predicates, Cl (conservation of lengths) and Ca (conservation of angles),
are thus properties of this new object s. The author uses French to show the
definite articles that are not appreciated in English. Since this is possible in
Spanish, we do the translation directly. 11 Machine Translated by Google 211
The different considerations and examples given above can be brought together
to theorize about the concept of representation. Behaviorists wanted to get
rid of that concept when they should have considered it a central concept of
psychology, like the concepts of force and motion in mechanics, or those of
evolution and the cell in the life sciences. They thought it was impossible
to have access to representation, but isn’t this the current state of science?
Newton had no access to the forces of attraction, neither did Darwin to the
succession of species, nor Mendel to genes. S C I Mathematical, scientific
and technical texts, and more generally texts of a certain level (philosophy,
literature, etc.), are full of such variations in the meaning of words, although
the authors try to make them unambiguous. Science is reduction, and the
following ideas are a drastic reduction of the psychological phenomenon, but at
least they offer possibilities to describe and analyze some important processes
of representation. The meaning of the in the symmetry in sentences 3 and
4 is the meaning of the universal quantifier. The la in the fortress or in
line d in sentences 1 and 2 has a deictic value: “this fortress”, “this line
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d”. Obviously the correspondence between the signified and the signifiers is
not a one-to-one correspondence at a lexical level. The succession of leaps
in the operative and predicative forms of mathematical knowledge inevitably
causes difficulties for students. Teachers are not sufficiently aware of these
jumps. Four different components of representation can be distinguished,
not as independent of each other, but as distinct in nature: (1) the stream
of consciousness, (2) language and other sets of symbols, (3) concepts and
categories, and ( 4) sets and subsets of schemes. Representation When we
move from statement 3 to statement 4, a new transformation takes place:
retention of lengths and angles becomes an object of thought, isometry. Now,
the predicate is the inclusion relation between two sets: the set of symmetries
S and the set of isometries I: Machine Translated by Google 212 Besides
that, thought is frequently accompanied, or even directed, by linguistic and
symbolic processes. Vygotsky has made this point very well. In the field of
mathematics, numerical and algebraic notations constitute a very important
part in the conceptualization and reasoning processes, although these are not
concepts by themselves; musical notation is not music either, but symphonies
would not be possible without it. What would mathematical thought be
without language and symbols? Obviously, the predicative form of knowledge
is essential, even if this is not the first form of knowledge. Every individual has
some experience of stream of consciousness. This is the most obvious proof of
the existence of representation as a psychological phenomenon, even if it does
not provide us with a fair and sufficient conception. This almost permanent
flow of images (visual, auditory, kinesthetic and somesthetic) accompanies
both wakefulness and sleep, as well as a certain awareness of one’s own gestures
and words, sometimes only outlined in the mind. Usually, we cannot analyze
this flow of percepts, ideas, images, words and gestures, but it testifies that
the representation works in a spontaneous and even irrepressible way. The
stream of perception is an integral part of the stream of consciousness, also
the stream of imagination, whether or not it is associated with perception. of
concepts and categories in the selection of information. The importance given
here to consciousness is not contradictory with the existence of the unconscious
phenomenon, or with the fact that there are privileged moments of sudden
consciousness, not reducible to the ordinary flow of consciousness. Concepts
and categories The fact that perception is a component of representation is
important for psychological theory because it is in the study of perception that
one sees the essential role language and symbols Concepts and categories form
the system with which we select information, with the purpose of directing
our activity in the most relevant way. This meaning of representation is not
as direct as the first two, because it rests on the thesis that perception is an
important component of representation, even when we do not have words to
be associated with the objects and relationships upon which the organization
depends. of our activity. The word ”concept” is taken here in a broader
sense than usual; It is normally restricted to the explicit objects of thought,
whereas here it is extended to concepts-in-act that are very often implicit in
the course of activity. This is why I use the expression “operational invariants”
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(instead of “concept” and “theorem”) as much as I can. The distinction
between conceptualization and symbolization is essential, to the point that
the understanding of words and statements by different people, particularly
between students and teachers, is not simply a binary significant/signified relationship, but a ternary one with the interpretation privileged by the Without
words and symbols, representation and experience cannot be communicated.
