Thinking Skills and Creativity 33 (2019) 100585

Contents lists available at ScienceDirect

Thinking Skills and Creativity

journal homepage: www.elsevier.com/locate/tsc

Creative thinking in mathematics curriculum: An analytic

T
framework⋆
⁎
Linor L. Hadar , Mor Tirosh
Faculty of Education, Beit-Berl College, Israel

A R T IC LE I N F O ABS TRA CT

Keywords: Current approaches in mathematics education promote the teaching of creative thinking (CT) to
Creative thinking develop a deep conceptual understanding of mathematics, and many nations are including ex-
Textbooks plicit CT learning goals in their curriculum. The goals alone are not enough; appropriate curri-
Curriculum materials culum materials should be used in the classroom if students are to acquire CT skills. This paper
Mathematics
describes the development of a framework to analyze CT in primary school mathematics curri-
culum materials. It then applies the framework to primary school math textbooks in the context
of the Israeli curriculum. The framework includes nine categories under three overarching
themes: lateral thinking, divergent thinking and convergent-integrative thinking. Although the
CT themes are covered, the textbooks provide much more opportunity to engage with CT in
mathematics in grade one than in any other grade, but the curriculum sets the same CT objectives
for all students. Further, the textbooks have emphases that do not necessarily correspond with the
emphasis in the official curriculum. The proposed framework permits the assessment of curri-
culum materials, an important mediating factor between curriculum and instruction. It can show
which activities and materials promote the development of students’ CT and which do not.

1. Introduction

Current approaches in mathematics education promote the teaching of creative thinking (CT) to develop a deep conceptual
understanding of mathematics (Aizikovitsh-Udi, A, & mit, 2011; Mann, 2006). Some even argue that the essence of mathematics is
thinking creatively, not simply arriving at the right answer (Dreyfus & Eisenberg, 1996). As teaching CT in mathematics is difficult
and demanding, incorporating CT in specifically designed curriculum materials can support teachers and increase the likelihood of
student engagement (Zohar, 2008). As a central resource for teaching and learning, curriculum materials give students opportunities
to engage with contents and skills (Houang & Schmidt, 2008). As major conveyors of curricula, they play a dominant role in learning
(Fan, Zhu, & Miao, 2013), influencing what and how mathematics is taught (Tarr, Chávez, Reys, & Reys, 2006).
Many countries are incorporating explicit CT learning goals in their math curricula (Gallagher, Hipkins, & Zohar, 2012). To reach
their goals, educators need to consider curriculum materials' ability to stimulate and support CT (Williams, 2002) by looking at their
assigned tasks, as these tasks form the basis of student learning (Doyle, 1988). For example, in mathematics, activities/tasks asking
students to perform a procedure in a routine manner represent an opportunity for student thinking – but this is not CT. Activities/
tasks requiring them to identify mathematical structures lead to a different, more creative, set of opportunities (Stein & Kaufman,
2010; Stein, Remillard, & Smith, 2007). In either case, students learn, but they engage differently with the learning contents. The

⋆
This paper includes two parts: a development of framework and implementation of the framework to primary school mathematics textbooks.
⁎
Corresponding author.
E-mail addresses: linor@a-hadar.co.il (L.L. Hadar), mor.no10@gmail.com (M. Tirosh).

https://doi.org/10.1016/j.tsc.2019.100585
Received 6 September 2018; Received in revised form 9 July 2019; Accepted 10 July 2019
Available online 16 July 2019
1871-1871/ © 2019 Elsevier Ltd. All rights reserved.
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

latter scenario supports CT; the former does not.

In this paper, we describe the construction of a framework to analyze how curriculum materials support CT in mathematics. The
framework classifies the types of activities/tasks based on their potential to stimulate CT. We apply the framework to primary school
mathematics textbooks in the context of the Israeli curriculum. The framework can be a useful tool to detect and quantify variations
in CT of different curriculum materials, with implications for the authorization of curriculum materials and textbook selection.

2. Theoretical background

2.1. Creative thinking

Creative thinking refers to the ability to generate novel ideas or solutions in a problem-solving process. This definition builds on
Guilford (1967) division of creativity into nine constructs: fluency, flexibility, novelty, synthesis, analysis, reorganization/redefini-
tion, complexity, and elaboration. Others have proposed variations of Guilford’s definition (e.g., Sternberg & Lubart, 1999; Torrance,
1988), and it or a variation of it is frequently used in CT research (e.g., Livne & Milgram, 2006; Tabach & Friedlander, 2017).
One area of research concerns the cognitive processes involved in CT (Kurtzberg & Amabile, 2001). This research asks what
individuals do in their minds when they think creatively. One answer comes from the notion of human insight (Perkins, 2013).
Researchers (e.g., De Bono, 1991; Sternberg & Lubart, 1999) propose creative thinkers employ lateral thinking, in which thoughts
flexibly leap from one aspect to another rather than simply following preexisting paths. Lateral thinking requires the ability to think
associatively, employing different perspectives in thinking outside the box to create novel ideas. Divergent thinking (De Bono, 1991) is
a second answer to the question about what individuals do when they think creatively. In divergent thinking, individuals generate
multiple solutions to a problem that does not have a right or wrong answer (Volle, 2017). A third answer is Sternberg and Lubart
(1999) description of convergent-integrative thinking in which individuals identify key elements of a problem, figure out how the pieces
fit together. Convergent-integrative thinkers see new relationships, combine different ideas, determine patterns, and form new links
between formerly disparate entities (Sill, 1996).

