Encyclopedia of Mathematics Education

A 28 Algebra Teaching and Learning

to coalesce as a community (Wagner and Kieran algebra by memorizing rules for symbol manip-
1989), research began to center on the ways in ulation, and by practicing equation solving and
which students construct meaning for algebra, on expression simplification, has largely been
the nature of the algebraic concepts and proce- replaced by perspectives that take into account
dures used by students during their initial a multitude of factors and sources by which
attempts at algebra, and on various novel students derive meaning for algebraic objects
approaches for teaching algebra (e.g., Bednarz and processes.
et al. 1996). While the study of students’ learning
of algebra favored a cognitive orientation for Several researchers have studied the specific
some time, sociocultural considerations have question of meaning making in school algebra
added another dimension to the research on (e.g., Kaput 1989; Kirshner 2001). More recently,
school algebra since the end of the 1990s the various ways of thinking about meaning
(Lerman 2000). making in algebra have been expanded
(see, e.g., Kieran 2007) to suggest a triplet of
The years since the late 1980s have also sources: (a) meaning from within mathematics,
witnessed a broadening of the content of school which includes meaning from the algebraic
algebra. While functions had been considered structure itself, involving the letter-symbolic
a separate domain of mathematical study during form, and meaning from other mathematical
the decades prior, the two began to merge at this representations, including multiple representa-
time in school algebra curricula and research. tions; (b) meaning from the problem context;
Functions, with their graphical, tabular, and and (c) meaning derived from that which is
symbolic representations, gradually came to be exterior to the mathematics/problem context
seen as legitimate algebraic objects (Schwartz (e.g., linguistic activity, gestures and body
and Yerushalmy 1992). Concomitant with language, metaphors, lived experience, and
this evolution was the arrival of computing image building). Further theoretical development
technology, which began to be integrated in of this area has been carried out by Radford
varying degrees into the content and emphases (2006) with his conceptualization of a semiotic-
of school algebra. A further change in perspective cultural framework of mathematical learning,
on school algebra was its encompassing in an which has been applied to the learning of algebra.
explicit way what has come to be called algebraic Through words, artifacts, and mathematical
reasoning: that is, a consideration of the signs, which are referred to as semiotic means
thinking processes that precede – and eventually of objectification, the cultural objects of algebra
accompany – activity with algebraic symbols, are made apparent to the student in a process by
such as the expression of general rules with which subjective meanings are refined.
words, actions, and gestures. This widening of
perspective on algebraic activity in schools Characterizing Algebraic Activity
reflected a double concern aimed at engaging Some research studies have used the nature of
primary school students in the early study of algebraic activity as a lens for investigating
algebra and at making algebra more accessible various constituents of students’ learning
to all students. experiences in algebra. Several models have
been proposed for describing algebra and its
A Focus on Algebraic Meaning activities (see, e.g., Bell 1996; Mason et al.
As the vision of school algebra widened consid- 2005; Sfard 2008). For example, a model devel-
erably over the decades – moving from a letter- oped by Kieran (1996) characterizes school
symbolic and symbol-manipulation view to one algebra according to three types of activity:
that included multiple representations, realistic generational, transformational, and global/
problem settings, and the use of technological meta-level.
tools – so too did the vision of how algebra is
learned. The once-held notion that students learn The generational activity of algebra is
typically where a great deal of meaning building

Algebra Teaching and Learning A29

occurs and where situations, properties, patterns, What Does Research Tell Us About the A
and relationships are interpreted and represented Learning of School Algebra?
algebraically. Examples include equations There is a considerable body of research on the
containing an unknown that represent problem learning of algebra that has been accumulating
situations, expressions of generality arising from since the late 1970s. What does this research have
geometric patterns or numerical sequences, to say?
expressions of the rules governing numerical
relationships, as well as representations of Many students beginning the study of algebra
functions by means of graphs, tables, or literal come equipped with an arithmetical frame
symbols. This activity also includes building of mind that predisposes them to think in terms
meaning for notions such as equality, equiva- of calculating an answer when faced with
lence, variable, unknown, and terms such as a mathematical problem. A considerable amount
“equation solution.” of time is required in order to shift their thinking
toward a perspective where relations, ways of
The transformational activity of algebra, representing relations, and operations involving
which involves all of the various types of symbol these representations are the central focus.
manipulation, is considered by some to be Teaching experiments have been designed to
exclusively skill based; however, this interpreta- explore various approaches to developing in
tion would not reflect current thinking in the students an algebraic frame of mind. Approaches
field. In line with a broader view, mathematical that have generally been found to be successful
technique is seen as having both pragmatic include those that (a) emphasize generalizing and
and epistemic value, with its epistemic value expressing generality by using patterns, func-
being most prominent during the period when tions, and variables; (b) focus on thinking about
a technique is being learned (see Artigue 2002). equality in a relational way, starting with number
In other words, the transformational activity sentences with multiple terms on both sides and
of algebra is not just skill-based work; it moving toward more complex examples involv-
includes conceptual/theoretical elements, as for ing the “hiding” of the same number on both sides
example, in coming to see that if the integer by a box and then by a letter, so as to generate
exponent n in xnÀ1 has several divisors, a literal-symbolic equation with an unknown on
then the expression can be factored several each side; and (c) use problem situations that are
ways and thus can be seen structurally in more amenable to more than one equation repre-
than one way. However, for such conceptual sentation and engage pupils in comparing the
aspects to develop, technical learning cannot be two (or three) resulting equation representations
neglected. to determine which one is better in that it is more
generalizable.
Lastly, there are the global/meta-level
activities, for which algebra may be used as Research also tells us that students have
a tool but which are not exclusive to algebra. difficulty with conceptualizing certain aspects of
They encompass more general mathematical school algebra, for example, (a) accepting
processes and activities that relate to the purpose unclosed expressions such as x + 3 or 4x + y as
and context for using algebra and provide valid responses, thinking that they should be
a motivation for engaging in the generational able to do something with them, such as solving
and transformational activities of algebra. They for x; (b) counteracting well-established
include problem solving, modeling, working with natural-language-based habits in representing
generalizable patterns, justifying and proving, certain problem situations such as the
making predictions and conjectures, studying well-known students-and-professors problem;
change in functional situations, and looking for (c) moving from the solving of word problems
relationships or structure – activities that could by a series of undoing operations toward the
indeed be engaged in without using any representing and solving of these problems with
letter-symbolic algebra at all. transformations that are applied to both sides of

A 30 Algebra Teaching and Learning

the equation; and (d) failing to see the power of Learning entry in this volume). Nevertheless, the
algebra as a tool for representing the general student older than 15 years of age has not been
structure of a situation. entirely neglected. Research with this age group
has investigated the learning of more advanced
These findings illustrate only a very small algebraic topics, including the study of structure
portion of the information to be gleaned from and equivalences, conceptual and technical work
the large corpus of research literature that is involving quadratics and higher-degree expres-
available (see also Kieran 1992, 2006, 2007) – sions, and proof and proving of number-theoretic
literature that has shifted in focus over the years, relations. It has been found, for example, that
due to theoretical developments regarding the while many students experience difficulty in
learning of algebra, as well as curricular change “seeing structure,” they have shown improve-
and the growing use of technological tools. While ments over their younger counterparts in
the research of the 1970s and 1980s was oriented representing word problems with equations.
primarily toward issues related to the transition Older students have also been found to prefer to
from arithmetic to algebra, later work illustrates work with literal symbolic representations than
the interest in patterning, students’ generalizing with the graphical. Computer algebra system
and use of multiple representations, as well as the (CAS) technology has figured in several studies
ways in which technology environments (e.g., with the older student. Much of this research,
spreadsheets, graphing calculators, calculator- which has been grounded in the instrumental
based rangers, computer algebra systems, cell approach to tool use (Artigue 2002), has been
phone technology, and specially designed soft- able to provide evidence for the role played by
ware environments) can support algebra learning. the CAS tool in the co-emergence of students’
The studies on the role of technology have technical and conceptual knowledge in algebra.
reported that students receive the most benefit
when the technological tools they use have Research Involving the Teacher of Algebra
a pedagogical role in the classroom and are avail- With the main focus in most of the research in
able not just for drill and practice or for checking school algebra during the 1970s and 1980s on the
work; however, exploiting their potential peda- learner, as well as on teaching approaches aimed
gogical role requires special curricular materials at improving student learning, little was revealed
that are designed for such tool use. Another shift about the teacher of algebra. From the few reports
that has been witnessed concerns what is meant available, one could only discern that, just as with
by algebra problem solving. In the past, this teachers of other mathematical subjects, algebra
phrase tended to refer almost exclusively to teachers viewed themselves primarily as pro-
word problems – an area of algebra learning that viders of mathematical information and tended
continues to challenge many algebra students. to follow the textbook in their teaching. However,
However, a much broader interpretation of the a research interest in the teacher of algebra
phrase exists today that includes many types of and the nature of algebra teaching practice took
nonroutine algebraic tasks, even those in purely shape in the early 1990s and has continued to
symbolic form with no connection at all to this day – research that has begun to deepen our
so-called “real-world” problems. knowledge of this domain. Doerr (2004) has
stated that this research tends to fall into three
With respect to the body of research areas: teachers’ subject matter knowledge and
literature on algebra learning, it is noted that the pedagogical content knowledge, teachers’ con-
12–15-year-old student has received the bulk of ceptualizations of algebra, and teachers learning
the attention of algebra researchers; however, to become teachers of algebra. However,
since the turn of the millennium, there has been according to Doerr, progress in teacher-oriented
a significant interest in the development of alge- research has been hampered by the lack
braic reasoning in younger students of elemen- of development of new methodological and
tary and middle school age (see, e.g., Kaput et al.
2007, as well as the Early Algebra Teaching and

Algebra Teaching and Learning A31

theoretical approaches to effectively investigate by the teacher in order to promote students’ alge- A
the practices of teachers of algebra. braic reasoning – support involving both task
design and whole-class teacher questioning.
Although the research on teachers’ practice However, as has also been noted, research involv-
with respect to promoting algebraic reasoning ing the development of such support in teachers
may still be relatively sparse, it has nevertheless has been hindered, at least up to the early 2000s,
been able to point to two aspects in particular by a lack of appropriate methodological and the-
as being critical to student learning. These are oretical tools. While some advances have clearly
the roles played by task design and teacher been made in this area, including teachers’ shar-
questioning in encouraging algebraic thinking. ing of their effective approaches with other
Many of the recent studies on innovative teachers, further work is needed. A general
approaches to teachers’ professional develop- framework for thinking about models for devel-
ment in the area of school algebra have focused oping teaching practice that can support students’
directly on the issue of tasks and their design. algebraic reasoning, and which offers a
These studies have been able to show that the perspective for moving forward in this area, has
content-related design and careful sequencing of been described in a recent article on connecting
tasks have a clear impact upon the ways in which research with practice (Kieran et al. in press). In
students come to conceptualize a variety of alge- that article, the researchers attempt to close the
braic ideas and operations, including the follow- distinctive gap between research and practice that
ing: relational views of equality; meaningful exists in much of the mathematics education
interpretations of algebraic symbols; awareness research literature by viewing teachers as
of the theoretical-technical interface in algebraic key stakeholders in research – stakeholders who
work; perception of form, structure, and general- coproduce professional and scientific knowledge –
ity; and the pedagogically effective use of instead of as “recipients of research” and some-
technological tools in algebraic activity. times even as “means” to generate or disseminate
knowledge. Their elaboration of the notion of the
The role of teacher questioning in developing teacher as key stakeholder with examples drawn
students’ algebraic reasoning has been found to from five international projects, all of them
be no less important than that of good task design. involving teachers researching their own or
For example, data drawn from the eighth-grade their colleagues’ practice, offers several viable
TIMSS Video Study (Stigler et al. 1999) illustrate models and a useful lens for considering how
the ways in which teachers’ well-conceived teachers and researchers might collaborate in
questions during whole-class discussions can further developing the teaching and learning of
encourage students to make explicit their school algebra.
problem-solving approaches and to generalize
them into literal-symbolic form. However, this Cross-References
research also shows that despite the use of tasks
designed to help students engage in classroom ▶ Early Algebra Teaching and Learning
discussions that focus on making conjectures ▶ Functions Learning and Teaching
and reasoning mathematically, simply using ▶ Information and Communication Technology
such tasks will not spontaneously promote the
desired discussions. Skillful teacher guidance is (ICT) Affordances in Mathematics Education
needed in order to help students engage in the ▶ Mathematical Representations
algebraic reasoning that is intended by the tasks. ▶ Teaching Practices in Digital Environments
▶ Technology and Curricula in Mathematics
For Further Research
The following closing remarks return to a central Education
issue regarding the practice of teaching algebra. ▶ Theories of Learning Mathematics
The research that was synthesized just above
emphasized the need for a certain kind of support

A 32 Algorithmics

References Lerman S (2000) The social turn in mathematics education
research. In: Boaler J (ed) Multiple perspectives on
Artigue M (2002) Learning mathematics in a CAS mathematics teaching and learning. Ablex, Westport,
environment: the genesis of a reflection about pp 19–44
instrumentation and the dialectics between technical
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7:245–274 Developing thinking in algebra. Sage, London

Bednarz N, Kieran C, Lee L (eds) (1996) Approaches to Radford L (2006) The anthropology of meaning. Educ
algebra: perspectives for research and teaching. Stud Math 61:39–65
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Bell A (1996) Problem-solving approaches to algebra: two function in and with algebra. In: Dubinsky E,
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DC, pp 261–289
Doerr HM (2004) Teachers’ knowledge and the teaching
of algebra. In: Stacey K, Chick H, Kendal M (eds) Sfard A (2008) Thinking as communicating. Cambridge
The future of the teaching and learning of algebra: University Press, New York
The 12th ICMI Study. Kluwer, Dordrecht,
pp 267–290 Stacey K, Chick H, Kendal M (eds) (2004) The future of
the teaching and learning of algebra: the 12th ICMI
Freudenthal H (1977) What is algebra and what has it Study. Kluwer, Dordrecht
been in history? Arch Hist Exact Sci 16(3):
189–200 Stigler JW, Gonzales PA, Kawanaka T, Knoll S,
Serrano A (1999) The TIMSS videotape classroom
Kaput JJ (1989) Linking representations in the symbol study: methods and findings from an exploratory
systems of algebra. In: Wagner S, Kieran C (eds) research project on eighth-grade mathematics
Research issues in the learning and teaching of algebra, instruction in Germany, Japan, and the United States,
vol 4, Research agenda for mathematics education. NCES publication no 1999074. U.S. Government
National Council of Teachers of Mathematics, Reston, Printing Office, Washington, DC
pp 167–194
Wagner S, Kieran C (eds) (1989) Research issues in
Kaput JJ, Carraher DW, Blanton ML (eds) (2007) Algebra the learning and teaching of algebra, vol 4,
in the early grades. Routledge, New York Research agenda for mathematics education.
National Council of Teachers of Mathematics,
Kieran C (1992) The learning and teaching of school Reston
algebra. In: Grouws DA (ed) Handbook of research
on mathematics teaching and learning. Macmillan, Algorithmics
New York, pp 390–419
Jean-Baptiste Lagrange
Kieran C (1996) The changing face of school algebra. Laboratoire de Didactique Andre´ Revuz,
In: Alsina C, Alvarez J, Hodgson B, Laborde C, University Paris-Diderot, Paris, France
Pe´rez A (eds) Eighth international congress on
mathematical education: selected lectures. S.A.E.M. Keywords
Thales, Seville, pp 271–290
Algorithmics; Algorithms; Programming;
Kieran C (2006) Research on the learning and teaching of Computer sciences; Programming language
algebra. In: Gutie´rrez A, Boero P (eds) Handbook of
research on the psychology of mathematics education. Definition
Sense, Rotterdam, pp 11–50
“Algorithmics” can be defined as the design and
Kieran C (2007) Learning and teaching algebra at the analysis of algorithms (Knuth 2000). As
middle school through college levels: building mean- a mathematical domain, algorithmics is not prin-
ing for symbols and their manipulation. In: Lester FK cipally concerned by human execution of
Jr (ed) Second handbook of research on mathematics
teaching and learning. Information Age, Greenwich,
pp 707–762

Kieran C, Krainer K, Shaughnessy JM (in press) Linking
research to practice: teachers as key stakeholders in
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Perspectives on school algebra. Kluwer, Dordrecht,
pp 83–98

Algorithmics A33

algorithms, for instance, for arithmetic computa- functions, etc.) are chosen in order to support A
tion (see 2010/index/chapterdbid/313187 for this approach. Students’ access to a formal
a discussion), but rather by a reflection on how algorithmic language is generally not an issue
algorithms are built and how they perform. because the tasks proposed for students gener-
Algorithms exist and have been studied since ally imply short programs with a simple
the beginning of mathematics. However, the structure.
emergence of algorithmics as a mathematical
domain is contemporary to digital computers, In a few countries and regions, curricula
the work on computability by Church (1936), for algorithmics have been implemented and,
Turing (1937), and other mathematicians being in parallel, research studies have been
often considered as seminal. Computer science, conducted. For instance, at the end of the year
also emerging at the same time, is concerned with 1980, a curriculum has been written and tested
methods and techniques for machine implemen- for 7th- and 8th-grade students in a region of
tation, whereas algorithmics focuses on the Germany (Cohors-Fresenborg 1993). Concepts
properties of algorithms. of algorithmics were taught by making students
solve calculation problems using a concrete
Typical questions addressed by algorithmics “register machine.”
are the effectiveness of an algorithm (whether or
not it returns the expected result after a finite These research studies are few and do not
number of steps), the efficiency (or complexity) really tackle questions at the core of algorithmics
of an algorithm (an order of the number of steps like effectiveness and complexity, reflecting the
for a given set of data), and the equivalence of fact that at school levels investigated by research
algorithms (e.g., iterative and recursive equiva- studies, students’ consideration of algorithmics is
lent forms). Djiskra (1976, p. 7) notes that “as still limited by the difficult access to a symbolic
long as an algorithm is only given informally, it is language.
not a proper object for a formal treatment” and
therefore that “some suitable formal notation” is Students’ Understanding of Algorithmic
needed “to study algorithms as mathematical Structures and Languages
objects.” This formal notation for algorithms or In France, programming algorithms has been
“language” is a vehicle for abstraction rather than proposed as a task for secondary students in
for execution on a computer. various curricula. Because the time devoted for
these tasks was short, students’ understanding of
Algorithms in Mathematics Education algorithmic structures and languages appeared
Research to be the real challenge, algorithmics in the
Research in mathematics education and com- sense of Knuth (2010) being inaccessible to
puters most often concentrates on the use of tech- beginners without this prerequisite. Didactic
nological environments as pedagogical aids. research studies were developed focusing on
Authors like Papert and Harel (1991), Dubinsky this understanding.
(1999), or Wilensky and Resnick (1999) pro-
posed computer programming as an important Samurcay (1985) was interested by 10th-grade
field of activity to approach mathematical notions students’ cognitive problems relatively to variables
and understanding. This strand of research does in iteration. The method was to ask students to
not consider the design and analysis of algorithms complete iterative programs in which instructions
as a goal in itself. The hypothesis is that building were missing. Missing instructions were of three
algorithms operating on mathematical objects types: the initialization of the iterative variable, an
and implementing these in a dedicated program- assignment of the iterative variable in the loop
ming language (LOGO or ISTL) is able to body, and the condition for exiting the loop.
promote a “constructive” approach to scientific Important misunderstandings of the semantics of
concepts. The language’s features (recursivity, variables were identified. For instance, regarding
the initialization, some students think that the
initial value has necessarily to be entered by

