Research on and activities for mathematically gifted students

Singer, Florence Mihaela

Sheffield, Linda Jensen

Freiman, Viktor

Brandl, Matthias

2018

Springer

This Topical Survey offers a brief overview of the current state of research on and activities for mathematically gifted students around the world. This is of interest to a broad readership, including educational researchers, research mathematicians, mathematics teachers, teacher educators, curriculum designers, doctoral students, and other stakeholders. It first discusses research concerning the nature of mathematical giftedness, including theoretical frameworks and methodologies that are helpful in identifying and/or creating mathematically gifted students, which is described in this section. It also focuses on research on and the development of mathematical talent and innovation in students, including connections between cognitive, social and affective aspects of mathematically gifted students. Exemplary teaching and learning practices, curricula and a variety of programs that contribute to the development of mathematical talent, gifts, and passion are described as well as the pedagogy and mathematics content suitable for educating pre-service and in-service teachers of mathematically gifted students. The final section provides a brief summary of the paper along with suggestions for the research, activities, and resources that should be available to support mathematically gifted students and their teachers, parents, and other stakeholders.

Gifted children -- Education

Mathematics -- Study and teaching

Teaching

English

978-3-319-39450-3

978-3-319-39449-7

ICME-13 Topical Surveys

Florence Mihaela Singer
Linda Jensen Sheffield
Viktor Freiman
Matthias Brandl

Research On and
Activities For
Mathematically
Gifted Students

ICME-13 Topical Surveys
Series editor
Gabriele Kaiser, Faculty of Education, University of Hamburg, Hamburg, Germany

More information about this series at http://www.springer.com/series/14352

Florence Mihaela Singer
Linda Jensen Sheffield Viktor Freiman
Matthias Brandl
•

Research On and Activities
For Mathematically Gifted
Students

Viktor Freiman
Faculté des sciences de l’éducation
Université de Moncton
Moncton, NB
Canada

Florence Mihaela Singer
University of Ploiesti
Ploiesti
Romania
Linda Jensen Sheffield
Northern Kentucky University
Highland Heights, KY
USA

ISSN 2366-5947
ICME-13 Topical Surveys
ISBN 978-3-319-39449-7
DOI 10.1007/978-3-319-39450-3

Matthias Brandl
Didaktik der Mathematik
Universität Passau
Passau, Bayern
Germany

ISSN 2366-5955

(electronic)

ISBN 978-3-319-39450-3

(eBook)

Library of Congress Control Number: 2016940119
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Main Topics You Can Find in This
ICME-13 Topical Survey

• Nature of Mathematical Giftedness
• Mathematical Promise in Students of Various Ages
• Research into Practice: Pedagogy, Programs and Teacher Education

v

Contents

Research On and Activities For Mathematically Gifted Students .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Nature of Mathematical Giftedness . . . . . . . . . . . . . . . . . . . . .
2.1 What Is Mathematical Giftedness? . . . . . . . . . . . . . . . . .
2.2 A Discovery or a Creation?. . . . . . . . . . . . . . . . . . . . . .
2.3 What Theoretical Frameworks and Methodologies
Are Helpful? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Mathematical Promise in Students of Various Ages . . . . . . . . .
3.1 How Are Cognitive, Social, and Affective
Aspects Connected? . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 What Does Brain Research Say? . . . . . . . . . . . . . . . . . .
3.3 What Are the Differences Between Mathematical
Novices and Experts? . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 How Are Mathematical Creativity and Giftedness
Connected? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Research into Practice: Programs and Pedagogy . . . . . . . . . . . .
4.1 How Might Teaching Practices Affect Mathematical
Promise and Talents? . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 How Might Curriculum Contribute to Mathematical
Development? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 What In-School Programs Might Develop Mathematical
Talent? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 What Extra-Curricular Programs Might Enhance
Mathematical Promise? . . . . . . . . . . . . . . . . . . . . . . . . .
5 Research into Practice: Teacher Education . . . . . . . . . . . . . . . .
5.1 What Teacher’s Traits Are Important? . . . . . . . . . . . . . .
5.2 What Should Be Included in Teacher Education? . . . . . . .
5.3 What Are Some Examples of Programs for Supporting
Teachers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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vii

viii

Contents

6 Summary and Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34
35

Erratum to: Research On and Activities For Mathematically
Gifted Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E1

Research On and Activities
For Mathematically Gifted Students

1 Introduction
In 1980, An Agenda for Action: Recommendations for School Mathematics for the
1980s from the National Council of Teachers of Mathematics in the United States
noted that “The student most neglected, in terms of realizing full potential, is the
gifted student of mathematics. Outstanding mathematical ability is a precious
societal resource, sorely needed to maintain leadership in a technological world”
(NCTM 1980, p. 18). Over 35 years later, the world has certainly become more
technological. In this Topical Survey, we explore whether our gifted mathematics
students around the world are closer to realizing their full potential and suggest
strategies and needed research to make that happen.
There is a continuous debate around the conception of giftedness and its definition. Over the time and places, several terms are being used in the context of
gifted learners: mathematically gifted and talented, (highly) able, (intellectually)
precocious, bright, mathematically advanced, among many others. While we use
many of these terms in our survey, reflecting choices made by the researchers and
practitioners who have contributed to the field, the term ‘mathematically promising’
introduces by the NCTM Task Force in the mid-90s, seems to us the most
appropriate to grasp the complexity of the topic in its largest and broader sense.
Because the domain of mathematical giftedness is, as an interdisciplinary
domain, still under development, we organize our discourse based on a list of
questions to which we try to give evidence-based answers. Examples drawn from
longitudinal studies, micro-analyses of classroom interactions, various educational
programs and projects, along with findings from recent cognitive and neuroscience
studies that offer insights into how the mathematically promising mind works are
brought together to offer a synthesis of state of the art on research in the education

The original version of this chapter was revised: Incorrect author name has been corrected.
The erratum to this chapter is available at DOI 10.1007/978-3-319-39450-3_2
© The Editor(s) (if applicable) and The Author(s) 2018
F.M. Singer et al., Research On and Activities For Mathematically Gifted Students,
ICME-13 Topical Surveys, DOI 10.1007/978-3-319-39450-3_1

1

2

Research On and Activities For Mathematically Gifted Students

of the gifted. We hope that this approach will stimulate a constructive debate and
will lead to new international research and development in this emerging field.

2 Nature of Mathematical Giftedness
There is a common truism to say that there is no single definition of mathematical
giftedness. For this reason, we do not give much place to present theoretical definitions, we just restrain to a minimum needed for understanding the topic; we
mostly focus on results from empirical research. Different actors involved in the
educational systems directly or indirectly, as teachers, parents, researchers, or students have different views on giftedness. We explore these different views through
various lenses, and the first question we address in this respect is:

2.1

What Is Mathematical Giftedness?

To answer, we start with a few words about general giftedness, and then we focus
on what is specific to mathematics.
2.1.1

General Giftedness

There are numerous definitions of a gifted child. Some emphasize the child’s current level of achievement based on an overlap and interaction among three clusters
of traits: above average ability, task commitment, and creativity (Renzulli 1986);
whereas for others, the key is the child’s potential to perform at a level significantly
beyond age-peers (e.g. Gagné 2003). According to Gagné (2003), personal characteristics such as motivation and temperament, as well as environment and the
interplay between these aspects and innate giftedness, play an important role in the
development of talent. Thus, certain traits should be evident in potentially gifted
young children, while others are developed through instruction. Therefore, we are
talking about a hidden potential with a genetic component of disposition on the one
side, versus observable performance or achievement or expertise on the other side.
Thus, the talent development process is a progressive transformation of gifts into
talents.
There exists now extensive literature on the identification of gifted children. In
terms of cognitive behaviors, a fast pace of learning, exceptional memory, extended
concentration span, ability to understand complex concepts, enhanced observational
ability, curiosity, and an advanced sense of humor should be apparent (e.g. Harrison
2003). Most research in giftedness has so far concentrated on intellectual and
academic aspects. However, high intellect and creativity are frequently accompanied by personality factors that impact the life of gifted children, such as: advanced
moral judgment; heightened self-awareness; heightened sensitivity to the expectations and feelings of others; perfectionism; introversion; high expectations of self

2 Nature of Mathematical Giftedness

3

and others; idealism and a sense of justice; and higher levels of emotional depth and
intensity (e.g. Winner 2000; Clark 2002).

2.1.2

Mathematical Giftedness

Mathematical giftedness is sometimes seen as a specific part or kind of giftedness.
However, narrowing the focus from the look on general giftedness to the subset of
mathematical giftedness does not necessarily imply taking only into account a subset
of the items describing general giftedness. When discussing mathematical giftedness,
many complexities occur and some domain-specific aspects are to be emphasized.
According to Krutetskii, “mathematical giftedness” is the name given to a
unique aggregate of mathematical abilities that opens up the possibility of successful performance in mathematical activity (Krutetskii 1976, p. 77). His comprehensive investigation of mathematical ability was designed to explore the nature
and structure of mathematical abilities over a 12-year period. He defined ability as a
personal trait that enables one to perform a given task rapidly and well, and contrasts this to a habit or skill, which relates to the qualities or features of the activity a
person is carrying out. Krutetskii (1976) uses the term ‘mathematical cast of mind’
to describe the mathematically gifted students’ tendency to view the world through
a mathematical lens. That means that gifted and talented mathematics students have,
among other capacities, the ability for rapid and broad generalization of mathematical relations and operations, and flexibility of mental processes.
Given the domain-specificity of mathematical giftedness, it always implies a
collection of certain mathematical abilities and personal qualities. However, looking at the subject at different times and within different cultural contexts we get
changing definitions corresponding to the “law of cultural differentiation” (Irvine
and Berry 1988). Furthermore, talking about mathematical giftedness is unavoidably tied to talking about mathematics. In complex-system theoretical terms, the
open and viable mental construct “mathematical giftedness” is connected to its
environmental system “mathematics” consisting both of the mathematical truths
(theorems, definitions, axioms, etc.) and the individual researchers making these
up. Now, different notions of mathematics determine different concepts of mathematical giftedness, or as Freudenthal put it, “The definition of mathematics varies.
Each generation and each subtle mathematician within each generation formulates a
definition that corresponds to his or her skills and insights.” (quoted in Käpnick
1998, p. 53). Hence, predominant philosophical notions about mathematics influence a conception of mathematical giftedness, which means there is a structural
connection between mathematical giftedness and mathematics itself.
Lists representing necessary aspects of mathematical giftedness, generally split
into abilities specific to mathematics on the one hand (such as: mathematical sensibility, exceptional memory, rapid content mastery and structuring, atypical
problem solution, preference for abstraction, interest and enjoyment of mathematics, success in identifying patterns and relationships, lengthy concentration span,
generalizing and reversion of mathematical processes) and general personality traits

4

Research On and Activities For Mathematically Gifted Students

on the other (intellectual curiosity, willingness of exertion, joy and interest in
problem solving, perseverance and frustration tolerance, ability to engage in
independent self-directed activities, and affinity for challenging tasks).
Giftedness then can manifest as school giftedness and creative productive giftedness. The first is manifested in the facility to take standardized tests and acquire
knowledge, while the second refers to the ability to create new products or processes.
In addition, the term good student might be used to describe a high achiever who is
not gifted and who is often focused on pleasing teachers or parents (e.g. Brandl
2011). Being gifted in mathematics does not necessarily lead to high attainment in
this subject while high attainment in mathematics does not necessarily mean being
mathematically gifted (e.g. Szabo 2015; Brandl and Barthel 2012; Brandl 2011;
Öystein 2011). Hong and Akui (2004) offer a similar distinction when they introduce
the constructs of academically gifted and creatively talented students. The ones in the
first category are those performing well in “school” mathematics, the high achievers,
while the other ones are highly interested and active individuals, yet they are not high
achievers. It seems that the category doing both is less visible in school settings.

