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Integrating Arithmetic and Algebra in a

Collaborative Learning and Computational

Environment Using ACODESA

Fernando Hitt, Carlos Cortés, and Mireille Saboya1

Abstract In the transition from arithmetic to algebra and in light of the

disjunction between the natural and symbolic approach to algebra and the choice

of a natural way of learning, this paper discusses the development of a cognitive

control structure in pupils when they are faced with a mathematical task.

Researchers sought to develop, in novice pupils in both Quebec (12-13 years old)

and Mexico (14-15 years old), an arithmetic-algebraic thinking structure that

would promote mathematics competencies in a method based on collaborative

learning, scientific debate and self-reflection (ACODESA, acronym which comes

from the French abbreviation of Apprentissage collaboratif, Débat scientifique,

Autoréflexion), and immersed in an activity theory approach. This paper

promotes the equal use of both paper and pencil and technology in order to solve

a mathematical task in a sociocultural and technological environment.

Introduction

Over the course of the last century, the mathematics curriculum took arithmetic

as a proper subject for study at primary school level education, and algebra as a

proper subject for secondary school level. This dissociation influenced research

in mathematics education, which, in turn, reverberated through the academic

programs implemented. Examining the work of psychologists before the 1940s,

1 Fernando Hitt and Mireille Saboya

Département de mathématiques (GRUTEAM), Université du Québec à Montréal, Montreal,

Canada

e-mail: hitt.fernando@uqam.ca

e-mail: saboya.mireille@uqam.ca

Carlos Cortés

Facultad de Ciencias, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico

e-mail: jcortes@zeus.umich.mx

© Springer International Publishing

G. Aldon, F. Hitt, L. Bazzini, U. Gellert (eds.), Mathematics and Technology,

Advances in Mathematics Education, DOI

270

F. Hitt, C. Cortés, and M. Saboya

Brownell (1942) noted that psychologists used “puzzles” in the study of

intelligence to analyse processes related to the “insight” involved when

individuals solved such problems. Brownell proposed a radical change focusing

on the study of the resolution of the arithmetic verbal problems used in

textbooks.

Brownell’s work (Ibid.) attracted the attention of psychologists and educators

interested in studying the skills involved in solving arithmetic word problems as

a means of understanding the phenomena linked to learning mathematics.

The experience of solving puzzle type problems (where, for example, from 9

matches one is required to build four equilateral triangles, and then, from 6

matches, one is asked to build the same number of equilateral triangles) gave rise

to the analysis of solving problems that had single or multiple solutions. From a

psychological point of view, these factors led, in the case of a problem with one

solution, to an analysis of convergent thinking linked to direct efforts towards

achieving a goal. In the case of problems with multiple solutions, this led to an

analysis of both divergent thinking (Guilford 1967) and creativity (Bear 1993). In

fact, Guilford’s model (Ibid.) stressed the importance of developing divergent

before convergent thinking. Gradually, Brownell's approach led to an extensive

research strand focusing on the phenomena related to the resolution of arithmetic

problems in primary school and the development of arithmetic thinking.

What is arithmetic thinking?

Brownell’s characterisation (Ibid.) of exercises, problems and puzzles

encouraged psychologists and mathematics teachers to focus their research on the

study of arithmetic problem solving. Polya (1945) expanded problem solving to

other levels of education, thus promoting the emergence of a new paradigm.

Some mathematics educators followed this trend and contributed their own new

theoretical approaches (RME through the influence of Freudenthal; Mason,

Burton and Stacey 1982; Schoenfeld 1985; Santos 2010). Returning to primary

school level, for example, Vergnaud’s work (1990) on solving arithmetic

problems led to the identification of both arithmetic in problem solving and

conceptualisation in primary school, and led to the theoretical approach related to

"conceptual fields".

Similar approaches led to some research products in order to characterise

arithmetic thinking. For example, Verschaffel and De Corte (1996), taking into

account the research conducted in the 1990s, propose arithmetic thinking related

to: a) number concepts and number sense; b) the meaning of arithmetic

operations; c) control of basic arithmetic facts; d) mental and written arithmetic;

and, e) word problems using digital literacy and arithmetic skills.

While progress was made in the study of learning problems linked to the

resolution of arithmetic problems, research continued toward an understanding of

the problems related to learning algebra (Booth 1988). The notion of variable

began to be studied (Sutherland 1993), thus promoting investigation into the

Integrating Arithmetic and Algebra in a Collaborative Learning

271

learning of covariation between variables (Carlson 2002) and the identification of

the role of the variable as an unknown, as a large number, and as a variation

between variables from a functional point of view (Trigueros and Ursini 2008).

Given the organisation of the curriculum, which designated arithmetic for

primary school and algebra for secondary school, researchers began to talk about

the problems related to the transition from one level to the other. At the same

time, the emergence of the notion of epistemological obstacle in the French

school (Brousseau 1976/1983) possibly reinforced this idea of a “break” between

arithmetic and algebra. Vergnaud (1988) points out that the transition from

arithmetic to algebra is linked to an epistemological obstacle. Other approaches,

related to the notion of the unknown in solving linear equations, led to the notion

of a "cut" between arithmetic thinking and algebraic thinking (Filloy and Rojano

1989) or even the cognitive obstacle (Herscovics and Linchevski 1994). These

studies announced the need to characterise algebraic thinking.

What is algebraic thinking?

As mentioned in the previous section, the research paradigm linked to the

“thinking break” between arithmetic and algebra was essential for the

characterisation of algebraic thinking. Under this paradigm, Kieran (2007)

characterises algebraic thinking using a model called GTGm: Generational

algebraic activities involve the forming of expressions and equations;

Transformational activities such as factoring, expanding, and substituting; and,

Global/meta-level mathematical activities such as problem solving and

modelling. An analysis of this model reveals that, in the past, much of the

secondary school level research focused on the teaching of algebra in section G

and T of Kieran’s model. It is likely that the Gm section is linked to the

Freudenthal School’s research results regarding realistic mathematics, at the heart

of which approach is mathematical modelling.

