Encyclopedia of Mathematics Education

T 586 Teacher Education Development Study-Mathematics (TEDS-M)

Swars SL, Smith SZ, Smith ME, Hart LC (2009) Definition
A longitudinal study of effects of a developmental
teacher preparation program on elementary prospec- TEDS-M is the first empirical cross-national
tive teachers’ mathematics beliefs. J Math Teach Educ study of teacher preparation to collect data on
12(1):47–66 the organization, curriculum, processes, and
outcomes of teacher education from national
Thompson AG (1992) Teachers’ beliefs and conceptions: probability samples of institutions, teaching
a synthesis of the research. In: Grouws DA (ed) staff, and students in 17 countries (Botswana,
Handbook of research on mathematics teaching and Canada, Chile, Chinese Taipei, Georgia, Ger-
learning. Macmillan Library Reference USA/Simon & many, Malaysia, Norway, Oman, Philippines,
Schuster/Prentice Hall International, New York/ Poland, Russian Federation, Singapore, Spain,
London, pp 127–146 Switzerland, Thailand, and the United States).
TEDS-M was designed to focus on the outcomes
Tschannen-Moran M, Woolfolk Hoy A, Hoy WK of the mathematics preparation of teachers at the
(1998) Teacher efficacy: its meaning and measure. primary and lower secondary levels and to serve
Rev Educ Res 68:202–248 as a valuable tool to help inform and develop
mathematics teacher preparation policy for
Wilson SM, Cooney TJ (2002) Mathematics teacher future mathematics teachers.
change and development: the role of beliefs. In:
Leder GC, Pehkonen E, To¨rner G (eds) Beliefs: The TEDS-M study was carried out under the
A hidden variable in mathematics education? Kluwer, aegis of the International Association for the Eval-
Dordrecht, pp 127–148 uation of Educational Achievement (IEA) and
was made possible by a major grant from the US
Teacher Education Development National Science Foundation. The College of
Study-Mathematics (TEDS-M) Education at Michigan State University (MSU)
and the Australian Council of Educational
Maria Tatto Research (ACER) were the joint international
College of Education, Michigan State University, study centers (ISCs) for TEDS-M under the
East Lansing, MI, USA executive direction of Principal Investigator
Maria Teresa Tatto of MSU. To design and carry
Keywords out the study, the ISCs worked in collaboration
with the International Association for the Evalua-
Assessment; Knowledge; Mathematics; Peda- tion of Educational Achievement (IEA) Data
gogy; Teacher education; Botswana; Canada; Processing and Research Center (DPC), the IEA
Chile; Chinese Taipei; Georgia; Germany; Secretariat in Amsterdam, Statistics Canada, and
Malaysia; Norway; Oman; Philippines; Poland; the TEDS-M national research centers in the 17
Russian Federation; Singapore; Spain; participating countries. Together, these teams of
Switzerland; Thailand; United States researchers and institutions conceptualized the
study, designed and administered the instruments,
The source for this entry is: Tatto MT, Schwille J, Senk SL, collected and analyzed the data, and reported the
Ingvarson L, Rowley G, Peck R, Bankov K, Rodriguez M, results.
Reckase M (2012) Policy, practice, and readiness to teach
primary and secondary mathematics in 17 countries. Find- The TEDS-M study findings in detail can be
ings from the IEA teacher education and development study found in Tatto et al. 2012, and in the IEA website
in mathematics (TEDS-M). International Association for [http://www.iea.nl/?id=20] along with additional
the Evaluation of Student Achievement, Amsterdam. The reports, and the publicly available data from the
study was funded by a major grant to MSU from the US study. This entry focuses on the following persis-
National Science Foundation NSF REC 0514431 (M.T. tent questions addressed by TEDS-M: What char-
Tatto, PI), any opinions, findings, and conclusions or rec- acterizes the institutions and the curriculum of
ommendations expressed in this material are those of the teacher education programs? What are the
author(s) and do not necessarily reflect the views of the
National Science Foundation.

Teacher Education Development Study-Mathematics (TEDS-M) T587

characteristics of future primary and secondary to teach (5) no higher than grade 10 or (6) up T
teachers who are expected to teach mathematics? to the end of secondary schooling (Tatto et al.
What are the outcomes of teacher education 2012).
concerning professional teachers’ knowledge
and beliefs in mathematics? Curriculum
In the TEDS-M study participating institutions
Institutions provided detailed information about the aca-
The TEDS-M found that the nature of preservice demic and professional content of their
teacher preparation institutions is diverse within preservice teacher education programs. This
and across countries. There is a wide array of included information about the number of subject
programs residing in public and private institu- areas graduates would be qualified to teach (i.e.,
tions; some in universities and some in colleges specialists versus generalists) and the number of
outside universities. Some offer programs only in hours of instruction allocated to each area.
education, some are comprehensive with regard Regarding specialization, one distinct pattern
to the fields of study offered. Some offer univer- emerged. While most programs prepare future
sity degrees, and some do not. Teacher education primary teachers to teach more than two subjects,
programs are typically categorized according to those preparing future secondary teachers, for the
whether the opportunities to learn that they offer most part, prepare them to teach one or two sub-
are directed at preparing future teachers for pri- jects. Regarding the relative emphasis given to
mary schools or for secondary schools. However, specific areas of the teacher education program –
this categorization proved to be an oversimplifi- as indicated by the number of hours allocated to
cation within the context of TEDS-M and likely each – the data revealed that teacher education
within the larger international context. The terms programs generally offer courses in four
primary and secondary do not mean the same areas: (a) liberal arts, (b) mathematics and
thing from country to country. There is no uni- related content (academic mathematics, school
versal agreement on when primary grades end mathematics, and mathematics pedagogy),
and secondary grades begin. Therefore, programs (c) educational foundations, and (d) pedagogy.
were defined by types, according to their pur- Specifically regarding mathematics-related
poses using two organizational variables – grade courses, TEDS-M found that in general, those
span (the range of school grades for which programs that intend to prepare teachers to
teachers in a program were being prepared to teach higher curricular levels such as lower and
teach) and teacher specialization (whether the upper secondary provide, on average, opportuni-
program was preparing specialist mathematics ties to learn mathematics in more depth than
teachers or generalist teachers). Primary program those programs that prepare teachers for the pri-
types were grouped according to whether mary level. Thus, on average, future lower and
they prepare specialist teachers of mathematics upper secondary teachers had greater opportunity
or generalist teachers and then subdivided to learn mathematics, both at the tertiary level as
into three groups according to the highest well as the school level, than future primary
grade level for which they offer preparation: teachers. The exception to this pattern was
(1) program types that prepare teachers to teach found within the primary mathematics specialist
no higher than grade 4, (2) program types that group where higher opportunity to learn tertiary
prepare teachers to teach no higher than grade 6, mathematics was reported more frequently
and (3) program types that prepare teachers to than within any of the other program groups.
teach no higher than grade 10. The specialist Regarding school mathematics in particular,
teachers of mathematics constituted group (4). preservice teacher education programs in the
At lower secondary level, program types were countries participating in the study included all
placed in two groups, according to whether grad- or a combination of some of the following topics:
uates from those program types would be eligible numbers; measurement; geometry; functions,

T 588 Teacher Education Development Study-Mathematics (TEDS-M)

relations, and equations; data representation, teachers being prepared for lower secondary
probability, and statistics; calculus; and valida- school (known in some countries as “middle
tion, structuring, and abstracting. But these pro- school”) than among those being prepared to
grams typically rationed the quantity and depth of teach Grade 11 and above. Not surprisingly, the
future primary teachers’ opportunities to learn countries with programs providing the most
school level mathematics (with primary teachers comprehensive opportunities to learn challenging
predominantly studying topics such as numbers, mathematics had higher scores in the TEDS-M
measurement, and geometry above any other tests of knowledge. In TEDS-M, primary level
topics). As programs prepare teachers for higher and secondary level teachers in high-achieving
grades, the proportion of areas reported as having countries such as Chinese Taipei, Singapore, and
been studied increases. Importantly, TEDS-M the Russian Federation had significantly more
found that the Asian countries and other countries opportunities than their primary and secondary
whose future teachers did well on the TEDS-M counterparts in the other participating countries
assessments did offer algebra and calculus as part to learn university and school level mathematics.
of future primary and lower secondary teacher This tendency seems to be closely related to the
education. And while the secondary curriculum expectation that primary schools can be staffed
across a large number of countries calls for with generalist teachers, defined in this study as
instruction in basic statistics, the study found teaching three or more subjects. Although this
a general gap in this area in teacher education as assertion may seem reasonable, the question of
reported by future teachers. This variability is how much content knowledge teachers need to
mirrored in the opportunities to learn in the math- teach effectively is still an issue of much debate.
ematics pedagogy domains between primary and The TEDS-M findings signal an opportunity to
lower secondary groups. In other areas TEDS-M examine how these distinct approaches play out
found that opportunity to learn how to teach in practice. If relatively little content knowledge
diverse students was highly variable with many is needed for the lower grades, then a lesser
countries reporting few or no opportunities to emphasis on mathematics preparation and
learn in this domain. Opportunity to learn general nonspecialization can be justified. The key ques-
pedagogy was high among all primary programs tion is whether teachers prepared in this fashion
and most secondary programs. Most programs can teach mathematics as effectively as teachers
preparing future primary teachers provide oppor- with more extensive and deeper knowledge, such
tunities to make connections between what they as that demonstrated by specialist teachers.
learn in their programs and future teaching Although TEDS-M has not provided definitive
practice; but in the secondary program groups conclusions in this regard (this question necessi-
these opportunities were not as prevalent. The tates studying beginning teachers and their influ-
TEDS-M findings regarding overall opportunities ence on student learning), this question is
to learn in mathematics teacher education reflect currently under investigation by a study called
what seems to be a cultural norm in some FIRSTMATH, as a follow-up of TEDS-M, also
countries, namely, that teachers who are expected funded by NSF and based at Michigan State Uni-
to teach in primary – and especially the lower versity (Tatto 2010). What TEDS-M does show is
primary – grades need little in the way of that within countries, future teachers intending to
mathematics content beyond that included in the be mathematics specialists in primary schools had
school curriculum. The pattern among secondary higher knowledge scores on average than their
future teachers is generally characterized by generalist counterparts, and similarly, future
more and deeper coverage of mathematics teachers intending to teach upper secondary
content; however, there was more variability in had higher scores on average than those intending
opportunities to learn mathematics and to teach lower secondary grades (see Tatto
mathematics pedagogy among those future et al. 2012).