2.2

A Discovery or a Creation?

Differing definitions of mathematics as a domain of knowledge, on the one hand, and
mathematically gifted, on the other, may lead to different responses to this question.
We will approach both aspects, showing some strengths and weaknesses of each
perspective. Research focusing the question if mathematical giftedness is a discovery
or a creation can have a major impact in improving the teaching strategies that
address the development of mathematical abilities in students of various ages.

2.2.1

Mathematical Giftedness as a Discovery

There is a largely accepted assumption that giftedness might equal intelligence that
is measurable by IQ-tests. From here comes the belief that about 2 % of people can
be seen as (highly) gifted. However, there are various problems related to considering IQ tests as unique reference for identifying giftedness. To list just a few
critiques: the IQ-score is not stable or constant over time but changes in connection
to the further development of the tested person. In addition, there is not a single
IQ-test, but different IQ-tests that collect different data and make different statements, as they are designed for specific norm groups. More than one hundred years
ago, Binet (1909), who is often acknowledged as the inventor of the modern
intelligence test, protested against what he termed the brutal pessimism of
philosophers who asserted that an individual’s intelligence is a fixed quantity, a
quantity that cannot be increased. Binet felt that with practice and training one
might become more intelligent than he/she was before. He claimed that not always
the people who start out the smartest would end up the smartest.

2 Nature of Mathematical Giftedness

5

Features of mathematical ability at an early age. Research literature in this area
describes various early signs of mathematical giftedness in children (e.g. Diezmann
and Watters 2000). In a 2-year study that examined 15 mathematically gifted and
talented students aged from 10 to 13 years, Bicknell (2008) characterizes mathematical giftedness through the eyes of parents, students, and teachers. Most of the
parents recognized their child’s abilities in mathematics at an early age. The parents’ descriptions of their children at pre-school age give an image of what might be
seen as innate abilities in these children. The characteristics identified by parents
include “impressive concentration” and the ability to work independently for a
relatively long period of time on a particular task. Young children no older than
2–3 years were self-initiating games involving numbers and numerical patterns,
showing a real fascination for numbers and how they behave in mathematics
operations. The types of activities the parents observed in their children at an early
age indicated an interest in mathematically driven games, such as: building with
construction blocks, creating symmetrical patterns, ordering objects, and completing puzzles and jigsaws in unconventional ways, spending hours of concentration in
such activities. Others made connections between ballet movements and angles in
geometrical rotations, or have shown a relatively sound concept of number and in
some cases an interest in concepts such as time and space (Bicknell 2008).
But not all young potentially gifted children show these signs of mathematical
giftedness. In a questionnaire given to more than 100 parents of mathematically gifted
students most of the parents’ answers support these findings; however, some pointed
out that they did not recognize signs of giftedness before their child began school
(Nolte 2012). Moreover, descriptions of mathematical precocity in 2- or 3-year old
children should not necessarily lead to the idea that their mathematical abilities were
innate. These could have been due to parental and other environmental factors.
Features of mathematical ability at the school level. Once gifted children began
school, their level of interest and ability in mathematics compared to their peers
became more apparent. The teachers observed in these children the different pace of
mathematics learning, an intuitive mathematical knowledge in problem solving,
their keen interest in mathematics, the sense of humor and ability to think in more
abstract terms than their age peers, as well as more mental flexibility and a discourse based on logical thinking. Perseverance and excitement with mathematical
problems were also observed. Other aspects of mathematics that, according to the
students, confirmed their mathematical giftedness include success in competitions;
competence with basic mathematical facts; speed of computational skills; problem
solving abilities; capacity to work on ‘special projects’, or on more/different work
(than their classmates) to complete independently (e.g. Bicknell 2008; Subotnik
et al. 2012). The conclusions should be nuanced however, because not all of the
students identified as mathematically gifted would categorize themselves as gifted,
and they even would not rate mathematics as their favorite subject.
Sometimes students’ abilities manifest quite differently across mathematical
domains and both students and teachers have recognized these differences. For
example, some students are stronger in visual patterns, and transformational geometry involving rotation or translation, while some others have good mathematical

6

Research On and Activities For Mathematically Gifted Students

computational skills. However, this might not be directly related to mathematical
giftedness, but to different cognitive styles. In a study investigating the relationship
between three ability-based cognitive styles (verbal deductive, spatial imagery, and
object imagery) and performance on geometry problems that provided different types
of clues, Anderson et al. (2008) found both spatial imagery and verbal reasoning
cognitive styles were helpful in solving some types of geometry problems, (but not
object imagery, which has been found to relate more with art creativity).
The existence of a specific individual giftedness potential is not sufficient for
high performance as the phenomenon of “underachiever” shows. Formal or informal learning provides a means of transforming this potential into talents or systematically trained abilities (achievement). Nevertheless, some researchers claim
this potential is necessary for excellent results in assessment (e.g. Heller and Ziegler
2007; Bicknell 2008).
Features of mathematical ability at the university level. Research on university
gifted students is quite limited. A possible explanation is that gifted students might
have learned to hide their giftedness and, thus, it might be difficult to identify them
at the university level (Albon and Jewels 2008). A different explanation might be
that IQs do not remain stable over time and the fact that the interplay between
interests, activities, environment, and mathematical explorations affect students’
mathematical achievement leads to question whether there is a need to distinguish
between giftedness and expertise as students enter the university level and beyond.
One area that has been studied is that of students entering college or university at
a younger age compared to their colleagues. A survey of empirical research shows
that, in general, early entrants earn higher grade point averages than regular students, are more likely to graduate, and are likely to earn other academic honors and
pursue graduate studies (Olszewski-Kubilius 2013).
Adult mathematical achievement. Those gifted children most likely to develop
their talent to the level of an expert will be those who have high drive and the ability
to focus and derive flow from their work, those who grow up in families that
combine stimulation with support; and those who are fortunate to have inspiring
teachers, mentors and role models. Those gifted children often discover their talent
in adulthood when they are catalyzed by a crystallizing experience, a life-changing
event in which a gift is discovered and self-doubts are dispelled (e.g. Winner 2000).

2.2.2

Mathematical Giftedness as a Creation

What does influence the development of abilities and performance? Are some
children born with special characteristics that allow them to become mathematically
gifted, or is mathematical talent and expertise something that can be developed or
created in the large majority of both male and female students from all ethnic and
socio-economic groups regardless of traits inherited at birth? A term that gains more
terrain is mathematical promise, which has been developed by the National Council
of Teachers of Mathematics (NCTM) as a function of maximizing variables such as
abilities, motivation, beliefs, and experiences or opportunities (Sheffield et al.

2 Nature of Mathematical Giftedness

7

1999). NCTM (1995) suggested using the term promising rather than gifted, purposely broadening the definition to include a much greater range of students and to
open the possibility of creating students with outstanding mathematical abilities and
not simply identifying students with mathematical pre-existing expertise and passion (Sheffield et al. 1999). Gagné (e.g. 2003) talks about having “gifts” as a
prerequisite for developing talents while others, including those that favor the
definition of mathematical promise do not.
Dweck (2006) has shown that middle grades students who believe in a “fixed”
mindset, that is a belief that they are born with certain “fixed” abilities, do more poorly
in learning mathematics than those who believe in a “growth” mindset, understanding
that their brain changes and develops. Those last students were more challenged to
learn and more successful in middle grade mathematics. This is true for gifted students
who believe that they have a math brain as well as those who believe that they do not.
In discussing the “myth of the mathematically gifted child”, Boaler (2015) makes a
powerful case for the harm caused by this idea of genetic determinism and teaching
mathematics as a subject that is used to separate children into those who have the math
gene and those who don’t. Instead, she calls for teaching mathematics as a lens to view
the world that is available to all students through study and hard work.
Furthermore, it holds that students who are interested in mathematics will be
more likely to develop mathematical talents. Kruteskii (1976) also stressed the
necessity of an interest in mathematics in order to be successful in this subject, and
“if the teacher is able to awaken his interest in it and his inclination to study it, that
pupil ‘carried away’ by mathematics can quickly achieve great success” (Kruteskii
1976, p. 347). Some preliminary results from a look into different fostering settings
of mathematically gifted students also indicate that the “perspective of mathematics
of somebody who is interested in mathematics differs essentially from the one
averaged over an “ordinary” class; it seems also to be more positive than that one
averaged over a “high-attaining” class.” (Brandl 2014, p. 1162). On the contrary, by
choosing subgroups according to typical characteristics of mathematical giftedness
from Tall’s (2008) formal-axiomatic-world (e.g. deep interest in mathematics,
inclination for the beauty of mathematics, loving to play around within elements of
mathematics voluntarily) out of a sample of very high attaining students, significant
correlations between the different subgroups’ fields of mathematical interests have
been found (Brandl and Barthel 2012). Additionally, the members of these subgroups who manifested special interests represented the top part of the sample,
considering their marks in mathematics.

2.3

What Theoretical Frameworks and Methodologies
Are Helpful?

Development models for mathematical giftedness are, in some cases, inspired by
development models for general giftedness such as Gagné’s Differentiated Model of

8

Research On and Activities For Mathematically Gifted Students

Giftedness and Talent (Gagné 2009), Renzulli’s tripartite model, in which giftedness is the product of three interacting clusters of traits: above average intellectual
ability, high levels of creativity and high levels of task commitment (Renzulli
1986), Ziegler`s Actiotope Model of Giftedness (Ziegler 2005) or Heller’s Munich
Model of Giftedness (e.g. Heller and Ziegler 2007). Dai (2010) scrutinized some of
the held assumptions about the nature of giftedness and explained why a contextual,
developmental framework of approaching giftedness is a more viable alternative to
the traditional psychometric framework.
As other researchers in this area (e.g. Leikin 2011), Szabo (2015) claims that, in
the last decade, only a few studies are focused on analyzing traits of mathematically
gifted and high achieving students in a conceptual perspective. Similarly, very few
studies analyze the connection between those students’ biological and cognitive
capacities and their mathematical performance. More research is needed on theoretical frameworks or models for explaining mathematical giftedness and promise.
In the meanwhile, from a pragmatic view, in some countries, various institutions,
NGOs or communities of parents or teachers included, started to develop general
frameworks for designing differentiated learning experiences for gifted students,
which have the role to complement the official standards and benchmarks.