2.2. Creative thinking in mathematics

The importance of engaging students with CT is widely recognized (Hwang, Chen, Dung, & Yang, 2007; Kwon, Park, & Park,
2006). Therefore, the educational goals of many countries promote CT to develop the next generation of innovators. For example, the
new curriculum in Singapore includes a specific focus on “inventive thinking” as a national goal for education (Ministry of Education
Singapore, 2016). British Columbia, Canada, relates to creative and critical thinking as one of the three core competencies on the new
curriculum (The Professional Learning Association, 2017).
Mathematics is no exception. The need to foster CT is stressed in research and appears in policy documents in many countries
(Tabach & Friedlander, 2017). For example, the Israeli national syllabus for elementary school mathematics recommends students be
involved in problem-solving activities that require “thinking and creativity” (Ministry of Education, pedagogic secretariat, depart-
ment of curriculum, 2006). This curricular goal is hard to interpret and implement, as the processes involved in CT in mathematics
are not defined in a generally accepted way (Tabach & Friedlander, 2017).
Ervynck (1991) suggests it includes the ability to formulate mathematical objectives and find their innate relationships. Runco
(1993) relates it to divergent and convergent thinking, problem finding, problem solving, seeing new relationships, and making
associations between techniques, ideas and areas of application. Haylock (1987) proposes that it involves seeing new relationships
between techniques and areas of application and making associations between possibly unrelated ideas. Silver (1997) characterizes
CT by mental flexibility, curiosity, a well-developed imagination, interest in finding solutions, the creation of metaphors, and goal-
oriented thinking. Balka (1974) addresses convergent thinking, characterized by determining patterns and breaking from established
mindsets, and divergent thinking, characterized by formulating mathematical hypotheses, evaluating unusual mathematical ideas,
sensing what is missing from a problem, and splitting general problems into specific subproblems. Similarly, Lev-Zamir (2014)
proposes that CT in mathematics requires forming hypotheses, proving and convincing, building an argument, and justifying
mathematical ideas. Several researchers have applied the concepts of fluency, flexibility, and originality to mathematics CT (Kim,
Cho, & Ahn, 2004; Leikin, 2009).
The multiple definitions of CT processes in mathematics can be classified into the three general dimensions. Divergent thinking in
mathematics requires thinking of multiple ways to solve problems (Campbell, 1997), suggesting additional solutions, and applying
mathematical ideas in different contexts and in varied ways (Owen-Wilson, 2016). Convergent-integrative thinking requires integrating
ideas to determine mathematical patterns or structures (Barnes, 2006), connecting mathematical ideas with other areas as the basis
for new mathematical understandings, and connecting mathematical ideas with wider contexts (Aizikovitsh-Udi & Star, 2011). Lateral
thinking requires creating and exploring associative innovative ways to solve problems and suggesting new mathematical under-
standings rather than following preexisting patterns (Stopel & Oxman, 1996; Miman, 2016).
CT is a critical component of advanced mathematical thinking, as it promotes innovative strategies (Ervynck, 1991). CT goes
beyond mathematics; it also applies to the ability to deal with complex problems and changing circumstances more generally
(Miman, 2016; Stopel & Oxman, 1996).

2
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

2.3. Creative thinking in mathematics curriculum materials

Encouraging CT is essential if children are to develop a deep conceptual understanding of mathematics (Mann, 2006; Sheffield,
2013), but CT is difficult to develop. Teachers cannot directly teach students to create new and unique solutions (Sarrazy & Novotná,
2013). However, they can create situations in which students think creatively (Švecová, Rumanová, & Pavlovičová, 2014). CT
requires a stimulus (Ulfah, Prabawanto, & Jupri, 2017), a well-chosen activity in which students engage while they are learning
(Kwon et al., 2006), one that requires thinking away from pre-established ideas and generating unusual (original) ideas (Volle,
2017).Therefore, students’ use of CT depends on the learning situation (Sarrazy & Novotná, 2013). A possible support for teachers in
this process is by incorporating CT activities in curriculum materials (Zohar, 2008). Teachers’ choice of a specific curriculum material
can influence what students learn, how they learn, and the cognitive level at which they learn (Grouws et al., 2013; Stein et al.,
2007). To encourage CT, teachers need to be aware of the kinds of materials, activities/tasks that support CT.
The first goal of this paper is to develop a framework for the analysis of CT in primary school math curriculum materials.
Specifically, we ask: What kinds of CT opportunities can be provided in mathematics curriculum materials? And what are the
manifestations of CT in these materials? The second goal is the application of the framework to primary school (grade 1–6)
mathematics textbooks in the context of the Israeli curriculum. We ask what curricular trends and emphases appear when we apply
the framework.

3. The development of a framework for the analysis of CT in primary school mathematics curriculum materials

The framework’s development began with a systematic literature search (Wolgemuth, Hicks, & Agosto, 2017). We conducted
searches using Science Direct, Google scholar and EBSCO host. Within EBSCO host, we searched Academic Search Elite, E-journals,
ERIC, and PsycINFO. We used several cross-searches for the term CT combined with mathematics. As our focus was the opportunity
provided in mathematics curriculum materials, we included only papers addressing the ways CT can be promoted, stimulated or
supported by mathematics activities/tasks/problems. We excluded publications referring to the measurement of students’ CT abil-
ities, or teachers’ views on creativity. The search resulted in 103 publications.
At a second stage we read the publications to identify definitions, activities/tasks, examples, and explanations of CT relevant to
the analysis of mathematics curriculum materials. We used grounded theory methodology (Corbin & Strauss, 2015) to guide the
process of identifying definitions of CT in mathematics curriculum materials. Grounded theory is considered a suitable approach to a
synthesis of studies (Dixon-Woods, Agarwal, Jones, Young, & Sutton, 2005). We treated all publications as data. This diversity of
voices provided the basis for our framework. This methodology involved a bottom up approach that inductively identifies categories
based on open coding (Corbin & Strauss, 2015). The phase entailed reading, coding, and categorizing based on a constant comparison
method until we arrived at a stable and comprehensive set of nine categories capturing the phenomenon. These include:

1 Tasks requiring alternative solutions for a given solution: Certain activities/tasks ask students to find an alternative solution to a
problem for which one solution is presented. For example: The raspberry juice preparation instructions suggest a 1:6 ratio for
raspberry concentrate and water. John prepared one jar. He poured 1 glass of raspberry concentrate and 6 glasses of water.
Suggest an alternative option for preparing the raspberry juice so that the ratio will stay the same (e.g. Mann, 2006; Owen-Wilson,
2016).
2 Tasks with more than one solution: Activities/tasks are open-ended, have more than one solution. Students are asked to give only
one solution. For example, they are asked which two numbers add up to ten and to provide one solution (e.g. Fatah, Suryadi,
Sabandar, & Turmudi, 2016; Leikin & Lev, 2012; Shriki, 2015; Williams, 2002).
3 Tasks requiring more than one pathway to be solved: Students are explicitly required to solve a mathematical problem in more
than one way, sometimes followed by a request to interact and explain their approach. Unlike the activities in above, the solution
is not provided, and the task it is not open to different solutions (e.g., Cook, 2001; Jorlen, 2013; Leikin, 2009; Owen-Wilson,
2016). An example is: 30 chocolate bars cost 10 cents. How much will you spend if you buy chocolate bars for 35 children in your
classroom? How many ways can you solve the problem?
4 Tasks taking mathematical knowledge outside mathematics: These activities/tasks ask students to use mathematical knowledge in
a context outside mathematics. For example: Step outside the classroom and collect items in the shape of a polygon (e.g., Lev-
Zamir, 2014; Patkin & Gazit, 2012; Shriki, 2015; Švecová et al., 2014).
5 Tasks requiring identification and implementation of mathematical principles: Students are asked to identify a mathematical
principal, pattern or structure, to follow it, and to provide additional examples. This category includes systematizing numerical
results while searching for mathematical patterns or the formulation of mathematical arguments to explain discovered patterns.
For example: Which number follows the series 2,2, 4,4, 6,6, 2,2, 4,4, 6,6, 8,8, 2,2, 4,4 6,6, 8,8_____? (e.g., Aizikovitsh-Udi & Amit,
2011; Presseisen, 2001; Williams, 2002)
6 Tasks requiring connections between mathematical ideas: In these activities/tasks, students are asked to connect several math-
ematical ideas, identify inherent relationships, integrate concepts, create original concepts, or to interconnect various re-
presentations, operations, and assumptions through comparison. For example: Which geometric shape will you be able to build
from the number of sides you get if you multiply the number of a cube’s sides by the number of a triangular pyramid’s sides and
divide it by the number of an octagon’s sides? (e.g., Aizikovitsh-Udi & Amit, 2011; Sriraman, 2009; Williams, 2002).
7 Tasks using mathematical procedures to solve problems from other contexts: This category includes authentic everyday activities
that should be solved using mathematical tools or procedures. Students are directed to use mathematics in daily life and to apply