A 34 Algorithmics

a reading instruction; others systematically initial- in relationship with analogous obstacles in
ize variables to zero. They are clearly influenced accessing the algebraic symbolism at middle
both by preconception of how a computer works school level. Programming simple algorithms
and by previous examples of algorithms that did involving these nonnumerical objects seemed
not challenge these preconceptions. The author promising for overcoming such obstacles.
concludes that more research studies are essential
in order to understand how students conceptualize Nguyen (2005) questioned the introduction of
the notions associated to iteration and to design elements of algorithmics and programming in the
adequate didactical situations. secondary mathematical teaching, showing that
on one hand, there is a fundamental solidarity
Samurcay, Rouchier (1990) studied students’ between mathematics and computer science
understanding of recursive procedures distinguishing based on the history and the current practice of
between two aspects: self-reference (relational these two disciplines and that on the other hand,
aspect) and nesting (procedural aspect). They the ecology of algorithmics and programming in
designed teaching sessions with the aim to help secondary teaching is not obvious. Focusing on
pupils to construct a relational model of recur- the teaching/learning of loop and of computer
sion, challenging students’ already existing pro- variable notions in France and in Vietnam, he
cedural model. After sessions of introducing the proposed an experimental teaching unit in order
students to the LOGO graphic language without that 10th-grade students learn the iterative struc-
recursion, they designed ten lessons: first intro- ture. He chose to make students build suitable
ducing the students to graphic recursive proce- representations of this structure by solving
dures, making them distinguish between initial, tasks of tabulating values of polynomial using
central, and final recursion and then helping them a dedicated calculator, emulated on the computer,
to generalize recursive structures by transferring and based on the model of calculator existing in
recursive procedure to numerical objects for tasks the secondary teaching of the two countries with
of generating sequences. Observing students, the additional capacity to record the history of the
they conclude that introducing recursion is keys pressed.
a nonobvious “detour” from already existing
procedural model of iteration and a promising The experimental teaching was designed as
field for research. a genesis of the machine of Von Neumann: the
students had to conceive new capabilities for the
Lagrange (1995) considered the way 10th- and calculator especially erasable memories and
11th-grade students understand representations controlled repetition in order to perform iterative
of basic objects (strings, Booleans) in calculations and programming through the
a programming language. Analyzing students’ writing of the successive messages (programs)
errors in tasks involving simple algorithmic to machines endowed with different characteris-
treatments on these objects, he found that tics. This allowed for the emergence of the
misunderstandings result from assimilation to notion of iterative variables and treatments. In
“ordinary” objects and treatments. For instance, the framework of the Theory of Didactical
when programming the extraction of a substring Situations, a milieu and a fundamental situation
inside a string, students often forgot to assign the are then offered for the construction of the
result to a variable; the reason is that they iterative structure.
were not conscious of the functional nature of
the substring instruction, being influenced by Algorithmics and Programming
the “ordinary” action oriented language. Another Competencies
example is that students generally did not In parallel to mathematics education research,
consider the assignment to a Boolean value, not studies have been carried out in the field of
understanding that in an algorithmic language, psychology of programming. Most studies in
“conditions” are computable entities. Similar the field address professional programming
difficulties found in this study were analyzed and discuss opportunities and constraints of

Algorithmics A35

programming languages and design strategies for consistent with Djiskra’s (ibid.) epistemological A
experts (e.g., see Petre and Blackwell 1997). view that a suitable formal notation is needed
Some studies focused on programming problem to study algorithms as mathematical objects.
solving by beginners with tasks very close to It is also a stimulating challenge that the
students’ activity in early algorithmics courses. abovementioned research studies just started
For instance, Rogalski and Samurc¸ay (1990) to take up.
focused on the acquisition of programming
knowledge “as testified by students’ ability to Cross-References
solve programming problem”, that is to say,
to pass from “real” world objects and situations to ▶ Algorithms
an effective program implementation. Rogalski
and Samurc¸ay (1990) insist on “the variety of References
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A 36 Algorithms

Samurcay R (1985) Signification et fonctionnement du An example of a simple well-known algorithm
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world. J Sci Educ Technol 8(1):3–19 have been compared, consider the count of the
number of changes. If the count is zero then
Algorithms the sequence is sorted into order. If the count is
greater than zero repeat from step 1.
Mike O. J. Thomas We note that two algorithms to accomplish the
Department of Mathematics, The University of same task may vary or be entirely different. For
Auckland, Auckland, New Zealand example, there are a number of different algo-
rithms for sorting numbers into order, often much
Keywords more efficiently than the bubble sort, such as the
quicksort algorithm.
Algorithm; Computing; History of mathematics; Another common example referred to as the
Instrumental understanding; Representation Euclidean algorithm for finding the greatest com-
mon divisor (gcd) of two integers n and m may be
Definition stated as:
1. If n ¼ m then output n as the gcd (n, m) and end.
The word algorithm probably comes from 2. If n > m the initialize a ¼ n and b ¼ m.
a transliterated version of the name al-Khwarizmi Otherwise, initialize a ¼ m and b ¼ n.
(c. 825 CE), the Arabic mathematician who 3. Apply the division theorem to a and b by
described how to solve equations in his publica- finding integers q and r such that a ¼ q.b + r,
tion al-jabr w’al-muqabala. An algorithm com- where 0 r b.
prises a step-by-step set of instructions in logical 4. If r ¼ 0 then output b as the gcd (n, m) and stop.
order that enable a specific task to be accom- Otherwise set a ¼ b and b ¼ r. Go to step 3.
plished. Due to its nature it can be programmed Based on Khoussainov and Khoussainova
into a computer, although some problems may (2012), p. 29.
not be computable or solvable by an algorithm. In this case we illustrate how algorithms to
In his famous paper, Turing (1936) showed, accomplish the same task may be equivalent but
among other things, that Hilbert’s Entscheidung- presented differently, and thus not necessarily
sproblem can have no solution. He did this by appear to be the same. Consider, for example,
proving “that there can be no general process for a second version of the Euclidean algorithm
determining whether a given formula U of the (based on the version found at http://www.math.
functional calculus K is provable, i.e., that there rutgers.edu/$greenfie/gs2004/euclid.html):
can be no machine which, supplied with any one 1. If m < n, exchange m and n.
U of these formulae, will eventually say whether 2. Divide m by n and get the remainder, r.
U is provable” (1936, p. 259). If r ¼ 0, report n as the gcd.

Algorithms A37

3. Replace m by n and replace n by r. Return to definition; and that a factorization of the form
ax2 + bx + c ¼ a(x À p)(x À q)[¼0] is possible.
the previous step.
One drawback of the step-by-step nature of an
What do we notice about these two versions? algorithm is that it leaves no room for deviation A
from the method. Hence, it cannot encourage or
While they are the same algorithm, that is, they promote the versatile thinking (Thomas 2008;
Graham et al. 2009) that is needed in order to
accomplish the same task in the same way, the understand some mathematical constructs and
hence to solve certain mathematical problems.
first one appears more complex. This is because it For example, it may be both useful and enlight-
ening to switch representations or registers to
uses function notation (gcd (n, m)); it is not self- comprehend an idea better (Duval 2006) or
to view a written symbolism (described as
contained, but refers to a previous result (the a procept) as either a process or an object (Gray
and Tall 1994) depending on the context. One
division theorem); and it introduces more vari- example of this is appreciating the relationship
between the roots of the quadratic equation above
ables (an extra a, b) than the second. These dif- and the graph of the function. Another is the
calculation of integrals through the use of limits
ferences may be the result of attempts to be of Riemann sums. Algorithms can be constructed
for processes that allow students to find
rigorous or to make the algorithm more amenable a Riemann sum or its limit, but there is evidence
that far fewer students understand the nature of
to computerization. the limit object itself (Tall 1992; Williams 1991).

It is perfectly possible to be able to carry out Cross-References

an algorithm, such as the quicksort or Euclidean ▶ Algorithmics
▶ Mathematical Approaches
algorithms above, without understanding how it
References
works. In this case an individual would demon-
Duval R (2006) A cognitive analysis of problems of
strate what Skemp (1976) called instrumental comprehension in a learning of mathematics. Educ
Stud Math 61:103–131
understanding, whereas knowing the reasons
Godfrey D, Thomas MOJ (2008) Student perspectives on
why it works would constitute relational under- equation: the transition from school to university.
Math Educ Res J 20(2):71–92
standing. It would also be a mistake to think that
Graham AT, Pfannkuch M, Thomas MOJ (2009) Versatile
mathematics may be reduced to a series of algo- thinking and the learning of statistical concepts. ZDM
Int J Math Educ 45(2):681–695
rithms. The idea of an algorithm is closely related
Gray EM, Tall DO (1994) Duality, ambiguity and flexi-
to what, in mathematics education terms, are bility: a proceptual view of simple arithmetic. J Res
Math Educ 26(2):115–141
often called procedures, since these may be
Hiebert J, Lefevre P (1986) Conceptual and procedural
accomplished using algorithms. They contrast knowledge in mathematics: an introductory analysis.
In: Hiebert J (ed) Conceptual and procedural knowl-
with other crucial elements of mathematics, edge: the case of mathematics. Erlbaum, Hillsdale,
pp 1–27
such as objects, constructs, or concepts. While

both procedures and concepts are important in

learning mathematics (Hiebert and Lefevre

1986), teaching algorithms is often easier than

addressing concepts and so this approach may

prevail in school (and sometimes university)

teaching. For example, the formula for solving

a quadratic eqpuaffitffiffiiffioffiffiffinffiffiffiffiax2 + bx + c ¼ 0 with real
b2 À4ac
roots x ¼ ÀbÆ 2a leads to an algorithm for

solving these equations. However, it may be

the case that students who can successfully find
the roots of a quadratic equation ax2 + bx + c ¼ 0

only have instrumental understanding and do not

understand well why the formula works, what an

equation is (Godfrey and Thomas 2008), or even

what a solution of an equation is. They may not

appreciate, for example, that the formula arises
from completing the square on ax2 + bx + c ¼ 0;
that if p and q are real roots of ax2 + bx + c ¼ 0,
then ap2 + bp + c ¼ 0 and aq2 + bp + c ¼ 0 by

A 38 Anthropological Approaches in Mathematics Education, French Perspectives

Khoussainov B, Khoussainova N (2012) Lectures on providing a unitary theory of didactic phenomena
discrete mathematics for computer science. World as defined in what follows. As for JATD, it has
Scientific, Singapore emerged from the theory of didactic situations
(Brousseau 1997) and the anthropological theory
Skemp RR (1976) Relational understanding and instru- of the didactic by focusing on the very nature
mental understanding. Math Teach 77:20–26 of the communicational epistemic process
within didactic transactions. ATD and JATD
Tall DO (1992) The transition to advanced mathematical share a common conception of knowledge as
thinking; functions, limits, infinity, and proof. In: a practice and a discourse on practice together –
Grouws DA (ed) Handbook of research on mathemat- i.e., as a praxeology – along with a pragmatist
ics teaching and learning. Macmillan, New York, epistemology which gives a prominent place to
pp 495–511 praxis. Their well-thought-out anthropological
stance leads the researcher to study didactic
Thomas MOJ (2008) Developing versatility in mathemat- facts wherever they are located in social
ical thinking. Mediterr J Res Math Educ 7(2):67–87 practices. Although these theorizations are by
necessity expounded tersely, we hope their
Turing AM (1936) On computable numbers, with an forthright presentation will allow the reader to
application to the Entscheidungsproblem. Proc catch the gist of them.
London Math Soc (2) 42:230–265
The Anthropological Theory of the Didactic
Williams SR (1991) Models of limit held by college The (seemingly) heavy theoretical load of the
calculus students. J Res Math Educ 22:237–251 presentation that follows should not be
misinterpreted. On the one hand, every and all
Anthropological Approaches in notions delineated hereinafter do refer to con-
Mathematics Education, French crete didactic practice (from which they gradu-
Perspectives ally emerged) and have led, in our view, to some
major scientific breakthroughs (Bosch et al.
Yves Chevallard1 and Ge´rard Sensevy2 2011; Bronner et al. 2010; Chevallard 1990,
1Apprentissage Didactique, Evaluation, 2006, 2007, to appear; Chevallard and Ladage
Formation, UMR ADEF – Unite´ Mixte de 2008; Ruiz-Higueras et al. 2007). On the other
Recherche, Marseile, France hand, we strongly believe that, according to
2Institute of Teacher Education, University a well-known remark by Lewin (1952), p. 169,
of Western Brittany, Brittany, France “there is nothing more practical than a good
theory,” which is exactly what we aim at
Keywords providing the interested reader with (Chevallard
1980, 1991, 1992).
Anthropology; Didactics; Joint action;
Mathematics; Praxeology; School epistemology

Characteristics

This entry encompasses two interrelated though Didactic Systems
distinct approaches to mathematics education: Didactics can be defined as the (historically
the anthropological theory of the didactic (ATD incipient) science of knowledge diffusion and
for short) and the joint action theory in didactics acquisition in society. The founding problem of
(JATD). Historically, the germs of ATD are to be didactics has long been reduced to two charac-
found in the theory of didactic transposition ters: some object of knowledge O and some
(Chevallard 1991), whose scope was at first lim- human subject x supposed to “study” O. This
ited to the genesis and the ensuing peculiarities of problem lays in what ATD calls the “relation of
the (mathematical) “contents” studied at school; x to O,” written in symbol R(x, O). If x knows
from this perspective, ATD should be regarded as nothing about O, her relation to O is void:
the result of a definite effort to go further by R(x, O) ¼ Æ. How can this relation change,

Anthropological Approaches in Mathematics Education, French Perspectives A39

grow, and possibly achieve proficiency, so that and outside classrooms, in the family, on the A
one can say that x “knows” O? Such is the key telephone yesterday, on the Internet today, etc.
question in classical didactics. But the traditional
two-character play to which it applies has long The Didactic
since been challenged by the theory of didactic In ATD, the adjective “didactic” applies to any
transposition, that forerunner of ATD which action induced by the intention to help
questions the nature of object O, its genesis. and someone study something. In any didactic system
alterations during the process that leads to the S(x, y, O), x and y act to help x study O. It is
face-to-face encounter of x and O. The meeting customary to say that x performs didactic moves
between x and O takes place in some institution – (or didactic gestures) with respect to the didactic
another keyword of ATD – which imposes upon stake O, to help herself study O, and that y
O a number of conditions that reshape O and “fix” achieves didactic moves or gestures with respect
the conditions under which x will study O. ATD to x and O with the same intention. Most didactic
is critical of the common view of the two- moves occurring in a system S(x, y, O) involve
character didactic scene. Its main tenet holds both x and y, who work together to produce some
that in order to “explain” x, O, and R(x, O), one determined didactic effect: didactic tasks are
has to take into account a greater number of generally cooperative tasks, jointly performed
conditions. First of all, the “binomial” arrange- by x and y (or by X and y, etc.). Considered within
ment made up of x and O is generally part of the larger frame of society, the fuzzy set of
a “trinomial” layout, including a third character, didactic moves, which ATD calls the didactic
y, to constitute a didactic system S(x, y, O), (as one can speak of the religious, the economic,
where y is a person supposed to help x to study etc.), is thus everywhere around us, and it is the
O. When y is missing, the system reduces to an specific object of study of the science we
autodidactic system, S(x, O), the basic “binomial call didactics.
arrangement” we started with. What is the use of
such a formal description? Before answering this Praxeological Analysis
question, let’s generalize a bit our symbolic By definition, any didactic study refers to some
gobbledygook: instead of a person x, let’s con- stake O and some category of “students” x. Both
sider a group of persons X; instead of y, let’s O and x impose conditions on the moves that can
introduce a team of y, Y, so that a didactic system appropriately be performed by x and y. Tradition-
is now denoted by S(X, Y, O). Didactic systems ally, O pertains to some discipline such as math-
S(x, y, O) are particular cases of this general ematics, physics, or geography. All these entities
form. When no y helps X, we’ll denote the are human made: they are, up to a point, artifacts,
corresponding system by S(X, Æ, O) or simply i.e., “works of art,” this expression being
S(X, O). Any class of students X studying some understood here in its most primitive meaning.
object O under the supervision of some “official” We shall say for short that O is “a work.”
teacher y can be written as S(X, y, O). Two Analyzing any work O amounts to making clear
students x1 and x2 working together on their its structure, its functioning, and its distinct uses.
homework O are part of the didactic system It has become common practice in ATD to
S({x1, x2}, Æ, O) or S({x1, x2}, O). When describe the set of conditions borne by the
a student x receives help from her mother y, disciplinary field to which O belongs – as partak-
they form together a system S(x, y, O). The ing of a four-scale hierarchy of disciplinary sub-
symbolic notation used thus allows us to “see” fields. Firstly, O is situated within some domain
not only the didactic systems insistently shown of the discipline – say, algebra – which in turn is
to us – within classrooms, basically – but also dissociated into a number of sectors, each of
the more or less informal, but no less which is made up of themes (or topics) that finally
crucial, didactic systems that may appear separate into subjects. Any work O that can be
almost everywhere in society: at school in offered for study in a system S(X, y, O) may fall