3 Mathematical Promise in Students of Various Ages
As noted earlier, not all high achievers are gifted and not all gifted children are high
achievers. Just as students with identified gifts may not be high performers, high
attaining students do not need to be (highly) gifted (e.g. Brandl and Barthel 2012;
Öystein 2011). For example, depending on the school environment, students might
receive very good grades in examinations, but examination tasks may be only
aimed at computational or algorithmic abilities and not demanding any type of
non-routine or innovative approach. This may not mean that the successful student
is a gifted or a talented future mathematician.
In her recent book, Boaler (2015) explains that teaching needs to reflect the new
science of the brain and communicate that everyone has the potential to learn
mathematics effectively, not only those believed to hold a “gift”. From the same
perspective, some authors try to identify profiles or traits of mathematically
promising students (Budak 2012; Trinter et al. 2015), focus the identification of
such students (Vilkomir and O’Donoghue 2009), or discuss how to foster the
promise of high achieving mathematics students (Hoeflinger 1998; Zmood 2014).
We use the concept of mathematically promising students in order to cover a large
span of baseline abilities, backgrounds, and environments as well as to capture the
vision that most students have capabilities that, with adequate training, can help
them reach much higher levels of mathematics performance.

3 Mathematical Promise in Students of Various Ages

3.1

9

How Are Cognitive, Social, and Affective Aspects
Connected?

Traditionally, the identification of gifted students has been linked to intelligence
tests and consequently, the identification of mathematically gifted students has
often built on the general giftedness identification. Many teachers use checklists of
typical traits to identify these students. Still, others focus on the development of
mathematical talent rather than its identification.

3.1.1

Cognitive Indicators

As noted earlier, there are difficulties in assuming that giftedness equals intelligence
as measured by IQ tests. Subsequently, researchers have tried to describe the
complex features that could define domain-specific giftedness.
Following up on Kruteskii’s work, many authors and research teams developed
lists of cognitive characteristics of a gifted child (e.g. Diezmann and Watters 2000).
An example of an age-group-specific list is the one from Käpnick (1998) who
investigated primary school children’s characteristics of giftedness: remembering
mathematical facts; structuring mathematical facts; mathematical sensitivity and
mathematical fantasy; transferring mathematical structures; intermodal transfer; and
reversing lines of thought. Similarly, but more detailed aspects taken from a
three-dimensional mathematical giftedness pattern represent the guideline for a
process-based identification of mathematically gifted third and fourth graders
during regular lessons, in Winkler and Brandl (2016). Assmus (2016) tested
Käpnick’s items with second graders and found the following characteristics of
mathematical giftedness in early primary school children: ability to memorize
mathematical issues by drawing on identified structures, ability to construct and use
mathematical structures, ability to switch between modes of representation, ability
to reverse lines of thought, ability to capture complex structures and work with
them, ability to construct and use mathematical analogies, mathematical sensitivity,
and mathematical creativity.
Other indicators of mathematical giftedness may include: unusual curiosity
about numbers and mathematical information; ability to understand and rapidly
apply mathematical concepts; high ability to identify patterns and to think
abstractly; flexibility and creativity in approaching problem solving; ability to
transfer mathematical concepts to an unfamiliar situation; persistency and resilience
in solving challenging problems (Stepanek 1999). Sriraman (2005) focused on
mathematical processes through which various authors define mathematical giftedness at K-12 level. These processes include, among others, the ability to: abstract,
generalize, and discern mathematical structures; manage data; master principles of
logical thinking and inference; think analogically and heuristically; visualize
problems and/or relations; distinguish between empirical and theoretical principles;
think recursively. However, researchers note that these indicators should not be

10

Research On and Activities For Mathematically Gifted Students

used as rules for qualifying students as being mathematically gifted. Not every
mathematically gifted student will display all these characteristics, or they may
emerge at different times depending on the student’s development. Much of identifying gifted students relies on ongoing assessments and teacher observations, as
well as the level of problems against which students show these traits (Nolte 2012).

3.1.2

Social and Affective Indicators

A combination of internal and situational factors may cause problems for mathematically promising students. Among the issues that may affect them are their
asynchronous development, their socialization problems, and their own problems
with self-learning.
Asynchronous Development in Gifted Children. A common issue with gifted
children is asynchronous development, i.e. non-uniform development through the
intellectual, emotional, social, and physical domains. For example, a student whose
mathematical abilities are far beyond other areas of development may have difficulties in adaptation to situational contexts. Suggestions for programs for these
students will be explored in the next section.
Need for Social Acceptance. In the absence of understanding and support, highly
gifted children may find themselves very different from their age mates and may
face negative reactions in situations where conformity is valued. Many learn to
mask their abilities in order to relieve their social problems (Gross 2003), especially
in areas where mathematical expertise is not valued. This could hinder the further
development of their unique abilities, and lead to a loss of self-esteem. However, a
large body of literature has found gifted children to be superior not only intellectually, but also physically, emotionally, and socially (e.g. Cross et al. 2008).
Problems with Self-Learning. Gifted children are often inclined to learn things
on their own, and are tempted to solve by novel methods problems that may be
beyond their current abilities, introducing large amounts of error and frustration.
Unassisted, such children may down-regulate their ambitions, develop a fear of
making mistakes, and reduce productive risk-taking behaviors (Freehill 1961).
Twice-Exceptional Children are those who possess giftedness or exceptional
ability in one or more areas in combination with special needs, a learning disability
or other handicap in other areas. They may achieve high scores on certain intelligence tests but may not do well in school. These children represent a category
among the gifted that is especially at risk without knowledgeable intervention.
Self-esteem issues are often disproportionately high in children with learning disabilities or with notable asynchronous development, as they tend to judge themselves by what they cannot do rather than by what they can. This problem is
relieved somewhat by sharing with them assessments of their abilities so that they
develop more appropriate levels of self-esteem (e.g. Nolte 2013; Nordheimer and
Brandl 2016).

3 Mathematical Promise in Students of Various Ages

3.2

11

What Does Brain Research Say?

Mathematical giftedness started to be conceptualized in the recent decades within a
context that is sensitive to modern biology. The findings from educational neuroscience help understanding how gifted students might better be taught and helped to
plenarily develop.
The application of neuroscience research to mathematically gifted students is
somewhat controversial, however, and a deeper analysis is needed to allow drawing
more detailed and specific conclusions. Reviewing studies of patient calculation
based on magnetic resonance imaging, Dehaene et al. (2003) have proposed that
specific regions of the brain play distinct functional roles in arithmetic. Several
mathematical concepts, including number sense, are constructed based on
spatial-numerical mapping (briefly, associating numbers with positions on a line). It
was assumed that this association is a basic cornerstone for arithmetic skills, but
recently, Cipora et al. (2015) concluded that the relationship between
spatial-numerical associations and arithmetic skills are rather weak or caused by
mediating variables. Nevertheless, interventions based on relationships between
space and numbers can be beneficial for arithmetic skills because space is a powerful tool to understand arithmetic concepts.
Synthesizing some recent studies, Geake (2006) inferred that doing mathematics
critically involves the lateral frontal cortices to support working memory; the
temporal cortices (and hippocampus) to reconstruct knowledge from long term
memory; the orbitofrontal cortices and the anterior cingulate for decision making, in
turn mediated by regions within the limbic sub-cortex; areas of the fusiform gyri
and temporal lobes for sequencing of symbolic representations; the parietal lobes
for spatial reasoning about conceptual inter-relationships; and the cerebellum for
mental rehearsal.
There have been several neuroimaging studies of the brain function of mathematically gifted children compared with normal age-matched peers. O’Boyle et al.
(2005) in a fMRI study found areas of the brain that were involved in both
pre-algebraic and geometrical thinking of able young mathematicians. It seems that
mathematical thinking requires the coordinated participation of several neural
systems, which in the brains of gifted mathematicians seem more extensive
throughout both right and left hemispheres (e.g. Geake 2009; O’Boyle et al. 2005).
Data derived from several psychophysiological studies support an important
relationship between the specialized capacities of the right hemisphere and mathematical ability, but this may depend on the type and complexity of the task (e.g. Jin
et al. 2007). However, fMRI studies comparing the functioning of gifted versus other
brains (Lee et al. 2006) showed that gifted individuals did not use more, or different,
brain structures; rather, increased activation of the entire frontal-parietal network
was noted, perhaps indicating higher-than-average activity distributed across the
brain when performing difficult tasks. There are different findings depending on the
difficulty of the problem. If the problem is easy, there is less activation of the frontal
lobe. The discussion is sometimes in terms of ‘neural efficiency’, where gifted

12

Research On and Activities For Mathematically Gifted Students

functioning involves a more integrated brain with greater cooperation between the
hemispheres (O’Boyle 2008), with reduced activity in certain areas as compared
with average brains when performing similar tasks—possibly implying that gifted
brains spend less time on such tasks.
Mathematical giftedness as a form of intelligence related to enhanced mathematical reasoning has been tested using a variety of numerical and spatial tasks. For
example, gifted adolescents displayed enhanced connectivity patterns during a task
involving mental rotation of complex three-dimensional block figures when compared to an average control group. Findings are consistent with previous studies
linking increased activation of the frontal and parietal regions with high fluid
intelligence, and may be a unique neural characteristic of the mathematically gifted
brain, at least for this type of task. As mentioned in the previous section, Anderson
et al. (2008) found that both spatial imagery and verbal deductive cognitive abilities
were important for solving geometry problems, whereas object imagery was not.
This correlates with the observation that object imagery is important in arts professions, while spatial imagery is helpful in math, science and engineering.
A number of studies investigating the brain characteristics of mathematically
gifted youth indicate that they possess different functional organization as compared
to those of average mathematics ability (O’Boyle et al. 2005; O’Boyle
2005; Raghubar et al. 2010). Specifically, data from a variety of behavioral and
psychophysiological experiments tend to suggest enhanced processing reliance on
the right cerebral hemisphere and heightened interhemispheric communication, as
unique functional characteristics of the mathematically gifted brain. Notably, these
brain differences may have important implications for the nature and timing of
mathematics instruction.
Case studies of extremely gifted individuals often reveal unique patterns of
intellectual precocity and related brain activity. Presenti et al. (2001) using PET
measures and brain imaging, found calculation in an adult mathematical prodigy
(Rüdiger Gamm), to be uniquely mediated by right prefrontal and right medial
temporal cortex. A long chain of arithmetical operations and data handling would
put a considerable strain on normal working memory, yet many types of experts
show increased capacities for the temporary storage of task-relevant materials.
In another study, comparing memory and speed of processing in 160 16- to
18-year-old general gifted and excelling in mathematics male students, Leikin et al.
(2013) examined the memory and speed of processing abilities associated with
general giftedness (G) and excellence in mathematics (E). Working-memory was
found to be related to both G and E factors. The results reveal that G factor is
related to high short term memory and that E factor is associated with high
visual-spatial memory. Gifted students who excel in mathematics (G-E group)
outperformed all in speed of processing tasks. The findings of this study partly
support previous observations and suggest that memory and speed of processing
abilities seem to be important factors in explaining mathematical giftedness. With
this in mind, educational programs for G-E students, should address the observation

3 Mathematical Promise in Students of Various Ages

13

that these students have high abilities in visual-spatial memory and in information
processing and implement the use of visual aids in teaching mathematics in gifted
classes (e.g. Leikin et al. 2013).

