JOSEP M. FORTUNY, JOAQUIM GIMÉNEZ, CLAUDI ALSINA
401
INTEGRATED ASSESSMENT ON MATHEMATICS 12-16
ABSTRACT. In this paper we give an account of an integrated package of assessment,
describing its conceptual posing as well as its formats , functions, use and effects. The
package introduces a diversity of materials searching for connecting and integrating
different contributions in a multidimensional scheme.
INTRODUCTION
One of the purposes in actual curricular reforms, is to include a design of assessment
which contributes to regulate the docent practice and should enhance mathematics
learning. In this context, we have tried to implement the experience on assessment
carried out during the past decade with the new trends on the subject (Niss 1993 and
NCTM 1993), and with the contrasting experience carried out already on the basis on a
recent development of a complete curricular project in the field of Mathematics for the
age group 12-16 designed for the Department of Education of the Catalan Government
by Alsina, Fortuny, Giménez (1992). The BDM 12-16 (translated as Good Morning
Maths 12-16 ) Project is developed on the basis of activities designed for a progressive
and interactive discovery of mathematical contents. These contents can be assimilated
by means of complementary activities which offer the students a chance to practice and
diversify the situations previously introduced.
The main aim of our research is to design and validate an "integrated package of
assessment", exemplifying the different necessary components for a reflection in a
regular teaching practice. The validation of such design uses a systematical observation
of teacher / student interrelations (Ball 1991). Our proposal try to give to the teacher
basic indications for building an accurate student' profile and help the diversity
treatement, by reflecting different tasks' results .
INTEGRATED PACKAGE
The integrated package of mathematical assessment , named IPMA, has been
developed throughout the whole cycle 12-16 into distinct formats. The scheme (see
figure 1) shows how the integrated package of assessment was designed with four key
elements (PJ) Work Projects, (AS) Progress activities, (AU) Self-regulators elements of
attitudes and values, and (OB) Observation. It also shows which were the criteria for
making the student's work (with balance of adjustement) together with the
corresponding justifications and decisions.
Educational Studies in Mathematics 27:401-412, 1994.
 1994 Kluwer Academic Publishers. DRAFT
402
JOSEP M. FORTUNY, JOAQUIM GIMÉNEZ, CLAUDI ALSINA
JOSEP M. FORTUNY, JOAQUIM GIMÉNEZ, CLAUDI ALSINA
403
With such elements, teacher and students control the general competences of
mathematical ability (Bell 1993), the specific range of abilities is revisted (Resnick
1987, De Lange in press) in different type of tasks, and interaction and attitudes is also
analyzed (Leder 1993). Diagnosis activities are not presented here as separated, because
of extension (see Fortuny,Giménez and Alsina 1994). The control of diagnostic aspect
is included as a starting point from the result of progress activities.
PROGRESS ACTIVITIES AND RANGE OF ABILITIES
By progress activities we mean our approach to the process of controling over the
students' learning process . We relate a 5x3 matrix of different type of activities
(concepts, algorithms, problem solving, visual-language and checking reasoning) with
their range of elaboration (low, intermediate, high). We suggest to the teachers the use
of different strategies of presentation of items in a test-form or more open activity
(NCTM 1992). Also different languages could be used ; verbal, pictures, combination
of them, etc. Progress activities give the oportunity to present different type and range
as we show in the following example (see AS matrix in figure 1 ).
.
Range indicates the variability of complexity of mental operations implied in a
mathematical activity, and describe the degree of inter-connections among concepts and
structures. A possible representation of range relates directed and radial conceptual
systems and procedural networks (Resnick 1987, Kaput 1987, Schoenfeld 1992, De
Lange in press). The range should be considered as an expression of the amount and
progressive quality of such networks. Range of abilities requires different type of
abilities as: mathematising, interpreting, processing, algorithmisation, recursivity...
The existance of variability in the range does not mean taking into account
different levels of difficulty, but considering the distinct complexity of the mental
processes involved activity. According to this principle, we used such ranges in the
implementation process in the classroom and in service teaching courses. The activities
of lower range presupose, among others the application of technical routines, standard
algorithms, solving problems of a given type; whereas the activities of an intermediate
range are connected to problem solving and to working with processes such as relating
or integrating. Finally, the activities of a high range requires the analysis of complex
situations under different prospectives and moreover, they imply decision tanking,
structuring, creativity and critical analysis (Alsina,Fortuny, Giménez 1992, De Lange in
press).
