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ORIGINAL ARTICLE
School mathematics and creativity at the elementaryand middle-grade levels: how are they related?
Michal Tabach • Alex Friedlander
Accepted: 5 October 2012 / Published online: 20 October 2012
� FIZ Karlsruhe 2012
Abstract In this study, we evaluated students’ creativity,
as expressed in the solution methods of three problems for
groups of students in different grades. Posing the same
problems to students of similar (advanced) mathematical
abilities in different grades allowed us to look for possible
connections between creativity and mathematical knowl-
edge. The findings indicate that at the elementary school
level, the number of solution methods and creativity scores
increased with age. The collective methods space of the
eighth graders seemed to narrow almost exclusively to
algebraic methods, but the increase in the number of
solutions was renewed in the ninth grade.
1 Introduction
School mathematics is characterized by a certain tension
between two seemingly opposing goals: learning proce-
dures and applying algorithmic problem-solving strategies
using routine problems, on the one hand, and learning
concepts and employing creative strategies for solving non-
routine problems, on the other hand. Achieving the first
goal allows students to solve many problems in an efficient,
concise, and quick manner, whereas promotion of crea-
tivity allows students to achieve the most important char-
acteristic of advanced mathematical thinking (Ervynck
1991). Our work with students and teachers led us to the
impression that a larger repertoire of procedures and more
extended time spent practicing technical skills might be
detrimental to becoming engaged in creative problem
solving. The aim of this study was to investigate the con-
nections between students’ mathematical knowledge and
their repertoire of solutions to problems that have the
potential to evoke creative thinking.
2 Theoretical background
2.1 School mathematics between algorithmic
and creative thinking
Mathematics curricula in many countries include general
policy statements. However, examining these statements
may reveal an inherent dichotomy between the recom-
mendation to develop students’ problem-solving skills and
a concern for promoting their ability to perform algorithms.
For example, the NCTM Standards (NCTM 2000) state, on
the one hand, that ‘‘the curriculum also must focus on
important mathematics—mathematics that is worth the
time and attention of students and that will prepare them
for continued study and for solving problems in a variety of
school, home, and work settings’’ (pp. 14–15). On the other
hand, the same document also states that ‘‘topics such as
recursion, iteration, and the comparison of algorithms have
emerged and deserve increased attention because of their
relevance’’ (ibid). The same dilemma is found with the
Israeli National Syllabus (Israeli National Syllabus 2006)
for elementary school mathematics, where the development
of both efficiency in performing algorithms and problem-
solving strategies are required and systematically empha-
sized throughout all mathematical topics and grade levels.
Mathematical creativity is evaluated with reference to
students’ own previous experiences and compared with the
M. Tabach (&)
Tel Aviv University, Tel Aviv, Israel
e-mail: tabachm@post.tau.ac.il
A. Friedlander
Weizmann Institute of Science, Rehovot, Israel
e-mail: Alex.Friedlander@weizmann.ac.il
123
ZDM Mathematics Education (2013) 45:227–238
DOI 10.1007/s11858-012-0471-5
performance of other students with a similar educational
history (Liljedahl and Sriraman 2006). In other words, for a
student to be mathematically creative he must ‘‘step out’’ of
his previous experience. Hence, an extension of knowledge
might limit the potential for creativity. In this paper, we
focus on the inherent tension between the development of
students’ own problem-solving strategies and their need to
master routine procedures.
2.2 Creative mathematical thinking and achievements
Several studies have tried to capture the relations between
students’ creative mathematical thinking and achieve-
ments. The expectation to find such connections in school
mathematics originates from two sources. One of them
relates to the fact that, in advanced mathematics, creativity
is considered to be one of the main generators that ensures
the growth of the field (Sriraman 2004). The second reason
for looking at possible connections between creative
mathematical thinking and achievements stems from
research done on general creativity and school academic
achievements (Murphy 1973; Torrance 1962), reporting on
high correlation between the two.
In the last decade several studies have investigated cor-
relations between mathematical creativity and achievement.
Bahar and Maker (2011) investigated the relationship
between the creative mathematical performance of first to
fourth graders and their achievements, and found significant
correlations between the two. These results are consistent
with those of Sak and Maker (2006), who studied connec-
tions among age, grade, mathematical knowledge and cre-
ative mathematical thinking for first to fifth grade students.
They reported an increasing contribution of mathematical
knowledge to children’s creative mathematical thinking.
They claim that their findings imply that ‘‘the more a child
learns about the mathematical domain, the more creatively
he or she performs in this domain’’ (p. 288).
