compartmentalization of representation in tasks related to ... - CiteSeerX

COMPARTMENTALIZATION OF REPRESENTATION IN TASKS RELATED TO ADDITION AND SUBTRACTION USING THE NUMBER LINE Myria Shiakalli*, Athanasios Gagatsis** * Primary School Teacher, PhD in Mathematics Education ** Department of Education, University of Cyprus The present paper aims at identifying the difficulties that arise in the conversion from one mode of representation of the concepts of addition and subtraction to another, and examining the phenomenon of compartmentalization, i.e. deficiency in the coordination of at least two modes of representation of a concept. Three tests were administered to 231 first and 241 second grade students. All the tests consisted of the same addition and subtraction tasks, but differed in the types of conversions that they involved. Data analysis showed lack of connections among the tasks of the three tests for both age groups. This finding reveals that different types of conversions among representations of the same mathematical content were approached in a distinct way, indicating the existence of the phenomenon of compartmentalization. INTRODUCTION Representations are used extensively in mathematics and translation ability is highly correlated with success in mathematics education. Mathematics teaching, school textbooks and other teaching materials in mathematics submit children to a wide variety of representations. The representational systems are fundamental for conceptual learning and determine, to a significant extent, what is learnt (Cheng, 2000). Understanding an idea entails (a) the ability to recognise an idea, which is embedded in a variety of qualitatively different representational systems; (b) the ability to flexibly manipulate the idea within given representational systems and (c) the ability to translate the idea from one system to another accurately (Gagatsis & Shiakalli, 2004). This is due to the fact that a construct in mathematics is accessible only through its semiotic representations and in addition one semiotic representation by itself cannot lead to the understanding of the mathematical object it represents (Duval, 2002). Understanding any concept entails the ability to coherently recognise at least two different representations of the concept and the ability to pass from the one to the other without falling into contradictions (Duval, 2002; Gagatsis, & Shiakalli, 2004; Griffin, & Case, 1997). Kaput (1992) found that translation disabilities are significant factors influencing mathematical learning. Strengthening or remediating these abilities facilitates the acquisition and use of elementary mathematical ideas. To diagnose a student’s learning difficulties or to identify instructional opportunities teachers can generate a variety of useful kinds of questions by presenting an idea in one representational mode and asking the student to illustrate, describe or represent the same idea in another mode. A central goal of mathematics teaching is thus taken to be that the students be able to pass from one 2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5, pp. 105-112. Prague: PME. 5 - 105

Shiakalli & Gagatsis representation to another (Hitt, 1998; Janvier, 1987). Despite this fact, many studies have shown that students face difficulties in transferring information gained in one context to another (e.g., Gagatsis, Shiakalli, & Panaoura, 2003; Yang, & Huang, 2004). In this paper, the term representations, in a restricted sense, is interpreted as the tools used for representing mathematical ideas (Gagatsis, Elia, & Mougi, 2002). By translation or conversion process, we refer to the psychological process involving the moving from one mode of representation to another (Janvier, 1987). The ability to identify and represent the same concept in different representations, and flexibility in moving from one representation to another, are crucial in mathematics learning, as they allow students to see rich connections, and develop deeper understanding of concepts (Even, 1998). Difficulties in the translation from one mode of representation of the same concept to another and inability to use a variety of representations for a mathematical idea can be seen as an indication for the existence of compartmentalization. The particular phenomenon reveals a cognitive difficulty that arises from the need to accomplish flexible and competent translation back and forth between different kinds of mathematical representations (Duval, 2002). The focus of the present study is to identify the difficulties that arise in the conversion back and forth between the symbolic and the number line representation of the concepts of addition and subtraction and to examine the phenomenon of compartmentalization, which may affect mathematics learning in a negative way. What is usually meant by the number line as a didactic means is a straight line with a scale, which is used for representing and carrying out arithmetic operations. Some researchers in mathematics education consider the number line to be (a) an important manipulative, that is a concrete model that students can use as a visual aid to solve mathematical problems (Hegarty, & Kozhevnikov, 1999; Raftopoulos, 2002); (b) a critical component in teaching arithmetic in general (Klein, Beishuizen, & Treffers, 1998; Selter, 1998) and the ordinal aspects of number. Related research explores the conditions under which the number line can enhance arithmetic understanding in first and second grade students (Fueyo, & Bushell, 1998; Klein et al., 1998). The effectiveness of the number line in helping students improve their performances in whole number addition and subtraction tasks, is, however questioned by other researchers in the field (Ernest, 1985; Hart, 1981; Liebeck, 1984). Previous empirical studies have not clarified compartmentalization in a comprehensive or systematic way. The existence of this phenomenon in students’ behaviour could not be revealed or verified by students’ success rates neither by students’ protocols in tasks involving different representations. These types of data can just be considered as an indication of the existence of compartmentalization. Thus, we theorize that the Factor Analysis along with the Implicative Statistical Method of Analysis, which reveals the similarity connections between students’ responses in the administered tasks, can be beneficial for identifying the existence of compartmentalization in students’ behaviour. Our basic conjecture is that 5 - 106

