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Encyclopedia of Mathematics Education

Encyclopedia of Mathematics Education

Keywords: Mathematics Education

E 230 External Assessment in Mathematics Education

Equations Technology-Based
Representational

Media Layer

Tables Written Graphs
Symbols
Spoken Pictures or
Language Diagrams

Traditional Experienced Concrete
Representational Based Models

Media Layer Metaphors

External Assessment in Mathematics Education, Fig. 1 A merged Kaput-Lesh diagram for thinking about
representational fluency

knowledge that we develop consist of models Notice that, in the literature on the diffusion of
for describing, explaining, designing, or innovations, complex systems tend to evolve
developing complex systems. So, models best when measurable goals are clear to all
(often embedded in purposeful artifacts or relevant subjects – and when strong steps are
tools) are among the most important kinds of taken to encourage diversity (of interactions),
knowledge that we need to develop and assess. selection (of successful interactions), commu-
Consequently, the question we must ask is as nication (about successful interactions), and
follows: How do we validate models? And, the accumulation (of successful interactions).
answer is that both qualitative and quantitative In mathematics education, many of the most
methods are useful for validating models. But important and powerful types of conceptual
the product isn’t simply a quantitative or qual- understandings occur in one of two closely related
itative claim. It’s a validated model – and forms. The first focuses on students’ abilities
trends and patterns involving development. to mathematize (e.g., quantify, dimensionalize,
• Is the unbiased objectivity of an assessment coordinate) situations which do not occur in
really assured by using “outside” specialists pre-mathematized forms and the second focuses
whose only familiarity with the relevant on representational fluency – or abilities that are
subjects come from pre-fabricated off-the- needed to translate from one type of description
shelf tests, questionnaires, interviews, and to another. For example, in the case of represen-
observation protocols which are not modified tational fluency, Kaput’s (1989) research on early
to emphasize the distinctive characteristics algebra and calculus concepts emphasized the
of the subjects and their interactions? And, importance of translations within and among the
if these “outside measures” are used for three types of representations which are desig-
purposes of accountability, can they really nated in the three ovals shown at the top of
avoid having powerful influences on the Fig. 1 (i.e., equations, tables, and graphs), and in
treatments themselves? a series of research studies known collectively as
• Can comparability of treatments really be The Rational Number Project, Lesh, Post, and
guaranteed by taking strong steps aimed at Behr (1987) emphasized the importance of trans-
trying to ensure that all teachers and all lations within and among the five types of repre-
students do exactly the same things, in exactly sentations which are designated in the five ovals
the same ways, and at exactly the same times? shown at the bottom of Fig. 1 (i.e., written

External Assessment in Mathematics Education E231

symbols, spoken language, pictures or their employees. So, how do specialists (or E
diagrams, concrete models, and experience- teams of specialists) get recognized and rewarded
based metaphors). for the quality of their work? For example, how do
professors validate their work? Or, how do doc-
From the perspective of psychometric theory, toral students validate the work on their Ph.D.
two of the main difficulties with test items that dissertations? Answers to these questions should
involve representational fluency result from the provide guidelines for the assessment of develop-
fact that when tasks involve description of ment related to students, teachers, curriculum
situations (a) there always exist a variety of dif- innovations, and other “subjects” in mathematics
ferent levels and types of descriptions and (b) education research. Space limitations do not allow
responding to one such task often leads to detailed answers to such questions to be given
improvements of similar tasks. So, according to here. But, when attention focuses on the systems
psychometric theory, where tasks are considered of knowledge being developed by students,
to have a single-level of difficulty which is teachers, and curriculum innovations, (a) it’s
unaffected by instruction and where the relative important to focus on the half-dozen-to-a-dozen
difficulty of two tasks also is considered to be “big ideas” which the subjects are intended to
unaffected by instruction, such tasks are develop, (b) it’s often useful to recognize that
discarded as being unreliable. Similarly, when a large part of what it means to “understand”
tasks focus on students’ abilities to conceptualize these big ideas tends to involve the development
situations mathematically, there once again of models (or interpretation systems) for making
exist a variety of different levels and types sense of relevant experiences, (c) these models
of mathematical descriptions, explanations, or often are embodied and function within purpose-
interpretations that can be given. So, once again, ful tools and artifacts, and (d) these tools and
the same two difficulties occur as for representa- artifacts often can be assessed in ways that simul-
tional fluency. taneously allow the underlying models to be
assessed. Procedures for achieving these goals
Especially when tests are used for accountabil- have been described in a variety of recent publi-
ity purposes and teachers are pressured to teach to cations about design research (e.g., Lesh and
these tests, it is important for such tests to include Kelly 2000; Lesh et al. 2007; Kelly et al. 2008),
tasks that involve actual work samples of desired and it is straightforward to adapt most of these
outcomes of learning – instead of restricting atten- procedures to apply to assessment purposes.
tion to indirect indicators of desired achieve-
ments. For example, if the development of References
a given concept implies that a student should be
able to do skill-level tasks T1, T2, . . . Tn, then Brown AL (1992) Design experiments: theoretical and
tasks T1, T2, . . . Tn tend to be indicators similar methodological challenges in creating complex inter-
to wrist watches or thermometers – in the sense ventions in classroom settings. J Learn Sci 2(2):141–178
that it is possible to change the readings on wrist
watches or thermometers without in any way Carmona G (2012) Assessment design: network-based
influencing the time or the weather. But, how environments as and for formative assessment and
can assessments of complex achievements be evaluation of student thinking (research paper).
achieved inexpensively, during brief periods of Annual Meeting of the American Educational
time, and in a timely fashion that provides useful Research Association, Vancouver
information for relevant decision makers? In
modern businesses where continuous adaption is Carmona G, Krause G, Monroy M, Lima C, A´ vila A,
necessary, and especially in knowledge industries Ekmekci A (2011) A longitudinal study to investigate
or in academic institutions, decision makers sel- changes in students’ mathematics scores in Texas
dom use multiple-choice tests or questionnaires to (research presentation). Annual Meeting of the Amer-
assess the quality of the kinds of complex work ican Educational Research Association, New Orleans
that constitute the most important activities of
Chudowsky N, Pellegrino JW (2003) Large-scale assess-
ments that support learning: what will it take? Theory
Pract 42(1):75

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Collins A (1992) Toward a design science of education. Lesh RA, Kelly EA (2000) Multitiered teaching experi-
In: Scanlon E, O’Shea T (eds) New directions in ments. In: Kelly AE, Lesh RA (eds) Handbook of
educational technology, vol 96. Springer, New York research design in mathematics and science education.
Lawrence Erlbaum, Mahwah
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Sci 13(1):15–42 formance in school mathematics. American Associa-
tion for the Advancement of Science, Washington, DC
Ebel RL (1962) Content standard test scores. Educ
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Lindquist ET (ed) Educational measurement. Problems of representation in teaching and learning
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Glaser R (1963) Instructional technology and the science education. Lawrence Erlbaum, Mahwah,
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F

Fieldwork/Practicum in Mathematics Guyton and McIntyre 1990) but commonly com-
Education prise three components:
• A university or college-based curriculum,
Paula Ensor
University of Cape Town, Cape Town, usually involving theoretical foundation courses
South Africa (educational psychology, philosophy and soci-
ology of education, historical approaches and
Keywords policy etc.)
• “Method” or “didactics” courses (devoted
Practicum; Field experience; School experience; specifically to the teaching of a specific
Teaching practice; Block school experience; subject, such as mathematics)
Initial teacher education; Preservice teacher • A school teaching experience, termed
education “teaching practice,” “field experience,” or
“practicum,” during which student teachers
Definition are placed in schools
The organization of the field experience
The practicum, teaching practice, or field experi- component varies considerably, (McIntyre et al.
ence refer to that component of those preservice 1996; Knowles and Cole 1996) including in the
(or initial) teacher education programs which following ways:
place student teachers in schools for a stipulated • The contractual arrangement with schools – in
period of time, for the purposes of classroom some countries universities are required to pay
observation and/or the teaching of lessons, schools to provide for field experience, in
usually under supervision. others this is not the case; in some countries
schools are obliged by regulation to accept
Features student teachers, in others not
• Who undertakes the supervision of student
Preservice mathematics teacher education teachers in schools (school teachers, inspec-
programs offered by high education institutions tors, educational advisors)
internationally vary greatly in composition across • The length of the field experience, which can
countries (see Comiti and Loewenberg Ball 1996; range from a few weeks to a whole year and
can be organized in discrete in blocks or
continuously throughout the year
• The nature of the partnership between the
university-based and school-based supervisors

S. Lerman (ed.), Encyclopedia of Mathematics Education, DOI 10.1007/978-94-007-4978-8,
# Springer Science+Business Media Dordrecht 2014

F 234 Fieldwork/Practicum in Mathematics Education

• The choice of school settings and the degree to interaction between foundational disciplines
which classroom practice in these schools and “methods” courses within initial teacher
aligns with “good practice” as espoused by education programs, between these programs
the teacher education provider and the practicum, and between initial teacher
education provision and the classroom prac-
• The degree of teaching responsibility given to tice of beginning teachers (see Dewey 1904,
student teachers Hirst 1990, and McIntyre 1995). Jaworski and
Gellert (2003) suggest a four-model contin-
• How explicitly requirements for the field uum to describe the level of integration or
experience are set out in advance and how insulation of the theoretical and practical
decisions about these requirements are made aspects of initial teacher education.
• The tacit, or craft dimension in the professional
• The degree of alignment between the vision development of teachers, or “professional craft
and values of the teacher education provider knowledge” (forms of knowledge which are not
and the schools in which student teachers are realizable in language and which are acquired
located via modeling and mentoring in the site of
practice (see Polanyi 1983 and Sho¨n 1983)).
Research on the Field Experience
As indicated, initial teacher education involves Cross-References
two distinct sites of learning and practice, each
with specialized identities, practices, forms of ▶ Mathematics Teacher Education Organization,
knowledge and relationships, and preferred Curriculum, and Outcomes
modes of pedagogy: the university or college
teacher education provider on the one hand and References
the schools involved in the practicum on the
other. (These in turn are oriented towards a third Bergsten C, Grevholm B, Favilli F (2009) Learning to
site: the schools into which student teachers will teach mathematics: expanding the role of practicum
move after graduation to take on their duties as as an integrated part of a teacher education pro-
beginning teachers.) The practicum constitutes gramme. In: Even R, Loewenberg Ball D (eds) The
a potential bridge between these two sites. professional education and development of teachers of
mathematics. The 15th ICMI study. Springer,
Research on the practicum within mathemat- New York, pp 57–70
ics education and within education studies more
generally is not extensive and focuses in the main Comiti C, Loewenberg Ball D (1996) Preparing teachers
on issues such as the degree of change in student to teach mathematics: a comparative perspective. In:
teachers’ knowledge, beliefs, decision-making Bishop A, Clements K, Keitel C, Kilpatrick J,
strategies, reflectiveness, and teaching practices Laborde C (eds) International handbook of mathemat-
as a result of the practicum experience (Bergsten ics education. Kluwer, Dordrecht, pp 1123–1115
et al. 2009). Some research evaluates interven-
tions aimed at reducing the insulation between Dewey J (1904) The relationship of theory to practice. In:
teacher educator provider and school, in order to Cochran-Smith M, Feiman-Nemser S, McIntyre DJ,
align school experiences more closely with the Demers KE (2008) Handbook of research on teacher
goals of initial teacher education. education, 3rd edn. Routledge, New York, pp 787–799

Three research areas which contribute towards Guyton E, McIntyre DJ (1990) Student teaching and
inquiry in initial teacher education, and the school experiences. In: Houston WR, Haberman M,
practicum in particular, are: Sikula J (eds) Handbook of research on teacher
• Teacher socialization (and in particular, the education. Macmillan, New York, pp 514–553

degree to which the field experience reinforces Hirst PH (1990) The theory-practice relationship in
or alters the predispositions towards teaching teacher training. In: Booth M, Furlong J, Wilkin
of student teachers) (Zeichner and Gore 1990). M (eds) Partnership in initial teacher training. Cassell,
• The issue of the relationship between “theory London, pp 74–86
and practice” which informs the study of the
Jaworski B, Gellert U (2003) Educating new mathematics
teachers. In: Bishop A, Clements MA, Keitel C,

Frameworks for Conceptualizing Mathematics Teacher Knowledge F235

Kilpatrick J, Leung FKS (eds) Second international knowing something is best evidenced in the perfor- F
handbook of mathematics education. Part two. mance of teaching. The Oxford philosopher John
Kluwer, Dordrecht, pp 829–875 Wilson (1975) endorsed and extended Aristotle’s
Knowles JG, Cole AL (1996) Developing practice through position on teacher knowledge with the argument
field experiences. In: Murray FB (ed) The teacher that comprehension of the logic of concepts offered
educator’s handbook. Jossey-Bass, San Francisco, guidance on how to teach them. In other words, not
pp 648–688 only do we need to know what we teach in the sense
McIntyre D (1995) Initial teacher education as practical of understanding it, but such a profound quality of
theorising: a response to Paul Hirst. Br J Educ Stud knowing actually acts as a guide to the pedagogy,
43(4):365–383 i.e., the “how to teach,” of subjects such as mathe-
McIntyre DJ, Byrd DM, Foxx SM (1996) Field and matics. This position has recently been developed
Laboratory experiences. In: Sikula J, Buttery TJ, by Watson and Barton (2011) in terms of pedagog-
Guyton E (eds) Handbook of research on teacher ical application of “mathematical modes of inquiry.”
education, 2nd edn. Macmillan, New York, pp 171–193 However, the seminal work of Lee Shulman and his
Polanyi M (1983) The tacit dimension. Peter Smith, colleagues in the 1980s underpins the dominant
Gloucester frameworks currently in use for conceptualizing
Sho¨n DA (1983) The reflective practitioner: how profes- mathematics teacher knowledge.
sionals think in action. Temple Smith, London
Zeichner KM, Gore J (1990) Teacher socialisation. In:
Houston WR, Haberman M, Sikula J (eds) Handbook
of research on teacher education. Macmillan,
New York, pp 329–348

Lee Shulman

Frameworks for Conceptualizing In a presidential address to the American Educa-
Mathematics Teacher Knowledge tional Research Association, Shulman argued
that in recent (American) research on teaching,
Tim Rowland insufficient emphasis had been placed on the
Faculty of Education, University of Cambridge, subject matter under consideration: he called
Cambridge, UK this omission “the missing paradigm.” Shulman’s
highly influential perspective on teacher
Keywords knowledge arose from empirical research, the
Knowledge Growth in a Profession project,
Mathematics teacher knowledge; Subject matter conducted at Stanford University in the mid-
knowledge; Pedagogical content knowledge; 1980s. His tripartite conception of teachers’
Mathematical knowledge for teaching; Knowledge knowledge of the content that they teach includes
Quartet not only knowledge of subject matter but
also pedagogical content knowledge, as well
Introduction as knowledge of curriculum. Subject matter
knowledge (SMK) refers to the “amount and
Discussion of the relationship between knowledge organization of the knowledge per se in the
and the profession of teaching is particularly con- mind of the teacher” (Shulman 1986, p. 9) and
voluted because knowledge is itself the commodity is later (Grossman et al. 1989) further analyzed
at the heart of education and the very goal of into substantive knowledge (the key facts,
teaching. For a starting point in theorizing knowl- concepts, principles, and explanatory frame-
edge and teaching, one can turn to Aristotle’s works in a discipline) and syntactic knowledge.
(384–322 BC) aphorism “it is a sign of the man The latter is knowledge about the nature of
who knows, that he can teach” (Metaphysics, inquiry in the field and the mechanisms through
Book 1). This can be interpreted that “really” which new knowledge is introduced and accepted
in that community; in the case of mathematics,
it includes knowledge about inductive and

F 236 Frameworks for Conceptualizing Mathematics Teacher Knowledge

deductive reasoning, the affordances and limita- and “horizon content knowledge” (HCK). CCK is
tions of exemplification, and problem-solving essentially “school mathematics,” applicable in
heuristics and proof. Pedagogical content knowl- a range of everyday and professional contexts
edge consists of “ways of representing the subject demanding the ability to calculate and to solve
which makes it comprehensible to others. . .[it] mathematics problems. SCK, on the other hand,
also includes an understanding of what makes is knowledge of mathematics content that mathe-
the learning of specific topics easy or difficult matics teachers need in their work, but others do
. . .” (Shulman 1986, p. 9). not. This would include, for example, knowing
why standard calculation routines work, such as
In addition to his taxonomy of kinds of teacher “invert and multiply” for fraction division. Exam-
knowledge, Shulman (1986) also draws out three ples of SCK offered by Ball et al. (2008) include
forms of such knowledge. These are propositional the evaluation of various student responses to col-
knowledge, consisting of statements about what is umn subtraction problems, claiming that the kinds
known about teaching and learning; case knowl- of knowledge required to diagnose incorrect strat-
edge, being salient instances of theoretical con- egies or to understand correct but nonstandard
structs which serve to illuminate them; and ones are essentially mathematical rather than ped-
strategic knowledge, where propositional and agogical. On the other hand, they suggest that
case knowledge are applied in the exercise of knowing about typical errors in advance, thereby
judgment and wise action. Shulman’s analysis enabling them to be anticipated, is a type of ped-
remains the starting point for most subsequent agogical content knowledge which they call
analyses of, and further investigation into, the pro- “knowledge of content and students” (KCS).
fessional knowledge base of mathematics teachers, Thus, the argument goes as follows: SCK is acces-
particularly in the Anglo-American research orbit. sible to the competent mathematician, by refer-
ence to their knowledge of mathematics (see also
Mathematical Knowledge for Teaching Watson and Barton 2011). KCS, on the other hand,
is conceived as a body of knowledge deriving from
Deborah Ball entered the research field on the empirical research in the behavioral and social
cusp of Shulman’s work at Stanford, and sciences, including mathematics education.
her contribution to research in the field of
mathematics teacher knowledge has been exten- Note that the MKfT model is not a simple
sive and far reaching. Videotapes and other records elaboration of Shulman’s three content
of her own elementary classroom teaching have categories, since curriculum knowledge is no
been an important source of data in the investiga- longer a separate category. In effect, it has been
tions of her research group at the University partitioned into two: horizon content knowledge,
of Michigan. The “practice-based theory of which becomes the third component of SMK, and
knowledge for teaching” (Ball and Bass 2003) knowledge of content and curriculum, which is
that emerges from the Michigan studies unpicks, now one of three components of PCK. In fact,
refines, and reconfigures the three kinds of content Ball et al. (2008, p. 391) draw out two aspects of
knowledge – subject matter, pedagogical, and curriculum knowledge, as conceived by
curricular – identified by Shulman (1986). This Shulman, that are often overlooked. The first,
Mathematical Knowledge for Teaching (MKfT) lateral curriculum knowledge, relates to cross-
framework (Ball et al. 2008) has been adopted by curricular mathematical connections, invoking
many researchers as a theoretical framework for conceptions and applications that enrich stu-
interpreting their own classroom data, as well as dents’ experience and appreciation. The second,
a language for articulating their findings. vertical curriculum knowledge, entails knowing
what mathematical experiences precede those in
In the MKfT deconstruction of Shulman, SMK a given grade level and what will follow in the
is separated into “common content knowledge” next, and subsequent, grades. Ball et al. then
(CCK), “specialized content knowledge” (SCK), relabel vertical knowledge as horizon content

