ICMI Study 25
TEACHERS OF MATHEMATICS WORKING AND LEARNING IN COLLABORATIVE GROUPS
Theme D
Lisbon, Portugal, 3-7 February 2020
EVOLUTION OF CRITERIA FOR REPRESENTATIONAL ADEQUACY FOR
TEACHING INTEGERS THROUGH COLLABORATIVE INVESTIGATION
Ruchi S. Kumar
Tata Institute of Social Sciences, Mumbai, India
ruchi.kumar31@gmail.com, ruchi.kumar@tiss.edu
Representations have been considered as an important tool for doing and teaching mathematics but
there is paucity of research about the criteria used by teachers to select, use evaluate and generate
representations. This paper discusses the criteria for representational adequacy that teachers
evolved while engaging in collaborative investigation workshops. Analysis of the teachers’ talk
during the workshops indicated evolution of three distinct criteria, such as translatability between
representations, meaningfulness and coherence of representations with the nature of mathematics.
Analysis has helped to arrive at a grounded framework for criteria used by teachers to assess
representational adequacy. The implications for how collaborative investigation can help in
developing the teachers knowledge of representations and about representations has been discussed.
Rationale
Representations are one of the important tools used for teaching. They can be in various forms like
symbolic, visual or contextual in nature. Teachers’ knowledge, goals and beliefs have been
considered important to understand selection and use of representations in the classroom (Thompson,
1992; Shoenfeld, 1999). However little research has been done on how teachers select and use
representations (Stylianou, 2010) and to identify the underlying criteria that guide this selection. This
study explores teachers’ engagement with representations in the context of collaborative
invesitigation which led to emergence of criteria for determining representational adequacy at two
different levels.
Literature review and theoretical framework
Pape and Tchoshanov (2001) state that “representation is an inherently social activity” (p.120)
allowing learners to construct and interpret both internal and external representations individually as
well as in social contexts. Study by Cobb, Yackel and Wood, (1992) illustrated that representations
are not “transparent” and communication using representations requires significant amount of
negotiation among members, to be able to develop shared meanings. It is not easy for a person to see
the “mapping” between a concrete material and an arithmetic operation without elaborating and
showing how they are connected (Pape & Tchoshanov, 2001). Several studies have identified the
challenge that teachers face in transforming mathematical ideas into representations (Ball, 1990,
1992) thus pointing towards a possible knowledge gap in making “translations” between multiple
representations. Knowledge of representations and their affordances and limitations has been
considered as part of mathematical knowledge for teaching (Ball, Thames and Phelps, 2008).
Teachers’ use of representations is also considered as part of “transformation” of the knowledge
which involves “representation of ideas to learners… in the form of analogies, examples, illustrations,
explanations and demonstrations” (Rowland, Huckstep and Thwaites, 2003, p.2).
Kumar
Though the importance of teachers’ knowledge and use of representation is acknowledged in research
literature, this paper contributes a grounded framework of teachers’ criteria for representational
adequacy that identifies ideas considered important by teachers in selecting, evaluating and using the
different representations for teaching. I argue that knowledge of these criteria is a part of “knowledge
about representations” which is distinct from “knowledge of representation”. The former is closely
related to the construct of “meta-representational competence” in the research literature (diSessa,
2004). For example, knowing the area and column representation of multiplication is knowledge of
representation, while knowing in which context using a particular method makes sense is the
knowledge about representation. In this paper, a theoretical framework has been arrived at as a result
of the analysis of teacher talk during the collaborative investigation (Smith and Bill, 2004) of the
topic of integers in the workshop.
Collaborative activities like lesson study, professional learning communities have been identified as
fruitful in developing teacher’s knowledge. Studies indicate that collaborative discussion through
anticipating and discussing student thinking while making lesson plans in lesson study leads to
substantial teacher learning (Fernandez, 2005). Silver, Clark, Ghousseini, Charalambous and Sealy
(2007) have reported how collaborative conversations helped to connect different pedagogical,
content and student related issues. Meyer and Wilkerson (2011) elaborate on how opportunities to
develop teachers’ knowledge arise through the discussion of concepts and instructional strategies
prior to making a lesson plan rather than through the use of an existing lesson plan and focusing on
its implementation. This indicate the potential for collaborative investigation for teacher learning.