3
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

math knowledge to solve real-life problems. For example: When it is born, a snake has a length of 7 cm. It sheds its skin 3 times
each year. With each shedding, it grows 1 cm. I found a 25-cm-long snake. How old is this snake? (e.g., Aizikovitsh-Udi & Amit,
2011; Barnes, 2006; Jorlen, 2013; Shriki, 2015).
8 Tasks requiring the posing of mathematical problems: In these activities/tasks, students are asked to create their own mathe-
matical problem, to create a mathematical story and think how it can be solved. For example, students are presented with a
collection of pictures taken outside the classroom and representing different geometrical shapes. They are asked to create ideas for
geometrical tasks based on the photos (e.g. Aizikovitsh-Udi & Star, 2011; Patkin & Gazit, 2012; Švecová et al., 2014; Ulfah et al.,
2017; Zohar, 2009).
9 Tasks requiring the exploration of mathematical ideas: These activities/tasks stem from mathematical discussions. Students use
associative and innovative ways to explore and solve them, develop new understandings, and not follow preexisting patterns. For
example: Here are two girls. You need to find out who is younger. What questions do you need to ask? (e.g. Jorlen, 2013; Owen-
Wilson, 2016; Sriraman, 2009).

The third stage in the framework's development included interviews with six experts in mathematics education: two professors of
mathematics education, two primary school mathematics supervisors, and two leader-teachers in primary school mathematics. We
asked: What is CT in mathematics? What are some examples of CT in mathematics? What are the criteria to explore CT in mathe-
matics curriculum materials?
We used grounded theory methodology to analyze the interviews (Corbin & Strauss, 2015). The open coding phase involved a
dual procedure: in a bottom-up procedure, the interviews informed the construction of the initial categories; in a top-down proce-
dure, the literature review provided scaffolds for potential categories (Charmaz, 2006). Specifically, we categorized the aspects we
found in the open coding phase into seven of the nine categories from the literature review. This analysis led to a refined definition of
each category. Table 1 provides excerpts from the interviews.
The fourth step included quality testing (Stylianides, 2009). We evaluated whether and to what extent each category and its
activities/tasks can be applied to mathematics CT. We selected 12 experts in mathematics teaching: three professors of mathematics
education, two primary-school mathematics supervisors, a certified mathematics teacher-instructor, and six experienced primary-
school mathematics teachers.
We created a questionnaire (supplement Table S1) which included: a list of the nine categories of CT in mathematics curriculum
materials and a representative activity in each category. Using a 5-point scale (1= not at all; 5= very much), each expert rated the
extent to which each category was an expression of CT in mathematics curriculum materials. We also asked the experts to suggest
additional categories not in the questionnaire. For each category, we calculated the agreement mean and Fleiss-kappa to determine
the measure of agreement among evaluators (Fleiss, 1971). Both showed substantial agreement for each category (Table 2).
Four experts suggested additional categories: critique mathematical conventions; search for new kinds of mathematics; engage
with questions without a solution; conceptualize mathematical ideas. Based on our literature review, critiquing mathematical con-
ventions and conceptualizing mathematical ideas are in the category- identify and implement mathematical principles (Presseisen,
2001; Zohar, 2009). Search for new mathematics and engage with questions without a solution are in the category- explore math-
ematical ideas (Cook, 2001; Lev-Zamir, 2014; Shriki, 2015). Accordingly, we refined the definition of these two categories to include
these suggestions for additional activities.
We now had our final list of categories, definitions of the activities/tasks in each category, and examples of activities. Based on the

Table 1
Categories and excerpts from the interviews.
Category Excerpts
(N)

Provide alternative solution for a given solution.

(0)
Tasks with more than one solution. “Discussing different possibilities and different solutions”.
(3) “Open questions enable children to provide varied opinions”.
Use more than one pathway to solve the task. “There are many pathways to solve a given task which are acceptable and should be
(4) acknowledged”.
“The ability to think of different and new pathways to solve a task”.
Take mathematical knowledge outside mathematics. (0)
Identify and implement mathematical principles. “Sorting and making generalizations to understand mathematical rules or structures”.
(2)
Find connections between mathematical ideas. (2) “While thinking we connect between mathematical ideas”.
“Interdisciplinarity between mathematics and the other contexts beyond mathematics”.
Use mathematical procedures to solve problems from other “Flexible thinking, connecting mathematics to everyday living”.
contexts. (3)
Pose mathematical problems. (2) “Creating mathematical discussions. The students can create their own problems and
create their own data”.
Explore mathematical ideas. (3) “Discover mathematical ideas; they create new mathematical ideas”.
“We are dealing with an open subject, and it enables us to come up with new
mathematical ideas”.

4
 Table 2
Final list of categories, quality measures, definitions, and examples.
Theme Category Agreement Possible manifestation of the category Task example
means
(SD)
Fleiss-kappa
L.L. Hadar and M. Tirosh

Divergent thinking Provide alternative solution for a 4.5 Provide a different solution for a solved The first two numbers of a sequence are shown. Shari and David were asked to add the
given solution (0.52) task. next three numbers to the sequence:
.91 2, 4, __, __, __
Shari: Here is my sequence 2,4,8,16,32
David: Here is my sequence 2,4,5,10,11
a. Explain the rule Shari used to determine the numbers she added to the sequence.
Explain David’s rule. Using Shari’s and David’s rules, add three additional numbers to each
of their sequences.
b. Choose another rule that can be used to determine the numbers in the original sequence
and add three numbers to the sequence. Describe your rule.
Perform tasks with more than one 4.5 Provide a correct solution (there is more Sharon has an arithmetic game that includes a game board and cards numbered from one
solution (0.79) than one). to six. She places her cards in the squares on the board. Rebecca looks at the board and
.91 Choose a correct solution between a says to Sharon, “You placed the cards on the board so that the sum of the three numbers on
couple of possible solutions. each diagonal is nine.” How might Sharon have placed her cards on the board? (note:
Choose a correct way to solve a task there is more than one possibility).
(there are a few correct ways).