Frameworks for Conceptualizing Mathematics Teacher Knowledge F237

knowledge and include it within SMK. The of kinds of knowledge without undue concern for F
importance of this Janus-like quality in mathe- the boundaries between them. The second dimen-
matics teachers is clear. On the one hand, they sion, transformation, concerns knowledge in action
need to know what knowledge their students as demonstrated both in planning to teach and in
can be expected to bring with them as a result the act of teaching itself. A central focus is on the
of previous instruction, including restricted representation of ideas to learners in the form of
conceptions and even misconceptions. On the analogies, examples, explanations, and demonstra-
other hand, Dewey (1903, p. 217) cautioned tions. The third dimension, connection, concerns
teachers against fostering “mental habits and the ways by which the teacher achieves coherence
preconceptions which have later on to be bodily within and between lessons: it includes the
displaced or rooted up in order to secure a proper sequencing of material for instruction and an
comprehension of the subject,” thereby impeding awareness of the relative cognitive demands of
progress in the later grades. different topics and tasks. The final dimension,
contingency, is witnessed in classroom events that
The Knowledge Quartet were not envisaged in the teachers’ planning. In
commonplace language, it is the ability to “think
In a study of London-based graduate trainee pri- on one’s feet.” Rowland, Huckstep, and Thwaites
mary teachers, Rowland, Martyn, Barber, and Heal (2005) include a more detailed conceptual account
(2000) found a positive statistical connection of these four dimensions and of the “grounded
between scores on a 16-item audit of content theory” approach to analyzing the video recordings
knowledge and competence in mathematics teach- of the 24 lessons.
ing on school-based placements. A team at the
University of Cambridge then surmised that if The Knowledge Quartet is a lens through which
superior content knowledge really does make the observer “sees” classroom mathematics instruc-
a difference when teaching elementary mathemat- tion. It is a theoretical tool for observing, analyzing,
ics, it ought somehow to be observable in and reflecting on actual mathematics teaching.
the practice of the knowledgeable teacher. The Devised first with researchers in mind, it has sub-
Cambridge team therefore set out to identify, and sequently been applied to support and facilitate the
to understand better, the ways in which elementary improvement of mathematics teaching. In particu-
teachers’ mathematics content knowledge, or the lar, it offers a four-dimensional framework against
lack of it, is made visible in their teaching. which mathematics lessons can be discussed,
with a focus on their subject matter content
The Knowledge Quartet (KQ) was the and the teacher’s related knowledge and beliefs.
outcome of research in which 24 lessons taught A book aimed at mathematics teachers and teacher
by elementary school trainee teachers were educators (Rowland et al. 2009) explains how
videotaped and scrutinized. The research team to analyze and give feedback on mathematics
identified aspects of trainees’ actions in the teaching, using the Knowledge Quartet.
classroom that could be construed as being
informed by their mathematics subject matter Both the Mathematical Knowledge for
knowledge or pedagogical content knowledge. Teaching framework and the Knowledge Quartet
This inductive process initially generated a set are practice-based theories of knowledge for
of 18 codes (later expanded to 20), subsequently teaching. However, while parallels can be
grouped into four broad, superordinate categories drawn between the origins of the two frame-
or dimensions – the “Quartet.” works, the two theories look very different. In
particular, the theory that emerges from the
The first dimension of the Knowledge Quartet, Michigan studies aims to unpick and clarify the
foundation, consists of teachers’ mathematics- formerly somewhat elusive and theoretically
related knowledge, beliefs, and understanding, undeveloped notions of SMK and PCK. In the
incorporating Shulman’s classic 3-way taxonomy Knowledge Quartet, however, the distinction
between different kinds of mathematical

F 238 Functions Learning and Teaching

knowledge is of lesser significance than the References
classification of the situations in which math-
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by Petrou and Goulding in Rowland and
Ruthven (2011). Ball DL, Thames MH, Phelps G (2008) Content
knowledge for teaching: what makes it special?
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significantly shape ways of thinking about how
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conceptual connections between these separate contextual application of modes of mathematical
discourses can be discerned. enquiry. In: Rowland T, Ruthven K (eds) Mathemati-
cal knowledge in teaching. Springer, London/
New York, pp 65–82

Wilson J (1975) Education theory and the preparation of
teachers. NFER, Windsor

Functions Learning and Teaching

Mogens Allan Niss
Institut for Natur, Systemer og Modeller,
IMFUFA, Roskilde Universitet, Roskilde,
Denmark

Definition and Brief History

The notion of function has three different, yet inter-
related, aspects. Firstly, a function is a purely

Functions Learning and Teaching F239

mathematical entity in its own right. Depending on introduced in the curricula of many countries F
the level of abstraction, that entity can be intro- from the late nineteenth century onwards,
duced, for example, as either a correspondence that following the reform program proposed by Felix
links every element in a given domain to one and Klein (NCTM 1970/2002), p. 41; Schubring
only one element in another domain, called the co- 1989, p. 188). Today, some version of the notion
domain, or as a certain kind of relation, i.e., a class of function permeates mathematics curricula in
of ordered pairs (in a Cartesian product of two most countries. However, the different aspects of
classes), which may be represented as a graph, or the notion of function also make it highly diverse,
as a process – sometimes expressed by way of an multifaceted, and complex, which introduces
explicit formula – that specifies how the dependent challenges to the conceptualization as well as to
(output) variable is determined, given an indepen- the teaching and learning of functions.
dent (input) variable, or as defined implicitly as
a parametrized solution to some equation (alge- Against this background, the concept of function
braic, transcendental, differential). Secondly, func- in mathematics education has given rise to a huge
tions have crucial roles as lenses through which body of research. The origins of this research seem
other mathematical objects or theories can be to date back to debates in the 1960s about the right
viewed or connected, for instance, when perceiving (or wrong) way to define a function. Thus, Nicholas
arithmetic operations as functions of two variables; (1966, p. 763) compares and contrasts three defini-
when a sequence can be viewed as a function tions (which he labels “variable,” “set,” and “rule”),
whose domain is the set of natural numbers; when which, in his view, generate a dilemma, because
maximizing the area of a rectangle given a constant they are not logically equivalent. The first empirical
perimeter or perceiving reflections, rotations, studies also seem to stem from the late 1960s.
and similarities of plane geometrical figures as Empirical studies focused on the formation of the
resulting from transformations of the plane; or concept of function, which has also preoccupied the
when Euler’s j-function (for a natural number n, far majority of subsequent research, as is reflected
j(n) is the number of natural numbers 1,2,. . ., by the seminal volume on this topic edited by
n that are co-prime with n) allows us to capture Dubinsky and Harel (1992) and in the relatively
and state fundamental results in number theory and recent overview of significant research offered by
cryptography, etc. Thirdly, functions play crucial Carlson and Oehrtman (2005).
parts in the application of mathematics to
and modelling of extra-mathematical situations Challenges to the Teaching and Learning
and contexts, e.g., when the development of of Function
a biological population is phrased in terms of The reason why the concept of function itself
a nonnegative function of time, when competing has attracted massive attention from researchers
coach company tariff schemes are compared by is that students (and many pre- or in-service
way of their functional representations,, or when teachers, see Even 1993) have experienced,
the best straight line approximating a set of exper- and continue to experience, severe difficulties
imental data points is determined by minimizing at coming to grips with the most significant
the sum-of-squares function. aspect of this concept in both intra- and
extra-mathematical contexts. More specifically,
These aspects of the notion of function make researchers have focused on identifying and
this notion one of the most fundamental and sig- analyzing the learning difficulties encountered
nificant ones in mathematics, and hence in math- with the concept of function; on explaining
ematics education. This is reflected both in the these difficulties in historical, philosophical, and
history of mathematics (the term “function” cognitive terms; and on proposing effective
going back at least to Leibniz (Boyer (1985/ means to counteract them in teaching. In so
1698), p. 444) and in the history of mathematics doing, researchers have introduced a number of
education, where the notion of function as terms and distinctions (e.g., between “action” and
a unifying concept in mathematics was “process” (Dubinsky and Harel 1992b)).

F 240 Functions Learning and Teaching

One important issue that arises in this linear or affine functions, exponential functions,
context is the fact that functions can be given recursively defined functions, and above all the
several different representations (e.g., verbal, real and complex functions that appear in calcu-
formal, symbolic (including algebraic), diagram- lus and analysis), which have been the subject of
matic, graphic, tabular), each of which captures study in an immense body of research.
certain, but usually not all, aspects of the concept.
This may obscure the underlying commonality – Another demanding facet of the concept of
the core – of the concept across its different function is the process-object duality (cf., e.g.,
representations, especially as translating from several chapters in Dubinsky and Harel (1992a))
one representation to another may imply loss of that is characteristic of many functions, espe-
information. If, as often happens in teaching, cially the ones that students encounter in second-
learners equate the concept of function with just ary and undergraduate mathematics teaching. In
one or two of its representations (e.g., a graph or its process aspect, a function yields outputs as
a formula), they miss fundamental features of the a result of inputs. In its object aspect, a function
concept itself. This is also true of the many dif- is just a mathematical entity which may engage in
ferent equivalent symbolic notations for the very relationships with other objects, or be subjected
same function (e.g., y ¼ x2 À 1/x, f: x 1x2 À 1/x, to various sorts of treatment (e.g., differentiation
in both cases provided x 6¼ 0; f: Aˆ \{0} at a point). Oftentimes the transition from a pro-
1Aˆ defined by f(x) ¼ x2 À 1/x; cess view to an object view of function is a severe
f(x) ¼ (x À 1)(x2 + x + 1)/x, x IˆAˆ \ {0}; f ¼ {(x, challenge to students.
x2 À 1/x)| x IˆAˆ \ {0}}; (x,y) Iˆf Uˆ y ¼ x2 À 1/x U` x
ˆIAˆ U` x ¼6 0; x ¼ y2 À 1/y, y IˆAˆ \ {0}, just to indicate Overcoming Learning Difficulties
a few). Interpreting and translating between func- In response to the observed learning difficulties
tion representations in intra- or extra-mathe- attached to functions and analyses of these diffi-
matical settings proves to be demanding for culties, mathematics educators have invested
learners. Of particular significance here is the efforts in proposing, designing, and implementing
translation between visual and formal represen- intervention measures so as to address and coun-
tations of the same function, which for some teract these difficulties specifically. The overarch-
learners are difficult to reconcile. ing result is that it is possible to counteract the
learning difficulties at issue, but this requires
Functions come in a huge variety of sorts, intentional and focused work on designing rich
types, and cases, ranging from familiar ones and multifaceted learning environments and
(such as linear or quadratic functions of one teaching-learning activities that are typically
variable) to abstract and complex ones (such as extensive and time-consuming. In other words,
the integral as a real-valued functional operating the desired outcomes are not likely to occur by
on the space of Riemann-integrable functions of n default with most students, they have to be aimed
variables). The plethora of functions of very at, and they come at a price: time and effort.
different kinds means that students’ concept of
function is also delineated by the set of function A few examples: One focal point has been to
specimens and examples of which the students help students develop a process conception of
have gained experience. This is an instance of function (in contrast to an action conception), by
the well-known distinction between concept way of technology (Goldenberg et al. 1992). Tech-
definition and concept image playing out in nology has also been used to consolidate students’
a very manifest manner in the context of function concept images so as not to “overgeneralize” the
(Vinner 1983), in particular in teaching and prototypical function examples that initially
learning that focuses on abstract functions. This underpinned their conception. Helping students to
distinction also proves important when zooming develop an object conception of function (by way
in on special classes of functions (such as of reification) has preoccupied many researchers,
e.g., Sfard (1992).

Functions Learning and Teaching F241

Future Research Harel G (eds) The concept of function: aspects of F
While research in this area in the past has focused epistemology and pedagogy, vol 25, MAA notes.
on the learning (and teaching) of the concept of Mathematical Association of America, Washington,
function in contexts when functions are already DC, pp 85–106
meant to be present, or presented to students, very Even R (1993) Subject-matter knowledge and pedagogical
little – if any – research has dealt with situations content-knowledge: prospective secondary teachers
in which students are requested or encouraged and the function concept. J Res Math Educ
to uncover or introduce, themselves, functions 24(2):94–116
or functional thinking into an intra- or Goldenberg P, Lewis P, O’Keefe J (1992) Dynamic
extra-mathematical context. Furthermore, there is representation and the development of a
a need for future research that focuses on designing process understanding of function. In: Dubinsky E,
teaching-learning environments that help generate Harel G (eds) The concept of function: aspects of
transfer of the notion of function from one setting epistemology and pedagogy, vol 25, MAA notes.
(e.g., real functions of one variable) to another Mathematical Association of America, Washington,
(e.g., functions defined on sets of functions). DC, pp 235–260
NCTM (1970) A history of mathematics education in
References the United States and Canada. National Council of
Teachers of Mathematics, Reston
Boyer CB (1985/1698) A history of mathematics. Nicholas CP (1966) A dilemma in definition. Am Math
Princeton University Press, Princeton Mon 73(7):762–768
Schubring G (1989) Pure and applied mathematics
Carlson M, Oehrtman M (2005) Research sampler: 9. Key in divergent institutional settings in Germany: the
aspects of knowing and learning the concept of role and impact of Felix Klein. In: Rowe DE,
function. Mathematical Association of America. McCleary J (eds) The history of modern mathematics,
http://www.maa.org/t_and_l/sampler/rs_9.html vol 2, Institutions and applications. AP, San Diego,
pp 171–220
Dubinsky E, Harel G (eds) (1992a) The concept of func- Sfard A (1992) Operational origins of mathematical
tion: aspects of epistemology and pedagogy, vol 25, objects and the quandary of reification – the case of
MAA notes. Mathematical Association of America, function. In: Dubinsky E, Harel G (eds) The concept of
Washington, DC function: aspects of epistemology and pedagogy,
vol 25, MAA notes. Mathematical Association of
Dubinsky E, Harel G (1992b) The nature of the America, Washington, DC, pp 59–84
process conception of function. In: Dubinsky E, Vinner S (1983) Concept definition, concept image and
the notion of function. Int J Math Educ Sci Technol
14(3):293–305

G

Gender in Mathematics Education a simple synonym of sex” (p. 97); this is also
evident in the mathematics education literature.
Helen Forgasz
Faculty of Education, Monash University, In this encyclopedia entry, the term “gender”
Clayton, VIC, Australia is used in the sense that Leder (1992) clarified
it with respect to mathematics learning. Gender
Keywords is considered a social construct, and gender
differences are considered to be contextually
Gender differences; Sex differences; Equity; bound and not fixed, that is, they are not geneti-
Women’s movement; Feminism; Social justice; cally determined. Sex differences are only
Explanatory models; Technology; Neuroscience described in this entry with respect to issues
associated with biological distinctions.

Issues of Definition Historical Overview of Gender and
Mathematics Education Research

According to Haig (2004), it was the feminist Research on gender issues in mathematics
scholars of the 1970s who adopted gender “as education began in earnest during the 1970s. This
a way of distinguishing ‘socially constructed’ work was mainly situated in the English-speaking,
aspects of male–female differences (gender) developed world (USA, UK, Australia), as well
from ‘biologically determined’ aspects (sex)” as in some European countries. The common
(p. 87). In the mathematics education literature, research findings were the following: (i) on aver-
the gradual shift from “sex differences” to age, females’ achievement levels were lower than
“gender differences” occurred during the period males’, particularly when it came to challenging
from the late 1970s into the 1980s. Fennema’s problems (it should be noted that it was recognized
(1974) seminal work in the field was reported as that the gender difference was small compared to
“sex” differences in mathematics achievement, within sex variations), (ii) females’ participation
and in the renowned Fennema and Sherman rates in mathematics were lower than males’ when
studies on affective factors (e.g., Fennema and mathematics was no longer compulsory, and (iii)
Sherman 1977), the findings were also described on a range of affective/attitudinal measures with
as “sex” differences. As noted by Haig (2004), in respect to mathematics or to themselves as math-
more recent times, the “distinction is now only ematics learners, females’ views were less “func-
fitfully respected and gender is often used as tional” (leading to future success) than males’.

