Using tree diagrams to develop combinatorial reasoning
of 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
reasoning of children and adults in early schooling. Konrad Krainer; Naďa Vondrová. CERME
9 - Ninth Congress of the European Society for Research in Mathematics Education, Feb 2015,
Prague, Czech Republic. pp.2480-2486, Proceedings of the Ninth Congress of the European
Society for Research in Mathematics Education. <hal-01289344>
HAL Id: hal-01289344
https://hal.archives-ouvertes.fr/hal-01289344
Submitted on 16 Mar 2016
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
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),
2
UFPE, Edumatec, Recife, Brasil, [email protected]
3
Prefeitura da Cidade do Recife, Recife, Brasil, [email protected]
Recife, Brasil, [email protected]
Combinatorial reasoning is very important in mathematical development, providing students with contact
with essential problem solving. One form of symbolic
representation of combinatorial situations is a tree diagram. Two experimental studies were designed and
implemented, the first with adults in initial schooling 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 reasoning. 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,
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 students understand how to obtain the total number of
combinations.
The role of symbolic representations in mathematical 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 development of combinatorial reasoning because this
representation helps systematization by pointing out
the necessary steps in choosing elements to compose
combinations.
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 essential 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.
CERME9 (2015) – TWG16
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 children, aged 11 to 13, and the authors observed improvements in student performance, especially regarding
the choice of strategies for efficient resolution. Thus,
it is emphasized that this software, through tree dia-
2480
Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)
Figure 1: Árbol screen of a Cartesian product problem
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 permutations). 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
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?
According to Vergnaud (1997), to study and understand how mathematical concepts develop in students’
minds through their experience in school and outside TWO STUDIES ON COMBINATORIAL
school, one needs to consider a concept as a three-uple REASONING DEVELOPMENT
of three sets: the set of situations (S) that make the
concept useful and meaningful; the set of operational With the aim of investigating the role of symbolic repinvariants (I) that can be used to deal with these sit- resentations – in particular the use of tree diagrams
uations; and the set of symbolic representations (R) – on the development of combinatorial reasoning, we
that can be used to represent invariants, situations designed and implemented two experimental studies.
and procedures. Thus, the immense role played by The first one involved adults in initial schooling and
symbols cannot be ignored in mathematical teach- in the second study took part Elementary School stuing and learning, as means to articulate invariants dents (5th grade, 10 year olds).
– conceptual properties and relations – situations and
strategies used in problem solving.
Method of the 1st Study: Adults in early
schooling using tree diagrams and lists
Vergnaud (1997) also points out that combinatori- The adults taking part in the study were 24 students
al problems are part of what he calls multiplicative of classes corresponding to the 4th and 5th years of regstructures. In the same direction, Pessoa and Borba ular Elementary School with no previous systematic
(2010) defend that as connected concepts, different instruction on Combinatorics. They were separated
types of combinatorial problems should be taught in into three groups, each group consisting of eight
the classroom and present examples of these distinct students. After solving an eight item pre-test (two
situations:
problems of each type), they were taught in groups
that varied in terms of symbolic representations used:
Cartesian product: At the square dance three boys and G1 – lists and tree diagrams; G2 – tree diagrams; and
four girls want to dance. If all the boys dance with all G3 – lists. After the learning session they solved an
the girls, how many pairs will be formed?
eight problem post-test.
2481
Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)
Method of the 2nd Study: Children using
virtual or written tree diagrams
This 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 problems), 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 posttest). 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 software 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 incorrect answers or partially correct answers with very
Problem type
Incorrect
Answer
few correct combinations presented. They preferred
to use lists in their answers but, usually, were not systematic 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 differences 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 correct answers at post-test.
The first example (a Cartesian product), asked to indicate 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
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
2482
Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)
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 permutations 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 example 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 representation 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 combinatorics
In 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, immediate 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 (either 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 posttest 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 experimental groups because, after nine weeks, the children still
were able to recognize the distinct combinatorial relations involved in the problems and also were still able
to successfully present correct combinations.
The children in the control groups presented no significant 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
2483
Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)
5
4
Pre-test - EG1
3
Post-test EG1
2
Pre-test EG2
1
Post-test EG2
0
CP
C
A
P
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
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 somewhat 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 construction 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 software, 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 procedure but understood the relations involved in distinct
types of combinatorial problems.
Taking in consideration problem type, Graph 1 indicates 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 improvement in permutations and, just as what was
observed with adults, the children possibly needed
more time to better understand the tree diagram construction 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.
Figure 4: Incorrect answer at pre-test and correct answer at immediate post-test of a child from the first experimental group
2484
Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)
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
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 arrangements. 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 arrangements 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 arrangements.
Progress was not generally observed in permutation
problems but Figure 6 shows how a child (from the second 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 correctly 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.
2485
Using tree diagrams to develop combinatorial reasoning of children and adults in early schooling (Rute Borba, Juliana Azevedo and Fernanda Barreto)
Figure 7: Correct answer at delayed post-test using listing of a child from the first experimental group
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 combinatorial reasoning by use of robust symbolic representations (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 combinations, 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 procedural knowledge (how to use tree diagrams), but had
developed a better understanding of combinatorial
situations.
Combinatorial reasoning is a very relevant aspect in
mathematical development and schooling is an important 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 especially useful at initial schooling by its visual aspect that
enables both children and adults to perceive how elements 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,
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 effort 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.
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
Care is required in using tree diagrams in combinations and permutations. In combinations care is needed in not considering twice equivalent cases and in
permutations the tree may have many steps of choice
that must all be considered.
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.
2486