Teacher Education Development Study-Mathematics (TEDS-M) T589

The Characteristics of Future Primary and indicated a positive relationship between the T
Secondary Teachers Who Are Expected to strength of quality assurance arrangements and
Teach Mathematics country mean scores in the TEDS-M tests of
The TEDS-M study found that different mathematics content knowledge and mathematics
countries’ policies designed to shape teachers’ pedagogy knowledge. Countries with strong qual-
career trajectories have a very important ity assurance arrangements, such as Chinese Tai-
influence on who enters teacher education and pei and Singapore, scored highest on these
who eventually becomes a teacher. These measures. Countries with weaker arrangements,
policies can be characterized as of two major such as Georgia and Chile, tended to score lower
types (with a number of variations in between): on the two measures of future teacher knowledge.
career-based systems where teachers are recruited These findings have implications for
at a relatively young age and remain in the public policymakers concerned with promoting teacher
or civil service system throughout their working quality. Quality assurance policies and arrange-
lives, and position-based systems where teachers ments can make an important difference to
are not hired into the civil or teacher service but teacher education. These policies can be designed
rather are hired into specific teaching positions to cover the full spectrum, from polices designed
within an unpredictable career-long progression to make teaching an attractive career to policies
of assignments. In a career-based system, there is for assuring that entrants to the profession have
more investment in initial teacher preparation, attained high standards of performance. The
knowing that the education system will likely TEDS-M findings point to the importance of
realize the return on this investment throughout ensuring that policies designed to promote teacher
the teacher’s working life. While career-based quality are coordinated and mutually supportive.
systems have been the norm in many countries, The TEDS-M data shows that countries such as
increasingly the tendency is toward position- Chinese Taipei and Singapore that do well on
based systems. In general, position-based sys- international tests of student achievement, such
tems, with teachers hired on fixed, limited-term as TIMSS, not only ensure the quality of entrants
contracts, are less expensive for governments to to teacher education but also have strong systems
maintain. At the same time, one long-term policy for reviewing, assessing, and accrediting teacher
evident in all TEDS-M countries is that of requir- education providers. They also have strong mech-
ing teachers to have university degrees, thus, anisms for ensuring that graduates meet high stan-
securing a teaching force where all its members dards of performance before gaining certification
have higher education degrees. These policy and full entry to the profession.
changes have increased the individual costs of
becoming a teacher while also increasing the Aside from qualifications, TEDS-M found
level of uncertainty of teaching as a career. that future teachers being prepared to teach at
the primary and secondary school levels were
A major part of TEDS-M involved examining predominantly female, although there were
the participating countries’ policies for assuring more males at the higher levels and in particular
the quality of future teachers. The study found countries. They seemed to come from well-
great variation in these policies, especially with resourced homes, and many reported having
respect to the quality of entrants to teacher access to such possessions as calculators,
education programs, the methods for assessing dictionaries, and DVD players, but not personal
the quality of graduates before they can gain computers – now widely considered essential for
entry to the teaching profession (e.g., periodic professional use. The latter was especially the
formative and summative examinations both writ- case among teachers living in less affluent coun-
ten and oral, a thesis requirement, and others), and tries such as Botswana, Georgia, the Philippines,
at the organizational level, the accreditation of and Thailand. The TEDS-M survey found that a
teacher education programs. The TEDS-M data relatively small proportion of the sample of

T 590 Teacher Education Development Study-Mathematics (TEDS-M)

future teachers who completed the survey did not grade 10, and (6) up to the end of secondary
speak the official language of their country schooling) and by the end of the teacher prepara-
(which was used in the TEDS-M surveys and tion programs, future teachers in some countries
tests) at home. Most future teachers described had substantially greater mathematics content
themselves as above average or near the top of knowledge and mathematics pedagogical content
their year in academic achievement by the end of knowledge than others. On average, future
their upper secondary schooling. Among the primary teachers being prepared as mathematics
reasons the future teachers gave for deciding to specialists had higher MCK and MPCK scores
become teachers, liking working with young than those being prepared to teach as primary
people and wanting to influence the next genera- generalists. Also, on average, future teachers
tion were particularly prevalent. Many believed being prepared as lower and upper secondary
that despite teaching being a challenging job, teachers (e.g., group 6) had higher MCK and
they had an aptitude for it (see Tatto et al. 2012). MPCK scores than those being prepared to be
only lower secondary teachers. In the top-scoring
The Outcomes of Teacher Education: countries within each program group, the major-
Mathematics Professional Knowledge ity of future teachers had average scores on math-
and Beliefs ematics content knowledge and mathematics
Mathematics Content and Mathematics pedagogy content knowledge at or above the
Pedagogy Content Knowledge for Teaching higher anchor points (see Tatto et al. 2012). In
Regarding the mathematics and mathematics ped- countries with more than one program type per
agogy content knowledge of future teachers, the education level, the relative performance on
TEDS-M study provides the first solid evidence, MCK and on MPCK of the future teachers with
based on national samples, of major differences respect to their peers varied. For instance, the
across countries in the (measured) mathematics mean mathematics content knowledge score of
knowledge outcomes of teacher education. The future primary teachers in Poland ranked fourth
answer to the TEDS-M research question about among five countries preparing lower primary
the teaching mathematics knowledge that the generalist teachers, but first among six countries
future primary and secondary teachers had preparing primary mathematics specialists. An
acquired by the end of their teacher education important conclusion of the TEDS-M study is
was clear, for the most part, this knowledge varied that the design of teacher education curricula
considerably among individuals within every can have substantial effects on the level of
country and across countries. The difference in knowledge that future teachers are able to acquire
mean mathematics content knowledge (MCK) via the opportunities to learn provided to them
scores between the highest and lowest-achieving (see Tatto et al. 2012).
country in each primary and secondary program
group was between 100 and 200 score points, or Beliefs
one and two standard deviations. This difference The TEDS-M study assessed beliefs about the
is a substantial one, comparable to the difference nature of mathematics (e.g., mathematics is a set
between the 50th and the 96th percentile in the of rules and procedures, mathematics is a process
whole TEDS-M future teacher sample. Differ- of enquiry), beliefs about learning mathematics
ences in mean achievement across countries in (e.g., through teacher direction, through student
the same program group on mathematics peda- activity), and beliefs about mathematics achieve-
gogical content (MPCK) were somewhat smaller, ment (e.g., mathematics is a fixed ability)
ranging from about 100–150 score points. (Philipp 2007; Staub and Stern 2002). We found
Therefore, within each program group (e.g., pre- that in general, educators and future teachers in all
paring teachers to teach (1) no higher than grade countries were more inclined to endorse the pat-
4, (2) no higher than grade 6, (3) no higher than tern of beliefs described as conceptual in
grade 10, (4) as specialists, (5) no higher than orientation and less inclined to endorse the

Teacher Education Development Study-Mathematics (TEDS-M) T591

pattern of beliefs described as computational or Tatto MT (1998) The influence of teacher education on T
direct-transmission. Several countries showed teachers’ beliefs about purposes of education, roles
endorsement for the belief that mathematics is a and practice. J Teach Educ 49:66–77
set of rules and procedures. The view that
mathematics is a fixed ability was a minority Tatto MT (1999) The socializing influence of normative
one in all countries surveyed, yet its existence is cohesive teacher education on teachers’ beliefs about
still a matter of concern because it implies a less instructional choice. Teach Teach 5:111–134
inclusive approach to teaching mathematics to all
learners. The TEDS-M data shows important Tatto MT (2010) The first five years of mathematics
cross-country differences in the extent to which teaching study (FIRSTMATH). http://firstmath.educ.
such views are held. The program groups within msu.edu/
countries endorsing beliefs consistent with a
computational orientation were generally among Tatto MT, Schwille J, Senk SL, Bankov K, Rodriguez M,
those with lower mean scores on the knowledge Reckase M, Ingvarson L, Rowley G, Peck R (2012)
tests. In some high-scoring countries on our Policy, practice, and readiness to teach primary and
knowledge tests, however, future teachers secondary mathematics in 17 countries. Now available
endorsed the beliefs that mathematics is a set of for download from the TEDS-M website at MSU:
rules and procedures as well as a process of http://teds.educ.msu.edu/. It is likewise available on-
enquiry (see Tatto et al. 2012). The TEDS-M line from the IEA webpage at: http://www.iea.nl/
findings thus showed endorsement for both of (homepage, recent publications)
these conceptions within mathematics teacher
education. This finding suggests the importance Further Reading
for teacher education institutions to find an
appropriate balance on these conceptions when Adler J, Ball D, Krainer K, Lin F-L, Novotna´ J (2005)
designing and delivering the content of their Reflections on an emerging field: researching
programs (Tatto 1996, 1998, 1999). mathematics teacher education. Educ Stud Math
60(3):359–381
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▶ Mathematics Teacher Education Organization, pedagogy in teaching and learning to teach: knowing
Curriculum, and Outcomes and using mathematics. In: Boaler J (ed) Multiple
perspectives on the teaching and learning of
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Philipp RA (2007) Mathematics teachers’ beliefs and Ball DL, Even R (2004) The International Commission on
affect. In: Lester FK (ed) Second handbook of research Mathematical Instruction (ICMI) – the fifteenth ICMI
on mathematics teaching and learning. National study, the professional education and development of
Council of Teachers of Mathematics/Information teachers of mathematics. J Math Teach Educ
Age, Charlotte, pp 257–315 7:279–293

Staub FC, Stern E (2002) The nature of teachers’ Even R, Ball D (eds) (2009) The professional education
pedagogical content beliefs matters for students’ and development of teachers of mathematics, vol 11,
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teaching diverse students: understanding the
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Llinares S, Krainer K (2006) Mathematics (student)
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Margolinas C, Coulange L, Bessot A (2005) What can the
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T 592 Teacher Supply and Retention in Mathematics Education

Tatto MT (2007) Educational reform and the global recruit, prepare, and retain the best possible
regulation of teacher education on teachers’ beliefs teachers. While policy decisions about pupil-
about instructional choice. Int J Educ Res 45:231–241 teacher ratios, initial teacher education pathways,
and teaching conditions influence countries’
Tatto MT, Lerman S, Novotna´ J (2009) Overview of overall supply and demand balance, a universal
teacher education systems across the world. In and relatively unfulfilled demand for high-quality
Even R, Ball D (eds) The professional education and mathematics teachers prevails. For some
development of teachers of mathematics. The 15th developing countries, this demand is evident
ICMI study. New ICMI study series, vol 11. Springer, across all sectors. For countries that – in various
New York, pp 15–24 ways – produce sufficient numbers of generalist
teachers for primary schools, the focus is on a
Tatto MT, Lerman S, Novotna´ J (2010) The organization search for ways to ensure sufficient numbers of
of the mathematics preparation and development of well-qualified mathematics specialist teachers
teachers: a report from the ICMI study 15. J Math for upper primary and/or secondary schools
Teach Educ 13(4):313–324 (Tatto et al. 2012).