However, in Indian context, there exists no institutional structure or support for teachers to work
collaboratively and teachers are expected to “follow” the facilitator or circulars issued by
administrators about what to teach and how to teach. In this reported study, the space for collaboration
among teachers and between teachers and teacher educators was established through collaborative
investigation activity in workshops.
The study
This study involved 4 in-service middle school math teachers (more than 15 years experience)
working with a team of 3 researchers and a faculty. These teachers were nominated by their principals
as “effective teachers” to participate in the study. Hence the sample was purposive in nature. All the
teachers who participated had more than 15 years experience of teaching and were in the age range
of 39 to 50 years. Teachers engaged in collaborative investigation through six one-day workshops
spanning a period of four months for designing representations for teaching integers. During these
workshops, teachers shared the explanations and representations used by them for teaching integers
at sixth grade, shared the common errors they have seen among students. Teachers then
collaboratively used a framework of integer meanings to analyse textbook chapter and think of
examples of contexts and models for teaching integer arithmetic and analyse them for
appropriateness. Research team also collaborated with teachers by suggesting ideas as well as giving
critical inputs about the meanings of representations and whether it will make sense to students. After
a collaborative discussion on a ten day plan for teaching integers, teachers made individual unit plans.
After teaching, they shared their experience of teaching integers with each other. The nature of
collaboration among teachers was discursive in nature as teachers built on thoughts and ideas shared
by them and research team in the process of evaluating and designing representations for teaching
Kumar
integers. The main research question addressed in this paper is: What are the criteria for
representational adequacy that teachers used to select, evaluate and generate representations for
teaching integers through collaborative investigation?
Methodology
The researcher was a participant observer in the workshops. Around 40 hours of audio data from all
6 days of the workshop were transcribed. Transcripts were then read and preliminary codes were
developed from the data through open coding by the researcher as described by Miles and Huberman
(1990). She also wrote memos summarizing the findings of each day while making analytical notes
about the codes and identifying the significant events which denoted evidences of teachers’ reflection
or learning. These were discussed with a faculty member and in the second round of transcript
reading, the researcher and the faculty member coded each turn in the transcript independently using
the following categories: speaker, mathematical purpose, pedagogical purpose, integer meaning,
operation meaning, type of representation, and specific model/context. After initial coding, codes
were reviewed by the researcher and faculty member to resolve ambiguities. Through analysis of
these codes recurrent themes which illustrated the criteria used by teachers to evaluate representations
were identified like translatability, meaningfulness and consistency. The events belonging to these
themes were compared and essential features were identified, and framework was developed as
illustrated in Table 1 and 2 in this paper. In each turn of the teacher talk, where a representation was
being discussed, the implicit and explicit reasons given by a teacher to evaluate a representation
positively or negatively was coded as one or more of the criteria of translatability, meaningfulness
and consistency based on initial analysis. The analysis transcripts related to criteria revealed that there
exists two levels of criteria application -surface level and deeper level. The integration of two or more
criteria reflected a deeper level application of criteria thus allowing it to be distinguished from surface
level application.
Findings
Through analysis of transcripts from the workshop, three criteria for representational adequacy were
identified which are translatability, meaningfulness and consistency with the nature of mathematics.
These criteria were used by teachers to evaluate different types of representations including contexts,
models and symbolic representations for teaching integers.
Translatability criteria refers to the feasibility as perceived by the teachers of translating one form of
representation into another, for example, from symbolic form to a model. Using the criteria of
meaningfulness requires one to acknowledge that representations are not transparent and that
misconceptions about these meanings may exist among students. It may refer to different meanings
ascribed in different contexts and may also refer to the required revision in meaning when learning a
mathematical concept of a higher level. For example, students need to revise their understanding of
whole numbers to include fractions and integers on a number line. Mathematical consistency indicates
the consistent nature of mathematics and the way it is exhibited in the discourse. This can be related
to concerns for consistency in explanations whether using smaller numbers or bigger numbers.
Teachers often did not prefer certain explanations as they felt that they can be used to explain
operations with smaller numbers (like 3- 4) but cannot be used to explain for example, 335 – 448 as
they did not know how to represent such big numbers.
Kumar
During the one-day workshop, teachers discussed the different representations used by them for
teaching integers in sixth grade. They shared that they generally use contexts for only introducing the
concept of integers, neutralisation model for addition of integers, number line model for both addition
and subtraction and finally moving on to use of rules with symbolic expressions for addition and
subtraction. The analysis of the prescribed textbook indicated that the sequence and the use of
representations by teachers exactly matched what was there in the textbook.