OBSERVATION AND SELF - REGULATION
It seems important for one of the components of the general assessment process
to be determined by the multiple and systematic observations the teacher keeps under
each of the students (motivation, interaction, ... ). The teacher should follow different
steps when observing the students: (a) analyze everyday work with open activities,
problem solving, self observation, classroom journal, or other activities (b) find
characteristics of students according to appropriate categories, (c) write corresponding
grids accurately prepared.
PROJECT WORK
Project activities is an open activity which provides the opportunity to present general
framework of student' ability to design, uses of mathematization, etc. and contribute to
show planification ability and work organisation. A project is a work to be done on a
specific topic at non-school hours. This project must follow certain steps proposed by
the teacher, though its structure must not be totally constrained; students should be
allowed to make some kind of choice. Some advice at the starting point, formal
constraints and the composition of the final work are some of the difficulties the teacher
will encounter and which will try to overcome. He/she will either give relevant
information or provide hints to facilitate the students' autonomy. Moreover, the teacher
will either have to supervise, and probably revise, the preliminary plan or to tolerate an
absolute freedom of action. The proposal of work projects facilitates the control over
the procedural elements, contributes to the acquisition of those procedures and makes
the work with transverse elements of the curriculum (consumption, environment,
leisure, sport, etc.) easier.
RESEARCH QUESTIONS
The main questions investigate in this study are the folowing:
1. Could we find a set of simple instruments (package of assessment) of
assessment which integrate enough elements to enhance mathematics learning as
an integral part of teaching -learning process, to engage the students, and to
ptovide opportunities to reflect and improve their work ?
2. Do the teachers use such set of instruments to extent their knowledge and
integrate in their regular classroom ?
METHOD
In view of the questions, the research was carried out in three phases: Design of the
package, Control of classroom processes ( by different elements of orientation and
analysis of teacher's observations ), and recognition of results.
The design of the package is presented in the introduction and started in 199192. Four secondary school teachers in two different schools in Barcelona area
participated in the development and experimentation for two years (1992-1994).
INSTRUMENTS
We consider as elements of orientation, all the instruments that help the initial control
about planification (phase 1) , implementation (phase 2) and regulation (phase 3). With
the planification elements, one must constantly take control over one's aims by means
of explicitation of the pedagogical intentions (see figure 2 for "week planification
gride" as an example) .
For the second and third phases of implementation and regulation, different kinds of
instruments will be used such as: observational grids, detailed analysis of the research
tasks, valuation of class journals, cooperative assessment, valuation of projects , etc.
404
These are tools which help the teacher to draw what it is known as "the students'
profile" and to observe the development of the students' learning process. We also
register the students' attitude, their intentions and their mathematical contents. We do
not evaluate only a final project but we assess the students' progress by setting them
tasks of different grades of difficulty. The results will be also registered in several grids
form and discussed with students.Let's look an example. In progress activities above
mentioned, double reading of control (of types of skills and of their range ) allows a
double collective valuation (figure 3) which can be illustrated in two tables giving to the
teacher more information than usual.
As teacher observations, we collect the observations that a teacher did about the
abilities in specific problem situations. We are using as a method for observation the
reflective thinking of different parts of the package from accurate analysis and we will
explain some case studies.
RESULTS AND DISCUSSION
Teachers revealed three levels of observation in the continuous assessment : (a) local
aims as specific attainment of the level of communication and interpretation , (b)
general aims as description of the answers with regards to the consecution of specific general purposes, (c) intentional, attitudinal and confidence as improvement of
classroom environment, reinforcing the need of research and individual participation.
Observing open and progress activities
Open activities gaves us the possibility to know by the observation the specific
connotative and intentional purposes of the students. In different situations the teacher
could observe that communication is not totally appropriate for a properly description of
some shapes the students had drawn. Thus, we do not only show, in the corresponding
grid, the students' skills, but also how the power of metacognitive reasoning fits in the
framework. This way, teacher write down in the final observational grid sentences like
"Peter has gone through the process of interiorization and adaptation in a satisfactory
way".