Other studies are less conclusive about the creativity–
achievement connection. Baran et al. (2011) looked for
correlations between general creativity and mathematical
achievements of 6-year-old kindergarten children in
Ankara, and did not find a meaningful correlation between
children’s creativity scores and their mathematical ability
scores. Mann (2005) explored the relationship between
mathematical creativity and mathematical achievement of
seventh-grade students. He reported that mathematical
achievement was the strongest predictor of mathematical
creativity: 23 % of the variance of mathematical creativity
was attributed to students’ mathematical achievement
scores, but 65 % variance in creativity scores remained
unexplained. He concluded that ‘‘there is a relationship
between mathematical experiences (knowledge and skills)
and creativity in mathematics’’ (p. 74).
A possible reason for the variation in the findings of
these studies can be the use of different instruments to
measure creative mathematical thinking.
2.3 Measures of creative mathematical thinking
The practice and assessment of routine procedures and
algorithms are widely documented in the literature (see, for
example, Ayres 2000; Demby 1997; Kieran 1981). In
contrast, the essence and measurement of creative problem
solving are less clear. In this paper, we follow the
assumptions presented by Torrance (1974) and Leikin and
Lev (2007) that student creativity may be associated with
problem-solving performance and can be considered as a
combination of the originality, fluency, and flexibility
expressed in solving a problem. Note that regarding general
creativity, Torrance (1974) considered elaboration as an
additional component of creativity. However, since we
discuss here mathematical creativity in a school context,
we chose to relate to just the first three of these compo-
nents. Note also that since an analysis of problem solving
may relate to both outcome and strategy (Tsamir et al.
2010), we will consider in this study mainly the latter. In
other words, while presenting multiple-solution tasks
(MST) to students, with an explicit request to solve the task
in several ways, the solution methods used by students will
serve as data to assess their mathematical creativity (Leikin
2009).
According to Leikin’s model (2009), originality (Or) is
based on the level of conventionality and the insight of a
problem’s solution method, flexibility (Flx) is associated
with the ability to change ideas and to produce a variety of
solution methods, and fluency (Flu) is evaluated according
to the number of non-repeating solution methods of the
problem at hand. Leikin (2009) proposed measuring the
level of creativity (Cr) of a particular solution method by
calculating the product of a method’s measure of origi-
nality and its measure of flexibility. Thus, with multiple
solution methods of a problem, the creativity of the ith
method is measured by Cri = Flxi 9 Ori, and the total
level of creativity is obtained by summing the creativity
levels of each method.
Leikin and Levav-Waynberg (2008) (following Watson
and Mason 2005), defined a variety of solution spaces of
a problem as ‘‘the collections of solutions produced
by individuals, groups of individuals, or experts’’. For
example:
Collective solution spaces are solutions produced by
groups of participants.
Expert solution spaces are those suggested by expert
mathematicians. These represent the fullest sets of
solutions known at any given time. It is important to
228 M. Tabach, A. Friedlander
123
distinguish between conventional and unconventional
expert spaces. (p. 236)
Tsamir et al. (2010) further distinguished between
solution methods spaces and solution outcome spaces.
In the present study, we evaluated students’ creativity,
as expressed in the solution methods of three problems for
groups of students at different grade levels. Posing the
same problems to students of similar (advanced) mathe-
matical abilities at different grades allowed us to look for
possible connections between creativity and mathematical
knowledge. We addressed two related questions: (1) How
does the collective method space of a given problem
change across grades? (2) How does grade-related mathe-
matical knowledge affect the level of creativity?
3 Methods
3.1 Participants
Six groups totaling 76 students ranging from elementary
school (4th grade) to junior high (9th grade) of the same
school participated in this study. All students were con-
sidered by their mathematics teacher as mathematically
advanced, based on the students’ work in class and at
home. The students participated as a separate group in a
weekly two-lesson period of ‘‘extended mathematics’’ that
substituted two out of six mathematics lessons of ‘‘regular
mathematics’’ received by their peers. The numbers of
students were 12, 16, 13, 16, 11, and 8 in Grades 4–9
correspondingly. The students’ mathematical background
followed the following sequence:
• The fourth graders summarized their knowledge about
whole numbers and then began exploring other
numbers.
• The fifth and sixth graders became more familiar with
decimals and quotients.
• The seventh and eighth graders followed a 2-year
beginning algebra course.
• The ninth graders focused on functions.
3.2 Tools
Three problems were chosen for the study: the Chickens &
Cows problem (Fig. 1), the Movie Theatre problem
(Fig. 2), and the Age problem (Fig. 3). The choice of the
tasks was based on their potential for (a) multiple solutions,
(b) possible solutions by a routine procedure (for example,
by solving an equation of one or two variables), and
(c) possible solutions by non-routine strategies (for exam-
ple, trial and error or numerical reasoning). In this sense,
the problems can be categorized as MST—defined by
Leikin (2009) as ‘‘assignments in which a student is
There are chickens and cows on old McDonald’s farm – altogether 70 heads and 186 feet. How many chickens and cows are on the farm?