PME30 — 2006

Shiakalli & Gagatsis compartmentalization exists when the following conditions appear: First, students deal inconsistently or incoherently with different types of translation of the same mathematical knowledge from one mode of representation to another; and second, success in one mode of representation or type of conversion of a concept does not entail success in another mode of representation or type of conversion of the same concept. The basic hypotheses of the present paper, which is part of a larger research study (Shiakalli, 2004), are the following: (a) There is a compartmentalization of addition and subtraction tasks in symbolic representation without the use of the number line and of the corresponding number line addition and subtraction tasks; (b) There is a statistically significant difference in the performance of first and second grade students according to the direction of the translation – from the symbolic to the number line representation and vice-versa. METHOD Participants In the research participated 231 first grade students – average age 6.1 – coming from 10 different classrooms and 241 second grade students – average age 7.2 – coming from 11 different classrooms. The population of the study was the number of first and second graders attending the 74 public primary schools in the Limassol district in Cyprus. The subjects came from six public primary schools. Research instruments Three tests (Test A, Test B, Test C) were administered to the students, a week apart from each other. Test A includes 12 addition and 12 subtraction tasks with numbers up to 30 expressed symbolically (e.g., 5+3=___, 16–2= ___, 19+___=25, 9 – ___ = 2, ___ + 2 = 8, ___ – 3 = 14). Test B includes the same tasks, but this time students had to use the number line in order to represent the given mathematical sentence, to find the answer on the number line and then to complete the mathematical sentence (translation from the symbolic representation – mathematical sentence to the number line representation). For example, 5 + 3 = ____ 0

1

2

3

4

5

6

7

8

9 10

11 12

13 14 15 16 17 18 19 20 21 22 23 24 25

Test C includes the same tasks, however, students had to write the mathematical sentence represented on the number line (translation from the number line representation to the symbolic representation). Data analysis The Factor Analysis, the Repeated Measures Analysis and the Implicative Statistical Method of Gras using a computer software called C.H.I.C. (Classification Hiérarchique Implicative et Cohésive) (Bodin, Coutourier, & Gras, 2000) were used for the analysis of the collected data based on students’ performance in the tasks. Gras’s Implicative Statistical Model is a method of analysis that determines the similarity connections of factors (Gras, 1996). For this study’s needs, similarity diagrams were produced for each age group of students. The similarity diagram PME30 — 2006

5 - 107

Shiakalli & Gagatsis allows for the arrangement of the tasks into groups according to the homogeneity by which they were handled by the students. RESULTS The six factors, which occurred from the Factor Analysis explaining 53% of the total variance, indicate that the addition and subtraction tasks were grouped based on the type of representation. Factor 1 is related to the symbolic representation tasks (tasks included in Test Α), Factor 6 is related to the easy symbolic representation tasks (tasks included in Test Α), Factor 4 is related to the translation tasks from the symbolic representation to the number line representation (tasks included in Test Β), Factor 5 is related to the easy translation tasks from the symbolic representation to the number line representation (tasks included in Test Β), whereas Factor 3 is related to translation tasks from the number line representation to the symbolic representation (tasks included in Test C). Only Factor 2 is related to symbolic representation tasks as well as to translation tasks from the symbolic representation to the number line representation. In this case, the grouping of the variables was based on the difficulty of the tasks, which is attributed to the place of the unknown quantity (initial quantity is unknown) and not based on the type of the representation.

C 7

C 5

C 8

C 6

C 3

C 4

C 2

C 1

A 1 9

A 1 4

A 2 0

A 8

A 1 3

A 7

A 2

A 1

The results from the Factor Analysis are reinforced by the results from the Implicative Statistical Method of Analysis. Figure 1 presents the Similarity Diagram (Lerman, 1981) of the addition and subtraction tasks in Test A (symbolic representation) and Test C (number line representation) according to the second graders’ responses.

Test A Test C Figure 1: Similarity diagram of second grade students’ responses in Test A and Ar br e des similar ites : C:\ My Documents\ PhD\ Chic Data Final\ Cla ss B\ Class B - Test AC - Chic.cs

Test C Similarities presented with bold lines are important at significant level 99%. Two distinct similarity groups of tasks are identified in Figure 1. The first group involves similarity relations among the tasks of Test A, while the second group involves similarity relations among the tasks of Test C. This finding reveals that the mode of representation (symbolic representation, number line representation), seems to influence students’ performance, even though the tasks involved the same algebraic 5 - 108

PME30 — 2006

Shiakalli & Gagatsis relations. The same conclusion can be drawn from the Similarity Diagram of first grade students’ responses in Test A and Test C (Shiakalli, 2004). The results from the Repeated Measures Analysis showed that the performance of first and second graders differs according to the direction of the translation – symbolic to the number line representation (tasks in Test Β) or vice-versa (tasks in Test C). The mean of the performance of first graders at translation tasks from the symbolic to the number line representation is 4.242, whereas the mean of their performance at translation tasks from the number line representation to the symbolic representation is 5.052. The mean of the performance of second graders at translation tasks from the symbolic to the number line representation is 4.207, whereas the mean of their performance at translation tasks from the number line representation to the symbolic representation is 5.589. The performance of first and second graders is higher at translation tasks from the number line to the symbolic representation (tasks in Test C) when compared to their performance at translation tasks from the symbolic to the number line representation (tasks in Test Β) (