Visual aspects in curricular change in mathematics

In the early 1990s, an important curricular change in the field of mathematics

began. The visual aspects were highlighted in curriculum changes, promoting

displays of the mathematical aspects. A clear example can be seen in the US

Standards (NCTM 2000). In this context, geometric aspects in problem solving

began to be included in algebra. It was explicitly important to approach a concept

through the use of different representations of that concept. From a curricular

standpoint as well as from a general standpoint related to research in mathematics

education, mathematical visualisation has attracted the attention of researchers.

These changes began from a curricular perspective, with a new approach to

teaching algebra and the promotion of a geometric-algebraic approach to algebra.

Progressing along this research line, for example, Zimmerman and Cunningham

(1991) begin the preface of their book with the question: What is visualisation in

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F. Hitt, C. Cortés, and M. Saboya

mathematics? This study explicitly referred to an important role in the production

of external representations:

Mathematical visualization is the process of forming images (mentally, or with pencil

and paper, or with the aid of technology) and using such images effectively for

mathematical discovery and understanding. (p. 3)

Technology influenced enormously in these changes. Graphical

representations that caused major programming problems were resolved, thus

giving rise to the production of computer software and enabling an approach to

mathematics from the multiple representations user standpoint.

Early algebra and the emergence of a new paradigm

While the previous section discussed the “rupture” approach to characterise

arithmetic thinking and algebraic thinking, little by little other research programs

arose, which were initially tied to the idea of the “generalization of arithmetic”

(see Mason 1996; and Lee 1996). Along these lines, Radford (1996) comments

how these authors stressed an approach to the learning problem regarding

“algebra as a generalised arithmetic”, and goes on to discuss the role of the

unknown and the equation:

The above discussion suggests that the algebraic concepts of unknowns and equations

appear to be intrinsically bound to the problem-solving approach, and that the concepts

of variable and formula appear to be intrinsically bound to the pattern generalization

approach. Thus generalization and problem solving approaches appear to be mutual

complementary fields in teaching algebra. How can we connect these approaches in the

classroom? I think this is an open question (p. 111).

From this perspective, a new paradigm was born. Kaput (1995, 2000)

proposes a research program under the following guidelines, with the first two at

the heart of the learning of algebra and the other three completing this learning:

1. (Kernel) Algebra as a generalization and formalization of patterns and constraints,

with, especially, but not exclusively, Algebra as Generalized Arithmetic Reasoning

and Algebra as Generalized Quantitative Reasoning

2. Kernel) Algebra as syntactically guided manipulations of formalism

3. (Topic-strand) Algebra as the study of structures and systems, abstracted from

computations and relations

4. (Topic-strand) Algebra as the study of functions, relationships and joint variation

5. (Language aspect) Algebra as a cluster of (a) modelling and (b) Phenomena-

controlling languages. (2000, p. 3)

Kaput called this research program Algebrafying the K-12 Curriculum (2000),

while Carpenter et al. (2003, 2005) initiated the Early algebra research project in

1996. This new paradigm formed part of research programs in the XXI century.

Thus, the Early Algebra movement, in which Carpenter and Kaput played an

important early role, is well situated in the USA. The book Early Algebraization,

edited by Cai et Knuth (2011), shows the progress of research in that area in

other countries. In this book, one can appreciate a division between the

Integrating Arithmetic and Algebra in a Collaborative Learning

273

“enthusiastic” and “cautious” researchers with regard to the Early Algebra

movement.

Among the enthusiasts are Blanton and Kaput (2011), as are Britt and Erwin

(2011, p. 139), who even criticised Filloy and Rojano's approach by highlighting

Carraher, Schliemann, Brizuela and Earnest (2006) with regard to their Early

Algebra proposal. Similarly, Schliemann, Carraher and Brizuela (2012) show

how algebraic notation can be introduced in elementary school in order to

develop mathematical content, stating: “The 5th grade lessons focused on

algebraic notation for representing word problems, leading to linear equations

with a single variable or with variables on both sides of the equal sign.” (p. 115).

Among the cautious, are Cooper and Warren (2011), who argue that:

The results have shown the negative effect of closure on generalisation in symbolic

representations, the predominance of single variance generalisation over covariant

generalisation in tabular representations, and the reduced ability to readily identify

commonalities and relationships in enactive and iconic representations. (p. 187)

In this regard, Radford (2011, p. 304) states that: “... the idea of introducing

algebra in the early years remains clouded by the lack of clear distinction

between what is arithmetic and what is algebraic”. On this, the debate remains

open, for example, Lins and Kaput (2012) characterising the movement as:

…algebrafied elementary mathematics would empower students, particularly by

fostering a greater degree of generality in their thinking and an increased ability to

communicate that generality. (p. 58)

As spokespersons for the Early Algebra working group at ICME (2004), they

openly criticised past generations, whose results were exclusively related to “sad

histories”, and specified that, in contrast, the Early Algebra movement presents

research results linked to “happy stories” regarding the experience of learning

algebra content.

The third excluded strikes back!

In light of the research results described above, this study approached the

problems of learning algebra by introducing new variables that could not be left

out of the discussion. As new theoretical approaches about learning algebra are

born, so are different general learning paradigms. Research in the last century

was strongly cognitivist, with Harel, Selden, and Selden (2006), surprised to

learn that, in the PME studies on the period of 1995 - 2005, most of the

investigations related to Advanced Mathematical Thinking were much more

cognitive and less socio-constructivist or sociocultural. While communication in

the mathematics classroom emerges as an essential element, the literature begins

to show that researchers are inclined towards a socio-constructivist or

sociocultural approach.

