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The bar model as a visual aid for developing
complementary/variation problems
Eugenia Koleza
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Eugenia Koleza. The bar model as a visual aid for developing complementary/variation problems. Konrad Krainer; Naďa Vondrová. CERME 9 - Ninth Congress of the European Society
for Research in Mathematics Education, Feb 2015, Prague, Czech Republic. pp.1940-1946, Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education.
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The bar model as a visual aid for developing
complementary/variation problems
Eugenia Koleza
University of Patras, Department of Primary Education, Patras, Greece, [email protected]
In this paper, we report the preliminary findings of a
study that considered how third grade students represent multiplication and division problems using the bar
model as a tool. Initial results indicate that students are
able to create visuals, representing problem’s structure,
and based on these visual representations they can formulate division problems given a multiplication one.
Keywords: Multiplication, division, complementary
problems, bar model.
INRODUCTION
Use of diagrams is considered an efficient strategy in
teaching and learning mathematics and especially in
mathematical problem solving. “From the most elementary class to the most advanced seminar, in both
introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase
understanding, and prove results. They thus fulfill a
variety of important roles in mathematical practice”
(Mumma & Panza, 2012, p. 1). Diagrams make plain the
quantities in the story context and the relationships
that exist amongst them, limit abstraction and thereby aid in the problem-solving process (Bishop, 1989).
Draw a diagram is a well known strategy for mathematical problem solving (e.g., Polya, 1957; Schoenfeld,
1985), grounding in the belief that generating a diagram enables deeper understanding of the situation
described and facilitates the conceptualization of the
problem structure (van Essen & Hamaker, 1990). Not
all diagrams are beneficial or can easily be used by
the students. Vosniadou (2010) distinguishes between
external representations that are perceptually based
(grounded on everyday observations) depictions
and those that represent conceptual models (theory
–based). Pictures used in mathematics and science
textbooks, number lines and bar models that are
CERME9 (2015) – TWG13
usually used on mathematical problem solving, are
conceptual models.
In our study, we used the bar model as a visual support for the resolution of simple (one operation) multiplication problems and the formulation of the two
corresponding division problems.
Children’s difficulty to understand the close relation
between multiplication and division was a problem
noticed very early in mathematics teaching literature.
Nevertheless, in most western curricula, multiplication and division, as well as addition and subtraction are taught separately. Paraphrasing Herscovics
(1989), it is as if students are taught the syntax of
operations, without the semantics. In other words,
students know the algorithms (the rules of the ‘grammar’) but they do not understand the meaning and
their relations. This situation explains the difficulty
that pupils often face when solving equations in algebra with an operational view of equality (Wagner
& Parker, 1988). “The ‘one-thing-at-the-time’ design,
provide fewer opportunities for ‘making connections’
compared to those adopted in eastern cultures” (Sun,
2013, p. 13). “In Chinese elementary schools, addition
and subtraction are introduced simultaneously, and
subtraction is introduced as the reverse operation
of addition. Division is also introduced as a reverse
operation of multiplication” (Cai, 2004, p. 110). As Cai
refers (p. 112), “representing quantitative relationships in different ways will not only help students
develop deeper understanding mathematics, but
also will help them develop their flexibility of using
equations to solve application problems”. Giving an
example of a variation in a multiplication problem,
Sun (2011, p. 104) explains that “within the problem set,
there are two concepts of multiplication and division
behind three similar problems made with 4, 6, and 24.
Example problem: How many trees do 6 lines need so
that each line can have 4 trees? Variation problem 1:
1940
The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza)
How many trees will each line get if we plant 24 trees
in 6 lines? Variation problem 2: How many lines do we
plant if we plant 24 trees in order so that each line has
6 trees? Clearly, the intent of One Problem Multiple
Changes is to enable students recapitulate the general relationship of multiplication and division, and
the meaning of equal from the problem set 4 × 6 = 24,
24 ÷ 4 = 6, 24 ÷ 6 = 4 […]. The task draws students into
a space of relations as opposed to directing attention
to the object itself.”
