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“Quaderni di Ricerca in Didattica (Mathematics)”, n. 25, Supplemento n.2, 2015

G.R.I.M. (Departimento di Matematica e Informatica, University of Palermo, Italy)

503

Visual Strategy and Algebraic Expression: Two Sides of the

Same Problem?

Shai Olsher1 & Rina Hershkowitz2

1University of Haifa, Haifa, Israel, olshers@edu.haifa.ac.il,

2Weizmann Institute of Science, Rehovot, Israel, rina.hershkowitz@weizmann.ac.il.

Abstract: It has been advocated that the search for patterns and their organization in mathematical language

is a central component of mathematical thinking. Hershkowitz, Arcavi and Bruckheimer (2001) investigated

problem solving processes of a "visual-pattern-problem" which start with visual strategies for reorganizing

the visual pattern and ended in a formal algebraic expression as a solution. In this study we had decided to

use the same problem for investigating also a process in an opposite direction: starting from a given algebraic

expression, analyzing it in a way that uncovers the visual strategies which are behind the algebraic

components of the given expression.

Resumè : Il a été préconisé que la recherche de modèles et de leur organisation en langage mathématique est

un élément central de la pensée mathématique. Hershkowitz, Arcavi et Bruckheimer (2001) ont étudié les

processus de résolution de problèmes d'un "problème de modèle visuel" ("visual-pattern-problem"). Ceux-ci

commencent par des stratégies visuelles afin de réorganiser le modèle visuel pour aboutir enfin à une

expression algébrique formelle. Dans cette étude, nous avons décidé d'utiliser le même problème pour

enquêter aussi un processus dans la direction opposée: partant d'une expression algébrique donnée, l'analyser

d'une manière qui décèle les stratégies visuelles qui sont derrière les composants algébriques de l'expression

donnée.

Background

Many questions about the different roles of visualization in mathematics have been addressed in the

last few decades of mathematics education research (Arcavi, 2003). Researchers have studied the

ways in which children develop ways to describe visual patterns gradually grasping the basic

concepts of algebra. Among the ways are the use of computer software that enables to perform

visual manipulations in building visual patterns (Healy & Hoyles, 1999), and also software that

enables the constructing of a model by visualizing sets of patterns gradually replaced by numbers

and variables (Mavrikis, Noss, Hoyles, & Geraniou, 2013).

Hershkowitz et al. (2001) used a rich visual task (Figure 1) in order to invoke as many visual

solution strategies as possible, that were categorized according to the visual-counting strategies

used by the different solvers.

Figure 1: nxn square made of matches

This problem is representative of a whole class of “counting” situations in which the solutions' main

How many matches are needed to

build the following nxn square?

Find as many strategies as you can

“Quaderni di Ricerca in Didattica (Mathematics)”, n. 25, Supplemento n.2, 2015

G.R.I.M. (Departimento di Matematica e Informatica, University of Palermo, Italy)

504

steps can be described as follows: observation, recording and understanding of regularity, finding

and applying a “visual-counting” strategy, generalizing and capturing the generalization in a

symbolic-algebraic form (see Figure 2).

Figure 2: Algebraic solution examples to the visual problem

The algebra expresses the visual-counting-strategies, and compresses them into expressions. This

emphasizes that even though the problem is visual, without using algebra we would not have been

able to write the solution in a compact way.

Methodology

Participants in the study were eight mathematics education graduate students from different

programs in Israel. The participants took part in a group's activity that has two parts. In the first part

participants were asked to individually find an algebraic expression that would represent the

number of matches needed to build the nxn matches' square shown in Figure 1.

In the second part, participants were asked to go in the opposite direction; they were given six

different algebraic expressions that represent solutions for the number of matches in the nxn square.

The participants were then asked to choose an expression, to find a visual strategy for finding the

number of matches in the nxn square, where the algebraic expression they choose is its solution.

Then they had to explain to their peers the visual strategy they had found. Data sources included the

individual written report, written by each student along his/her work's process, video documentation

of the group activity, and field notes.

The analyses and interpretations of the students' reports were based to large extent on the findings

of Hershkowitz et al. (2001). Next, we focused on categorizing the different characteristics of the

visual strategies which are hidden behind the given algebraic expressions.

Results

Our initial analysis revealed three main characteristics in uncovering the visual pattern strategy

which is the origin of the algebraic expression as a whole or its components. In the following we

will illustrate these characteristics as they appeared.

The dual role of numbers in the algebraic expressions - a quantity and a

visual-construction-unit

While trying to uncover the visual strategy which was the origin to algebraic expressions, we found

out that the most popular component in the matches' square problem was "4". In some cases it was

standing for a quantity. For example, when dealing with the expression:

2

)1n(n

4+

⋅

, student C

referred to the "4":

C: "Where does this four come from? I now need to decompose the, to decompose it to four parts.

