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ANALYSIS OF MATHEMATICAL SOLUTIONS OF 7 YEAR OLD PUPILS WHEN SOLVING AN ARITHMETIC PROBLEM ON

DISTRIBUTION Edelmira Badillo*, Vicenç Font**, Mequè Edo* & Núria Planas* *Universitat Autònoma de Barcelona, **Universitat de Barcelona

We analyse mathematical solutions of 7 year old pupils when they individually solve an arithmetic problem. The analysis has used “the configuration of objects”, an instrument provided by the onto-semiotic approach to the mathematical knowledge, combined with the organisation of data into a “systemic network”. Results are illustrated by three cases. The aspects inferred from the overall analysis of the mathematical solutions include the use of iconic representations as a counting instrument, and the demonstrative nature of the arguments developed by the pupils.

INTRODUCTION The objective of this research is to analyse the written and verbal mathematical solutions of 7 year old children when solving an arithmetic problem in an individual context of work. Solutions by children when solving arithmetic problems tend to have a predominance of iconic and symbolic representations and a lack of explicit verbal argumentations (for example, Saundry & Nicol, 2006). For this reason, different research projects have shown an interest in studying the representations produced by children when solving problems (Edo, Planas & Badillo, 2009; Saundry & Nicol, 2006). To analyse the productions made by pupils we take their representations and other elements such as calculation procedures, argumentations, etc. This is why in order to analyse the underlying mathematical activity of pupils we use instruments of the onto-semiotic approach to the mathematical knowledge -OSA (Godino, Batanero & Font, 2007). We start with a brief review of the literature. We then present the theoretical and methodological instruments that have been applied to our analysis. After that, we describe part of the design of the study and present some of the most relevant data. We end with a discussion of results and final conclusions.

REVIEW OF THE LITERATURE

MANY   RESEARCH   WORKS   HAVE   SHOWN   THAT   STUDENTS   CAN   SOLVE   A  DIFFERENT   MULTIPLICATIVE   PROBLEM   MUCH   BEFORE   THE   INSTRUCTION  ABOUT   THE   MULTIPLICATION   AND   THE   DIVISION   HAS   BEEN   GIVEN  (MULLIGAN   Y   MITCHELMORE,   1997).   CARPENTER,   ANSELL,   FRANKE,  FENNEMA  AND  WEISBECK  (1993)  FOUNDED  OUT  THAT  EVEN  STUDENTS  OF  EARLY   YEARS   STUDENTS   COULD   LEARN   HOW   TO   SOLVE   MULTIPLICATIVE  

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PROBLEMS.   SUCH   STUDIES   HAVE   ALSO   BEEN   CARRIED   OUT   WITH  STUDENTS  WITH  SPECIAL  EDUCATIVE  NEEDS  (NUNES,  BRYANT,  BURMAN,  BELL,  EVANS  &  HALLETT,  2009) Much research has been done on primary age arithmetic problems on distribution in which the task is to share a number of elements that are to be shared out one by one among a variable number of participants; for example, to share out a number of cookies among different children on the basis of questions with multiple solutions (Davis y Hunting, 1990). Research has also been done on problems that work on the idea of the division of units and distribution. Charles and Nason (2000), in a study of the development of the concept of fractions among 8 year old children, proposed a type of problem in which the unit(s) is/are divided into parts. We examine the particular case in which the context requires the elements to be separated into groups, which involves a distribution in which not everything is a unit (or several units) that has to be divided into parts, but in which everything has to be separated into discrete sets, that can(not) have a different cardinal.

THEORETICAL AND METHODOLOGICAL FOUNDATION Some studies involving the OSA (Malaspina & Font, 2010), in which mathematical solutions have been analysed, first consider the mathematical practices and then the mathematical objects and processes that are activated in them. In this study we adapt such approach with the pupils’ practices being the reading of the text of the arithmetic problem and the production of a written answer. Due to space limitations, we will only analyze the mathematical objects that are activated by said practice. If we consider the mathematical objects activated in undertaking a practice that enables the resolution of a problem situation (e.g. tackling and solving an arithmetic problem), we observe the use of verbal, iconic, symbolic and other representations. These representations are the ostensive part of a series of concepts/definitions, propositions and procedures that intervene in the production of arguments to decide whether the practice is satisfactory. So, when a pupil performs and evaluates a mathematical practice s/he activates a conglomerate formed by problem situations, representations, concepts, propositions, procedures and arguments, which are articulated in the configuration of Figure 1 (Font & Godino, 2006, p. 69). To move from the individual analysis on pupils’ mathematical solutions to a more general analysis on the whole group, we used a systemic network. This is a classical instrument from the organisation and interpretation of qualitative data proposed by Bliss, Monk and Ogborn (1983).