Van Dooren W, Verschaffel L, Onghena P (2002) The In looking to address teacher quality, concerns
impact of preservice teachers’ content knowledge on about the sufficiency of mathematics and peda-
their evaluation of students’ strategies for solving gogical content knowledge, both at the recruit-
arithmetic and algebra word problems. J Res Math ment and graduate phase of teacher education, are
Educ 33(5):319–351 central. A trend is for countries to require teacher
graduates to meet additional criteria measured by
Teacher Supply and Retention in tests of mathematics knowledge or periods of
Mathematics Education probationary teaching in schools before gaining
professional certification. Efforts to increase
Glenda Anthony recruitment of potential mathematics teachers
Institute of Education, Massey University, have prompted the design of alternative teacher
Palmerston North, New Zealand education pathways which provide additional
mathematics content focus. An allied recruitment
Keywords issue is the trend for an increasing proportion of
career switchers to enter teaching. For many
Recruitment; Supply; Attrition; Retention; career switchers, their experiences of learning
Morale mathematics are distal and often confined to
mathematics service courses. Findings from
Definition a large-scale survey study of elementary and
middle school teachers in the USA (Boyd et al.
Issues and strategies concerning the supply and 2011) suggesting that career switchers may be
retention of high-quality mathematics teachers in less effective at teaching math than other teachers
primary and secondary classrooms. during their first year of teaching warrant further
investigation of how the use of mathematics in
Characteristics previous careers might impact on the quality of
students’ mathematical learning experiences and
The supply and retention of high-quality design of teacher education programs.
mathematics teachers are crucial to the success
of any education system. Faced with mounting Echoing findings from the UK and South
evidence that the most important in-school Africa, Ingersoll and Perda (2010) claim that the
influence on student achievement is teachers’ shortage of quality mathematics teachers in the
knowledge and skill (Hattie 2009), policy makers USA is not just an issue of recruitment – but also
are paying closer attention to strategies likely to an issue of retention. Common across many
education systems, high teacher attrition rates are

Teacher-Centered Teaching in Mathematics Education T593

linked to inadequate degree of classroom auton- References
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T
Tatto M, Schwille J, Senk S, Ingvarson L, Rowley G,
Peck R et al (2012) Policy, practice, and readiness to
teach primary and secondary mathematics in 17 coun-
tries: findings from the IEA Teacher Education and
Development Study in Mathematics (TED-M). Inter-
national Association for the Evaluation of Educational
Achievement (IEA), Amsterdam

Teacher-Centered Teaching in
Mathematics Education

Cross-References Michelle Stephan
College of Education, Middle Secondary
▶ Communities of Practice in Mathematics Department, The University of North Carolina at
Teacher Education Charlotte, Charlotte, NC, USA

▶ Mathematical Knowledge for Teaching Keywords
▶ Mathematics Teacher Identity
▶ Models of Preservice Mathematics Teacher Intellectual heteronomy; Direct instruction;
Explicit instruction; Special education
Education
▶ Teacher Education Development Study-

Mathematics (TEDS-M)

T 594 Teacher-Centered Teaching in Mathematics Education

Definition Contrast with Student-Centered Instruction
Recent research has suggested that teachers shift
Teacher-centered teaching is an approach to their practices towards more student-centered
teaching that places the teacher as the director instruction (Yackel and Cobb 1994; Hiebert et al.
of learning and is mainly accomplished by lec- 1997; Tarr et al. 2008) primarily to promote higher
ture, repetitive practice of basic skills, and con- and deeper engagement of students with the math-
structive feedback. ematics. Additionally, Mathematics Education in
Europe reports that many European countries have
Intellectual Heteronomy reconceptualized their mathematics instruction
Many researchers have contended that one of the towards more student-centered teaching (http://
most important contributions that education can eacea.ec.europa.eu/education/eurydice/documents/
make in individuals’ lives is to their development thematic_reports/132EN_HI.pdf).
of autonomy (Piaget 1948/1973). Autonomy is
defined as the determination to be self-governing While some researchers have shown that students
to make rules oneself rather than rely on the rules perform better on standardized tests when taught
of others to make one’s decisions (heteronomy). using teacher-centered instruction, others have
Kamii (1982) suggests that autonomy is the ability shown the opposite, leaving room for exploring
to think for oneself and make decisions indepen- which of the characteristics of both approaches can
dently of the promise of rewards or punishments. be used to maximize learning. The table below
In relation to education, Richards (1991) distin- illustrates the major differences between teacher-
guishes between two types of traditions in the and student-centered approaches and represents
mathematics education of children, what he a merging of two tables found at the sites: www.
terms school mathematics and inquiry mathemat- nclrc.org/essentials/goalsmethods/learncentpop.html
ics. School mathematics is what is typically and http://assessment.uconn.edu/docs/Teacher
thought of as a teacher-directed environment in CenteredVsLearnerCenteredParadigms.pdf.
which learning mathematics is a process of both
memorizing teacher-modeled rules and proce- Teacher centered Learner centered Theme
dures and solving routine problems that often
have little significance to the real world until mas- Focus is on instructor Focus is on both Role of
tery of the teacher’s solution methods is attained. students and instructor
Heteronomy is fostered here as students learn to instructor
replicate what the teacher has shown them.
Instructor talks; Students interact
Teacher-centered instruction has been around students listen with instructor and
for years and generally refers to a complex ped- passively one another, students
agogy that places the teacher as the mathematical are engaged
authority for learning. This approach to teaching
and learning has enjoyed prominence for decades Mathematical Students construct Lesson
despite recent pushes towards student-centered learning is knowledge through design
teaching. Teacher-centered classrooms can best transmitted from gathering and
be described as environments in which the teacher to student synthesizing
teacher emphasizes mastery of content and basic information and
skills and transfers knowledge primarily by integrating it with the
lecture and repetition. The students are viewed general skills of
as recipients of information and can master the inquiry,
skills by repeated practice and memorization. communication,
The term teacher-centered instruction is also critical thinking,
known as direct instruction and explicit problem solving, and
instruction in educational circles. so on

Emphasis is on Emphasis is on using

acquisition of and communicating

knowledge outside knowledge

of the context in effectively to address

which it will be used issues arising in real-

life contexts

(continued)

Teacher-Centered Teaching in Mathematics Education T595

Teacher centered Learner centered Theme to solve a diverse set of problems. She breaks the
modeling down into chunks that will be more
Lessons are designed Lessons are designed easily understood by the students. When the
modeling is complete and student questions
so that the around a problematic have been answered, the teacher will have them
practice solving very similar problems either
mathematics can be situation that independently or with peers. At this time, she
will walk around the room to monitor student
broken into small students must solve behavior and provide positive and/or negative
feedback. The guidance here is heavy with
manageable pieces without much pre- students practicing and making corrections to
their errors until mastery is attained. Most of the
lecture. Students talk is teacher directed with little student talk.
There are obviously variations in this lesson
mathematize the design and often direct instructors attempt to
make the lesson more engaging by relating
situation some of the mathematics to real life and by using
manipulative or notations. These manipulatives
Students work alone Students work in Role of and diagrams/notations are controlled by the
pairs, in groups, or students teacher and used as a modeling device.
alone depending on
the purpose of the Direct Instruction Lesson Design
activity In a direct instruction approach, the teacher might
begin with students working through a series of
Environment is Culture is prerequisite skills, like whole number operations
viewed as cooperative, and inequalities (e.g., 3 < 8). When the prerequi-
competitive and collaborative, and site skills are mastered, then the teacher explicitly
individualistic supportive states the objectives. To help students see the
importance of integers, she might show some
Instructor monitors Students talk without Assessment examples of real-world situations that illustrate
and corrects every constant instructor integer concepts. Then, the teacher models for
student utterance monitoring; both students how to solve problems. For example, it
instructor and is common to show students how to order integers
students provide on a horizontal number line by starting with a zero
analyze solutions, marked near the middle of the line and counting
particularly when the necessary spaces either left or right for each
questions arise integer. In a direct instruction environment, the
lesson is carefully structured and the teacher is the
Instructor answers Students answer center of the activity, showing students how to
students’ questions each other’s place integers on the number line and order inte-
questions using gers appropriately. During this part of instruction,
instructor as an the teacher asks questions as a way to monitor
information resource whether students are able to repeat the skills she
has shown them. Next, the class enters a period of
Assessment is used Assessment is used guided practice with the teacher monitoring stu-
to monitor learning to monitor learning dent progress and giving immediate feedback. For
and to inform students still struggling with the skill, she might
instruction give prompts or hints to help them along their

Classroom is quiet Classroom is noisy
and interactive

Characteristics T
Role of the Instructor
The role of the instructor in a teacher-centered
classroom is to impart knowledge onto the
student through lecture and modeling of the
mathematics concept(s). A typical lesson format
might consist of reminding students of the work
they did previously or eliciting prerequisite
knowledge that is needed to begin a new concept.
Once the groundwork has been laid by the
teacher, she states the objectives for the class
period and proceeds to lecture, drawing upon
a variety of sources. Generally speaking, the
teacher’s goal is to illustrate how and why
a basic skill or concept works by showing how

T 596 Teacher-Centered Teaching in Mathematics Education

Teacher-Centered Teaching in Mathematics Education, Fig. 1

learning. When students demonstrate accuracy Teachers would model the problem by placing
without teacher assistance, they are asked to two red chips on the negative side of the mat and
work independently to reach mastery of the skill. nine black chips on the positive side (Fig. 1).