Explain your solution.
Try to find different ways to solve the problem.
Fig. 1 The Chickens & Cowsproblem
A movie theatre offers two kinds of tickets:
Ron is a club member, and he pays $240 per year, and $10 for each movie he sees.
John is not a club member, and he pays $25 for each movie he sees.
Throughout the year, Ron and John went to the same movies and were surprised to find out that both paid the same total amount.
How many movies did each of them see that year?
Explain your solution.
Try to find different ways to solve the problem.
Fig. 2 The Movie Theatreproblem
School mathematics and creativity 229
123
explicitly required to solve a mathematical problem in
different ways’’ (p. 133). In the Chickens & Cows problem
the quantities of chickens and cows play a symmetrical role
in reaching the solution. In the Movie Theatre problem two
quantities are changed (the amounts paid by Ron and John)
as a function of the third quantity (the number of movies
watched). Hence, conventional symbolic solutions may be
found for these two problems. However, this is not the case
for the Age problem, since it involves three variables, and
as such it is not commonly found in textbooks for middle-
school algebra.
3.3 Collection of data
The three problems were given to the students during their
weekly lessons, within a time frame of about 30 min for
each problem. In each case, the context of the problem was
presented first orally, and the requirements for multiple
solutions and for detailed written documentation were
emphasized.
3.4 Data analysis
Following Leikin’s procedure (2009), we constructed an
expert method space for each problem separately. There-
after, the methods were categorized, and an initial scaling
value was assigned to each method. We will demonstrate it
here for the first problem.
For the Chickens & Cows problem, the methods were
categorized as either numerical or algebraic, and each
category was divided into several solution strategies. Thus,
a numerical strategy could be based on (a) an iterative
method—starting with an initial ‘‘guess’’, followed by
several iterations of guesses, (b) a pattern-based method—
looking for patterns and finding the answers by numerical
operations, and (c) a method of combined iterations and
pattern-based considerations. For each of these three
numerical methods, a student could (1) keep the total
number of heads constant, and calculate the corresponding
number of feet and make adaptations, (2) keep the total
number of feet constant, and calculate the corresponding
number of heads and make adaptations, or (3) alternate
between the two methods. Thus, we identified nine
numerical solution methods (see Table 1).
An algebraic method could be based on (a) graphs—
plotting two graphs in a coordinate system, (b) solving an
equation with one variable, and (c) solving a system of
equations with two variables. Since for each of the last two
methods, the variable can represent the number of heads or
How old are they? Explain your solution.
Try to find different ways to solve the problem.
54 years 51 years 21 years
Fig. 3 The Age problem
Table 1 Expert solution space, solution categories, and the scoring
scheme for determining the creativity level in the solution methods of
the Chickens & Cows problem
Methods Groups of
methods
Flx Or Flx 9 Or
Numerical Iterative
Heads constant 10 1 10
Feet constant 1 0.1 0.1
Heads and feet 1 0.1 0.1
Iterative andpattern based
Heads constant 10 1 10
Feet constant 1 0.1 0.1
Heads and feet 1 0.1 0.1
Pattern based
Heads constant 10 10 100
Feet constant 1 0.1 0.1
Heads and feet 1 0.1 0.1
Pseudo-algebra 10 1 10
Algebraic Graphic 10 1 10
One variable
x represents heads 10 1 10
x represents feet 1 0.1 0.1
Two variables
x, y heads 10 1 10
x, y feet 1 0.1 0.1
Total 78 16.8 160.8
Task-embedded
Cr level
(# methods) 9
(Cr total)
15 9 160.8 =
2,412
230 M. Tabach, A. Friedlander
123
the number of feet, we identified five algebraic solution
methods (see Table 1). In addition, we also identified a
pseudo-algebraic method (assigning fixed values to vari-
ables and making numerical calculations and reasoning in a
symbolical representation). Hence, the expert method
space for this task contained 15 methods.
Next, we followed Leikin’s (2009) scoring scheme (with
minor adjustments) and assigned to each solution method
two scores of 0.1, 1, or 10 according to its degree of
flexibility and originality. A sample of students’ solution
methods was coded by two experts and a consensus
between them was obtained.
Below, we elaborate on the scoring of the degrees of
flexibility and originality employed in each case. The
scoring of the solution methods used by a particular group
of students consisted of the following stages:
• The solution methods were collected, categorized, and
organized according to the expert’s solution methods
(see Table 1).
