relationship between high mathematical ability and mathematical creativity in secondary school ...

115 Pages·2013·2.11 MB·English

RELATIONSHIP BETWEEN HIGH MATHEMATICAL
ABILITY AND MATHEMATICAL CREATIVITY IN
SECONDARY SCHOOL CHILDREN

RESEARCH REPORT

Prof. Roza Leikin, Miriam Lev

University of Haifa
Faculty of Education
Department of Mathematics Education

November, 2013
*This Research report is based on the PhD dissertation by Miriam Lev under supervision of
Prof. Roza Leikin.
The work is under evaluation by the external reviewers.
Please do not cite this works or parts of this work without permission of the authors.
AKNOWLEDGMENT
We are thankful to the Israeli Ministry of Education for its generous financial support to this
research project.

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TABLE OF CONTENT
ABSTRACT ................................................................................................................................................. iv
PART 1 RATIONALE ................................................................................................................. 1
PART 2 METHODOLOGY ......................................................................................................... 3
2.1 Research goals ................................................................................................................... 3
2.2 Phases of the study ............................................................................................................ 3
2.3 Research sample ................................................................................................................ 8
2.4 The quantitative part of the study.................................................................................... 10
2.5 The qualitative part of the study...................................................................................... 14
PART 3 PILOT STUDY ............................................................................................................ 16
3.1 Research goals ................................................................................................................. 16
3.2 Population ....................................................................................................................... 16
3.3 Test .................................................................................................................................. 17
3.4 Findings ........................................................................................................................... 19
3.5 Summary and discussion ................................................................................................. 27
PART 4 FINDINGS – MAIN STUDY....................................................................................... 30
CHAPTER 1 PROBLEM 1 ............................................................................................................... 31
1.1 The problem .................................................................................................................... 31
1.2 Groups of solutions to Problem 1 produced by the study participants ............................ 32
1.3 Differences in problem-solving performance
(correctness and creativity components) associated with
general giftedness and excellence in school mathematics............................................... 35
1.4 Differences in problem-solving performance
(correctness and creativity components) of students with
superior school mathematical performance..................................................................... 40
CHAPTER 2 PROBLEM 2 ............................................................................................................... 47
2.1 The problem .................................................................................................................... 47
2.2 Groups of solutions to Problem 2 produced by the study participants ............................ 48
2.3 Differences in problem-solving performance
(correctness and creativity components) associated with
general giftedness and excellence in school mathematics............................................... 50
2.4 Differences in problem-solving performance
(correctness and creativity components) of students
with superior school mathematical performance ........................................................... 55

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CHAPTER 3 PROBLEM 3 ............................................................................................................... 62
3.1 The problem .................................................................................................................... 62
3.2 Groups of solutions to Problem 3 produced by the study participants ............................ 63
3.3 Differences in problem-solving performance
(correctness and creativity components) associated
with general giftedness and excellence in school mathematics ...................................... 65
3.4 Differences in problem-solving performance
(correctness and creativity components) of students
with superior school mathematical performance ............................................................ 70
CHAPTER 4 PROBLEM 4 ............................................................................................................... 76
4.1 The problem .................................................................................................................... 76
4.2 Groups of Solutions to Problem 4 produced by the study participants ........................... 77
4.3 Differences in problem-solving performance
(correctness and creativity components) associated with general giftedness and
excellence in school mathematics ................................................................................... 78
4.4 Differences in problem-solving performance
(correctness and creativity components) of students with superior school
mathematical performance .............................................................................................. 83
CHAPTER 5 THE VOICE OF THE STUDENTS .......................................................................... 89
PART 5 SUMMARY AND DISCUSSION ............................................................................... 98
5.1 Construct validity of the tool........................................................................................... 98
5.2 Creativity in different groups of participants .................................................................. 99
5.3 Contribution .................................................................................................................. 106
5.4 Study Limitations .......................................................................................................... 106
5.5 References ..................................................................................................................... 107