The teachers involved in the experiences (discussing the assessment process)
established that the aims of the activity we will observe were going to be the following:
Interpretation of data (indirect measurements in a scheme) and discussion (meaning of
coefficient and assignment of volume units), elaboration process (working with a scale
and planning with conditions) , communication processes (incorporation of criticisms),
consolidation and recursive understanding (writing a report stating the remaining ideas).
In the next sample activity (figure 4) the students should, first of all, be able to
recognize a real situation (the value of the "coefficient of building land ") and showing a
mathematical ability (geometrical design and construction).
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THE COEFFICIENT OF BUILDING LAND
43a
A great deal of new buildings are nowadays being bilt on many different sites. The local council controls the buildings in
course of construction in its area.
URBANIZE AND BUILD
A man has bought a piece of land, pentagon in shape, of the following proportions:
D AF= 10 m
FG= 40 m
GH= 15 m
HC= 20 m
EF= 20 m
DG= 20 m
HB= 35 m
E
G
F
H
A
He knows about the town-planning proposed by the local
Council. In this area the maximum volume apted for construction
is stablished by means of a coefficient of building land of 0.4;
that is to say, the maximum volume suitable for construction
cannot go beyond the value of the area of the piece of land
multiplied by 0.4.
C
B
1. What is the maximum volume suitable for construction?
2. Plan the distribution of several terraced houses with rectangular-shaped plants of 100 m2, and 3.1 m height. Some of
them can be two-storey houses but they must be built in accordancewith the coeficient of building land.
3. Write a critical report on the different distributions planned by your class-mates.
4. Write a report on the importance of the coefficient of building land.
Figure 2. Translation of a first part of an activity sheet in the project BDM 12-16.
Let's look at one teacher's observational exemple to illustrate one of this levels of
observation from the above presented activity . Eric's distribution (figure 5) was
generally rejected because of "non-rectangular buildings" , but Jordi admited that he had
tried to do something different. The student remarked: "If I had tried to use an oblique
rectangle I would also have made a mistake !" But other mistakes were generally
observed by the students in the class-discussion: "All the designs except Melisa's did
not fit into a realistic situation " (a lot of students drew their buildings too close to the
border).
406
Generally, the students had all tried to be original but they decided Melisa's project was
the best one because she had distributed the buildings as in a real situation". A lot of
people forgot the scale indication. With this "way of doing " the Students revealed
their consecution of local mathematical objectives, and the teacher can analyze it and
situate in the student profile. Progress activities could be also considered as a good
instrument. In the experimental situations, teachers confirmed that students with same
global scores were separated by two different categories: level and type of activities.
The experiences with 13 to 15 years old students gave the oportunity to show that the
range discriminate the students by levels.
407
In the example Florencia and Boyana (13 years old students in a private school in
Barcelona) have small differences on scores. Both can solve either a closed problem or
direct inference but cannot solve an open situation. Florencia using an algorithmical
way of doing in all the situations (see figure 6), tried to solve by equations the open
situation, she made a lot of mistakes in language & visualization situations (represent
the building figure as 2 dimensional) and cannot explain completely the 2nd range
problem about similar triangles. Nevertheless, Boyana is more "regular" , and made
mistakes in all third level questions solving the others. The teacher observed
immediately how the differences of quality means that Boyana is situated regularly in a
2nd level , but Florencia still remain in 1st level in some activities. Florencia has more
difficulties to establish theoretical frameworks and her language of explanations is less
fluent than other students. The teacher involved in such experiece said: " I never
thought that such differences could appeared".
About General Tasks
Use of classroom journals and note-books was considered very important observational
tools to recognize what happens in classroom activities. Nevertheless, they require a
disciplinate effort and dedication on the teacher part since each student must be
controlled, at least, twice a year. The following categories should be included in the
collection of information from the group-class: reflecting the whole work, formulation
and expression, integration, invention, group work, attitudes of progress, norms, selforganization, reflection of action, use of technological resources, including accurate
comments.
Following with the example of coefficient of builidng land, the teacher A wrote
down the following comments:
About Melisa - "Adjusted interpretation of data in drawing activities".
408
About Eric - "Extremely Interested . Original purposes. No incorporation of data assignment ".
About Jordi - "Bad interpretation of scale drawings".