5
Use more than one pathway to 4.75 Solve a mathematical problem in several There are 10 seats for children on each minibus.
solve tasks. (0.45) ways. A group of children are waiting together for minibuses to take them to school. The first
.95 Discuss students’ approach to solve the minibus is empty when it gets to the bus stop, but there is not enough room for all the
problem. children. Two children must wait for the next minibus.
How many children were originally at the bus stop? Provide more than one possibility.
Take mathematical knowledge 4.8 Use mathematical knowledge in other Take photographs of buildings and other things that remind you of three-dimensional
outside mathematics. (0.79) fields. geometric shapes we have studied.
.96 Collect mathematical knowledge outside Use your photographs to make a poster for a class geometry exhibition.
mathematics. Search for sources of Find three-digit numbers in today's newspaper and explain what they are used for.
information. Now that we know how to calculate a square meter, measure the size of your room and
Apply mathematical algorithms in the report its square meter.
immediate environment.
Collect data from the environment and
draw mathematical conclusions.
Convergent-integrative Identify and implement 4.0 Identify a mathematical principle, In this lab, we’ll use the applet shown in the picture. The applet can be used to create
thinking mathematical principles. (0.90) pattern, or structure. sequences, shapes and expressions.
.66 Follow a newly identified mathematical The first three elements of a sequence of shapes are shown in the picture.
pattern. a Add the fourth and fifth shapes to the sequence.
b Predict the number of squares in the seventh shape.
c Write a general expression for the number of squares in the "n" -th shape. Check if your
expression is correct.
d Check your prediction from part b. using the expression you found in part c.
(continued on next page)
Thinking Skills and Creativity 33 (2019) 100585
 Table 2 (continued)

Theme Category Agreement Possible manifestation of the category Task example

means
(SD)
Fleiss-kappa
L.L. Hadar and M. Tirosh

e Choose an “empty sequence” and try to create your own sequence that fits with the
expression you wrote.

Find connections between 4.6 Connect several mathematical ideas. A grid of geometric shapes is shown below. Each shape corresponds to a rational number.
mathematical ideas. (0.51) Identify inherent relationships. The sum of each row is shown to the right of the grid. The sum of each column is shown
.93 Integrate concepts. below the grid. Find the number that corresponds to each shape.
Create original concepts.
Make associations between techniques.
Make associations between areas of
application.

Use mathematical procedures to 4.5 Apply math knowledge to solve real-life The length of a honeybee is approximately 1.5 cm. Below is an enlargement of a
solve problems from other (0.90) problems. photograph of a honeybee.
contexts. .93 Cope with a problem from daily life Measure the length of the honeybee shown in the photo and record your answer here. ——
which requires a mathematical process. (Use the segment as a guide).
What is the scale of the photograph? Explain your reasoning.

6
Lateral thinking Pose mathematical problems. 4.8 Create a mathematical story. Write a word problem that can be modeled by this equation.
(1.08) Build an original mathematical problem. 9 + 5=____
.96
Lateral thinking Explore mathematical ideas. 4.6 Mathematical discussion leads to new Examine the number grid and describe some things about it that you find interesting.
(0.93) understandings.
.95 Raise mathematical ideas.
Evaluate unusual mathematical ideas.
Explore questions without solutions.
Formulate mathematical hypotheses.
Build an argument and justify
mathematical ideas.
Thinking Skills and Creativity 33 (2019) 100585
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

theoretical definition of CT overarching themes we were able to slot the final categories under these themes (Table 2).
The CT framework gives insights on two levels: the nine categories shed light on the different CT activities/tasks and the three
overarching themes provide a more general overview of the CT trends in mathematics curriculum materials.

4. Application of the framework to curriculum materials

The second goal of this paper was to apply the framework to Israeli primary school (grade 1–6) mathematics textbooks. This
application provides validation to the frameworks' ability to analyze mathematical tasks and reveals (sometime hidden) trends in
opportunities to learn CT in mathematics.
Israel follows a centralized curriculum. Curricula in all subjects are tied to a list of state-authorized textbooks; school decision
makers adopt textbooks from this list. In primary school mathematics, five textbook series encompass all the formal mathematical
contents and address all students. Mathematics classes are not differentiated by level, classes are heterogeneous, and all students
follow the same curriculum. The goal of the math curriculum at the primary level is for students to learn basic concepts and structures
in the numbers and geometry domains, as well as develop mathematical skills and abilities, such as number sense and geometric
insight, computational skills, the ability to use mathematical tools to solve word problems, and conceptual understanding and
knowledge of mathematical language. Attaining mathematical concepts is considered a cumulative process dependent on students’
ability to grasp mathematical concepts and link them to other school subjects and the real world (TIMMS & PIRLS, 2015).
We analyzed the most popular textbook series used by approximately 48% of the primary schools in Israel. We based our decision
on a survey of textbook usage (Polikoff, 2015). The textbook series is authorized and follows the state syllabus of mathematics
content for grades one to six. The textbooks are organized in sections by subject. Each section begins with definitions and worked
examples and concludes with multiple opportunities (i.e. exercises) for students to engage with practice problems. Exercises are
generally grouped into sets requiring the same performance for successful completion. Jones and Tarr (2007) define such sets as a
textbook-task (henceforth task), that is, an activity or an exercise written with the intent of focusing a student’s attention on a
particular idea. Following their characterization, an exercise providing a set of questions building on one another is a single task (see
Table 3, task example in category 1). An exercise providing a single problem is considered a task (see Table 3, task example in
category 2). The textbooks also offer more challenging tasks, which are specifically marked in the textbooks as "challenge-tasks”.
These are considered appropriate for advanced students. Overall, we analyzed 6,267 tasks (Table 3)

4.1. Procedures

Using a single task as our unit of analysis, we applied the CT framework and used qualitative coding to identify the types of CT
required to complete each task in the textbooks. Reliability procedure was performed before coding the entire tasks corpus. We used
the SPSS statistical software to randomly choose 120 tasks. This sample included tasks from all nine categories, as well as tasks that
did not address any of the categories. Both researchers coded each of these tasks. From these 120 tasks, the SPSS software randomly
constructed nine combination sets of 10 tasks each. Nine raters (five mathematic teachers, one mathematics supervisor, two pro-
fessors of mathematics education) were each assigned one combination set and coded the tasks according to the framework. Each set,
also included a list and a definition of the categories, examples of tasks in each category, and instructions on how to rate each task
(e.g., supplement Table S2). The Fleis-kappa inter-rater reliability measure was computed to determine the agreement between the
raters and the researchers' coding of the same task. We obtained Fleis-kappa scores between .91 and 1.00 for each task.
Reaching high reliability scores, one researcher coded the remaining tasks. Each task was coded by indicating which of the nine
CT categories were required to complete it. One line in the SPSS software was assigned to each task. Each column represented one CT
category. A task received a score of one when a specific CT category was identified, and a score of zero was given if it was not. Each
task was mapped to a string of zeros and ones corresponding to the nine CT categories. While coding, we noticed that many tasks
provide an opportunity to engage in more than one CT category. For example, in some tasks, students are first asked to solve a
mathematical problem in more than one way and then to write a word problem that models one of their solutions. Therefore, these
types of tasks received a score of one in more than one column. We identified the type of CT category required to complete each task
and the complexity of CT, i.e., the number of different CT categories identified in the task, creating a category complexity scale of 0-9.
Additional columns in the software were used to code the grade level addressed by the task and whether it was a regular or a
challenge-task.
In addition, more general CT scores were constructed for the three overarching themes: divergent thinking, lateral thinking,
convergent-integrative thinking. Each task received a score of zero or one in each of the themes. For example, a task addressing one

