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Using tree diagrams to develop combinatorial reasoningof children and adults in early schooling

Rute Borba, Juliana Azevedo, Fernanda Barreto

To cite this version:Rute Borba, Juliana Azevedo, Fernanda Barreto. Using tree diagrams to develop combinatorial rea-soning of children and adults in early schooling. CERME 9 - Ninth Congress of the European Societyfor Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME,Feb 2015, Prague, Czech Republic. pp.2480-2486. �hal-01289344�

2480CERME9 (2015) – TWG16

Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling

Rute Borba1, Juliana Azevedo2 and Fernanda Barreto3

1 Universidade Federal de Pernambuco (UFPE), Programa de Pós-Graduação em Educação Matemática e Técnológica (Edumatec),

Recife, Brasil, resrborba@gmail.com

2 UFPE, Edumatec, Recife, Brasil, azevedo.juliana1987@gmail.com

3 Prefeitura da Cidade do Recife, Recife, Brasil, fernandasabarreto@gmail.com

Combinatorial reasoning is very important in mathe-matical development, providing students with contact with essential problem solving. One form of symbolic representation of combinatorial situations is a tree di-agram. Two experimental studies were designed and implemented, the first with adults in initial school-ing and the second with Elementary School children. Comparisons were performed in order to observe the impact that the use of tree diagrams – virtual or built with pencil and paper – had on combinatorial reason-ing. From initial poor performance, both children and adults in initial schooling benefitted from instruction by use of tree diagrams that enabled them to perceive how elements can be combined in a systematic manner and helped them develop their combinatorial reasoning.

Keywords: Combinatorial reasoning, tree diagrams,

children and adults; initial schooling.

THE ROLE OF SYMBOLIC REPRESENTATIONS ON COMBINATORIAL REASONING

Combinatorial reasoning is a way of thinking very useful in general mathematical learning. According to Batanero, Godino and Pelayo (1996), Combinatorics is a key element of discrete mathematics, being essen-tial for the construction of formal thought. The nature of combinatorial situations – counting techniques of possible groupings of a given set of elements that meet certain conditions, without necessarily having to count them one by one – provides students contact with essential problem solving and may help their development in Mathematics and other subjects.

Diverse forms of symbolic representation may be used in solving combinatorial problems, such as: drawings, lists, tree diagrams, tables, formulas and other forms. These distinct symbolic representations may provide means of systematization and help stu-dents understand how to obtain the total number of combinations.

The role of symbolic representations in mathemat-ical development is pointed out by Vergnaud (1997) as a key element in conceptualisation. Considering Combinatorics, Fischbein (1975) emphasizes that the use of tree diagrams can enable advances in the de-velopment of combinatorial reasoning because this representation helps systematization by pointing out the necessary steps in choosing elements to compose combinations.

In a longitudinal study, Maher and Yankelewitz (2010) investigated the initial understanding of eight and nine year olds in a problem of Cartesian product. The authors defend that it is necessary to invite children to use various representations to express their ideas and ways of thinking, because representations give meaning to the problems and communicate ideas. Thus, children can find patterns, be systematic and generalize results.

Sandoval, Trigueiros and Lozano (2007) proposed the learning of Combinatorics by use of the software Árbol. The study was conducted with 25 Mexican chil-dren, aged 11 to 13, and the authors observed improve-ments in student performance, especially regarding the choice of strategies for efficient resolution. Thus, it is emphasized that this software, through tree dia-

Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)

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grams, favours the use by children at initial level of schooling, because it provides possible combinations in all types of combinatorial problems (Cartesian products, combinations, arrangements and permuta-tions). Figure 1 shows a screen of the software Árbol of a Cartesian product problem in which in it possible to visualise the total number of combinations in a tree diagram.