S. Lerman (ed.), Encyclopedia of Mathematics Education, DOI 10.1007/978-94-007-4978-8,
# Springer Science+Business Media Dordrecht 2014

G 244 Gender in Mathematics Education

The theoretical frameworks of this early in bringing the field of gender and mathematics
research were founded in those prevalent at the education to prominence. The International Orga-
time, that is, a positivist view that findings were nization for Women in Mathematics Education
generalizable beyond the contexts in which the [IOWME] sessions at the ICMI conferences in
research was conducted. Most of the research was Budapest (1988), Montreal (1992), and Seville
quantitative, although sometimes accompanied (1996) were watershed events. New scholarship
by qualitative dimensions. The dominant femi- was brought to light, and there were several nota-
nist perspective that could be inferred from the ble outcomes: Roger and Kaiser’s (1995) book
stances adopted was that of “liberal” feminism, introduced various feminist perspectives on gender
that is, that females’ relative underperformance issues in mathematics learning; Burton’s (1990)
in mathematics and their under-representation in book included important contributions to the field
challenging mathematics offerings at the school by authors from a range of international settings;
level, and in mathematics and science-related and in Keitel’s (1998) book, gender was consid-
courses of study at the tertiary level as well as ered within the broader framework of social justice
in related careers, had to be brought up to the and equity. ICMI’s support for a study on gender
levels of those found for males. and mathematics learning (Ho¨o¨r, Sweden, 1993)
was also significant. Ironically, it was at the Ho¨o¨r
In the west, the “Women’s Movement” conference that the all-male leadership of ICMI,
(second wave of feminism) was very active the organization representing the field of mathe-
across western societies in the 1970s and 1980s. matics education internationally, was openly chal-
Within the broader context of women’s inferior lenged; this may well have been the catalyst for
status in society, girls had been identified as edu- change. In subsequent years, women in mathemat-
cationally disadvantaged. Women’s legal rights ics education have played significant and active
and roles in the family and the workplace, as well roles in the leadership of ICMI.
as sexuality and reproductive rights, were all
under scrutiny. Legislation was enacted to The “golden era” of research on gender and
address women’s demands for a more equitable mathematics education appears to have ended in
society. Money was flowing for educational the mid-1990s. In the West, there was a sense that
research to address female disadvantage in math- the “female problem” in society had been solved.
ematics (and science), and intervention programs For gender and mathematics education, research
flourished – see Leder et al. (1996) for an over- and intervention funding dried up; governments
view of a range of these intervention programs, had other considerations at the top of their
their outcomes, and what was learned from them. agendas. Arguably, too, there was a backlash to
the focus on girls’ education, and attention
In the 1980s and 1990s, the “founding mother” switched to boys’ educational needs.
in the field, Elizabeth Fennema, was joined by
a number of eminent scholars. Among them were One positive and lasting outcome of the
Gilah Leder and Leone Burton whose books (e.g., era was the mandating of statistical data on
Fennema and Leder 1990; Burton 1990) and educational outcomes emanating from many
other scholarly journal articles, handbook contri- government sources to be reported by sex. At
butions (e.g., Leder 1992), and conference papers the international level, there is also easy access
formed the building blocks for ongoing research to the Trends in Mathematics and Science Studies
in the field. The history of women’s place in [TIMSS] data and the Program for International
mathematics (e.g., Henrion 1997), including Student Assessment [PISA] data. These data
mathematics education (e.g., Morrow and Perl provide researchers with the capacity to examine
1998), and the relationship to mathematics achievement and participation in mathematics
curricula (e.g., Kaiser and Rogers 1995; Perl for gender differences, both within and across
1978) were also documented. nations, for the age cohorts taking these tests. It
should be noted that affective data are also
The International Commission on Mathemati- included in the TIMSS and PISA databases.
cal Instruction [ICMI] had a significant role to play

Gender in Mathematics Education G245

The ability to refine investigations for achieve- men; radical feminism, targeting the power/politi- G
ment by mathematics content domain and/or by cal system that oppresses women; feminist
various other equity factors (e.g., socioeconomic standpoints, founded in the lives and experience
background, race, ethnicity, and religious of women) and theoretical frameworks from other
affiliation) is possible from some large-scale disciplines to underpin subsequent research
international data sources, as well as those avail- endeavors (e.g., postmodernism, rejection of the
able within nations, for example, national testing homogeneity of groups such as girls/boys, instead
and competition data. It was the reporting of focusing on the relative truths of individuals;
gender differences from large-scale data sources poststructuralism, gender is socially and culturally
that provided the initial impetus for research in created through discourse; queer theory, gender is
the field; this must continue. In the contemporary not fixed and does not define the individual;
world of the twenty-first century, it is these kinds postcolonialism, identifies parallels between
of data that have sparked concerns and interest women in a patriarchy and recently decolonized
in research on gender issues in mathematics countries; racism is implicated).
learning in the developing world and in Asia.
Fennema (1995) explained that feminist
Theoretical Considerations scholars had convincingly argued that male
In seeking explanations for observed gender perspectives dominated traditional research
differences in mathematics learning outcomes, approaches and interpretations and that this view
a number of explanatory models were postulated was incomplete as female perspectives were omit-
in the early period of research in the field. Several ted. To progress towards gender equity in mathe-
focused on explanations for specific aspects matics education, she urged researchers to embrace
of mathematics education including differential “new types of scholarship focused on new ques-
elective course enrolments (Eccles et al. 1985), tions and carried out with new methodologies”
mathematics achievement (Ethington 1992), (p. 35) including feminist methodologies, through
achievement on cognitively demanding tasks which the world is viewed and interpreted from
(Fennema and Peterson 1985), and explanations a female perspective.
for the relationships between race, socioeconomic
background, and gender differences in levels of Following the lead of feminist science
performance on standardized tests (Reyes and educators, Burton (1995) challenged the mathe-
Stanic 1988). Leder’s (1990) model was more matics establishment in questioning the objectivity
general. Two groups of factors – student-related of the discipline. She argued that mathematics was
and environmental – were identified as interacting contextually bound and that from this perspective
contributors to patterns of gender difference in could be viewed in more human terms; this, she
achievement and participation. The postulated contended, would challenge traditional pedagogi-
models shared several common elements: social cal approaches to the teaching of mathematics as
environment, significant others, learning context, well as the content taught. The stages of women’s
cultural and personal values, affect, and cognition way of knowing (more likely to be “connected”) as
(Leder 1992). different from men’s (more likely to be “separate”)
were identified by Belenky, Clinchy, Goldberger,
A major critique of the liberal feminist para- and Tarrule (1986). Becker (1995) adapted
digm framing the early research on gender and Belenky et al.’s model to the learning of mathe-
mathematics education was that it positioned matics. Kaiser and Rogers (1995) applied
females as “deficit.” In the pursuit of expanding McIntosh’s evolution of the curriculum model to
knowledge of gender issues in mathematics women and the mathematics curriculum. They
education, the explanatory models described identified five phases: womanless mathematics,
above were supplanted by a range of feminist women in mathematics, women as a problem in
perspectives (e.g., feminism of difference, embrac- mathematics, women as central to mathematics,
ing the ways in which women are different from and mathematics reconstructed. In line with
Burton’s challenge of a feminist epistemology of

G 246 Gender in Mathematics Education

mathematics, Belenky et al.’s gender-related (e.g., Forgasz, Vale, and Ursini 2010). Some
distinction between “connected” and “separate” evidence suggests that females may be disadvan-
knowing, and Kaiser and Rogers’ curriculum taged by computers and the mandated use of
model, research on feminist pedagogies was to CAS calculators in high-stakes examinations;
follow. research is ongoing with respect to the impact of
technologies such as the iPad.
In recent years, there does not appear to
be extensive scholarly writing on theoretical Another exciting development is the entry of
developments in the field. researchers from Asian, South American, and
developing countries including African nations
Methodological Considerations into the field (see Forgasz, Rossi Becker, Lee,
As noted above, positivism underpinned early and Steinthorsdottir 2010). The common issues
research studies on gender and mathematics highlighted – males’ superior mathematics
learning. Although often not acknowledged by achievement, participation, and attitudes towards
the researchers, post-positivism, in which context mathematics – and the methodological and episte-
is recognized as relevant in pursuit of the truth, is mological approaches adopted resonate with the
identifiable as the epistemological basis of many early research on gender and mathematics learning
more recent large- and smaller-scale quantitative undertaken in western, English-speaking nations.
studies. Mixed methods research, in which quan- UNESCO’s emphasis on gender mainstreaming
titative data are complemented or supplemented (see Vale 2010) has contributed strongly to the
by qualitative data, or vice-versa, has been efforts being made to the more general goal of
embraced in educational research more broadly achieving equity for women in many societies.
and in the field of gender and mathematics Interestingly, the more recent PISA and TIMSS
learning more specifically. With the advent of results from several Islamic nations (recent
cheaper, more reliable, digital technologies in entrants into these international comparative
recent times, innovative data-gathering instru- studies) reveal generally low overall achieve-
ments (e.g., mobile devices) and data-gathering ment levels, with a trend for girls to outperform
sources (e.g., Facebook) have been employed. boys. Clearly factors other than gender per se
contribute to these patterns; further research is
Recent Developments clearly needed.
The advent and pervasive presence, both outside
and within mathematics classrooms, of calcula- Finally, Fennema’s (1995) prognostication of
tors, computers, and ICTs and the mobile devices the importance of combining neuroscientific
to access them has introduced a new strand of research with gender equity considerations has
research into gender issues and mathematics learn- begun but within the framework of broader equity
ing. As evidenced by course participation rates and considerations including diversity and special
workforce figures, male dominance in the field of needs (see Forgasz and Rivera 2012). The research
computer science and in the world of ICTs is even has been conducted scientifically and not from
stronger than in mathematics and the physical feminist perspectives, however. Yet, these intrigu-
sciences. Surrounded by the high expectation ing interdisciplinary research findings with respect
that technological advancements will enhance to sex differences invite further exploration.
mathematics learning for all and recognizing
that another male domain was being introduced Cross-References
into the preexisting male domain of mathematics
education, researchers began questioning whether ▶ Inclusive Mathematics Classrooms
the widespread implementation of these technolo- ▶ Poststructuralist and Psychoanalytic
gies into mathematics classrooms and assessment
regimes would challenge or exacerbate gender Approaches in Mathematics Education
differences in mathematics learning outcomes ▶ Socioeconomic Class in Mathematics

Education

Giftedness and High Ability in Mathematics G247

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Mathematical discovery/invention; Mathematical
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Forgasz HJ, Vale C, Ursini S (2010) Technology for Mathematical giftedness is an extremely
mathematics education: equity, access, and agency. complex construct exhibited in mathematical
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G 248 Giftedness and High Ability in Mathematics

giftedness implies high mathematical abilities. general giftedness. However, most models of
Nonetheless, often mathematical giftedness is per- general giftedness can be applied to mathemati-
ceived as inborn personal characteristic, whereas cal giftedness associated with mathematical
high abilities in mathematics are perceived as abilities and skills.
a dynamic characteristic that can be developed.
The dynamic view of mathematical giftedness Mathematical Abilities and Skills
assumes that it can be realized only if appropriate
opportunities are provided to a person with high The precise acquisition of mathematical abilities
mathematical potential. Mathematical talent, involves a broad range of different general cogni-
which is realized giftedness, is expressed in high- tive skills, including spatial perception, visuospatial
level performance in mathematics that leads to ability, visual perception, visuomotor perception,
mathematical discoveries and, thus, is closely attention, and memory. Together these skills enable
connected with mathematical creativity. the acquisition, understanding, and performance of
various mathematical activities. Mathematical
There is no singular clear definition of activities deal with five main types of mathematical
mathematical giftedness. Moreover, until recently, objects: number and quantity, shape and space,
the construct of mathematical giftedness was pattern and function, chance and data, and arrange-
overlooked in mathematics education research ment, while successful mathematical performance
for several decades. Thus, this entry synthesizes involves modeling and formulating, manipulating
accounts from two fields of educational psychol- and transforming, inferring and drawing conclu-
ogy: gifted education and mathematics education. sions, argumentation, and communication.

General Giftedness During the past half century, only a small number
of systematic studies devoted to mathematical gift-
In the field of gifted education, gifted students are edness were performed. Krutetskii’s (1976) study on
identified by qualified specialists by virtue of high mathematical abilities in schoolchildren is sem-
outstanding abilities expressed in exceptional inal to the field and remained unique for several
performance. In the adult population, the criteria decades. It introduced components of high mathe-
for giftedness are restrictive, like in the case matical ability in schoolchildren which included the
of Nobel Prize laureates. General giftedness is abilities to grasp formal structures; think logically in
often measured by means of IQ tests, while spatial, numeric, and symbolic relationships; think
a number of theorists have developed broad, critically; generalize rapidly and broadly; and be
multidimensional formulations of giftedness and flexible with mental processes. According to
talent that are widely accepted (Gardner 2003; Krutetskii students with high mathematical abilities
Sternberg 2000). Gardner’s multiple intelligence are able to switch from direct to reverse trains of
theory differentiates between not necessarily thought and to memorize mathematical objects,
connected dimensions including verbal-linguistic, schemes, principles, and relationships. These stu-
logical-mathematical, and visuospatial intelli- dents appreciate clarity, simplicity, and rationality
gences. Sternberg claims that giftedness is and can be characterized by the general synthetic
a function of analytical, practical, and creative abil- component called mathematical cast of mind.
ities accompanied with personal wisdom. Several
models postulated giftedness as being the result of Mathematical Giftedness and Creativity
the complex interactions of cognitive, personal-
social, and sociocultural traits and environmental At times mathematical abilities are measured
conditions (e.g., Renzulli 2000; Milgram 1989). using the nonverbal portions of psychometric
tests like SAT-M. The main criticism of these
In the field of gifted education, mathematical tools is that they do not test creativity since
giftedness is usually regarded as a special
type of specific giftedness which is opposed to

Giftedness and High Ability in Mathematics G249

creativity is fundamental to the work of mathematics usually reflect students’ problem- G
a professional mathematician, while the ability solving proficiency on the topics that they have
to discover mathematical objectives and find studied in school; however, they are not an indicator
inherent relationships among them requires of mathematical giftedness, since they do not reflect
mathematical creativity. students’ independent mathematical reasoning.

The connection between mathematical Mathematical invention, which is an integral
giftedness and creativity leads to an eight-tiered part of the activities of research mathematicians,
(from 0 to 7) hierarchy of mathematical gift consists of four stages: initiation, incubation,
(Usiskin 2000) assuming that the abilities of illumination, and verification (Hadamard 1945).
professional mathematicians are at levels 5, 6, Special attention is given to illumination which
and 7. The most creative mathematicians that involves a large measure of intuitive thinking that
discover new mathematical theorems and invent leads to mathematical insight. Insight exists when
new mathematical concepts are at the highest a person acts adequately in a new situation, and as
(7th) by virtue of their creative ability. This such, insight is closely related to creative ability.
hierarchy implies that while mathematical Thus, success in insight-based problem solving
creativity implies mathematical giftedness, the can serve as an indication of mathematical
reverse is not necessarily true (Sriraman 2005). giftedness among school students.

The notion of mathematical giftedness and its Insight is viewed as a trait central to the
relationship to mathematical creativity is quite construct of general giftedness, in which gifted
clear with respect to research mathematicians; children outperform their average-achieving
however, it is rather vague with respect to high peers in problem solving because of their
school students. This duality reflects the increased tendency towards insight. Accordingly,
distinction between absolute and relative creativ- students with high ability in mathematics have
ity (Leikin 2009, cf. Big C and Little c defined by been found to understand an insight-based
Csikszentmihalyi 1996). Absolute creativity is problem immediately and to solve it quickly.
associated with mathematical discoveries at
a global level. Relative creativity refers to Development of Mathematical Ability
discoveries of a specific person within a specific
reference group. Better understanding of the nature of mathemat-
ical giftedness at the relative (e.g., school) level
When connecting between high mathematical can inform mathematics educators of the ways in
abilities and mathematical creativity, researchers which school mathematics should be taught to
express a diversity of views. Some researchers students who can become research mathemati-
argue that creativity is a specific type of cians. This understanding can lead to a special
giftedness; others feel that creativity is an instructional design and mathematical curricula
essential component of giftedness, while others that can be suitable for these students including
suggest that these are two independent human the choice of mathematical problems for
characteristics. Thus, analysis of the relationships MG students.
between mathematical creativity and giftedness
is an important question for future research. The construct of mathematical potential
accepts the dynamic perspective on mathematical
Mathematical Giftedness, Problem giftedness (Leikin 2009; Sheffield 1999).
Solving, and Insight The mathematical potential of a student includes
abilities (analytical and creative), affective
High-level problem-solving expertise (e.g., success factors (including motivation), and personal
in solving Olympiad problems) often serves characteristics (including commitment). These
as an indicator of mathematical giftedness in factors can be advanced and developed if
schoolchildren. High achievements in school a student is provided with challenging learning

G 250 Giftedness and High Ability in Mathematics

opportunities that take into consideration his/her study populations and study methodologies, the
ability, personality, and affect. These leaning question of the relationship between creativity
opportunities help a mathematically gifted stu- and giftedness in mathematics remains open
dent to realize his/her mathematical potential for future systematic investigation. One of
and become a talented mathematician. Some the more challenging questions for research
insight on learning opportunities provided to in mathematics education is the relationship
mathematically gifted students can be learned between creativity and expertise, as expressed in
from nearly five decades of experience of solving Olympiad problems.
Kolomgorov’s mathematical schools in Russia
(Vogeli 1997). Some studies have explored problem-solving
strategies used by mathematically advanced
Research on Mathematical Giftedness students as compared to strategies employed by
those who are not identified as being advanced in
In the past, giftedness, creativity and high abilities mathematics. These studies demonstrate that stu-
in mathematics did not receive sufficient research dents with higher abilities are more successful in
attention in the fields of gifted education and solving complex mathematical problems and that
mathematics education. Fortunately, during the their heuristics in solving mathematical problems
last decade, attention to the nature and nurture lead to this success. Still, the underlying
of mathematical giftedness and creativity has mechanisms for their success can be the focus
increased (Leikin et al. 2009); as a result, the Inter- of mathematics educational researchers. Special
national Group for Mathematical Creativity and qualities of mathematical understanding of
Giftedness was established (http://igmcg.org/). mathematical concepts in mathematically gifted
Several directions can be traced in the research students can be seen as an additional promising
conducted on mathematical giftedness. and fascinating direction for future research.