Analysis of interactions of the first workshop indicated that teachers preferred symbolic
representations and use of rules over other types of representations for teaching integers. Table 1
below presents a few example of excerpts/descriptions of teachers’ talk during the workshops and the
implicit criteria inferred from analysis of the teachers’ talk.
Table 1: Examples of Use of Criteria in Teachers’ Talk During the Workshops
No.
Representation
used
Excerpts from workshop
Criteria
1
Symbolic
expression
“For subtraction (of integers), they have to first convert it
into addition (3 – 4 = – 4 +3), which children forget to
do” (Day 1)
Translatability:
surface level
2
Neutralisation
model
“Here [with coloured buttons] they can see +1 and –1.
They make it 2. They don’t consider it zero… (how to
explain) why it becomes zero?” (Day 1)
Translatability +
Meaningfulness:
deeper level
3
Number line
model
“If (while adding)negative integer (as addend) then move
in left direction on the number line. If subtraction of
Meaningfulnessnegative integer then move in opposite direction (right)”
surface level
(Day 1)
4
Neutralisation
model
“We are calling them for buttons [i.e asking them to work
on it].... Ultimately we should tell them rules otherwise
big numbers they will face problems” (Day 1)
Mathematical
consistency:
surface level
5
Symbolic
expressions
“You cannot take away 5 from 2.... This is also a
problem.... We did that so that they avoid mistakes when
they are young” (Day 1)
Meaningfulness
+ Mathematical
consistency:
deeper level
6
Context
“But negative numbers are not there... marks scored in
[suppose] five class tests are there. Suppose in the first Translatability+
test [one gets] 15, second test 17 [and then] 14 and 20 so Meaningfulness
if the child scores 2 marks more in the second test than in + Consistency:
the first so +2 then in the third test scoring 3 marks less so
deeper level
[–3. So it is a] change question.” (Day 3)
In row 1 of Table 1, a teacher discussed how they teach students subtraction of integers using numeric
expressions by asking them to convert the subtraction problem into an addition problem. The teacher
made the claim of translating one form of expression to another as one would get the same answer
Kumar
but did not justify why it should be done or the equivalence of the expressions. It was also clear that
students are told this as a procedure rather than discussing the equivalence. This is a case where the
criterion of translatability has been used at a surface level. On the other hand, the excerpt in row 2 of
Table 1 indicate the use of translatability criterion at deeper level since the teacher is not only
concerned about translating the actions done using neutralisation model with buttons to numeric
expression but also about the meaning of addition of positive and negative integers being clear to
students. The teacher realised that models are not transparent as students could not understand why
+1 and –1 should cancel each other and thus some explanation is required. She implicitly understood
how meanings held by students are not consistent with addition as neutralization as they view addition
as increment based on their past experience and thus counted all the 1s irrespective of the signs The
discussion then moved to considering the positive and negative integers as representing increase or
decrease in quantity. When the increase and decrease is equal, there would be no change in the state
of the quantity. Teachers felt that this could be a worthwhile idea to explore with students by using
the red and black buttons in the neutralisation model to represent the increase or decrease in quantity.
In row 3 of the Table 1, the operation of addition and subtraction on Number line was translated by
a teacher as moving towards positive and negative integers respectively by identifying the + and the
– sign in the symbolic expression. The instruction usually given to students was in form of rules to
be memorised about the direction in which to move when given a particular symbolic problem. The
instructions were different when performing subtraction of positive versus that of a negative number.
In reconstructing the the meaning of addition and subtraction as movement towards right and left
direction on the number line, no consideration has been made for the meanings held by students about
addition as increase and subtraction as decrease in quantity as a result of engagement in whole number
arithmetic. Another issue in this explanation is the confusion that might arise due to not differentiating
between the sign of the operation with the sign of the integer, as will be in the case of 3 – (– 4).
Teachers used number line as a tool for communication about how to solve numerical problems rather
than a tool for expanding students’ understanding of numbers, addition and subtraction.
The row 4 of table 1 has an excerpt from a teacher from the beginning of the workshop where in the
teacher indicated his preference of rules over the use of other representations like models and context.
The reason for his faith in telling rules was that he felt that rules would work for even larger numbers
while he had seen use of models and contexts for only few easy numbers. His preference was also
due the speed at which one is able to solve the integer arithmetic problems using the rules while use
of models and contexts required more time. It is thus inferred that the teacher used the criterion of
mathematical consistency at surface level to consider use of rules as consistent without having the
knowledge of how even the models and contexts can be used consistently with different types of
numbers.