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F. Hitt, C. Cortés, and M. Saboya

Our theoretical approach to an introduction to learning algebra

The research objectives for this study are founded on a cultural approach, which

takes into account the theory of activity and which views communication in the

classroom as essential. The work of Engeström (1999) is taken as a culturally

unifying approach, as advocated by Vygotsky (1962), incorporating Leontiev’s

(1978) activity theory, in which communication is an essential element in the

building of knowledge as described by Voloshinov (1973). Our approach to

Radford’s processes of signification, is immersed in a mathematics classroom

teaching method named ACODESA (see Hitt 2007; and, Hitt and González-

Martín 2015), that also take into account a self-reflection component.

An analysis of the literature on the followers of the Early Algebra movement

shows that some research is aimed at building a “Fast Track” from arithmetic to

algebra. This study posits that a functional approach to algebra should be

followed, such as that developed in both Passaro (2009) and Hitt et al. (2015).

This study concurs with some followers of Early Algebra, in that the use of

patterns is able to generate generalisation processes in pupils, and, thus,

proposes, in the context of the use of patterns, the following:

Generalisation. Construction of the subsequent term in a series when the

previous terms are provided. Construction of an intermediate term when the

previous and subsequent terms are provided. Construction of a term when the

term in the series is a “large number” and when the first terms of the series

have been provided. Construction processes for “any term from the series.”

This study considers generalisation in the context of a pattern, where, rather

than as a way of moving quickly from arithmetic to algebra, it is an element used

to integrate into the pupils’ mathematical structure. This will enable the pupils to

develop the skills of prediction, argumentation and validation (Saboya, Berdnaz,

and Hitt 2015), and will assist them in their transition from arithmetic to algebra

and vice versa. Indeed, a research program is proposed here that would develop

arithmetic-algebraic thinking within a sociocultural context of knowledge

construction.

As described above, this chapter, seeks to make a modest contribution, in that

it represents the beginning of a research program. Our research is focused on

specific content related to the construction of polygonal numbers in a

sociocultural environment within Engeström’s post-Vygotskian model (1999),

and takes into account the results of those post-Vygotskian authors considered as

comprising the fifth generation, such as Nardi (1997):

The object of activity theory is to understand the unity of consciousness and activity.

Activity theory incorporates strong notions of intentionality, history, mediation,

collaboration and development in constructing consciousness. (p. 4)

As the use of technology in the learning of mathematics is a variable yet to be

mentioned here, Mariotti’s (2012) work on the role of artefacts as mediators in a

Integrating Arithmetic and Algebra in a Collaborative Learning

275

learning process is integral to the inclusion of technology in a sociocultural

environment.

The use of patterns, and especially the construct of generalisation, is related

to mathematical visualisation. Visualisation, as mentioned by Duval (2002), is

different from perception. In our case, then, the following applies:

Visualisation. Considering perception as something created by the individual

– a “transparent” mental image depicting the situation with which s/he are

faced – visualisation requires the transformation of representations associated

with the task at hand, and the ability to articulate other representations that

emerge in pupils’ resolution processes, as associated with the task.

This study is not only interested in institutional representations (which can be

associated with a register of representations, as described by Duval 1995). It is

also concerned with the non-institutional semiotic representations that can be

produced in a visualisation process (diSessa et al. 1991; Hitt 2013; Hitt and

González-Martín 2015; Mariotti 2012) and which emerge in a semiotic process

of signification (Radford 2003) when pupils follow a process of resolving a

mathematical activity immersed in a technological setting.

Institutional representation. Representation found in textbooks, computer

screens or those used by the mathematics teacher.

Non-institutional representation. Representation produced by pupils, as

linked to actions undertaken in a process of resolving a non-routine activity.

Since the method proposed here is related to polygonal numbers and the use

of technology, ideas related to the construction of polygonal numbers that date

back to the time of the Greeks were considered here. Furthermore, including

technology as one of the variables led to the inclusion of Healy and Sutherland

(1990) and Hitt (1994), who, in Excel environments and Excel and LOGO

environments, respectively, conducted investigations into the construction of

polygonal numbers by secondary school and pre-service teachers respectively.

Healy and Sutherland (Ibid.) mention that the result obtained by those

secondary level pupils (in the Excel environment) that expresses a relationship

linked to the calculation of a triangular number "n",

"trig.

Δ

n = na before + position", is a non-institutional representation linked to a

process of iteration. Hitt (ibid.) criticises these results, indicating that activities in

an exclusively Excel environment provoke “an anchor” which does not allow

them to switch to a classical algebraic context. Hitt (Ibid.) aimed to combine

working with paper and pencil with the use of an applet constructed using the

LOGO program. In light of these results and considering new theoretical and

curricular contributions, both approaches are of contemporary importance, in that

they generate diversified thinking for the production of non-institutional

representations and iteration processes. Secondly, they enable the careful design

of activities that promote the use of paper and pencil, while also fostering the

evolution of the non-institutional representations that emerge at the initial stage

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F. Hitt, C. Cortés, and M. Saboya

into institutional representations within meaningful processes (a broader

discussion on task design is provided in Chapter 3 of this volume).

Methodology

Our research was developed within two populations, with one group from

Quebec comprising 13 first grade secondary school pupils (aged 12-13 years

old), and the other from Mexico, which consisted of 14 third year secondary

school pupils (aged 14-15 year-old). Pupils agreed voluntarily to take part in the

experiment, which aimed to gain insight into the problem, as occurring in the two

populations individually, rather than comparing results.