… The most natural way for me to get to four parts is to look at each triangle (One quarter of the

square while divided to four parts by its diagonals).

In a different situation, student D was addressing the "4" as something else:

D: "… and I saw n four (nx4). OK?"

1. 4+3(n-1)·2+2(n-1)(n-1)

2.

2

n4n4 2+

3. 2·2(1+2+···+n)

4.

2

)1n(n

4+

⋅

“Quaderni di Ricerca in Didattica (Mathematics)”, n. 25, Supplemento n.2, 2015

G.R.I.M. (Departimento di Matematica e Informatica, University of Palermo, Italy)

505

Instructor: "When you say n four what do you mean?"

D: "n four is n times square" [referring to a square construct made of four matches.]

Visual counting strategies behind arithmetic operations

Arithmetic operations within the algebraic expression might serve as hints for the visual counting

strategies upon which the expression is constructed. For example, the division by 2 operator in the

expression:

2

n4n4 2+

. This expression was the product of the visual counting strategy which was

called by Hershkowitz et al. (2001), Shake and count. In this solution some of the visual constructs

counted "shared" matches which are thus counted twice, so the solution process has to take into

account fixing this double counting by division. When describing his way of constructing a visual

solution suitable for this expression, student B referred to the operator:

"We have here four n squared, plus four n, divided by two. So first the division by two gives me a

hint that something is counted here twice and at the end we divide".

Ambiguity in visual representations of algebraic expressions

One expected outcome was that each algebraic expression had several suggested visual solutions

that could be described with it. This ambiguity manifested, for example, in the "3" included in the

expression: 4+3(n-1)·2+2(n-1)(n-1). This expression was visualized commonly by what

Hershkowitz et al. (2001) categorized as from one square on…As described visually in in figure 3,

and explained by student A:

Figure 3: From one square on…

A: The first square [upper left] has four sides, four matches, and here we have n minus one

[showing left column], and here we have n minus one [showing top row], and each one of these, we

add three. …

[background] Wait. I did not understand. Which three matches?

A: This has four matches [upper left square]. Here we have n minus one squares. n minus one sides,

like, we multiply by three. One, and two, and three [counting sides of U pattern marked in red in

figure] one-two-three, one-two-three

Instructor: All right.

A: And also these [showing the turned U figures that make up the top row]. So we have four, plus

two times n minus one times three.

For this case, although the U and the כ patterns are visually different, it took only one minor

clarification for all of the students to accept that number "3" stands for them both, thus not

explicitly giving any information about the visual solution apart from this part of the expression is

constructed from three matches.

“Quaderni di Ricerca in Didattica (Mathematics)”, n. 25, Supplemento n.2, 2015

G.R.I.M. (Departimento di Matematica e Informatica, University of Palermo, Italy)

506

Expression form which is familiar as representing numerical counting

strategy

Some expressions are familiar as the sum of numerical sequence. For example, the expression:

2

)1n(n

4+

⋅

. This expression was called by Hershkowitz et al. (2001) Starting with symbols. In this

case, participants identified this expression as four times the sum of 1+2+3+…+n. This resulted in

many instances of an arithmetic series that could be identified in the pattern in various ways. Two

of which appear in Figure 4.

Figure 4: Two instances of 1+2+3+…+n (marked by dashed lines)

in the nxn square of matches

This exemplifies how visual reasoning can be guided, inspired and supported, a posteriori, by a

symbolic expression known to be the solution. Therefore, the visualization process may not only

involve the decomposition into units or the creation of auxiliary constructions, it may also be

guided by a known symbolic result.

Concluding Remarks

Can a visual strategy be inspired by a symbolic expression? Our data show that yes, students might

be familiar with the visual problem, and then may "see" the visual-counting strategy or even

strategies in the certain algebraic expression given to them.

It is worth to note that this thinking direction is much more complicated than the opposite direction

described at Hershkowitz et al. (2001) paper. At the previous paper the solver is going from the

visual problem to look for appropriate visual solution's strategies, and at the end transform them

into algebraic expressions. In this paper the solver has to go back and forth in tiny steps looking for

hints that evolve from the components of the algebraic expressions.

REFERENCES

Arcavi, A. (2003). The role of visual representations in the learning of Mathematics.

Educational Studies in Mathematics, 52, 215-241.

Healy, L., & Hoyles, C. (1999). Visual and symbolic reasoning in Mathematics: Making

connections with computers? Mathematical Thinking and Learning, 1:1, 59-84.

Hershkowitz, R., Arcavi, A., & Bruckheimer, M. (2001). Reflections on the status and nature

of visual reasoning - the case of the matches. International Journal of Mathematical

Education in Science and Technology, 32(2), 255-265.

Mavrikis, M., Noss, R., Hoyles, C., & Geraniou, E. (2013). Sowing the seeds of algebraic

generalization: Designing epistemic affordances for an intelligent micro-world. Journal of

Computer Assisted Learning, 29(1), 68-84.