THEORETICAL FRAMEWORK
meanings by the use of the bar model. In an initial
phase the bar model may serve as a model of the mathematical structure of a word problem. Later, through
a process of vertical mathematization, reflecting on
the relationship between their actions upon a diagram
and the effects of those actions, students may generalize and abstract those actions to successfully solve
problems of the same semantic structure. In this way,
bar model becomes, to those students, a model for the
mathematical structure. Bar model is particularly useful for problems that involve comparisons, part-whole
calculations, ratio and proportion.
The ‘model method’, also known as graphical heuristic, In this paper, we will restrict to the use of the bar
consists of the use of rectangular bars to represent model as a visual support in order to formulate mulnumbers rather than abstract letters to represent tiplication and division problems.
unknowns in word problems. This method is often
used in education systems of many countries under The Singapore education system’s approach concernvarious names: tape diagrams – in Japan (Murata, ing the use of diagrams, has close relation with the one
2008), strip diagrams – in US (Beckmann, 2004), or bar several soviet researchers (Bodanskii, Mikulina, in
models – in Singapore (Hoven & Garelick, 2007). This Davydov, 1991/1969) has used in their studies concernspecial kind of diagrams “are clearly designed to help ing algebra word problems. For example in Figure
children decide which operations to use. Instead of rely- 1 (on the left) is the diagram of the problem: “In the
ing on superficial and unreliable clues like key words, kindergarten, there were 17 more hard chairs than soft
the simple visual diagram can help children under- ones (labeled M). When 43 more hard chairs were addstand why the appropriate operations make sense” ed, there were 5 times more hard chairs than soft. How
(Beckmann, 2004, p. 43). Cai and colleagues (2005) con- many hard and soft chairs were there?” as presented in
sider the ‘model method’ as one of the big ideas related Bodanskii (1991/1969, p. 302), and on the right the bar
to algebraic thinking in the Singaporean elementary model representation.
curriculum. “Children solve word problems using the
‘model method’ to construct pictorial equations that In both cases, the common element is the concept of
represent all the information in word problems as a “unit”. Units are not simply single discrete entities, but
cohesive whole, rather than as distinct parts. To solve instead may be composed of one or more ‘shapes’ (in
for the unknown, children undo each operation. This the diagram) of various types (Davydov et al., 2000).
approach helps further enhance their knowledge of Taking as a “unit” the number of soft chairs, the sum
the properties of the four operations”(p. 8). In other 17+43=60 is translated as 6 units, giving, thus, the anwords a basic property of the bar model is that it can swer of the problem. The problem solver is able to
support an exploration and visualization of the ‘doing’ reason from the very diagram that was created as a
and ‘undoing’ processes in mathematics. According model of the situation given. The algebraic equation
to Hall and colleagues (1989) when the structure of that correspond to this visual solution- the model for
a problem is recognized, a formal representation of this kind of relations- is just a step further.
this relationship may be constructed. Departing from
meaningful tasks, students may construct personal The research question behind our study was:
Figure 1: Similar representations of the same problem
1941
The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza)
researcher who hasn’t conducted a teaching experiment independently, but who wishes to do so, should
engage in exploratory teaching first. It is important
that one become thoroughly acquainted, at an experiMETHOD
ential level, with students’ ways and means of operating in whatever domain of mathematical concepts and
In this study, 19 third graders (aged 8) were involved, operations are of interest” (Steffe & Thompson, 2000, p.