The relationship between mathematical Brain research is another direction in
creativity and mathematical giftedness is at the educational research that has been gaining the
focus of attention of several research groups attention of mathematics education researchers. In
nowadays (Leikin and Pitta, accepted). Some the field of general giftedness, several studies have
studies demonstrate that mathematical creativity demonstrated neuro-efficiency effect (lower brain
is a subcomponent of mathematical ability, electrophysiological activity associated with solv-
whereas others show that differences in creativity ing problems) in the gifted population. Following
in gifted students and in those who are not advances in brain research, a research group at the
identified as gifted are task dependent: the higher University of Haifa has revealed/demonstrated that
the mathematical insight required for the prob- the nature of general giftedness differs from that of
lem’s solution, the stronger between-group dif- excellence in mathematics (Waisman et al. 2012).
ferences in the students’ mathematical creativity. The group hypothesizes that excellence in school
Study of professional mathematicians is a rich mathematics is a necessary but not sufficient
source for understanding of the relationship condition for mathematical giftedness and that
between mathematical creativity and mathemati- generally gifted students who excel in school math-
cal giftedness. Starting from Hadamard (1945) ematics have high potential to become talented
these studies demonstrate special qualities of research mathematicians.
their reasoning in terms of the inventiveness
of their mathematical mind, as expressed in To conclude, research on mathematical
illuminations, mathematical imagery, and inner giftedness is a relatively new field in mathematics
need for rigorous proof. education. This fascinating field calls on mathe-
maticians and mathematics educators to gain
While differences between the research a better understanding of the nature and structure
findings can be related to the differences in of high mathematical abilities, of the ways in
which future talented mathematicians can be
identified in school and in which ways they can

Giftedness and High Ability in Mathematics G251

and should be educated in order to fulfill their Koichu B (eds) Creativity in mathematics and the G
mathematical potential and to further develop education of gifted students. Sense, Rotterdam,
mathematics as a scientific field. pp 383–409
Leikin R, Berman A, Koichu B (eds) (2009) Creativity in
Cross-References mathematics and the education of gifted students.
Sense, Rotterdam
▶ Creativity in Mathematics Education Milgram RM (ed) (1989) Teaching gifted and talented
▶ Critical Thinking in Mathematics Education children learners in regular classrooms. Charles C.
▶ Logic in Mathematics Education Thomas, Springfield
▶ Mathematical Ability Renzulli JS (2006) Swimming up-stream in a small river:
▶ Problem Solving in Mathematics Education changing conceptions and practices about the
▶ Visualization and Learning in Mathematics development of giftedness. In: Constas MA, Sternberg
RJ (eds) Translating theory and research into
Education educational practice: developments in content
domains, large-scale reform, and intellectual capacity.
References Lawrence Erlbaum, Mahway, pp 223–253
Sheffield LJ (ed) (1999) Developing mathematically
Csikszentmihalyi M (1996) Creativity: flow and the promising students. National Council of Teachers of
psychology of discovery and invention, 1st edn. Mathematics, Reston
Harper Collins, New York Sriraman B (2005) Are giftedness and creativity synonyms
in mathematics? J Second Gift Educ 17(1):20–36
Gardner H (1983/2003) Frames of mind. The theory of Sternberg RJ (ed) (2000) Handbook of intelligence.
multiple intelligences. Basic Books, New York Cambridge University Press, New York
Usiskin Z (2000) The development into the mathemati-
Hadamard J (1945) The psychology of invention in the cally talented. J Second Gift Educ 11:152–162
mathematical field. Dover, New York Vogeli BR (1997) Special secondary schools for the
mathematically and scientifically talented. An
Krutetskii VA (1976) The psychology of mathematical international Panorama. Teachers College Columbia
abilities in schoolchildren. (trans and eds: Teller J, University, New York
Kilpatrick J, Wirszup I). The University of Chicago Waisman I, Shaul S, Leikin M, Leikin R (2012) General
Press, Chicago ability vs. expertise in mathematics: an ERP study with
male adolescents who answer geometry questions. In:
Leikin R (2009) Bridging research and theory in The electronic proceedings of the 12th international
mathematics education with research and theory in congress on mathematics education (Topic Study
creativity and giftedness. In: Leikin R, Berman A, Group-3: activities and programs for gifted students).
Coex, Seoul, pp 3107–3116

H

Heuristics in Mathematics Education Mathematical Discovery books (1962, 1965).
In “How to Solve It,” Polya (1945) initiated the
Nicholas Mousoulides1 and Bharath Sriraman2 discussion on heuristics by tracing their study back
1University of Nicosia, Nicosia, Cyprus to Pappus, one of the commentators of Euclid, and
2Department of Mathematical Sciences, The other great mathematicians and philosophers like
University of Montana, Missoula, MT, USA Descartes and Leibniz, who attempted to build
a system of heuristics. His book also included
Keywords advice for teaching students of mathematics and
a mini-encyclopedia of heuristic terms. The role of
Discovery-based learning; Heuristics; Polya; heuristics and his 4-step model for problem
Problem solving solving impacted enormously on the teaching of
problem solving in schools.
Definition
The term “Heuristic” comes from the Greek
In this entry we examine Polya’s contribution to word “Evriskein,” which means “Discover.”
the role of heuristics in problem solving, in According to the definition originally coined by
attempting to propose a model for enhancing Polya in 1945, heuristics is the “study of means
students’ problem-solving skills in mathematics and methods of problem solving” (Polya 1962,
and its implications in the mathematics education. p. x) and refers to experience-based techniques for
problem solving, learning, and discovery that
Characteristics would enhance one’s ability to solve problems.
A heuristic is a generic rule that often helps in
Research studies in the area of problem solving, solving a range of non-routine problems. Heuris-
a central issue in mathematics education during the tics, such as Think of a Similar Problem, Draw
past four decades, have placed a major focus on the a Diagram or a Picture, Working Backward, and
role of heuristics and its impact on students’ Guess and Check, can serve different purposes
abilities in problem solving. The groundwork for such as helping the student to understand and rep-
explorations in heuristics was established by the resent the problem, simplify the problem, identify
Hungarian Jewish mathematician George Polya in similarities with other problems, and to identify
his famous book “How to Solve It” (1945) and was possible solutions. These heuristics, often used in
given a much more extended treatment in his combinations, can be used to solve different types
of problems, though there is no guarantee that
applying these heuristics will be successful.

Heuristics are an important aspect of mathe-
matical problem solving, especially if we refer to

S. Lerman (ed.), Encyclopedia of Mathematics Education, DOI 10.1007/978-94-007-4978-8,
# Springer Science+Business Media Dordrecht 2014

H 254 Heuristics in Mathematics Education

them as the capabilities for mathematical reason- to the simplified problem may help in solving
ing that enable insightful problem solving. the original problem.
Beyond those proposed by Polya, the appropriate
inclusion of more general heuristics like spatial While theories of mathematics problem
visualization, diagrammatic and symbolic repre- solving have placed a focus on the role of teach-
sentations in complex novel problems, and the ing heuristics for an enhanced problem-solving
recognition of mathematical structures in the performance, research from Begle (1979) to
teaching and learning of problem solving might Schoenfeld (1992) has a consistent outcome
result in enhanced student problem-solving that classroom teaching of problem-solving heu-
behavior (Goldin 2010). ristics does little to improve students’ problem-
solving abilities. There is, of course, a number of
Based on Polya’s contribution, extended and constraints related to the teaching and learning
more refined lists of heuristics have often been of heuristics. First, in a number of problem-
proposed by researchers, and quite often they solving approaches, problem solving is taught
have been included in official documents and through textbook sections in which students
mathematics curricula around the world. Among are presented with a strategy (e.g., finding
others, students should be exposed to and know a pattern), then are given practice exercises
when to use the following heuristics: (a) Try it using the strategy, and finally they are tested
out; take the role of other people and try to do on the strategy. When the strategies are taught
what they would do. Make use of objects and in this way, they are no longer heuristics, in the
other (electronic) media to represent the situation sense described by Polya.
or problem. (b) Use a diagram and/or a model of
the problem to create a diagrammatic description A second constraint is related to the nature
of the problem and to visualize the problem data. of heuristics. Despite their long history and
(c) Organize data in systematic lists and look for although heuristics have descriptive power in
patterns might help the solver to identify how describing experts’ problem-solving behaviors,
data is related to the problem question and to there is little evidence that these heuristics could
perceive patterns in the data. (d) Work back- also serve well as prescriptions to guide novices’
wards; looking at the required end result and next steps during ongoing problem solving.
working backwards can be especially useful in This problem lies, according to Begle (1979,
problems involving a series of steps. (e) Use p. 145–146), in the fact that heuristics are “both
before after concept; compare the situation problem- and student-specific often enough to
before and after the problem is solved. This com- suggest that finding one (or few) strategies
parison can shed light on the cause and lead to which should be taught to all (or most) students
a possible solution. (f) Use guess and check; are far too simplistic.” In line with Begle (1979),
make an educated guess of the answer and Schoenfeld (1992) concluded that a better
check its correctness. Use the outcome to “understanding” of heuristics is needed, since
improve the next guess and look for patterns in most heuristics are really just names for large
the guesses. (g) Make suppositions; studying categories of processes rather than being
the problem data and make suppositions well-defined processes in themselves. To over-
(assumptions without proof) to form the basis come this constrain, Sriraman and English (2010)
for further and better thinking will reduce the contended that “understanding” heuristics
number of possible solutions. (h) Restate the means to knowing when, where, why, and how
problem to better understanding the problem to use heuristics and other tools, including
and identifying important factors of the prob- metacognitive, emotional (e.g., beliefs), and
lem. (i) Simplify the problem; try to make social (e.g., group-mediated) tools.
a difficult problem simpler, by changing
complex numbers to simple or by reducing the A third constraint related to the appropriate
number of factors in the problem. The solution teaching of heuristics for enhanced problem-
solving skills is related to teachers’ skills. As
Burkhardt (1988) identified, the task of teaching

History of Mathematics and Education H255

heuristics is harder for teachers, because References H
(a) mathematically, teachers should provide con-
structive and formative feedback to students’ Begle EG (1979) Critical variables in mathematics
different approaches in solving problems; (b) education. MAA & NCTM, Washington, DC
pedagogically, teachers should carefully plan
their interventions and feedback and assist Burkhardt H (1988) Teaching problem solving. In:
students using the least possible help; and (c) Burkhardt H, Groves S, Schoenfeld A, Stacey K (eds)
personally, teachers should be equipped with Problem solving - A world view (Proceedings of the
experience, confidence, and self-awareness, in problem solving theme group, ICME 5). Nottingham,
order to work well with problems without Shell Centre, pp 17–42
knowing all the answers requires.
English L, Sriraman B (2010) Problem solving for the 21st
How to overcome the above constrains? In his century. In: Sriraman B, English L (eds) Theories of
review on heuristics, Schoenfeld (1992) concluded mathematics education: seeking new frontiers.
that better results could be obtained by (a) teaching Springer, Berlin, pp 263–290
specific (rather than general) problem-solving heu-
ristics that better link to structurally similar prob- Goldin G (2010) Problem solving heuristics, affect, and
lems, (b) teaching metacognitive strategies that discrete mathematics: a representational discussion.
could help students in effectively deploying their In: Sriraman B, English L (eds) Theories of mathemat-
problem-solving heuristics, and (c) improving stu- ics education: seeking new frontiers. Springer, Berlin,
dents’ views of the nature of problem solving in pp 241–250
mathematics, by enhancing their productive
beliefs, while eliminating their counterproductive Polya G (1945) How to solve it. Princeton University
beliefs. Further, as English and Sriraman (2010) Press, Princeton
noted, next research steps in the area of heuristics
in problem solving need to develop operational Polya G (1962) Mathematical discovery, vol 1. Wiley,
definitions that enable the mathematics education New York
community to answer more prescriptive, than
descriptive, questions like the following: “What Polya G (1965) Mathematical discovery, vol 2. Wiley,
does it mean to “understand” problem-solving New York
heuristics and other tools?” “How, and in what
ways, do these understandings develop and how Schoenfeld A (1992) Learning to think mathematically:
can we foster this development?” “How can we problem solving, metacognition, and sense making in
reliably observe, document, and measure such mathematics. In: Grouws DA (ed) Handbook of
development?” research on mathematics teaching and learning.
Macmillan, New York, pp 334–370
The legacy of Polya’s contribution to heuris-
tics in problem solving is not restricted to a list of Sriraman B, English L (eds) (2010) Theories of mathe-
strategies used by experts or novices when matics education: seeking new frontiers (Advances in
solving problems, but rather implies for the sig- mathematics education). Springer, Berlin
nificance of problem solving in mathematics and
the necessity to find appropriate teaching and History of Mathematics and
self-regulated methods to enhance students’ Education
problem-solving skills.
Evelyne Barbin1 and Constantinos Tzanakis2
Cross-References 1Faculte´ des sciences et des Techniques,
Centre Franc¸ois Vie`te, Nantes, France
▶ Inquiry-Based Mathematics Education 2Department of Education, University of Crete,
▶ Problem Solving in Mathematics Education Rethymnon, Crete, Greece

Keywords

Epistemology; History and Pedagogy of
Mathematics; International Commission on
Mathematical Instruction; Interdisciplinary
teaching; New Math reform; Original sources;
Problem solving; Activity-based teaching;
Mathematical proof; Construction of concepts
and theories; Modelization; Ethnomathematics

H 256 History of Mathematics and Education

History and Epistemology for organized by Ph.S. Jones during the 2nd Interna-
Mathematics Education: tional Congress on Mathematics Education
An Ever-Increasing Interest (ICME 2), and in 1976 the International Study
Group on the relations between the History and
In the last 30 years or so, integrating history of Pedagogy of Mathematics (known afterwards as
mathematics (HM) in mathematics education the HPM Group) was created as an international
(ME) has emerged as a worldwide intensively study group affiliated to the International
studied area of new pedagogical practices and Commission on Mathematical Instruction
specific research activities. However, the interest (ICMI). For the history and the activities of this
on the HM in the context of ME dates back to the group, which has been playing a leading role in
second half of the nineteenth century. Mathema- this area, see Fasanelli and Fauvel (2006).
ticians like F. Klein and A. De Morgan and
historians like P. Tannery and G. Loria showed In fact, the eight points which constituted the
an active interest on the role of the HM in educa- original focus and aim of the HPM Group and to
tion. Already at that time, history enters into some extent achieved so far, are still pertinent
textbooks, e.g., in France in Rouche´ and today (see Fasanelli and Fauvel 2006) to promote
Comberousse (Barbin et al. 2008, ch. 2.2.). international contacts and exchange information
At the beginning of the twentieth century, this in this area, to promote and stimulate interdisci-
interest was revived as a consequence of the plinary investigation, to further a deeper
discourse and the related debates on the founda- understanding of mathematics’ evolution, to
tions of mathematics. Poincare´ criticized assist in improving instruction and curricula by
Hilbert’s axiomatic approach and declared that relating mathematics teaching and its history to
the history of science should be the “principal the development of mathematics, to produce
guide for the educator.” Later, history became relevant material for the teachers’ benefit, to
a resource for the various epistemological facilitate access to this material and to historical
approaches, like Bachelard’s historical episte- sources, and to promote awareness of the rele-
mology (Bachelard 1938), Piaget’s genetic vance of the HM for mathematics teaching and its
epistemology (Piaget and Garcia 1989), and significance for the development of cultures.
Freudenthal’s phenomenological epistemology
(Freudenthal 1983), at the same time stimulating In the mid 1980s, the French network
the formulation of specific ideas and conclusions of the IREMs (Instituts de Recherche sur
on the learning process (Lakatos 1976; Brousseau l’Enseignement des Mathe´matiques) began to
1997; Ernest 1994) (see ▶ Learning Study in organize every 2 years a Summer University on
Mathematics Education). the History and Epistemology in Mathematics
Education. Since 1993, this was extended on
The interest on the history and epistemology a European scale constituting the European
of mathematics became stronger and more Summer University on the History and Episte-
competitive in the 1960s and 1970s in response mology in Mathematics Education (ESU), which
to the “New Math” reform. Those supporting the gradually has become a major international
reform were strongly against “a historical con- activity in the spirit of the HPM Group (Barbin
ception of education” (“a` bas Euclide” declares et al. 2010). This spirit goes beyond the use of
Dieudonne´), whereas, for its critics, history history in teaching mathematics and conceives
appeared like a “therapy against dogmatism,” mathematics as a living science with a long
conceiving mathematics not only as a language history, a vivid present and an unforeseen future,
but also as a human activity. together with the conviction that this conception
of mathematics should be not only the core of its
Since 1968 ME has constituted a standard teaching but also its image spread out to the
subject in regularly organized international outside world.
meetings. In 1972 a working group on the
“History and Pedagogy of Mathematics” was The gradually increasing interest of mathema-
ticians, historians and mathematics teachers and

History of Mathematics and Education H257

educators in this area, has led to various research to debate and controversy among mathemati- H
activities and didactical experiments, which were cians. Actually, fundamental notions like rigor,
analyzed, and their results were disseminated in evidence, and proof have been different in differ-
the context of regularly organized local and ent historical periods; that is, (meta) ideas and
international meetings and were presented in (meta) concepts that today are taken for granted
numerous publications in international journals, in their present form are actually the product of
collective volumes, and conference proceedings. a historical development; there is a historicity
Some standard works in chronological order inherent to them and maybe it is more appropriate
(with detailed extensive bibliography therein) to use plural number when referring to them
are NCTM 1969/1989, Commission Inter-IREM (Barbin and Benard 2007). This fact gave rise to
(1997), Swetz et al. (1995), Calinger (1996), ideas about learning processes of school mathe-
Fauvel and van Maanen (2000), Katz (2000), matics. From this point of view, history has what
Bekken and Mosvold (2003), Katz and has been called a “replacement role” (“roˆle vicar-
Michalowicz (2004), Barbin and Benard (2007), iant” in French) by offering to teachers the
Knoebel et al. (2007), Barbin et al. (2010), and possibility to approach and explore pieces of
Katz and Tzanakis (2011). mathematics, which are not included in the offi-
cial school curricula, and in this way to often
There are three different – though interrelated – replace what is usual with something different
types of contributions of didactical/educational and/or unusual.
research and the associated experimental work
on the role of HM in ME in the last 30 years, Since the 1990s, a lot of research has
epistemological, cultural, and didactical. been conducted on number systems, equations,
geometrical constructions, the role of technical
Epistemological Contributions instruments in mathematics, the history of proof,
etc. (Calinger 1996). In addition, the historical
Bachelard and Lakatos’ influence clearly appears study of the range of applicability of concepts has
in the research conducted on the role of problems led to a critical analysis of school programs (e.g.,
and on constructing and rectifying concepts and on the history of probability and statistics; see
theories. This research should be placed both in Barbin 2010; Katz and Tzanakis 2011, ch. 16).
the context of pedagogical constructivism of the
1980s and 1990s, as well as, in the area of More recently, many works propose to con-
“problem solving” (see ▶ Problem Solving in nect history and semiotics in order to analyze the
Mathematics Education). Here, history plays role of script and figures in the evolution of math-
a crucial role, since it provides specific pertinent ematics, concerning both invention of concepts
examples of problems on the basis of which and the mode of reasoning (see ▶ Mathematical
concepts were invented and/or transformed (see, Proof, Argumentation, and Reasoning). There are
e.g., Commission Inter-IREM 1997; Fauvel and several international meetings and publication in
van Maanen 2000, section 7.4.7; Katz and this context (see, e.g., Hanna et al. 2010).
Michalowicz 2004).
Cultural Contributions
More generally, history allows for a deeper
analysis of mathematical activities, thus motivat- In many works it is claimed that the main cultural
ing and stimulating research in relation to aspect of history is to provide a different image of
“activity-based teaching,” promoted in the mathematics both to teachers and – more
1990s. A lot of research work in this context importantly – to students, on which their more
consists of determining the issues at stake and positive relation with mathematical knowledge
the practices adopted concerning mathematical can be founded. In fact, history allows placing
reasoning. They show that rigor and the evalua- mathematics in the philosophical, artistic, liter-
tion of mathematical proof have been subjected ary, and social context of a certain period.