• A flexibility score of 10 (Flxi = 10) was assigned to the
first appropriate method within a group of methods; a
score of 1 (Flxi = 1) was assigned to a method that
belonged to the same group, but had a clear, minor
distinction; and a score of 0.1 (Flxi = 0.1) was assigned
to a method considered to be almost identical to a
previously used one. Thus, if one group of students
provided the Feet Constant iterative solution method for
the Chickens & Cows problem, but did not provide
another iterative method (e.g., Heads Constant), then the
group received a (maximal) flexibility score of 10 for the
given method. However, the same method may receive a
lower score in another group that provided more than one
iterative solution method. In other words, a higher score
is assigned according to its dissimilarity with other
solution methods provided by the same group—and
there is no attempt to evaluate the quality of the method
itself.
• The flexibility scores of the methods produced within a
group were summed. Owing to the variation in group
size, the flexibility score of a group of students (Flx)
was obtained by dividing the sum of scores by the
number of students in that group Flx ¼P
Flxi
N
� �
:
• The originality score of a solution method produced
within a group was based on its conventionality and its
level of insight (Ervynck 1991). A score of 10
(Ori = 10) was assigned to an insight-based or uncon-
ventional method; a score of 1 (Ori = 1) was assigned to
a mathematical model-based or partly unconventional
method, possibly learned in a different context; and a
score of 0.1 (Ori = 0.1) was assigned to an algorithm-
based or conventionally learned method.
• The originality scores of the methods produced within a
group were summed. Owing to the variation in group
size, the originality score of a group of students (Or)
was obtained by dividing the sum of scores by the
number of students in that group Or ¼P
Ori
N
� �
:
• According to Leikin (2009), the creativity of a partic-
ular solution method is the product of its flexibility and
originality (Cri = Flxi 9 Ori). The creativity scores of
the methods produced within a group were summed.
Owing to the variation in group size, the creativity
score of a group of students (Cr) was obtained by
dividing the sum of scores by the number of students in
that group Cr ¼P
Cri
N
� �
:
• The fluency score of a group was based on the total
number of different methods produced by the mem-
bers of that group. Similarly, in order to compare the
fluency measures between groups, we divided the
fluency score of a group by the number of students in
that group.
Tables 2 and 3 present the expert solution space, the
solution categories, and the scoring scheme for the Movie
Theatre and the Age problems.
4 Findings
In this section, we first discuss examples of methods used
by students for each problem and our categorization of
these methods. We start with the Chicken & Cows problem,
Table 2 Expert solution space, solution categories, and scoring
scheme for the creativity level in the solution methods of the MovieTheatre problem
Method Flx Or Flx 9 Or
Numerical
Iterative 10 1 10
Mixed iterative and
reasoning based
10 1 10
Reasoning based 10 10 100
Trial and error 10 1 10
Pseudo-algebraic 10 1 10
Algebraic
Graphic 10 1 10
One variable x represents
the # of movies
10 0.1 1
Two variables 10 1 10
Total sum 50 13.1 131
Task-embedded Cr level (# methods) 9
(Cr total)
5 9 131 = 655
School mathematics and creativity 231
123
followed by the Movie Theatre problem, and finally the
Age problem. Next, we analyze the collective methods
space of each group, and finally we analyze the creativity
scores of the groups.
4.1 Examples of solution methods for the Chicken &
Cows problem
The three examples presented in Fig. 4 use an iterative
approach but vary in the heads or feet chosen as a con-
stant. The first solution (4a) is an example of a Heads
First approach, the second (4b) is an example of a Feet
First solution, and the third (4c) belongs to a mixed
iterative and pattern-based solution method. Each of these
examples is based on an iterative approach starting with
an initial guess, and continues with a gradual modification
thereafter.
The next two examples (Fig. 5) illustrate a pattern-based
method and, as in the case of iterative methods, the solu-
tions were classified according to whether the heads or feet
variables were considered first.
Since the algebraic and graphic solutions are quite
standard and well-known, we chose to present in Fig. 6 an
example of a pseudo-algebraic solution.
4.2 Examples of solution methods for the Movie
Theatre problem
Figure 7 presents five examples of solution methods for
the Movie Theatre problem. The first three examples (7a–
7c) were categorized as numerical methods, whereas
Example 7d was considered a pseudo-algebraic solution
method.
• Example 7a presents a method based on a systematic
search. Interestingly, the student did not choose to
compare the total amount of money paid by a club
member and that of a non-member for a given number
of movies. He monitored the difference between the
ticket prices of a club member and that of a non-
member, and looked for a difference that corresponds to
the $240 membership fee.
• Example 7b presents a method based on iterative
reasoning. This student chose to sample 10 movies,
followed by 20 movies, and then adjust his ‘‘guess’’
according to the results.
• Example 7c shows a reasoning-based method based on
the difference between the amounts paid for each
separate movie and the club member payment.