belonging in a class of a primary school in Patras. 274). In other words, we wanted first to explore the role
During a period of eight 45-minutes sessions the this specific visual representation may have in helpstudents followed the regular lessons (teaching of ing students to understand how multiplication and
the multiplication table) that were enriched by the division are connected, and their eventual difficulties,
bar model as a way to solve and represent problems, in order to design our ‘teaching experiment’. Because
and mainly as a way to connect multiplication and “incomplete understanding […] can result in inapprodivision, in a unified scheme. This connection was priately designed artifacts or artifacts that result in
designed only for the needs of our experiment, given undesirable side effects.”(March & Smith, 1995, p. 254)
that these two operations are taught separately in the
Greek mathematics curriculum. Tasks given during DISCUSSION
instruction were arithmetic and algebraic tasks (given a problem, the task asked for his complementary The first of the 8 lessons (the corresponding textbook
ones-focus on relations) with three (group, restate and objective was the multiplication table of, 2, 5 and 10)
vary) of the five semantic relations (the other two were began with a teachers’ question.
change and compare) identified by Marshall (1995).
Data was collected and analyzed from field notes, phoT:
What is a problem? Who can make a probtos, pupils’ written work, and final tests. Learning
lem?
gains were assessed by means of a word-problem test
S:
I have 80 candies and I give 10 candies to
just after the end of instruction and a second one a
each one of 8 children
month later.
T:
Is this a problem? What are you asking?
In order to have a problem you must ask
Hereafter, the problems given in the tests.
for something. For example, 8 children
bought 10 candies each one. How many
Vary-problem in the short-term test: Grandma has in
bought in total? Lets make a picture of the
her sac 45 sweets in 5 bags. How many sweet are there
problem. (The teacher designs the bar
in each bag?
model). In order to have a problem, we
must ‘hide’ something.
Vary-Problem in the final test: Grandpa gave to each
one of his 5 grandkids 12 euros. How many euros he gave
in total?
Can the bar model support students in the resolution
and formulation of multiplication and (complementary) division problems?
For each problem was asked to the students: “Departing
from the problem, with the same story and the same
numbers make two other problems and ‘design’ them”.
Teaching students for short periods of time could
not serve as a basis for a solid understanding of their
thinking and how it might be influenced by the use of
a certain visual representation. Our ‘teaching experiment’ has not the characteristics described by Steffe
and Thompson (2000), but it was an exploratory one
which aimed only to the evaluation of the bar model as
a visual aid for the resolution and formulation of complementary multiplicative-structure problems. “Any
Figure 2
T:
S:
Can someone else make another problem, with the same story?
We can ’hide’ the number of candies that
each child will take.
(The teacher changes the previous model)
Figure 3
1942
The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza)
T:
S:
T:
T:
S:
T:
Another problem?
We can ‘hide’ the children.
So, I will delete the squares.
If I had 40 candies, and 10 children, how
many candies could have each child?
4
Can you ‘design’ the problem?
During the next 2 sessions students worked with the
multiplication table (according to the official schedule), and solved multiplication problems following a
‘didactic contract’ of 4 precise steps that had emerged
from the previous lesson: (1) we read first and we ‘design’ (make a representation of ) the problem, (2) in
order to make the 2 ‘inverse’ (complementary) problems, we “hide” a number from the design, (3) we write
(in words) the ‘inverse’ problems, and (4) we solve the
problems.
The complementary/inverse problems were introduced by the teacher as the two variations of an initial
problem, and the bar model was presented as a tool
of organizing problems’ data. Each complementary/
inverse problem was creating by ‘hiding’ a number
on the bar.
During this process, students faced two major difficulties: (1) Expressing in natural language the problem
that they had already represented by the bar model,
and (2) representing by the bar model the quotative/
measurement division problem-variation. For example, while the multiplication problem «If I save 8 euro
in a week, how many euros I will have after five weeks?»
and –‘hiding’ the 8- the corresponding partitive division problem “If I saved 40 euros in 5 weeks, how many
euros I saved in a week” were easily represented by
the bar model, it was not the same for the ‘quotative
division’ problem. The rule of ‘hiding’ could not be
applied: while in the “partitive division’ problem they
had to ‘hide’ the numbers, in the ‘quotative division’
one, they had to ‘deconstruct’ the bar model. In the
following pictures (Figure 4) we see (to the left) the
incomplete diagram made by a student (reproduced
in the middle), and the one suggested by the teacher
(to the right).