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ABSTRACT
This study relates to two critical points in mathematical education in Israel and abroad which
are the need to increase the importance of mathematical creativity when teaching mathematics
and the necessity of developing tools to assess mathematical creativity that is needed to
promote and develop creativity among schoolchildren. The study is also motivated by the
observation that the issue of mathematical creativity as well as of creativity in mathematics
education is not well explored. Characterizing the relationship between mathematical
giftedness and mathematical creativity is still missing in the professional literature (e.g.,
Leikin, 2013; Sriraman, 2005). Since Krutetskii's (1976) seminal study of mathematical
abilities, only a small number of studies have been done on the characterization of the
mathematical abilities of gifted students which have left a large number of open questions
(Leikin, 2009, 2013). This research attempted to address this deficiency.
We employed Multiple Solution Tasks (i.e., tasks that explicitly require solving a
mathematical problem in different ways - MSTs) in order to explore students’ creativity in
mathematics (Leikin & Lev, 2007; Leikin, 2013) so we examined the relationship between
mathematical creativity, general giftedness and excellence in school mathematics. The study,
which is a part of multidimensional examination of mathematical giftedness (e.g. Leikin,
Leikin, Lev, Paz-Baruch & Waisman, 2013) , attempted to get better understanding of the
concept of mathematical giftedness from the point of view of mathematical creativity.
This study had three main interrelated Goals:
1. To examine relationships between mathematical creativity, general giftedness and
excellence in secondary school mathematics.
2. To explore creativity of adolescents with superior mathematical abilities.
3. To explore the power of different types of Multiple Solution Tasks (MSTs) for the
identification of between-group differences related to mathematical creativity.
The research sample consisted of 184 students (16-18 years old) divided into four major
experimental groups according to varying combinations of the levels of excellence in school
mathematics (EM factor) and of general giftedness (G factor) (see Section 2.3). Additionally
7 students (16-18 years old) with superior mathematical abilities (S-MG) took part in the
study.
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Data collection and analysis: The data was collected by means of written tests and
individual interviews.
The Test consisting of five MSTs – was designed for the purpose of this study (see
Sections 2.2.1 and 2.2.2). To evaluate students' creativity the study employed a model for
evaluation of mathematical creativity using MSTs (see Part 3 and Section 1.2.7; Leikin, 2009,
2013). The students' solutions to the Test were analyzed with 5 criteria (correctness of the
solutions, fluency, flexibility, originality and creativity). The quantitative analysis of the tests
was performed in 2 steps: Step A focused on the similarities and differences in the four major
groups of participants -- G-EM, NG-EM, G-NEM, and NG-NEM. Step B was directed
towards the identification of specific characteristics of S-MG participants. At Step A we used
MANOVA for G factor and EM factor with consequent ANOVAs and pair-wise comparisons
(G vs. NG in EM and NEM groups and EM vs. NEM in G and NG groups, separately). At
Step B a non-parametric (Kruskal-Wallis) test was performed to examine specific
characteristics of S-MG students (as compared to G-EM and NG-EM) with consequent Mann-
Whitney tests for pair-wise comparison between the groups.
The Interviews with students from the groups of excelling in mathematics students were
conducted in order to analyze the strategies they had employed in solving problems whose
solutions required insight. During the semi-structured interviews the students were asked to
think aloud when solving the problems in multiple ways. The interviews were videotaped,
transcribed and analyzed using inductive content analysis (see Section 2.5 and Chapter 5 in
Part 3).
Main findings and hypothesis:
Based on the findings of the current study we raise several research hypotheses, which we
plan to examine in a new research:
1. The G and EM factors are interrelated but different personal traits. This hypothesis is
based on the following findings: (a) The G and EM factors have different main effects
on performance on different mathematical tasks. (b) The G and EM factors interact on
the flexibility criterion. (c) Identical tasks reveal different effects of the G and EM
factors.
2. Following the above mentioned findings, we argue that while general giftedness is not a
necessary condition for excellence in school mathematics, excellence in school
mathematics is not necessarily an indicator of general giftedness.
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3. Combination of the G and EM factors are expressed in both qualitative and quantitative
characteristics of creative thinking.
4. We hypothesize that in the fluency-flexibility-originality triad, fluency and flexibility
are of developmental nature (EM-related), whereas originality is a "gift" (G-related).
5. Originality appeared to be the strongest component in determining creativity while the
G factor is strongly associated with differences related to solving insight-based tasks.
We suggest that even though mathematical ability seems to be a part of general giftedness
rather it is a specific ability, Mathematical giftedness seems to require general giftedness
and can be characterized by high mathematical creativity associated with originality of
mathematical thought which is closely related to mathematical insight.
The study contributes in three interrelated dimensions.
In the theoretical dimension the study is first that demonstrates -- based on systematic
empirical investigation -- that general giftedness and excellence in mathematics can be
characterized by different creativity-related phenomena. The study proposes a definition of
mathematical giftedness as a combination of mathematical creativity and general giftedness. It
is the first that demonstrates – on the secondary school level – that mathematical insight is a
specific characteristic of mathematical giftedness.
In the methodological dimension the study extends the use of the Model of mathematical
creativity with MSTs, validates it and provides broad community of mathematics educators
with ready-to-use tests for the identification of mathematical creativity in senior high school
students.
In the practical dimension the study provides additional support to equity principle of
mathematics education that argues that each student has to be provided with equal learning
opportunities that meet individual differences that focus on personal needs, abilities and
motivation.