• The Quebec experiment used Excel and an applet called POLY (see

below), which had been designed exclusively for this activity (Cortés and

Hitt 2012). Two researchers, known here as R1 and R2, developed the

teaching experiment in a sociocultural setting. Two cameras and several

voice recorders were used in this experiment.

• The Mexican experiment used a calculator (TI-Nspire) instead of Excel

and the POLY applet. The activities were developed by one teacher,

known here as P1, and another researcher, known here as R3. One camera

was used in this experiment.

This study adheres to a teaching method known as ACODESA is divided into

5 steps (fully explained in Chapter 3):

• Individual work: production of official and non-official representations

related to the task.

• Teamwork on the same task. Process of prediction, argumentation and

validation.

• Debate (could become scientific debate). Process of argumentation and

validation.

• Self-reflection (individual work in a process of reconstruction)

• Process of institutionalisation.

Engeström’s (Ibid.) model was used to organise the ACODESA steps, taking

into account a sociocultural learning setting. In previous research undertaken by

these authors (see Hitt 2007; Hitt 2011; and, Hitt and González-Martín 2015), the

self-reflection step was implemented immediately after plenary discussion. Due

to problems of knowledge retention (Hitt and González-Martín idem; Karsenty

2003; Thompson 2002), for this experiment we decided that for the self-

reflection step, it would be interesting to implemented 45 days after the plenary

debate.

Integrating Arithmetic and Algebra in a Collaborative Learning

277

ACODESA steps

ACODESA and activity theory

Fig. 1 ACODESA, as immersed in activity theory in line with Engeström’s model

The two first activities were implemented as an introductory activity in order

to remind pupils of some of the Excel commands and provide a historical

approach to polygonal numbers. The ACODESA method was implemented after

the two previous activities.

1. Resolution of two arithmetic word problems in a paper and pencil

setting and a plenary discussion about how, according to the population,

to solve them with either Excel or a calculator. This was implemented to

remind pupils how to use Excel or a calculator.

2. Introduction to polygonal numbers from a historical point of view.

3. Invitation to the populations to solve the activity in line with the

ACODESA characteristics shown in Fig. 1.

The tasks were used in both countries with only a few changes.

1) Look carefully at these numbers. What is the fifth triangular number? Make a

representation. Explain how you proceeded.

2) In your opinion, how are the triangular numbers constructed? What do you

observe?

3) What is the 11th triangular number? Explain how you found it.

4) You have to write a SHORT email to a friend describing how to calculate the

triangular number 83. Describe what you would write. YOU DO NOT HAVE

TO DO THE CALCULATIONS!

5) And, how do you calculate any triangular number (we still want a SHORT

message here).

Fig. 2 First five questions in a paper and pencil environment

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F. Hitt, C. Cortés, and M. Saboya

The second part of the activity was designed to work with Excel or CAS and

to be verified with the POLY applet.

Develop the same ideas as in the previous section, but this time using Excel (or a

calculator). Here is what we are requesting of you:

What do you do to find the 6th, 7th, and 8th triangular numbers?

Is it possible to calculate the 30th triangular number, the 83rd triangular number,

and the 120th triangular number?

How do you do this? Explain

What kind of limitations and possibilities do you encounter when calculating

under this approach?

Please show the operations you must undertake when calculating a polygonal

number.

Fig. 3 Second part of the activity

Analysis of the Quebec results

The first introductory part of the session comprised the individual resolution of

the two word arithmetic problems using paper and pencil, and a plenary

discussion about how to solve the same problems using Excel. Researcher R1

conducted the plenary discussion, immediately after which Researcher R2

conducted a short historical introduction to polygonal numbers and then initiated

the first part of the ACODESA activity related to polygonal numbers. In this first

step, individual work was required, as was work in a pencil and paper

environment.

Once the pupils had undertaken the first individual explorations, R2 organised

the teamwork, with Team G1 comprising three girls, Team G2 comprising 3 girls,

Team G3 comprising 3 boys and a girl, and Team G4 comprising a boy and two

girls. Only one computer was permitted for each team.

Integrating Arithmetic and Algebra in a Collaborative Learning

279

Teamwork and the first results

After pupils exchanged ideas, R2 requested a plenary discussion, asking each

team to present its findings on how to calculate the 11th Triangular number (T11).

Three teams (G1, G2 and G4) presented their findings, while members of Team G3

mentioned that their strategy was similar to the first team (see Fig. 3).

An initial and surprising outcome was the emergence of three different

strategies. Their initial production (see Fig. 3) indicates that the pupils have

undertaken a process of visualisation. They realised that it is possible to move

along the diagonal, adding balls progressively, while one can add to the number

of balls along the diagonal in an arithmetic progression. Pupils presented the first

three iconic figures with their respective values and a process of generalisation in

order to calculate T11 (see Fig. 4).

1+2+3+4+5+6+7+8+9+10+11

Fig. 4 The representation used by Team G1

It seems that these pupils have undertaken a visualisation process in order to

construct a general numerical progression. The action of adding balls along on

the diagonal is transformed by adding the number of balls to the arithmetic

progression, thus abandoning the iconic representation used to calculate the 3rd

triangular number.

Team G2 presented the results of their calculation of T11 with a single figure,

indicating that, in the first column, one should place 11 balls, and then reduce the

number of balls in the next column by one (10) in order to reach, at the end, only

one ball, imagining the 1st column with 11 balls, the next with 10, and so on (see

Fig. 4). An initial process of visualisation and generalisation is then made with

only one drawing, thus inducing a numerical process which is inverted, as

compared to that presented by the first team in Fig. 4. Team G2’s visualisation

process is more compact than Team G1, in that the team members made a direct

iconic representation of T11, expressing their process arithmetically. This

example is generic in that they were able to represent any triangular number

under this visual representation.