Teacher’s proposition in order to face the difficulty
was to reformulate the problem as “How many 8s we
want in order to make 40?”
We present three different treatments of the “quotative division” problems representation during instruction
a) An arithmetic treatment: no use of the bar model
“30 ducks fly in groups of 5. How many groups of
ducks are there?”
Figure 5
b) The ‘quotative division’ as a subtraction: A “filling-in” use of the bar model. “I have 45 stamps
in pockets of 9 stamps. How many pockets I have?”
The thought behind the model was:
“I take off each time 9 stamps from the 45 stamps”
Figure 6
The ‘quotative division’ as the inverse of the multiplication: An ‘algebraic’ use of the bar model.
Figure 7
Figure 4
1943
The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza)
After instruction a final test was given. Hereafter, we
comment students’ competency in using the bar model
for the resolution of the ‘vary-problem’ given in this
final test (see page 4).
“Grand papa has 60 euros and he wants to distribute them to his 5 grandkids. How many euros will
have each one?”
From a total of 19 students,
a) 13 students formulated correctly the two division-problems. The two main strategies observed
during instruction, appeared also in the final test:
――
a mixed (‘filling-in’ and algebraic) strategy -MS(10
students), and
――
an ‘algebraic’ strategy -AS (3 students).
Figure 10
c) The rest 3 students were not able to use the bar
model at all. For example for the multiplication
problem the diagram proposed was
We give an example of each case.
(MS) Fey
Figure 11
or
Figure 8
(AS) Joseph
Figure 12
Figure 9
CONCLUSIONS
The 3 students who used the ‘algebraic’ strategy were
the higher achieving students of their class. This outcome confirms similar results by Booth and Koedinger
(2007) according to which, higher-achieving middle
school students do benefit from the diagrams while
low-achieving students perform better on story problems that do not have accompanying diagrams.
The results has shown, that the bar model is an effective model, but only for multiplication and partitive
division problems. Though most students (13/19) after
a relatively short term instruction (eight 45-minutes
sessions) could formulate and represent by the bar
model a multiplication and the corresponding division problems, further research is needed about the
kind of representation that is more appropriate for
the quantitative division problems.
b) 3 students formulated only the partitive-division
problem.
1944
The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza)
The fact that a representation may be “clear” for a
multiplication problem does not mean that it would
be a useful tool for the formulation of the corresponding division problem. For example, lets take the array
model.
This model in a concrete (on the left) or in a more abstract (on the right) version, under certain conditions
may be useful for students to understand the relation
between a multiplication and the two complementary-division problems. It is useful, if all information
is on the representation, but not if the students must
construct the representation by themselves, especially in case of big numbers.
The data collected do not permit us to know if the students who had used the algebraic strategy (AS) were
acting in a pure algorithmic way, or ‘with understanding’. Eventually a confrontation with the strategies
used by the same students in additive-structure problems may offer a more complete explanatory framework. A more accurate analysis of the relationship
between the instrument and students’ meanings is
required, and that is what is going to be done with
the data analysis of the whole teaching experiment.
REFERENCES
Beckmann, S. (2004). Solving algebra and other story problems
with simple diagrams: A method demonstrated in grade
4–6 texts used in Singapore. The Mathematics Educator,
14, 42–46.
Bishop, A. J. (1989). Review of research on visualization in
mathematics education. Focus on Learning Problems in
Mathematics, 11(1–2), 7–16.
Figure 13
Booth, J.L, & Koedinger, K.R. (2011). Are diagrams always helpful
tools? British Journal of Educational Psychology, 82(3),
On the other hand, the use of the bar model, as our
research has shown, is a high demanding task, because
the level of abstraction needed. For example, in the
problem: Sarah made 210 cupcakes. She put them into
boxes of 10 each. How many boxes of cupcakes were
there?, students must put on the representation information that does not exist.