In row 5 of Table 1, teachers realised that the emphasis during teaching in the primary grades that
bigger number cannot be subtracted from smaller number may lead to students reversing the minuend,
that is, write 5 – 2 when asked to subtract 5 from 2. The teacher’s discourse shifted from attributing
students’ error to students’ lack of understanding to recognising that instruction too can lead to errors
when due consideration of the mathematical concepts is not done while teaching in earlier classes.
Earlier, teachers were insisting that students need to be told to reverse the order while writing the
subtraction problem in numbers. Thus, the shift in discourse is also in terms of translating the problem
Kumar
“Subtract 5 from 2” into symbols to looking at what meaning could be made of this problem by
students. Teachers discussed how the symbolic representation of, 2 – 5 does not correspond to the
‘take away’ meaning that students are familiar with. This explained why students would reverse the
numbers in a subtraction problem to conform to the take away model of subtraction that they knew.
This brought the meaning of subtraction into focus as teachers became aware of the inconsistency of
“take away” meaning of subtraction for subtraction with integers. The meaning of subtraction as
‘finding the difference’ is consistent for both whole numbers as well as integers and thus could
probably be used to build students’ understanding of integer subtraction. Another issue that gets
highlighted in this mistake is of considering the minus sign as indication of subtraction operation and
not as integer. This is another difficulty faced by students which teachers realised through this
discussion. These discussions related to meanings attributed to minus sign, numbers and operations
created the need to reflect on the meanings ascribed by students as well as teachers themselves and
explore the alternative meanings of integers and their operations. On the other hand, it also made
them aware of the consistency that needs to be maintained in the meanings of subtraction and the
distinction between the subtraction operation and the integer. Thus, it was considered that the teacher
was using the criteria of meaningfulness and mathematical consistency at much deeper level during
this conversation.
In row 6 of the Table 1, a teacher is discussing a proposal of using scores in tests to be represented as
integers. She identified that it would not involve use of negative integers as signifying state and then
used change meaning to think of a way where change in marks could be represented using unary
integers which could be combined and represented using addition. The consistency in context
description was established by finding referents first for integers and then for operations of integers,
thus making a distinction between use of minus sign to denote integers and to denote the subtraction
operation. For example, there are different referents for negative integer (score of teams) and minus
sign for subtraction refers to the process of finding the difference between scores of teams making
the meaning of minus sign representing integers and that of subtraction as distinct. This led to
consistency in use of minus sign as integer and that for subtraction. This excerpt shows how the
teacher has internalised the criteria of translatability, meaningfulness and consistency by being able
to translate the actions in the context into numeric expressions using both sign of operations as well
as for integers and being able to meaningfully depict integers and their addition through the meaning
of integers as ‘change’.
Based on analysis of excerpts like above, a framework was arrived at to identify the features of using
a criterion at surface level or at deeper level which is presented in Table 2. The discursive nature of
collaboration among the teachers as well as researchers pushed the discourse in the workshops
towards use of criteria at deeper level. Although it is not possible within the scope of this paper to
illustrate how the discourse changed over the course of workshops, the nature of interactions in the
workshop were such that surface level application of criteria for a representation was either countered
by teachers or research group members with a deeper level application. Each of the teachers shared
their ideas about what would be an appropriate representation to use for teaching integers for the
particular topic. This was followed by evaluation or elaboration by other members either in support
or against the use of the particular representation. Initially teachers had divergent opinions about the
use of representations and favoured teaching the rules. The initial talk indicate overt concerns for
translatability of representations. However, over the course of collaboration teacher talk indicated the
Kumar
emergence of other criterias of meaningfulness and consistency as well as integration of criteria in
their voiced concerns about representations. When teachers reported their use of representations for
teaching integers using the plan developed in the workshops, their talk indicated more extensive use
of contexts and models than before. Their shared experience also included their reflections about the
use of representations and their role in developing understanding of integers and their operations.