COMBINATORIAL SITUATIONS

According to Vergnaud (1997), to study and under-stand how mathematical concepts develop in students’ minds through their experience in school and outside school, one needs to consider a concept as a three-uple of three sets: the set of situations (S) that make the concept useful and meaningful; the set of operational invariants (I) that can be used to deal with these sit-uations; and the set of symbolic representations (R) that can be used to represent invariants, situations and procedures. Thus, the immense role played by symbols cannot be ignored in mathematical teach-ing and learning, as means to articulate invariants

– conceptual properties and relations – situations and strategies used in problem solving.

Vergnaud (1997) also points out that combinatori-al problems are part of what he calls multiplicative structures. In the same direction, Pessoa and Borba (2010) defend that as connected concepts, different types of combinatorial problems should be taught in the classroom and present examples of these distinct situations:

Cartesian product: At the square dance three boys and four girls want to dance. If all the boys dance with all the girls, how many pairs will be formed?

Simple permutation (without repetition): Calculate the number of anagrams that can be formed with the letters of the word LOVE.

Simple arrangement: The semi-finals of the World Cup will be played by: Brazil, France, Germany and Argentina. In how many distinct ways can the three first places be formed?

Simple combination: A school has nine teachers and five of them will represent the school in a congress. How many groups of five teachers can be formed?

TWO STUDIES ON COMBINATORIAL REASONING DEVELOPMENT

With the aim of investigating the role of symbolic rep-resentations – in particular the use of tree diagrams

– on the development of combinatorial reasoning, we designed and implemented two experimental studies. The first one involved adults in initial schooling and in the second study took part Elementary School stu-dents (5th grade, 10 year olds).

Method of the 1st Study: Adults in early schooling using tree diagrams and listsThe adults taking part in the study were 24 students of classes corresponding to the 4th and 5th years of reg-ular Elementary School with no previous systematic instruction on Combinatorics. They were separated into three groups, each group consisting of eight students. After solving an eight item pre-test (two problems of each type), they were taught in groups that varied in terms of symbolic representations used: G1 – lists and tree diagrams; G2 – tree diagrams; and G3 – lists. After the learning session they solved an eight problem post-test.

Figure 1: Árbol screen of a Cartesian product problem

Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)

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Method of the 2nd Study: Children using virtual or written tree diagramsThis study was conducted with 40 students from the 5th grade of Elementary School, divided into four groups, that took part in a pre-test (eight Combinatorics prob-lems), followed by different forms of intervention and two post-tests (also with two of the four types of problems), which assessed the progress achieved a few days after the teaching session (immediate post-test) and nine weeks after the intervention (delayed post-test). Just as the adults, the children had no previous systematic instruction on Combinatorics. During the teaching session, the students worked in pairs. The first experimental group (EG1) worked with the soft-ware Árbol (Aguirre, 2005) in which diagram trees are constructed; the second experimental group (EG2) constructed tree diagrams with pencil and paper; the third group, a control group (CG1), worked, through drawings, multiplicative problems, according to the classification proposed by Nunes and Bryant (1996) (excluding Combinatorics); and the fourth group was an unassisted control group (CG2), that took part only in the pre and the post-tests.

HOW DO TREE DIAGRAMS HELP COMBINATORIAL REASONING?

Results of the 1st Study: Adults progress on combinatorial reasoning Initially (at pre-test) the adults presented only incor-rect answers or partially correct answers with very

few correct combinations presented. They preferred to use lists in their answers but, usually, were not sys-tematic in their usages and could not obtain the total correct number of combinations.

After the teaching session, all three groups progressed in their combinatorial reasoning. An Analysis of Variance (ANOVA) showed no significant differenc-es between groups (F (2, 21) = .78; p >.05). Thus, both forms of symbolic representations helped the adults to understand the combinatorial relations involved in the problems, but tree diagrams more clearly helped them to be systematic in their answers.

Table 1 shows that at post-test the adults presented more correct answers or much more answers very close to the correct ones.

Progress in understanding was, however, sometimes limited. Figure 2 shows two examples of partially cor-rect answers at post-test.