the stream of consciousness Machine Translated by Google It is essential for
psychologists to recognize the central role of activity in the development of
representation, competencies, and concepts. Because language and symbols
play an important role in conceptualization processes, many 2 To calculate
the height I must know the volume and the area, and divide the volume by
the Conclusion 213 1 To calculate the volume I must know the area and the
height, and multiply one by the other. 3 Volume is proportional to area when
height is held constant, and to height when area is held constant. This reading
requires much more than understanding multiplication and division operations
and the meaning of letters. It is not always mentioned in textbooks; however,
this is the real reason for the formula. area; this reading is more difficult
than the first, since it is its inverse. Schema and subschema systems operative
invariants. A compelling example of such processes is the understanding of
a formula such as that for the volume of rectangular prisms: Whatever role
symbols play in the conceptualization process, concepts and symbols should
not be confused. Where V= volume of the prism, A= area of the base and
H= height of the prism Therefore, the operational form of knowledge must
be considered as a component of representation. Schemas are essential: they
organize gestures and actions in the physical world, as well as interaction with
others, conversation, and reasoning. Consciousness often accompanies the
activity, but only partially: it is especially related to the goals and sub-goals,
evaluating the relevance of the captured information, controlling the effects of
the action. The structure of consciousness is different from the structure of
activity: we are aware of the most relevant properties of objects, but we are
more or less ignorant of how activity is generated and how subschemas are
activated by superschemas. . This hierarchical organization gives opportunity
for improvisation and contingency: schemas and sub-schemas are frequently
evoked by contingent aspects of situations; this is the reciprocal character of
their adaptive function. When students have to use it, they can read and
interpret this formula in various ways. Here are some of their interpretations:
V=A x H Representation is a dynamic activity, not an epiphenomenon that
could accompany the activity without nourishing or guiding it. Representation
is neither a dictionary nor just a library, but also a functional resource: it
organizes and regulates action and perception; at the same time, it is also the
product of action and perception. Machine Translated by Google Vergnaud, G.
(1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition
of math concepts and processes (pp. 127–174). London: Academic Press.
Piaget, J. (1968b). The formation of the symbol in the child: imitation, play
and dream, image and representation. Lausanne: Delachaux and Niestlé.
Vygotsky, L.S. (1962). Thought and language. Cambridge: MIT Press. There
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are specific features in mathematical concepts that need to be considered as
such. This is the main reason, both theoretical and practical, why it is more
fruitful to use the framework of conceptual fields than logical structures to
analyze the continuities and discontinuities of development in mathematics;
also to imagine the appropriate situations to prompt and help students move
through the multifaceted complexity of the field. References Revault d’Allonnes,
G. (1915). Schematism. Minutes of the 43rd session of the French Association
for the Advancement of Sciences (pp. 563-574). Paris: Masson et Cie. Finally,
the operational form of knowledge and the predicative form are intertwined at
all levels. There is no need to oppose one to the other; both are necessary to
analyze the difficulties experienced by children and the way in which they can
overcome them. Janet, P. (1928). Full text of the courses at the Collège de
France, Chair of Experimental and Comparative Psychology. The evolution
of memory and the notion of time. Paris: Chahine. 214 Revault d’Allonnes,
G. (1920). The mechanism of thought: mental schemes. Philosophical
Review, XC, 161–202. Piaget, J. (1968a). The birth of intelligence in children.
Neuchâtel: Delachaux and Niestle. Researchers identify conceptualization
and symbolization, as if the activity of enunciation and symbolization were
sufficient roots of knowledge, particularly of mathematical knowledge. This
is not the case. The analysis of the situations and the schemas shows that
the conceptualization process already takes place in the simplest forms of
the activity (even without language): the reason is that no action can be
efficient without the identification of some objects and their properties. Even
the most complex concepts, to gain meaning and operationality, need to be
contextualized and exemplified in situations. Therefore, from the point of view
of development, a concept is in short: a set of situations, a set of operative
invariants (contained in the schemes), and a set of linguistic and symbolic
representations.
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