A 40 Anthropological Approaches in Mathematics Education, French Perspectives

under any of these disciplinary levels: O may be d’eˆtre or ultimate purpose of these praxeologies.
“algebra” (domain) or “equations” (sector) or Analyzing what x and y can do in this respect –
“quadratic equations” (theme) or “incomplete their possible moves – is tantamount to producing
quadratic equations with no constant term” a didactic analysis, i.e., an analysis of the didac-
(subject). This four-level structure is one part of tic situations that the system S(x, y, O) goes
the story. For ATD purports that the ultimate through. Any didactic analysis implies some
building block of all works is a four-component degree of praxeological analysis of O (e.g., even
structure called a praxeology. The first compo- if O is praxeologically far from complete: the
nent of a praxeology is a type of tasks T (e.g., “to raisons d’eˆtre of O are almost always lacking in
solve a quadratic equation”). The second compo- today’s mathematics education). Provided this is
nent is a technique t(tau), i.e., a way of done appropriately, ATD offers a model to guide
performing the tasks of type T (or at least some didactic analysis. In the case of a praxeology
of them). The third component is a technology O ¼ (T, t, q, Q), to which much can be reduced,
q(theta), i.e., a way of explaining and justifying there comes a moment when, in some didactic
or even of “designing” the aforesaid technique t. situation, x meets the type of tasks T for the first
Last but not least (although often ignored), there time – also, there will come a moment when x
is a fourth component, the theory Q(“big theta”), tries to design and then master a technique
which should explain, justify, or generate what- t relating to T; this model of didactic moments is
ever part of technology q may sound unobvious essentially dictated by the aforementioned prax-
or missing. It is a crucial precept of classical eological model. To this model, ATD adds
didactics – one that severs didactics from the another decisive multilevel structure without
old “pedagogy” – that the didactic generated in which this theory would not fully deserve to be
any system S(x, y, O) depends essentially on the called “anthropological”: the scale of levels of
conditions ingrained in the stake O, acting in this didactic codetermination that can be sketched as
respect as a quasi-autonomous system. ATD follows: humankind, civilizations, societies,
posits that O is a combination of a number of schools, pedagogies, disciplines . . . O. The
praxeologies (T, t, q, Q) that usually share parts conditions to be taken into account in any didac-
of their theory Q and of their technology q. tic analysis should not be limited to conditions
Of course, O can be also a “mere” detail of carried by the stake O (or by the “discipline” to
a praxeology, e.g., some instrument used to carry which O “belongs”). Contrary to other
out a given technique t. More often than not, approaches, which regard higher-level conditions
praxeologies are identified by some emblematic as “neutralized variables,” ATD allows for
“detail”: studying “Pythagoras’ theorem,” for conditions originating at the levels of pedagogy,
instance, usually does not boil down to learning school, society, civilization, and even human-
a bare statement (“In any right-angled triangle, kind, in so far as they determine (think, e.g., of
the area. . .”) but amounts to studying at least class and gender) the didactic opportunities open
a whole praxeology whose technological compo- to x and y. At the same time, ATD classically holds
nent q crucially features Pythagoras’ theorem. that the manipulation of conditions of higher levels
(starting from the pedagogic level) is of little avail
Didactic Analysis if we ignore the conditions properly pertaining
The study of work O consists in providing some to O. This anthropological turn results in a deep
praxeological analysis of O. To do so, x change in didactics’ theory and practice.
(helped by y) engages in hard didactic work to
analyze the praxeological structure of O as well Toward an Anthropology of Didactic Inquiry
as the raisons d’eˆtre of O, i.e., the role O plays in In ATD, a school is any institutional arrangement
the functioning of the praxeologies of which it is devoted to study, i.e., in which it is legitimate to
an ingredient and, by the same token, the raisons study some works O – a family is thus usually

Anthropological Approaches in Mathematics Education, French Perspectives A41

a school for some of its members. The contract For example, if a parent holds her hands out to A
linking a society and a school can be of one of two a young child, who is learning to walk, as an
kinds. The traditional contract decides in advance incentive to make her walk towards these hands,
which works O will be studied, a work O being in while the young child tries to take some steps to
this case some praxeological entity supposed reach these hands, this is an epistemic joint act.
to allow one to answer questions of a given One cannot understand each behavior (parent/
type – e.g., “What are the roots of this quadratic teacher or child/student) without taking into
equation?” Such works are classified beforehand account the joint process and the knowledge
as belonging to mathematics, physics, biology, (walking) that gives its form to the enacted ges-
music, etc. But a different kind of contract can be tures. In this perspective, in the JATD, knowledge
considered, more akin to the mores of scientific is always seen as a power of acting, in a specific
research, in which the stake O is no longer a tool situation, within a given institution. When a person
for answering questions but is itself a question Q, knows something, she becomes able to do
which is a (human made) work as well. In this case, something that she was previously unable to do.
the study of Q ceases to be a priori governed by
a discipline determined in advance: these primary The Didactic Game
questions Q are not clearly introduced as questions We aim to describe the didactic interactions
of mathematics, of biology, etc. It is the role of x to between the teacher and the students as a game of
find out the secondary questions Q’, the tertiary a particular kind, a didactic game (Sensevy 2011a).
questions Q”, etc., and the other works O that will What are the prominent features of this game?
prove useful to answer Q. The study of Q, i.e., the
inquiry into question Q led by x (with some help It involves two players, X and Y.
from y), is now “co-disciplinary” in that it Y wins if and only if X wins, but Y cannot give
generally requires combined contributions from the winning strategy to X directly.
several, known as well as unknown, disciplinary Y is the teacher (the teaching pole). X is the
fields. As of today, ATD is increasingly concerned student (the studying pole). Under this descrip-
with the analysis of conditions of every level that tion, the didactic game is a collaborative game,
may hinder or facilitate the advent of such an a joint game, within a joint action. To identify the
anthropological pedagogy of inquiry. very nature of the didactic game, we have to
consider it as a conditional game, in which
The Joint Action Theory in Didactics the teacher’s success is conditioned by the stu-
One cannot understand the didactic system S(X, dent’s success. This structure logically entails
Y, O) without taking into account the relation- a fundamental characteristic of the didactic
ships between the three subsystems (teacher, game. In order to win the game, the teacher can-
student, the piece of knowledge at stake) as not act directly. For example, in general, she
a whole. With this respect, the JATD (Sensevy cannot ask a question to the student and immedi-
2012) puts the emphasis on the “actional turn” ately answer this question. She needs a certain
in didactics. Emerging from a comparative kind of “autonomy” from the student. In order to
approach in didactics (Ligozat and Schubauer- win, Y (the teaching pole) has to lead X (the
Leoni 2009), the JATD institutes a specific unit studying pole) to a certain point, a particular
of analysis that we call an epistemic joint act. The “state of knowledge” which allows the student
linguistic criterion of the description of such an to play the “right moves” in the game, which can
act is that it is impossible to describe it without ensure the teacher that the student has built the
describing at the same time the teacher’s action, right knowledge. At the core of this process,
the student’s action, and the way the knowledge there is a fundamental condition: in order to be
at stake shapes these actions. This assertion sure that X has really won, Y must remain tacit
is a very general and anthropological one. on the main knowledge at stake. She has to be
reticent. On her side, the student must act

A 42 Anthropological Approaches in Mathematics Education, French Perspectives

proprio motu; the teacher’s help must not allow Sensevy et al. 2012b). Documenting this joint
the student to produce a “good” behavior without action needs a specific methodological instrumen-
calling on the adequate knowledge. This proprio tation process (Sensevy and Forest 2012;
motu clause is necessarily related to the reticence Tiberghien and Sensevy 2012).
of the teacher (Sensevy 2011b). Indeed the
proprio motu clause and the teacher’s reticence Epistemic Games
compose the general pattern of didactic transac- In a nutshell, the notion of learning game is a way
tions and give them their strongly asymmetrical of modelizing what the teacher and the student
nature. jointly do in order for the student to learn
something. The notion of epistemic game is
Learning Games a way of modelizing this something, i.e., what
We call learning game the didactic game we has to be learned.
modelize by using the concepts of didactic
contract and didactic milieu (Sensevy et al. Speaking of epistemic game rather than of
2005). Consider this example: at primary school, “knowledge” or “subject content” is a way of
students have to reproduce a puzzle by enlarging actualizing the JATD’s actional turn. An episte-
it, in such a way that a segment which measures mic game is a modelization of what we can call
4 cm on the model will measure 7 cm on the a knowledge practice (the practice of
reproduction. The pieces of this puzzle constitute a mathematician, a fiction writer, an historian,
the milieu that the students face for this “enlarge- etc.). We argue that these practices have to be
ment problem.” The didactic contract (Brousseau carefully scrutinized in a comprehensive way that
1997; Sensevy 2012) refers to the strategic system may express their fundamental principles, rules,
the student uses in order to work out the problem at and strategies. For example, if one intends to
stake. This strategic system has been shaped some extent to have students as mathematicians,
mainly in the previous joint didactic action. In one has to modelize this practice (that of the
our example, it is mainly an “additive” contract, mathematician) so that the teachers may monitor
in that students try to add 3 to every dimension of students’ activity in a relevant way by relying on
the puzzle. The milieu (Brousseau 1997; Sensevy this model. In this respect, an epistemic game is
2012a) refers to the set of symbolic forms that the a model (Sensevy et al. 2008), which attempts to
didactic experience transforms in an epistemic grasp the fundamental dynamic structure of
system. In our example, the fact that the puzzle a knowledge practice and which can help
pieces are not compatible has first to be an incen- the designers of a curriculum in the didactic
tive to refute the additive strategy. Modelizing the transposition process.
teaching process by using the concept of learning
game enables the researcher to identify the Cooperative Engineering
teacher’s game on the student’s game. When In order to contribute to the elaboration of new
teaching a piece of knowledge, the teacher may forms of schooling, the JATD aims at theorizing
rely on the contract properties (by having the stu- a specific process of design-based research,
dents recognize the previous taught knowledge called cooperative engineering (Sensevy 2012)
necessary to deal with the problem at stake) or in which teachers and researchers jointly act to
on the milieu structure (by orienting the students build teaching-learning sequences grounded on
so that they experience some epistemic features of learning games nurtured by specific epistemic
this milieu, in our example, the fact that the puzzle games. This process rests on the dilution of
pieces do not fit together). The JATD considers dualisms between theory and practice, ends and
such a joint work as a didactic equilibration means. In this way, teachers and researchers may
process, relying on the research of equilibrium, temporally occupy the same position, that of
in the teacher’s discursive work, between didactician engineer, by sharing the same educa-
expression and reticence (Mercier et al. 2000; tional ends and by working out together the means
which will allow to reach these ends and to

Anthropological Approaches in Mathematics Education, French Perspectives A43

reconceptualize them. In this respect, the JATD Chevallard Y (to appear) Teaching mathematics in
endeavors to overcome the classic distinction
between applied and fundamental research by pro- tomorrow’s society: a case for an oncoming
posing concrete curriculum designs.
counterparadigm. Regular lecture at ICME-12
Concluding Remarks
Beyond different results and uses, ATD and (Seoul, 8–15 July 2012). http://www.icme12.org/ A
JATD suggest a new school epistemology and
urge a thorough reconstruction of the form of upload/submission/1985_F.pdf
schooling, more open to the basics of cooperative
studying and learning that they jointly advocate. Chevallard Y, Ladage C (2008) E-learning as a touchstone

Cross-References for didactic theory, and conversely. J E-Learn Knowl

▶ Design Research in Mathematics Education Soc 4(2):163–171
▶ Didactic Contract in Mathematics Education
▶ Didactic Engineering in Mathematics Education Lewin K (1952) Field theory in social science:
▶ Didactic Situations in Mathematics Education
▶ Didactic Transposition in Mathematics Education selected theoretical papers by Kurt Lewin. Tavistock,
▶ Didactical Phenomenology (Freudenthal)
London
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A 44 Argumentation in Mathematics

Argumentation in Mathematics a branch of logic which was familiar to the medi-
eval scholastics. Modal logic today is a useful
Kristin Umland1 and Bharath Sriraman2 language for proof theory, the study of what can
1The University of New Mexico, Albuquerque, and cannot be proved in mathematical systems of
NM, USA deduction. Issues of completeness of mathemati-
2Department of Mathematical Sciences, The cal systems, the independence of axioms from
University of Montana, Missoula, MT, USA other axioms, and the consistency of formal
mathematical systems are all part of proof theory.
Keywords One also finds the use of logical argumentation to
prove the existence of God in the theological
Argumentation; History of logic; Logic; works of Descartes, Leibniz, and Pascal.
Modal logic; Proof theory
The importance of the role of formal logic in
Definition mathematical argumentation continued to increase
and reached its apex with the work of David
Argumentation refers to the process of making an Hilbert and other formalists in the nineteenth and
argument, that is, drawing conclusions based on first half of the twentieth century. The Principia
a chain of reasoning. Go¨tz Krummheuer suggests Mathematica, by Alfred North Whitehead and
that argumentation can be thought of as a social Bertrand Russell, was a three-volume work that
process in which the cooperating individuals attempted to put the foundations of mathematics
“adjust their intentions and interpretations by on a solid logical basis (Whitehead and
verbally presenting the rationales for their actions” Russell, 1927). However, this program came
(Cobb and Bauersfeld 1995, p. 13). In mathemat- to a definitive end with the publication of
ics, unlike any empirically based discipline, the Go¨del’s incompleteness theorems in 1931, which
validity of an argument in its final form is judged subsequently opened the door for more complex
solely on whether it is logically consistent. views of mathematical argumentation to develop.

Characteristics of Argumentation Given this historical preview of the
development of logic and its role in mathematical
The origins of logic, a key component of mathe- argumentation, we now turn our attention to
matical argumentation, can be traced back to contemporary views of mathematical argumenta-
Aristotelian logic and his use of syllogisms, tion, and in particular its constituent elements.
with thinkers making improvements to this Efraim Fischbein claimed that intuition is an
method over time as they were confronted with essential component of all levels of an argument,
paradoxes. Argumentation was primarily the with qualitative differences in the role of intuition
domain of theologians and medieval and between novices [students] and experts [mathema-
postmedieval scholastics for over 1,700 years ticians]. For novices, it exists as a primary compo-
after Aristotle. Some well-known examples of nent of the argument. Fischbein (1980) referred to
theological argumentation are the Italian this use of intuition as anticipatory, i.e., “. . .while
prelate St. Anselm of Canterbury’s (1033–1109) trying to solve a problem one suddenly has the
“ontological argument” in the Proslogion, which feeling that one has grasped the solution even
was later revised by Leibniz and Go¨del. Today, before one can offer an explicit, complete justifi-
sophisticated versions of the ontological argu- cation for that solution” (p. 10). For example, in
ment are written in terms of modal logic, response to why a given solution to a problem is
correct, the novice may respond “just because . . . it
has to be.” The person using this type of intuition
accepts the given solution as the truth and believes
nothing more needs to be said. In a more advanced
argument, intuition plays the role of an “advanced
organizer” and is only the beginning of an

Argumentation in Mathematics A45

individual’s argument. In this sense, a personal independent of the other axioms of Euclidean A
belief about the truth of an idea is formed and geometry; category theory is a refinement of set
acts as a guide for more formal analytic methods theory that resolves set theoretic paradoxes; and
of establishing truth. For example, a student may the axioms of nonstandard analysis are a reorgani-
“see” that the result of a theorem is obvious, but zation of analysis that eliminates the use of the law
realize that deduction is needed to establish truth of the excluded middle.
publicly. Thus intuition serves to convince oneself
about the truth of an idea while serving to organize However, the mathematical community has on
the direction of more formal methods. numerous occasions placed epistemic value on
results before they were logically consistent
In an attempt to determine how mathematicians with other related results that lend credence to
establish the truth of a statement in mathematics, its logical value. For instance, many of Euler and
Kline (1976) found that a group of mathematicians Ramanujan’s results derived through their phe-
said they began with an informal trial and error nomenal intuition and self-devised methods of
approach guided by intuition. It is this process argumentation (and proof) were accepted as true
which helped these mathematicians convince in an epistemic sense but only proved much later
themselves of the truth of a mathematical idea. by mathematicians using a more rigorous form of
After the initial conviction, formal methods were mathematical argumentation to meet contempo-
pursued. “The logical approach to any branch of rary standards of proof. If one considers Weyl’s
mathematics is usually a sophisticated, artificial mathematical formulation of the general theory
reconstruction of discoveries that are refashioned of relativity by using the parallel displacement
many times and then forced into a deductive sys- of vectors to derive the Riemann tensor, one
tem.” (p. 451). There definitely exists a distinction observes the interplay between the intuitive and
between how mathematicians convince them- the deductive (the constructed object). The
selves and how they convince others of the truth continued evolution of the notion of tensors in
of mathematical ideas. Another good exposition of physics/Riemannian geometry can be viewed as
what constitutes argumentation in mathematics is a culmination or a result of the flaws discovered
found in Imre Lakatos’ (1976) Proofs and Refuta- in Euclidean geometry. Although the sheer
tions, in the form of a thought experiment. The beauty of the general theory of relativity was
essence of Lakatos’ method lies in paying attention tarnished by the numerous refutations that arose
to the casting out of mathematical pathologies in when it was proposed, one cannot deny the
the pursuit of truth. Typically one starts with a rule present day value of the mathematics resulting
and clearly identifies the hypothesis. This is from the interplay of the intuitive and the logical.
followed by an exploration of the possibility of Many of Euler’s results on infinite series have
its truth or falsity. The process of conjecture- been proven correct according to modern
proof-refutation results in the refinement of the standards of rigor. Yet, they were already
hypothesis in the pursuit of truth in addition to established as valid results in Euler’s work. This
the pursuit of all tangential hypotheses that arise suggests that mathematical argumentation can be
during the course of discourse. The Lakatosian thought of as successive levels of formalizations
exposition of mathematical argumentation brings as embodied in Lakatos’ thought experiment.
into focus the issue of fallibility of a proof, Such a view has been expressed in the writings
either due to human error or inconsistencies of prominent mathematicians in Hersh’s (2006)
in an axiomatic system. However, there are 18 Unconventional Essays on Mathematics.
self-correcting mechanisms in mathematics, i.e.,
proofs get fixed or made more rigorous and axi- Cross-References
omatic systems get refined to resolve inconsis-
tencies. For example, non-Euclidean geometries ▶ Argumentation in Mathematics Education
arose through work that resolved the question of ▶ Deductive Reasoning in Mathematics Education
whether the parallel postulate is logically

A 46 Argumentation in Mathematics Education

References Mathematics Classrooms and Argumentation
In the context of a mathematics classroom, we
Cobb P, Bauersfeld H (1995) The emergence of mathe- will take a “mathematical argument” to be a line
matical meaning. Lawrence Erlbaum and Associates, of reasoning that intends to show or explain why
Mahwah a mathematical result is true. The mathematical
result might be a general statement about
Fischbein E (1980) Intuition and proof. Paper presented some class of mathematical objects or it might
at the 4th conference of the international group simply be the solution to a mathematical prob-
for the psychology of mathematics education, lem that has been posed. Taken in this sense,
Berkeley, CA a mathematical argument might be a formal or
informal proof, an explanation of how a student
Hersh R (2006) 18 unconventional essays on the nature of or teacher came to make a particular conjecture,
mathematics. Springer, New York how a student or teacher reasoned through a
problem to arrive at a solution, or simply a
Kline M (1976) NACOME: implications for curriculum sequence of computations that led to a numeri-
design. Math Teach 69:449–454 cal result. The quantity and nature of mathemat-
ical arguments that students and teachers
Lakatos I (1976) Proofs and refutations: the logic of produce in mathematics classrooms varies
mathematical discovery. Cambridge University Press, widely. Observational studies of mathematics
Cambridge classrooms indicate that in some there
is essentially no dialogue between students
Whitehead AN, Russell B (1927) Principia mathematica. and students and teacher that would constitute
Cambridge University Press, Cambridge an argument that is more complex than a
series of calculations. In some classrooms, the
Argumentation in Mathematics teacher produces the majority of arguments, in
Education others the teachers and students coproduce
arguments, while in a very few, students spend
Bharath Sriraman1 and Kristin Umland2 time working together to develop arguments
1Department of Mathematical Sciences, which they then present or even defend to the
The University of Montana, Missoula, MT, USA entire class.
2The University of New Mexico, Albuquerque,
NM, USA The different ways in which mathematical
arguments are enacted in classrooms reflect
Keywords different philosophies about the kinds of math-
ematical arguments that belong in there and
Argumentation; Beliefs; Heuristics; Lakatos; the different belief systems held by teacher
Proof related to how students develop the knowledge
and skill to produce such arguments. These
Definition philosophies and belief systems are largely
cultural in that teachers learn them implicitly
“Argumentation in mathematics education” can through their own schooling; such knowledge
mean two things: is often tacitly held. In some cases, however,
1. The mathematical arguments that students teachers believe that students should be engag-
ing in more complex argumentation but do
and teachers produce in mathematics not have the practical skills to structure class-
classrooms room episodes so that students are successful
2. The arguments that mathematics education in creating or defending more complex
researchers produce regarding the nature of mathematical arguments.
mathematics learning and the efficacy of
mathematics teaching in various contexts.
This entry is about the first of these two
interpretations.