Table 2: The Surface and Deeper Level Application of Criteria of Translatability,
Meaningfulness and Mathematical Consistency for Representational Adequacy
Criteria for
representational
adequacy
Translatability
criteria
Meaningfulness
criteria
Mathematical
Consistency
Surface level application
-getting same answer
-correspondence between
symbolic procedure and operation in
representation
-Translation is rule based
- No need felt for justification
Deeper level application
-Translations are conceptual
based
- Structures and processes in
representation have some meaning
-Justifications made explicit for
why representation works
-Awareness of non-transparency
-Representation as a tool for
of representations
communication
-Awareness and recognition of
-Meanings are under explored, range of meanings that correspond to the
may not be explicitly discussed
concept
--Meaning held by students not
- Meaningful connections among
explored or revised
translations
-Representation as a tool for
exploration of meaning
- Equivalence between
representations is assumed
-Usage of symbols not
consistent with meaning
- Mathematically consistent
-Equivalence between
representations established
Discussion and Conclusion
Teacher talk and criteria used for evaluating representations indicated that teachers tried translations
between models and symbolic representations through arbitrary rules which did not have any
justification and indicated surface level concerns for representational adequacy. These criteria
indicated the beliefs that teachers held about representations, about mathematics and about teaching
and learning of mathematics. Teachers believed that symbolic representations are more efficient than
other more concrete representations like models or contexts which has been reported by studies
elsewhere (Cai, 2006) as concrete or visual approaches were believed to be not useful for representing
larger numbers. Other studies have also found that teachers considered symbolic and numerical
representations as more central to learning and doing mathematics as compared to visual which are
termed as “informal” (Stylianou, 2010; Bergquist, 2005). Teachers’ collaborative exploration of
models and contexts during the workshops made them experience how it is possible to represent
larger numbers and operations using models and contexts and develop deeper levels of concern for
Kumar
representation selection through the arguments and justification given by their peers and teacher
educators. As discussed, teachers exhibited translatability criterion initially but developed the
criterion of meaningfulness and consistency only within the professional development workshop.
This is perhaps due to the professional development setting having different culture from what
teachers usually experience in their schools making them give rationale for the use of certain
representation. The nature of interactions in collaborative investigation supported teachers in making
shifts in their discourse and criteria for determining representational adequacy. Thus collaborative
investigation has implications for how teachers’ criteria for representational adequacy can be
extended through engaging teachers in collaboratively evaluating and designing representations for
teaching.
References
Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The elementary
school journal, 449-466.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it special?. Journal of
teacher education, 59(5), 389-407.
Cai, J. (2006). US and Chinese teachers’ cultural values of representations in mathematics education. In Mathematics
education in different cultural traditions-A comparative study of East Asia and the West (pp. 465-481). Springer US.
Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics
education. Journal for Research in Mathematics education, 2-33.
diSessa, A. A. (2004). Meta representation: Native competence and targets for instruction. Cognition and instruction,
22(3), 293-331.
Fernández, M. L. (2005, July). Exploring “lesson study” in teacher preparation. In Proceedings of the 29th Conference of
the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 305-312). PME Melbourne.
Kinach, B. M. (2002). A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics
methods course: Toward a model of effective practice. Teaching and Teacher Education, 18(1), 51-71.
Meyer, R. D., & Wilkerson, T. L. (2011). Lesson study: The impact on teachers’ knowledge for teaching mathematics.
In L.C. Hart, A. Alston & A. Murata (eds), Lesson study research and practice in mathematics education (pp. 15–
26). Springer Netherlands.
Mitchell, R., Charalambous, C. Y., & Hill, H. C. (2014). Examining the task and knowledge demands needed to teach
with representations. Journal of Mathematics Teacher Education, 17(1), 37-60.
Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies
in mathematics, 47(2), 175-197.
Pape, S. J., & Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical understanding. Theory
into practice, 40(2), 118-127.
Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge
quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255-281.
Schoenfeld, A. H. (1999). Models of the teaching process. The Journal of Mathematical Behavior, 18(3), 243-261.
Silver, E. A., Clark, L. M., Ghousseini, H. N., Charalambous, C. Y., & Sealy, J. T. (2007). Where is the mathematics?
Examining teachers’ mathematical learning opportunities in practice–based professional learning tasks. Journal of
Mathematics Teacher Education, 10(4–6), 261–277.
Smith, M. S., & Bill, V. (2004). Thinking through a lesson: Collaborative lesson planning as a means for improving the
quality of teaching. In Presentation at the annual meeting of the Association of Mathematics Teachers Educators, San
Diego: CA.
Stylianou, D. A. (2010). Teachers’ conceptions of representation in middle school mathematics. Journal of Mathematics
Teacher Education, 13(4), 325-343.
Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. Macmillan Publishing Co, Inc.