• Example 7d was categorized as pseudo-algebra, since
the student ‘‘prepared’’ the foundation for creating a
symbolic model of the situation. However, after
representing the amounts of each payment scheme
symbolically, she performed a sequence of numerical
calculations, rather than continuing algebraically by
writing and solving an equation.
4.3 Examples of solution methods for the Age problem
Figure 8 shows five examples of solution methods used to
solve the Age problem. Examples 8a and 8b present
numerical methods, Example 8c presents a pseudo-alge-
braic method, whereas the last two examples (8d, 8e)
present algebraic methods.
• The first example (8a) employs a clear systematic trial
and error method: the student started with an initial
Table 3 Expert solution space,
solution categories, and scoring
scheme for the creativity level
in the solution methods of the
Age problem
Methods Groups of methods Flx Or Flx 9 Or
Numerical Trial and error 10 1 10
Mixed trial and error and reasoning based 10 1 10
Trial and error Iterative 10 1 10
Reasoning based 10 10 100
Pseudo-algebraic One variable 10 1 10
Two variables 10 1 10
Three variables 10 1 10
Algebraic One variable 10 1 10
Two variables 10 1 10
Three variables 10 10 100
Algebraic and reasoning based One variable 10 1 10
Two variables 10 1 10
Three variables 10 10 10
Total sum 50 23 230
Task-embedded Cr level (# methods) 9 (Cr total) 5 9 230 = 1,150
232 M. Tabach, A. Friedlander
123
guess (1 ? 4 = 5) and proceeded systematically until
the sum of the ages of the boy and girl met the given
condition of 21.
• The second numerical method (8b) is based on several
considerations: the student obtained twice the age of the
mother by adding the sums of ages of the mother and
girl, and the mother and boy, and then subtracted the
sum of the boy and girl.
• The next method (8c) is categorized as a pseudo-
algebraic method: the student represented the ages by
letters, and wrote three correct equations to represent
the given information. However, from that point on, he
abandoned the symbolic representation, and continued
by employing logical reasoning.
• Example 8d started similarly to example 8c, but it
continued the solution in a symbolic representation.
• The solution method presented in the last example (8e)
started by employing logical reasoning and continued
by using symbols and equations.
4.4 The collective methods space
In this section we analyze changes in students’ collective
methods space from Grades 4 to 9 for each of the three prob-
lems (Chickens & Cows, Movie Theatre, and Age). Table 4
presents the percentage of types of solution methods by grade
level and by problem. The data indicate the following trends:
Total # of feet
# of feet per animal
Total # of heads
Trial #
Animal Check:
80 4 20 1 Cows 84 4 21 2 88 4 22 3 92 4 23 4
100 2 50 1 Chickens 100 + 80 = 180, which is not enough
98 2 49 2 84 + 98 = 182, which is not enough
96 2 48 3 88 + 96 = 184, which is not enough
94 2 47 4 92 + 94 = 186
Example 4a Iterative, Heads First
I chose 40 cows. Calculated feet: 40 × 4 = 160. I need 186 feet, which means 26 chicken feet or 13 chickens. I calculated the number of animals, 40 + 13 = 53, which is too small. I continued, as described in the table.
Cows Chickens 40 13 39 15 38 17 37 19 … … 24 45 23 47
Example 4b Iterative, Feet First
I started looking for tens that would sum up to 70, for example, 40 cows and 30 chickens. I checked the number of feet: 40×4 + 30×2. I also checked 40×2 + 30×4. In both cases it did not sum up to 186. After several more trials, I concluded that I cannot obtain 186 from tens. So I changed my strategy. I tried to divide 186 – so that I will have 6 ones and 18 tens. This is how I came up with 94 and 92, which gave me 47 and 23 heads, respectively.
Example 4c Mixed iterative and pattern-based solutions
Fig. 4 Examples of iterative
and mixed iterative and pattern-
based solutions
School mathematics and creativity 233
123
• Fourth–sixth-grade students produced exclusively
numerical methods—except for solutions of the Age
problem by several sixth graders. This is, of course,
expected, because in Israel the use of symbols and
algebra is only introduced to students in the seventh
grade.
• The transitional stage of the seventh graders is
reflected by the appearance of solutions using
pseudo-algebraic and some algebraic methods: the
percentage of numerical methods used in the Chickens
& Cows and the Movie Theatre problems is greater
than that of the algebraic methods, whereas more
algebraic than numerical solutions were found for the
Age problem.
• The main mathematical learning theme for the eighth
and ninth grade students is algebra. This is reflected in
our data by more frequent use of algebraic than
numerical methods for the Chickens & Cows problem
and for the Movie Theatre problem. However, whereas
eighth graders used almost exclusively algebraic meth-
ods, ninth graders employed both algebraic and
numerical methods. The solution methods employed
in the Age problem were also exceptional at these
grades, in that more numerical than algebraic methods
were used in this particular case.