The first example (a Cartesian product), asked to in-dicate possible couples by choosing a man, out of a group of four, and a woman, out of a group of six. The adult that answered in this manner presented only four couples, considering there were only four men available. The second example (a combination) asked to form pairs, out of a group of five people. The adult in this case incorrectly considered, for example, Luíza

Problem type IncorrectAnswer

Partially correct answer Correct answer

Pre-test Post-test Pre-test Post-test Pre-test Post-test

A 75 14,6 25 81.2 0 4,2

C 47,9 6,2 52,1 83.3 0 10.5

P 91,7 50 8,3 50 0 0

CP 91,7 12.5 8,3 75 0 12.5

A – Arrangements; C – Combinations; P – Permutations; CP: Cartesian products

Table 1: Percentage of types of answers in each problem type, at pre and at post-test

Figure 2: Partially correct answers at post-test

Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)

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and Ricardo as a different pair from Ricardo and Luíza. In this way, 20 pairs were listed instead of only 10.

In three of the problem types, correct answers were presented at post-test, but difficulties with permuta-tions still remained. When using lists, the adults listed some of the permutations but not all of them. The lists were sometimes limiting in the understanding that in permutations all elements are used in all possible orders.

Some preferred, after the teaching session, to use tree diagrams and those that used this symbolic representation tended to do so in a more systematic manner that led to correct answers. Figure 3 is an ex-ample of a correct answer presented at post-test of the problem of combination of two elements out of five. In this type of problem, and others, the adults benefitted from use of tree diagrams because this form of rep-resentation highlighted the need to be systematic in the combination of elements in distinct combinatorial situations. The adult in this case noticed that if Emília and Ricardo have been already marked as a pair (top left hand side), than the branches started with Ricardo must not include Ricardo and Emília as a distinct pair.

Results of the 2nd Study: Children’s advances in combinatoricsIn the study with 5th grade children, each of the eight problems was scored zero to four, depending on how correct the answer was. If no combinatorial relation was observed the answer was scored zero; if only one combination case was presented, the score was one; if

a limited number was presented, the score was two; if a larger number was presented, but not the total number of cases, the answer was scored three; and, finally, the score four was given to answers that were completely correct – with the total number of cases asked for. Thus, the total possible score was 32.

Table 2 shows children’s performance at pre-test, im-mediate post-test and delayed post-test. The initial means were very similar because the children were distributed in the groups by pairing their scores and all four groups had very low initial scores.

By use of paired t-tests, significant differences were observed when performance at pre-test and immediate post-test were compared, for both experimental groups (EG1: t (8) = -2.920; p = .0019; EG2: t (8) = -3,447; p = .0009). Thus, the teaching session that used tree diagrams (ei-ther virtual or in pencil and paper) was effective in developing children’s combinatorial reasoning.

No significant differences were observed between performance at immediate post-test and delayed post-test for the two experimental groups (EG1: t (8) = -0.472; p = .649; EG2: t (8) = -1.541; p =.162). This indicates that learning was retained by children of both experimen-tal groups because, after nine weeks, the children still were able to recognize the distinct combinatorial rela-tions involved in the problems and also were still able to successfully present correct combinations.

The children in the control groups presented no sig-nificant differences in performance, neither when

Figure 3: Correct answer at post-test with tree diagram

Groups Pre-test Immediate post-test Delayed post-test

EG1 – Software Árbol 4,6 12,1 13,22

EG2 – Pencil and paper 4,8 14,8 16,44

CG1 – Multiplicative problems 4,7 4,1 4,0

CG2 – Unassisted 4,9 2,8 4,2

Table 2: Means of groups at pre-test and two post-tests

Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)

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pre and immediate post-test were compared (CG1: t (9) = 0.751; p = .472; CG2: t (9) = 0.391; p = .705), nor when immediate and delayed post-test were compared (CG1: t (9) = 0.0750; p = .946; CG2: t (9) = -0.782; p = .454). Thus, only solving other multiplicative problems (not including combinatorial ones) or just taking part in other regular school activities was not sufficient to improve combinatorial reasoning.