Table 3
Number of tasks analyzed by grade and level.
Total Sixth Fifth Fourth Third Second First Grade
level
Task level

5,903 1,139 1,198 716 761 989 817 Ordinary tasks

364 59 98 36 56 60 55 Challenge-tasks
6,267 1,198 1,296 797 1,055 1,049 872 Total

7
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

Table 4
Framework and study variables.
Level Variable/measure Definition Scale

Organizing theme Theme type Type of CT theme (lateral, divergent, convergent-integrative) 0=not required in the task
required to complete the task 1=required to complete the task
Theme complexity Number of CT themes identified in the task. scale of 0-3
0=none of the CT themes are required to
complete the task
to
3=all CT themes are required to complete the
task
Individual categories Category type Type of CT category required to complete the task 0=not required in the task
1=required to complete the task
Category complexity Number and types of different CT categories identified in the scale of 0-9
task. 0=none of the CT categories are required to
complete the task
to
9=all of the CT categories are required to
complete the task
General Grade level Grade level addressed by task 1-6
General Task level Ordinary or challenge-task

(or more) of the categories included in divergent thinking received a score of one in this theme. The CT theme type of each task was the
type of CT theme (lateral, divergent, convergent-integrative) required to complete the task. The CT theme complexity score was the
number of CT themes identified in the task. This created a theme complexity scale from zero (the task did not refer to this theme) to
three (the task provided opportunity for students to engage with all three). Table 4 outlines the variables.
We used descriptive statistics to identify trends in CT of the tasks in the textbooks, within and between grade levels.

4.2. Findings

4.2.1. Overview and category type across and within grade and task level
We computed the percentage of tasks with references to at least one category of CT across grade levels: 24.9% of all tasks
(n = 1563) give students the opportunity to engage with at least one of the CT categories; 75% (n = 4704) do not. We also computed
this percentage within grade levels (Fig. 1).
Students in the first grade have the most opportunity to engage with CT when using the mathematics textbooks (39.4% of the
grade level textbook tasks, n = 344). The opportunity drops in the second grade (23.10%, n = 242) and through the third and fourth
grades (20.7%, n = 218,165), with a slight increase in the fifth (22.3%, n = 289) and sixth (25.5%, n = 305). Despite this increase,
there is still a 14% difference between the grade one and six textbooks. Further, 23.8% (n = 1404) of the ordinary tasks and 43.7%
(n = 159) of the challenge tasks give students the opportunity to engage in CT in mathematics.
Next, we computed the incidence of each CT category as percent of the total tasks in the textbook across grade levels (Fig. 2)
The most prominent CT category across grade levels is use mathematical procedures to solve problems from other contexts (8.7%,
n = 545). The second is use more than one pathway to solve problems (4.10%, n = 260) and the third is perform tasks with more
than one solution (3.9%, n = 245). Virtually no tasks require an alternative solution for a given solution or ask students to pose
mathematical problems.

Fig. 1. Inclusion of CT tasks in textbooks for grades one to six.

8
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

Fig. 2. Incidence of CT categories across grade levels.

We computed the same percentage within each grade level (Table 5).
The analysis shows that first grade students have more opportunities to engage with CT than older students in most categories.
Sixth-grade students have slightly more opportunities in certain categories: pose mathematical problems, use mathematical proce-
dures to solve problems from other contexts, and find connections between mathematical ideas. The most dominant category in all
grade levels, especially in the sixth grade (13% of tasks), is use mathematical procedures to solve problems from other contexts. This
reflects the Israeli curriculum’s emphasis on the “connection of the numbers to the environment and other content areas” (2006:12).
Further, the idea of more than one answer is emphasized in the first grade and de-emphasized in later grades. The category
explore mathematical ideas is also much more prominent in the first grade (4%); it is almost absent in the sixth grade (0.3%). Finally,
the idea that mathematics is something that students should explore appears only in the very early stages of mathematical learning.
The analysis of the task levels shows that in most categories, CT tasks are more prominent in the challenge-tasks than in the
ordinary tasks (Fig. 3).
The most prominent category across grade levels - use mathematical procedures to solve problems from other contexts – is also
more prominent in challenge-tasks (14.8% of challenge tasks, n = 54), than ordinary tasks (8.3%, n = 491). This large disparity is
also apparent in the category perform tasks with more than one solution and in the category use more than one pathway to solve
problems. The former appears in 11% (n = 40) of the challenge-tasks but only 3.5% (n = 205) of the ordinary tasks; the latter
appears in 9.6% (n = 35) of the challenge-tasks and 3.8% (n = 225) of the ordinary tasks. The gap between the opportunity to engage
with CT in ordinary and challenge-tasks is much smaller in the other categories.

Table 5
Incidence CT categories within grade levels.
Category Grade Grade Grade Grade Grade Grade
one two three four five six
(n) (n) (n) (n) (n) (n)

Provide alternative solution for a given solution. 0.2% 0.0% 0.3% 0.4% 0.4% 0.6%
(2) (0) (0) (3) (5) (2)
Perform tasks with more than one solution. 9.5% 3.1% 3.6% 3.1% 3.1% 2.8%
(82) (32) (32) (25) (40) (34)
Use more than one pathway to solve problems. 6.8% 4.5% 4.8% 4.1% 3.4% 2.2%
(59) (47) (51) (33) (44) (26)
Take mathematical knowledge outside mathematics. 1.9% 1.0% 0.8% 0.5% 1.1% 0.8%
(17) (10) (8) (4) (14) (9)
Identify and implement mathematical principles. 5.4% 3.2% 3.2% 1.9% 2.5% 1.9%
(47) (34) (34) (15) (32) (23)
Find connections between mathematical ideas. 3.6% 2.4% 1.1% 1.4% 1.7% 3.8%
(31) (25) (12) (11) (23) (46)
Use mathematical procedures to solve problems from other contexts. 10.2% 6.8% 5.5% 7.3% 8.5% 13.3%
(89) (71) (58) (58) (110) (159)
Pose mathematical problems. 0.8% 1.0% 0.8% 0.5% 0.6% 0.5%
(7) (10) (8) (4) (8) (6)
Explore mathematical ideas. 4.0% 1.5% 1.5% 2.1% 1.9% 0.3%
(35) (16) (16) (17) (24) (4)
% of tasks relating to CT across categories* 39.4% 23.1% 20.7% 20.7% 22.3% 25.5%

* Percentages of individual categories do not necessarily sum to the total % of tasks in each grade level because of the multiple scoring system.

9
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

Fig. 3. Incidence of CT categories in challenge and ordinary tasks.