H 258 History of Mathematics and Education

Thus, teachers could link mathematics to idea for an interdisciplinary teaching that has
philosophy, or history; e.g., the history of the been promoted since 2000. The movement for
concept of perspective, which is also interesting interdisciplinarity has been officially incorpo-
for the teachers of plastic arts, stimulated many rated in curricula through pedagogical innova-
works (e.g., Commission inter-IREM 1997). tions in secondary education, and in this context
Similarly, the relation between mathematics the interest in the history of science has been
and literature leads to cultural insights if seen in stimulated (Barbin et al. 2010).
a historical perspective. This could consist of the
intrusion of mathematics into a roman through There are two areas which have been recently
this roman’s human characters, but mathematics developed on the intersection among mathemat-
could also inspire the subject or the structure of ics, culture, and societies: on the one hand, the
a roman. history of ME, which forms part of the HM in
general, has contributed to research in education
Other research activities have shown the way proper and has led to several international meet-
HM leads to the history of science. In particular, ings (see ▶ History of Mathematics Teaching and
reading a text often requires placing it in relation Learning); on the other hand, the research on
to the author’s scientific preoccupations, preju- ethnomathematics initiated by U. d’Ambrosio
dices, and concerns. Sometimes, the solution of makes appeal on history, given that the investi-
a problem requires establishing passages or gated methods and practices can be traced back to
developing analogies among different disci- old ones that were transmitted to the present era
plines. It is interesting for ME to study the circu- (see ▶ Ethnomathematics).
lation of problems, concepts, methods, or modes
of writing (scripts) between mathematics and History of Mathematics in the Classroom
other sciences (e.g., see the work on vectors
Barbin 2010). There is a gradually increasing number of works
introducing a historical aspect in the mathematics
Research on the history of complex numbers classroom (Fauvel 1990); as a consequence,
constitutes a privileged domain to unfold the dif- the activities and publications in the context
ferent aspects of cultural or interdisciplinary of the HPM Group have been enriched.
development (see ▶ Interdisciplinary approaches A comprehensive presentation is given by Fauvel
in Mathematics Education). This case articulates and van Maanen (2000); for subsequent develop-
and connects mathematics to physics and philoso- ments, see Katz (2000), Katz and Michalowicz
phy; additionally, it questions mathematical inven- (2004), Shell-Gellasch and Jardine (2005),
tion and the link between reality and the status of Knoebel et al. (2007), and Katz and Tzanakis
mathematical truth (Fauvel 1990; Barbin 2010). (2011). This does not mean that the research
conducted concerns a line of approach of teach-
In the last few years, the relation between ing history to students as an independent subject,
mathematics and other disciplines has been but rather, to orient the teacher towards enriching
subsumed in education by the concept of his/her teaching by taking into account ideas
modelization. On this issue, proposals have based on epistemology and history or directly
been put forward that seem to be incompatible introducing historical elements. The aim of
with each other if one ignores the different “introducing a historical perspective in mathe-
conventions adopted for this concept in its short matics teaching” is not to approach a subject in
history. The conception of mathematics as an the classroom or at home in a way completely
“experimental science” – also used in education – detached from conventional teaching. Rather, it
has given rise to historical reflections, e.g., on the should be meant as the stimulation of historical or
comparison between mathematical and physical epistemological reflections of the teacher in
experiments. connection with his/her teaching (Barbin 2010)

This multifaceted aspect of the character of
mathematical notions and concepts revealed
through the work done on the HM supports the

History of Mathematics and Education H259

to give important dates for a concept, to explain constitute the starting point of new ideas and H
its historical significance, to refer and/or read trends on developing further a historically
original texts, to solve “historical problems,” etc. inspired and epistemologically driven approach
to teach specific pieces of mathematics and/or to
Reading original texts allows for a “cultural design mathematics curricula.
shock” by directly immersing mathematics into
history. Therefore, the majority of researches Cross-References
insist on the necessity to read original texts, not
in relation to our present knowledge and under- ▶ Ethnomathematics
standing, but in the context where they were ▶ History of Mathematics Teaching and Learning
written. It is this line of approach which becomes ▶ Interdisciplinary Approaches in Mathematics
a source of “epistemological astonishment” by
questioning knowledge and procedures that Education
“have been taken for granted” so far. Thus, ▶ Learning Study in Mathematics Education
reading original texts has a strong virtue of ▶ Mathematical Proof, Argumentation, and
“reorientation” (“vertu de´paysante” in French).
A lot of works in the last 20 years present numer- Reasoning
ous resources for reading original texts and the ▶ Problem Solving in Mathematics Education
variety of activities related to this reading. It
gives the opportunity to introduce methods that References
may not be taught today and/or to compare
different methods of solution (e.g., Fauvel and Bachelard G (1938) La formation de l’esprit scientifique.
van Maanen 2000; Knoebel et al. 2007; several Vrin, Paris
chapters especially in Swetz et al. 1995; Barbin
et al. 2010; Katz and Tzanakis 2011). Barbin E (ed) (2010) Des de´fis mathe´matiques d’Euclide a`
Condorcet. Vuibert, Paris
The introduction of a historical dimension in
ME requires appropriate teacher training, how- Barbin E, Be´nard D (2007) Histoire et Enseignement des
ever (Fauvel and van Maanen 2000, ch. 4). Since Mathe´matiques: Rigueurs, Erreurs, Raisonnements.
the creation of the HPM Group, a large number of Institut National de Recherche Pe´dagogique, Lyon
studies have been devoted on the conditions of such
preservice and in-service training. To this end, Barbin E, Kronfellner M, Tzanakis C (eds) (2010) History
a large number of monographs and anthologies and epistemology in mathematics education:
addressed to students and teachers have appeared proceedings of the 6th European Summer University.
in the last 30 years. A direct approach in this con- Holzhausen, Vienna
text – though not the only one – is to give under-
graduate courses based on historical material (e.g., Barbin E, Stehl´ıkova´ N, Tzanakis C (eds) (2008) History
Katz and Tzanakis 2011; Knoebel et al. 2007). and epistemology in mathematics education proceedings
of the 5th Summer University. Vydavatelsky´ servis,
The analysis of numerous teaching experi- Plzenˇ
ments has led to specific pedagogical and
didactical questions. On one hand, such Bekken O, Mosvold R (eds) (2003) Study the masters: the
experiments – like any other pedagogical innova- Abel-Fauvel conference. NCM, Go¨teborg
tion – aim to have a value by themselves.
But, they are not easily reproducible since they Brousseau G (1997) Theory of didactical situations in
depend on the teacher’s culture and the resources mathematics. Kluwer, Dordrecht
at his/her disposal. On the other hand, these
experiments should be evaluated/assessed in rela- Calinger R (ed) (1996) Vita Mathematica: historical research
tion to their own objectives, which do not often and integration with teaching. MAA Notes 40. The
correspond to the conventional conceptions of Mathematical Association of America, Washington DC
evaluation and assessment. These two issues
Commission inter-IREM Episte´mologie et Histoire des
Mathe´matiques (1997) History of mathematics, histo-
ries of problems. Ellipses, Paris

Ernest P (ed) (1994) Constructing mathematical
knowledge: epistemology and mathematics education.
The Falmer Press, London

Fasanelli F, Fauvel J (2006) History of the
International Study Group on the relations between
the history and pedagogy of mathematics: the first
25 years 1976–2000. http://www.clab.edc.uoc.gr/
HPM/HPMhistory.pdf. Accessed 1 Sep 2012

H 260 History of Mathematics Teaching and Learning

Fauvel J (ed) (1990) History in the mathematics General Education and Professional
classroom. The IREM papers. The Mathematical Training in Mathematics
Association, Leicester
The first known systematic teaching of mathemat-
Fauvel J, van Maanen J (eds) (2000) History in mathemat- ics started in the Third Millennium in states of
ics education – the ICMI study. Kluwer, Dordrecht Mesopotamia, where scribal schools – edubba,
the houses of tablets – prepared the scribes
Freudenthal H (1983) Didactical phenomenology of math- who had to work for the state administration and
ematical structures. Boston/Lancaster, Dordrecht were required to master writing and accounting
techniques. Similar processes are observed in
Hanna G, Jahnke N, Pulte H (eds) (2010) Explanation and Ancient Egypt. Thus, for a long period, the goal
proof in mathematics: philosophical and educational of teaching was professional training.
perspectives. Springer, New York
Mathematics became a subject of general
Katz V (ed) (2000) Using history to teach mathematics: education for the first time in the city states of
an international perspective. MAA notes 51. The Greece, when a new class of free citizens
Mathematical Association of America, Washington, governing their state emerged. This form of
DC general education practiced two distinct patterns:
(1) rhetoric and dialectic as qualifications for
Katz VJ, Michalowicz KD (eds) (2004) Historical political activity and (2) mathematics as
modules for the teaching and learning of mathematics. a certain complement. This two-sided general
The Mathematical Association of America, education became later conceptualized as the
Washington, DC trivium and the quadrivium, together constituting
the septem artes liberales, which became
Katz V, Tzanakis C (eds) (2011) Recent developments a characteristic of general education in Europe.
on introducing a historical dimension in mathematics Professional training, as related to manual work,
education. MAA notes 78. The Mathematical Associ- turned to be practiced by the lower social
ation of America, Washington, DC strata. In countries of Islamic civilization, insti-
tutionalized education was limited to basic
Knoebel A, Laubenbacher R, Lodder J, Pengelley D teaching of reading. Acquiring practical knowl-
(2007) Mathematical masterpieces –further edge or studying for a learned profession
chronicles by the explores. Springer, New York depended on an individual’s decisions.

Lakatos I (1976) Proofs and refutations. Cambridge In European states, two parallel systems were
University Press, Cambridge institutionalized yet in premodern times – general
education and vocational training provided in
Piaget J, Garcia R (1989) Psychogenesis and the history of private or corporate forms (guilds). Gradually
science. Columbia University Press, New York these were transformed into parallel forms of
classical secondary schools and socially lower-
Shell-Gellasch A, Jardine D (eds) (2005) From ranking schools that provided training for
calculus to computers: using the last 200 years of commercial and technical professions (their
mathematics history in the classroom. MAA notes curriculum assigned an important role to applied
68. The Mathematical Association of America, mathematics). Largely by the end of the
Washington, DC nineteenth century, these schools rose in
social status and quality and began to rival the
Swetz F, Fauvel J, Bekken O, Johansson B, Katz V (eds) classical schools. Internationally, the situation
(1995) Learn from the masters. The Mathematical was addressed in a variety of ways. One way
Association of America, Washington, DC was to run parallel types of schools, differing in
the degree of teaching languages and sciences.
History of Mathematics Teaching Another way was to integrate both parallel types
and Learning

Gert Schubring1 and Alexander P. Karp2
1Fakult€at fu€r Mathematik, Institut fu€r
Didaktik der Mathematik, Universit€at Bielefeld,
Bielefeld, Germany
2Teachers College, Columbia University,
New York, NY, USA

Keywords

Curriculum; General education; Mathematics in
the global world; Professional education;
Teacher education; Textbooks

History of Mathematics Teaching and Learning H261

into one “middle” school but organizing the In the same vein, arithmetic had to be H
tracking of students according to their supposedly complemented by basic notions of geometry.
better abilities or professional expectations. The German pedagogue F. W. A. Fro¨bel
(1782–1852) developed didactic materials for
With the enormous expansion of the such geometry teaching. Yet, including geometry
educational system from the 1960s in industrial- into primary school teaching remained highly
ized countries because of social changes and controversial throughout the nineteenth century;
technological advances in the professions, governments feared that pupils – and their
“Mathematics for All” became the goal for the teachers – would be too highly educated. There-
entire pre-college education. Likewise, from fore, the initiatives of F. A. W. Diesterweg
the 1980s onward, “Mathematics for All” became (1790–1866) for including geometry into teacher
a popular conception in developing countries, training at Prussian seminaries were interrupted.
calling for equal access to quality teaching of This strict confinement was due to the social
mathematics for everybody (Schubring 2012). status of primary schools: nearly everywhere,
they constituted a separate school system for the
Mathematics in Primary Schooling lower social classes, with schools, curriculum,
and teacher education all a world apart from
Primary schools were often the last to be institu- secondary schools. Yet, it was in institutions for
tionalized within educational systems, and when teacher training that pedagogical and methodo-
they began to be established in the seventeenth– logical approaches for teaching (elementary)
eighteenth centuries, mathematics was not their mathematics first began to be developed.
major focus. Eventually, arithmetic became one
of the “three Rs” (along with reading and Only during the twentieth century did primary
writing), providing basic education for daily schools become the first step in a consecutive
use, which included rudimentary techniques of system, which all students had to pass to continue
calculating. The rule of three, with its various on in secondary schooling. In this process, the
applications in converting measures, indicated syllabus was reformed, and basic arithmetic was
the highest level of teaching for a long time. replaced by fundamental concepts of school
Typically, primary school teachers for this mathematics. In large measure owing to the
subject were poorly prepared. New Math and Modern Mathematics Movements
in the 1960s, the primary school syllabus became
The situation began to improve in the second an integral part of the entire school mathematics
half of the eighteenth century, because of the coursework.
Enlightenment. Significantly, teacher education
first became a concern for state initiatives. The Mathematics in Secondary Schooling
term “normal school,” predominantly used in
many countries from the nineteenth century on, Secondary schools differentiated from the uni-
first referred to such state-run teacher education versities by the first half of the sixteenth century
institutions in Austria, in Naples, and from 1795 and thus shared with them the same social and
in France. From the 1780s, teacher seminaries professional orientations: to prepare upper social
were analogous institutions in various German strata for university studies and hence for learned
states. The ideas of the Swiss pedagogue professions. As a consequence, classical lan-
J. H. Pestalozzi (1746–1827) had an enormous guages dominated the secondary schools – of
influence in Europe, from the early nineteenth both the Catholic and Protestant educational sys-
century onward; he called to transform dull drill tems in Western Europe – that were rivalling each
and rote learning into approaches for active other. In the Jesuit colleges, mathematics was
methods and to convert the practice of reckoning reduced – according to their general curriculum,
into a deeper understanding of elementary the Ratio Studiorum of 1599 – to brief teaching in
mathematics.

H 262 History of Mathematics Teaching and Learning

the last grade (the philosophy grade); the was the threefold type of secondary schooling in
Protestant gymnasia at first taught mathematics Germany: the humanistisches Gymnasium, with
as arithmetic in the lower grades and slowly Greek and Latin; the Realgymnasium, with only
introduced geometry in the upper grades. Since Latin; and the Oberrealschule, with no classical
school attendance was not compulsory, the language but modern languages. Mathematics
parents were left to decide when their sons was a major subject in all three types but had
would enter the Gymnasium (or college) and their different profiles.
which preparation they would get before enter-
ing. Sources for France, for example, show that In the second half of the twentieth century,
most students left the colleges before the philos- the lower and middle grades of the secondary
ophy grade, thus without having experienced any schools typically provided a common curriculum
teaching of mathematics. in mathematics for all students. The upper
grades, however, often differentiated according
During the eighteenth century, various to curricular profiles (there, mathematics could
developments led to establishing more teachings be optional or a certain course of mathematics
of mathematics, generally in somewhat rivalling was obligatory).
and parallel types of schools, like the Realschulen
for the middle classes and the Ritterakademien Curriculum
for the nobility in various German states, or
in colleges as in several Catholic states (by It is often believed that the mathematics curricu-
attaching engineer training to existing colleges). lum has essentially been the same in all countries
As a consequence, mathematics achieved over the centuries. This belief is based on the
a stronger status in the Gymnasia. Some non- similarity of some superficially descriptive
Jesuit orders such as the Oratoire in France also terms, like algebra and geometry. In reality,
taught more mathematics. The next critical step history shows enormous differences in the curric-
came mid-eighteenth century with the foundation ulum among countries, particularly because of
in France of military schools to prepare diverse epistemological conceptions of school
engineers; there – based on concepts of the mathematics and methodological approaches to
Enlightenment – mathematics became the the subject.
principal teaching subject.
From the beginning of a somewhat broadly
One of the impacts of the French Revolution organized teaching in premodern times, there
was the establishment of the first system of public was already a clear difference between
education. Latin and mathematics became the two a Euclidean approach to geometry and an
pillars of general education in French secondary anti-Euclidean one, first propagated by Petrus
schools. Other countries followed this pattern. In Ramus (1515–1572); later, influenced by him,
particular, Prussia offered three components of algebraizing approaches appeared and, even
neo-humanist general education: classical lan- later, during the French Revolution, the analytic
guages, history and geography, and mathematics ones. The opposition between geometric and
and the sciences. Yet, this strong role of mathemat- algebraic-analytic approaches characterizes the
ics was not permanently assured: during the nine- spectrum of school mathematics curricula at the
teenth century, France almost returned to the Jesuit secondary level.
model, while in Germany, only Prussia continued
with mathematics as a major teaching subject, and Since secondary schools used to be dominated
the classical languages dominated other German by classical languages, at least until the end of the
states (Schubring 1991). Italy, after its unification nineteenth century, mathematics followed this
in 1861, basically assigned mathematics a pattern and likewise emphasized classical
secondary role. geometry – in some countries (England, Italy)
even by directly using Euclid’s Elements. The
Characteristic of the various functions that analytic approach was in general short lived,
mathematics can assume in a school curriculum

History of Mathematics Teaching and Learning H263

appearing only at the beginning of the nineteenth the textbook and compelled teachers to function H
century. as the “organ” of the textbook.