492–511.
Cai, J. (2004): Developing algebraic thinking in the earlier
grades: A case study of the Chinese mathematics curriculum. The Mathematics Educator, 8(1), 107–130
Cai, J. Lew, H. C., Morris, A., Moyer, J. C., Ng, S. F.,
& Schmittau, J. (2005). The Development of Students’
Algebraic Thinking in Earlier Grades: A Cross-Cultural
Comparative Perspective. ZDM, 37, 5–15.
The mathematics symbolization “…..?....”, is not evident
for the young students.
Corte, T. de Jong, & J. Elen (Eds.), Use of representations in
reasoning and problem solving (pp. 36–54). London, UK:
Routledge.
Davydov, V. (Ed.). (1991/1969). Soviet studies in mathematics
education: Vol. 6. Psychological abilities of primary school
children in learning mathematics. Reston, VA: NCTM.
Davydov, V. V., Gorbov, S. F., Mikulina, G. G., & Saveleva, O.V.
Figure 14
(2000). Mathematics: Class 2. Ed. by J. Schmittau. NY:
State University of New York.
A combination of an array and a bar model representation could be eventually easier to use for students.
For example, in the problem: Sarah had 12 apples to
hand out to her class. Each group of students in the class
got 3 apples. How many groups were there in the class?
This kind of representation would be:
Hall, R., Kibler, D., Wenger, E., & Truxaw, Ch. (1989). Exploring
the episodic structure of algebra story problem-solving.
Cognition and Instruction, 6(3), 223–283.
Herscovics, N. (1989). Cognitive obstacles encountered in
the learning of algebra. In S. Wagner & C. Kieran (Eds.),
Research issues in the learning and teaching of algebra (pp.
60–86). Hillsdale, NJ: Lawrence Erlbaum.
Hoven, J., & Garelick, B. (2007). Singapore math: Simple or complex? Educational Leadership, 65, 28–31.
March, S.T., & Smith, G.F. (1995). Design and Natural Science
Research on Information Technology. Decision Support
Figure 15
Systems, 15(4), 251–266.
1945
The bar model as a visual aid for developing complementary/variation problems (Eugenia Koleza)
Mumma, J., & Panza, M. (2012). Diagrams in mathematics: history and philosophy. Synthese, 186, 1–5.
Murata, A. (2008). Mathematics teaching and learning as a mediating process: The case of tape diagrams. Mathematical
Thinking and Learning, 10, 374–406.
Pólya, G. (1957). How to solve it. Princeton, NJ: Princeton
University Press.
Schoenfeld, A. H. (1985). Mathematical problem solving.
Orlando, FL: Academic Press.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment
methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp.
267–306). Mahwah, NJ: Lawrence Erlbaum Associates.
Sun, X. (2011). An insider’s perspective: “Variation Problems”
and their cultural grounds in Chinese curriculum practice.
Journal of Mathematical Education, 4(1), 101–114.
Sun, X. (2013). The fundamental idea of mathematical tasks
design in China: the origin and development. In ICMI STUDY
22: Task design in mathematics education. Oxford, UK:
University of Oxford.
van Essen, G., & Hamaker, C. (1990). Using self-generated
drawings to solve arithmetic word problems. Journal of
Educational Research, 83(6), 301–312.
Vosniadou, S. (2010). Instructional considerations in the use of
external representations: The distinction between perceptually based depictions and pictures that represent conceptual models. In L. Verschaffel, E. De Corte, T. de Jong
& J. Elen (Eds.), Use of representations in reasoning and
problem solving (pp. 36–54). London, UK: Routledge.
Wagner, S., & Parker, S. (1988). Advancing algebra. In B. Moses
(Ed.), Algebraic thinking, grades K-12. Reston, VA: NCTM.
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