4.2.2. Category complexity across and within grade and task level
As noted, 75.1% (n = 4707) of the tasks do not require any CT category to complete them; 24.1% (n = 1510) tasks require one CT
category, 0.7% (n = 47) of the tasks require two, and 0.1% (n = 4) require three. The two most frequent combinations are: first, use
math procedures to solve problems from other contexts, combined with perform tasks with more than one solution (20% of the 47
tasks requiring two CT, n = 9); second, perform tasks with more than one solution, combined with use more than one pathway to
solve problems (20%, n = 9). The next is the combination of use math procedures to solve problems from other contexts and use more
than one pathway to solve problems (12,7%, n = 6). The rest of the combinations are sporadic. The combination of three categories is
also sporadic, but three out of four possible combinations include the categories perform tasks with more than one solution and
explore math ideas.
The analysis of category complexity within grade and task levels shows that 46.8% (n = 22) of the tasks requiring two categories
to complete them appear in grade one, 6.4% (n = 3) in grade two, 8.5% (n = 4) in grade three, 10.6% (n = 5) in grade four, 21.3%
(n = 10) in grade five, and 6.4% (n = 3) in grade six. Further, 80.9% of these tasks (n = 38) are ordinary and 19.1% (n = 9) are
challenge. These findings suggest a lack of intentional thinking about the different CT opportunities in the tasks, either for grade or
task level.

Fig. 4. Incidence of CT Themes Within Grade Levels.

10
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

4.2.3. Theme type and complexity across and within grade and task level
Across grade levels, the most prominent CT theme is convergent-integrative thinking (13.9%, n = 870). The second is divergent
thinking (9.1%, n = 571), and the third is lateral thinking (2.5%, n = 155). The distribution within grade levels is similar (Fig. 4). In
each grade level, the most frequent theme is convergent-integrative thinking, following by divergent thinking; fewer tasks feature
lateral thinking.
The distribution of ordinary and challenge-tasks in each theme shows a predominance of the latter: 20.3% (n = 74) of the
challenge-tasks require convergent-integrative thinking to complete, compared to 13.5% (n = 796) of the ordinary tasks; 21.4%
(n = 78) of the challenge-tasks require convergent thinking, compared to 8.4% (n = 493) of the ordinary tasks. The distribution of
challenge and ordinary tasks for lateral thinking is more equal: 3.3% (n = 12) of the former require lateral thinking, compared to
2.4% (n = 143) of the latter.
Analysis of theme complexity shows that 24.4% (n = 1529) require one CT theme to be completed. Only 0.5% (n = 32) require
two CT themes, and 0.01% (n = 1) require three. We also find that in 68.8% of the tasks requiring more than one category, students
are also required to apply different creative themes (32 cross-themes tasks / 47 cross-category tasks). In 65% (n = 21) of these cross-
themes tasks, students are required to use convergent-integrative thinking together with divergent thinking to complete them. In
28.1% (n = 9) of the tasks, students must use divergent thinking together with lateral thinking, and in 0.06% (n = 2) of them, they
must use convergent-integrative thinking together with lateral thinking.
The analysis of theme complexity within grade and task levels shows 53% (n = 17) of the tasks requiring two themes appear in
grade one, 9.4% (n = 3) in grade two, 12.5% (n = 4) in grade three, 9.4% (n = 3) in grade four, 12.5% (n = 5) in grade five, and
3.1% (n = 1) in grade six. Further, 84.4% of these tasks (n = 27) are ordinary and 15.6% (n = 5) are challenge-tasks.

5. Discussion

This paper introduces a framework for the analysis of CT in mathematics curriculum materials and applies this framework to
Israeli primary school mathematics textbooks. The application of the framework to the analysis of mathematics textbooks serve two
purposes (a) it provides validation for the applicability of the framework to the analysis of mathematical tasks and (b) it exposes open
or hidden perceptions of CT in primary school mathematics textbooks. These, in turn, may greatly influence student's opportunity to
learn mathematics.
Current approaches promote the teaching of CT as part of the cognitive domain of learning mathematics (Aizikovitsh-Udi & Amit,
2011). Various taxonomies have been used to analyze the cognitive domain in curriculum materials. For example, the TIMMS
framework distinguishes between levels of understanding (Grønmo, Lindquist, Arora, & Mullis, 2015), while PISA ranks cognitive
demand in a low-to-high hierarchy (Jones & Tarr, 2007; Porter, 2006; Wijaya, van den Heuvel-Panhuizen, & Doorman, 2015). The
QUASAR (Stein, Smith, Henningsen, & Silver, 2016), and Surveys of Enacted Curriculum (Porter, 2006) are frameworks analyzing the
level and general category of the cognitive domain, and the SEC framework has been adapted to characterize the level of cognitive
demand of textbook tasks (Polikoff, Zhou, & Campbell, 2015). None of the frameworks specifically addresses CT in mathematics.
When CT is considered an essential 21-century skill, the educational community should be able to evaluate how curriculum attend to
this goal. To our knowledge, this paper is the first to create a taxonomy for the analysis of CT in mathematical curriculum materials. It
encourages specific attention to CT as part of the cognitive domain of learning mathematics, as well as to how curriculum promotes
these skills.
Indeed, curricula documents in many countries include explicit CT learning goals (Gallagher et al., 2012). For example, the Israeli
mathematics curriculum states that “the development of mathematical understanding deals with… [the] understanding of different
ways of solving a problem… and with linking numbers to the environment and other areas of knowledge” (2006:11). The types of CT
mentioned in the Israeli curriculum include divergent thinking, convergent-integrative thinking, and lateral thinking (2006: 11). The
document outlines seven core mathematical competencies required of students in the primary grades; three of the seven include an
emphasis on CT.
In an era of standardization, one of the strongest influences on the successful implementation of standards is the quality and
alignment of the curriculum materials used by teachers (Polikoff, 2015). The application of the framework to the analysis of primary
school mathematic textbooks shows that although the three CT themes are covered by the textbooks, they are not aligned with the
curriculum in several areas. First, the textbooks provide much more opportunity to engage with CT in mathematics in grade one than
in any other grade, but the curriculum does not differentiate between grades. The textbooks, but not the curriculum, convey the
message that CT is more suited for younger children, and older children should engage in different kind of mathematics.
Second, the textbooks have their own emphases that do not necessarily correspond with the emphasis in the curriculum. For
example, while the mathematics curriculum specifically requires “the learner to create new mathematical ideas” (2006:11), the
corresponding categories (pose mathematical problems, explore mathematical ideas) have very minor emphases in the textbooks. Of
course, some CT objectives may merit more attention than others. Our findings raise the question of whether teachers, publishers, or
administrators should decide what to emphasize or if they should rely on textbooks. In our view, unless teachers are made aware of
the shortcomings of existing curriculum materials and address them by supplementing them, their teaching may be weaker than
desired. The framework promotes awareness and provides teachers a way to identify CT in mathematics.
Third, the textbooks give advanced students more opportunities to engage with CT (as exemplified in the challenge-tasks) but the
curriculum sets the same CT objectives for all students. This tendency of the textbooks backs up research showing that teachers
believe high-advantage pupils should receive instruction with a greater focus on thinking activities while low-advantage pupils
should receive more didactic instruction (e.g., Torff & Sessions, 2006). This is a worrying trend, as studies on actual student

11
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

achievements show that while stronger students perform better than weaker students, students at all levels improve their thinking
skills when instruction is focused on this goal (Zohar & Dori, 2003; Hadar & Genser, 2015).
One of the limitations of the study is that it applies to one set of textbooks. However, this should be considered a case study, that
shows the applicability of the framework to the analysis of curriculum materials, and the kinds of trends that it might expose. The
framework should be applied to other textbooks in different countries. It is also important to extend the analysis into the classroom to
understand how curriculum materials influence teachers’ instructional decisions.
The characterization of CT offered by the framework in this paper suggests the potential for and the limitations of mathematics
curriculum materials in terms of the opportunities to learn various types of cognitively demanding mathematics. For teachers, the
framework can be used as a compass to show how specific activities encourage students' CT. For decision-makers, it offers a tool to
examine and assess curriculum materials. We claim that there is a need for more detailed information on the quality of curriculum
materials (Polikoff, 2015; Hadar, 2017; Hadar & Lefcourt Ruby, 2019), opening the door to future research.