Argumentation in Mathematics Education A47

Approaches to Argumentation in definitions and accepted rules. The class was A
Mathematics Education also trained in identifying hidden assumptions
An exemplary case study of student’s success- and terms that need no definition. That is, stu-
fully creating and defending mathematical dents were trained to start with agreed upon pre-
arguments is found in Fawcett’s (1938) classic mises (be they axioms, definitions, or generally
book The Nature of Proof, in which students are accepted criteria outside of mathematics) and
guided to create their own version of Euclidean produce steps that lead to the sought conclusions.
geometry. This 2-year teaching experiment with Included in this training is the analysis of other
high school students highlighted the role of argu- arguments on the basis of how well they do the
mentation in choosing definitions and axioms and same. This is a deductivist approach to argumen-
illustrated the pedagogical value of working tation (Sriraman et al. 2010) and allows for only
with a “limited tool kit.” The students in a single method of proof. Direct proof is given to
Fawcett’s study created suitable definitions, students with little regard to the way in which
chose relevant axioms when necessary, and cre- they will internalize the method. In the book
ated Euclidean geometry by using the available Proofs and Refutations (1976), Lakatos makes
mathematics of Euclid’s time period (Sriraman the point that this sort of “Euclidean methodol-
2006). The glimpses of the discourse one finds in ogy” is detrimental to the exploratory spirit of
Fawcett’s study also illustrate the Lakatosian ele- mathematics. Not only can an overreliance on
ments of the possibilities in an “ideal” classroom deduction dampen the discovery aspect of
for argumentation. In the case of Lakatos, the mathematics; it can also ignore the needs of
argumentation (or classroom discourse) occurs students as they learn argumentation that
in his rich imagination in the context of constitutes a proof.
a teacher classifying regular polyhedra and
constructing a proof for the relationship between In Patterns of Plausible Inference, Polya
the vertices, faces, and edges of regular polyhedra (1954) lays out heuristics via which the plausibil-
given by Leonhard Euler as V + F À E ¼ 2. The ity of mathematical statements may be tested for
essence of the “Lakatosian” method lies in pay- validity. By doing so, he gives a guide for stu-
ing attention to the casting out of mathematical dents as they go about exploring the validity of
pathologies in the pursuit of truth. Typically one a statement. “I address myself to teachers of
starts with a rule and clearly identifies the mathematics of all grades and say: Let us teach
hypothesis. This is followed by an exploration guessing” (Polya 1954, p. 158). This is quite
of the possibility of its truth or falsity. The pro- different from the deductive view which holds
cess of conjecture-proof-refutation results in the fast to inferences that can be logically concluded,
refinement of the hypothesis in the pursuit of where inconclusive but suggestive evidence has
truth in addition to the pursuit of all tangential no place. While we do not doubt that the
hypotheses that arise during the course of dis- deductivist approach leaves room for guessing, it
course. Mathematics educators have attempted is not its primary emphasis. This is not to say,
to implement the technique of conjecture-proof- either, that the heuristic approach would abandon
refutation with varying degrees of success in the demonstrative proof. In Polya’s (1954) heuristic
context of number theoretic or combinatorial approach, students are exposed to ways familiar to
problems (see Sriraman 2003, 2006). An impor- mathematicians when they are judging the poten-
tant aspect of argumentation in the context of tial validity of a statement and looking for proof.
Fawcett’s (1938) study is that while the proofs Lakatos (1976) makes a similar case. In his fic-
themselves are student created, the format they tional class, the students argue in a manner that
take on is largely orchestrated by the teacher. mirrors the argument the mathematical commu-
The first objective of the class in Fawcett’s nity had when considering Euler’s formula
study was to emphasize the importance of for polyhedra. He states that an overly deductive
approach misrepresents the ways the mathematics

A 48 Assessment of Mathematics Teacher Knowledge

community really works. Fawcett shows, how- Cross-References
ever, a way in which a deductivist classroom can
model the mathematical community to a certain ▶ Argumentation in Mathematics
extent. Like in the mathematics community, dis- ▶ Deductive Reasoning in Mathematics
agreements arise and the need for convincing
fosters the need for proof (Sriraman et al. 2010). Education
▶ Quasi-empirical Reasoning (Lakatos)
Over the past several decades, philosophers of
mathematics have been attempting to describe References
the nature of mathematical argumentation (e.g.,
Lakatos, Hersh, others; see ▶ Argumentation in Fawcett HP (1938) The nature of proof: a description and
Mathematics). Many mathematics education evaluation of certain procedures used in senior high school
researchers have called for teachers to engage to develop an understanding of the nature of proof.
students in the practice of doing mathematics as National Council of Teachers of Mathematics, Reston
mathematicians, which has mathematical
argumentation at its core, for example, Deborah Lakatos I (1976) Proofs and refutations: the logic of math-
Ball http://ncrtl.msu.edu/http/craftp/html/pdf/ ematical discovery. Cambridge University Press,
cp903.pdf, Schoenfeld (1985). Cambridge

In the USA, this call was brought to the national Polya G (1954) Patterns of plausible inference. Princeton
conversation through the inclusion of the process University Press, Princeton
standards in the NCTM Principles and Standards
for School Mathematics and has evolved to Schoenfeld AH (1985) Mathematical problem solving.
become more specific and concrete in the recent Academic, Orlando
Standards for Mathematical Practice in the Com-
mon Core State Standards for Mathematics. Such Sriraman B (2003) Can mathematical discovery fill the
standards, when coupled with the picture painted in existential void? The use of conjecture, proof and refu-
US classrooms, show a wide gulf between the tation in a high school classroom. Math Sch 32(2):2–6
vision the mathematics education community has
for how mathematical argumentation might look Sriraman B (2006) An Ode to Imre Lakatos: bridging the
and what actually transpires in classrooms, at least ideal and actual mathematics classrooms. Interchange
in the USA. However, the US mathematics educa- Q Rev Educ 37(1&2):155–180
tion researcher community is not alone; other
countries’ educational systems also grapple with Sriraman B, Vanspronsen H, Haverhals N (2010) Com-
similar issues, although the framing and details mentary on DNR based instruction in mathematics as
vary as they reflect cultural attitudes about the a conceptual framework. In: Sriraman B, English
appropriate nature of mathematical argumentation L (eds) Theories of mathematics education: seeking
in mathematics classrooms. new frontiers. Springer, Berlin, pp 369–378

Caveat emptor: Neither Lakatos nor Polya were Assessment of Mathematics Teacher
mathematics educators in the contemporary sense Knowledge
of the word. The former was a philosopher of
science who was trying to address his community Vilma Mesa and Linda Leckrone
to pay attention to the history of mathematics, School of Education, University of Michigan,
whereas the latter an exemplary mathematician Ann Arbor, MI, USA
that became interested in pedagogy. Both Lakatos
and Polya’s work has found an important place in Keywords
the canon of literature in mathematics education
that addresses discourse, argumentation, and Teacher assessment; Teacher knowledge; Math-
proof and hence made central in this encyclopedia ematical knowledge for teaching
entry.
Definition

Practices and processes used to assess the
mathematical knowledge of teachers. This

Assessment of Mathematics Teacher Knowledge A49

information is frequently used to establish hersconference/ravitch.html for a brief history in A
certification of new teachers, to give promotion the US), from requiring a demonstration of moral
and recognition to current teachers, to determine character, to demonstration of competency in ele-
the need and content of professional development mentary subjects (e.g., arithmetic, reading, history,
for current teachers, and to provide information to and geography), to more specialized processes that
researchers about teacher knowledge. Methods to may include demonstrations of teaching specific
assess the mathematical knowledge of teachers mathematics topics. In countries without a central-
include paper and pencil or oral examinations ized system for regulating certification, more than
with multiple choice, short answer, or open-ended one process can exist. The processes of certification
questions; portfolios; interviews; and demonstra- vary across educational systems, with some requir-
tions of teaching. ing various examinations in several selection stages
(e.g., written, oral, microteaching in Korea, http://
The need to systematically assess mathemat- www.MEST.go.kr) and some requiring a written
ics teacher knowledge was initiated by the test only (e.g., PRAXIS, in the USA, http://www.
publication in the early 1980s of reports about ets.org/praxis/about/praxisii).
the low quality of mathematics in schools
(e.g., the Cockcroft report in the UK, Nation at Promotion and Recognition of Practicing
Risk in the US; see Howson et al. 1981). These Teachers
reports stirred the need for reform in mathematics Assessment of mathematics teacher knowledge is
classrooms, and in particular to attend to relevant to the employment status of current
“salary, promotion, tenure, and retention deci- teachers. An educational system may use evi-
sions [of teachers, which] should be tied to an dence that teachers hold or have gained sufficient
effective evaluation system that includes knowledge and competence to retain the teachers
peer review” (National Commission on Excel- in their current jobs, to recognize them, and to
lence in Education 1983, Recommendation promote them. Current teachers may also use the
D.2 Teaching). Reports of the low attainment processes to guide their professional develop-
of students in international comparisons of ment. These processes include peer reviews or
mathematics achievement in the studies observation of instruction by administrators, or
conducted by the IEA, the OECD, and the in more formal cases, teachers may document
UNESCO also have heightened awareness of their knowledge and create a portfolio that is
the need to assess teachers’ knowledge and to evaluated by a national board (e.g., National
find its connections to student performance. Board for Professional Teaching Standards,
Several decades later, many educational systems http://www.nbpts.org/for_candidates/certificate_
have passed resolutions that impose stringent areas1?ID¼3&x¼57&y¼8).
requirement to certify teachers, to maintain
them in the profession, and that have led Research on Teacher Knowledge
researchers to investigate methods to measure Assessment of mathematics teacher knowledge
this knowledge with a goal of producing valid has lately been associated with measures of math-
results that are useful in policymaking. ematical knowledge for teaching (MKT). The
impetus for this work can be traced to Shulman’s
Certification of New Teachers (1986) categorization of teachers’ knowledge
Assessment of mathematics teacher knowledge can into content, pedagogical, and curricular. Prior
be associated with processes of certification or to Shulman’s publication, a standard way to mea-
licensing of teachers. Certification ensures that sure teachers’ knowledge was by the number of
people who wish to work as mathematics teachers subject-matter courses teachers had been exposed
have sufficient knowledge and competence to to during training or the number of hours of
practice the profession. Certification processes professional development in which they have
have changed over time (see Ravitch, n.d., http:// engaged as practicing teachers.
www2.ed.gov/admins/tchrqual/learn/preparingteac

A 50 Assessment of Mathematics Teacher Knowledge

Assessment of Mathematics Teacher Knowledge, Table 1 Examples of assessments of teacher knowledge

Assessment, Level, purpose What is assessed Format of assessment
location
Preservice teachers, Mathematics knowledge Three tests: multiple choice (for all areas),
Teacher certification open ended (mathematics and general), oral
education General pedagogical and microteaching (general and specific)
test, South knowledge
Korea
Specific knowledge for
teaching mathematics

PRAXIS, Preservice teachers, Varies from state to state, Multiple choice, usually 2-h long.
USA certification primarily content, although Requirements vary by state
pedagogy is offered

NBPTS, USA Practicing teachers, Knowledge of Four portfolio entries, two of which are

National Board mathematics, students, and video, followed by six short answer

Certification teaching assessment exercises, which are 30 min each

MKT, USA, Elementary teachers, Mathematical knowledge Primarily multiple choice, occasional short
for teaching, with six answer with optional interviews depending
other research, and professional subcategories on purpose of test (validation, research, and/
countries improvement or professional improvement)

COACTIV, Secondary teachers whose Content knowledge and Two short answer paper and pencil tests
Germany students participated in three areas of pedagogical (70 min for pedagogical content knowledge
PISA, research content knowledge and 50 min for content). There 2 h more
available for follow-up questions

TEDS-M, Preservice teacher Content knowledge, 60-min paper and pencil, some open-ended
international education programs, pedagogical content questions
research, comparative knowledge, and beliefs
studies

The need for measuring teacher knowledge US-practicing elementary teachers (Hill et al.
has become more prominent as demands for 2008). The instrument is not meant to be used for
establishing links between teacher behaviors and certification or promotion, rather for establishing
student achievement have increased. Initial attempts a connection between teacher knowledge, student
to establish connections using characteristics such as achievement, and quality of instruction (Hill et al.
the number of mathematics courses taken or the 2005). Because teaching is a highly contextual-
number of hours of professional development as ized practice, current research on the instrument
proxies for teacher knowledge led to inconclusive focuses on validity of the instrument in other
results (Blo¨meke and Delaney 2012). countries (see the 44th issue of ZDM Mathemat-
ics Education, 2012 on assessment of teacher
Research in this area has proposed that math- knowledge).
ematics teacher knowledge includes six areas,
three related to subject-matter knowledge Similar efforts to measure teacher knowledge
(common content knowledge, knowledge at the with the purpose of connecting it to student
mathematical horizon, and specialized content achievement have been pursued in other
knowledge) and three related to pedagogical countries. In Germany the impetus for the
content knowledge (knowledge of content and Cognitive Activation in the Classroom
students, knowledge of content and teaching, (COACTIVE) project (Krauss et al. 2008) was
and knowledge of curriculum, Ball et al. 2008). German students’ lower than expected perfor-
Research since the late 1990s has focused on mance in the Program of International Student
the construction of an instrument that can Assessment (PISA) compared to other European
measure specialized content knowledge. This countries. Other recent efforts to assess mathe-
instrument has been successfully validated with matics teacher knowledge in other countries

Authority and Mathematics Education A51

come from the international Teacher Education Hill HC, Ball DL, Schilling SG (2008) Unpacking A
and Development Study (TEDS), which is pedagogical content knowledge: conceptualizing and
designed to describe the quality of teacher measuring teachers’ topic-specific knowledge of
education programs in the 16 participating students. J Res Math Educ 39(4):372–400
countries (http://www.iea.nl/teds-m.html). As
part of the data collected, an instrument to assess Howson G, Keitel C, Kilpatrick J (1981) Curriculum
mathematics teachers’ knowledge and beliefs development in mathematics. Cambridge University
was used. Press, Cambridge

In Table 1 we present an overview of different Krauss S, Neubrand M, Blum W, Baumert J, Kunter M,
types of processes to assess mathematics teacher Jordan A (2008) Die Untersuchung des professionellen
knowledge. Wissens deutscher Mathematik-Lehrerinnen und
-Lehrer im Rahmen der COACTIV-Studie.
Future research on this area of assessment of J Mathematik-Didaktik 29(3/4):223–258
teacher knowledge will be in three fronts:
calibration of the instrument for different National Commission on Excellence in Education
contexts, validation of the construct with local (1983) A nation at risk. U.S. Government Printing
definitions of instructional quality, and connec- Office, Washington, DC
tions between the measures of teacher knowledge
obtained and student performance within Shulman LS (1986) Those who understand: a conception
educational systems and as part of the interna- of teacher knowledge. Educ Res 15(2):4–14
tional studies of student achievement.
Authority and Mathematics
Cross-References Education

▶ Competency Frameworks in Mathematics Michael N. Fried
Education Graduate Program for Science & Technology
Education, Ben-Gurion University of the Negev,
▶ Frameworks for Conceptualizing Mathematics Beer-Sheva, Israel
Teacher Knowledge
Keywords
▶ Mathematics Teacher as Learner
▶ Mathematics Teacher Education Organization, Authority of mathematics; Democratic values;
Expert authority; Sociological perspectives
Curriculum, and Outcomes
▶ Teacher Education Development Study- Definition

Mathematics (TEDS-M) The role of authority relations in mathematics
▶ Teacher Supply and Retention in education.