3.3

What Are the Differences Between Mathematical
Novices and Experts?

Mathematical expertise implies the existence and use of two types of knowledge:
explicit knowledge of facts, principia, formulae pertaining to the domain, and
implicit knowledge of how to operate with them, i.e. declarative and procedural
knowledge. Research on the cognitive sub-processes involved in the expert problem
solving of the gifted, as compared to the problem solving of the average person, has
attributed the difference between these two populations to selectivity in their
encoding, comparison and combination sub-processes. Gorodetsky and Klavirb
(2003) extend this list by adding two sub-processes that are imported from the
literature on experts and novices: namely, retrieval and goal directness. Based on
these five sub-processes, middle high school students (gifted and average) solved
insight problems, without and with analogical learning, and were asked to report on
the solution process they undertook. Though both the gifted and the average were
able to arrive at correct solutions, the study shows that they employed different
sub-processes in doing so (Gorodetsky and Klavirb 2003).
Usiskin (2000) has devised an eight-tiered hierarchy, which ranges from Level 0
to Level 7, to classify mathematical talent. In this hierarchy, Level 0 (No Talent)
represents adults who know very little mathematics, and Level 1 (Culture level)
represents adults who have rudimentary number sense as a function of cultural
usage with mathematical knowledge comparable to that of students in grades 6–9.
Clearly, a very large proportion of the general population would fall into the first
two levels. Thus, the remaining population is spread throughout Levels 2 through 7
on the basis of mathematical talent, from Level 2 representing the honor high
school student who is capable of majoring in mathematics, up to Level 7 with the
Fields Medal winners in mathematics, or geniuses like Leonard Euler, Karl
Friedrich Gauss, Srinivasa Ramanujan and others (Usiskin 2000).
While Usiskin’s levels start from a social framing, Glaser (1988) characterizes
expertise on six cognitive dimensions: knowledge organization, complexity of
problem-solving representation, goal-oriented procedural knowledge, automatic
procedures, and metacognition. The model of the gifted and talented learner as an
expert knower and thinker can be used to differentiate the regular curriculum in the
sense that the transition from the novice to expert knower can be mediated by
adequate strategies and resources the teachers of the gifted are supposed to organize
and develop. Still, more research is needed for identifying evidence-based pathways
that lead to increasing expertise of mathematically promising students.

14

3.4

Research On and Activities For Mathematically Gifted Students

How Are Mathematical Creativity and Giftedness
Connected?

The discussion about the differences between novices and experts cannot avoid the
relationship between expertise and creativity. There are conflicting views about this
relationship. Analyzing the students’ level of expertise may depend on how
expertise and creativity are defined. Thus, for example, Diezmann and Watters
(2000) claim that for a student to be creative, he/she needs some intellectual
autonomy and expertise. Expertise is therefore seen as a necessary precondition for
the manifestation of creativity. On the other hand, Craft (2005) claims that every
student is capable of creative manifestations regardless of level of expertise.
In a recent study focused on how expertise interacts with creativity in problem
solving and posing, Singer and Voica analyzed the results of activities undertaken
by mathematics students enrolled in a pre-service teacher-training program and
found that, in the process of problem solving and problem posing, expertise and
creativity support and mutually develop each other. Consequently, a possible
method of training excelling students is through practicing tasks appropriate to their
level of mathematical abilities, but containing nonstandard challenging components
for which that person does not have yet internalized models of solving, in order to
train metacognitive self-regulation capabilities through creative leaps. The authors
revealed that, in the process of building a solution for a nonstandard problem,
expertise and creativity interact and enable bridges to the unknown, mutually
developing each other. This interaction leads to an increase in students’ expertise
(Singer and Voica 2016).
In professional mathematics, “creative” mathematicians constitute a very small
subset within the field. From the hierarchical classification of mathematical talent
outlined by Usiskin, it appears that in the professional realm, mathematical creativity implies mathematical giftedness, but the reverse is not necessarily true.
Usiskin emphasized that students have the potential of moving up into the professional realm (Level 5) with appropriate affective and instructional scaffolding as
they progress beyond K–12 into the university setting.
Hoyles (2001) analyzed the role that a computer-based learning environment can
play in the navigation between skills and creativity in teaching mathematics. She
concluded that technology-based inquiry opens opportunities for the advancement
of students’ mathematical creativity.
Much of the empirical research explores the learning processes of mathematically talented students through problem-solving strategies, but problem posing has
also been linked to mathematical creativity. As early as 1973, Jensen (Sheffield)
studied relationships among numerical aptitude, mathematical creativity, and
mathematical achievement, using a problem-posing instrument to measure one
aspect of mathematical creativity (Jensen 1973). This connection between mathematical creativity and giftedness identification in relation to problem posing has
also been studied more recently (e.g. Singer et al. 2015). Other studies revealed that
problem posing may stimulate creativity, possibly even more than problem solving

3 Mathematical Promise in Students of Various Ages

15

(e.g. Voica and Singer 2013). In addition, Voica and Singer (2014) found three
characteristics that can offer an indication of mathematical giftedness in problem
posing contexts: a thorough understanding of conveyed concepts, an ability to
generalize reasoning, and a capacity to frame and reframe content in order to devise
new problems.
For years, creativity has been studied using four related components outlined by
Torrance: fluency, flexibility, originality, and elaboration. Starting from here, various frameworks for studying creativity in relation to giftedness and high
achievement have been generated, usually adapted to specific types of tasks. In a
problem-solving context, Leikin (2009, 2013) uses multiple-solution tasks as a lens
to observe creativity. The dimensions used in her model are originality, fluency and
flexibility; these were aggregated into a creativity score by a research-based and,
subsequently refined, scoring technique. Leikin and Kloss (2011) examined students’ problem solving performance on Multiple Solution Tasks (MSTs) and
demonstrated that correctness in problem solving is highly correlated with fluency
and flexibility, whereas originality is shown as a special mental quality.
A different approach to creativity, one based on organizational theory, has been
taken by Singer and her research team (e.g. Singer and Voica 2013; Pelczer et al.
2013; Voica and Singer 2013). Their framework relies on the concept of cognitive
flexibility. Cognitive flexibility is described by cognitive variety, cognitive novelty,
and changes in cognitive framing. Cognitive variety manifests in the formulation of
different new problems/properties from an input stimulus; cognitive novelty captures the innovative aspect in the posed problem—its distance from the starting
element; while changes in the participant’s mental frame refer to shifts in the
“on-focus” elements during the problem posing. Thus, cognitive flexibility arises as
a complex, non-linear interplay between these dimensions. Consequently, the
construct of cognitive flexibility opens up the possibility to capture different ways
of being creative, namely through the differing loads on the three dimensions. The
use of the cognitive-flexibility framework in analyzing data offers more possibilities
to capture implications of a social-communicative nature.
By putting fourth to sixth graders, and also university students in
problem-posing contexts, Singer and Voica found that, in problem posing situations, high achievers, including gifted mathematics students, develop cognitive
frames that make them cautious in changing the parameters of their new problems,
even when they make interesting generalizations. The students’ capacity to generate
coherent and consistent problems in the context of problem modification may
indicate the existence of a generalization strategy that seems to be specific to
mathematical creativity, differentiating it from creative manifestations in other
domains (Singer and Voica 2015). The domain-specificity of mathematical creativity was also identified in other studies (e.g. Kattou et al. 2015). They investigated whether creativity is domain-general or domain-specific by relating fourth
through sixth graders’ performance on two tests: the Creative Thinking Test and the
Mathematical Creativity Test. Their data analysis from 476 students converged on
the conclusion that creativity is domain-specific. Other studies found that creativity

16

Research On and Activities For Mathematically Gifted Students

is not only domain-specific, but it even seems to be task specific within content
areas (e.g. Baer 2012).
New studies on the relationships between creativity and giftedness extend their
area of research from students’ cognitive dimensions to attitudes and values, based
on the anticipated roles that individuals with high potential play in society. As some
researchers underlined, the biggest challenge in gifted education is to extend the
traditional investment in the production of intellectual capital to include an equal
investment in social capital, innovation and the development of leadership capabilities (e.g. NSB 2010; Renzulli 2012). The goal should be not simply to ensure
that mathematically gifted students fulfill their potential by becoming productive
pure and applied mathematicians, but also to ensure that mathematical creativity is
enhanced to prepare innovative, thoughtful leaders in all fields with their new,
atypical methods and insights.

4 Research into Practice: Programs and Pedagogy
As important as it is to define and recognize students with mathematical gifts,
talents, and promise, it is even more important to develop, support and enhance
those traits. In this section, we look at examples of teaching practices, tasks, curricula, and in-school and extracurricular programs that are based on the research
described earlier and are designed not only to develop mathematical talent and
creativity but also to increase mathematical passions by engaging students in
problem solving, problem posing, and innovation.

4.1

How Might Teaching Practices Affect Mathematical
Promise and Talents?

As noted in Sect. 3, brain plasticity—the ability of the brain to grow and change
with learning and experience—has been well documented. Jensen (2000) even
stated: “We now know that the human brain actually maintains an amazing plasticity throughout life. We can literally grow new neural connections with stimulation, even as we age. This fact means nearly any learner can increase their
intelligence, without limits, using proper enrichment” (p. 149). We may not know
whether or not there are limits on how much students might increase their mathematical expertise or just how much a growth mindset as described by Dweck
(2006) and others might help, but we do know that teaching practices can greatly
increase a student’s mathematical performance and passion.
In the United States, the Common Core State Standards for Mathematics
(NGA/CCSSO 2010) list eight Standards for Mathematical Practice that have been
shown to be effective in developing all students mathematical talents. These

4 Research into Practice: Programs and Pedagogy

17

standards for students are: Make sense of problems and persevere in solving them;
Reason abstractly and quantitatively; Construct viable arguments and critique the
reasoning of others; Model with mathematics; Use appropriate tools strategically;
Attend to precision; Look for and make use of structure; Look for and express
regularity in repeated reasoning. All students should be engaged in these practices
throughout their mathematical education, but the CCSSM make no mention of
special provisions for gifted, talented, promising or high-achieving students. Noting
this oversight, in a joint publication from the NCTM, the National Association for
Gifted Children (NAGC), and the National Council of Supervisors of Mathematics
(NCSM), titled Using the Common Core State Standards for Mathematics with
Gifted and Advanced Learners, the addition of a ninth Standard for Mathematical
Practice is suggested. “In order to support mathematically advanced students and to
develop students who have the expertise, perseverance, creativity and willingness to
take risks and recover from failure, which is necessary for them to become mathematics innovators, we propose that a ninth Standard for Mathematical Practice be
added for the development of promising mathematics students—a standard on
mathematical creativity and innovation: Solve problems in novel ways and pose
new mathematical questions of interest to investigate. The characteristics of the new
proposed standard would be that students are encouraged and supported in taking
risks, embracing challenge, solving problems in a variety of ways, posing new
mathematical questions of interest to investigate, and being passionate about
mathematical investigations.” (Johnsen and Sheffield 2012, pp. 15–16).
To implement these effective practices, Sheffield (2009) recommends that
teachers pose problems that allow all students, including the most talented, to
struggle; expect coherent explanations and critiques of unique and creative solutions; give formative and summative assessments that provide opportunities for
students to reason, create problems, generalize patterns, solve problems in unique
ways, and connect various aspects of mathematics; and generally act as a role model
who is comfortable with making mistakes and demonstrating the joy of solving
difficult problems.