An instructional strategy that is becoming A conversation might look like this:
increasingly acknowledged as useful for lessons T: What is the temperature at the beginning of the
that follow direct instruction is called concrete-
semiabstract-abstract (CSA) design. Take, day? (T displays a mat with chips on the
for example, the teaching integer operations. overhead projector while each student has
A CSA approach would have students represent the same at their desks.)
integers with two colored chips (see also Bennet Ted: À2.
and Musser 1976; Maccini and Hughes 2000; T: So since the temperature is 2 below zero, is
Flores 2008), black being positive and red negative, we would put two red chips on the
representing negative integers. Basically, negative side. Go ahead, get two red chips and
a black chip and a red chip cancel each other out put them on your own mat. Now, what does
and represent what is called a zero pair. What is the next part of the word problem say? Maya?
crucial to understanding this scenario is that zero Maya: The temperature rose by 9 F by the
added to any amount will not change the original afternoon.
amount. For example, if a student has 5 black T: The temperature rose by 9 F by the afternoon.
chips (a + 5 value) and adds 2 red and 2 black That means we should put nine black chips on
chips, she has to recognize that, although the total the positive part of the mat. Please put those on
number of chips has increased to 9, the value of 5 there (Maya puts nine black chips on the pos-
remains the same since the 4 chips represent zero. itive side). Now, to find out the temperature at
the end of the day, we need to take out zero
In the example below, students have already pairs. A zero pair is one red chip and one black
reached mastery of the integer concept of order- chip. They equal zero because one positive
ing positive and negative numbers and are being and one negative chip cancel each other out,
introduced to the operation of addition for the make 0. So we can just take them away. I’ll
first time. Following the CSA design, the teacher take away one set (physically removes one red
first shows students a workmat separated into two and one black chip together). You do the next
areas, a negative and a positive area. At the con- one Grace. (Grace comes to the overhead and
crete phase of instruction, students are taught takes a red and black chip off the mat). How
how to model an integer word problem with many chips do we have left?
chips. For example, consider the problem, “In Gwen: 7.
State College, Pennsylvania, the temperature on T: Right, seven chips. What color are they?
a certain day was À2 F. The temperature rose by (Students say “black.”) Right. Are black
9 F by the afternoon. What was the temperature chips positive or negative?
that afternoon?” (Maccini and Hughes 2000). Students: Positive.

Teacher-Centered Teaching in Mathematics Education T597

T: OK, so it is a positive 7 F at the end of the day. Further Areas of Research T
Let’s try another one. There are a number of studies that show
As students work out more and more exam- students who have received direct instruction
outperform students who received student-
ples, either with the teacher or in small groups, the centered instruction. Typically, these tests
teacher walks around and gives immediate feed- revolve around mathematical achievement on
back concerning the correctness of their methods calculational proficiency. However, critics of the
and answers. In the next phase, the semi-concrete, teacher-centered approach also cite studies show-
students are given a worksheet that is structured ing that students who received student-centered
so that students continue their previous activity instruction perform equally well on calculational
but instead of using actual chips, they are to problems and outperform their teacher-centered
draw their chips on the work paper and solve the peers on critical thinking problems. It is clear that
problem with their drawings. Again, the teacher research shows disparate results and the
provides positive and/or corrective feedback as mathematics education field must work towards
the students solve these problems. reconciling these differences. One suggestion
that seems to be popular in the special education
Finally, in the abstract phase, students are field, and supported by statements from the
asked to write symbolic equations for integer National Mathematics Advisory Panel (2008),
word problems and use rules for addition/subtrac- is to merge explicit and student-centered instruc-
tion of integers to solve them. Mastery of the tion together (Hudson et al. 2006; Scheuermann
skills at each of the phases is required before et al. 2009). Proponents of this approach are typ-
moving on to the next phase. If a student produces ically from special education and advocate mak-
an incorrect answer or method, the teacher ing instruction more realistic and hands on (like
reteaches by modeling the methods again. The the integer example), but simultaneously scaf-
students practice the modeled methods until mas- folding students’ learning by explicit and direct
tery is attained. means.

The Role of the Students. As can be seen in the Cross-References
example above, the teacher does a great job of
modeling and explaining to the students the steps ▶ Learner-Centered Teaching in Mathematics
behind integer addition. She has placed integers in Education
a real-world context, using manipulatives to help
students make sense of the concept. The students, References
for their part, are required to follow her steps and
answer questions as best they can throughout the Bennet A, Musser G (1976) A concrete approach to
modeling. The teacher has broken down integer integer addition and subtraction. Arith Teach
operations into one small chunk, working with 23:332–336
addition first. Once students master addition prob-
lems, through repetition and feedback, the teacher Flores A (2008) Subtraction of positive and negative
will move to subtraction. The role of the student is numbers: the difference and completion approaches
to practice the skill enough to master the content. with chips. Math Teach Middle Sch 14(1):21–23
The classroom environment is fairly quiet with
little interaction between students, unless the Hiebert J, Carpenter T, Fennema E, Fuson K, Murray H
teacher allows them to practice with one another. (1997) Making sense: teaching and learning mathe-
matics with understanding. Heineman, Portsmouth
Assessment. Assessment is typically
conducted as a way to monitor student success Hudson P, Miller S, Butler F (2006) Adapting and merging
in performing the skills that have been taught. In explicit instruction within reform based mathematics
this way, assessment occurs on a regular basis classrooms. Am Second Educ 35(1):19–32
and immediate feedback is given to students.
The goal is to reach mastery on basic skills and Kamii C (1982) Number in preschool and kindergarten.
move on to more sophisticated ones. National Association for the Education of Young
Children, Washington, DC

T 598 Teaching Practices in Digital Environments

Maccini P, Hughes C (2000) Effects of a problem solving digital technologies in the last years, main and
strategy on the introductory algebra performance of recent issues, theoretical perspectives, and
secondary students with learning disabilities. Learn considerations for the future.
Disabil Res Pract 15(1):10–21
Characteristics
National Mathematics Advisory Panel (2008) Foundations
for success: the final report of the national mathematics Much of the research related to the use of digital
advisory panel. U.S. Department of Education, technologies in mathematics education has
Washington, DC focused on learners and on the particular effects
that a given technology might have on the nature
Piaget J (1948/1973) To understand is to invent. and quality of student learning. Over the past
Grossman, New York decade, in a shift that has occurred more
generally in mathematics education research,
Richards J (1991) Mathematical discussions. In: Von researchers have begun to pay more attention to
Glasersfeld E (ed) Radical constructivism in the existing practices of teachers, as they relate to
mathematics education. Kluwer, Dordrecht, pp 13–52 the use of technology, and the changes that
such practices might or must undergo in order to
Scheuermann A, Deshler D, Schumake J (2009) The effects more effectively make use of available technolo-
of the explicit inquiry routine on the performance of gies. This change of focus is driven in part by
students with learning disabilities on one-variable equa- the fact that despite the availability of and insti-
tions. Learn Disabil Q 32(2):103–120 tutional support for digital technologies, the
everyday practice of most teachers has changed
Tarr J, Reys R, Reys B, Cha´vez O, Shih J, Osterlind little with respect to the use of technology
S (2008) The impact of middle-grades mathematics (Laborde 2008). This entry focuses on the chang-
curricula and the classroom learning environment on ing uses of digital technologies over the past 30
student achievement. J Res Math Educ 39(3):247–280 years and provides an overview of the theoretical
perspectives that have been developed over the
Yackel E, Cobb P (1994) Sociomathematical norms, past decade to study ways of understanding and
argumentation and autonomy in mathematics. J Res supporting changing teaching practices.
Math Educ 27:458–477
Teachers’ Changing Uses of Digital
Teaching Practices in Digital Technologies
Environments Since its introduction in schools – in the 1980s –
the use of ICT (information and communication
Nathalie Sinclair1 and Ornella Robutti2 technology) in teaching mathematics has had two
1Faculty of Education, Burnaby Mountain main functions: (a) as a support for the organiza-
Campus, Simon Fraser University, Burnaby, tion of the teacher’s work (producing work
BC, Canada sheets, keeping grades) and (b) as a support for
2Dipartimento di Matematica, Universita` di new ways of doing and representing mathemat-
Torino, Torino, Italy ics. The past decade has seen an evolution of
technology itself with the introduction of new
Keywords communication and representational infrastruc-
tures (Hegedus and Moreno-Armella 2009).
Teaching practices; Representational infrastruc- The representational infrastructures used in
tures; Communication infrastructures; Graphical, mathematics education can involve specific
numerical, symbolic, and geometric environ- software for teaching topics such as statistics,
ments; Community of practice; Community algebra, and modelling as well as graphical,
of inquiry; TPACK theory; Instrumental
approach; Documentational genesis; Instrumen-
tal orchestration

Definition

The teacher’s activities and methodologies with
the use of digital technologies: changing uses of

Teaching Practices in Digital Environments T599

numerical, symbolic, and geometric environ- students to compare and connect their own work T
ments that are used to represent mathematical with that of others. In the latter case, teachers can
objects. Over time, teachers have moved from use projectors or interactive whiteboards to
content-specific graphical and mathematical pro- enable whole classroom sharing of digital repre-
grams toward more generic and multi- sentations, thus retaining control of the use of the
representational environments (Thomas 2006). technology and reducing the need for student
instrumentation – such a modality has become
The communication infrastructures (such as increasingly frequent in both primary and
electronic mail, web platforms, and social net- secondary school classrooms.
works) have become useful both for teacher pro-
fessional development and for teaching practice Teacher Practice and Technology: Theoretical
in the classroom. In the first case, teachers can Perspectives
become members of communities of colleagues Over the past decade, there has also been a shift in
in the same school, in a network of schools or in focus from the learner to the teacher, echoing the
a teacher education program (as community of broader increase of attention in mathematics edu-
practice, in the sense of Wenger 1998), or in cation research. Early research involved studying
a research program (as community of inquiry, the variables, such as attitudes and levels of pro-
Jaworski 2006). They can participate in these ficiency, which affect teachers’ use of a given
communities in synchronous and asynchronous technology (Thomas 2006). Subsequent attention
activities aimed at sharing materials, designing was placed on the interaction that might occur
curricular plans, doing teaching experiments, between teachers’ proficiency with and attitude
collecting data for assessment, and discussing toward technology use and their proficiency with
results. In the second case, they can organize and attitude toward mathematics. For example,
their classroom activities in ways that combine in the case of DGS (and other dynamic mathe-
face-to-face interactions with distance ones matical environments), the dynamic/visual con-
mediated by these infrastructures. ception of a given mathematical object or
relationship that the technology offers might not
The use of digital environments in classroom accord with the static/algebraic conception that
in recent years has changed from a more a given teacher has developed – or that the text-
“private” to a “public” use that integrates the book and assessment items assume. The resulting
private use (Hegedus and Moreno-Armella mismatch will have an important effect on the
2009; Robutti 2010), as predicted in Sinclair way a given technology is used (see Laborde
and Jackiw (2005). This shift, which echoes the 2001; Sinclair and Robutti 2012) and on the
historical shift from the use of individual related learning process.
handheld slate to blackboards, can be described
again in terms of the technology available. White By extending the well-known PCK framework
the computer laboratory and handheld technol- to TPACK (Koehler and Mishra 2009),
ogy (calculators) settings featured individual or researchers have also drawn attention to the way
small group interactions with the technology that new technology resources interact with
could not easily be shared with the whole class teacher’s pedagogical and content knowledge.
and with new infrastructures combine the public- This framework highlights the fact that technol-
private uses or reverse the dominant interaction. ogy use cannot change (or be changed) in isola-
In the former case, handheld devices can tion of other aspects of teacher practice. This
be connected to the teacher’s computer, which echoes the extensive research that documents
projects student-generated work to a large public the way in which the use of technology changes
screen or to an interactive whiteboard. Further, the learner and the learner’s understanding. With
Robutti (2010) documents that “blended” a dual focus on the teacher and the learner, Borba
approach, in which the public screen not only and Villarreal (2005) have coined the phrase
displays the student work in real time, providing “humans-with-media,” a term that emphasizes
immediate feedback, it enables individual