Mathematics Education

References Characteristics

Ball DL, Thames M, Phelps G (2008) Content knowledge As a topic for inquiry, authority enters into math-
for teaching: what makes it special? J Teach Educ ematics education by way of two main argu-
59(5):389–407 ments. The first is that because sociology,
anthropology, and politics are relevant to under-
Blo¨meke S, Delaney S (2012) Assessment of teacher standing mathematics education, as is discussed
knowledge across countries: a review of the state of elsewhere in this encyclopedia, authority must be
research. ZDM Math Educ 44:223–247 as well, being a central construct in all these

Hill HC, Rowan B, Ball DL (2005) Effects of teachers’
mathematical knowledge for teaching on student
achievement. Am Educ Res J 42:371–406

A 52 Authority and Mathematics Education

attendant fields; indeed, any treatment of power, powers, a shaman for example. Legal authority
hierarchy, and social regulation and relations is authority within an “established impersonal
must refer to the notion of authority in some order,” a legal or bureaucratic system; the system
way. The second argument, more specific to within the legal authority acts is considered ratio-
mathematics education per se, is that, owing to nal, and, accordingly, so too are the grounds of
the perception of mathematics as certain and authority and the obedience it commands. These
final, the discipline itself is authoritarian, at “ideal types” are not necessarily descriptions of
least in a manner of speaking. Whether or not given individual authority figures. Weber’s claim
authority can be attributed to mathematics strictly is that authority can be analyzed into these types:
speaking is moot of course; however, because of the authority of any given individual is almost
this perception of mathematics, authority, often always an amalgam of various types.
in matters having little to do with mathematics,
tends to be transferred to those who are consid- Expert authority, which is an essential aspect
ered mathematical experts. The latter links the of teachers’ authority, does not appear in Weber’s
second argument with the first, but it also shows writings, but it is clear that because the grounds of
how difficulties with authority can arise in class- such authority are rational and sanctioned by
room situations, for the authority of the mathe- official actions, for example, the bestowing of
matics teacher may trump the authority of the an academic degree or a license, Weber could
discipline, however that is understood. reasonably categorize it as a form of “legal
authority.” Still, it is different enough and impor-
The Social Science Context tant enough for educational purposes to distin-
guish expert authority as a distinct type with its
In the social sciences generally, the locus legitimacy founded on the possession of knowl-
classicus for the treatment of authority is surely edge by the authority figure (regardless of
Max Weber’s The Theory of Social and whether the knowledge is true or truly possessed).
Economic Organization (Weber 1947). There,
Weber describes “authority” (Herrschaft) as Students’ lives are influenced by a broad web
“. . .the probability that a command with a given of authorities, but the teachers’ authority is the
specific content will be obeyed by a given group most immediate of these and arguably the most
of persons” (p. 139). Weber’s definition stresses important. It has been suggested too that
that true authority involves more than power of teachers’ authority manifests elements not only
one person or body over another, more than mere of expert authority, but also traditional, legal, and
coercion: it involves “. . .a certain minimum of even charismatic authority (Amit and Fried
voluntary submission” on the part of the con- 2005). It is not by accident, then, that early socio-
trolled and an interest in obedience on the part logical studies of education, such as Willard
of the authority (p. 247). The crucial point is that Waller’s classic 1932 sociological study of edu-
for authority to be authority, it must be recog- cation (see Amit and Fried 2005) and Durkheim’s
nized as legitimate by those who submit to it; it works on education (Durkheim 1961), underlined
is this that distinguishes it from mere power the authority of teachers, nor is it surprising that
(Macht) (p. 139). these sociological studies particularly empha-
sized the function of authority as a socializing
Weber identifies three grounds of legitimacy force and its connection, accordingly, with
and three concomitant “ideal types” of authority: moral instruction and discipline.
traditional, charismatic, and legal authority.
Traditional authority is the authority of parents Because of the strength of teachers’ authority,
or of village elders. Charismatic authority is the it can conflict with modes of teaching and learn-
authority of one endowed with superhuman ing which mathematics education has come to
value. Such a conflict arises naturally between
teachers’ authority and democratic values.
This was studied by Renuka Vithal (1999), who

Authority and Mathematics Education A53

concluded that the teachers’ authority, although to the authoritarian nature of mathematics A
opposed to democracy, could actually live itself. This is not a new phenomenon. Judith
with democracy in a relationship of complemen- Grabiner (2004), writing about Colin Maclaurin
tarity. She suggests that the very fact of (1698–1746), has argued that mathematics in the
the teacher’s authority, if treated appropriately, eighteenth century attained an authority greater
could provide an opportunity for students to even than that of religion, since mathematics was
develop a critical attitude toward authority perceived as having the power to achieve agree-
(see also Skovsmose 1994). ment with a universality and finality unavailable
to religion. How far that authority was transferred
To take full advantage of authority as Vithal to a mathematician like Maclaurin can be judged
suggests, or in any other way, it is essential to by the remark of a contemporary referring to
understand the mechanisms by which relations of actuarial work carried out by Maclaurin, not
authority are established and reproduced. Indeed, strictly mathematical work, that “The authority
these may be embedded not only in social struc- of [Maclaurin’s] name was of great
tures already in place when students enter use. . .removing any doubt” (quoted in Grabiner
a classroom, but in subtle aspects of classroom 2004, p. 847).
discourse. Herbel-Eisenmann and Wagner
(2010), for example, have looked at lexical- The tension between teachers’ authority and
bundles, small segments of spoken text, reflecting democratic modes of teaching has already been
one’s position in an authority relationship. These noted. But that had little to do with mathematics
lexical-bundles are as much a part of the students’ as such. The authority of mathematics combined
discourse as the teachers’, recalling how author- with the authority subsequently transferred to
ity relations are always a two-way street, as practitioners and teachers of mathematics,
Weber was at pains to stress. however, creates a tension arising directly from
the nature of mathematical authority. This is
Paul Ernest’s study of social semiotics (Ernest because what is essential about mathematical
2008) gives much support to Herbel-Eisenmann authority is precisely its independence from
and Wagner’s approach. Ernest shows how the any human authority: a great mathematician
analysis of classroom-spoken texts brings out the must yield even to a child who has discovered
existence of overlapping forms of teachers’ a flaw in the mathematician’s work. But Keith
authority, different roles in which teachers’ Weber and Juan Mejia-Ramos (in press)
authority is manifest. In particular, he says, the have shown that mathematicians themselves
teacher is both one in authority, a “social regula- are influenced by human authorities or by
tor” determining how a class is run, and also an authoritarian institutions – all the more so with
authority, a “knowledge expert” (p. 42) determin- students.
ing, for example, what tasks are set to the
students. How this plays out in a specific mathematical
context can be seen in Harel and Sowder’s (1998)
The Authoritarian Nature of category of proof schemes based on external con-
Mathematics viction, which includes a subcategory called
“authoritarian proofs.” Typical behavior associ-
The role of teachers as expert authorities, as task ated with this proof scheme is that students
controllers, to use Ernest’s term, has very much to “. . .expect to be told the proof rather than take
do with mathematical content and how it is part in its construction” (Harel and Sowder 1998,
passed on to students. We are brought, thus, to p. 247). The authority of a teacher presenting
the second argument concerning authority and a proof can thus take precedence over the internal
mathematics education, for the degree of the logic behind the authority of the discipline: the
overlap Ernest refers to is very much related whole notion of “proof” is vitiated when this
happens, since the truth of a claim becomes

A 54 Autism, Special Needs, and Mathematics Learning

established not because of argument but because References
of a teacher’s authoritative voice.
Amit M, Fried MN (2005) Authority and authority rela-
Reminiscent of Vithal’s (1999) argument tions in mathematics education: a view from an 8th
above, the challenge of mathematics teachers grade classroom. Educ Stud Math 58:145–168
must then be, paradoxically, to use their authority
to release students from teachers’ authority. Jo Boaler J (2003) Studying and capturing the complexity of
Boaler (2003) suggests as much when she practice – the case of the ‘Dance of Agency’. In:
remarks favorably about a teacher in her study Pateman NA, Dougherty BJ, Zilliox JT (eds) Proceed-
that she “employed an important teaching ings of the 27th annual conference of PME27 and
practice—that of deflecting her authority to the PME-NA25, CRDG, vol 1. College of Education,
discipline [of mathematics]” (p. 8). Honolulu, pp 3–16

Since the authority of mathematics as Durkheim E (1961) Moral education (trans: Wilson EK,
a discipline becomes ultimately the posses- Schnurer H). The Free Press, New York
sion of the student as the teacher deflects
her own authority, we see that the problem Ernest P (2008) Towards a semiotics of mathematical text
of authority in mathematics education is how (Part 3). Learn Math 28(3):42–49
to devolve authority. The problem of author-
ity, in this way, becomes the mirror problem Grabiner J (2004) Newton, Maclaurin, and the authority of
of agency. mathematics. Am Math Mon 111(10):841–852

Future Avenues of Research Harel G, Sowder L (1998) Students’ proof schemes:
results from exploratory studies. In: Schoenfeld AH,
One of the important conclusions from Vithal’s Kaput J, Dubinsky E (eds) Research in collegiate
(1999) work as well as Amit and Fried’s (2005) mathematics III. American Mathematical Society,
work is that authority may be more than Providence, pp 234–282
a necessary evil in mathematics education. But
exactly how authority can be used to create Herbel-Eisenmann BA, Wagner D (2010) Appraising
a more democratic classroom and a more auton- lexical bundles in mathematics classroom discourse:
omous student needs to be investigated: what the obligation and choice. Educ Stud Math 75(1):43–63
actual mechanisms are through which this is
achieved. This will be particularly important Skovsmose O (1994) Towards a critical philosophy of
for teacher education, since it is teachers who mathematics education. Kluwer, Dordrecht
command authority in the most explicit way.
This presupposes that researchers have ways of Vithal R (1999) Democracy and authority:
tracing authority relations in the classroom. In a complementarity in mathematics education? ZDM
this regard, a second necessary avenue of 31(1):27–36
research is the identification of how authority
relations are reproduced, research of the sort Weber M (1947) The theory of social and economic orga-
represented here by Herbel-Eisenmann and nization (trans: Henderson AR, Parsons T). William
Wagner (2010). Hodge, London

Weber K, Mejia-Ramos JP (2013) The influence of
sources in the reading of mathematical text: a reply
to Shanahan, and Misischia. J Lit Res 45(1):87–96

Autism, Special Needs, and
Mathematics Learning

Richard Cowan and Liz Pellicano
Department of Psychology and Human
Development, Institute of Education,
University of London, London, UK

Cross-References Keywords

▶ Sociological Approaches in Mathematics Autism; Autistic spectrum condition; Autistic
Education spectrum disorders (ASD); Asperger syndrome;
Individual differences; Special needs; Inclusion

Autism, Special Needs, and Mathematics Learning A55

Definition seem to solve date calculation problems by using A
a combination of memory for day-date combina-
Autism spectrum conditions are lifelong tions, addition and subtraction, and knowledge of
neurodevelopmental conditions that are calendrical patterns, such as the 28-year rule, i.e.,
characterized by often striking difficulties in 2 years 28 years apart are the same unless the
social communication and repetitive and rigid interval contains a non-leap century year such as
patterns of behavior (American Psychiatric 2100 (Cowan and Frith 2009). The degree of skill
Association APA 2000). Current estimates they exhibit may result from practice. Several autis-
indicate that 1 in every 100 children is on the tic calendar calculators do not appear to know how
autism spectrum, meaning that all schools and to multiply or divide. Most children with autism do
colleges are likely to include pupils who lie not show any exceptional numerical ability.
somewhere on the autism spectrum.
There are remarkably few studies of the
Characteristics mathematical progress of children with ASD
and most have relied on standardized tests that
Although autism is now considered a highly use arithmetic word problems to assess mathe-
heritable disorder of neural development matical skill. The results should be interpreted
(Levy et al. 2009), specific genes, and the ways cautiously as standardized tests can be extremely
that these genes interact with the environment, limited in the skills they assess (Ridgway 1987)
are not yet fully understood (Frith 2003). The and difficulties with arithmetic word problems
diagnosis of autism therefore relies on may reflect autistic children’s problems in verbal
a constellation of behavioral symptoms, which can comprehension rather than difficulties in their
vary substantially from individual to individual. computational skill. Nevertheless a recent review
This variability includes marked differences in concludes that most children with autism show
the degree of language skills: some individuals do arithmetical skills slightly below those expected
not use oral language to communicate, while others from their general ability with some doing
use grammatically correct speech, but the way that markedly worse in arithmetic and others doing
they use language within social contexts can be odd markedly better (Chiang and Lin 2007). The
and often one sided. Also, a substantial minority, reasons for this variation have not been examined
roughly a third, meet the criterion for intellectual and individual differences in mathematical
disability (Levy et al. 2009). Furthermore, there is learning by children with ASD are as little under-
wide variation in developmental outcomes: while stood as the reasons for individual differences in
some individuals with autism will go on to live typically developing children.
independently and gain qualifications, many indi-
viduals are unable to live on their own or enjoy Some of the core features of autism – including
friendships and social contacts (Howlin et al. 2004). rigid and repetitive ways of thinking and behaving
and heightened responses to environmental
The unusual abilities of some people with ASD features (such as the sound of the school bell) –
show, such as Dustin Hoffman portrayed in the can make learning difficult for many children.
film Rain Man, have captured public attention. Guidance on teaching children with autism
The most common ASD ability is calendar calcu- therefore emphasizes the need for educators both
lation, the ability to name weekdays corresponding to help the individual child/young person to
to dates in the past or present. Some mathemati- develop skills and strategies to understand situa-
cians have delighted in calendar calculation (e.g., tions and communicate needs and to adapt the
Berlekamp et al. 1982), but autistic calendar cal- environment to enable the child to function and
culation does not reflect any substantial mathemat- learn within it (Jordan and Powell 1995; Jones
ical abilities. Instead, autistic calendar calculators 2006; Freedman 2010; Charman et al. 2011; see
also websites run by the National Autistic Society
and the Autism Society of America). As we have

A 56 Autism, Special Needs, and Mathematics Learning

stressed, children with autism differ enormously. References
For this reason, mathematical educators must be
adept at understanding each student’s individual American Psychiatric Association (APA) (2000) Diagnostic
needs and use innovative methods of modifying and statistical manual of mental disorders (DSM-
the curriculum, exploiting autistic students’ IV-TR), 4th edn., text revision. American Psychiatric
strengths and interests, to make mathematics Association, Washington, DC
accessible and rewarding for such students.
Berlekamp ER, Conway JH, Guy RK (1982) Winning
Cross-References ways, vol 2. Academic, New York

▶ 22q11.2 Deletion Syndrome, Special Needs, Charman T, Pellicano E, Peacey LV, Peacey N,
and Mathematics Learning Forward K, Dockrell J (2011) What is good practice
in autism education? Autism Education Trust, London
▶ Blind Students, Special Needs, and
Mathematics Learning Chiang H-M, Lin Y-H (2007) Mathematical ability of
students with Asperger syndrome and high-functioning
▶ Deaf Children, Special Needs, and autism. Autism 11:547–556
Mathematics Learning
Cowan R, Frith C (2009) Do calendrical savants use
▶ Down Syndrome, Special Needs, and calculation to answer date questions? A functional
Mathematics Learning magnetic resonance imaging study. Philos Trans
R Soc B 364:1417–1424
▶ Giftedness and high ability in mathematics
▶ Inclusive Mathematics Classrooms Freedman S (2010) Developing college skills in students
▶ Language Disorders, Special Needs and with autism and Asperger’s syndrome. Jessica
Kingsley, London
Mathematics Learning
▶ Learner-Centered Teaching in Mathematics Frith U (2003) Autism: explaining the enigma. Blackwell,
Oxford
Education
▶ Learning Difficulties, Special Needs, and Howlin P, Goode S, Hutton J, Rutter M (2004) Adult
outcome for children with autism. J Child Psychol
Mathematics Learning Psychiatry 45:212–229
▶ Mathematical Ability
▶ Word Problems in Mathematics Education Jones G (2006) Department for education and skills/
department of health good practice guidance on the
education of children with autistic spectrum disorder.
Child Care Health Dev 32:543–552

Jordan R, Powell S (1995) Understanding and teaching
children with autism. Wiley, Chichester

Levy SE, Mandell DS, Schultz RT (2009) Autism. Lancet
374:1627–1638

Ridgway J (1987) A review of mathematics tests.
NFER-Nelson, Windsor

B

Bilingual/Multilingual Issues Characteristics
in Learning Mathematics
Theoretical Perspectives
Judit N. Moschkovich The study of bilingual and multilingual mathe-
Education Department, University of California matics learners requires theoretical notions
Santa Cruz, Santa Cruz, CA, USA that simultaneously address not only the cogni-
tive and domain-specific aspects of learning
Keywords mathematics but also the linguistic and cross-
cultural nature of this work. Therefore, research
Bilingual; Bilingualism; Code-switching; addressing these issues draws on work from out-
Communication; Language; Learners; Linguis- side mathematics education. For example, educa-
tic; Linguistics; Monolingual; Multilingual; tional anthropology and cultural psychology
Multilingualism; Sociolinguistics; Students; have been used to ground cross-cultural aspects
Switching Languages of this work. Similarly, linguistics, especially
approaches to bilingualism and multilingualism,
Definition has been used to ground linguistic aspects of
this work. In particular, psycholinguistics and
Bilingual and multilingual issues in learning sociolinguistics are two theoretical perspectives
mathematics refer to questions regarding frequently used in the study of bilingual and
bilingual and multilingual learners as they multilingual issues in learning mathematics.
learn mathematics. Research in mathematics
education focusing on bilingual and multilin- Bilingualism
gual issues in learning mathematics is primar-
ily concerned with the study of bilingual and “Bilingualism” (Pen˜a and Bedore 2010) is an
multilingual mathematics learners. Below is example of a concept that has different meanings
an overview of some key issues, ideas, and depending on the theoretical perspective used.
findings that focus on research on learners Definitions of bilingualism range from native-
rather than on teaching practices, although like fluency in two languages to alternating
the two are clearly connected. use of two languages, to participation in
a bilingual community. A researcher working

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

B 58 Bilingual/Multilingual Issues in Learning Mathematics

from a psycholinguistic perspective would define arithmetic. There is evidence that adult bilinguals
a bilingual person as an individual who is in sometimes switch languages when carrying out
some way proficient in more than one language. arithmetic computations and that adult bilinguals
This definition would include someone who has may have a preferred language for carrying out
learned a second language in school with some arithmetic computation, usually the language of
level of proficiency but does not participate in arithmetic instruction. Language switching can
a bilingual community. In contrast, a researcher be swift, highly automatic, and facilitate rather
working from a sociolinguistic perspective would than inhibit solving word problems in the
define a bilingual person as someone who partic- language of instruction, provided the student’s
ipates in multiple language communities and is proficiency in the language of instruction is
“the product of a specific linguistic community sufficient for understanding the text of a word
that uses one of its languages for certain functions problem. These findings suggest that classroom
and the other for other functions or situations” instruction should allow bilingual and multilin-
(Valde´s-Fallis 1978, p. 4). The second definition gual students to choose the language they prefer
frames bilingualism not as an individual but as for arithmetic computation and support all stu-
a social and cultural phenomenon that involves dents in learning to read and understand the text
participation in the language practices of one or of word problems in the language of instruction
more communities. Some researchers propose (Moschkovich 2007).
using “monolingual” and “bilingual” not as labels
for individuals but as labels for modes of com- Another common practice among bilinguals
municating (Grosjean 1999). is switching languages during a sentence or
conversation, a phenomenon linguists call
A common misunderstanding of bilingualism “code-switching” (Mercado 2010). Bilingual and
is the assumption that bilinguals are equally multilingual mathematics students may use two
fluent in their two languages. If they are not, languages during classroom conversations. In
then they have been described as not truly mathematics classrooms, children will use one or
bilingual or labeled as “semilingual” or “limited another language. Which language children use
bilingual.” In contrast, current scholars of principally depends on the language ability and
bilingualism see “native-like control of two or choice of the person addressing them. After the
more languages” as an unrealistic definition. age of five, young bilinguals (beyond age 5) tend
Researchers have recently strongly criticized to “speak as they are spoken to”. If Spanish–
the concept of semilingualism (Cummins 2000) English bilinguals are addressed in English, they
and propose we leave that notion behind. reply in English; if they are addressed in Spanish,
they reply in Spanish; and if they are addressing
Research Findings a bilingual speaker, they may code-switch.