4.5 Level of creativity
Table 5 and Fig. 9 present the collective creativity scores
by grade level for each of the three problems. As previ-
ously mentioned, the creativity scores were calculated to
control for group size variations (from 8 to 16 students per
group). We would like to note two trends indicated by this
table—the first relating to variations in creativity measures
across grade levels, and the second relating to variations in
creativity levels across problems.
In general, for all three components of creativity, the
scores increased with grade level:
(a) The flexibility level increased with grade level for all
three problems, except for the Age and the Movie
Theatre problems in Grade 8.
If there are 70 chickens, they will have 70 × 2 = 140 feet, so I still need 186 – 140 = 46 feet. A cow has two feet more than a chicken, hence, 46:2 = 23, which is the number of cows. The number of heads is constant, so I will change 23 chickens for cows.
Example 5a Pattern-based, Heads First
There are 186 feet, which comes to 93 pairs of feet. We have only 70 heads, hence, 93 – 70 is the number of animals with two pairs of feet (four feet).
Example 5b Pattern-based, Feet First
Fig. 5 Examples of pattern-
based solutions to the Chicken& Cows problem
Cows are x and chickens are y. x = 4, y = 2 50y + 20x = 100 + 80, which is close. I added cows and subtracted chickens: 49y + 21x = 98 + 84 = 182 48y + 22x = 96 + 88 = 184 47y + 23x = 94 + 92 = 168. This is the only answer, since if we will increase the number of chickens and decrease the number of cows so that we still have 70 heads, the number of feet will decrease. If I increase the number of cows and decrease the number of chickens, the number of feet will increase.
Fig. 6 Example of a pseudo-
algebraic solution to the
Chicken & Cows problem
234 M. Tabach, A. Friedlander
123
(b) A similar trend can be observed for originality—a
general pattern of growth, with some exceptions in
Grades 7 and 8.
(c) The fluency scores displayed a similar pattern of
variation: general growth and exceptions in Grades 7
and 8.
Finally, the overall creativity measures increased (see
Fig. 9), with a notable decrease in Grade 8 for all three
problems, and in Grade 7 for the solution of the Chickens &
Cows problem. However, for both overall creativity and for
its components, a notable increase was observed in Grade 9.
Within a grade level, the creativity measures varied
considerably, according to the task at hand, and we could
not detect any characteristic patterns for each of the three
problems.
5 Discussion
All the students that participated in this study attended the
same school (one group per grade level), and the same
tasks were given to all groups. The study design, however,
posed some limitations regarding the scope of the
Create the following table, and look for a difference of 240 Ronen Yoav # 10 25 1 20 50 2 … … … 150 375 15160 400 16
Example 7a Numerical, iterative method
Let’s try 10 movies: John 25 × 10 = 250; Ron 10 × 10 = 100, 100 + 240 = 340, Ron pays more. Let’s try 20 movies: John 25 × 20 = 500; Ron 20×10=200, 200 + 240 = 440, Ron pays less. I need a number in between. Let’s try 15: John 15 × 25 = 375; Ron 15×10 + 240 = 370, Ron pays less, but the gap narrows. Let’s try 16 movies: John 16 × 25 = 400; Ron 16×10 + 240 = 400 that’s it!
Example 7b Numerical, mixed iterative, and reasoning-based methods
The number of movies must be even, because 25 is odd. If they go to 20 movies, Ron has an advantage of $60. A club member saves $15 for each movie (25 – 10 = 15), so it is 16 movies.
Example 7c Numerical, reasoning-based method
x – the number of movies that each student viewed. Annual expressions for each student: Ron – 240 + 10x ; John – 25x Now I am trying to guess and substitute in the two expressions to obtain the same number. 1. substitute 15: 240 + 10 × 15 = 390 ; 25 × 15 = 375 2. substitute 16: 240 + 10 × 16 = 400 ; 25 × 16 = 400
Example 7d Pseudo-algebraic method
Fig. 7 Examples of solution
methods for the Movie Theatreproblem
School mathematics and creativity 235
123
conclusions that could be drawn from the findings. One
could argue that since this study was not longitudinal, it did
not confirm that some of the younger students will exhibit
in the future the same behavior as the older students that
participated in the study. On the other hand, the students’
similar learning environment and the identical test items
allowed us to attribute most of the differences among grade
levels to the extent of the mathematical knowledge and
experience of each group.
An analysis of the collective solution methods and
creativity scores of the groups revealed the following:
• The collective methods space of the fourth graders was
limited and their creativity scores were relatively low—
possibly resulting from the relatively high demands of
the tasks.
• The fifth and sixth graders produced a more extensive
collective methods space and achieved higher creativity
scores.