DESIGN OF THE STUDY The participant sample was made up of 21 primary school pupils (7 years of age) at a school in Barcelona, Spain. The mathematical task presented to the pupils, which was to be solved individually and in writing, was: 1) an arithmetic problem involving

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distribution in which everything had to be separated into discrete sets of various elements, which could (not) have a different cardinal; 2) an open-ended situation; and 3) a feasibly resolvable task using the pupils’ prior knowledge. The problem was: “If you have 18 wheels, how many toys with wheels can you have?”

Figure 1. Configuration of objects

The problem was read aloud and the pupils were expected to solve it with paper and pencil during a one-hour class. When they finished the task, they were individually asked, “what did you do?” Their answers were recorded in audio and transcribed.

DATA ANALYSIS Two types of analysis were performed, an analysis of each of the cases and then a global one of all of the mathematical solutions. For the former, an analysis (Figure 1) was made of each pupil’s solutions. Table 1 illustrates part of an example. Following the categories suggested by Malaspina and Font (2010), the data was analysed as indicated in Table 2. Each pupil’s mathematical practice was analysed individually, paying attention to 1) representations, 2) concepts, 3) properties, 4) procedures, and 5) arguments. We present the analysis of one of the mathematical practices by one of the pupils, Pupil 15. Figure 2 shows the systemic network obtained from the overall analysis. It is organised into categories and aspects (using the terminology by Bliss, Monk and Ogborn, 1983). We use braces ({) to represent inclusive aspects and lines to group exclusive categories (|). The analysis of the mathematical practices leads to two main categories. First we have pupils that put the emphasis on the cardinal of the set. Here there are three subcategories: 1) those which give a single answer (e.g. Pupil 10 says, “if I had 18 wheels I’d have 6 toys with wheels”); 2) those whose answers suggest more than one answer (e.g. Pupil 15 writes, “I could have four toys with wheels”); and 3) those who give more than one answer (the only case is Pupil 18 who gives four different answers “…you could have 9 motorbikes, you could have 6 tricycles…”). Second we have pupils who point to the set and only refer to it by extension (e.g. Pupil 12 says,

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“I have made a car, a bike, a car, a scooter and another scooter”) or give the cardinal for the subsets (e.g. Pupil 6 says, “four cars and a bicycle make 18 wheels”).

Written production Verbal production

I have drawn a car that has four wheels, a motorbike that has two, another motorbike that has two and a train that has ten wheels. And here I have explained what I have drawn and how many wheels they all have. And here I have added them up and this is the answer.

Table 1. Data from Pupil 15

For each of the five subcategories above, we organised the data on the basis of the mathematical objects following the configuration of objects in Figure 1. We considered each object as an aspect in the systemic network. On this occasion, we have grouped the procedures and properties as a single aspect and we have left the argument aspect for another occasion. Given the richness of the responses, for each aspect we have introduced meanings that have been used as categories; we do not go into the details of all of them. We now illustrate three significant cases. The case of Pupil 15 Pupil 15 solves the problem well by giving the cardinal of one of the possible sets and concluding, “I could have 4 toys with wheels”. We consider that she is suggesting there is more than one answer, as she uses the verb tense “could”. First, we examine the richness of her representations. She starts with an iconic representation of the toys in perspective (Table 1) and then translates this into a symbolic numerical representation (4+2+2+6+4=18) and a verbal one (one car, two motorbikes and one train have 18 wheels). In relation to concepts, this pupil breaks down the set of wheels (18) into parts or subsets (she draws a 4-wheeled car, two 2-wheeled motorbikes and a 10-wheeled train). She is then able to treat each of the subsets as an element (a toy) in a new set (the set of toys). Finally, she implicitly distinguishes between a set and the cardinal of a set, because in her answer she refers to the cardinal of the set of toys.