Appendix A. Supplementary data

Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.tsc.2019.
100585.

References

Aizikovitsh-Udi, E., & Amit, M. (2011). Developing the skills of critical and creative thinking by probability teaching. Procedia-Social and Behavioral Sciences, 15,
1087–1091.
Aizikovitsh-Udi, E., & Star, J. (2011). The skill of asking good questions in mathematics teaching. Procedia-Social and Behavioral Sciences, 15, 1354–1358.
Balka, D. S. (1974). Creative ability in mathematics. Arithmetic Teacher, 21(7), 633–636.
Barnes, M. K. (2006). How many days’ til my birthday? Helping kindergarten students understand calendar connections and concepts. Teaching Children Mathematic,
12(6), 290–295.
Campbell, P. F. (1997). Connecting instructional practice to student thinking. Teaching Children Mathematics, 4(2), 106–110.
Charmaz, K. (2006). Constructing grounded theory: A practical guide through qualitative analysis. London: Sage.
Cook, M. (2001). Mathematic: The thinking arena for problem solving. In A. Costa (Ed.). Developing minds. A source book for teaching thinking (pp. 286–291). Alexandria,
VA: ASCD.
De Bono, E. (1991). Lateral and vertical thinking. In J. Henry (Ed.). Creative management (pp. 16–23). London: Sage.
Dixon-Woods, M., Agarwal, S., Jones, D., Young, B., & Sutton, A. (2005). Synthesising qualitative and quantitative evidence: A review of possible methods. Journal of
Health Services Research & Policy, 10(1), 45–53.
Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23(2), 167–180.
Dreyfus, T., & Eisenberg, T. (1996). On different facets of mathematical thinking. In R. J. Sternberg, & T. Ben-Zeev (Eds.). The nature of mathematical thinking (pp. 253–
284). Mahwah, NJ: Lawrence Erlbaum.
Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.). Advanced mathematical thinking (pp. 42–53). Dordrecht: Kluwer Academic.
Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: Development status and directions. ZDM Mathematics Education, 45(5), 633–646.
Fatah, A., Suryadi, D., Sabandar, J., & Turmudi (2016). Open-ended approach: An effort in cultivating students’ mathematical creative thinking ability and self-esteem
in mathematics. Journal on Mathematics Education, 7(1), 11–20.
Fleiss, J. L. (1971). Measuring nominal scale agreement among many raters. Psychological Bulletin, 76(5), 378.
Gallagher, C., Hipkins, R., & Zohar, A. (2012). Positioning thinking within national curriculum and assessment systems: Perspectives from Israel, New Zealand and
Northern Ireland. Thinking Skills and Creativity, 7(2), 134–143.
Corbin, J., & Strauss, A. (2015). Basics of qualitative research: Techniques and procedures for developing grounded theory (4th ed.). Thousand Oaks, CA: Sage.
Grønmo, L., Lindquist, M., Arora, A., & Mullis, I. (2015). TIMSS 2015 mathematics framework. Retrieved fromhttps://timssandpirls.bc.edu/timss2015/downloads/T15_
FW_Chap1.pdf.
Grouws, D. A., Tarr, J. E., Chávez, Ó., Sears, R., Soria, V. M., & Taylan, R. D. (2013). Curriculum and implementation effects on high school students’ mathematics
learning from curricula representing subject-specific and integrated content organizations. Journal for Research in Mathematics Education, 44(2), 416–463.
Guilford, J. P. (1967). The nature of human intelligence. New York: McGraw-Hill.
Hadar, L. L. (2017). Opportunities to learn: mathematics textbooks and students’ achievements. Studies in Educational Evaluation, 55, 153–166.
Hadar, L. L., & Genser, L. (2015). Promoting critical thinking for all ability levels in an online English as a second language course. International Journal of Computer-
Assisted Language Learning and Teaching (IJCALLT), 5(1), 71–88.
Hadar, L. L., & Lefcourt Ruby, T. (2019). Cognitive opportunities in textbooks: The cases of grade four and eight textbooks in Israel. Mathematical thinking and learning,
21(1), 54–77.
Haylock, D. W. (1987). A framework for assessing mathematical creativity in school children. Educational Studies in Mathematics, 18(1), 59–74.
Houang, R. T., & Schmidt, W. H. (2008). TIMSS international curriculum analysis and measuring educational opportunities. 3rd IEA International Research Conference
Retrieved fromHttp://Www.Iea.Nl/Fileadmin/User_upload/IRC/IRC_2008/Papers/IRC2008_Houang_Schmidt.PdfTaipei, Chinese Taipei.
Hwang, W.-Y., Chen, N.-S., Dung, J.-J., & Yang, Y.-L. (2007). Multiple representation skills and creativity effects on mathematical problem solving using a multimedia
whiteboard system. Educational Technology & Society, 10(2), 191–212.
Jones, D. L., & Tarr, J. E. (2007). An examination of the levels of cognitive demand required by probability tasks in middle grades mathematics textbooks. Statistics
Education Research Journal, 6(2), 4–27.
Jorlen, A. (2013). Five types of creative thinking. Retrieved fromhttp://adamjorlen.com/2013/06/10/five-types-of-creative-thinking/.
Kim, H., Cho, S., & Ahn, D. (2004). Development of mathematical creative problem-solving ability test for identification of the gifted in math. Gifted Education
International, 18(2), 164–174.
Kurtzberg, T. R., & Amabile, T. M. (2001). From Guilford to creative synergy: Opening the black box of team-level creativity. Creativity Research Journal, 13(3-4),
285–294.
Kwon, O. N., Park, J. H., & Park, J. S. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Education Review, 7(1),
51–61.
Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. Creativity in Mathematics and the Education of Gifted Students, 9, 129–145.
Leikin, R., & Lev, M. (2012). Mathematical creativity in generally gifted and mathematically excelling adolescents: What makes the difference? Mathematics Education,
45(183-), 179.
Lev-Zamir, H. (2014). Creativity in mathematics teaching viewed from the teacher’s perspective. In A. Gazit, & D. Patkin (Eds.). The narrative of mathematics teachers’
elementary school: Features of education knowledge, teaching and personality (pp. 145–163). Tel Aviv: Mofet Institute [Hebrew].
Livne, N. L., & Milgram, R. M. (2006). Academic versus creative abilities in mathematics: Two components of the same construct? Creativity Research Journal, 18(2),