• The seventh graders first attempted to use algebraic tools
after they had started the beginning algebra course. How-
ever, they displayed a limited knowledge in this domain,
and some of them tended to avoid algebra altogether.
You can know that the girl is 3 years older than the boy. So let’s try: 1 + 4 = 5 ; 2 + 5 = 7 ; 3 + 6 = 9 ; 4 + 7 = 11; 5 + 8 = 13 ; 6 + 9 = 15 ; 7 + 10 = 17 ; 8 + 11 = 19 ; 9 + 12 = 21 That’s it.
Example 8a Numerical, trial and error – iterative method
(51 + 54 – 21):2 = 42 , 54 – 42 = 12, 51 – 42 = 9
Example 8b Numerical, reasoning-based method
x – mother; y – girl; n – boy x + y = 54; n + y = 21; x + n = 51 I noticed that the mother and boy are three years younger than the mother and girl. That is, the girl is three years older. The girl and boy together are 21, so the boy is 9 and the girl is 12. We continued calculating and found that the mother is 42.
Example 8c Pseudo-algebraic, three-variable method
x- mother; y-girl; n-boy y + n = 21 x + n = 51 x + y = 54 / + x + n 2x + y + n = 105 / – y – n 2x = 84 x = 42
Example 8d Algebraic, three-variable method
Since the mother and girl are 54 and the mother and boy are 51, the girl is 3 years older than the boy. The boy – x x + 3 + x = 21 2x + 3 = 21 / –3 2x = 18/:2 x = 9
Example 8e Algebraic and reasoning-based, one-variable method
Fig. 8 Examples of solution
methods for the Age problem
236 M. Tabach, A. Friedlander
123
• The eighth graders showed a strong preference for
algebraic methods, displayed a relatively narrow
methods space, and, as a result, obtained lower
creativity scores.
• The ninth graders displayed a more balanced methods
space from a numerical versus algebraic perspective,
and obtained higher creativity scores.
As noted before, the students’ solution methods for the
Age problem followed a somewhat different pattern as
compared with the other two problems. This finding can be
explained by the fact that an algebraic solution to this
problem is based on a system of three equations with three
variables—an uncommon feature for problems usually
taught in school. In this sense, most of the eighth and ninth
graders could not produce an algorithmic solution to this
problem, and as a result, their challenge was similar to that
posed by the other two problems for the elementary school
students.
Finally, we would like to relate our findings to the two
questions posed at the beginning of this paper. The
questions referred to possible changes in creativity levels
across grade levels, and to the possible effects of
mathematical knowledge on the level of creativity. Our
findings provide evidence for an affirmative answer to
both questions.
Table 4 Methods spaces in percentages grouped by age for each problem (C&C, Chickens & Cows; MT, Movie Theatre)
Grade: 4 5 6 7 8 9
Problem C&C MT Age C&C MT Age C&C MT Age C&C MT Age C&C MT Age C&C MT Age
Numerical
Trial and error 75 90 20 59 80 42 23 64 25 66 34 6 6 28 10 17 41
Mixed 10 10 12 10 29 46 9 25 6 18 6 3 5 11
Reasoning based 25 70 29 10 29 31 27 30 11 12 38 42 17 3 45
Pseudo-algebraic 15 17 12 6 6 3 5
Symbolic
Symbolic 5 24 6 82 72 42 56 56 33
Symbolic and
reasoning
44 11
Italicized values represent frequencies of 25–50 %
Bold values represent frequencies of more than 50 %
The remaining frequencies of less than 25 %
Table 5 Creativity scores by grade level for each of the three problems
Grade Chickens & Cows Movie Theatre Age
Mean Flx Or Flu Cr Flx Or Flu Cr Flx Or Flu Cr
4 1.9 0.9 0.3 46 3 0.3 1 9 3 1.2 1 36
5 2.1 0.8 0.4 53 4 1.3 1 52 3.6 1.2 2.2 47
6 2.5 0.9 0.5 56 4 1.3 1.1 52 6.7 1.7 2.2 100
7 2.6 0.8 0.4 49 5 1.1 1.2 76 5 2 1.3 120
8 5.5 0.5 0.6 33 3.6 0.3 1.9 11 3.8 1.8 2.2 88
9 7.6 1.9 1 169 10 1.3 3.2 66 8.3 3.8 1.5 192
Fig. 9 Graph of general creativity scores by grade level
School mathematics and creativity 237
123
Our findings lead us to assume that an increase in
mathematical knowledge (i.e. grade level) has the potential
to raise the level of creativity as well—with possible
exceptions because of the temporary influence of learning a
new domain (in our case, algebra). Thus, the observed
increase in creativity scores throughout the upper elemen-
tary school (Grades 4–6) can be attributed to students’
increasing familiarity with the arithmetical domain. The
first encounters with algebra in the seventh grade did not
hinder the general trend of growth in creativity, but in
terms of flexibility, the transition to algebraic solution
methods was relatively limited. The decrease in all crea-
tivity measures in the eighth grade resulted from the almost
exclusive use of algebraic solution methods at this stage
where algebra was very intensively used in school. How-
ever, the pattern of growth in creativity appeared to end in
the ninth grade—by the end of the first algebra course and
the beginning of a more advanced stage of studying this
subject. At this stage the almost exclusive use of algebraic
solution methods, noted before, was abandoned, and a
more balanced use of both numerical and algebraic meth-
ods was observed. Thus, we can assume that the learning of
algebra might have a temporary limiting effect on crea-
tivity, but in the long run, it has the potential to enrich the
students’ repertoire of solution methods.