11+10+9+8+7+5+4+3+2+1

Fig. 5 The representation used by Team G2

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F. Hitt, C. Cortés, and M. Saboya

Team G3 mentioned that they had a similar approach to Team G1. A boy from

Team G4 (named G4-1 hereafter) then approached the blackboard. From his first

representation onwards, this pupil substituted the iconic ball-based representation

for a more practical one, explaining that whenever one passed from one

triangular number to the next, one had to add the appropriate number (see Fig. 6,

below and left).

Fig. 6 The representations used by Team G4 to calculate T11

While giving his explanation, he suddenly changed the strategy without

discussing this with his team members. He changed the representation he was

using to calculate T11 for one which enabled him to construct both an iterative

process to calculate T11 and a generic algorithm for any Triangular number (see

Fig. 6). It seems that, through a process of signification (Radford 2003), the pupil

was constructing a sign that enabled him to arrive at an iterative process for the

calculation of triangular numbers.

An analysis of the pupils’ written productions reveals that there were pupils

in each team who undertook iconic calculations solely counting ball by ball. One

female member of Team G3 said nothing in response to the “leader” of the group

indicating that they had done something similar to Team G1, when in fact she had

actually done something similar to Team G2.

Process of generalisation

R2 then requested that teamwork continue, approaching team G4 and mentioning

that, when undertaking a calculation, they should show their working. A female

member of Team G4 (known as G4-2) interjected by saying that she did not

understand how to calculate T83, thus initiating a dialogue between G4 and R2,

with G4-1 and G4-2 mainly involved in the discussion.

R2: You must calculate it ... and show what you did.

(...)

G4-2: I do not understand.

G4-2: I do not understand.

R2: Well, here we do not tell you the number of....

G4-1: Is this number related to the diagonal?

G4-2: The number on the side?

(...)

G4-1: I take 83 on the side or on the diagonal and then you can count 1, 2, and

3 up to 83.

Integrating Arithmetic and Algebra in a Collaborative Learning

281

R2: It's interesting, you have two different strategies.

In this excerpt, trying to assimilate that which was presented to the whole

group by her teammate G4-1, G4-2 ceases to refer to the iterative process, instead

only associating the number of balls, either vertically, at the base, or on the

diagonal, and, thus, jumping from one triangular number to the next.

Once the pupils had worked in teams, R2 again requested a process of

recapitulation in a large group discussion, asking pupils how they would perform

the calculation of the triangular number T83.

Dialogue between the researcher and pupils

Interpretation

Pupil 1: Uh ... you have to add up all the numbers

from 1 to 80 for the triangular number of 80 ... er

from 1 to 83 for the triangular number of 83.

R2: Yes, but if you give me a number, which can

change, it can always change. What kind of

operation do I have to do?

G4-1: You added together 1 + 2 + 3 + 4 + 5 + 6 etc.,

until you arrive at your number. Ben! Then the

answer is the ... your answer is the triangular

number.

R2: Ok. So there I would do 1 + 2 + 3 + ...

G4-1: ... 4 + 5 + 6 ...

R2: until my number.

Pupil 1: Etc, until your number. You add up all this

and it gives you your triangular number.

R2: How do I write my number I do not know?

G4-1: Question Mark!

R2: Question Mark? Do you all agree? Yes? That's

going to be my number I do not know?

G4-1: + x.

Pupil 3: Yes + x.

R2: x? Do I put something else? Yes, do I? (Point to a

pupil)

Pupil 4: Any letter.

R2: Any letter, yes. There? A heart? Can we put a

heart?

G4-1: You can put anything that is not a number.

Addition of 1 + 2 + 3 + ... + 83

R2 tried to promote a

generalisation.

Using words, the pupils could

describe the last number "until you

arrive at your number."

R2 repeated the question, but

continued to say "until your

number."

As there had been no change, R2

directly asked “how do I write the

number I do not know?”

Here pupils have shown that

they have mastered the situation,

suggesting several symbolisms.

Even the ! (heart) proposed by

R2 did not bother the pupils.

Fig. 7 Dialogue between R2 and the group in a plenary session

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F. Hitt, C. Cortés, and M. Saboya

This extract is extremely important to the research conducted in this study.

Pupils proposed a calculation of T83 that is identical to that proposed by Team G1

for the triangular number T11 – namely T83 = 1 + 2 + 3 + ... + 83. R2 tried to

verbalise the calculation in terms of a generalisation for any triangular number.

Pupils had no difficulty with this kind of process of generalisation. The symbolic

process was executed naturally, with the assignation of a variable seeming not to

disturb pupils at all. Even when the researcher proposed the use of a heart (!) as

a variable, this did not appear to disrupt the pupils in any way.

While, right up until this point, it is possible to say that pupils have been

following various processes of generalisation, the question remains as to who,

precisely, undertook this process. Throughout this process, pupils generally

seemed to show that there was consensus. How stable was the pupils' knowledge

as it emerged from a process of communication in the mathematics class? Can

these pupils retain these results in the future?

Once this part of the activity had been completed, R2 asked the pupils to

return to work in teams, suddenly announcing "I can calculate any triangular

number with three operations. Can you?" This was a question that resonated with

some pupils, as described below.

Generalisation and emergence of the concept of the variable

At this stage, pupils could use Excel in order to continue the activity. The idea of

using a single computer introduced an unanticipated variable. The owner of the

computer determined the user. For example, in Team G3, the owner of the

computer (a boy) was the only user. This reminded researchers of Hoyles’ (1988)

recommendation that attention should be paid to the constitution of a team when

mixing boys and girls in a computational setting.