Table 2 also indicates a slightly better improvement in EG2 when compared to EG1. This performance some-what higher may be related to the fact that students in this second group solved the situations using the same representation (writing with pencil and paper) adopted in the pre-test and post-tests, while students in the first experimental group solved the problems aided by the software and at post-tests had to use a pencil and paper representation. In addition, EG2 may have been benefited by having to think about the combinatorial relations concurrently with the con-struction of the tree diagrams, while EG1 had the tree diagrams built by the software, and it was necessary to think of these relations only when selecting the valid cases constructed. A positive result, observed mainly by the students of the group that learned with the soft-ware, was that at post-test the children used different types of strategies – tree diagrams, lists and diagrams

– showing that students did not merely learn a proce-dure but understood the relations involved in distinct types of combinatorial problems.

Taking in consideration problem type, Graph 1 indi-cates that there were huge improvements at post-test in Cartesian products, combinations and arrangements.

Means in these types of problems were higher than four (considering that there were two problems of each type and that for each problem the maximum score was four, and the total maximum was eight). This indicates that children in the experimental groups tended to obtain correct answers in at least one problem of these types. The graph shows no im-provement in permutations and, just as what was observed with adults, the children possibly needed more time to better understand the tree diagram con-struction of this problem type, in which all elements are used in distinct orders.

Figure 4 shows a child, from the first experimental group, solving similar combination problems at pre and at immediate post-test. At pre-test the problem involved selecting two pets out of three animals (a dog, a bird and a turtle) and at immediate post-test the problem involved the choice of two teachers out of four (Ricardo, Tânia, Luíza and Sérgio). Initially the child incorrectly answered that there was only one way of choosing two pets out of three animals. At post-test the child used a tree diagram and correctly answered that there were six different ways.

0

1

2

3

4

5

CP C A P

Pre-test - EG1

Post-test EG1

Pre-test EG2

Post-test EG2

CP: Cartesian product; C: Combination; A: Arrangement; P: Permutation

Graph 1: Means of experimental groups at pre-test and immediate post-test according to problem type

Figure 4: Incorrect answer at pre-test and correct answer at immediate post-test of a child from the first experimental group

Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)

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Figure 5 shows a child, from the second experimental group, solving similar arrangements problems at pre and at immediate post-test.

The pre-test problem involved choosing two out of five letters (X, Y, Z, K and W) for license plates. The child listed only five options out of the 20 possible arrange-ments. The list was not systematic and the child did not consider that the order of choice implied in different options (the licence plate XY is different from the YX one). The immediate post-test problem involved the choice of a representative and vice-representative of a class out of six students (Luciana, Marcos, Priscila, João, Talita and Diego). The child listed 15 arrange-ments and answered that there were 30 in all. The child was very systematic in the listing produced. First all the choices with Luciana (represented by the initial L) as representative were listed, followed by Marcos (represented by the initial M) as representative and Diego as representative (represented by the initial D). The child at this point generalized that for each student as representative there were five options, so with all six children there would be 30 distinct ar-rangements.

Progress was not generally observed in permutation problems but Figure 6 shows how a child (from the sec-ond experimental group) solved this type of problem at pre and at post-test. The pre-test problem involved

ordering three people (Maria, Luís and Carlos) in a line. The child presented only one option.

At post-test, the child used a tree diagram and was successful in doing so. The problem involved putting a ring (anel), a necklace (colar) and a pair of earrings (brincos) on three trays of a jewel box. The child cor-rectly considered all possible branches of the tree and concluded there were six different ways of putting the jewellery in the box.

One interesting aspect is that many children from the experimental groups preferred to use listings at post-tests, despite having had the experience – by use of software or pencil and paper – of using tree diagrams. What was observed was that learning with tree diagrams enabled systematic listing, not present at pre-test. Figure 7 is an example of a child, of the second experimental group, that at post-test correctly listed the 24 possible couples, chosen from a group of six boys (Gabriel, Thiago, Matheus, Rebato, Otávio and Felipe) and four girls (Taciana, Eduarda, Letícia and Rayssa).