Overcoming a static curriculum, which uti- In fact, for extended periods, only two text-
lized the synthetic methodology of geometry books were broadly used for teaching: Euclid’s
and was unconnected to scientific progress, Elements of Geometry (about À300) in Europe
became the motto of the reform movement, first and parts of the Islamic civilization and the Jiu
in Germany and France, and then, directed by Zhang Suan Shu, the Nine Chapters of Arithmetic
Felix Klein and the IMUK (ICMI), of the first Technique (about À200) in China and East Asia.
international reform movement in the early twen- Although both were likely not been composed as
tieth century. The reformers wanted functional textbooks for teaching, they were used as such.
thinking to permeate the entire curriculum. Euclid’s text or an uncountable number of diverse
The introduction of the function concept and the adaptations of it constituted the standard material
elements of calculus became the characteristics for secondary schools in many European coun-
of this reform movement. tries, particularly in Catholic colleges where at
least its Book I was required.
From then on, school mathematics tried to
keep up a better pace with the progress of math- The printing press stimulated the publication
ematics. The main goal of the second interna- of an enormous number of arithmetic textbooks
tional reform movement, from 1959, which for practitioners in the vernacular as well as
was known as the New Math or the Modern new textbooks for the university and secondary
Mathematics Movement, was to align school school level. Noteworthy were textbooks
and modern mathematics even more tightly, algebraizing mathematics, such as Antoine
constructing the curriculum on the basic struc- Arnauld’s Nouveaux E´le´mens de ge´ome´trie
tures of mathematics. Although later many ideas (1667) and subsequent works by members of the
of this movement were rejected, school mathe- Oratoire in France (Prestet, Reynaud, Lamy).
matics finally became structured, from the Another trend was textbooks for a mundane
primary grades, according to fundamental con- public (Clairaut 1741 and 1746).
cepts of mathematics in arithmetic, algebra,
geometry, calculus, and, as a recent innovation, The establishment of systems of public
probability theory and statistics. instruction created new dimensions (Schubring
2003). Following its centralistic policy, France
Textbooks for Mathematics first assigned only one and then later a very lim-
ited number of textbooks for the entire country.
Throughout the millennia, textbooks constituted Few authors, like S. -F. Lacroix, became entre-
the main resource for the teaching of mathemat- preneurs, dominating the schoolbook market.
ics. In the epochs before the invention of the Other countries, like neo-humanist Prussia,
printing press, the uniqueness of the manuscript, emphasized the autonomy of the teacher with
not being reproducible, led to teaching practice regard to method and let him choose his textbook.
consisting of its oral reading to the students. In Textbook writing was provided according to the
fact, a genuine qualification for teaching was not respective values of education either mostly by
even desired: knowledge was regarded as university mathematicians (France, Italy, and
“classic” and canonical; its static character was in some periods Russia) or mostly by school
enhanced by the few existing educational institu- teachers (Germany). In a few cases, some text-
tions. Striving for new knowledge was even books, like Legendre’s book on geometry,
considered suspect, and original productivity continued to be used internationally. Predomi-
appeared primarily in the form of commentaries nantly, however, textbooks were now published
on canonical textbooks. Moreover, the overall exclusively for use in their respective countries.
culture of orality enforced the leading role of The former type of single book for a teacher and
his students gave way to more differentiated sets
including schoolbooks for students, methodical

H 264 History of Mathematics Teaching and Learning

guides for teachers, collection of problems and for Protestant schools, largely the graduates of
exercises, and booklet with solutions. the Theology Faculty came teaching to the
schools when they could not find a parish (then
Mathematics Learning for Girls they taught mainly classical languages). For
teaching arithmetic, Gymnasia used to hire
Even when primary education was available for a practitioner. The first specialized teachers of
girls, they were for a long time excluded from mathematics at the Gymnasia are known in the
attending public secondary schools. The feminist early eighteenth century only (in the kingdom of
movement in the second half of the nineteenth Saxony).
century was instrumental in establishing eventu-
ally separate schools for girls. The history of such France did not establish teacher education
schools in various countries has been poorly stud- even after the Revolution and left it to the
ied. These first schools offered fewer career individual’s preparation for a concours. Later
opportunities for girls than for boys; in particular, on, the E´cole normale supe´rieure prepared
mathematics only played a minor role, given the candidates for this concours, the agre´gation. It
persistently strong prejudices negating women’s was Prussia that reformed its Philosophy Faculty
ability to understand mathematics. At best, they from 1810 by charging it with the scientific
were attributed intuition instead of abstract formation of teachers, particularly in mathe-
thinking. The curriculum for these schools thus matics. From the 1820s, this education was
focused on intuitively accessible geometric complemented by a subsequent probationary
concepts. Secondary schools created for girls in year for training in the teaching practice
Italy in 1923 featured “drawing” as the only (Schubring 1991). While various profiles of scien-
subject with some remote mathematical kinship. tific formation emerged for future mathematics
In Nazi Germany, the curriculum for girls teachers in different countries, the basic problem
became reduced even further, focusing only on remained: How would qualification in mathematics
those geometric forms which might be of use in be complemented by qualification in teaching prac-
the household. tice? A good overview of the situation during the
first half of the twentieth century is provided by the
By the social reforms of the 1960s and the international reports of the IMUK/CIEM at the 1932
expansion of the secondary schools, the girls’ Congress of Mathematicians (see L’Enseignement
schools merged with the boys’ schools almost Mathe´matique vol. 32, 1933, 5–22).
everywhere, and both girls and boys were
taught the same curriculum. No longer did Only during the 1970s did a broader concept
the curriculum maintain a female inferiority in of professional qualifications become established
mathematical thinking. in numerous countries, now including pedagogi-
cal qualifications and studies in mathematics edu-
Teachers of Mathematics cation in the university, followed by probationary
training in schools. In some countries, the educa-
The professionalization and special training of tion of teachers for the primary grades was ele-
mathematics teachers are recent developments. vated to university level; yet, it remained largely
For a long time, teachers used to be self-taught unspecialized for mathematics and included
persons, practitioners, or generalists. The first preparation for teaching various subjects.
time teacher training became institutionalized in
primary schools (see Section “Mathematics in Mathematics in the Global World
Primary Schooling”). For Catholic secondary
schools, the various religious orders practiced Mathematics has been created and developed
rudimentary forms of training for their novices; around the world, and each culture made its
own distinctive contribution to its development
(D’Ambrosio 2006). The modern system of

History of Mathematics Teaching and Learning H265

mathematics education, however, for all the modern mathematics education practices – making H
variety that it exhibits across different countries, use of European textbooks, exams, methods of
owes a great deal to structures and conceptions teaching, and either European teachers or at least
that emerged during the first half of the sixteenth teachers who had been trained in Europe.
century in Western Europe due to specific
economic, social, and cultural developments and Note that the process of borrowing from
were decisively shaped later, during the time of other countries was not always unproblematic.
the Enlightenment. The manner in which West- Researchers have pointed out that, for example,
ern Europe’s influence spread and took root “without the brutal intrusion of Western
around the world varied from country to country. powers, development of the Chinese culture in the
political, social and scientific arenas may have
In Japan, the Meiji Restoration (officially achieved a totally different but harmonious exis-
announced in 1868) led to Westernization and tence” (Chan and Siu 2012, p. 471). Even in Russia,
ushered in the broad use of foreign textbooks and where active employment of Western European
the recruitment of foreign teachers (Ueno 2012). teaching materials and teachers began as early as
In China, as has been noted, the development of the first half of the eighteenth century, foreign
mathematics and of the teaching of mathematics influences in education were not infrequently later
has a history that is many centuries long, and at perceived as hostile (Karp 2006). Mathematics
certain stages China was far ahead of the countries education was often part of political discussions.
of Europe. By the nineteenth century, however,
China was clearly and appreciably falling behind The complicated process in which national
in science and technology, which led to its defeat systems of mathematics education were formed
in a number of wars. The response to these defeats in developing countries is part of more recent
was modernization, which may be to a certain history. Only very gradually did a national work
extent equated with Westernization: new educa- force of teachers and centers for their prepara-
tional institutions began to appear and new pro- tions began to appear in these countries, along
grams and methods of teaching, borrowed from with textbooks and teaching materials. The colo-
the West, began to be used (Chan and Siu 2012). nial powers left these countries largely illiterate
and mathematically illiterate. The development
The Ottoman Empire’s system of mathematics and often even the establishment of an education
education developed in a largely similar way. system based on practices available in the world
While in the countries that made up this empire and aimed not at an elite, but at all students, was
interest in astronomy and mathematics, and and in many instances remains a crucial problem.
consequently in an education in these subjects Such international organizations as UNESCO, as
based on Arab sources, was noted by travelers well as separate countries, including countries
as early as the eighteenth century, a crucial step belonging to hostile political blocs, have
was taken with the establishment of national mil- provided assistance with the development of
itary schools, in which the teaching of mathemat- education, including mathematics education. In
ics was conducted in accordance with European the process, distinctive local features were quite
models (Abdeljaouad 2012). frequently ignored (Karp 2013). Meanwhile, the
preservation of indigenous and culturally specific
Another pattern is exemplified, for example, features is particularly important in the context of
by Tunisia, which at one time belonged to increasing tendencies toward globalization.
the Ottoman Empire – European-type schools
were later set up here by French colonial author- Research into History of Mathematics
ities (Abdeljaouad 2014). Such a pattern was also Education as a Field
characteristic of many other countries in Africa,
Asia, and Latin America: European colonial author- The history of mathematics education as
ities established schools for European settlers, as a scholarly field is still in the process of
well as for a narrow segment of local elites, thereby
nonetheless introducing into these countries more

H 266 History of Mathematics Teaching and Learning

formation. To be sure, many significant Special mention should be made of the impor-
studies were conducted as early as the nineteenth tance of research in the history of mathematics
or early twentieth centuries (see International education in developing countries. Usually little
Bibliography); for example, the first doctoral dis- is known about education in these countries
sertations in mathematics education in the United during the pre-colonial period, yet mathematics
States completed in 1906 under the supervision of was in one way or another a part of culture
David Eugene Smith were devoted precisely to everywhere. Nor have interactions between
the history of mathematics education. local cultures and various European cultures
been sufficiently studied, even though education
In recent decades, the development of in the colonies of different European countries
mathematics education as a scientific discipline was by no means identical. Nor was the formation
(Kilpatrick 1992) has led to a growing interest in of mathematics education during the postcolonial
its history. This is attested to by the creation of period in these countries everywhere alike.
special ICME Topic Study Groups, the appear- Research in these directions must continue.
ance of a special journal devoted to the history of
mathematics education, the organization of spe- Cross-References
cial scientific conferences, etc. (Furringhetti
2009). All such activity facilitates the formation ▶ Curriculum Resources and Textbooks in
of shared standards of research and methodology. Mathematics Education

The history of mathematics education, like ▶ Gender in Mathematics Education
any historical discipline, is based first and fore- ▶ History of Research in Mathematics Education
most on the analysis of primary sources. It is ▶ Mathematics Teacher Education Organization,
important, however, to conceive of these sources
in a sufficiently broad manner, not limiting Curriculum, and Outcomes
research to “administrative history” (Schubring ▶ Professional Learning Communities in
1988) – that is, the history of decrees concerning
education or even standards and curricula. Mathematics Education
Objects and sources of study include textbooks,
students’ notebooks, exam questions and References
answers, complaints and their analysis, biograph-
ical documents, diaries, letters, memoirs, journal- Abdeljaouad MM (2012) Teaching European mathemat-
istic, and even imaginative writing. ics in the Ottoman Empire during the eighteenth and
nineteenth centuries: between admiration and
Perhaps even more important is not stopping rejection. ZDM Int J Math Educ 44:483–498
at a purely descriptive approach: that is, to seek
not only to establish the events that have taken Abdeljaouad MM (2014) Mathematics education in
place but also to understand their position in Islamic Countries in the modern time: case study of
the context of other events and social historical Tunisia. In Karp A, Schubring G (eds) Handbook on
processes. The very choice of what to teach or the history of mathematics education. Springer,
offer on exams is evidently determined not only New York, pp 405–428
by strictly mathematical but also by social
considerations, whose meaning and content Chan Yip-Cheung, Siu Man-Keung (2012) Facing the
must be elucidated (Karp 2011). The role and change and meeting the challenge: mathematics
place of the mathematics teacher and of the curriculum of Tongwen Guan in China in the second
subject of mathematics itself; the interaction half of the nineteenth century. ZDM Int J Math Educ
between higher and secondary education; the 44:461–472
mutual influences among various cultures in
teaching; the causes of, attempts at, and outcomes D’Ambrosio U (2006) Ethnomathematics: link between
of reforms – these and other areas of research are traditions and modernity. Sense, Rotterdam
today the most worthy of study.
Furringhetti F (2009) On-going research in the history of
mathematics education. Int J Hist Math Educ
4(2):103–108

International bibliography on the history of teaching and
learning mathematics. http://www.tc.edu/centers/
ijhmt/index.asp?Id. Accessed 17 Sep 2012

History of Research in Mathematics Education H267

Karp A (2006) “Universal responsiveness” or “splendid Main Text H
isolation”? Episodes from the history of mathematics
education in Russia. Paedagog Hist 42(4–5):615–628 Although mathematics has been taught and
learned for millennia, not until the past century
Karp A (2011) Toward a history of teaching the mathe- or so have the nature and quality of teaching and
matically gifted: three possible directions for research. learning mathematics been studied in any
Can J Sci Math Technol Educ 11:8–18 a serious manner. Clay tablets from ancient
Babylonia (c 1900 BC to c 1600 BC), for
Karp A (2013) From the local to the international in example, show that students in the scribal school
mathematics education. In Clements MA (Ken), were expected to solve problems involving
Bishop AJ, Keitel C, Kilpatrick J, Leung FKS (eds) quadratic polynomials (Høyrup 1994, pp. 4–9),
Third international handbook of mathematics but no available evidence indicates how much
education. Springer, New York, pp 797–826 drill and practice either they received or their
instructors thought they needed. As of 1115 BC,
Kilpatrick J (1992) A history of research in mathematics applicants to the Chinese civil service had to pass
education. In: Grows DA (ed) Handbook of research an examination in arithmetic (Kilpatrick 1993,
on mathematics teaching and learning. Macmillan, p. 22), but as far as anyone knows, no one
New York, pp 3–38 ever investigated how well their examination
performance predicted their job performance. In
Schubring G (1988) Theoretical categories for investiga- Plato’s Meno, he relates how, in the fifth century
tions in the social history of mathematics education BC, Socrates helped a slave boy discover that
and some characteristic patterns. Institut f€ur Didaktik doubling the side of a square apparently squares
der Mathematik der Universitat Bielefeld: Occasional its area. Plato does not, however, say how well
paper # 109 the boy fared with similar geometry problems
once his teacher was no longer around.
Schubring G (1991) Die Entstehung des Mathematikleh- Mathematics education is a long-established
rerberufs im 19. Jahrhundert. Studien und Materialien field of practice; research in mathematics educa-
zum Prozeß der Professionalisierung in Preußen tion, a relatively recent enterprise.
(1810-1870), 2nd edn. Deutscher Studien, Weinheim
Over the centuries, teachers of mathematics in
Schubring G (2003) Ana´lise Histo´rica de Livros de various countries have offered reflective accounts
Matema´tica. Notas de Aula. Editora Autores of their work, often writing textbooks constructed
Associados, Campinas around teaching techniques they developed
out of their own experience. Only during the
Schubring G (2012) From the few to the many: historical nineteenth century, however, as national educa-
perspectives on who should learn mathematics. In: tional systems were established and the training
“Dig where you stand.” 2. Proceedings of the second of teachers moved into colleges and universities,
international conference on the history of mathematics did people begin to identify themselves as math-
education. UIED, Lisbon, pp 443–462 ematics educators and begin to conduct research
as part of their scholarly identity (Kilpatrick
Ueno K (2012) Mathematics teaching before and after the 1992, 2008). Not until 1906 were the first
Meiji Restoration. ZDM Int J Math Educ 44:473–481 doctorates in mathematics education granted –
to Lambert L. Jackson and Alva W. Stamper,
History of Research in Mathematics students of David Eugene Smith at Teachers
Education College, Columbia University (Donoghue
2001). Within the next few decades, research
Jeremy Kilpatrick in mathematics education gradually began to
University of Georgia, Athens, GA, USA be conducted in several countries as lectures
in mathematics education were offered and
Keywords

Research; Mathematics education; History;
Academic field; Psychology; Mathematics

Definition

An account of activities and events concerned with
the development of disciplined inquiry in mathemat-
ics education as a flourishing academic enterprise.