Once teamwork had commenced, pupil G4-1 called R1 over, saying that his

group had formed a strategy to calculate any triangular number. He mentioned

that their strategy involved taking any triangular number, for example 101,

adding 1, dividing by two and multiplying the result by 101. R1 told them to use

another number, such as100. Then, G4-1, stating that he would add one to 100

and divide the result by two, suddenly stopped, turned to his companion (G4-2)

and asked whether it made sense to get a decimal number when dividing by two.

R1 suggested that they discuss their strategy again and use the Poly applet to

verify their results. R1 remarked that the team had used Excel to calculate the

triangular numbers up to T84 in a column and that they had the operation

(indicated in Fig. 8) in their workbook.

Integrating Arithmetic and Algebra in a Collaborative Learning

283

Fig. 8 Technology and paper and pencil – the construction of a strategy

This is another key episode in this research. Pupils already had an algorithm

to calculate T83, which was to add 1 + 2 + 3 + ... + 83. It was G4-1who put the

iterative algorithm on the board (see Fig. 6), using it with Excel to calculate up to

T84. The hypothesis explored in this study is that, using T83 and T84, the pupil

built his new algorithm.

Fig. 9 Construction of a general strategy to calculate any triangular number

Both the design of the activity and the pupils’ processes show the possibility

of reconciling work with pencil and paper and technology. In addition, and very

importantly, the team destabilization generated by R1’s question about using an

even triangular number (T100) caused the team to reflect on that which is

expressed by the use of the letter “x” in the arithmetic operations (see Fig. 9). It

seems that pupil G4-1 had written 100 + 1 divided by two (“no matter what you

get”), in this case x, the result (x) must be multiplied by 100. Finally, the pupils

used the POLY applet to ensure that the conjecture obtained might work with

other triangular numbers.

The POLY applet is able to show a series of polygonal numbers or to give a

specific polygonal number (as in Fig. 10). Due to screen limitation problems, if a

polygonal number is too big, the applet can only provide the numeric result and

thus excludes the figure.

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F. Hitt, C. Cortés, and M. Saboya

Fig. 10 Examples using POLY (series of triangular numbers and the fifth pentagonal

number, including partition into triangles if required)

Discovery of an algebraic expression for calculating any

polygonal number

The session was ending and R2 called for a plenary discussion. G4-1 asked to

present what his team had found. G4-1 exemplified their strategy with T46 and

explained their algorithm. After R2 explicitly asked him to explain how they had

discovered their strategy, he repeated the algorithm, but did not mention how the

discovery was made. It seemed that he did not understand R2’s question.

When the bell rang and R2 had finished the session, a girl’s voice, almost

drowned out by the noise made by the pupils, indicated that she would have liked

to know how the three operations could be used to calculate any triangular

number. R2 mentioned that there was no time for that explanation and that she

would show it to her later. However, as G4-1 mentioned that he knew this, R1 and

R2 asked him to write it on the blackboard, even though the entire class was on

their feet and ready to leave the classroom, whereupon G4-1 wrote the following

algebraic expression (see Fig. 11).

R2: x + 1, x ... So it is your triangular number? [Trying to interpret

what the pupil wrote]

G4-1: x is not my triangular number, it is my base number, plus one,

divided by two, it's going to give y. y times x gives the triangular

number.

Fig. 11 Symbolising in a communication process

The researchers reacted very positively to this development at the end of this

stage of ACODESA, deciding at that time to interview G4-1 to obtain more

information about the process of constructing the algebraic expression. The

interview provided a few elements of which researchers were already aware.

While G4-1 insisted that it was through the POLY applet that he had come to

discover the formula, both pupil productions (Fig. 9) and R1’s discussion with

Integrating Arithmetic and Algebra in a Collaborative Learning

285

Team G4 seem to indicate that the discovery took place while working with either

Excel and pencil and paper, with Poly allowing them to check their conjecture.

Self-reflection phase without technology. What happened 45

days later?

Aware of the problem with student retention of the mathematics that they learn

and also aware, thus, that “consensus is ephemeral”, the authors decided to make

the self-reflection phase as different as possible to that which had had been

undertaken in other experiments. Also, given that a talented pupil had been

discovered in the sample, it was decided that an additional challenge would be

added to the self-reflection activity (which, while generally the same, excludes

technology) exclusively for him. So, in addition to the reconstruction process

related to triangular numbers, he was asked to work with pentagonal numbers,

something which was not dealt with in the classroom experiment.

It seems that G4-1 did not pay attention to the examples given about triangular

numbers, as he wrote that he already knew the formula to calculate any triangular

number. He applied a wrong formula and did not check his results against the

examples provided. However, and to the researchers’ surprise, in a paper and

pencil task (the use of technology is not allowed in this phase) that followed a

similar process of finding relationships among the first four examples provided

for pentagonal numbers, he constructed both the fifth pentagonal number and a

general expression that allowed him to calculate any pentagonal number.

rang ×rang +rang ×0, 5 −0, 5

( )

()

=nombre pentagonal

Fig. 12 G4-1’s production in the self-reflection phase

The results obtained are presented below. The data was collected on an

individual basis before the teamwork and self-reflection phase 45 days later, with

only eight pupils sampled from the other phases, plus others that could not be

taken into account. Pupils 1 to 13 (the last being G4-1) were identified in order to

observe progress and setbacks.

Table 1 shows two setbacks and four advances, not counting pupil G4-1, who

used an incorrect algebraic expression to calculate the triangular numbers. The

female pupil (subject 11) followed her teammates from Team G3 passively.

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F. Hitt, C. Cortés, and M. Saboya

Table 1 Results from the ACODESA self-reflection phase after 45 days.

I. Anchor

to the

drawing

II. Drawing

+ addition

III.