There are several research findings relevant to Another common misunderstanding is that
bilingual and multilingual issues in learning code-switching is somehow a sign of deficiency.
mathematics. Overall, there is strong evidence However, empirical research in sociolinguistics
suggesting that bilingualism does not impact math- has shown that code-switching is a complex
ematical reasoning or problem solving. There are language practice and not evidence of deficiencies.
also relevant findings regarding two common prac- In general, code-switching is not primarily
tices among bilingual and multilingual mathemat- a reflection of language proficiency, discourse pro-
ics learners, switching languages during arithmetic ficiency, or the ability to recall (Valde´s-Fallis
computation and code-switching. 1978). Bilinguals use the two codes differently
depending on the interlocutor, domain, topic,
Older bilingual students may carry out arith- role, and function. Choosing and mixing two
metic computations in a preferred language, usu- codes also involves a speaker’s cultural identities.
ally the language in which they learned
Research does not support a view of code-
switching as a deficit itself or as a sign of any

Bilingual/Multilingual Issues in Learning Mathematics B59

deficiency in mathematical reasoning. Researchers similar responses to syntactic aspects of algebra B
in linguistics agree that code-switching is not word problems.
random or a reflection of language deficiency –
forgetting a word or not knowing a concept. There- Some early research used vague notions of
fore, we cannot use someone’s code-switching to language and narrow conceptions of mathematics
reach conclusions about their language profi- as arithmetic or word problems and focused on
ciency, ability to recall a word, knowledge of two scenarios, carrying out arithmetic computa-
a particular mathematics word or concept, mathe- tion and solving word problems (Moschkovich
matical reasoning, or mathematical proficiency. 2002, 2010). Later studies developed a broader
It is crucial to avoid superficial conclusions view of mathematical activity, examining not
regarding code-switching and mathematical cog- only responses to arithmetic computation but also
nition. For example, we should not conclude that reasoning and problem solving, detailed protocols
bilingual and multilingual students switch into of students solving word problems, the strategies
their first language because they do not remember children used to solve arithmetic word problems,
a word, are missing vocabulary, or do not under- and student conceptions of two digit quantities.
stand a mathematical concept. Rather than viewing (The volume “Linguistic and cultural influences
code-switching as a deficiency, instruction for on learning mathematics” edited by Cocking and
bilingual mathematics learners should consider Mestre includes both types of research studies.)
how this practice serves as a resource for commu-
nicating mathematically. Bilingual speakers have More recent research uses broader notions of
been documented using their two languages and mathematics and language, in particular by
code-switching as a resource for mathematical using sociocultural, sociolinguistic, and ethno-
discussions, for example, first giving an explana- mathematical perspectives. A central concern
tion in one language and then switching to the has been to shift away from deficit models of
second language to repeat the explanation bilingual and multilingual students to theoretical
(Moschkovich 2002). frameworks and practices that value the resources
these students bring to the mathematics class-
History room from their previous experiences and
their homes. More recently, researchers have
Research on bilingual mathematics learners dates studied language, bilingualism, and mathematics
back to the 1970s. Early research focused on learning in many different settings (for examples
the disadvantages that bilinguals face, focusing, see Adler 1998; Barton et al. 1998; Barwell et al.
for example, on comparing response times 2007, 2009; Barwell 2003b and 2009; Clarkson
between monolinguals and bilinguals (for exam- and Galbraith 1992; Dawe 1983; Kazima 2007;
ples and a review see Moschkovich 2007) or the Roberts 1998; Setati 1998).
obstacles the mathematics register in English
presents for English learners (for some examples This work can provide important resources for
see Cocking and Mestre 1988). Studies focused addressing issues for bilingual and multilingual
on the disadvantages bilingual learners faced students in other settings, as long as differences
did not consider any possible advantages among settings are considered. One difference is
of bilingualism, for example the documented how languages are used in the classroom. Barwell
“enhanced ability to selectively attend to infor- (2003a) provides some useful distinctions among
mation and inhibit misleading cues” (Bialystok different language settings, using the terms
2001, p. 245). Studies that focused on the differ- monopolist, pluralist, and globalist. In monopolist
ences between bilinguals and monolinguals may classrooms, all teaching and learning take place in
also have missed or de-emphasized any similari- one dominant language; in pluralist classrooms,
ties, for example, that both groups may have several languages used in the local community
are also used for teaching and learning; in globalist
classrooms, teaching and learning are conducted
in an internationally used language that is not used
in the surrounding community.

B 60 Bilingual/Multilingual Issues in Learning Mathematics

Another difference to consider across settings mathematical topics. There are serious chal-
is the nature of the mathematics register in stu- lenges that research still needs to address, given
dents’ first language. For example, the mathemat- the complexity of defining a construct such as
ics register in Spanish is used to express many language “proficiency”: (a) the lack of instru-
types of mathematical ideas from everyday to ments sensitive to both oral and written modes
advanced academic mathematics. This may not for mathematical communication and (b) the
be the case for the home languages of students in scarcity of instruments that address features of
other settings. Barwell (2008) makes two crucial the mathematics register for specific mathemati-
observations: (1) “all languages are equally capa- cal topics. Studies should not assess language
ble of developing mathematics registers, although proficiency in general but rather specifically for
there is variation in the extent to which this has communicating in writing and orally about
happened” and (2) “the mathematics registers of a particular mathematical topic. Students have
different languages. . . stress different mathemati- different opportunities to talk and write about
cal meanings.” These differences in mathematics mathematics in each language, in informal
registers, however, should not be construed as or instructional settings, and about different
a reflection of differences in learner’s abilities to mathematical topics. Assessments of language
reason mathematically or to express mathematical proficiency, then, should consider not only
ideas. Furthermore, we should not assume that proficiency in each language but also proficiency
there is a hierarchical relationship among lan- for using each language to talk or write about
guages with different ways to express academic a particular mathematical topic.
mathematical ideas, for example using one word
versus using (or inventing) multiple word phrases. Future Issues and Questions

Issues in Designing Research Research on bilingual and multilingual issues in
mathematics learning is still in a developing
One challenge researchers face when designing stage. A central issue is grounding research in
research with bilingual and multilingual learners mathematics education on theoretical perspec-
is that these labels are used in ambiguous ways tives and findings from relevant fields such as
and with multiple meanings. Research studies linguistics and anthropology. Future work should
need to specify how the labels bilingual or mul- avoid reinventing wheels or, worse, reifying
tilingual are used, when applied to learners or myths or misunderstandings about bilingualism/
classrooms. These labels do not describe exactly multilingualism. This is best accomplished
what happens in the classroom in terms of how through repeated and extended interactions
teachers and students use languages. Studies between scholars who study mathematics learn-
should document students’ language profi- ing and scholars who study bilingual and multi-
ciencies in both oral and written modes and lingual learners. Future studies should avoid
also describe students’ histories, practices, and deficit-oriented models of bilingual and multilin-
experiences with each language across a range gual learners and consider any advantages that
of settings and mathematical tasks. bilingualism might provide for learning
mathematics.
“Language proficiency” is a complex con-
struct that can reflect proficiency in multiple Cross-References
contexts, modes, and academic disciplines.
Current measures of language proficiency may ▶ Cultural Diversity in Mathematics Education
not give an accurate picture of an individual’s ▶ Discourse Analytic Approaches in
language competence. We do not have measures
or assessments for language proficiency related to Mathematics Education
competence in mathematics for different ages or

Blind Students, Special Needs, and Mathematics Learning B61

▶ Immigrant Students in Mathematics Education Moschkovich J (2002) A situated and sociocultural B
▶ Language Background in Mathematics perspective on bilingual mathematics learners. Math
Think Learn 4(2–3):189–212
Education
▶ Language Disorders, Special Needs and Moschkovich JN (2007) Using two languages while
learning mathematics. Educ Stud Math 64(2):121–144
Mathematics Learning
▶ Mathematical Language Moschkovich JN (2010) Language and mathematics
▶ Mathematical Representations education: multiple perspectives and directions for
▶ Semiotics in Mathematics Education research. Information Age Publishing, Charlotte

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Barwell R (ed) (2009) Multilingualism in mathematics Keywords
classrooms: global perspectives. Multilingual Matters,
Bristol Blind mathematics learners; Perception and
cognition; Visualization; Auditory representa-
Barwell R, Barton B, Setati M (2007) Multilingual issues tions; Tactile representations
in mathematics education: introduction. Educ Stud
Math 64(2):113–119 Characteristics

Bialystok E (2001) Bilingualism in development: lan- Blindness, in itself, does not seem to be an
guage, literacy and cognition. Cambridge University impediment to learning mathematics. Indeed, his-
Press, Cambridge, UK tory shows that there have been a number of very
successful blind mathematicians, perhaps the most
Clarkson PC, Galbraith P (1992) Bilingualism and math- well known being Euler (1707–1783), who became
ematics learning: another perspective. J Res Math blind in the latter part of his life, and Saunderson
Educ 23(1):34–44 (1682–1739) who lost his sight during his first year.
Jackson (2002), in his consideration of the work of
Cocking R, Mestre J (eds) (1988) Linguistic and cultural these and more contemporary blind mathemati-
influences on learning mathematics. Lawrence cians, suggests that the lack of access to the visual
Erlbaum, Hillsdale, pp 221–240 field does not diminish a person’s ability to

Cummins J (2000) Language, power, and pedagogy.
Multilingual Matters, Buffalo

Dawe L (1983) Bilingualism and mathematical reasoning
in English as a second language. Educ Stud Math
14(4):325–353

Grosjean F (1999) Individual bilingualism. In: Spolsky B
(ed) Concise encyclopedia of educational linguistics.
Elsevier, London, pp 284–290

Kazima M (2007) Malawian students’ meanings for prob-
ability vocabulary. Educ Stud in Math 64(2):169–189

Mercado J (2010) Code switching. In: Encyclopedia of
cross-cultural school psychology. Springer US, Berlin,
pp 225–226

B 62 Blind Students, Special Needs, and Mathematics Learning

visualize – but modifies it, since spatial imagination mathematics, with different notations used in
amongst those who do not see with their eyes relies different countries. The coding systems are com-
on tactile and auditory activity. This would suggest plex and can take considerable time to master
that to understand the learning processes of blind (Marcone and Penteado 2013). An additional
mathematics learners, it is important to investigate complication is that Braille is a strictly linear
how the particular ways in which they access and notation, whereas conventional mathematical
process information shapes their mathematical notations make use of visual features – fractions
knowledge and the learning trajectories through provide a case in point. The linear versions of
which it is attained. conventional notations require additional sym-
bols, making expressions in Braille lengthy;
Vygotsky’s work with disabled learners, in gen- compounded by the fact that Braille readers can
eral, and those with visual impairments, in partic- only perceive what is under their fingers at
ular, during the 1920s and 1930s represented an a particular moment in time, it can be very
early attempt to do just this. Rather than associating difficult for them to obtain a general view of
disability with deficit and focusing on quantitative algebraic expressions. Digital technologies are
differences in achievements between those with facilitating conversions between Braille and text
and without certain abilities, he proposed that and offering the blind learner spoken versions of
a qualitative perspective should be adopted to written mathematics, but research is needed to
research how access to different mediating investigate how such alternative notation forms
resources impacts upon development (1997). The might impact differently on mathematical under-
key to understanding and supporting the practices standings and practices.
of blind learners, he argued, lies in investigating
how the substitution of the eyes by other tools both Use of spoken rather than written materials
permits and shapes their participation in social and suggests that the ears can also be used as
cultural activities, such as mathematics learning. substitutes for the eyes. But auditory learning
materials need not be limited to speech. Leuders
For the study of mathematical topics that (2012) argues that auditory perception represents
involve working with spatial representations and an important modality for processing mathemat-
information, the hands represent the most obvi- ical structures that has been under-explored.
ous substitute for the eyes, and hence it is not Here, too, digital technologies are bringing new
surprising that research involving blind geometry forms of representing and exploring mathemati-
learners has focused on how explorations of cal objects; one example is a musical calculator
tactile representation of geometrical objects con- which enables students to hear as well as see
tribute to the particular conceptions that emerge. structures of rational and irrational numbers
While vision is synthetic and global, with touch (Fernandes et al. 2011).
the whole emerges from relationships between its
parts, a difference which Healy and Fernandes In short, although the practice of blind
(2011) suggest might explain the tendency mathematics learners is a topic that has been
amongst blind learners to describe geometrical relatively under-researched in the field of
properties and relations using dynamic rather mathematics education, the evidence that does
than static means, which simultaneously corre- exist suggests that in the absence of the visual
spond to and generalize their physical actions field, information received through other sen-
upon the objects in question. sory and perceptual apparatuses provides alter-
native forms of experiencing mathematics.
Hands also play an important role in blind Deepening our understandings of how those
students’ access to written materials, with Braille who do not see with their eyes learn and do
codes substituting text in documents for blind mathematics may hence contribute to furthering
readers. There are, however, a number of partic- our understanding of the relationships between
ular challenges associated with learning and perception and mathematical cognition more
doing mathematics using Braille. First, there is generally.
no one universally accepted Braille code for

Bloom’s Taxonomy in Mathematics Education B63

Cross-References Definition

▶ Deaf Children, Special Needs, and An approach to classifying reasoning goals with B
Mathematics Learning respect to mathematics education.

▶ Equity and Access in Mathematics Education Overview
▶ Inclusive Mathematics Classrooms
▶ Mathematical Representations Bloom’s Taxonomy is arguably one of the most
▶ Political Perspectives in Mathematics Education recognized educational references published in
▶ Psychological Approaches in Mathematics the twentieth century. As noted in a 40-year
retrospective by Benjamin Bloom (1994),
Education “it has been used by curriculum planners, admin-
▶ Visualization and Learning in Mathematics istrators, researchers, and classroom teachers at
all levels of education” (p. 1), and it has been
Education referenced in academic publications representing
virtually every academic discipline. Given the
References prevalence of testing in mathematics and the
regular use of mathematics as a context for study-
Fernandes SHAA, Healy L, Martins EG, Rodrigues MAS, ing student reasoning and problem solving,
Souza FR (2011) Ver e ouvir a Matema´tica com uma Bloom’s Taxonomy has been applied and
calculadora colorida e musical: estrate´gias para incluir adapted by mathematics educators since its
aprendizes surdos e aprendizes cegos nas salas de publication.
aulas. In: Pletsch MD, Damasceno AR (eds) Educac¸a˜o
Especial e inclusa˜o escolar: reflexo˜es sobre o fazer Historical Development
pedago´gico. EDUR = Editora da Universidade Federal Originally designed as a resource to support the
do Rio de Janeiro, Serope´dica, pp 97–111 development of examinations, Bloom et al.
(1956) wrote their taxonomy to insure greater
Healy L, Fernandes SHAA (2011) The role of gestures in accuracy of communication among educators in
the mathematical practices of those who do not see a manner similar to the taxonomies used in
with their eyes. Educ Stud Math 77:157–174 biology to organize species of flora and fauna.
The ubiquitous reference to Bloom’s Taxonomy
Jackson A (2002) The world of blind mathematicians. Not is a triangle with six levels of named educational
Am Math Soc 49(10):1246–1251 objectives for the cognitive domain: knowledge,
comprehension, application, analysis, synthesis,
Leuders J (2012) Auditory teaching material for the and evaluation (Fig. 1; Office of Community
inclusive classroom with blind and sighted students. Engagement and Service 2012).
In: Proceedings of the 12th international congress on
mathematical education, Seoul Because of this reductivist use of Handbook 1:
Cognitive Domain in which the taxonomy
Marcone R, Penteado MG (2013) Teaching mathematics appeared (Bloom et al. 1956), few will recall
for blind students: a challenge at the university. Int that the knowledge category included multiple
J Res Math Educ 3(1):23–35 “knowledge of” subcategories such as knowl-
edge of conventions, knowledge of trends and
Vygotsky L (1997) Obras escogidas V–Fundamentos da sequences, and knowledge of methodology. The
defectolog´ıa [The fundamentals of defectology] (trans: writing team recognized that even knowledge
Blank JG). Visor, Madrid ranges in complexity and is quite nuanced and
detailed in ways that belie its perfunctory
Bloom’s Taxonomy in Mathematics contemporary placement on the base of the
Education

David C. Webb
School of Education, University of Colorado
Boulder, Boulder, CO, USA

Keywords

Cognition; Evaluation; Educational objectives;
Student achievement; Assessment

B 64 Bloom’s Taxonomy in Mathematics Education

BLOOMS TAXONOMY

Using old concepts to create new ideas; EVALUATION Assessing theories; Comparison of ideas;
Design and Invention; Composing; Imagining; SYNTHESIS Evaluating outcomes; Solving; Judging;
Inferring; Modifying; Predicting; Combining ANALYSIS Recommending; Rating
APPLICATION
Using and applying knowledge; COMPREHENSION Identifying and analyzing patterns;
Using problem solving methods; KNOWLEDGE Organisation of ideas;
Manipulating; Designing; Experimenting recognizing trends

Recall of information; Understanding; Translating;
Discovery; Observation; Summarising; Demonstrating;
Listing; Locating; Naming
Discussing

Bloom’s Taxonomy in Mathematics Education, Fig. 1 Bloom’s Taxonomy

triangle. It is also worth noting that the Handbook A revision of the Taxonomy, which took into
includes many examples of “illustrative test account recent advances in educational psychol-
items,” suggesting both its intended use as ogy and potential applications in curriculum and
a resource for evaluation and the importance of instruction, was published by Anderson et al.
using content-specific examples to communicate (2001); however, since the influence of the
objectives for student learning. Few of these revised Taxonomy is difficult to determine,
illustrative test items, however, were in the it is not discussed here in reference to
domain of mathematics. mathematics education.

The authors of Bloom’s Taxonomy Influence on Mathematics Education
acknowledged that it was imperfect and subject Much of the influence of the Taxonomy on math-
to adaptation and critique. Since these criticisms ematics education has been on evaluation and
are relevant to the use and misuse of the more specifically in the design and interpretation
Taxonomy in mathematics education, they are achievement tests (e.g., Webb 1996). Since these
presented here to frame the section that follows. aspects of school mathematics often influence the
curricular goals, there has also been some
Postlethwaite (1994) summarized the major indirect influence on curriculum development
criticisms as: and classroom assessment in mathematics
1. The distinctions between any two levels of the (Sosniak 1994).

Taxonomy may be blurred. Many assessment frameworks for mathematics
2. The Taxonomy is not hierarchical; rather it is have utilized the Taxonomy for guidance regarding
the distribution of items on achievement tests.
just a set of categories. In Korea in the late 1950s, “teacher-made
3. The lockstep sequence underlying the

Taxonomy based on one dimension (e.g.,
complexity or difficulty) is na¨ıve (p. 175).