4.1.1

Problem Solving and Problem Posing

Problem solving is often cited as a major goal in any mathematics program. This
statement from Adding It Up, is typical: “We see problem solving as central to
school mathematics. Problem solving should be the site in which all of the strands
of mathematics proficiency converge. It should provide opportunities for students to
weave together the strands of proficiency and for teachers to assess students’ performance on all of the strands” (National Research Council 2001, p. 421). Making
sense of problems and persevering in their solutions is the first of the CCSS
Standards for Mathematical Practice.
Problem solving is often defined as seeking a solution to a mathematical situation for which students have no immediately obvious process or method.
For gifted mathematics students, that means that a question that may be a problem

18

Research On and Activities For Mathematically Gifted Students

for other students may not engender difficulty for the gifted student. It is important,
therefore, that students not only learn to solve problems, but also to rephrase and
pose new questions that are authentic problems for themselves, challenging them to
persevere and struggle to find a solution.
Several studies on problem solving and problem posing have shown the efficacy
of this approach in the development of mathematical creativity and talent. For
example, in a 4-year longitudinal study with primary students, Singer found that a
pattern of training organized under the name of dynamic structural learning could
substantially raise students’ creative approaches in mathematical problem solving
and posing (Singer 2007). The dynamic structural learning is based on distributing
the training procedures across several categories including systematic training of
transfer (e.g. transfer from objects to various unconventional ad-hoc notations, then
to conventional representations, and then to abstract reasoning and back; from
thinking aloud to “thinking in mind” and vice-versa; etc.), randomized training of
the developed capacities, which is realized by means of various mental games, and
structured training of specific competencies, which aims at assimilating the
invariants by constantly resorting to models and diagrams (Singer 2007). In addition, an effective context for both broadening and deepening student’s knowledge
can be offered by problem posing sessions. The problem-posing research field is an
emerging force within mathematics education, which offers a variety of contexts for
studying and developing abilities in mathematically promising students (Singer
et al. 2013).

4.1.2

Discourse and Questioning

When solving problems, students should discuss their processes, justifying their
reasoning, and critiquing their own and their peers methods and solutions. Chapin
and colleagues designed and researched what they labeled “talk moves” as part of
Project Challenge, a Jacob K. Javits grant program of the United States Department
of Education that was looking for projects to increase the number of ethnic and
linguistic minority students in programs for gifted and talented students. Not only
did they find that this oral discourse paid off with more complex, sophisticated, and
mathematical reasoning, but students moved from 4 % being classified as “Superior”
or “Very Superior” on the Test of Mathematical Abilities, Second Edition (TOMA)
at the beginning of the program to 41 % being classified at this level after 2 years in
the program. They hypothesized that the discussions allowed misconceptions to
surface and be corrected, developed students’ ability to reason, gave more students
the opportunity to observe, model, build on and add to the development of complex
ideas, and provided motivation and engagement (Chapin et al. 2009).
To instigate a rich discussion, teachers and students themselves need questions
that assist students in focusing on the big ideas in the problems rather than funneling students to rotely follow a fixed procedure. Sheffield (2006) suggests the use
of “who, what, when, where, why and how” questions that are commonly used to
teach students to write informational articles. These questions include “Who used a

4 Research into Practice: Programs and Pedagogy

19

different method or has a different solution? Who has a new or unique question or
suggestion? What generalizations or conjectures might I make from the patterns?
What proof do I have? What if I change one or more parts of the problem? Why does
that work? If it does not work, why not? How does this compare to other problems
or patterns that I have seen? How many ways might I use to represent, simulate,
model, or visualize these ideas?”
These questions are an integral part of Project M3: Mentoring Mathematical
Minds, Project M2: Mentoring Young Mathematicians, and Math Innovations
curricula, which will be discussed later in this review, which have been shown to be
successful in developing mathematical promise in students with various initial
levels of ability.

4.2

How Might Curriculum Contribute to Mathematical
Development?

In any mathematics program, in addition to the teaching practices, the curricula and
the problems that students encounter often determine whether students have the
opportunity to develop their mathematical expertise to the fullest extent possible.
One important aspect is the opportunity for students to deepen their understanding
of mathematics. Deepening is usually associated with studying a curricular topic at
greater depth or with greater complexity than prescribed by the curriculum or
school textbooks. Deepening mathematical understanding can include, for example,
justifying or proving the reasons behind arithmetic operations, solving problems in
a variety of ways, or posing and solving related problems. Students might also work
on fields of problems. This approach supports competencies via problem solving
with a high level of complexity.

4.2.1

Challenging Mathematical Tasks

A common approach around the world to support students in deepening their mathematical understanding is the use of challenging or rich tasks for the realization of
mathematical potential. Whether these are called rich tasks, problems, investigations
or challenges, these are interesting and motivating mathematical difficulties that a
person can overcome. In the publication of the sixteenth International Commission on
Mathematical Instruction (ICMI) Study on Challenging Mathematics, Barbeau
defines a challenge as “a question posed deliberately to entice its recipients to attempt
a resolution, while at the same time stretching their understanding and knowledge of
some topic” (Barbeau and Taylor 2009, p. 5). Stretching understanding, making
continuous progress in the face of difficulties, and creating new knowledge is especially important for gifted students who are too often faced with repetitive tasks,
memorized algorithms or arithmetic skills that they have already mastered. It is also

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Research On and Activities For Mathematically Gifted Students

important for challenges to engage the mind, encourage students to explore the beauty
and structure of mathematics, and exult in the mastery of new ideas.
Many authors recognize the centrality of mathematical challenge for the realization of mathematical promise and as a characteristic of the activities in which
gifted mathematicians are involved. Solutions of challenging tasks often involve
explanations, multiple strategies, models and tools, questioning, conjecturing, and
ongoing evaluation. Challenges might be multiple-solution tasks, proof tasks,
concept-defining tasks, inquiry-based tasks, or other complex tasks that engage
students in new mathematical explorations. Mathematical challenge depends on the
type and conceptual characteristics of the task, for example, conceptual density,
mathematical connections, the building of logical relationships, or the balance
between known and unknown elements (e.g. Leikin 2011).
Rich learning tasks are not rich on their own. It depends on what is done with
them. Instead of using the straightforward four-step heuristic that is common in
many U.S. mathematics textbooks (1. Understand the problem; 2. Devise a plan; 3.
Carry out the plan; and 4. Check), Sheffield (2003) has developed a model for a
more open heuristic for solving and posing problems where one may start at any
point and proceed in any order that makes sense, repeating steps as necessary as
they become engrossed in problem solving and problem posing. Steps include
relating the task to what students already know; investigating the problem; evaluating the findings; communicating the results; and creating new questions to
explore. This heuristic encourages multiple solutions, models and methods as well
as problem posing that have been shown to be successful in developing students’
mathematical promise.

4.2.2

Curriculum and Textbooks

A mathematics curriculum broadly may be thought of as the total of all the students’
mathematical learning experiences. Curriculum may include a body of knowledge
to be transmitted as well as the process by which this happens. Thus curriculum
writers generally attempt to write materials for teachers explaining expectations for
teaching practices as well as student books with specific mathematics content.
Most mathematics textbooks and units are written for a broad range of students.
One example of an exception to this is Project M3: Mentoring Mathematical Minds
(www.projectm3.org) with units that were developed to nurture mathematical talent
in elementary students, utilizing the “talk moves”, questioning strategies, an open
heuristic and challenging tasks as described earlier. With effect sizes ranging from
0.69 to 0.97 on the Open-Response Assessments, results indicated that these units,
designed to address the needs of mathematically promising students, positively
affected their achievement (Gavin et al. 2009). Following the success of Project M3,
Project M2: Mentoring Young Mathematicians (www.projectm2.org) was developed with support from a grant from the US National Science Foundation. Units in
this program were designed for heterogeneous classes of students from kindergarten
through second grade. One purpose of the M2 program was to determine whether

4 Research into Practice: Programs and Pedagogy

21

using the same “talk moves”, questioning strategies, open heuristic and challenging
tasks described earlier could increase the numbers and levels of mathematically
talented students. Following participation in the program, results showed a significant difference at the 0.001 level in favor of M2 students in the percent of
students performing one and two standard deviations above the mean between
students in the M2 program and students in the comparison groups even though
groups were not significantly different before the beginning of the program (Gavin
et al. 2009, 2013; Sheffield et al. 2012).

4.3

What In-School Programs Might Develop Mathematical
Talent?

Countries vary widely in programs to identify, support, create and enhance students
with mathematical expertise and passion. A brief overview of the status of gifted
education across the world is provided in this section. The World Council for
Gifted and Talented Children (WCGTC/ www.world-gifted.org) provides worldwide advocacy and support, has affiliated federations in Africa, Asia-Pacific,
Europe, and Ibero-America with organizations and resources specific to those areas,
and holds international conferences every two years. The United States has no
federal policy on gifted education, but The National Association for Gifted Children
(NAGC) www.nagc.org/ regularly surveys gifted programs across the US, and
posts information on their website. In Africa, Mhlolo (2014) reported on a survey of
15 African countries to determine the extent to which mathematically talented
students were identified, tracked and nurtured, and in Europe, Mönks and Pflüger
(2005) surveyed 21 European countries concerning legislation, identification, provisions, teacher training, research and priorities in gifted education. A few of the
in-school programs and activities specific to the support and enhancement of
mathematical promise and talents are described here.

