School mathematics and creativity at the elementary and middle-grade levels: how are they related?

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: [email protected]

A. Friedlander

Weizmann Institute of Science, Rehovot, Israel

e-mail: [email protected]

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.


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.


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




Pattern based




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


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


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


Three variables 10 1 10

Algebraic One variable 10 1 10



Algebraic and reasoning based One variable 10 1 10



Total sum 50 23 230

Task-embedded Cr level (# methods) 9 (Cr total) 5 9 230 = 1,150


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


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


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


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


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


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.

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School mathematics and creativity at the elementary and middle-grade levels: how are they related?