T 600 Teaching Practices in Digital Environments

the way in which the technology is considered intentions. Drijvers et al. (2010) introduce also
part of these communities and can influence the “didactical performance,” which involves the
teaching and learning processes. ad hoc decisions taken while teaching about how
to perform in the chosen didactic configuration
More recently, however, researchers have and exploitation mode.
sought to better theorize their understanding of
teaching practices using technology in such a way Gueudet and Trouche (2009) call “documen-
to move beyond the logical demarcation of types tational genesis” the way teachers go from
of teacher knowledge perpetuated by TPACK. being untutored operators of materials (any
Two main approaches have emerged. Ruthven’s kind of teaching resource, including digital
(2009) Structuring Features of Classroom Prac- technologies) to being proficient users of
tice framework identifies five structuring features them. As teachers develop ways of using these
of classroom practice that shape the choices that materials, they turn into documents that have
teachers make when integrating new technolo- stable usage schemes. This approach enables
gies: working environment, resource system, researchers to attend to the broad range of
activity structure, curriculum script, and time materials involved in a particular lesson, as
economy. So, for example, in terms of resource well as the relationship between a teacher’s
system, teachers must decide how they will build preparation of it and its implementation in the
a coherent set of elements that function in classroom.
a complementary manner in the classroom – this
might involve choosing a digital tool that uses Central Considerations for the Future
the same kind of notation that is used in the As Stacey (2002) argues, “new technology ren-
textbook or encouraging students to take notes ders some traditional examination questions
on their laptops, where their technology-based obsolete and others problematic” (p. 11). As
explorations are taking place, instead of in such, even in situations where there is a high
their notebooks. Ruthven et al. (2009) used adoption of technology in teaching (e.g., the use
this framework to identify the various adapta- of CAS in university-level courses in Canada)
tions of teaching practices in their study of (Buteau et al. 2009), assessment continues to be
teachers’ use of graphing software at lower pencil and paper driven. However, many have
secondary level. argued that until teachers develop practices in
which technology is used both in formative and
The second approach draws on the notion of summative assessment, the putative effects of
“instrumental genesis,” which has been exten- these technologies (increasing the focus on con-
sively used to study the way in which tool and ceptual understanding, enabling broader forms of
person coevolve and which has focused on the mathematical expression, empowering student
ways in which learners go from being untutored agency and creativity, etc.) will be greatly
operators of a given tool to being proficient users. compromised.
Guin and Trouche (2002) extend this notion to
“instrumental orchestration,” which focuses Cross-References
more specifically on technology integration in
teaching and learning. In particular, instrumental ▶ Information and Communication Technology
orchestration involves practices that take into (ICT) Affordances in Mathematics Education
account both the constraints involved in using
a tool and the way in which students’ use of the ▶ Instrumentation in Mathematics Education
tool develops. Orchestration is described in ▶ Learning Practices in Digital Environments
terms of two variables: (1) “didactical configura- ▶ Technology and Curricula in Mathematics
tion” is the arrangement of artifacts in the
environment, and (2) “exploitation mode” is the Education
way the teacher decides to exploit a didactical
configuration for the benefit of her didactical

Technology and Curricula in Mathematics Education T601

References Clements MA, Keitel C, Leung F (eds) Third interna- T
tional handbook of mathematics education. Kluwer,
Borba MC, Villarreal ME (2005) Humans-with-media and Dordrecht
the reorganization of mathematical thinking: informa- Stacey K (2002) Challenges to mathematics assessment
tion and communication technologies, modeling, from new mathematical tools. In: Edge D, Yeap BH
visualization and experimentation. Springer, New (eds) Mathematics education for a knowledge-based
York era. Proceedings of second east Asian regional confer-
ence on mathematics education, vol 1. Association of
Buteau C, Jarvis D, Lavicza Z (2009) Technology use in Mathematics Educators, Singapore, pp 11–16
post-secondary mathematics instruction. CMS Notes Thomas M (2006) Teachers using computers in mathe-
41(4):6–7 matics: a longitudinal study. In: Novotna´ J, Moraova´
H, Kra´tka´ M, Stehl´ıkova´ N (eds) Proceedings of the
Drijvers P, Doorman M, Boon P, Reed H, Gravemeijer K 30th conference of the international group for the
(2010) The teacher and the tool: instrumental orches- psychology of mathematics education, vol 5. PME,
trations in the technology-rich mathematics classroom. Prague, pp 265–272
Educ Stud Math 75(2):213–234 Wenger E (1998) Communities of practice: learning,
meaning, and identity. Cambridge University Press,
Gueudet G, Trouche L (2009) Teaching resources and Cambridge
teachers professional development: towards
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Guerrier V, Soury-Lavergne S, Arzarello F (eds) Pro- in Mathematics Education
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Rosamund Sutherland1 and Teresa Rojano2
Guin D, Trouche L (2002) Mastering by the teacher of the 1Graduate School of Education, University
instrumental genesis in CAS environments: necessity of Bristol, Bristol, UK
of instrumental orchestrations. ZDM 34(5):204–211 2Center for Research and Advanced Studies
(Cinvestav), Mexico City, Mexico
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representation and communication infrastructures. Keywords
ZDM 41(4):399–412
Technology; Mathematics; Curriculum
Jaworski B (2006) Theory and practice in mathematics
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learning in teaching. J Math Teach Educ 9:187–211
The relationship between technology and
Koehler MJ, Mishra P (2009) What is technological mathematics curriculum from the perspective
pedagogical content knowledge? Contemp Issues of research, mathematical practices in the
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of geometry tasks with cabri-geometry. Int J Comput
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proof: the case of dynamic geometry. In: Bishop AJ,

T 602 Technology and Curricula in Mathematics Education

having evolved from Babbage’s automatic mathematics curriculum (e.g., the graph plotting
calculating machine. In the 1970s computer package Autograph or the statistics education
programming began to be taught in schools in package Fathom), sometimes they have been
some countries around the world, although this designed to make mathematics accessible to new
was not explicitly linked to the mathematics groups of students (e.g., SimCalc which was
curriculum, despite the fact that programming is designed to democratize the learning of calculus),
strongly related to the idea of variable and and sometimes they have been adapted from
algorithm. At a similar time the Logo program- technologies that had not been designed for
ming language was developed by Seymour educational purposes (e.g., spreadsheets, see,
Papert and colleagues with its best known e.g., Sutherland and Rojano 1993).
feature being an on-screen turtle which could
be controlled by programming commands Nowadays technologies for learning
(Papert 1980). During the 1980s Logo began to mathematics are increasingly available on mobile
be used in schools, and evidence from empirical devices, which include calculators and tablet
studies suggested that Logo could engage young computers, and such devices linked to the Inter-
students in exploring mathematical ideas such net can provide students with access to a wide
as ratio and proportion, geometry, variables, range of mathematical digital technologies. How-
functional variation, recursive processes, mathe- ever, research clearly shows that whatever the
matical generalization, and its symbolization designer’s intentions students can use technolo-
(Hoyles and Noss 1992). Interestingly different gies developed for learning mathematics for
perspectives began to emerge in terms of the nonmathematical purposes (Bartolini Bussi and
relationship of technology use and the mathemat- Mariotti 2008). For example, students might use
ics curriculum, with some people arguing that dynamic geometry tools to draw shapes or pic-
technology use in the mathematics classroom tures on the screen instead of constructing math-
should fit with the existing curriculum and others ematical objects using geometrical properties.
arguing that technology should be used to Theories of learning with mathematical digital
introduce students to complex mathematical technologies provide explanations for why this
ideas that had previously been inaccessible to is the case and at the same time offer a framework
them, for example, introducing the underlying for developing classroom practices that exploit
ideas of calculus to primary school students the potential of technology for mathematical
(Kaput 1994). The Logo programming language learning. For example, the theory of instrumental
had been developed as a means of transforming genesis (Artigue 2002) distinguishes between the
school mathematics and the curriculum. By technology (artifact) and the instrument, separat-
contrast in the 1980s, dynamic geometry environ- ing what relates to the intention of the designer
ments were developed (e.g., The Geometer’s (the technology) and what is constructed by the
Sketchpad and Cabri) to support the learning of user and relates to the context of use (the instru-
geometry within the curriculum, although ment). This theory has been used to explain the
dynamic geometry environments have evolved discrepancy between the students’ behavior and
so that they can be used within different curricu- the teacher’s intentions and points to the impor-
lum areas (e.g., functions and trigonometry). tance of the design of mathematical activities and
the role of the teacher within technology-
By the early 1990s a wide range of technolo- enhanced learning environments. Other theories
gies were available to be used within school math- such as the theory of semiotic mediation and
ematics, including graph plotting packages, the theory of constructionism also provide
spreadsheets, and computer algebra packages frameworks for designing technology-enhanced
which have been developed for university and learning environments for mathematics
professional mathematicians (e.g., Mathematica). (Drijvers et al. 2010).
Sometimes these technologies have been designed
to fit with particular aspects of the school The extended presence of computers,
calculators, and mobile devices in schools, as