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Figure 2. Systemic network from the overall analysis of the pupils’ answers2

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This pupil applies the property that “a number can be broken down into the sum of smaller numbers”, in order to break down 18 (into three different addends) and 10 (into two different addends): 4+2+2+2+6+4. We consider this pupil to be aware of the application of this property because she writes (6+4) and draws a two-carriage train with 6 and 4 wheels, though in her verbal answer she refers to a train with 10 wheels.

Mathematical Object Mathematical practice

Problem situation If you have 18 wheels, how many toys with wheels could you have?

Representation

• Iconic with perspective

- Verbal (one, four, two, ten) • Symbolic - Numbers (4, 2, 6, 18)

- Signs (+, =)

Concepts

• Addition (Previous) • Implicit terms of the addition (addends and results) • Number (Previous) • Subtraction (implicit) • Set • Elements of a set

Properties • A number can be broken down as the sum of smaller numbers (this is applied to 10 and to 18)

Procedures • Combination of numbers to obtain 18 • Add and subtract (mentally) • Determination of a set by extension

Arguments

• Explicit thesis: I could have 4 toys with wheels (to make 18) • Graphic argument: draws the 4 toys • Verbal argument: describes the elements of the set (a 4-wheeled car, two 2-

wheeled motorbikes and a 10-wheeled train) • Numerical-written argument: 4 + 2 + 2 + 6 + 4 = 18

Table 2. Configuration of objects in Pupil 15’s answer

In relation to procedures, she uses the previous property to break down number 18. She seems to take a first number (she draws a 4-wheeled car), then adds another number (she draws a two-wheeled motorbike), and as the result is less than 18, she adds another addend (a two-wheeled motorbike); given that the result is still less than 18, she adds another addend (a ten-wheeled train). She iconically determines the set by extension. Finally, the explicit thesis of her demonstrative argument (she could have 4 toys with wheels) is justified by the ostensive presentation of the set (iconic representation and verbal description) and by the numerical-written verification (4+2+2+2+6+4=18), of which she is aware because she says, “…and here I have added them up…”.

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The case of Pupil 19 Pupil 19 solves the problem implicitly in that he draws 6 tricycles (Figure 3). We consider this to be implicit because the pupil expresses the cardinal of the set of wheels (18), which leads to start the solving process using symbolic representations. This pupil starts his answer with a symbolic-numerical representation based on the sum (3+3+3+3+3+3=18) and translates this to another symbolic expression based on multiplication (3x6=18, see Figures 2 and 3). Later, he switches to an iconic representation without perspective.

Figure 3. Representations used by Pupil 19

In relation to concepts, this pupil comes to the concept of multiplication and seems to be clear of its concept as a repeating addition. We consider this because he uses a mathematical property: “18 can be broken down as the repeated addition of number three”. In relation to procedures, he uses the previous property to break down number 18. He likely takes a first number, 3, then adds another 3, and given that result is still less than 18, he adds another addend (3), and so on successively until he reaches number 18. He iconically determines the set by extension. Finally, the explicit thesis of his argument (6 tricycles make 18 wheels) is justified by the ostensive presentation of the set (iconic representation). He gives a verbal description of the procedure he used to get to number 18 (I did three plus three…). The case of Pupil 20 Pupil 20 solves the problem implicitly, as he draws two cars, a truck and a scooter (Figure 4). He also gives the cardinal of the subsets as a verbal response (2 cars, 1 scooter and 1 truck). We find his type of representation significant, and we have named it in the systemic (Figure 2), iconic and symbolic (Figure 4) networks. The drawings are not in perspective but the pupil represents the total number wheels on each toy using numerical symbols ( ). The only conversion he makes is to switch from an iconic and symbolic representation of the set of toys to a verbal and written description of the cardinal of the subsets. In relation to concepts, he breaks down the set of wheels (18) into parts or subsets (he draws 2 cars with 4 wheels, 1 scooter with 2 wheels and a truck with 8 wheels).