H 268 History of Research in Mathematics Education

graduate programs in mathematics education a similar but less formal inquiry into the working
became established in universities. habits of mathematicians in America that
went somewhat deeper into the methods and
Mathematics Education images they used (Kilpatrick 1992). Other
as an Academic Field early researchers were psychologists who were
developing an interest in how children think
The education of teachers, which had often been about and learn mathematical ideas. Beginning
a hit-or-miss affair, did not become a field of in 1875, with Wilhelm Wundt’s establishment of
professional studies until the nineteenth century. a laboratory in Leipzig and William James’s
Although teacher-training schools had begun in establishment of one at Harvard, dozens of
France and Prussia late in the seventeenth century, psychological laboratories were established in
only in the eighteenth century were normal Europe, Asia, and North America (Kilpatrick
schools – very much influenced by the ideas of 1992). Psychologists such as Alfred Binet,
the Swiss pedagogue and reformer Johann his colleague Jean Piaget, Max Wertheimer,
H. Pestalozzi – established in European countries Otto Selz, and Lev Vygotsky investigated
(Cubberley 1919). In 1829, the American mental ability and productive thinking using
geographer William C. Woodbridge, who in the mathematical tasks. Psychology was becoming
previous 4 years in Europe had observed schools in the so-called master science of the school:
Prussia and Switzerland and had visited Pestalozzi, “Psychology . . . became the guiding science of
tried unsuccessfully to establish in Hartford, Con- the school, and imparting to would-be teachers
necticut, a teachers seminary modeled after the the methodology of instruction, in the different
Prussian version. In 1831, he observed: “In those school subjects, the great work of the normal
of the countries of Europe where education has school” (Cubberley 1919, p. 400). Together,
taken its rank as a science, it is almost as singular mathematicians and psychologists began the
to question the importance of a preparatory semi- efforts that would lead to research in mathematics
nary for teachers, as of a medical school for phy- education.
sicians” (quoted by Cubberley 1919, p. 374).
Education in general had slowly been entering Comparative Studies of School
the university since the eighteenth century, begin- Mathematics
ning with a chair of education established at the
University of Halle in 1779, but not until the late In 1908, the International Commission on the
nineteenth and early twentieth centuries were such Teaching of Mathematics (ICTM) was formed
chairs established elsewhere, and only then did at the Fourth International Congress of
school mathematics start to become an object of Mathematicians in Rome. Its purpose was “to
scholarly study (Kilpatrick 2008). report on the state of mathematics teaching at
all levels of schooling around the world”
Many of the early researchers in mathematics (Kilpatrick 1992, p. 6). In 1912, at the Fifth
education were mathematicians who had become International Congress in Cambridge, England,
interested in how mathematics is done. For exam- some 17 countries presented reports, and by
ple, the editors of L’Enseignement Mathe´ 1920, the countries active in the ICTM had
matique, Henri Fehr and Charles-Ange Laisant, produced almost 300 reports (Schubring 1988;
sent a questionnaire to over 100 mathematicians Furinghetti 2008). The international comparisons
to learn how they did mathematics. The report based on these reports, however, were essentially
of their survey, which was published in 11 restricted to descriptions by a handful of
installments in the journal from 1905 to 1908, mathematicians or educators in each country of
was essentially a list of verbatim responses to activities that they were aware of. They did not
their questions. In contrast, the French mathema- engage in large-scale, systematic surveys of the
tician Jacques Hadamard later undertook

History of Research in Mathematics Education H269

school mathematics curriculum, nor did they certain tasks such as judging the size of rectan- H
visit classrooms to record instructional practices. gles did not improve their performance in – that
Nonetheless, they had begun the process of is, did not transfer to – judging the size of
looking across countries to get a better perspec- triangles (Kilpatrick 1992). Thorndike’s research
tive on mathematics education around the world. studies dealt a major blow to arguments that
mathematics ought to be taught and learned
In the last half century, researchers have because the logical thinking it promoted trans-
undertaken a variety of international comparative ferred to other realms. He argued that his research
assessments of students’ mathematical knowl- showed that transfer was much more limited than
edge and of teachers’ knowledge of pedagogy mathematics teachers appeared to assume.
and mathematics. They have also compared
mathematics teaching across countries using Thorndike not only published important books
video records of lessons. (For an analysis of the on the psychology of arithmetic and the
levels at which these comparisons have been psychology of algebra in which he promoted the
made, see Artigue and Winsløw 2010). Consid- psychology he termed connectionism; he also
erable progress has been made in both the published a series of arithmetic textbooks that
thoroughness with which such comparative was widely used in schools. Connectionism
studies have been done and the sophistication of became the forerunner of the behaviorism that
the data collection and analyses. Although these came to dominate much of research in mathematics
studies can be criticized for being too oriented education in the United States from the 1930s
toward Western practice and inadequately sensi- through the 1950s (Clements and Ellerton 1996).
tive to Asia-Pacific cultures (Clements and Although other psychologists, such as Charles
Ellerton 1996), they have had, in many countries, H. Judd, Guy T. Buswell, and William A.
considerable influence on curriculum, teaching, Brownell, performed research studies that called
and educational policy. For an account of the Thorndike’s work into question, thereby develop-
development of international collaboration in ing a psychology of the school subjects that
mathematics education during the past century, mathematics educators found more congenial
see Karp (2013). (Kilpatrick 1992), connectionism and its successor
behaviorism exerted a much stronger influence on
Becoming Scientific research methodology in mathematics education
for many years and not just in the United States.
In trying to make their field scientific, educational
psychologists looked to the natural sciences for Elsewhere in the first decade of the twentieth
models, and in much the same way, some century, some psychologists were looking at errors
mathematics educators seeking to establish their and difficulties that children were having in
field as a science took those sciences as models. arithmetic. Paul Ranschburg in Budapest,
They studied mathematics learning under con- in particular, began the study of differences in
trolled laboratory conditions, testing hypotheses calculation performance between normal children
about the effects of various “treatments,” and and low achievers in arithmetic. In 1916, he coined
making careful measurements of the learning the term Rechenschwa€che (dyscalculia) for severe
achieved. Influential examples were studies inability to perform simple arithmetic calculations
by the psychologist Edward L. Thorndike in (Schubring 2012). Like Thorndike, Ranschburg
the early years of the twentieth century. Using attributed children’s successful performance to
a control group whose performance was their possession of Vorstellungsketten (chains of
compared with that of an experimental group association), but his research method relied more
(with students assigned randomly to one of the on observation of differences between existing
two groups), Thorndike demonstrated that prac- groups (normal and low achieving) than on
tice by the experimental group in performing experimentation.

Psychologists gradually stopped being so
concerned about emulating the natural sciences

H 270 History of Research in Mathematics Education

and began to develop their own techniques for Didactics of mathematics, however, was not
studying learning, and researchers in mathemat- the only research effort to address mathematics
ics education followed. For example, in the teaching. In a number of studies conducted in the
movement known as “child study” (Kilpatrick first half of the twentieth century, components
1992), which had appeared in Germany and the of teaching or characteristics of teachers were
United States at the end of the nineteenth century, linked to learners’ performance in efforts to
researchers looked at the development of understand what might constitute effective
concepts in young children using techniques of teaching. Researchers eventually moved from
observation and interview. Although mathemat- such simple “process-product” models to more
ics was not often the focus of child study sophisticated efforts that attempted to capture
research, it did give rise to a number of descrip- more of the complexity of the teaching-learning
tive, naturalistic studies. Less than a century later, process, including the knowledge and beliefs
research on the learning of mathematics had of the participants as well as their activities
burgeoned. A survey in the 1970s, for example, during instruction. For an account of the
located some 3,000 published studies of gradual elaboration of research models for study-
mathematics learning (Bauersfeld 1979). ing mathematics teaching, see Koehler and
Grouws (1992).
Studying the Teaching of Mathematics
In later developments, researchers attempted
As mathematics educators began to study to go deeper into questions of what constitutes
children’s mathematics learning and thinking, classroom practice in mathematics and how that
they increasingly recognized that laboratory is experienced by teachers and learners. In
studies present a restricted view of those particular, they studied how discourse is struc-
processes; however, they are conceived. Children tured in mathematics classes, how norms are
do most of their learning of mathematics in established in classrooms for learning and doing
school classrooms along with other children, mathematics, and how teachers and learners build
and their thinking about mathematical concepts relationships based on getting to know each other
and problems is much influenced by others, (Franke et al. 2007). Research on teaching and
including their teacher. The psychologist Ernst teachers has become a major strand of current
Meumann, who had studied with Wundt in Leip- research in mathematics education, and those
zig, was one of the first to address what he called studies now extend from preschool to tertiary
“experimental pedagogy” and in 1914 published instruction.
a volume in which he looked at the didactics of
teaching specific school subjects (Schubring An especially fertile development of recent
2012). Meumann was the forerunner of decades has been the growth of research on
researchers who were later in the century to technology and digital environments for
establish a critically important field of research, mathematics teaching and learning. Physical
especially in Germany and France: the didactics tools have been used for centuries to assist the
of mathematics (Artigue and Perrin-Glorian teaching and learning of mathematics, and an
1991; Biehler et al. 1993). Although the didactics examination of how those tools have been used
of mathematics began with a psychological can help put into perspective the use of comput-
orientation, it came under the influence of ing technology today (Roberts et al. 2013).
other fields – anthropology and philosophy, in In an early review of how electronic technologies
particular – as it was increasingly located in had been studied in mathematics education
university departments of mathematics and research, Kaput and Thompson (1994) lamented
began to become established as one of the the paucity of technology-related research publi-
mathematical sciences. cations. That situation has changed dramatically
since that review, as numerous recent books (e.g.,
Guin et al. 2005; Hoyles and Lagrange 2010)
and journals (e.g., International Journal for

History of Research in Mathematics Education H271

Technology in Mathematics Education; Journal such as the Canadian Mathematics Education H
of Computers in Mathematics and Science Study Group (CMESG), the French Association
Teaching) attest. pour la Recherche en Didactique des Mathe´
matiques (ARDM), and the Mathematics Educa-
A Flourishing Academic Enterprise tion Research Group of Australasia (MERGA).
For a comprehensive survey of international
The last half century has witnessed a growing or multinational organizations in mathematics
flood of research activity in mathematics education, see Hodgson et al. (2013). Many of
education that has been an integral part of its these organizations hold regular conferences on
growth and development: research and publish research journals. Main-
stream journals that have been publishing research
Today an astonishing profusion of books, for more than four decades, such as Educational
handbooks, proceedings, articles, research reports, Studies in Mathematics and the Journal for
newsletters, journals, meetings, and organizations is Research in Mathematics Education, have lately
devoted to mathematics education in all its aspects. been joined by more specialized research journals
A search of the scholarly literature on the Web for such as the Journal of Mathematics and Culture,
the phrase mathematics education yields 125,000 started in 2006, and the Journal of Urban Mathe-
hits; a search of the entire Web yields almost 9 matics Education, started in 2008. For an account
times that number. (Kilpatrick 2008, p. 38). of the growth of journals and research conferences
in mathematics education, see Furinghetti et al.
One measure of the maturation of the field of (2013). The sheer volume of research activity
mathematics education is that researchers have being reported in these journals and at these
begun to study its history. A major milestone was conferences is staggering. A comprehensive
the founding in 2006 of the International Journal portrayal of research activity in mathematics
for the History of Mathematics Education. education today is no longer possible; the terrain
The history of the field had been discussed at is simply too extensive and diverse to be
various international conferences beginning in captured in toto.
2004, and a series of biennial conferences
devoted to the topic began in Iceland in 2009. Cross-References

As the field of mathematics education has ▶ Anthropological Approaches in Mathematics
grown, research in the field has grown even Education, French Perspectives
faster. The subject matter of research studies has
broadened to include such topics as the school ▶ Constructivist Teaching Experiment
mathematics curriculum, assessment in mathe- ▶ Design Research in Mathematics Education
matics, the education of mathematics teachers ▶ History of Mathematics Teaching and Learning
and their professional development, the sociopo- ▶ International Comparative Studies in
litical context of learning and teaching mathe-
matics, teaching mathematics to students in Mathematics: An Overview
special education programs, and the politics of ▶ Teacher as Researcher in Mathematics
mathematics education. The methods used to
conduct research now go well beyond experimen- Education
tation to include case studies of teachers and ▶ Theories of Learning Mathematics
students, surveys of attitudes and beliefs, and
ethnographies of cultural practices. References

Organizations of researchers have been formed Artigue M, Perrin-Glorian M-J (1991) Didactic engineer-
that range from those of international scope, such ing, research and development tool: some theoretical
as the International Group for the Psychology of problems linked to this duality. Learn Math
Mathematics Education (IGPME, or PME), to 11(1):13–18
organizations within one or several countries,

H 272 Hypothetical Learning Trajectories in Mathematics Education

Artigue M, Winsløw C (2010) International comparative Kilpatrick J (1992) A history of research in mathematics
studies on mathematics education: a viewpoint from education. In: Grouws D (ed) Handbook of research on
the anthropological theory of didactics. Recherche en mathematics teaching and learning. Macmillan,
Didact des Maths 31:47–82 New York, pp 3–38

Bauersfeld H (1979) Research related to the learning Kilpatrick J (1993) The chain and the arrow: from the history
process. In: Steiner H-G, Christiansen B (eds) of mathematics assessment. In: Niss M (ed) Investiga-
New trends in mathematics teaching, vol 4. Paris, tions into assessment in mathematics education: an
Unesco, pp 199–213 ICME Study. Kluwer, Dordrecht, pp 31–46

Biehler R, Scholz RW, Straesser R, Winkelmann B (eds) Kilpatrick J (2008) The development of mathematics
(1993) The didactics of mathematics as a scientific education as an academic field. In: Menghini M,
discipline. Kluwer, Dordrecht Furinghetti F, Giacardi L, Arzarello F (eds) The
first century of the International Commission on
Clements MA, Ellerton NF (1996) Mathematics education Mathematical Instruction (1908–2008): reflecting and
research: past, present and future. UNESCO Principal shaping the world of mathematics education. Istituto
Regional Office for Asia and the. Pacific, Bangkok della Enciclopedia Italiana, Rome, pp 25–39

Cubberley EP (1919) Public education in the United Koehler MS, Grouws DA (1992) Mathematics teaching
States: a study and interpretation of American practices and their effects. In: Grouws D (ed)
educational history. Houghton Mifflin, Boston Handbook of research on mathematics teaching and
learning. Macmillan, New York, pp 115–126
Donoghue EF (2001) Mathematics education in the
United States: origins of the field and the development Roberts DL, Leung AYL, Lins AF (2013) From the slate to
of early graduate programs. In: Reys RE, Kilpatrick the web: technology in the mathematics curriculum. In:
J (eds) One field, many paths: U.S. doctoral programs Clements MA, Bishop AJ, Keitel C, Kilpatrick J,
in mathematics education. American Mathematical Leung FKS (eds) Third international handbook of math-
Society, Providence, pp 3–17 ematics education. Springer, New York, pp 525–547

Franke ML, Kazemi E, Battey D (2007) Mathematics Schubring G (1988) The cross-cultural “transmission” of
teaching and classroom practice. In: Lester FK Jr (ed) concepts: the first international mathematics curricular
Second handbook of research on mathematics teaching reform around 1900, with an appendix on the
and learning. Information Age, Charlotte, pp 225–256 biography of F. Klein. Occasional paper no 92,
corrected edition. Institut f€ur Didakdik der
Furinghetti F (2008) Mathematics education in the ICMI Mathematik, Universit€at Bielefeld, Bielefeld
perspective. Int J Hist Math Educ 3:47–56
Schubring G (2012) ‘Experimental pedagogy’ in Germany,
Furinghetti F, Matos JM, Menghini M (2013) From math- elaborated for mathematics: a case study in searching the
ematics and education, to mathematics education. roots of PME. Res Math Educ 14:221–235
In: Clements MA, Bishop AJ, Keitel C, Kilpatrick
J, Leung FKS (eds) Third international handbook Hypothetical Learning Trajectories
of mathematics education. Springer, New York, in Mathematics Education
pp 273–302
Martin Simon
Guin D, Ruthven K, Trouche L (eds) (2005) The didactical Steinhardt School of Culture, Education, and
challenge of symbolic calculators: turning Human Development, New York University,
a computational device into a mathematical New York, NY, USA
instrument. Springer, New York
Keywords
Hodgson BR, Rogers LF, Lerman S, Lim-Teo SK
(2013) International organizations in mathematics Learning; Teaching; Constructivism; Teacher
education. In: Clements MA, Bishop AJ, Keitel C, thinking; Learning progressions
Kilpatrick J, Leung FKS (eds) Third international
handbook of mathematics education. Springer, Definition
New York, pp 901–947
Hypothetical learning trajectory is a theoretical
Hoyles C, Lagrange JB (eds) (2010) Mathematics model for the design of mathematics instruction.
education and technology – rethinking the terrain: the
17th ICMI Study. Springer, New York

Høyrup J (1994) In measure, number, and weight: Studies
in mathematics and culture. State University of
New York Press, Albany

Kaput JJ, Thompson PW (1994) Technology in
mathematics education research: the first 25 years in
the JRME. J Res Math Educ 25:667–684

Karp A (2013) From the local to the international in
mathematics education. In: Clements MA, Bishop
AJ, Keitel C, Kilpatrick J, Leung FKS (eds) Third
international handbook of mathematics education.
Springer, New York, pp 797–826