References
Ayres, P. (2000). An analysis of bracket expansion errors. In T.
Nakahara, & M. Koyama (Eds.), Proceedings of the 24thconference of the International Group for the Psychology ofMathematics Education (Vol. 2, pp. 25–32). Hiroshima, Japan.
Bahar, A. K., & Maker, C. J. (2011). Exploring the relationship
between mathematical creativity and mathematical achievement.
Asia-Pacific Journal of Gifted and Talented Education, 3(1),
33–48.
Baran, G., Erdogan, S., & Cakmak, A. (2011). A study on the
relationship between six-year-old children’s creativity and math-
ematical ability. International Education Studies, 4(1), 105–111.
Demby, A. (1997). Algebraic procedures used by 13- to 15-year-olds.
Educational Studies in Mathematics, 33, 45–70.
Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.),
Advanced mathematical thinking (pp. 42–53). Dordrecht:
Kluwer.
Israeli National Syllabus in Mathematics for Elementary School
(2006). http://cms.education.gov.il/EducationCMS/Units/Tochni
yot_Limudim/Math_Yesodi/PDF (in Hebrew). Accessed 3 Oct
2012.
Kieran, C. (1981). Concepts associated with the equality symbol.
Educational Studies in Mathematics, 12, 317–326.
Leikin, R. (2009). Exploring mathematical creativity using multiple
solution tasks. In R. Leikin, A. Berman, & B. Koichu (Eds.),
Creativity in mathematics and the education of gifted students(pp. 129–135). Rotterdam: Sense Publishers.
Leikin, R., & Lev, M. (2007). Multiple solution tasks as a magnifying
glass for observation of mathematical creativity. In J. H. Woo, H.
C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31stinternational conference for the psychology of mathematicseducation (Vol. 3, pp. 161–168). Seoul: The Korea Society of
Educational Studies in Mathematics.
Leikin, R., & Levav-Waynberg, A. (2008). Solution spaces of
multiple-solution connecting tasks as a mirror of the develop-
ment of mathematics teachers’ knowledge. Canadian Journal ofScience, Mathematics, and Technology Education, 8(3),
233–251.
Liljedahl, P., & Sriraman, B. (2006). Musings on mathematical
creativity. For The Learning of Mathematics, 26(1), 20–23.
Mann, E. (2005). Mathematical creativity and school mathematics:
Indicators of mathematical creativity in middle school stu-
dents (Doctoral dissertation). http://www.gifted.uconn.edu/siegle/
Dissertations/Eric%20Mann.pdf. Accessed 3 Oct 2012.
Murphy, R. T. (1973). Relationship among a set of creativity,
intelligence, and achievement measures in a high school sample
of boys. In Proceedings of the 81st annual convention, AmericanPsychological Association, Vol. 81, pp. 631–632.
National Council of Teachers of Mathematics (NCTM) (2000).
Principles and standards for school mathematics. Reston, VA:
National Council of Teachers of Mathematics.
Sak, U., & Maker, C. J. (2006). Developmental variations in
children’s creative mathematical thinking as a function of
schooling, age, and knowledge. Creativity Research Journal,18(3), 279–291.
Sriraman, B. (2004). The characteristics of mathematical creativity.
The Mathematics Educator, 14, 19–34.
Torrance, E. P. (1962). Guiding creative talent. Englewood Cliffs, NJ:
Prentice Hall.
Torrance, E. P. (1974). Torrance tests of creative thinking. Bensen-
ville, IL: Scholastic Testing Service.
Tsamir, P., Tirosh, D., Tabach, M., & Levenson, E. (2010). Multiple
solution methods and multiple outcomes—is it a task for
kindergarten children? Educational Studies in Mathematics,73(3), 217–231.
Watson, A., & Mason, J. (2005). Mathematics as a constructiveactivity: Learners generating examples. Mahwah: Lawrence
Erlbaum.
238 M. Tabach, A. Friedlander
123
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