Bloom’s Taxonomy in Mathematics Education B65

achievement tests and . . . entrance examinations,” Bloom’s Taxonomy in Mathematics Education,
including those in mathematics, were analyzed for Table 1 TIMSS 2003 mathematics framework (cogni-
the distribution of test items across the six catego- tive domains)
ries of Taxonomy (Chung 1994, p. 165). Since its
inception in 1958, the International Association for TIMSS math cognitive Subcategories B
the Evaluation of Educational Achievement has domains Recall
used the Taxonomy to support curriculum analysis, Knowing facts and Recognize/identify
test construction, and data analysis, which precipi- procedures Compute
tated its widespread use internationally (Lewy and Use tools
Ba´thory 1994, p. 147). A familiar international Using concepts Know
achievement test to most mathematics educators is Classify
the Trends in Mathematics and Science Study Solving routine problems Represent
(TIMSS). The TIMSS framework for mathematics Formulate
(Mullis et al. 2003) includes four cognitive domains Reasoning Distinguish
along with several subcategories (Table 1): Select
Model
When taking into account both the TIMSS Interpret
domains and subcategories, several similarities Apply
are found with respect to Bloom’s Taxonomy: Verify/check
(a) the hierarchical representation of knowledge Hypothesize/conjecture/
to more complex forms of mathematical reason- predict
ing, (b) a large base of knowledge-related Analyze
subcategories in the first two TIMSS domains, Evaluate
(c) application in the Taxonomy is synonymous Generalize
with the TIMSS domain solving routine Connect
problems, and (d) the Taxonomy domains of Synthesize/integrate
analysis, synthesis, and evaluation are all named Solve nonroutine problems
in the reasoning domain. Even though Bloom’s Justify/prove
Taxonomy is not explicitly named in the
narrative for the TIMSS framework, it is evident reasoning goals of reproduction, connections,
that the Taxonomy influenced the organization and analysis as a horizontal set of mathematical
and subcategories of the TIMSS framework. competencies. Yet, in spite of the various ways in
This serves as one example, although there are which cognitive domains or competencies are
many, of the ways in which the Taxonomy has represented, results from studies of teachers’
permeated the way evaluation in mathematics classroom assessment practices suggest that
education is conceived and communicated. the general perception of mathematics teachers
is that knowledge of skills and procedures is
With respect to Postlethwaite’s summary of a prerequisite for student engagement in any of
major criticisms, the TIMSS framework does the other cognitive domains (Dekker and Feijs
caution the reader in incorrectly perceiving 2005; Webb 2012).
these four domains as hierarchical or organized
as a lockstep sequence. Mullis et al. (2003) state, One of the more outspoken critics of Bloom’s
“cognitive complexity should not be confused Taxonomy was the Dutch mathematician Hans
with item difficulty. For nearly all of the cogni- Freudenthal, who was noteworthy for his
tive skills listed, it is possible to create relatively contributions to both mathematics and mathemat-
easy items as well as very challenging items” ics education. By the mid-1970s, Freudenthal had
(p. 25). Likewise, to counter the perception that argued that the simplification of reasoning into
reasoning goals are hierarchical, the Mathematics the taxonomic categories had a detrimental effect
Framework for the Program for International on test development. As summarized by Marja
Student Assessment (OECD 2003) organized the

B 66 Bloom’s Taxonomy in Mathematics Education

Bloom’s Taxonomy in Levels of Thinking Level III Assessment
Mathematics Education, analysis Pyramid
Fig. 2 Dutch assessment
pyramid Level II Over time,
connections assessment

questions
should “fill”
the pyramid.

Level I
reproduction

algebra geometry difficult
easy Questions Posed
Domains number staptriosbticasbi&lity
of Mathematics

van den Heuvel-Panhuizen (1996), “In a nutshell, Contemporary Classroom Applications
Bloom sees the capacity to solve a given problem A primary motivation in publishing and dissem-
as being indicative of a certain level, while, inating Bloom’s Taxonomy was the need to
in Freudenthal’s eyes, it is the way in which advance the design of achievement measures to
the student works on a problem that determines assess more than recall of skills, facts, and
the level. The latter illustrates this viewpoint procedures. A similar argument could be made
using the following example: for investigating different examples of mathe-
A child that figures out 8 + 7 by counting 7 further matical reasoning with teachers. In one 3-year
study conducted by Webb (2012) with middle-
from 8 on the abacus, acts as it were on a senso- school mathematics teachers, analyses of over
motoric level. The discovery that 8 + 7 is 10,000 assessment tasks used by 19 teachers
simplified by 8 + (2 + 5) ¼ (8 + 2) + 5 witnesses revealed that greater than 85 % of the tasks
a high comprehension level. Once this is assessed knowledge of skills and procedures.
grasped, it becomes mere knowledge of the
method; as soon as the child has memorized To motivate teachers to use tasks assessing
8 + 7 ¼ 15, it is knowledge of facts. At the a broader range of mathematical reasoning
same moment figuring out 38 + 47 may still goals, teachers categorized the assessment tasks
require comprehension; later on, knowledge of they used using the Dutch assessment pyramid
method can suffice; for the skilled calculator it (Fig. 2; Shafer and Foster 1997; adapted from
is mere knowledge of facts” (Freudenthal 1978, Verhage and de Lange 1997).
p. 91, as cited in van den Heuvel-Panhuizen
1996, p. 21). Even though the pyramid hints at the triangular
At issue in Freudenthal’s remarks are not the representation of Bloom’s Taxonomy in Fig. 1, the
categories themselves, but the way in which additional dimension of Questions Posed (i.e.,
a taxonomy implies levels, orders of sophistica- from easy to difficult) illustrates that questions
tion, and artificially imposed limits on educators’ that elicit student reasoning at different levels are
perceptions of children’s mathematical reasoning not necessarily more difficult. This was the identi-
(also see Kreitzer and Madaus 1994). cal argument made in the TIMSS framework. This
work has since been extended into the design of

Bloom’s Taxonomy in Mathematics Education B67

professional development activities that support education, part II. University of Chicago Press, B
teacher change in classroom assessment (Her and Chicago, pp 164–173
Webb 2005; Webb 2009). As Black and Wiliam Dekker T, Feijs E (2005) Scaling up strategies for change:
(1998) and Hattie and Timperley (2007) meta- change in formative assessment practices. Assess Educ
analyses have given greater attention, respectively, 17(3):237–254
to formative assessment and instructional Freudenthal H (1978) Weeding and sowing. Preface to
feedback, there will be a continued need among a science of mathematical education. Reidel,
mathematics educators to communicate goals for Dordrecht
student learning. Over 50 years ago, Bloom’s Hattie J, Timperley H (2007) The power of feedback. Rev
Taxonomy offered a compelling and influential Educ Res 77(1):81–112
example to address this need. Her T, Webb DC (2005) Retracing a path to assessing for
understanding. In: Romberg TA (ed) Standards-based
Cross-References mathematics assessment in middle school: rethinking
classroom practice. Teachers College Press,
▶ Abstraction in Mathematics Education New York, pp 200–220
▶ Critical Thinking in Mathematics Education Kreitzer AE, Madaus GF (1994) Empirical investigations
▶ Deductive Reasoning in Mathematics of the hierarchical structure of the taxonomy. In:
Anderson LW, Sosniak LA (eds) Bloom’s taxonomy:
Education a forty-year retrospective, Ninety-third yearbook of
▶ Mathematical Knowledge for Teaching the national society for the study of education, part
▶ Mathematical Modelling and Applications in II. University of Chicago Press, Chicago, pp 64–81
Lewy A, Ba´thory Z (1994) The taxonomy of educational
Education objectives in continental Europe, the Mediterranean,
▶ Mathematical Proof, Argumentation, and and the middle east. In: Anderson LW, Sosniak LA
(eds) Bloom’s taxonomy: a forty-year retrospective,
Reasoning Ninety-third yearbook of the national society for the
▶ Mathematics Classroom Assessment study of education, part II. University of Chicago
▶ Problem Solving in Mathematics Education Press, Chicago, pp 146–163
▶ Questioning in Mathematics Education Mullis IVS, Martin MO, Smith TA, Garden RA, Gregory
KD, Gonzalez EJ, Chrostowski SJ, O’Connor KM
References (2003) TIMSS assessment frameworks and specifica-
tions, 2nd edn. International Association for the
Anderson LW, Krathwohl DR, Airasian PW, Cruikshank Evaluation of Educational Achievement, Chestnut Hill
KA, Mayer RE, Pintrich PR, Raths J, Wittrock MC OECD (2003) The PISA 2003 assessment framework.
(2001) A taxonomy for learning, teaching, and Mathematics, reading, science and problem solving
assessing: a revision of Bloom’s taxonomy of knowledge and skills. OECD, Paris
educational objectives. Longman, New York Office of Community Engagement and Service
(2012) Models and theories. Miami University,
Black P, Wiliam D (1998) Assessment and classroom Oxford, Ohio. Retrieved from: http://www.units.
learning. Assess Educ 5(1):7–74 muohio.edu/servicelearning/node/316
Postlethwaite TN (1994) Validity vs. utility: personal
Bloom BS (1994) Reflections on the development and use experiences with the taxonomy. In: Anderson LW,
of the taxonomy. In: Anderson LW, Sosniak LA (eds) Sosniak LA (eds) Bloom’s taxonomy: a forty-year
Bloom’s taxonomy: a forty-year retrospective, retrospective, Ninety-third yearbook of the national
Ninety-third yearbook of the national society for the society for the study of education, part II. University
study of education, part II. University of Chicago of Chicago Press, Chicago, pp 174–180
Press, Chicago, pp 1–27 Shafer MC, Foster S (1997) The changing face of
assessment. Princ Pract Math Sci Educ 1(2):1–8
Bloom BS, Engelhart MD, Furst EJ, Hill WH, Sosniak LA (1994) The taxonomy, curriculum, and their
Krathwohl DR (eds) (1956) Taxonomy of educational relations. In: Anderson LW, Sosniak LA (eds)
objectives: the classification of educational goals. Hand- Bloom’s taxonomy: a forty-year retrospective,
book I: cognitive domain. David McKay, New York Ninety-third yearbook of the national society for the
study of education, part II. University of Chicago
Chung BM (1994) The taxonomy in the Republic of Press, Chicago, pp 103–125
Korea. In: Anderson LW, Sosniak LA (eds) Bloom’s van den Heuvel-Panhuizen M (1996) Assessment and
taxonomy: a forty-year retrospective, Ninety-third realistic mathematics education. CD-b Press, Center
yearbook of the national society for the study of for Science and Mathematics Education, Utrecht.
Retrieved from: http://igitur-archive.library.uu.nl/dis-
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Verhage H, de Lange J (1997) Mathematics education and In: Proceedings of the international congress of
assessment. Pythagoras 42:14–20 mathematics education (Topic Study Group 33), Seoul
Webb NL (1996) Criteria for alignment of expectations
Webb DC (2009) Designing professional development and assessments in mathematics and science educa-
for assessment. Educ Des. 1(2):1–26. Retrieved tion. Research monograph no 6. National Institute for
from: http://www.educationaldesigner.org/ed/volume1/ Science Education, University of Wisconsin-Madison
issue2/article6 and Council of Chief State School Officers, Washing-
ton, DC. Retrieved from: http://facstaff.wcer.wisc.
Webb DC (2012) Teacher change in classroom assess- edu/normw/WEBBMonograph6criteria.pdf
ment: the role of teacher content knowledge in the
design and use of productive classroom assessment.

C

Calculus Teaching and Learning limiting processes, which intrinsically contain
changing quantities. The differential and integral
Ivy Kidron calculus is based upon the fundamental concept
Department of Applied Mathematics, Jerusalem of limit. The mathematical concept of limit is
College of Technology, Jerusalem, Israel a particularly difficult notion, typical of the kind
of thought required in advanced mathematics.

Keywords Characteristics

Calculus key concepts; Intuitive representations; Calculus Curriculum
Formal definitions; Intuition of infinity; Notion of There have been efforts in many parts of the world
limit; Cognitive difficulties; Theoretical dimen- to reform the teaching of calculus. In France, the
sions; Epistemological dimension; Research in syllabus changed in the 1960s and 1970s due to the
teaching and learning calculus; Role of technol- influence of the Bourbaki group. The limit concept
ogy; Visualization; Coordination between semi- with its rigorous basis has penetrated even into the
otic registers; Role of historical perspective; school curriculum: in 1972, the classical definition
Sociocultural approach; Institutional approach; of the derivative in terms of the limit of a quotient
Teaching practices; Role of the teacher; Transi- of differences was introduced. Another change
tion between secondary school and university occurred in the French calculus curriculum in
1982, this time influenced by the findings of math-
Definition: What Teaching and Learning ematics education research, and the curriculum
Calculus Is About? focused on more intuitive approaches. As
a result, the formalization of the limit has been
The differential and integral calculus is omitted at secondary school. This is the situation
considered as one of the greatest inventions in in most countries today: at high school level, there
mathematics. Calculus is taught at secondary is an effort to develop a first approach to calculus
school and at university. Learning calculus concepts without relying on formal definitions and
includes the analysis of problems of changes proofs. An intuitive and pragmatic approach to
and motion. Previous related concepts like the calculus at the high senior level at school
concept of a variable and the concept of function (age 16–18) precedes the formal approach intro-
are necessary for the understanding of calculus duced at university.
concepts. However, the learning of calculus
includes new notions like the notion of limit and At university level, calculus is among the more
challenging topics faced by new undergraduates.

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

C 70 Calculus Teaching and Learning

In the United States, the calculus reform move- consider some facets of the dynamic interaction
ment took place during the late 1980s. The recom- between formal and intuitive representations as
mendation was that calculus courses should they were discussed in these early studies. We
address fewer topics in more depth, and students encounter the first expression of the dynamic
should learn through active engagement with interaction between intuition and formal reason-
the material. The standard course syllabus ing in the terms concept definition and concept
was revised, and new projects arose which image. For example, the intuitive thinking,
incorporated technology into instruction. the visual intuitions, and verbal descriptions of
the limit concept that precede its definition are
In most countries, the transition towards more necessary for understanding the concept. How-
formal approach that takes place at university is ever, research on learning calculus demonstrates
accompanied with conceptual difficulties. that there exists a gap between the mathematical
definition of the limit concept and the way one
Early Research in Learning Calculus: The perceives it. In this case, we may say that there is
Cognitive Difficulties a gap between the concept definition and the
The cognitive difficulties that accompany the concept image (Tall and Vinner 1981; Vinner
learning of central notions like functions, limit, 1983). Vinner also found that students’ intuitive
tangent, derivative, and integral at the different ideas of the tangent to a curve are in conflict
stages of mathematics education are well with the formal definition. This observation
reported in the research literature on calculus might explain students’ conceptual difficulties
learning. These concepts are key concepts that to visualize a tangent as the limiting case
appear and reappear in different contexts in of a secant.
calculus. The students meet some of these central
topics at school, then the same topics appear Conceptual problems in learning calculus are
again, with a different degree of depth at univer- also related to infinite processes. Research dem-
sity. We might attribute the high school students’ onstrates that some of the cognitive difficulties
cognitive difficulties to the fact that the notions that accompany the understanding of the concept
were presented to them in an informal way. In of limit might be a consequence of the learners’
other words, we might expect that the difficulties intuition of infinity. Fischbein et al. (1979)
will disappear when the students will learn the observed that the natural concept of infinity is
formal definition of the concepts. Undergraduate the concept of potential infinity, for example,
mathematics education research suggests other- the non-limited possibility to increase an interval
wise. The cognitive difficulties that accompany or to divide it. The actual infinity, for example,
the key concepts in calculus are well described in the infinity of the number of points in a segment,
Sierpinska (1985); Davis and Vinner (1986); the infinity of real numbers as existing, as given
Cornu (1991); Williams (1991); Tall (1992), as is, according to Fischbein, more difficult to grasp
well as in the book Advanced mathematical think- and leads to contradictions. For example, “If
ing edited by Tall (1991). The main source of one has 1/3 it is easy to accept the equality
difficulty resides in the fact that many students’ 1/3 ¼ 0.33. . . The number 0.333. . .represents
intuitive ideas are in conflict with the formal a potential (or dynamic) infinity. On the other
definition of the calculus concepts like the notion hand, students questioned whether 0.333. . . is
of limit. equal to 1/3 or tends to 1/3 answer usually that
0.333. . .tends to 1/3.”
In these early researches on learning
calculus, the theoretical dimensions are essen- Among the theoretical constructs that accom-
tially cognitive and epistemological. The cogni- pany the early strands in research on learning
tive difficulties that accompany the learning of calculus, we mention the process-object duality.
the key concepts in calculus like the limit concept The lenses offered by this framework highlight
are inherent to the epistemological nature of students’ dynamic process view in relation to
the mathematics domain. In the following we concepts such as limit and infinite sums and

Calculus Teaching and Learning C71

help researchers to understand the cognitive a rapid succession of new ideas for use in C
difficulties that accompany the learning of the calculus and its teaching. Calculus uses numeri-
limit concept. Gray and Tall (1994) introduced cal calculations, symbolic manipulations, and
the notion of procept, referring to the manner in graphical representations, and the introduction
which learners cope with symbols representing of technology in calculus allows these different
both mathematical processes and mathematical registers. Researches on the role of technology in
concepts. Function, derivative, integral, and the teaching and learning calculus are described, for
fundamental limit notion are all examples of example, in Artigue (2006); Robert and Speer
procepts. The limit concept is a procept: the (2001), and in Ferrera et al. in the 2006 handbook
same notation represents both the process of of research on the psychology of mathematics
tending to the limit and also the value of the limit. education (pp. 256–266). In the study by Ferrera
et al., some researches that relate to using a CAS
Research and Alternative Approaches to towards the conceptualization of limit are
Teaching and Learning Calculus described. For example, Kidron and Zehavi use
Different directions of research were investigated symbolic computation and dynamic graphics to
in the last decades. The use of technology offered enhance students’ ability for passing from visual
a new mean in the effort to overcome some of the interpretation of the limit concept to formal
conceptual difficulties: the power of technology reasoning. In this research a sort of balance
is particularly important to facilitate students’ between the conception of an infinite sum as
work with epistemological double strands like a process and as an object was supported by the
discrete/continuous and finite/infinite. Visualiza- software. The research by Kidron as reported in
tion and especially dynamic graphics were also the study by Ferrera et al. (2006) describes some
used. Some researchers based their research on situations in which the combination of dynamic
the historical development of the calculus. Other graphics, algorithms, and historical perspective
researchers used additional theoretical lenses enabled students to improve their understanding
that include the sociocultural approach, the of concepts such as limit, convergence, and the
institutional approach, or the semiotic approach. quality of approximation. Most researches offer
In the following we relate to these different an analysis of teaching experiments that promote
directions of research. the conceptual understanding of key notions like
limits, derivatives, and integral. For example,
The Role of Technology in a research project by Artigue (2006), the
A key aspect of nearly all the reform projects has calculator was used towards conceptualization
been the use of graphics calculators, or computers of the notion of derivative. One of the aims of
with graphical software, to help students develop the project was to enable grade 11 students to
a better intuitive understanding. Since learning enter the interplay between local and global
calculus includes the analysis of changing points of view on functional objects.
quantities, technology has a crucial role in
enabling dynamic graphical representations and Thompson (1994) investigated the concept of
animations. Technology was first incorporated as rate of change and infinitesimal change which are
support for visualization and coordination central to understanding the fundamental theo-
between semiotic registers. The possibility of rem of calculus. Thompson’s study suggests that
computer magnification of graphs allows the students’ difficulties with the theorem stem from
limiting process to be implicit in the computer impoverished concepts of rate of change. In the
magnification, rather than explicit in the limit last two decades, Thompson published several
concept. In his plenary paper, Dreyfus (1991) studies which demonstrate that a reconstruction
analyzed the powerful role for visual reasoning of the ideas of calculus is made possible by its
in learning several mathematical concepts and uses of computing technology. The concept of
processes. With the new technologies there was accumulation is central to the idea of integration
and therefore is at the core of understanding many