Hypothetical Learning Trajectories in Mathematics Education H273

It consists of three components, a learning goal, To elaborate the last point, the second and H
a set of learning tasks, and a hypothesized third components of the hypothetical learning
learning process. The construct can be trajectory, the learning activities and the hypo-
applied to instructional units of various lengths thetical learning process, are interdependent and
(e.g., one lesson, a series of lessons, the learning co-emergent. The learning activities are based on
of a concept over an extended period of time). anticipated learning processes; however, the
learning processes are dependent on the nature
Explanation of the Construct of the planned learning activities. Clement and
Simon (1995) postulated the construct hypothet- Sarama (2004a, p. 83) reaffirmed this point.
ical learning trajectory. Simon’s goal in this
heavily cited article was to provide an empirically Although studying either psychological develop-
based model of pedagogical thinking based on mental progressions or instructional sequences
constructivist ideas. (Pedagogical refers to separately can be valid research goals, and studies
all contributions to an instructional intervention of each can and should inform mathematics
including those made by the curriculum devel- education, the power and uniqueness of the learning
opers, the materials developers, and the teacher.) trajectories construct stems from the inextricable
The construct has provided a theoretical frame for interconnections between these two aspects.
researchers, teachers, and curriculum developers
as they plan instruction for conceptual learning. They went on to define learning trajectories
as follows.
Simon (1995. P. 136) explained the compo-
nents of the hypothetical learning trajectory: We conceptualize learning trajectories as descrip-
tions of children’s thinking and learning in a
The hypothetical learning trajectory is made up of specific mathematical domain and a related,
three components: the learning goal that defines the conjectured route through a set of instructional
direction, the learning activities, and the hypothet- tasks designed to engender those mental processes
ical learning process – a prediction of how the or actions hypothesized to move children through a
students’ thinking and understanding will evolve developmental progression of levels of thinking,
in the context of the learning activities. created with the intent of supporting children’s
achievement of specific goals in that mathematical
There are a number of implications of this domain (c.f. Clements 2002; Gravemeijer 1999;
definition including the following: Simon 1995) (p. 83).
• Good pedagogy begins with a clearly
According to Simon (1995), a hypothetical
articulated conceptual goal. learning trajectory was part of a mathematics
• Although students learn in idiosyncratic ways, teaching cycle that connects the assessment of
student knowledge, the teacher’s knowledge,
there is commonality in their ways of learning and the hypothetical learning trajectory. The
that can be the basis for instruction. Therefore, cycle is meant to capture a progression in which
useful predictions about student learning can an instructional intervention is made based
be made. on the hypothetical learning trajectory. Student
• Instructional planning involves informed knowledge/thinking is monitored throughout.
prediction as to possible student learning This monitoring leads to new understandings of
processes. student thinking and learning, which, in turn,
• Based on prediction of students’ learning leads to modifications in the hypothetical
processes, instruction is designed to foster learning trajectory. The mathematics teaching
learning. cycle also stresses that, in the context of teaching,
• The trajectory of students’ learning is not teachers develop additional knowledge of
independent of the instructional intervention mathematics and mathematical representations
used. Students’ learning is significantly and tasks. All modifications in teacher
affected by the opportunities and constraints knowledge contribute to changes in the revised
that are provided by the structure and content hypothetical learning trajectory. Thus, an
of the mathematics lessons. implication of the mathematics teaching cycle is

H 274 Hypothetical Learning Trajectories in Mathematics Education

that a big part of good teaching is the ability to (CCSSO/NGA 2010) in the United States has
analyze student learning in order to revise the leaned heavily on the learning progressions work
instructional approach. to date. A key issue as research on learning pro-
gressions develops is whether a central idea in
The mathematics education research commu- Simon’s hypothetical learning trajectory will be
nity picked up the hypothetical learning trajectory maintained. That is, will the learning process con-
construct, and 9 years after the original article, tinue to be seen as interrelated with the instruc-
Clements and Sarama (2004b) edited a special tional approach or will various stakeholders in
issue of Mathematics Thinking and Learning on mathematics education seize on particular learning
hypothetical learning trajectories. Although the progressions as the way that students learn. The
hypothetical learning trajectory construct grew quote above from Daro et al. seems to imply that
out of constructivist ideas, it has been adapted for there is a set of learning steps, and then instruction
use with social learning theories (e.g., McGatha is built to foster that sequence of steps. This stands
et al. 2002). in contrast to a view that any particular sequence of
steps is in part a product of the instructional expe-
Two lines of research grew out of the original riences provided to the students. Clements and
work on hypothetical learning trajectories. The Sarama pointed to an important implication of the
first, conducted by Simon and his colleagues, is perspective based on Simon’s original definition:
an attempt to explicate the mechanisms of con-
ceptual learning, that is, to provide a framework Thus, a complete hypothetical learning trajectory
for generating hypothetical learning processes in includes all three aspects. . . . Less obvious is that
conjunction with learning activities. (See Tzur their integration can produce novel results. . . . The
and Lambert 2011; Simon et al. 2010; Tzur enactment of an effective, complete learning
2007; Simon et al. 2004; Simon and Tzur 2004; trajectory can actually alter developmental pro-
Tzur and Simon 2004). Whereas research gressions or expectations previously established
grounded in constructivist ideas has a tradition by psychological studies, because it opens up new
of modeling students’ thinking at various points paths for learning and development.
in their conceptual learning, postulation of the
hypothetical learning trajectory construct called Cross-References
for modeling the learning process itself, the means
by which the students’ thinking changes as they ▶ Constructivism in Mathematics Education
interact with the instructional tasks and setting.
References
The second line of research, which grew out
of the original hypothetical learning trajectory CCSSO/NGA (2010) Common core state standards for
work, is research on learning trajectories in mathematics. Council of Chief State School Officers
mathematics (also referred to as “learning and the National Governors Association Center
progressions”; see discussion of learning progres- for Best Practices, Washington, DC. http://
sions in this volume). Learning progressions corestandards.org
research is an attempt to provide an empirical
basis for instructional planning. Clements DH (2002) Linking research and curriculum
development. In: English LD (ed) Handbook of inter-
Trajectories involve hypotheses both about the national research in mathematics education. Erlbaum,
order and nature of the steps in the growth of Mahwah, pp 599–630
students’ mathematical understandings and about
the nature of the instructional experience that Clements DH, Sarama J (2004a) Learning trajectories
might support them in moving step-by-step toward in mathematics education. Math Think Learn 6:
their goals of school mathematics (Daro et al. 81–89
2011, p 12).
Clements DH, Sarama J (2004b) Hypothetical learning
Not only have a significant number of trajectories (special issue). Math Think Learn 6(2).
researchers gotten involved in this line of Erlbaum, Mahwah
research, but the Common Core Standards
Daro P, Mosher FA, Corcoran T (2011) Learning
trajectories in mathematics: a foundation for standards,

Hypothetical Learning Trajectories in Mathematics Education H275

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Think Learn 1:155–177 approach to studying students’ learning through their
McGatha M, Cobb P, McClain K (2002) An analysis of mathematical activity. Cognit Instr 28:70–112
students’ initial statistical understandings: developing Tzur R (2007) Fine grain assessment of students’ mathe-
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H

I

Immigrant Students in Mathematics is largely based on the work undertaken
Education to address this theme for that survey team
(Civil 2012). The pressing need to address
Marta Civil the mathematics education of immigrant
School of Education, University of North students is reflected in the following quote by
Carolina at Chapel Hill, Chapel Hill, NC, USA Gates (2006):

Keywords In many parts of the world, teachers – mathematics
teachers – are facing the challenges of teaching in
Diversity; Multicultural; Multilingual; multiethnic and multilingual classrooms
Immigration; Ethnomathematics containing immigrant, indigenous, migrant, and
refugee children, and if research is to be useful it
has to address and help us understand such chal-
lenges. (p. 391)

Definition This quote mentions four diverse groups –
immigrant, indigenous, migrant, and refugee
“Immigrant students” refers to the case when children. The research reviewed for this article
students (or their parents) were born in will focus on immigrant students. However, the
a country other than the one they are currently research with indigenous communities contains
living in and attending school. much relevant information to the teaching and
learning of immigrant students. One example is
Characteristics the work from an ethnomathematics perspective
that emphasizes engaging indigenous communi-
The topic of the mathematics education of ties in the development of the teaching and
immigrant students has become quite prominent learning of mathematics, hence bringing in
in different parts of the world. An indication of the communities’ knowledge, experiences, and
this is that one of the survey teams at the 11th approaches as valuable resources (Meaney 2004;
International Congress on Mathematical Educa- Lipka et al. 2005).
tion (ICME) in 2008 focused on mathematics
education in multicultural and multilingual It is important to acknowledge that there
environments. One of the four themes under is large diversity among immigrant students.
this survey team was “the mathematics teaching This article focuses on some general charac-
and learning of immigrant students.” This article teristics that are likely to impact the mathe-
matics teaching and learning of low-income,
immigrant students, whose first language is
different from the language of schooling in

S. Lerman (ed.), Encyclopedia of Mathematics Education, DOI 10.1007/978-94-007-4978-8,
# Springer Science+Business Media Dordrecht 2014

I 278 Immigrant Students in Mathematics Education

the receiving country. It is organized around Different Forms of Mathematics
five themes: educational policy and immigra-
tion, different forms of mathematics, teacher The relationship between mathematics and
education in an immigration context, multilin- culture/context has been widely described (Bishop
gualism and mathematics teaching and learn- 1991; Nunes et al. 1993; DiME 2007; Presmeg
ing, and immigrant parents’ perceptions of 2007; Abreu 2008). This body of research stresses
mathematics education. that mathematics is not culture-free and illustrates
the complexity of the relationship between differ-
Educational Policy and Immigration ent forms of mathematics, in particular between
in- and out-of-school mathematics. Immigrant
It is important to understand the educational students are quite likely to bring with them differ-
policies in place with regard to the education of ent ways of doing mathematics. These differences
immigrant students. Whether those are grounded may be obvious, such as using different algorithms
on seeing immigrants as a resource or as for arithmetic operations, or subtler, such as
a problem is likely to affect the schooling expe- emphasis of topics studied. Immigrant students
riences of immigrant children. In their account of may have also experienced different pedagogical
the many faces of migration in the world, King, approaches from those in the receiving country
Black, Collyer, Fielding, and Skeldon (2010) (e.g., teacher lecturing vs. group work). Depending
discuss two prominent models of integration, on their context of immigration, they may bring
multiculturalism and assimilation. They note approaches that are more related to out-of-school
that while multiculturalism may have been the mathematical practices. Issues related to the gap
model in some European countries, more recently between in-school mathematics and out-of-school
“a swing back to assimilation has occurred, with mathematics and transitions across contexts are
greater demands on immigrants to learn the host- well documented (de Abreu et al. 2002; Nasir
country language and subscribe to core national et al. 2008; Meaney and Lange, 2013).
values” (p. 92). The research addressing the
mathematics teaching and learning of immigrant The research surveyed in Civil (2012) from
students underscores the potential negative different countries points to some general find-
impact of some educational policies and of ings concerning these different forms of mathe-
a general public discourse that frames immigra- matics. One such finding is that schools and
tion as a problem (Civil 2012). Such a framing is teachers are often not familiar with the mathe-
likely to affect teachers who may view the matical knowledge that immigrant children may
diversity of approaches to doing mathematics bring with them. A belief that mathematics is
that immigrant students often bring (e.g., dif- universal and culture-free may lead teachers to
ferent algorithms) as problematic rather than as not see these different forms and focus only on
an opportunity to learn. More research is the different languages at play (home and school)
needed to examine the possible connections as the main issue that affects immigrant students’
among educational policies, public views on learning of mathematics. Another finding is
immigration, and the mathematics education related to the concept of valorization of knowl-
of immigrant children. The complexity of the edge (Abreu and Cline 2007). That is, different
situation calls for interdisciplinary teams, forms of mathematics may be given different
where in addition to the expected expertise in valorization, and it is often the case that immi-
mathematics and mathematics education, there grant children’s mathematical knowledge may
is expertise on the political and policy (social, not be valued as much as the “expected” mathe-
educational, language, in particular with respect matical knowledge in the given school context.
to immigrant students) scene in the context Even in the cases in which teachers are aware of
(country, region) of work. these different forms of mathematics, they may
not have the appropriate background knowledge,
preparation, or support to develop learning

Immigrant Students in Mathematics Education I279

experiences that reflect and build on these differ- dispositions (Hollins and Torres Guzman 2005). I
ent forms. Thus, this finding underscores the need In mathematics teacher education, although there
to make sure that teacher education programs is a large body of research addressing teachers’
prepare teachers to not only acknowledge differ- mathematical knowledge and beliefs about
ent approaches to doing mathematics but also to teaching and learning mathematics, there seems
learn how to build on those in ways that are to be little research about teachers’ beliefs about
inclusive for immigrant students. equity, in particular in the areas that are likely
to apply to immigrant students (race, culture,
Teacher Education in an Immigration ethnicity, language, and socioeconomic back-
Context ground) (Forgasz and Leder 2008). Efforts in
mathematics teacher education need to empha-
The research surveyed on this topic addresses size that mathematics is not culture-free and may
teachers’ attitudes, beliefs, and knowledge with have to be more upfront in engaging teachers
respect to the teaching and learning of immigrant to discuss topics that are likely to create discom-
students. Overall, teachers feel unprepared to fort and may lead to resistance to diversity
address the mathematical learning needs of (Rodriguez and Kitchen 2005; Sowder 2007).
immigrant students (Favilli and Tintori 2002). As
mentioned earlier, language seems to be the main Multilingualism and Mathematics
factor of concern for teachers. However, in the Teaching and Learning
different studies surveyed, researchers point to
other areas that should be addressed when working As mentioned before, for education policy-
with teachers of immigrant students. One such area makers and many teachers and school personnel,
is the need to pay more attention to the cultural limited knowledge of the language of instruction
nature of learning (Abreu 2008). Another area is seems to be the main (if not the only) obstacle that
the need to confront deficit views towards immi- immigrant students need to overcome. Thus,
grant students. These views are often grounded on different educational systems across a variety of
public discourse about immigration rather than on countries attempt to address “the language prob-
a direct knowledge of the students and their lem” through systems that segregate immigrant
families and can lead to teachers not valuing the students for all or part of the day to focus on
mathematical knowledge that immigrant students learning the language of instruction. Researchers
bring with them (Alrø et al. 2005; Gorgorio´ and de in the teaching and learning of mathematics with
Abreu 2009; Planas and Civil 2009). immigrant students raise questions about the
implications of these language policies on the
One approach to engaging teachers in learning learning of mathematics (Alrø et al. 2005; Civil
about their immigrant students and their families 2011; Barwell 2012; Setati and Planas 2012).
is based on the concept of funds of knowledge Barwell (2012) provides an overview of some of
(Gonza´lez et al. 2005). Through ethnographic the key themes in multilingual mathematics
home visits, teachers learn about their students’ classrooms through a discussion of four tensions.
and families’ experiences, knowledge, and back- One such tension is around school language and
grounds. They can then build on this knowledge home languages. Researchers in mathematics
in their classroom teaching. In Civil and Andrade education in multilingual classrooms call for
(2002), this approach is applied to the teaching a focus on the strengths that multilingualism
and learning of mathematics. provides rather than on the fact that immigrant
students may lack proficiency in the language
Although there is considerable research in of schooling (Moschkovich 2002; Barwell
teacher education and diversity in general terms 2009; Clarkson 2009). Research shows the com-
(not necessarily specific to mathematics), still we plexity behind code-switching and language
know little about how effective teachers for
diverse students developed their knowledge and

I 280 Immigrant Students in Mathematics Education

choice in mathematics classrooms (Adler 2001; the schools, and conversely, parents do not
Moschkovich 2007; Planas and Setati 2009) and always see the point in some of the school
how code-switching is actually a resource approaches to teaching mathematics.
towards students’ learning of mathematics rather
than a deficit. This body of research also points Some Implications
to the need to develop models of teaching in
multilingual mathematics classrooms that are Based on the literature reviewed, here are some
not based on a monolingual view of teaching key points to keep in mind when addressing the
and learning mathematics (Clarkson 2009). mathematics teaching and learning of immigrant
students. Efforts should be made to focus on the
A focus on language as an obstacle may prevent knowledge and experiences that immigrant
teachers from seeing the mathematical knowledge students and their families bring rather than on
that immigrant students bring with them. An what they lack (e.g., limited knowledge of the
important question to consider is to which extent language of instruction). Seeing diversity as
placement decisions in mathematics classes are a resource rather than as a problem could enhance
based on students’ knowledge of this subject or the learning opportunities in mathematics for
on their level of proficiency in the language of all students in the classroom. Through a deeper
instruction (Civil 2011; Civil et al. 2012). understanding of their students’ communities and
families (e.g., their funds of knowledge), teachers
Immigrant Parents’ Perceptions of can work towards using different forms of doing
Mathematics Education mathematics as resources for learning.

Immigrant parents also concur in naming language The diversity of languages plays a prominent
as the main obstacle towards their children’s learn- role in the mathematics education of immigrant
ing of mathematics. A question to raise is whether students. Barwell’s (2012) four tensions can
a focus on language as the main obstacle to over- serve as a document for discussion with teachers
come may prevent parents from assessing their to see multilingual classrooms as complex and
children’s overall mathematical experience in rich environments for the learning and teaching
school (Civil 2011; Civil and Mene´ndez 2011). of mathematics. This may call for the need for
mathematics teachers to seek the expertise of
The research reviewed in Civil (2012) with language teachers and/or linguists to further
immigrant parents in some European countries understand the strengths of multilingualism in
and in the USA points to some common themes communicating about mathematics.
despite the diversity in countries of origin. Three
related perceptions stand out (see also Abreu Finally, the research reviewed on the mathe-
2008 for some similar themes): (1) a lack of matics education of immigrant students makes
emphasis on the “basics” (e.g., learning of the clear the need for a holistic approach to their
multiplication facts) in the receiving country, education. Such an approach should include mul-
(2) a higher level of mathematics teaching in tiple voices and participants (parents, teachers,
their country of origin, and (3) schools as less school administrators, community representa-
strict in the receiving country (i.e., discipline, tives, and the students themselves).
homework). Underlying these perceptions is the
concept of valorization of knowledge, which Cross-References
affects teachers as well as parents.
▶ Ethnomathematics
These perceptions underscore the need for ▶ Indigenous Students in Mathematics
schools and teachers to establish meaningful
communication with immigrant parents. Parents Education
tend to bring with them different ways to do ▶ Urban Mathematics Education
mathematics that are often not acknowledged by

Immigrant Students in Mathematics Education I281

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Education at school in the home and host cultures, there are
very likely different ways of finding the answers
Wee Tiong Seah to the same questions (e.g., using a computer
Faculty of Education, Monash University, algebra system, or not) and/or different ways of
Frankston, VIC, Australia

The level of cross-border human movements –
temporary or permanent – in the current world
order is unprecedented. The transnational mobil-
ity of teachers – whether these are young and
newly qualified teachers looking for a different
lifestyle for a few years or teachers who migrate
permanently to a new nation – is thus a relatively
new phenomenon which challenges in many


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