After analyzing such particular observational process , the teachers wrote down in their
journals some general comments , which we have translated as follows :
A - The general planning was quite successful. Nobody gave the sheet without an answer. (Acceptation
as an intentional issue) . Everybody suggested that the suitable volume was the "area of the pentagon
multiplied by 0,4) and nobody made a mistake on the "scale reproduction" needed in requirement 2.
B - Some students recognize immediately the missing data, but 60% couldn't solve that part of the
problem because they didn't use a scheme to establish relations on measurements.
C - About 80% used some values as equivalences of sides (As FG =40, then ED = 40) but they were
blocked by the figures because they were not "adjusted by scale of data".
D - Only about 10 % recognized a scheme as a solution to give to their colleagues, but some students
found it difficult to use other representations than the picture to find the missing data .
The specific grid of assessment of the Students' journal used was contrasted with
teachers' beliefs according three general criteria : Conceptual integration, co-operation
and social interaction, and communicative aspects. Taking different the above
information as base, we used an observational grid with inputs derived from the three
following categories : intentional, procedural and attitudinal elements (see more
explanations in Fortuny, Giménez 1993). For the collection of data we proposed a grid
and a guideline which ask the teacher to write only important comments on those
aspects. In fact, observational grids and criteria offered were found difficult for teachers,
but the acceptance in our teaching training seminars and research refers to the
recognition of its being a meaningful help for the updating and self-analysis of the
acomplishment of objectives.
Montse - "I like grids. At the begining, it was impossible for me, short space, difficult sentences and
categories,... but you must force yourself to explain briefly what you observe. Now, it's not difficult at
all,.... I had never thought of intentional issues before. Now I think it's important to take them into
consideration. It was interesting to use methods used by researchers. Qualitative observations can be
done with grids. It's easy to use.them
José Antonio - Observation help me to concrete posterior interviews with children. It looks to me as a
score of the Student. Perhaps I like verbal explanations... But it's nice to spend less time to explain and
promote discussion and confrrontation about the individuals earlier than the interviews. I think I will use
better next year, even for parents' communication.
Anna - There are a lot of difficulties in the introduction of observation as a tool for reviewing the
negociation process.
Intentions, attitudes and Projects .
We use the valuation of a Project as a way to ilustrate how the attitudes and intentions
are related to instructional activity. Next figure shows the attainment of a Project about
building a sports ground carried out with 15-year-old students. On the right hand side of
the sheet the remarks made by the teacher can be read.
409
Credit 6
Research Project
2. You have to present your project onthe closing
date. No more than foursheets, excluding illustrations
and graphs. You must follow the format specified in 3.
Athletics track
1. Topic: Athletics track
- You can do a study in-depth of the 400 m. athletic track.
Starting Date:
- You must use mathematicalconcepts such as
proportionality,statistics,geometry,angles,graphs.
Closing Date:
- Take initiatives. Analize throughly all the questions.
General Advice
STUDENTS'
INSTRUCTIONS
Start off by collecting information about the physical
aspects of thetrack ( measures, materials,...)and about
its dynamic uses (recordskind of races, ... )
Content:
1. Topic. General adv ice and
Starting point
starting
point.
1. Build a scale model (1:200) of the track, drawing the
2. Conditions
carry out
arrival to
andstarting
points for all the races.
the project.
2. Draw some graphs illustratinghow the records in the
differentraces
have changed.
3. Structure
of the memoir.
3. Analyze the sports facilitieslocated in the centre zone.
4. Assessment
grid and criteria.
4. Think about the best places to observe the races.
Conditions to carry out the project
Once you have decided on your focusof inquiry, tell the
teacher.
First of all, you must draw a plan ofhow to develop the
whole project andshow it to the teacher.
3.Structure of the Memoir
Write the memoir following these steps:
1.Title page
A. GENERAL DESIGN AND STRATEGIES
The exact title of the project and thename of the student o
1. Identify information
who has done it must be specified. Write a
o
2. Logic and systematic work
brief-but-vivid account of theaims and discoveries of
o
3. Extension & in-depth work
the project; also a table with the contents.