4.3.1

Ability Grouping, Self-contained Classes and Specialized Schools

In a survey of over 1000 school districts in the United States, Callahan et al. (2014)
reported that while over 90 % of the districts claimed to identify gifted students,
services for these students varied. About half the elementary programs responding
said that they had special homogeneous classes for gifted students pulling them
from heterogeneous classes from 1 to 4 h a week, about two-thirds of the middle
schools reported the existence of some special homogeneous classes and 90 % of
the high schools reported using Advanced Placement® as the predominant option.
Mönks and Pflüger (2005) found that 12 of the 21 European countries surveyed
reported that giftedness (or a synonym such as high-ability, or talented) was
explicitly named in the law of the country, 13 countries reported differentiation for

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Research On and Activities For Mathematically Gifted Students

these students, and 15 reported having a special curriculum. In a survey of 15
sub-Saharan African countries, Mhlolo (2014) found that the most common method
of identifying mathematically talented students was through participation in a
mathematics Olympiad in 12 of the 15 countries with no method of identification
named in the others. Many mentioned that the Olympiads were used to recognize
mathematical achievement but nothing was done to support or nurture it. Gifted
students were placed in regular classrooms per national policies of non-streaming of
students. No country mentioned having special public schools for gifted students or
providing any special training for teachers of the gifted.
One method of serving mathematically talented students around the world is
through ability-grouping, self-contained classes and specialized secondary schools
for the mathematically talented where students are tested and must show some level
of general giftedness and/or mathematical expertise before being accepted into the
program. One issue that has triggered many debates all-over the world is ability
grouping classroom versus heterogeneous classroom. The debate on the necessity
of ability grouping is legitimate, and both proponents and opponents of heterogeneous mathematics education use valid arguments to justify their positions (Leikin
2011).
Some studies suggest that ability grouping may be essential for the education of
gifted both from cognitive and affective perspectives and that therefore, schools
ought to supply special education to mathematically gifted students and prevent
talent loss (Milgram and Hong 2009). Olszewski-Kubilius (2013), past President of
the National Association for Gifted Children in the US, reports that a 2013 National
Bureau of Economic Research study of students who were grouped by ability found
that the performance of both high and low performing students significantly
improved in mathematics and reading. On the other hand, ability grouping is still
questionable both in light of the equity principle and of some research findings.
Some of this may be due to different definitions of ability grouping. In the United
States, tracking which identifies students at a given point in time and places them on
a semi-permanent, rigid, defined path is distinguished from flexible grouping where
students are not given permanent levels and may move up or down based on
performance.
At the high school level, several countries have special schools for talented
mathematics students. In Special Secondary Schools for the Mathematically
Talented: An International Panorama, over 100 special schools from twenty
countries are described ranging from the Super Science High Schools in Japan and
the Science High Schools and Gifted High Schools in Korea to the special schools
in Europe, Russia, North and South America, the Middle East, China, South Asia,
and Australia. These schools have shown to be effective in offering exciting
frameworks for the education of gifted students (e.g. Vogeli 2016; Vogeli and Karp
2003). Many of today’s leading mathematicians and mathematics educators have
come and will continue to come from these specialized secondary schools for the
mathematically talented (Vogeli 2016). In the United States, the National
Consortium of Secondary STEM Schools (www.ncsss.org) is designed to allow

4 Research into Practice: Programs and Pedagogy

23

these schools to share and build on each other’s successes and challenges. The new
volume from Vogeli (2016) facilitates this sharing on a worldwide level.

4.3.2

Acceleration and Grade Skipping

Acceleration is usually defined as learning topics within the curriculum at a faster
pace. In a comprehensive study of acceleration that includes 13 different types of
acceleration among which grade-skipping, moving ahead in one subject area,
advanced placement, curriculum compacting, dual enrollment in high school and
college classes, and entering college early, Colangelo et al. (2004, p. xi) found that
“acceleration is educationally effective, inexpensive, and can help level the playing
field between students from rich schools and poor schools”. Even for moderately
gifted students, recent research shows that approximately 40–50 % of traditional
classroom material could be eliminated for targeted gifted students in one or more
of content areas, among which is mathematics (Reis et al. 1998). Care must be
taken not to skip critical material, however, and to ensure that students are engaged
and passionate about the mathematics they are learning, and are not simply rotely
memorizing algorithms or accelerating so they can finish taking required mathematics classes early.
A caveat must be added as well for students whose mathematics program has
been accelerated by simply moving faster through a traditional program. In the
United States where the numbers of STEM majors in college decreased between
1984 and 2010 while the number of students taking calculus in high school skyrocketed, Bressoud et al. (2012, p. 2) note that “What the members of the mathematical community—especially those in the Mathematical Association of America
(MAA) and the National Council of Teachers of Mathematics (NCTM)—have
known for a long time is that the pump that is pushing more students into more
advanced mathematics ever earlier is not just ineffective: It is counter-productive.
Too many students are moving too fast through preliminary courses so that they can
get calculus onto their high school transcripts. The result is that even if they are able
to pass high school calculus, they have established an inadequate foundation on
which to build the mathematical knowledge required for a STEM career.”

4.4

What Extra-Curricular Programs Might Enhance
Mathematical Promise?

Programs for gifted mathematics students that are offered during the school day are
often supplemented and enhanced by extra-curricular programs. At other times,
extra-curricular programs are the only way that mathematical expertise is identified,
challenged and enhanced.

24

4.4.1

Research On and Activities For Mathematically Gifted Students

Recreational Mathematics

As noted earlier, pullout programs for a few hours a week for identified gifted
elementary students in the United States who spend the majority of time in
heterogeneous classes are a common service option as is differentiation or cluster
grouping within a heterogeneous class. Other activities such as math clubs, competitions, online courses, project learning, and work with mentors can be found both
in school and out of school. In all of these settings, teaching practices and challenging tasks are important for the fostering of mathematical talent.
Brandl (2014) analyzed different settings of fostering mathematical talent with
respect to the students’ attitude towards mathematics. The findings suggest that, on
the one hand, the environment of an ordinary mathematics class often seems not to
be supportive enough for promoting mathematics as something beautiful, challenging and joy-bringing; on the other hand, selection with respect only to performance and high achievement can lead to a reported “psychological hindrance of
a narcissistic wound/shock that comes from being confronted with just
best-of-students in class and the eventual loss of this status for oneself” (Brandl
2014, p. 1164). So, non-selective interest-based courses seem to be more promising.
A variety of extra-curricular options exist to engage students in mathematical
explorations. Many of these focus on the enjoyment of recreational mathematics. In
many parts of the world, students routinely turn to reading both fiction and
non-fiction as an enjoyable pastime, but “doing math” for fun has not been as
popular. These activities exist to give students an enjoyable mathematical option.
The term math “club” is often used to describe any extra-curricula mathematics
program designed as a fun way to challenge students to encounter interesting
mathematics. Math circles are generally programs where college and university
mathematicians share their expertise and love of mathematics with K-12 students
and teachers. They follow a variety of styles from informal activities and games to
more traditional enrichment classes. Some have a strong emphasis on preparing
students for competitions while others avoid all competitions. Math Circles
appeared in Russia in 1930 and have existed in Bulgaria and Romania for over a
century. Math circles migrated to the United States in the 1990s, often with
immigrants who enjoyed these activities themselves as teenagers, and have grown
in popularity in the US in the last 25 years. The National Association of Math
Circles (NAMC, www.mathcircles.org) provides resources and support for Math
Circles and other similar informal mathematics education programs around the
world. Math houses (www.mathhouse.org) in Iran serve a similar function to the
math circles with activities and competitions for students and workshops and
meetings for their teachers from the primary through the university level.
There are numerous resources for recreational math (for example, see lists from
the Mathematics Association of America (www.maa.org/programs/students/
student-resources) and the NCTM (www.nctm.org/Classroom-Resources/BrowseAll/), including videos and websites with challenging problems, games, and other
interactive resources. There are also extended extra-curricular programs for
engaging mathematics such as the summer and week-end math camps that are listed

4 Research into Practice: Programs and Pedagogy

25

by the American Mathematical Society (www.ams.org/programs/students/empmathcamps), or the activities and camps related to the Kangaroo contests (e.g.
www.mathkangaroo.org; mathplus.math.toronto.edu/).
Online enrichment problems such as those at CAMI (www.umoncton.ca/cami), a
website (in French) developed to provide all students with challenging and rich
mathematical problems online following a Problem-of-the-Week model to which all
students could submit their solutions and receive a feedback from mentors—
pre-service teachers from a local university, are another way to challenge and
engage students. Results from the use of this site with over a million hits from more
than 100,000 visitors between 2005 and 2010 found that students seemed to
appreciate problems that were different from those they encountered in the regular
classroom and solving them was more challenging, motivating and enjoyable, even
if sometimes problems seemed to be too difficult and frustrating. Personal feedback
from mentors was also greatly appreciated (Freiman 2009; Freiman et al. 2009;
Freiman and Lirette-Pitre 2009; Freiman and Manuel 2015).

4.4.2

Competitions

There are a variety of math competitions around the world. The most known of these
is the International Mathematical Olympiad (www.imo-official.org), which is the
World Championship Mathematics Competition for High School students and is
held annually in a different country. The first IMO was held in 1959 in Romania,
with 7 countries and has gradually expanded to over 100 countries from 5 continents.
Other international math competitions can be found at www.artofproblemsolving.
com/. Preparing students for this type of competition is somewhat different from
some of the programs based on problem solving or recreational mathematics. With
these competitions it is often necessary to acquire mathematical knowledge, and
learn algorithms, theorems, and mathematical “tricks” explicitly. The difference lies
in the speed of working on the questions. Students are successful if they “see”
immediately the mathematical core of the question and if they can embed the
question in the mathematical background. This is different than other types of
problem solving or problem posing where the students have time for going a long
way around, for trying different approaches and by this acquire more and more
knowledge about the subject as well as about metacognitive aspects.
Not all competitions are of that same type, however. There are game competitions such as: Set (www.setgame.com); Math Pentathlon (www.mathpentath.org);
Kangaroo, which is the largest mathematical competition in the world, with more
than six million participants from 72 countries in 2015 (http://www.aksf.org/); or
Calculation Nation (calculationnation.nctm.org/), puzzles like Ken-Ken (www.
kenken.com) and competitions that give students an extended period of time to
solve interesting problems such as the USA Mathematical Talent Search (www.
usamts.org) that gives students at least a month to work out problems with written
explanations and encourages the use of any materials including books, calculators,
and computers. Like other programs and activities in this section, regardless of the

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Research On and Activities For Mathematically Gifted Students

type of competition, they are designed to encourage and reward high-level mathematical performance, and hopefully to entice more students to continue with
mathematics-intensive studies and careers.
Despite the variety of programs for the education of mathematically gifted
students, there is lack of systematic empirical studies on various programs to gain
better understanding of their suitability for the realization of students’ mathematical
potential. Theoretical characterizations of effective courses and programs for
mathematically talented students, as well as empirical studies to test their effectiveness are still to be developed. More information is needed on just how extensive, widespread and effective are programs that are designed to serve or develop
students with mathematical promise or expertise. Unfortunately, in many parts of
the world, no such programs exist, and in many cases, the programs that do exist are
small, insignificant or not optimally effective. This results in untold loss of needed
mathematical knowledge and aptitude in citizens and educators around the world, in
addition to the loss of expertise in science, technology, engineering and many other
math-intensive fields.