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After, he gives the cardinal of the subsets (2 cars, 1 scooter and 1 truck). Meanwhile, he implies the mathematical property: “18 can be broken down into the sum of smaller numbers” in order to break down 18 (into four addends, three of which are different): four, four, two and eight. In relation to concepts, this pupil uses the previous property to break down the eighteen. We consider that he takes a first number (he draws a four-wheeled car), adds another number 3, then adds another (he draws another four-wheeled car), and as the result is less than 18 he adds another addend (and eight-wheeled truck). He iconically determines the set by extension. The explicit thesis of his demonstrative argument (2 cars, 1 scooter and 1 truck) is justified by the ostensive presentation of the set (iconic representation and verbal-written description).

Figure 4. Representations used by Pupil 20.

CONCLUSION All of the pupils make an iconic representation of the set of toys. It could be said that this is because of the need at this age to work using contextualised scenarios. However, there is also the need to use drawing as a counting instrument, as has been shown by Saundry and Nicol (2006). In our study, this use of iconic representations as a counting instrument is made clear in the representation that we have called iconic and symbolic (Figure 4). This is a type of representation (used by Pupils 8 and 20), that can be considered an intermediate step between flat representations (used by Pupils 6, 7, 9, 11, 13, 14, 18, 19) and representations in perspective (used by Pupils 1, 2, 3, 4, 5, 8, 10, 12, 15, 16 17, 21). There are three pupils (3, 17, 19) who separate the set of 18 wheels into discrete sets with an equal cardinal and start solving the problem using written symbolic-numerical representations (they break 18 down into equal addends). In all three cases, they translate this representation into another written symbolic expression in which they use the concept of multiplication (Figure 5), to end with a conversion to an iconic representation of the set of toys. In cases 3 and 17 this is in perspective and in case 19 without perspective (Figure 3).

Figure 5. Breakdown of equal addends into multiplication

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Pupil 18 uses multiplication in his four different answers (which are given verbally and iconically). But he does not use written symbolic-numerical representations, so we suppose he reached his answer by making mental calculations. From his verbal responses, we infer that he makes an implicit use of the commutative property (“you can have 6 tricycles with three wheels or you can have 3 limousines with 6 wheels and you also get 18”) and that, unlike the three previous pupils, he does not need to explicitly break 18 down into equal addends to break 18 down into the product of two factors. He does not need to add first in order to get to multiplication. All of the pupils implicitly or explicitly use the property of breaking 18 down into addends (we include the extreme case of Pupil 18 who breaks 18 down using 4x4+2). When the addends are equal, this facilitates the use of the concept of multiplication, and on the other hand, facilitates the process of giving the cardinal for a set of toys of a certain type (e.g. Pupil 17’s answer, “6 tricycles”), which implies that the term “toy”, which is more abstract, is not used. However, the two pupils that explicitly use that term in their answers (e.g. Pupil 15, “I could have 4 toys with wheels”), break number 18 down into different addends. In this last case we have the close relationship between properties and concepts. The use of a certain mathematical property (a type of breakdown of number 18) conditions the use of certain mathematical concepts (addition or multiplication). Meanwhile, the use of multiplication -a concept that is considered, in curricular terms, to be more difficult than addition- involves, in this case, less abstraction in solving the problem. We observe two fundamental procedures. One is related with the application of the mathematical property/ies that guarantee the breakdown of number 18 into addends. The pupils mentally apply addition and subtraction, and even multiplication, to reach that breakdown. The other is the determination by extension of the set (via an iconic representation). This latter method, used by all pupils, is the one that enables them to defend, explicitly or implicitly, their answer. These are demonstrative arguments that consist of the ostensive presentation of the set (the drawings of the toys). We consider the theoretical categories provided by the OSA to facilitate an in-depth analysis of the pupils’ solutions and to reveal the complexity of objects (concepts, representations, properties, etc.) that are activated when solving arithmetic problems. The systemic network has also been a powerful instrument of organization that has led to construct a taxonomy of the pupils’ responses when they solve a problem in which the whole has to be separated into discrete sets of various elements, which can also have (or not) a different cardinal. ACKNOWLEDGEMENTS The research is part of Projects EDU2009-07113/EDUC and EDU2009-08120/EDUC, from the Spanish Ministry of Science and Innovation.

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NOTES 1. *The represented drawings are in perspective. ** Without in perspective.

2. The pupils that draw an iconic representation and give a verbal description of the set of toys (almost all), are part of the aspect “makes a translation/conversion”. When the pupil also makes another type of translation/conversion, s/he is part of “makes several translations/conversions”.

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