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PART 1 INTRODUCTION
1.1 Rationale
This study relates to two critical points in mathematical education in Israel and abroad which
are the need to increase the importance of mathematical creativity when teaching mathematics
and the necessity of developing tools to assess mathematical creativity that is needed to
promote and develop creativity among schoolchildren. There are number of open questions in
the mathematics education literature about the connection between excellence in mathematics
and mathematical creativity; the relationship between general giftedness and mathematical
creativity; mathematical creativity as a part of general giftedness or mathematical creativity
and general giftedness as two independent characteristics. The study is also motivated by the
observation that the issue of mathematical creativity as well as of creativity in mathematics
education is not well explored. Mann (2006) maintained that analysis of the research
attempting to define mathematical creativity demonstrates that the absence of an accepted
definition for mathematical creativity hinders research efforts. Characterizing the relationship
between mathematical giftedness and mathematical creativity is still missing in the
professional literature (e.g., Leikin, 2013; Sriraman, 2005). Since Krutetskii's (1976) seminal
study of mathematical abilities, only a small number of studies have been done on the
characterization of the mathematical abilities of gifted students which have left a large
number of open questions (Leikin, 2009, 2013). This research attempted to address this
deficiency. It draws on the views of Ervynck (1991), Krutetskii (1976), Polya (1973), and
Silver (1997) who claimed that solving mathematical problems in multiple ways is closely
related to personal mathematical creativity. Solutions were analyzed in terms of their
flexibility, fluency and originality as suggested by Torrance (1974). The evaluation of
originality was by the creativity levels proposed by Ervynk (1991) combined with the
conventionality of the solution with respect to knowledge acquired by students during their
studies (Leikin, 2009, 2013).
This study employed Multiple Solution Tasks (i.e., tasks that explicitly require solving a
mathematical problem in different ways - MSTs) in order to explore students’ creativity in
mathematics (Leikin & Lev, 2007; Leikin, 2013). It continued a series of studies that
validated the research tool used herein (Leikin & Lev, 2007; Leikin, 2009, 2013; Levav-
Waynberg & Leikin, 2012a, 2012b; Guberman & Leikin, 2012; Leikin & Lev, 2013). These
studies also examined the relationship between mathematical creativity and the level of
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mathematical ability of the participants. All studies led to two main hypotheses that are
currently being examined in this large-scale study: (1) between-group differences are task
dependent and (2) in the originality-fluency-flexibility triad, fluency and flexibility are of a
dynamic nature, whereas originality is of the “gift” type.
My preliminary research (Lev, 2007 Master's Thesis under the guidance of R. Leikin)
investigated mathematical creativity with the use of problem solving in different ways and
revealed that there is a connection between giftedness and creativity expressed in
mathematical problem solving in different ways. The research was a qualitative study so that
the findings produced the hypotheses for this quantitative research and the analysis of the
findings showed that a change in the schema for assessing mathematical creativity was
needed. Leikin (2009, 2013) developed schema for assessing mathematical creativity with
MSTs based on Leikin & Lev (2007) and it was employed in this study.
The study (which is a part of multidimensional examination of mathematical giftedness,
e.g. Leikin, Leikin, Lev, Paz-Baruch and Waisman, 2013) attempted to get better
understanding of the concept of mathematical giftedness from the point of view of
mathematical creativity.

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PART 2 METHODOLOGY
2.1 Research goals
This study had three main interrelated goals:
1. To examine relationships between mathematical creativity, general giftedness (G factor)
and excellence in school mathematics (EM factor).
2. To explore creativity of adolescents with superior mathematical abilities.
3. To explore the power of different types of Multiple Solution Tasks (MSTs) for the
identification of between-group differences related to mathematical creativity.
2.2 Phases of the study

Designing
Research
the Research
sample
Tool

Using the
Research Pilot study

Tool

Categories Improving
and the Research

Solution Tool
Spaces

Evaluation -
Scoring
Scheme

Interviews

Figure 2.1 Phases of the study
An additional goal of the study was to build a reliable instrument for examining
mathematical creativity in high school students based on multiple solution tasks. Using this
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tool, the connection between general giftedness, mathematical excelling and mathematical
creativity was then examined.
2.2.1 The development of the research tool
The development process of the tool had a number of phases:
 Designing a pool of items for the research tool.
 Constructing three variants of tests from the pool of items.
 Piloting the tests in three types of groups: (1) classes for gifted children, (2) classes for
students that learn mathematics at the highest level (HL) and (3) at the regular level (RL).
 Collecting the expert solution spaces for each problem in order to implement Leikin's
(2009, 2013) scoring scheme that was proved to be efficient in several studies (Levav-
Waynberg & Leikin, 2009; Leikin, Levav-Waynberg & Guberman, 2011; Leikin & Kloss,
2011).
 Analyzing the data and choosing the final variant of the test (see Part 3).
Designing a pool of items for the research tool
A multiple-solution task (MST) is an assignment in which a student is explicitly asked to
solve a mathematical problem in different ways. For the pool of problems, we searched for
MSTs in literature that deals especially with problem solving (e.g., Chiu, 2009; Kwon, Park,
& Park, 2006).
Constructing variants of tests from the pool of items
Three variants of the test were designed in order to examine the possibility of varying the
problems. The problems chosen for each test displayed a range of solutions in the expert
solution space. Experts approved the problems by considering the diversity of their solutions.
Each class participating in this study was presented with at least two variants of the test, so
that students seated next to each other were given different problems to solve.
The problems in the test differed with respect to:
 Mathematics topic to which the problem belongs in the school curriculum.
 Complexity.
 Non-conventionality of the problem and of its solutions, thus requiring insight in order to
produce the solutions (following Ervynk 1991).
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