5 Research into Practice: Teacher Education
Teachers, of course, are critical to implanting the programs and practices that have
been shown to be effective with mathematically promising students. In addition to
traits and professional development that are important for all teachers of gifted
students, there are supplementary expectations from teachers of mathematically
gifted students.

5.1

What Teacher’s Traits Are Important?

Greenes et al. (2010) noted that research findings indicated that teachers of gifted
and talented should be flexible thinkers who are appreciative of creative approaches, curious, persistent, and confident when solving difficult problems. In addition,
for teachers of mathematically talented students, she also recommended that they
should understand a wide range of mathematics concepts and skills, have a
“toolbox” of problem solving heuristics, love math, and be armed with challenging
and engaging mathematical problems. Chamberlin and Chamberlin (2010) summarized the needed competencies of teachers of gifted students as knowledge of
student needs; skill in promoting higher-level thinking, creativity and problem
solving; development of a differentiated curriculum with multiple resources,
enrichment and acceleration; creation of a safe, flexible, learner-centered environment; and avoidance of rote memorization and overreliance on gifted students as
tutors.

5 Research into Practice: Teacher Education

27

Leikin (2011) used one example of a teacher of mathematically gifted she called
‘outstanding’ to reveal several similar characteristics that his students valued the
most, with the addition of a category on the teacher’s relationship to the students.
Among the characteristics that students valued in a teacher, one category represented
relationship to the subject, namely, genuine interest, deep knowledge, including
knowledge that went beyond mathematics, and openness to challenge from the students. The second category featured relationship to the students, including kindness,
trustworthiness and pride along with patience and sensitivity to students’ interests,
needs, difficulties and differences. The third category referred to teaching style and
methods such as being flexible, knowledgeable, creative, and open to improvisation,
as well as having a sense of humor, love of and enjoyment from mathematics.
According to Holton et al. (2009), the teacher’s role is central in promoting the
mathematical understanding and learning of students by choosing appropriate tasks
and providing expert assistance. First, the teachers must be able recognize the
importance of balancing teaching mathematical rules and algorithms, on the one
side, and more complex and sophisticated mathematical processes and open
problems on the other side, as part of a more creative work. Second, they need to be
aware of students’ cognitive and social processes including a theoretical knowledge
of how students learn. Third, be able to adjust teaching to the result of interaction
with the students and to encourage both oral and written communication among
students. Finally, they need to be aware of the nature and importance of mathematical challenge.
Several researchers have noted difficulties that teachers have in challenging
promising mathematics students. For instance, regarding the use of challenging
tasks by teachers, Holton et al. (2009) point at different barriers, such as the lack of
attention to the process side in students’ mathematics learning, and particularly, in
supporting cooperative processes of discovery learning or teamwork, and also low
expectation-levels from what are the students’ abilities (p. 216). Along with the lack
of teachers’ own motivation to do challenging tasks, other, more systemic obstacles
related to the social and educational policies and economic conditions can also
affect teachers’ willingness to use challenging tasks with their students.
Leikin (2011) found that simply providing teachers with challenging mathematics activities is not sufficient for their implementation. She noted that teachers
have to be provided with multiple opportunities to advance their knowledge and to
develop commitment and beliefs in their own and students’ abilities for high-level
mathematical performance. Consequently, she lists the following questions that
need additional research:
• Should the teachers of gifted be gifted? Should the teachers be creative in order
to develop students’ creativity?
• How might teachers’ creativity be characterized both from the mathematical and
from the pedagogical points of view?
• What are the desirable qualities of teachers’ knowledge, beliefs and personality
that make them creative and gifted teachers?

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Research On and Activities For Mathematically Gifted Students

Answers to these questions may lead to more effective teaching, which is oriented towards helping students to reach their cognitive potential and to develop a
balanced personality. This leads us to the next question.

5.2

What Should Be Included in Teacher Education?

The recommendations in this section for teachers’ preparation and professional
development are based on the proposed teacher’s traits and knowledge described in
the previous section.

5.2.1

Teachers of All Students

In the United States, the National Association for Gifted Children (NAGC) and the
Council for Exceptional Children (CEC) collaborated to develop the NAGC-CEC
Teacher Preparation Standards in Gifted Education to assist state departments of
education in developing standards for teachers of K-12 gifted students and to help
colleges and universities develop pre-service and in-service teacher education
programs to prepare and support teachers of gifted students (NAGC-CEC 2013).
These standards were followed by the Knowledge and Skill Standards in Gifted
Education for All Teachers. Those standards stated that all K-12 teachers should be
able to recognize learning differences, cognitive/affective characteristics and needs
of gifted and talented students of all backgrounds and design appropriate learning
modifications to enhance creativity, acceleration, as well as depth and complexity
of learning, by using a repertoire of instructional strategies to advance their learning
(NAGC-CEC 2014).
Tomlinson (1999) analyzed differentiation in the classroom and pre-service teacher
preparation, viewing it as a way to meet the needs of gifted and other academically
diverse students. In the conducted experimental studies, Tomlinson and colleagues
found that although participants affirmed the existence and importance of recognizing
student differences and concomitant needs, they used ambiguous criteria for identifying these differences and needs, expressed incomplete views of differentiating
instruction, exhibited shallow wells of strategies for enacting differentiation, and were
influenced by factors which complicated and discouraged understanding and
addressing student differences and needs. Presenting these teachers with workshops on
strategies for more differentiated instruction and coaching them when implementing
these strategies in schools may be a promising way of better teacher preparation; yet,
several issues arose when translating training into the real practice due, for example, to
the lack of collaboration between the coach, the teacher and the university supervisor.
This risk was addressed by Singer and Sarivan (2009), who developed a strategy
of preparing teachers called multirepresentational training (MRT) that has been
applied in graduate and undergraduate mathematics courses for primary prospective
teachers. The MRT model is based on two directions of action: providing a variety

5 Research into Practice: Teacher Education

29

of representations as powerful tools for learning, and developing representational
models to stimulate abstraction and synthesis (Singer 2007, 2009). The contextualized learning paths developed through the MRT help students-prospective
teachers reach a depth of understanding that enables them to reiterate their learning
acquisition in the different and complex problem-solving contexts of teaching in the
real setting (Singer and Sarivan 2009, 2011). This method worked as a shortcut for
internalizing mathematical content knowledge, mathematics pedagogical content
knowledge, and general pedagogical knowledge in an integrated effective way.

5.2.2

Teachers of Mathematically Gifted and Talented Students

The NCTM Task Force Report on teaching mathematically promising students
suggest including information on dealing with mathematically promising students
in in-service and pre-service programs for teachers at all levels. Regardless of the
type of program being offered to promising students, teachers should have access to
professional development, research information, and resources to deal with such
issues as identification or recognition of students with mathematical promise, high
levels of expectations for all students along with challenging top students to even
higher levels of success, pedagogical and questioning techniques to extend students’ thinking, and selection and/or development of appropriate curriculum and
assessment tools that provide opportunities for students to create problems, generalize patterns, and connect various aspects of mathematics, development of
teachers’ own mathematical power to make connections and the mathematical
sophistication to see the big picture, making appropriate instructional decisions for
these promising students, and awareness of, access to and ability to use technology
and other tools. In addition, teachers should continue to strengthen their own
mathematical content knowledge and demonstrate the joy of being a lifelong learner
of mathematics (Sheffield et al. 1999).
Holton et al. (2009) note the importance of modeling the incorporation of
challenging tasks programs for teachers asking them to construct meaning and
make explicit connections among mathematical ideas and to prior knowledge.
Leikin and Winicky-Landman (2001) note that both teachers’ Pedagogical Content
Knowledge and Mathematical Content Knowledge are enhanced when the teachers
cope with mathematical challenges as learners. Greenes and Mode (1999) also note
the importance of teachers of mathematically gifted strengthening their own
mathematical content knowledge through individual, partner, and small-group
problem solving during pre-service courses and in-service workshops. They suggest
that prospective teachers should individually assess and mentor students who have
been identified as mathematically promising and develop individualized learning
plans for them that identify specific areas of mathematical talent, interests and
needs; set goals to be achieved; and identify resources, challenging problems, and
strategies for accomplishing the goals as well as assessments to monitor progress.
They also recommend that mathematics teachers observe and plan with each other;
analyze, critique, and maintain a file of students’ most outstanding mathematics

30

Research On and Activities For Mathematically Gifted Students

work; tackle challenging mathematics problems themselves; identify and provide
engaging extracurricular mathematics activities; join professional societies and seek
out opportunities for ongoing professional development; and learn strategies for
academic and career counseling.
For teachers of promising secondary mathematics students, Greenes et al. (2010)
specifically recommend a sequence of five workshops or courses that go beyond
their initial teaching certification. These courses include a problem-solving lab;
classes on assessing mathematical talent and differentiating instruction; instructional strategies for this category of students; and a concept study focusing on a
variety of ways in which students might develop deep understanding of a key
mathematical concept.
More research is needed to validate different strategies for enhancing the quality
of professional development of the teachers of the gifted, and specifically on the
ways in which such strategies can be embedded into the initial preparation for the
teaching profession of all (future) teachers.

5.3

What Are Some Examples of Programs for Supporting
Teachers?

As mentioned earlier, Mhlolo (2014) in his survey of Sub-Saharan African countries found none that offered teacher training specifically for teachers of gifted and
talented students, but Mönks and Pflüger (2005) in their survey of European
countries found 16 of the 21 countries offered some type of teacher training on
gifted education. In the United States, the NAGC-CEC Teacher Preparation
Standards in Gifted and Talented Education (2013) are included in accreditation
standards for colleges and universities that choose to have their graduate programs
for preparing teachers of the gifted accredited through the Council for the
Accreditation of Educator Preparation (CAEP). Callahan et al. (2014) found that the
amount of professional development that in-service teachers received related to
gifted and talented students varied widely from district to district, ranging from 0–
15 min to 4 days per year. However, those surveys refer to general gifted education
professional development that is not necessarily focused on mathematics. The
present section includes a few examples of professional development programs for
teachers of mathematically promising students in different countries.