12
L.L. Hadar and M. Tirosh Thinking Skills and Creativity 33 (2019) 100585

199–212.
Mann, E. L. (2006). Creativity: The essence of mathematics. Journal for the Education of the Gifted, 30(2), 236–260.
Miman, A. (2016). Mathematics!? Why to teach it at all? Time for education arrives. [Hebrew]. Retrieved fromhttp://www.edunow.org.il/edunow-media-story-176807.
Ministry of Education Singapore (2016). 21st century competencies. Retrieved from:https://www.moe.gov.sg/education/education-system/21st-
centurycompetenciesRetrieved on January 23, 2018.
Ministry of Education, pedagogic secretariat, department of curriculum (2006). The new math curriculum. [Hebrew]Jerusalem, Israel: Author.
Owen-Wilson, L. (2016). The second principle. Retrieved fromhttp://thesecondprinciple.com/creativity/creativity-essentials/types-of-creative-thinking/.
Patkin, D., & Gazit, A. (2012). Math roots, power and numerical insights. Powerful Number, 2000(21), 22 [Hebrew].
Perkins, D. N. (2013). Knowledge as design. NY: Routledge.
Polikoff, M. S. (2015). How well aligned are textbooks to the common core standards in mathematics? American Educational Research Journal, 52(6), 1185–1211.
Polikoff, M. S., Zhou, N., & Campbell, S. E. (2015). Methodological choices in the content analysis of textbooks for measuring alignment with standards. Educational
Measurement Issues and Practice, 34(3), 10–17.
Porter, A. C. (2006). Curriculum assessment. In J. Green, G. Camilli, & P. Elmore (Eds.). Handbook of complementary methods in education research (pp. 141–159).
Mahwah, NJ: Lawrence Erlbaum Associates.
Presseisen, B. Z. (2001). Thinking skills: Meanings and models revisited. In A. Costa (Ed.). Developing minds: A source book for teaching thinking (pp. 47–53). Alexandria,
VA: ASCD.
Runco, M. A. (1993). Creativity as an educational objective for disadvantaged students. National Research Center on the Gifted and Talented, The University of
Connecticut.
Sarrazy, B., & Novotná, J. (2013). Didactical contract and responsiveness to didactical contract: A theoretical framework for enquiry into students’ creativity in
mathematics. ZDM Mathematics Education, 45(2), 281–293.
Sheffield, L. J. (2013). Creativity and school mathematics: Some modest observations. ZDM Mathematics Education, 45(2), 325–332.
Shriki, A. (2015). Creativity: A multi-faceted phenomenon. In A. Gazit, & D. Patkin (Eds.). Creative problems solving in mathematics: Strategies, dilemmas and mistakes (pp.
19–69). Tel Aviv: Mofet Institute [Hebrew].
Sill, D. J. (1996). Integrative thinking, synthesis, and creativity in interdisciplinary studies. The Journal of General Education, 45(2), 129–151.
Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM Mathematics Education, 29(3), 75–80.
Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM Mathematics Education, 41(1-2), 13–27.
Stein, M. K., & Kaufman, J. H. (2010). Selecting and supporting the use of mathematics curricula at scale. American Educational Research Journal, 47(3), 663–669.
Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. Lester (Ed.). Second handbook of research on mathematics teaching
and learning (pp. 319–370). Charlotte, NC: Information Age Publishing.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2016). Implementing standards-based math instruction: A casebook for professional development. NY: Teachers
College Press.
Sternberg, R. J., & Lubart, T. I. (1999). The concept of creativity: Prospects and paradigms. In R. J. Sternberg (Ed.). Handbook of creativityNew York: Cambridge
University Press (pp. 15-3).
Stopel, M., & Oxman, V. (1996). Geometric constructions as a tool for developing creative thinking. Shanan Annual Journal, 2, 95–108 [Hebrew].
Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11, 258–288.
Švecová, V., Rumanová, L., & Pavlovičová, G. (2014). Support of pupil’s creative thinking in mathematical education. Procedia - Social and Behavioral Sciences, 116,
1715.
Tabach, M., & Friedlander, A. (2017). Algebraic procedures and creative thinking. ZDM Mathematics Education, 49(1), 53.
Tarr, J. E., Chávez, Ó, Reys, R. E., & Reys, B. J. (2006). From the written to the enacted curricula: The intermediary role of middle school mathematics teachers in
shaping students’ opportunity to learn. School Science and Mathematics, 106(4), 191–201.
The Professional Learning Association (2017). The state of educators’ professional learning in British. Columbia. Oxford: OH.
TIMMS, & PIRLS (2015). TIMMS encyclopedia. TIMMS & PIRLS international study center Retrieved from: http://timssandpirls.bc.edu/timss2015/encyclopedia/
countries/israel/the-mathematics-curriculum-in-primary-and-lower-secondary-grades/ on July, 2019.
Torff, B., & Sessions, D. (2006). Issues influencing teachers’ beliefs about use of critical-thinking activities with low-advantage learners. Teacher Education Quarterly,
33(4), 77–91.
Torrance, E. P. (1988). The nature of creativity as manifest in its testing. NY: Cambridge University Press.
Ulfah, U., Prabawanto, S., & Jupri, A. (2017). Students’ mathematical creative thinking through problem posing learning. Journal of Physics Conference Series,
895(1), 1–7.
Volle, E. (2017). Associative and controlled cognition in divergent thinking: Theoretical, experimental, neuroimaging evidence, and new directions. In R. E. Jung, & O.
Vartanian (Eds.). The Cambridge handbook of the neuroscience of creativity (pp. 333–345). New York: Cambridge University Press.
Wijaya, A., van den Heuvel-Panhuizen, M., & Doorman, M. (2015). Opportunity-to-learn context-based tasks provided by mathematics textbooks. Educational Studies in
Mathematics, 89(1), 41–65.
Williams, G. (2002). Identifying tasks that promote creative thinking in mathematics: A tool. Mathematics Education in the South Pacific, 2, 698–705.
Wolgemuth, J. R., Hicks, T., & Agosto, V. (2017). Unpacking assumptions in research synthesis: A critical construct synthesis approach. Educational Researcher, 46(3),
131–139.
Zohar, A. (2008). Teaching thinking on a national scale: Israel’s pedagogical horizons. Thinking Skills and Creativity, 3(1), 77–81.
Zohar, A. (2009). High level thinking strategies –Document for curriculum planners and for developers of learning materials. Jerusalem: Ministry of Education [Hebrew].
Zohar, A., & Dori, Y. J. (2003). Higher order thinking skills and low-achieving students: Are they mutually exclusive? Journal of the Learning Sciences, 12(2), 145–181.

13