B. MATEMATICAL CONTENT
2. Main text
o
4. Mathematical formulation
Write an acurate description of the development of the o
5. Use of the mathematical language
project stating the
o
6. Applying techniques
different subparts, and an appropiatenumeric and graphic
C. ACURATENESS
documentation.
o
7. Acurateness in the use of Mathematics
3. Conclusions
o
8. Correcting the results
Write a summary of the discoveries
D. CLARITY AND COMMUNICATION
and include also some comments on yourlimitations.
o
9. Clear and concise explanations
4. Mathematical methods
o
10. Structure
Enclose a conceptual list or map of the mathematical
E. MATHEMATICAL ATTITUDE
contents and instrumentsyou have made use of.
o
11. Research spirit
5. Resources
o
12. Mathematizing contexts
Give a reference of the books and other material used.
F. AUTONOMY
6. Acknowlegment
o
13. Decision-Taking
o
14. Organization
G. GENERAL VALUATION
o
15. Valuating the conclusions
o
16. Restrictions and Perspectives
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
o o o o
Final Mark
Comments
Fig. 5. Presentation of a work project for children aged 12-14 (BDM 12-16)
The use of tests and activities for self observation are specially recommended for the
self-regulation of attitudes and values of the Students towards the classroom. Grids for
collecting the results were also considered as useful instruments for teachers.
CONCLUSIONS
Some general trends emerges from the experience and research:
(1) A package of assessment includes an specific learning process as a "way of
doing" in which observational tasks were facilitated by such important issues.
Interdisciplinar activities and treatement of diversity considered, give opportunities to
improve teaching.
(2) The package is in accordance with a continuous process of assessing, and
synthonized with new curriculum in Spain and other countries.
(3) Student takes an important role assuming responsablities in building the
learning process. Teachers must be careful in their posing of general aims to be assessed
and have a new role, which is difficult to accept.
(4) The assessment in the Package (from the point of view of the student) is well
accepted by teachers as a function of student's profile on time.
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EFFECTS
As one of the effects of the application made, we consider : the learning profile
and treatement of diversity. In our package, we call a profile the graph assigning a level
(1,2,3) on each part A= f (A,PJ,AS,AU,SR,OB). In each category and block of
activities, different pairs are given according the categories as vector space : A = (A1 ,
A2 , A3,...). This profile change with time, in such a form that we could observe the real
progress of the student by comparing graphs in a usual mathematical way, for each
category. The analysis which will adjust the students' profile will have to: (a) State the
weak spots of each student in relation to his/her skills. (b) Consider, if any, the
particular difficulties of a group of them. (c) Take into account the possibility of
establishing different routes in solving a task carried out in groups of any number of
students. (d) Recognize the necessity of a study in-depth on the reflections made when
building the syllabus of the course. (e) Promote reflections on the contributions of the
initial syllabus design, on the cognitive and attitudinal trades of the students and on
their values.
Together with the profile, the teacher should provide a brief-but-vivid account
for the decision he or the whole teaching staff have taken. It is time now for the teacher
and the teaching team to decide on possible paths to follow. The results which favoured
the assignment of levels may give raise to the re-planning of the units, changing either
its shape or its organization. This way, the teachers can propose new alternative routes
for the students in each of the different groups. In our didactic proposal, the tasks were
set considering different levels of difficulty. Thus, we include the possibility of setting
the students in one of the three possible groups, which often will be working on
different tasks. We aim the students to encounter the content (conceptual and
procedural) in the most appropiate way, taking their competence into account. The
teacher himself must decide whether tasks requiring a higer level of competence than
that of the students will motivate them or, on the contrary, will give raise to fustration.
In brief, the mentioned work of assessment is designed as an integrated package.
This implies that diverse formats and phases were used: (1) a temporal process and an
implement, (2) modification , (3) following the classroom management. One can
decide to work with flexible groups following different routes, can simply take into
account the students' levels for a new design or can restructure the units, following a
guide steps. In case of failure in the attainment of objectives, we do not suggest that the
teacher should do the same problem again. We propose that the same conceptual or
procedural problem should be analyzed under a different prospective, using a new
strategy or a new situation.
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Josep M. Fortuny
Department of Didactics of Mathematics
Universitat Autònoma de Barcelona
Edifici G 5-138
08193 Bellaterra (Barcelona) Spain
Joaquim Giménez
Engeniering Computer Department
Universitat Rovira i Virgili.
Ctra Valls s/n
43007- Tarragona. Spain
Claudi Alsina
Department of Mathematics
Universitat Politècnica de Catalunya
Diagonal 648
Barcelona . Spain
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