5.3.1

Brief Snapshots into Professional Development Programs

A pilot study of a professional-development course for science and mathematics
teachers offered as part of a 2-year in-service program for promoting excellence in
education was conducted in Israel by Karsenty and Friedlander (2008). The course

5 Research into Practice: Teacher Education

31

aimed to expose teachers to theoretical aspects of gifted education in general, and
particularly in science and mathematics; to develop leadership qualities based on
classroom contexts provided by teachers; and to increase teachers’ domain-specific
pedagogical content knowledge. Five types of activities were developed as answers
to the above questions: (1) analysis of lessons and interviews with gifted students in
order to learn about their cognitive and affective characteristics; (2) analysis of
investigative activity (launch-explore-summarize) from the students’ point of view;
(3) analysis of mathematical tasks for advanced students (mathematical content,
context, level of openness, representations and sequence of sub-tasks); (4) design,
adaptation or adoption of tasks: learning about strategies for adapting routine
mathematical tasks to the needs of the gifted (e.g. What if not? problem-posing
strategy); and (5) classroom implementation of activities that are then reported and
analyzed (Karsenty and Friedlander 2008). These activities have been shown to
have a strong potential for impact on prospective teachers.
A large gamut of studies addresses the issues of professional development for
teachers of mathematically promising students in the United States. For example,
Adelson et al. (2007) discussed teachers’ professional development for specific
programs, like Project M3: Mentoring Mathematical Minds, which was based on an
enriched and accelerated curriculum focused on developing conceptual understanding in mathematics (see also Sect. 4.2.2). The teachers involved in this project
participated in a two-week summer training program in order to increase their
mathematical content knowledge and to learn how to implement teaching strategies
to promote deeper reasoning, problem solving and problem posing, and verbal and
written mathematical communication. Teachers also attended four to six professional development sessions throughout the academic year prior to teaching each
unit of the curriculum. A professional development team member visited each
school every week the Project M3 units were taught, to ensure fidelity of treatment
and offer individualized assistance to teachers. Students made highly significant
progress, while teachers reported that this embedded professional development
contributed to strengthen their own mathematical content knowledge and understanding of their students, and led to an enhancement of their repertoire of teaching
practices and strategies.
In a study by Chamberlin and Chamberlin (2010), pre-service teachers enrolled
in a mathematics teaching methods course, without having special training in
teaching the gifted, were asked to choose tasks they found appropriate to use with
gifted students and to implement them in a real classroom setting. According to
participants’ reports, teachers seemed to have broadened their view of giftedness,
recognized the need to adapt instruction for gifted students, made efforts to align
problem-solving tasks with gifted students’ readiness and interests, realized the
necessity of knowing students to differentiate instruction, and emphasized
student-centered approaches.
These studies, as well as several others, point to the need for teachers to have
experience with the use of challenging tasks and the appropriate feedback to be
given to students.

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5.3.2

Research On and Activities For Mathematically Gifted Students

Professional Development Related to Mentoring Students’
Online Problem Solving

Noting that the analysis of children’s mathematical production by pre-service
teachers has become an important part of mathematics education courses, little is
known about the impact of participation of pre-service teachers in online activities
with schoolchildren and even less about their capacity to guide young learners by
means of asynchronous feedback (LeBlanc and Freiman 2011). In a study in
Canada, pre-service teachers served as mentors in a context of a virtual mathematics
learning community, CASMI (Freiman and Lirette-Pitre 2009) to assess students
solving mathematically rich, contextual, and open-ended problems posted on a
website that is meant to challenge all children. This activity aimed to help
pre-service teachers appreciate the diversity of solutions and learn how to guide
schoolchildren in a personalized and caring manner, nurturing their curiosity,
interest and perseverance that are very important for all children and especially for
the gifted ones. It was concluded that participation in the online project allowed
pre-service teachers to experience new mathematical problem-solving approaches
that stress the use of multiple strategies and communication means by schoolchildren. At the same time, in a context of asynchronous assessment with no opportunity to give feedback in another way than written comments, a good
understanding of a child’s reasoning appears to be not an easy task. Participants
overlooked some plausible, even ingenious solutions, alternative views, as well as
various misinterpretations and misconceptions. LeBlanc and Freiman (2011) point
at the potential of ‘feed-forward’ pedagogy while stressing the need to reinforce
pre-service teachers’ own conceptual understanding of mathematics and develop a
better understanding of how children think and explain their thinking by practicing
more their ability to understand the problem itself for an effective guiding of
students.
From a different perspective, but in the same idea of personalizing learning and
feedback via on-line tools, the System of Testing, Analyzing and Reporting for
students (STARs) is based on collaborative databases of questions (items) and
allows assessing students’ mathematical competence by using tests from the databases, followed by an individualized feedback, obtained by processing student’s
answers based on a multi-criteria analysis. Further, the student can receive new sets
of questions situated in his/her identified range of proximal development. For this
system to be effective, the questions need to capture essential elements of mathematics understanding and mathematics creativity (Singer and Singer 2010). To date,
the implementation of this system created opportunities for some teacher professional development studies. In one of the studies, in-service mathematics teachers at
secondary school level, involved in Mathematics Olympiads training, participated in
a two-week summer institute focused on teachers’ ability to pose multiple choice
problems that would assess student understanding. Two major aspects were identified. On the one hand, a certain resistance of teachers to shift from the mathematical
content of a posed problem toward interpretations of students’ thinking in relation to
that problem was identified; often, the posed-problem formulation was elliptic or full

5 Research into Practice: Teacher Education

33

of ambiguity, while the background topic was irrelevant for students’ motivation. On
the other hand, a change of participants’ behavior happened on two dimensions, both
observable in group interactions: an openness to discuss and analyze the quality of
their own posed problems, as well as an emerging awareness on the need of conceptual understanding of their students’ thinking. The training program, thus, contributed to the development of a reflective attitude toward addressing tasks to
students (Pelczer et al. 2014). It seems that an in-service training program that
systematically combines group interactions with individual problem-posing tasks
exploited during further interactions could significantly influence the building of
tasks that are focused on students learning with understanding.

5.3.3

Administrative Changes and Cross-Country Studies

In the former USSR, in special schools for mathematically gifted and talented,
mathematics was often taught by professional mathematicians who themselves had
attended such type of schools (e.g. Freiman and Volkov 2004; Karp 2016). In
Canada, with a provincially governed school system, New Brunswick is developing
an inclusive model of schools where students of all abilities must be provided with
appropriate learning opportunities within a regular classroom setting; here, a more
generalist approach to teacher preparation for such context was adopted (see more
in Freiman 2010).
In Israel, The Division of Gifted Education of Israeli Ministry of Education
encourages teachers to get special education, though there is still a shortage of
corresponding programs. Among examples of such initiative, Applebaum et al.
(2011) mention special teaching certification programs (in three teacher training
colleges and two Universities) and the first M.A. program (in Haifa University)
devoted to the education of gifted students. However, these programs are mainly
interdisciplinary and are not focused on specific school subjects. Applebaum et al.
(2011) investigated prospective teachers’ conceptions in these two programs about
teaching mathematically talented students in Canada and Israel by addressing the
issues of teachers’ capacity to solve challenging tasks and their views on mathematics education of mathematically promising students. By proposing a particular
open-ended task to the participants from both countries (Canada and Israel) and
analyzing responses, along with conducting a survey and a group discussion with
participants about their own capacity to deal with the task, the strategies they use to
solve the problem, the nature of the task suitable for mathematically talented students, and their own preparations to work with these students, it was found that
teachers cope with the challenging task with varying levels of success. The majority
used ‘non-systematic’ strategies, without analysis of the efficiency of the strategies.
It was also found that Israeli teachers used both non-systematic strategies and
systematic ones (that they have previously learned in a different context), whereas
most Canadian prospective teachers used mainly non-systematic strategies. The
discussion and questionnaire confirmed that participants in both countries
acknowledged the importance of challenging and open-ended tasks, sustaining also

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Research On and Activities For Mathematically Gifted Students

the need for a special curriculum for mathematically talented students. However,
they themselves did not feel prepared for dealing with such tasks in their classroom.
Results imply that teachers need better mathematical preparation in terms of solving
open-ended challenging tasks that would enable them not to limit students’
problem-solving processes with the finding of one suitable solution. Acquiring such
cognitive and meta-cognitive skills will help teachers in guiding their students on
the way to deeper and more meaningful mathematical knowledge (Applebaum et al.
2011). At a more general level, it implies that teacher education programs should,
first, expose teachers to the complexities of teaching mathematically promising
students, which might be beneficial for all students in the math class; second,
develop teachers’ capacities to investigate challenging tasks by proposing such
tasks and explorations during their training; and third, amplify teachers’ didactical
inventory of teaching strategies to allow identification and fostering of students’
abilities using inquiry-based challenging tasks.

6 Summary and Looking Ahead
The discussion about giftedness and intellectual power in children of various ages is
ultimately a discussion about future leadership and a precious human capital
resource. Consequently, the research in this area has a strong social impact.
Assuming this impact, the present survey started by offering a basic description
of the nature of mathematical giftedness, then offered a multidimensional analysis
of mathematical promise in students of various ages, covering cognitive, social, and
affective aspects, recent research in cognitive science and neuroscience, the relationships between novice knowledge and expertise, as well as the interplay between
giftedness and creativity. The next two chapters moved from research into practice,
focusing on programs and pedagogy for educating mathematically promising students and respectively on teacher education. The programs and pedagogy under
discussion referred to practices that could best encourage mathematical promise and
talents, approaching problem solving and problem posing, discourse and questioning, creativity and innovation, challenging mathematical tasks, curriculum and
textbooks, in-school programs and activities (with reference to ability grouping,
self-contained classes and specialized schools, acceleration and grade skipping),
and extra-curricular programs and activities (with reference to recreational mathematics and competitions). Given the social impact discussed above, the initial and
continuous professional development of teachers is of a major importance. Our
survey recorded features of effective teachers; structure and content of
teacher-education for teachers of mathematically gifted and talented students (but
also for all teachers because a promising student can be everywhere); and examples
of successful programs for preparing and supporting teachers of mathematically
gifted students in a variety of country-specific contexts.
Our overview recorded the advancement in the research and practices in this
emerging interdisciplinary field of mathematical promise in youth. Still, many

6 Summary and Looking Ahead

35

questions remain unanswered and they can orient further research to be carried out
in the following areas:
• The nature of classroom culture and the role of the teacher in fostering mathematical expertise
• The types of curriculum that support individualization and differentiation of
learning
• The use of neuroimaging techniques to inform the learning and teaching of
gifted and talented students
• The development and use of digital tools to facilitate personalized effective
learning
• The nature of professional development that supports teachers’ capacity to foster
mathematical promise in as many students as possible
• The impact of various frameworks of giftedness treatment for later professional
careers.
Acknowledgments We would like to give special thanks to Professor Dr. Marianne Nolte of the
University of Hamburg for her invaluable contributions and assistance in the development of this
paper.

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Erratum to: Research On and Activities
For Mathematically Gifted Students

Erratum to:
Chapter “Research On and Activities For Mathematically
Gifted Students” in: F.M. Singer et al., Research On and
Activities For Mathematically Gifted Students, ICME-13
Topical Surveys, DOI 10.1007/978-3-319-39450-3_1
In the original version of the book, the incorrect author name “Printer” has been
changed to read as “Trinter” in reference list and citation.

The updated online version for this chapter can be found at
DOI 10.1007/978-3-319-39450-3_1
© The Editor(s) (if applicable) and The Author(s) 2018
F.M. Singer et al., Research On and Activities For Mathematically Gifted Students,
ICME-13 Topical Surveys, DOI 10.1007/978-3-319-39450-3_2

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