This was also observed amongst adults – the preference of use of listing was maintained but the child or adult used systematic lists, after instruction. This seems to be strong evidence that the child or adult did not simply learn a procedure but understood what relations were involved in distinct combinatorial problems.

Figure 5: Incorrect answer at pre-test and correct answer at immediate post-test of a child from the second experimental group

Figure 6: Incorrect answer at pre-test and correct answer at immediate post-test of a child from the second experimental group in a

permutation problem

Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)

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USING TREE DIAGRAMS IN INITIAL SCHOOLING

The two studies presented show that despite initially not understanding combinatorial problems, children and adults in early schooling can develop their com-binatorial reasoning by use of robust symbolic rep-resentations (Vergnaud, 1997), such as tree diagrams, that aid the systematic enumeration of combinations.

Tree diagrams – built by software use or in writing – may help students’ understanding of combinatorial situations, because this form of representation may aid systematic choice of elements to compose com-binations, as pointed out by Fischbein (1975), and also attested by Maher and Yankelewitz (2010); and Sandoval, Trigueiros and Lozano (2007). The use of tree diagrams also enabled improvement in listings that at post-test were used in a systematic manner by adults and children. The better use of listings shows that the participants had not merely developed proce-dural knowledge (how to use tree diagrams), but had developed a better understanding of combinatorial situations.

One aspect that must be pointed out is that the use of the software Árbol helped children develop their combinatorial reasoning but it required an extra ef-fort when the students answered the problems by use of pencil and paper. In this case, what had been learnt using the software, had to be transferred to pencil and paper solutions. This aspect must be considered in teaching situations and future studies may look into how the use of technology may enable the use of varied forms of symbolic representation in solving combinatorial problems.

Care is required in using tree diagrams in combina-tions and permutations. In combinations care is need-ed in not considering twice equivalent cases and in permutations the tree may have many steps of choice that must all be considered.

Combinatorial reasoning is a very relevant aspect in mathematical development and schooling is an im-portant factor in this progress. How Combinatorics is taught can aid combinatorial reasoning development and tree diagrams may be used as tools that effectively represent combinatorial situations and the relations involved. This symbolic representation is especial-ly useful at initial schooling by its visual aspect that enables both children and adults to perceive how el-ements can be combined in a systematic manner and help them develop their combinatorial reasoning.

REFERENCES

Aguirre, C. (2005). Diagrama de Árbol. [Computer software].

Mexico: Multimidea.

Batanero, C., Godino, J. D., & Pelayo, V. N. (1996). Razonamiento

Combinatorio en Alumnos de Secundaria. Educación

Matemática, 8, 26–39.

Fischbein, E. (1975). The Intuitive Sources of Probabilistic

Thinking in Children. Dordrecht, The Netherlands: Reidel.

Maher, C., & Yankelewitz, D. (2010). Representations as tools for

building arguments. In Maher, C., Powell, A., & Uptegrove,

E. (Eds.), Combinatorics and Reasoning: Representing,

Justifying and Building Isomorphisms (pp. 17–26). New

York, NY: Springer.

Nunes, T., & Bryant, P. (1996). Children doing mathematics.

Oxford, UK: Blackwell Publishers Ltd.

Pessoa, C. A. S., & Borba, R. (2011). An analysis of primary to

high school students’ strategies in combinatorial problem

solving. In B. Ubuz (Ed.), 35th Conference of the Group

for the Psychology of Mathematics Education (pp. 1–8).

Ankara, Turkey: PME.

Sandoval, I., Trigueiros, M., & Lozano, D. (2007). Uso de un

interactivo para el aprendizaje de algunas ideas sobre

combinatoria en primaria. In 12 Comitê Interamericano de

Educação Matemática. Querétaro, México.

Vergnaud, G. (1997). The nature of mathematical concepts.

In T. Nunes & P. Bryant (Eds.), Learning and Teaching

Mathematics (pp. 5–28). East Sussex, UK: Psychology

Press Ltd.

Figure 7: Correct answer at delayed post-test using listing of a child from the first experimental group