C 72 Calculus Teaching and Learning

ideas and applications in calculus. Thompson which the calculus concepts were developed and
et al. (2013) describe a course that approaches then defined, appropriate historically inspired
introductory calculus with the aim that students teaching sequences were elaborated.
build a reflexive relationship between concepts of
accumulation and rate of change, symbolize that Recent approaches in learning and teaching
relationship, and then extend it. In a first phase, calculus refer to the social dimension like the
students develop accumulation functions from approach to teach calculus called “scientific
rate of change functions. In the first phase, debate” which is based on a specific form of
students “restore” the integral to the fundamental discussion among students regarding the validity
theorem of calculus. In the second phase, students of theorems. The increasing influence taken by
develop rate of change functions from accumula- sociocultural and anthropological approaches
tion functions. The main idea is that accumulation towards learning processes is well expressed in
and rate of change are never treated separately: research on learning and teaching calculus. Even
the fundamental theorem of calculus is present the construct concept image and concept defini-
all the time. Rate is an important, but difficult, tion, which was born in an era where the theories
mathematical concept. Despite more than 20 years of learning were essentially cognitive theories,
of research, especially with calculus students, diffi- was revisited (Bingolbali and Monaghan 2008)
culties are still reported with this concept. and used in interpreting data in a sociocultural
study. This was done in a research which inves-
Tall (2010) reflects on the ongoing develop- tigated students’ conceptual development of the
ment of the teaching and learning calculus since derivative with particular reference to rate of
his first thinking about the calculus 35 years ago. change and tangent aspects.
During these years, Tall’s research described
how the computer can be used to show dynamic In more recent studies, the role of different
visual graphics and to offer a remarkable power theoretical approaches in research on learning
of numeric and symbolic computation. As calculus is analyzed. Kidron (2008) describes
a consequence of the cognitive difficulties that a research process on the conceptualization of
accompany the conceptual understanding of the the notion of limit by means of the discrete
key notions in calculus, Tall’s quest is for continuous interplay. The paper reflects many
a “sensible approach” to the calculus which years of research on the conceptualization of the
builds on the evidence of our human senses and notion of limit, and the focus on the complemen-
uses these insights as a meaningful basis for later tary role of different theories reflects the
development from calculus to analysis and even evolution of this research.
to a logical approach in using infinitesimals.
Reflecting on the many years in which reform of The Role of the Teacher
calculus teaching has been considered around the In the previous section, different educational
world and the different approaches and reform environments were described. Educational envi-
projects using technology, Tall points out that ronments depend on several factors, including
what has occurred is largely a retention of tradi- teaching practices. As mentioned by Artigue
tional calculus ideas now supported by dynamic (2001), reconstructions have been proved to play
graphics for illustration and symbolic manipula- a crucial role in calculus especially at the
tion for computation. secondary/tertiary transition. Some of these recon-
structions deal with mathematical objects already
The Role of Historical Perspective and Other familiar to students before the teaching of calculus
Approaches at university. In some cases, reconstructions
The idea to use a historical perspective in result from the fact that only some facets of
approaching calculus was also demonstrated in a mathematical concept can be introduced at the
other studies not necessarily in a technological first contact with it. The reconstruction cannot
environment. Taking into account the long way in result from a mere presentation of the theory and
formal definitions. Research shows that teaching

Calculus Teaching and Learning C73

practices underestimate the conceptual difficulties located in several countries (Brazil, Canada, C
associated with this reconstruction and that teach- Denmark, France, Israel, Tunisia) and use different
ing cannot leave the responsibility for most of the frameworks. Some have shown that calculus
corresponding reorganization to students. conflicts that emerged from experiments with
first-year students could have their roots in
Research shows that alternative strategies can a limited understanding of the concept of
be developed fruitfully especially with the help of function, as well as suggesting the need for
the technology but successful integration of a more intensive exploration of the dynamical
technology at a large-scale level is still a major nature of the differential calculus. Results of the
problem (Artigue 2010). Technology cannot be survey suggest that there is some room for
considered only as a kind of educational assistant. improvement in school preparation for university
It was demonstrated how it deeply shapes what study of calculus.
we learn and the way we learn it.
The transition to advanced calculus as taught
Artigue points out the importance of the at the university level has been extensively inves-
teacher’s dimension. Kendal and Stacey (2001) tigated within the Francophone community, with
describe teachers’ practices in technology-based the research developed displaying a diversity of
mathematics lessons. The integration of technol- approaches and themes but a shared vision of the
ogy into mathematics teachers’ classroom prac- importance to be attached to epistemological and
tices is a complex undertaking (Monaghan 2004; mathematical analyses.
Lagrange 2013). Monaghan wrote and cowrote
a number of papers in which teachers’ activities Analyzing the transition between the
in using technology in their calculus classrooms secondary school and the university, French
were analyzed but there were still difficulties that researchers reflect on approaches to teaching
the teachers had experienced in their practices and learning calculus in which the consideration
that were difficult to explain in a satisfactory of sociocultural and institutional practices plays
manner. Investigating the reasons for the discrep- an essential role. These approaches offer comple-
ancy between the potentialities of technology in mentary insights to the understanding of teaching
learning calculus and the actual uses in the class- and learning calculus. The theoretical influence
room, Lagrange (2013) searches for theoretical of the theory of didactic situations which led to
frameworks that could help to focus on the a long-term Francophone tradition of didactical
teacher using technology; the research on the engineering research has been designed in the last
role of the teacher strengthened the idea of decade to support this transition from secondary
a difficult integration in contrast with research calculus to university analysis.
centered on epistemological and cognitive aspects.
An activity theory framework seems helpful to New Directions of Research
give insight on how teachers’ activity and profes- New directions of research in teaching and learning
sional knowledge evolve during the use of technol- calculus were investigated in the last decades. We
ogy in teaching calculus. observe the need for additional theoretical lenses as
well as a need to link different theoretical frame-
The Transition Between Secondary and works in the research on learning and teaching
Tertiary Education calculus. In particular, we observe the need to add
A detailed analysis of the transition from additional theoretical dimensions, like the social
secondary calculus to university analysis is offered and cultural dimensions, to the epistemological
by Thomas et al. (2012). A number of researchers analyses that were done in the early research. In
have studied the problems of the learning of calcu- some cases, we notice the evolution of research
lus in the transition between secondary school and during many years with the same researchers
university. Some of these studies focus on the facing the challenging questions concerning the
specific topics of real numbers, functions, limits, cognitive difficulties in learning calculus. The
continuity, and sequences and series. They were questions are still challenging.

C 74 Calculus Teaching and Learning

The theoretical dimension is essential for a mathematical instrument. Springer, New York,
research on calculus teaching and learning, but pp 231–294
we should not neglect the practice. As pointed out Artigue M (2010) The Future of Teaching and Learning
by Robert and Speer (2001), there are some Mathematics with Digital Technologies. In: Hoyles C,
efforts towards a convergence of theory-driven Lagrange JB (eds) Mathematics Education and
and practice-driven researches. Further research Technology – Rethinking the Terrain. The 17th ICMI
on how to consider meaningfully theoretical and Study. Springer, New York, pp 463–476
pragmatic issues is indicated. Bingolbali E, Monaghan J (2008) Concept image
revisited. Educ Stud Math 68:19–35
As mentioned earlier, reconstructions have been Cornu B (1991) Limits. In: Tall D (ed) Advanced mathe-
proved to play a crucial role in calculus, essentially matical thinking. Kluwer, Dordrecht, pp 153–166
these reconstructions that deal with mathematical Davis RB, Vinner S (1986) The notion of limit: some
objects already familiar to students before the seemingly unavoidable misconception stages. J Math
teaching of calculus. Further research should Behav 5:281–303
underline the important role of teaching practices Dreyfus T (1991) On the status of visual reasoning in
for such successful reorganization of previous mathematics and mathematics education. In:
related concepts towards the learning of calculus. Furinghetti F (ed) Proceedings of the 15th PME
international conference, Assisi, Italy, vol 1, pp 33–48
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Collaborative Learning in Mathematics Education C75

Tall D, Vinner S (1981) Concept image and concept Three dimensions seem to define collaborative C
definition in mathematics with particular reference learning (CL) and help distinguish among its
to limit and continuity. Educ Stud Math 12: many different models: the structure of the
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rewards), the teacher and student roles, and the
Thomas MOJ, De Freitas Druck I, Huillet D, Ju MK, types of tasks.
Nardi E, Rasmussen C, Xie J (2012) Key mathematical
concepts in the transition from secondary to university. The CL structure defines how student groups
In: Pre-proceedings of the 12th international congress are formed (usually by teacher assignment) and
on mathematical education (ICME-12) Survey how group members are expected to interact.
Team 4, pp 90–136, Seoul. http://www.math.auck- Research generally recommends mixed ability
land.ac.nz/ thomas/ST4.pdf grouping. Carefully designed assessment and
reward structures document student learning
Thompson PW (1994) Images of rate and operational and provide incentives for students to work
understanding of the fundamental theorem of calculus. productively together. All models of CL
Educ Stud Math 26(2–3):229–274 involve group accountability, but some models
also include some individual rewards, while
Thompson PW, Byerley C, & Hatfield N (2013) others may pit groups against each other in
A conceptual approach to calculus made possible a competitive reward structure.
by technology. Computers in the Schools, 30:
124–147 The teacher’s role is to determine the CL
structure and task, then serve as facilitator. In
Vinner S (1983) Concept definition, concept image and some CL models, students are assigned specific
the notion of function. Int J Math Educ Sci Technol group roles (e.g., recorder, calculator); other
14:239–305 models require students to tackle portions of the
task independently, then pool their efforts toward
Williams S (1991) Models of limits held by college a common solution. Individual accountability
calculus students. J Res Math Educ 22(3):219–236 requires that each student be responsible not
only for his/her own learning but also for sharing
Collaborative Learning in the burden for all group members’ learning.
Mathematics Education
CL tasks must be carefully chosen: amenable
Paula Lahann and Diana V. Lambdin to group work and designed so that success
School of Education, Indiana University, depends on contributions from all group
Bloomington, IN, USA members. Particular attention to task difficulty
ensures all students can engage at an appropriate
Keywords level.

Collaborative learning; Cooperative learning; CL is grounded in a social constructivist
Project-based learning model of learning (Yackel et al. 2011). Some
CL models involve peer tutoring (e.g., Student
Collaborative learning (CL) involves a team of Team Learning: Slavin 1994). In the more
students who learn through working together common investigative CL models (e.g., Learning
to share ideas, solve a problem, or accomplish Together: Johnson and Johnson 1998), the
a common goal. In mathematics education, CL’s emphasis is on learning through problem solving,
popularity surged in the 1980s, but it has since but higher-order skills such as interpretation,
continued to evolve (Artzt and Newman 1997; synthesis, or investigation are also required.
Davidson 1990). The terms collaborative/cooper-
ative learning are often used interchangeably, Project-based learning (PBL) – a twenty-first-
although some claim the former requires century group-investigation CL model – involves
giving students considerable autonomy (more cross-disciplinary, multifaceted, open-ended
appropriate for older students), while the latter tasks, usually set in a real-world context, with
is more clearly orchestrated by the teacher results presented via oral or written presentation.
(appropriate for all ages) (Panitz 1999).

C 76 Communities of Inquiry in Mathematics Teacher Education

PBL tasks often take several weeks because Communities of Inquiry in
students must grapple with defining, delimiting, Mathematics Teacher Education
and planning the project; conducting research;
and determining both the solution and how best Barbara Jaworski
to present it (Buck Institute 2012). A stated Loughborough University, Loughborough,
PBL goal is to help students develop “twenty- Leicestershire, UK
first-century skills” relating to collaboration,
time management, self-assessment, leadership, Keywords
and presentation concurrently with engaging
in critical thinking and mastering traditional Mathematics teacher education; Community;
academic concepts and skills (e.g., Inquiry
mathematics).
Definition
Research has found student learning is
accelerated when students work collaboratively Mathematics teacher education (MTE) consists of
on tasks that are well structured, carefully processes and practices through which teachers or
implemented, and have individual accountability. student teachers learn to teach mathematics. It
There is also evidence that affective outcomes, involves as participants, primarily, student teachers,
such as interest in school, respect for others, teachers, and teacher educators; other stakeholders
and self-esteem, are also positively impacted such as school principals or policy officials with
(Slavin 1992). regulatory responsibilities can be involved to dif-
fering degrees. Thus a community in MTE consists
References of people who engage in these processes and prac-
tices and who have perspectives and knowledge in
Artzt A, Newman CM (1997) How to use cooperative what it means to learn and to educate in mathemat-
learning in the mathematics class, 2nd edn. ics and an interest in the outcomes of engagement.
National Council of Teachers of Mathematics, An inquiry community, or community of inquiry, in
Reston MTE is a community which brings inquiry into
practices of teacher education in mathematics –
Buck Institute (n.d.). “What is PBL?” Project based where inquiry implies questioning and seeking
learning for the 21st century. http://www.bie.org/. answers to questions, problem solving, exploring,
Accessed 24 July 2012 and investigating – and in which inquiry is the basis
of an epistemological stance on practice, leading to
Davidson N (ed) (1990) Cooperative learning in mathe- “metaknowing” (Wells 1999; Jaworski 2006). The
matics: a handbook for teachers. Addison-Wesley, very nature of a “community” of inquiry rooted in
Menlo Park communities of practice (Wenger 1998) implies
a sociohistorical frame in which knowledge grows
Johnson DW, Johnson R (1998) Learning together and and learning takes place through participation and
alone: cooperative, competitive, and individualistic dialogue in social settings (Wells 1999).
learning, 5th edn. Allyn and Bacon, Boston
Characteristics
Panitz T (1999) Collaborative versus cooperative
learning: A comparison of the two concepts which Rather than seeing knowledge as objective,
will help us understand the underlying nature of pre-given and immutable (an absolutist stance:
interactive learning. ERIC Document Reproduction
Service No. ED448443

Slavin R (1992) Research on cooperative learning: consensus
and controversy. In: Goodsell AS, Maher MR, Tinto
V (eds) Collaborative learning: a sourcebook for higher
education. National Center on Postsecondary Teaching,
Learning, Assessment, University Park

Slavin R (1994) Cooperative learning: theory,
research, and practice, 2nd edn. Allyn and Bacon,
Boston

Yackel E, Gravemeijer K, Sfard A (eds) (2011) A journey
in mathematics education research: insights from the
work of Paul Cobb. Springer, Dordrecht

Communities of Inquiry in Mathematics Teacher Education C77

Ernest 1991) with learning as a gaining of such example, the design of inquiry-based mathemati- C
knowledge and teaching as a conveyance of cal tasks for students – their critical attitude to their
knowledge from one who knows to one who practice generates new knowledge in practice
learns, an inquiry stance sees knowledge as and new practice-based understandings (Jaworski
fluid, flexible and fallible (Ernest 1991). These 2006).
positions apply to mathematical knowledge and
to knowledge in teaching: teachers of mathemat- In a community of practice, Wenger (1998)
ics need both kinds of knowledge. Knowledge is suggests that “belonging” to the community
seen variously as formal and external, consisting involves “engagement,” “imagination,” and “align-
of general theories and research-based findings ment.” Participants engage with the practice, use
to be gained and put into practice; or as craft imagination in weaving a personal trajectory in the
knowledge, intrinsic to the knower, often tacit, practice and align with norms and expectations
and growing through action, engagement, and within the practice. The transformation of
experience in practice; or yet again as growing a community of practice to a community of
through inquiry in practice so that the knower and inquiry requires participant to look critically at
the knowledge are inseparable. Cochran-Smith their practices as they engage with them, to ques-
and Lytle (1999) call these three ways of concep- tion what they do as they do it, and to explore new
tualizing knowledge as knowledge for teaching, elements of practice. Such inquiry-based forms of
in teaching, and of teaching. With regard engagement have been called “critical alignment”
to knowledge-of-teaching, they use the term (Jaworski 2006). Critical alignment is a necessity
“inquiry as stance” to describe the positions for developing an inquiry way of being within
teachers take towards knowledge and its relation- a community of inquiry.
ships towards practice. This parallels the notion
of “inquiry as a way of being” in which teachers Like teachers, teacher educators in mathe-
take on the mantle of inquiry as central to how matics (sometimes called didacticians, due to
they think, act, and develop in practice and their practices in relation to the didactics of
encourage their students to do so as well mathematics) are participants in communities
(Jaworski 2006). of inquiry in which they too need to develop
knowledge in practice through inquiry. Their
An inquiry community in mathematics teacher practices are different from those of teachers,
education therefore involves teachers (including but there are common layers of engagement in
student teachers who are considered as less expe- which teachers and teacher educators side by
rienced teachers) engaging together in inquiry into side explore practices in learning and teaching
teaching processes to promote students’ learning of mathematics in order to develop practice and
of mathematics and, moreover, involving students generate new knowledge. Teacher educators
in inquiry in mathematics. The main purpose of also have responsibilities in linking theoretical
inquiry is to call into question aspects of a source perspectives to development of practice and to
(such as mathematics) which encourages a deeper engaging in research formally for generation of
engagement as critical questioning takes place and academic knowledge. Thus it is possible to see
knowledge grows within the community. When three (nested) layers of inquiry community in
the source is mathematics, inquiry in mathematics generating new understandings of teaching to
allows students to address mathematical questions develop the learning of mathematics: inquiry
in ways that seek out answers and lead to new by students into mathematics in the classroom,
knowledge. Thus mathematics itself becomes inquiry by teachers into the processes and prac-
accessible, no longer perceived as only right or tices of creating mathematical learning in class-
wrong, and its revealed fallibility is an encourage- rooms, and inquiry by teacher educators into the
ment to the learner to explore further and under- processes by which teachers learn through
stand more deeply. Similarly as teachers explore inquiry and promote the mathematical learning
into aspects of mathematics teaching – for of their students (Jaworski and Wood 2008)
(Fig. 1).