Abandoning

drawing +

addition

IV. Abandoning

drawing +

another strategy

“Algebraic

expression”,

triangular or

pentagonal numbers

1, 2

3, 4, 5, 6, 7

8, 9, 10, 11

12

13

45 days without technology

4, 8

1, 5, 7, 12

11, 13

While her strategies were different, she did not discuss them with her

teammates. The owner of the computer in team G3 became the leader, which

corresponds to Sela and Zaslavsky’s research (2014) with four people working

together. This male pupil, when using the computer, showed his colleagues

various things not related to the task, thus creating a situation unrelated to the

requested task. Teams G1 and G2 were more homogeneous and presented more

balanced participation, with both teams composed of girls. The computer owner

from team G4 was strongly committed to the task and adapted very quickly to the

rhythm of his colleagues, with one of the setbacks for the team posed by G4-2

(G4-3 was not present during the self-reflection phase). More careful study is

required to analyse the role of technology in sociocultural learning. In fact, this,

bearing in mind Hoyles (1988) on Girls and computers and the results reported

by Sela and Zaslavsky (2014), leads to the realisation of the importance of

creating teams consisting of a maximum of two or three subjects, and of trying to

balance the use of technology in each team.

Experiment conducted in Mexico

The experiment conducted in Mexico proceeded as follows. Once the initial

problems were resolved in order to introduce pupils to the TI-Nspire calculator,

the teacher asked the pupils to work on the first five questions individually (see

Fig. 2). Four teams with three pupils and one team with two pupils were formed,

and in which each pupil had a calculator (TI-Nspire).

Individual work

Two types of strategies emerged from the individual work – one linked to the

drawings as shown in the examples, and the other to the formation of a table of

values. Again, spontaneous representations linked to functional representations

appeared in the communication process.

Integrating Arithmetic and Algebra in a Collaborative Learning

287

1. Adding 1 to the first

range.

2. With 1 added to the first

range and both the line at

the base of the x and the

diagonal \ − | x .

Fig. 13 Spontaneous representations used by one pupil

Teamwork

One team made up of Diana, Karla and Omar underwent a reconciliation of

strategies, with Diana and Karla making the calculation by counting balls from

the drawings, while Omar used a table of values. Through this strategy

integration process, Diana and Karla left their strategy and decided to use the

table of values.

The pupils’ individual productions and the film of the plenary session are the

only evidence of the individual work carried out by the subjects. A useful

research technique was that pupils were asked to write with red ink when

working in teams (see Fig. 14).

Fig. 14 Karla’s individual work and teamwork

There is no evidence of how the formula was obtained. The formula appeared

in Karla’s productions and was written in red ink during the teamwork phase.

This shows that, in her team, she adopted the table of values and proposed an

algebraic expression. At this secondary level pupils had already learned about the

notation of variables, and can clearly be seen to use the variables x and y. The

formula enables pupils to estimate the calculation of polygonal numbers. Our

hypothesis is that having used y = x2 / 2 (related to base * height / 2) and having

realised that it did not work with the examples given in the table, these pupils

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F. Hitt, C. Cortés, and M. Saboya

decided to approximate the results, thus giving y = 0.52 * x2. They did not

present their proposal in the large group discussion.

A surprising result is that many Mexican pupils participating in this research

immediately associated the triangular arrangement of the triangular numbers,

base * height / 2, with the calculation of the area of a triangle, thinking that this

was the algebraic expression required.

This was the first thing that emerged during the large group presentation,

having been captured from the beginning of the pupil discussion.

Plenary discussion

Monica, facing the blackboard, gave as example the T8 (see Fig. 15), describing

the following as necessary in order to calculate it:

Monica: To calculate the area of the triangle would

require 8 times 8, giving 64, which, divided

by 2, is 32...

Another pupil: [a girl is heard addressing her

classmate in a very low voice] But no, that

would be 36!

Fig. 15 Monica presenting the calculation of T8

R3 intervened, saying that, at this point, it was necessary to review her theory.

She was then interrupted by the pupil Rob.

Rob: ... But... this, the triangular number 8 would be 36 and not 32 – then it

cannot be.

R3: Please come to the blackboard Rob.

Rob: This is number 6 [pointing to the figure just produced by María]. We

apply the formula that says the base multiplied by height would be 6 by 6

giving 36 divided by 2, giving us 18, while the triangular number would

be 21. This, therefore, is not the formula [points to the base * height / 2

formula].

P1: What is the difference between ... if you apply the formula that tells you,

apply the formula. Write it there.

María: [Writes the formula].

P1: In this case, what is the basis?

Rob: [Writing] 6 * 6/2 is 18 and there is the triangular number.

P1: What is the triangular number?

Pupils: 21 [answering chorus]

Rob: The difference would be 3.

P1: The difference would be...?

Integrating Arithmetic and Algebra in a Collaborative Learning

289

Rob: 3.

Rob calculates the Triangular 8, and, typing the formula, obtains 32.

P1: And what is it for the triangular number 8?

Rob: It's 36.

P1: The 8?

Rob starts counting the balls for the figure that was already on the

blackboard, and says that it is 36.

Pupils: The difference is 4.

Rob realizes that the formula does not work and that he has shown a counter-

example (similar to the comment made by the unidentified girl). However, so far

he is not able to build the exact formula for triangular numbers.

EUREKA ! Rectification of the formula in a scientific debate

In the midst of the discussion, a surprised voice is heard, saying, "and from 8 it is

four, then it would be half."

Gaby: Half of 8 is 4, and 4 is what is missing from 32 to 36 in the formula,

then we have to put the base multiplied by height divided by two more....

[PAUSE] plus the half of [PAUSE] plus the half of the triangular number,

half [PAUSE] half of the base.

While speaking, Gaby paused several times while she completed the

transformation of her numerical idea into a geometric-algebraic idea. It is clear

that the control element was provided by the arithmetic relationship and the

transformation from that into a geometric relationship. However, something else

occurred in the process of communication when Gaby was verbalising what she

was thinking: there was a process of deduction. At this very moment, the pupils

were rejecting their initial conjecture in favour of a new one, using the arguments

to refute the conjecture.