Technology and Curricula in Mathematics Education T603

well as three decades of using these technologies transfer, and mathematics across different subject T
in education at an experimental level, has resulted areas (Julie et al. 2010), to countries that explicitly
in an increasing interest in the relationship introduce in the mathematics curriculum the use
between mathematics curriculum and technology of software such as dynamic geometry, Logo,
development among researchers, teachers, spreadsheets, graphing calculators, computer
parents, educational authorities, and curriculum algebra systems (CAS), and applets for the
designers and developers. Nevertheless, the teaching of specific mathematical domains either
potential impact that such technologies may in a compulsory way (e.g., Hong Kong, Russia,
have in the official curriculum has been and still France) or in an optional way (e.g., South Africa,
is a controversial issue in these communities. Mexico, Brazil, and Central American countries).
Despite this controversy, research experiences A common denominator in many of these
with a variety of computer programs and tools examples is a disparity between implementation
have already influenced curriculum changes of the use of such technology in the mathematics
in many countries, and this has happened in classroom (which tends to be teacher-centered)
different ways, such as (1) connecting different and the pedagogical strategies suggested in the
mathematics curricular areas, both at the same curriculum documents (such as learner-centered
and at different school levels, due to the possibil- and exploratory or experimental approaches).
ity to work with multiple digital representations Overall it is widely recognized that at the level
(which are dynamically linked to each other) of of the classroom, mathematics teachers are not
one concept or situation (e.g., the concept of exploiting the potential of technologies for
function); (2) giving students an early access to learning mathematics despite what might be
powerful mathematical ideas (e.g., mathematics specified in the curriculum and despite the
of variation); (3) incorporating new topics in the research evidence that indicates the ways in
curriculum (e.g., 3D geometry); (4) making it which technologies could be used in mathematics
possible for students to analyze large authentic education (Assude et al. 2010).
data sets in statistics; and (5) removing classic
topics. With regard to this last point, in the early As part of the non-static relationship of
1990s, manipulative aspects of algebra were sub- technology with curriculum, technological
stantially reduced in the UK national curriculum evolution and innovation are also potential
at secondary school level (Sutherland 2007) but factors of mathematics curricular changes. The
have since been reintroduced and are increasingly progress made in improving dynamic geometry
emphasized due to an appreciation of the programs offers students access to advanced
importance of symbolic manipulation with geometric ideas in three dimensions, mainly
paper and pencil for developing symbol sense. changing the point of view of a 3D scene and in
In this respect there is a continuing debate about this way, visually obtaining full information of a
the relative importance of paper-and-pencil 3D object. In computer algebra systems (CAS), it
mathematics versus computer-based mathemat- is possible to either de-emphasize manipulative
ics in terms of developing mathematical knowl- skills and focus students’ work on conceptual
edge and understanding, with many people tasks (Kieran 2007) or promote conceptual and
arguing that digital technologies for mathematics technical aspects of mathematics (Lagrange
do not replace paper-and-pencil technologies. 2003). Applets and other programs run on digital
tablets which allow students to physically touch
The relationship between technology and the and manipulate representations of mathematical
mathematics curriculum is constantly in flux, objects. Recent versions of spreadsheets offer a
changing over time and varying between friendly environment for mathematical modeling
countries, from countries like the USSR in the tasks using (hot-linked) graphical, symbolic, and
1980s, which considered informatics as “a new numeric representations of phenomena of the
mathematics” and introduced meta-content, physical world, which opens up the possibility
such as discovery, collaboration, generalization, of promoting mathematical modeling approaches

T 604 Technology and Curricula in Mathematics Education

more generally in the curriculum. For example, are being designed, with, for example, the
the open-source software GeoGebra favors the dynamic geometry software GeoGebra
connection between Euclidean, Cartesian, and developed as a free software within the open-
analytic geometry. source software movement. A promising area of
future research is to investigate the ways in which
Beyond the potential or real influence of the connectivity can transform mathematical
use of digital technologies in the official practices in schools and in particular whether
mathematics curriculum, the actual implementa- students can use social media to create collabo-
tion of effective modes of technology use in the rative mathematical communities. Another
mathematics classroom is still a big challenge, a important area of research relates to the ways in
situation that represents an opportunity for prom- which digital technologies can be used to assess
ising future research. In this regard, the experi- the learning of mathematics, and many people
ence from longitudinal studies dealing with consider that assessment practices have to
alternative technology-enhanced mathematics change before digital technologies become fully
curricula provides an important antecedent. One integrated into the mathematics curriculum.
of the innovations in some of these experiences is
the use of technology for curriculum design and Cross-References
development with a functional approach to alge-
bra at secondary, tertiary, and university levels. ▶ Information and Communication Technology
For example, in the pioneering technology project (ICT) Affordances in Mathematics Education
Computer-Intensive Algebra (Penn State Univer-
sity), beginning algebra concepts were introduced ▶ Technology Design in Mathematics Education
in mathematical modeling contexts, and students ▶ Types of Technology in Mathematics
used specialized software to work with numeri-
cal, graphical, and symbolic representations Education
of functions of one and two variables (Fey,
et al. 1991). In a similar way, in the VisualMath References
curriculum project (Yerushalmy and Shternberg
2001), specialized software with multiple Artigue M (2002) Learning mathematics in a CAS
nonsymbolic representations of functions was environment. The genesis of a reflection about instru-
used, in a functional approach in which letters mentation and the dialectics between technical and
represent quantities that vary, and solving conceptual work. Int J Comput Math Learn 7:245–274
equations consists of identifying a particular
case of the comparison of two functions. It is Assude T, Buteau C, Forgasz H (2010) Factors influencing
worth mentioning that in such experimental technology-rich mathematics curriculum and
studies, technology is one of the main factors practices. In: Hoyles C, Lagrange J-B (eds)
of curriculum change, demonstrating the feasibil- Mathematics education and technology – rethinking
ity of its implementation in an educational the terrain. Springer, New York, pp 405–419
system.
Bartolini Bussi M, Mariotti A (2008) Semiotic mediation
Nowadays Internet connectivity potentially in the mathematics classroom: artifacts and signs after
changes the ways in which digital technologies a vygotskian perspective. In: English L (ed) Handbook
can be integrated into the mathematics curricu- of international research in mathematics education,
lum, both in terms of applications that are avail- 2nd edn. Routledge, New York, pp 746–783
able in “the cloud” and because teachers and
students can work collaboratively within virtual Drijvers P, Kieran C, Mariotti M (2010) Integrating
communities. Teachers are increasingly using technology into mathematics education. In: Hoyles C,
the Internet to access teaching resources and Lagrange J-B (eds) Mathematics education and tech-
organize their work. Internet connectivity is also nology – rethinking the terrain. Springer, New York,
changing the way in which digital technologies pp 81–87

Fey JT, Heid MK (with Good RA, Sheets C, Blume G,
Zbiek RM) (1991, 1999) Concepts in algebra: a techno-
logical approach. Janson, Dedham (republished in 1999
version, Everyday Learning Corporation, Chicago)

Hoyles C, Noss R (1992) Learning mathematics and logo.
MIT Press, Cambridge

Technology Design in Mathematics Education T605

Julie C, Leung A, Thanh NC, Posadas L, Sacrista´n AI, the successful integration of digital technology T
Semenov A (2010) Some Regional Developments in in mathematics education. By using the term
Access and Implementation of Digital Technologies “design,” the author means not only the
and ICT. In: Hoyles C, Lagrange J-B (eds) design of digital technology involved but also
Mathematics Education and Technology-Rethinking the design of corresponding tasks and activities
the Terrain. New ICMI Study Series Vol 13, and the design of lessons and teaching, in general.
pp. 261-383. Springer, NY
An appropriate design, according to the
Kaput J (1994) Democratizing access to calculus: New author, refers explicitly to the instrumental gene-
routes to old roots. Mathematical thinking and sis model which considers co-emergence of tech-
problem solving 77–156 nical mastery to use technology for solving
mathematical problems and the genesis of mental
Kieran C (2007) Research on the learning and teaching of schemes leading to conceptual understanding
school algebra at the middle, secondary, and college (Drjvers 2012). As such, the model seeks a
level: building meaning for symbols and their match between didactical and pedagogical func-
manipulation. In: Lester FK (ed) Second handbook of tionality in which digital tool is incorporated with
research on mathematics teaching and learning. the tool’s characteristics and affordances. It
Information Age Publishing, Charlotte, pp 707–762 also emphasizes a priority of pedagogical and
didactical considerations as main guidelines and
Lagrange J-B (2003) Learning techniques and concepts design heuristics over technology’s limitations
using CAS: a practical and theoretical reflection. and properties related to its affordances and
In: Fey JT (ed) Computer algebra systems in secondary constrains (Drjvers 2012).
school mathematics education. NCTM, Reston,
pp 269–283 This global definition of “design” related to
the technology use in mathematics education was
Papert S (1980) Mindstorms: children, computers, and given by Drijvers during his plenary talk at the
powerful ideas. Basic Books, New York ICME Congress in Seoul, Korea, in 2012, which
reflects 40 years of history after S. Papert’s talk,
Sutherland R (2007) Teaching for learning mathematics. also during the ICME Congress in Exeter, Great
Open University Press, Maidenhead Britain, in 1972, expressing ideas of the
micro-world vision which set up a long-term
Sutherland R, Rojano T (1993) A spreadsheet approach to guidelines in research and development of the
solving algebra problems. J Math Behav 12:353–383 technology design principles (Healy and Kynigos
2010). The turtle geometry microworld (and
Yerushalmy M, Shternberg B (2001) Charting a visual related programming language LOGO specifi-
course to the concept of function. In: Cuoco AA, cally designed for learning) grounded in the the-
Curcio FR (eds) The roles of representation in school ory of constructionism is one of the first examples
mathematics: 2001 yearbook of the national council of of technology design which – instead of being
teachers of mathematics. NCTM, Reston, pp 251–268 the aid to teach school mathematics – provides
an opportunity to make mathematics
Technology Design in Mathematics “more learnable” where computers are used “as
Education mathematically expressive media with which to
design an appropriate mathematics fitted to the
Viktor Freiman learner” (Healy and Kynigos 2010, p. 63; see also
Faculte´ des sciences de l’e´ducation, Universite´ de Papert 1980 and Kynigos 2012).
Moncton, Moncton, NB, Canada
In their analysis of technology development in
Keywords mathematics education based on 25 years of pub-
lication in the JRME (Journal for Research in
Technology design; Microworlds; Virtual com- Mathematics Education), Kaput and Thompson
munities; Web design (1994) argue that “technology can reinforce any

