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. 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