P1: Write it!

Pupil: I think that we, all together, are arriving at something, not alone!

The sociocultural construction of knowledge has occurred at this stage of

scientific debate, in accordance with ACODESA. The pupil openly expresses the

co-construction generated through the debate.

Gaby goes to the blackboard to write the idea that she had just thought of.

Gaby: How do I represent half the base?

Interestingly, at this point, Gaby has difficulties in transforming the geometric

argument “half the base” into algebraic terms:

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F. Hitt, C. Cortés, and M. Saboya

R3: … One half, or that in half.

Gaby writes

b

×

h

2

+

1

2

.

Many pupils worked together to undertake the final writing activity. In this

process, Gaby was guided by her colleagues because she did not fully understand

the process of algebrafying "half the base". Finally, she wrote:

b×h

2

+b

2

P1: And how do you represent the triangular number in that formula you are

writing there? How do you represent the triangular number?

Pupils: .... The base or the height...

Gaby writes the formula

b×h

2

+b

2

P1: Precisely, what she said, the height is the same as the...

Pupils: The base.

P1: Replace it and do not write the height.

Pupil: It would be base times base.

Gaby misspelled the formula and was corrected by her peers, after which she

wrote:

b

×

b

2

+b

2

.

P1: Ok then, base times base is what?

Pupils: Base squared.

Gaby writes finally the formula

b2

2+b

2

.

P1: Do you think that is the formula? Verify it with the triangular number 15

Gaby: Do I have to count the balls in a drawing?

P1: No, no, no you have already got the formula!

Self-reflection phase without technology. What happened 30 days later?

This phase, referred to here as self-reflection without technology, comprised a

questionnaire (with slight modifications) similar to that completed by the

participants 30 days previously. Only 10 of the 14 pupils participated, with the

main idea at this stage being a reconstruction of what had been undertaken in the

classroom.

The questionnaire for the self-reflection phase had three questions:

1st Question: Calculate the 27th triangular number.

2nd Question: Write the formula for calculating any triangular number.

3rd Question: Using your formula, calculate the 313th triangular number.

Integrating Arithmetic and Algebra in a Collaborative Learning

291

The results are as follows: from the ten pupils, two continued the process of

"drawing balls and counting", while four of the ten rebuilt a similar expression

related to the area of a triangle – b × h / 2. A pupil was able to reconstruct the

formula, but mistook the result of calculating T27 by finding a triangular number

to provide 27 as a result, with the closest being T7 = 28. It is possible that he

made a mistake when counting the balls. He followed the same strategy when

calculating T313. Among those who were able to reconstruct the right formula

were Alejandra, Omar and Rob.

Conclusions

These results reveal the importance of building arithmetic-algebraic thinking in

order to support algebraic thinking. The experiment conducted in Quebec with 1st

year secondary pupils (11-12 years old) revealed the following:

• It was performed in a sociocultural environment, with a gradual construction

of the concept of the variable, and patterns built from the visual work.

• The strategy consisted of visualisation processes that related drawings,

arithmetic addition series, iterations and formulas. Pupils used natural

language with letters representing variables.

• The validation process was supported by the use of technology.

• The availability of a device on the table, shows that its use is delicate

(Hoyles 1988). In the case of one team, it was the owner of the computer

who exclusively used it. In the team with a boy and two girls, the boy

mostly used his computer, the girls used it when he was at the blackboard.

• Even though there was more progress than setbacks in the self-reflection

phase, the results show that concluding that consensus had occurred should

be undertaken with caution.

The experiment conducted in Mexico with 9th grade secondary pupils (14-15

year old) revealed the following:

• It was performed around visualising a process related to the area of a

triangle, with use of variables to represent the variation (x, y, b, h).

• The validation process rested more on visual configurations.

• The technology was not widely used by the pupils. Plenary discussion and

co-construction attracted the pupils’ attention.

• Again, it showed that “consensus is ephemeral”, with only four out of ten

able to rebuild the formula and one of them mistaking the number of the

triangular with the result.

A surprising fact is that the institutional representation n (n + 1) / 2 did not

emerge in neither of the two populations. This demonstrates the importance of

pupils’ spontaneous representations in the construction of mathematical concepts

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F. Hitt, C. Cortés, and M. Saboya

(Hitt 2013; Hitt and González-Martín 2015). This reveals that evolution of

spontaneous representations is important in a signification process. During the

institutional stage under the ACODESA model, the teacher must collect different

ideas and productions and relate them to the institutional representations.

Our research is taking into account the importance made in the 40’s when

psychologists payed attention to the importance to moving from analysis of

pupils’ performances when solving puzzles to the analysis of pupils’ problem

solving activity (see Brownell 1947). Our approach, in this technological era

goes from an arithmetic context to an algebraic one, in a natural way using

technology as a tool in a process of generalization and, in a sociocultural context

of learning. In this way, pupils construct arithmetic relations (product of this is

what is shown in Fig. 9) that permit them to control their process of

generalization to an algebraic context in a milieu of creativity and autonomy (see

Fig. 12). The results are showing the importance to promote the production of

spontaneous representations and conversions among them even if they are not the

institutional representations. This contrast directly with Kirshner’s approach

(2000) concerning his ideas of exercises, probes and puzzles; and, about his

restricted approach to learning algebra focusing on the algebraic register about

visually salient rules (Kirshner 2004). Our research takes as central a task-design

where the but is related to enchained tasks, to be solved by the pupils in a

sociocultural milieu, and the teacher role is to promote students’ reflexion and

productions of spontaneous representations, not only the algebraic one.

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