Introduction and Historical Background

The role of the teacher, educational context, and
design are three key factors, called by Drjvers
(2012) decisive and crucial to promote or hinder

T 606 Technology Design in Mathematics Education

bias the user or designer brings to it” (p. 678) and technology as interactive tool or as a medium
do it by changing fundamentally the experience of in which one designs and builds interactive
doing and learning mathematics and this in three artifacts (technology as ‘tutee’)” (Kaput et al.
main aspects, interactivity, control, and connec- 2007, pp. 177–178).
tivity. At the early stages, computer environments
were reproducing a “human-human” interactivity Example of a Microworld
via so-called CAI (computer-assisted instruction)
by putting computers in the role of teacher Design of a dynamic visualization software envi-
presenting standard skill-based materials. ronment 3DMath (Christou et al. 2006) aimed at
Designers had a full control to engineer constrains enabling learners to construct, observe, and
and supports, create agents to perform actions for manipulate geometrical figures in a 3D-like
the learner (resources, aid, feedback, representa- space. General principles of the design meeting
tion systems), and thus could influence students’ these purposes are based on three major fields of
mathematical experiences (Kaput and Thompson educational theory: constructivist perspective
1994). Finally, connectivity was seen as linking about learning as personally constructed and
teachers to teachers, students to students, students achieved by designing and making artifacts that
to teachers, and, in a more general sense, the are personally meaningful, semiotic perspective
world of education to wider worlds of home and about mathematics as a meaning-making
work (Kaput and Thompson 1994). At the more endeavor that encourages multiple representa-
subtle level, the technology development tions of knowledge, and fallibilist nature of math-
involved “a gradual reshaping or expansion of ematics where knowledge is a construction of
human experience – from direct experience in human beings and is subject to revision (Christou
physical space to experience mediated by the et al. 2006). Also, according to the authors, core
computational medium” (Kaput and Thompson visual abilities must be taken into account: per-
1994, p. 679). ceptual constancy, mental rotation, perception of
spatial positions, perception of spatial relation-
This vision was directing mainstream of ships, and visual discrimination – accumulation
research and practice of designing technology- of representations makes possible the creation of
enhanced learning and teaching environments mental images.
over the past 20 years that resulted in building
of interactive microworlds that foster modelling Related to these perspectives, design princi-
and collaboration by “layering of mathematical ples for the 3DMath would (a) allow students to
and scientific principles and abstraction and see a geometric solid presented in several possible
embedding increasing problem-solving complex- ways; (b) introduce software-controlled speed
ity into the software” (Confrey et al. 2009, p. 20). and directions of rotations that can enable stu-
Such environments need to be engaging for the dents to devise strategies of movement and antic-
students so to help them to achieve goals they find ipated their results; (c) integrate intuitive
compelling by making, at the same time, mathe- interface allowing the learner to make and design
matics “visible to students and expressed in personally meaningful artifacts by means of rich
a language with which they can connect” semiotic resources enabling multiple perspectives
(Confrey et al. 2009, p. 20). Kaput et al. (2007) and representations; (d) help students to focus on
use the term “infrastructure” which implies not mental images; (e) be rich in the ability to manip-
only material support for activity but also social ulate and transform solids; (f) focus on observa-
systems at different size scales, like communities tion, construction, and exploration; (g) contribute
of practice (in the sense of Lave and Wenger to the development of visual abilities (dragging,
1991). Related to the users of mathematics and tracing, measuring, adding text, and diagrams);
mathematics education software, this implies and (h) add export of construction and control of
active participation in a practice as an intrinsic available (hidden) options (Christou et al. 2006).
property of membership, “whether one uses the

Technology Design in Mathematics Education T607

Virtual Learning Communities in
Mathematics Education

Virtual communities emerge in early 2000s Analysis
expanding affordances of the Internet technology
while allowing for designing collaborative learn- Enaction
ing environments in mathematics. For example, (enactment)
a Math Forum community brings together
teachers, students, parents, software developers, Design
mathematicians, math educators, professionals,
and tradespeople. While having different experi- Technology Design in Mathematics Education, T
ence, expertise, and interest in mathematics by Fig. 1 A DBR cycle for the CASMI community (Freiman
playing different roles, they all contribute in build- and Lirette-Pitre 2009, p. 248)
ing sustainable learning space with a variety of
educational resources that helps to scaffold each communicate relevant implications to practi-
other’s understanding of mathematics (Renninger tioners and to other educational designers
and Shumar 2004). From the point of view of the (Design-Based Research Collective 2003).
design, we note two key features, namely, the
content with extensive archives and links to infor- The DBR model allowed for implementing five
mation and the interactive tools that promote techno-pedagogical principles in the CASMI:
information exchange, discussion, and community friendly welcome allowing everyone (students,
building (Renninger and Shumar 2004). This type teachers, parents) to join the community at any
of the design lets participants to try out and select time; math challenge using authentic, complex,
different ways of working with the content and and contextualized problems to which every mem-
thus facilitate learning driven by their personal ber can submit a solution via an e-form on the
questions and interests (Renninger and Shumar website; formative individual feedback provided
2004). The website provides them with services by mentors (mostly university students) aiming to
that support learning, such as Problem of the Week encourage each participant to be persistent and
section with five interactive, nonroutine, challeng- continue to participate; acceptance of variety of
ing problems posted weekly accompanied further styles and strategies valuing different ways of
with solutions and explanations; Ask Dr. Math thinking as rich and valuable contribution to the
service allowing posing and answering frequently community; and open communication as a vehicle
asked questions from the members; and of the community to promote knowledge sharing
Teacher2Teacher discussion forum, examples of and knowledge building through collaboration and
lessons, projects, games, and a newsletter discussion (Freiman and Lirette-Pitre 2009).
(Renninger and Shumar 2004).
A newest development of collaborative
The design of virtual communities is a cyclic models of technology-enhanced learning envi-
process which reflects a design-based research ronments is grounded in what Gadanidis and
(DBR) model of the CASMI (Communaute´ Namukasa (2012), referring to the works of
d’Apprentissages Scientifiques et Mathe´matiques Levy (1998) and Borba and Villarreal (2005),
Interactifs, www.umoncton.ca/cami) community call “integral component of a cognitive ecology
(Fig. 1). The model illustrates an innovative of the human-with-technology thinking collec-
research approach suitable for studying complex tives” (p. 164). As new media affordances,
problems in real, authentic contexts in collabora- Gadanidis and Namukasa (2012) mention
tion with practitioners. Research and develop-
ment happens through continuous cycles of
design, enactment, analysis, and redesign which
would lead to sharable theories that help

T 608 Technology Design in Mathematics Education

democratization, as new media learning activities on mathematics software, needs to be
resources are available from any place with the able to use features of the software and to con-
Internet access; multimodality that connects sider the objects constructed in/by the software as
physical, linguistic, cognitive, and symbolic material/real.
experiences; collaboration that allows for new
ways of thinking collaborative, participatory, Interactive geometry sketches based on two
and distributed; and performance metaphor reflection tasks were designed in Sinclair’s
with multimedia authoring tools used to study using the Geometer’s Sketchpad and
create online content which are orchestrated saved as JavaSketches (Jackiw 2002, cited in
(programmed) as “stage” “scenes,” “actors” Sinclair 2006, p. 32). The findings from the
making of the Web a “performative medium” experimentation with teachers and students
(p. 167). reveal several issues related to some technical
problems with the sketches, student difficulties
Applications and Task Design with with the wording of questions and instructions, as
Technology well as interpretation of mathematical concepts
embedded in the applets; therefore more research
Designing tasks with interactive technology is yet is needed in order to develop strategies to gather
another direction of research and practice of information about the needs and the abilities of
mathematics education over the past decades. end users during the design process (Sinclair
According to the epistemic model developed by 2006, pp. 34–35).
Leung (2011), the task design “focuses on peda-
gogical processes in which learners are New Paths in Research and Practice with
empowered with amplified abilities to explore, Technology Design
re-construct (or re-invent) and explain mathemat-
ical concepts using tools embedded in A recent development in the design of technology
a technology-rich environment” (p. 327). Sinclair applications to support mathematics learning is
(2006) brought attention to the issues related to related to the mobile technology, to constructing
the use of interactive web-based applets (web- complex integrated systems by combining micro-
based sketches) whose design principles are worlds and virtual community features, and use
under-researched. A design of the computer- of games.
based tasks is grounded in a complex combina-
tion of a variety of theories in mathematics Mobile Learning Design
education on the use of manipulatives, teaching First reports are coming from pilot studies about
approaches for some specific topics, and structure design for mobile devices, such as cellular
of classroom discussions, which can be borrowed phones with the use of Sketch2Go and Graph2Go
and adapted for the use in the technological envi- (Botzer and Yerushalmy 2007). The first enables
ronment often in the constructivist perspective students to sketch graphs (constant, increasing,
which seeks in helping students to build their and decreasing functions) and get an immediate
own understanding by connecting new ideas and feedback of the drawn graph and present a graph
prior understanding; the activity theory is also of the rate of change thus reinforcing visual
used to explain learning as being dependent on exploration of (physical temporal) phenomena
personal experience and can be mediated by the and providing with qualitative indication of the
tools. In its turn, the activity theory can be linked ways in which sketch drawn by the user changes;
to the affordance theory in a way that Martinovic this motivates students to experiment with
et al. (2012) conceptualize as a “handshake” a given situation, analyze it, and reflect upon it.
which is prerequisite for an action by a subject, The second application (Graph2Go) is a graphing
for example, a student, during the explorative calculator which operates for given sets of func-
tion expressions and enables the dynamic

Technology Design in Mathematics Education T609

transformation of functions (including changing academic matters more fun, if not easier by T
parameters of algebraic expression, Botzer and embedding school-like exercises in a computer
Yerushalmy 2007, p. 314). game (Kafai 2006). For example, a game called
How the West Was Won offered the players to
Combining Multiple Platforms throw a dice than perform various arithmetic
The design of a virtual learning environment that operations on the numbers to determine how far
integrates synchronous and asynchronous media to advance a token on the board (Kafai 2006). By
with an innovative multiuser version of mentioning Math Blaster as another example of
a dynamic math visualization and exploration thousands of instructional games on the market,
toolbox is discussed by Stahl (2012) using the the author mentions that little is known about
example of the VMT with GeoGebra. The com- which features make an educational game good
bination of features of the computer-supported for learning and few studies are available on what
collaborative learning (CSCL) software are successful design features for good educa-
(VMT – Virtual Mathematics Team platform tional games (Kafai 2006). As about construc-
that engages learners in significant discourse tionist perspective, the main idea expressed by
and practicing teamwork) and dynamic mathe- Kafai (2006) is that rather than embedding “les-
matics software (such as Geometer’s Sketchpad, sons” directly in games, the goal should be
Mathematica, Cabri, or GeoGebra allowing users directed to providing students with opportunities
to manipulate geometric diagrams and equa- to construct their own games and thus new rela-
tions). The combination of both environments, tionships with knowledge in the process, as
VMT and GeoGebra, helped to overcome issues shows the study of primary children and
related to multi-user collaboration by means of preservice teachers designing games with
a client–server architecture. This allows “multi- representing fractions in different ways. Not
ple distributed users to manipulate constructions only this opportunity makes possible game
and to observe everyone else’s actions in real design environment in which the user can load
time (through immediate broadcast by the server fraction design tools with a set of objects and
and further logged in detail for replay and graphic tools for creating, representing, and oper-
research)” (Stahl et al. 2012, p. 5). ating on fractions (like splitting, fair sharing) and
fraction objects. Moreover, there were also tools
Design of Games for Teaching and Learning allowing students and preservice teachers –
Mathematics designers to share, annotate, and modify their
While exploring educational potential of com- designs using electronic discussion forum.
puter games in mathematics, Hui (2009) men- Researchers found that conversation and discus-
tioned several general categories of games, such sion among participants were essential in helping
as action, adventure/quest, fighting, puzzle, role- the designers build more sophisticated represen-
play, simulations, sports, and strategy games. For tations (Kafai 2006).
mathematics education, problem solving and
deductive reasoning as well as skills like numer- Cross-References
ical calculation and monetary skills are men-
tioned by the author as the most viable avenues ▶ Learning Environments in Mathematics
for acquisition and application of mathematics in Education
computer games (Hui 2009).
▶ Learning Practices in Digital Environments
Kafai (2006) discussed two different perspec- ▶ Technology and Curricula in Mathematics
tives on design of games for learning: making
games for learning instead of playing games for Education
learning. The instructionist perspective builds on ▶ Types of technology in Mathematics
a vision that making a game for practicing the
multiplication tables can make the learning of Education

T 610 The Learning Framework in Number

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Kynigos C (2012) Constructionism: theory of learning
or theory of design? Regular Lecture, Proceedings of
the 12th international congress on mathematical
education, Seoul

The Learning Framework in Number T611

Stages: Early Arithmetical Levels: Forward Number Word Sequences (FNWS)
Learning & Number Word After
0 - Emergent Counting
1 - Perceptual Counting 0 - Emergent FNWS.
2 - Figurative Counting 1 - Initial FNWS up to ‘ten’.
3 - Initial Number Sequence 2 - Intermediate FNWS up to ‘ten’.
4 - Intermediate Number 3 - Facile with FNWSs up to ‘ten’.
Sequence 4 - Facile with FNWSs up to ‘thirty’.
5 - Facile with FNWSs up to ‘one hundred’.
5 - Facile Number Sequence
Levels: Backward Number Word Sequences
Levels: Numeral Identification (BNWS) & Number Word Before
0 - Emergent Numeral
Identification. 0 - Emergent BNWS.
1 - Initial BNWS up to ‘ten’.
1 - Numerals to ‘10’ 2 - Intermediate BNWS up to ‘ten’.
2 - Numerals to ‘20’ 3 - Facile with BNWSs up to ‘ten’.
3 - Numerals to ‘100’ 4 - Facile with BNWSs up to ‘thirty’.
4 - Numerals to ‘1000’ 5 - Facile with BNWSs up to ‘one

hundred’.

The Learning Framework in Number, Fig. 1 The Learning Framework in Number (LFIN) (Adapted with permis-
sion from Wright et al. (2006), p. 20)

interrelated. Each progression takes a summary In this study, use of a process of videotaped, T
form referred to as a model and consisting of interview-based assessment enabled the profil-
a table, setting out progressive levels of knowl- ing of the knowledge of 45 children – 15 drawn
edge of the domain. The LFIN relates to young from each of three classrooms – on LFIN at the
children’s early arithmetical learning and was beginning, middle, and end of the school year.
the first example of such a framework (Wright Table 1 shows the progress of 15 students from
1986, 1991, 1994, 1998). Figure 1 shows LFIN a class in the Kindergarten year. This study
in summary form and includes models for not only showed the kinds of knowledge
four domains: Early Arithmetical Learning; progressions typical of students in the first and
Forward Number Word Sequences; Backward second years of school but also highlighted the
Number Word Sequences; and Numeral Identifi- relatively wide range of knowledge within
cation (Steffe 1992; Wright et al. 2006; Wright a given classroom.
2008). The origin of LFIN is independent of
that of learning trajectories (Simon 1995) and Applications
instructional progressions (Gravemeijer 2004).
Nevertheless, LFIN has been described (Clem- LFIN has been used in several research studies to
ents and Sarama 2009) and can be regarded chart the progress of very large cohorts of stu-
as a set of interrelated learning/instructional dents (Thomas and Ward 2002; Wright and
trajectories. Gould 2002). These studies were undertaken in
conjunction with large-scale, systemic
Origin implementations of new initiatives in early arith-
metic instruction, which adopted or drew on
LFIN was initially developed as part of LFIN as a guiding pedagogical model (Bobis
a research study of the knowledge progression et al. 2005). Table 2 is drawn from a study in
across a school year, of children in the first and which 23,121 students with ages ranging from
second years of school (Wright 1991, 1994). 4.5 to 9.9 were assessed to determine their

T 612 The Learning Framework in Number

The Learning Framework in Number, Table 1 Five-year olds – Kindergarten Year – School B, March August
November (Adapted from Wright (1994), p. 34)

Counting stage Forward N.W.S. Backward N.W.S. Numeral identification Spatial patterns

Boy/girl 0–5 0–5 0–5 0–4 0–3

1 G 1 1 1 3 3 3 1 0 1–3 1 1 1 0 0 0

2B 122 3 240 0 0 0 1 1 100

3B 222 5 553 5 5 2 3 4 111

4G 2 2–3 3 3 4 4 1–3 3 3 1 1 2 121

5B 233 3 352 3 3 1 2 2 122

5G 333 4 553 3 3 1 2 2 122

7 G 2–3 3 3 3–4 5 5 3 3 3 1 3 3 2 2 2

7B 233 3 553 3 3 1 2 3 122

7B 2 3 3 3 3–4 5 0 1–3 3 1 2 3 022

10 G 333 4 554 3 5 3 3 3 122

11 G 3 3 4–5 3 4 5 1–3 3 3 1 2 3 122

12 G 3 3 4–5 5 5 5 3–4 5 5 1 2 3 212

13 G 355 4 553 5 5 1 3 3 232

Notes: In column one, the order has been determined by considering the data from the November interviews (the value
appearing on the right in each cell) as follows: The counting stage is considered first, then the levels are considered, in
order from left to right
The Counting Stage corresponds to the Stage of Early Arithmetical Learning
Table entries in the form of a range, e.g., two to three, rather than a single level or stage, indicate that the precise level or
stage could not be determined

The Learning Framework in Number, Table 2 Number of children at each stage, for each age group (n ¼ 23,121)
(Adapted with permission from Wright and Gould (2002))

Emergent 4.5–4.9 years 5.0–5.9 years 6.0–6.9 years 7.0–7.9 years 8.0–8.9 years 9.0–9.9 years
Perceptual 552 1,406 388 113 39 18
Figurative 555 3,254 825 97
Counting on 56 2,032 262
Facile 13 887 1,749 1,321 441 182
Totals 0 322 1,445 2,413 1,563 772
29 765 681
1,176 5,898 259 682 3,070 1,750
5,873 5,354

stage on the domain of Early Arithmetical References
Learning.
Bobis J, Clarke B, Clarke D, Thomas G, Wright R, Young-
Finally, LFIN has been used extensively in Loveridge J, Gould P (2005) Supporting teachers in the
professional practice in at least eight countries, development of young children’s mathematical thinking:
as a guiding model for both classroom instruction three large-scale cases. Math Educ Res J 16(3):27–57
and intensive intervention with low-attaining
students (Wright et al. 2006). Clements D, Sarama J (2009) Learning and teaching early
mathematics: the learning trajectories approach.
Cross-References Routledge, New York

▶ Hypothetical Learning Trajectories in Gravemeijer K (2004) Local instruction theories as means
Mathematics Education of support for teachers in reform mathematics educa-
tion. Math Think Learn 6(2):105–128
▶ Number Teaching and Learning
Simon M (1995) Reconstructing mathematics pedagogy
from a constructivist perspective. J Res Math Educ
26:114–145

Steffe L (1992) Learning stages in the construction of the
number sequence. In: Bideaud J, Meljac C, Fischer J (eds)

The van Hiele Theory T613

Pathways to number: children’s developing numerical husband and wife team Pierre van Hiele and T
abilities. Lawrence Erlbaum, Hillsdale, pp 83–98 Dina van Hiele-Geldof. Dina died in 1959 and
Thomas G, Ward J (2002) An evaluation of the Early Pierre continued to develop and refine the Theory
Numeracy Project 2001: exploring issues in mathe- that is explored thoroughly in his 1986 book,
matics education. Ministry of Education, Wellington Structure and Insight.
Wright R (1986) A counting-based framework for observ-
ing and designing activities for prenumerical children. Much of the resurgence of interest in teaching
In: Proceedings of the eleventh biennial conference of of Geometry that began in the 1980s and 1990s
the Australian Association of Mathematics Teachers, can be traced to the ideas developed in the van
AAMT, Brisbane, pp 122–130 Hiele Theory. Detailed accounts and summaries
Wright R (1991) What number knowledge is possessed by of this early, but still highly relevant, work can be
children entering the kindergarten year of school? found in the following: Clements and Battista
Math Educ Res J 3(1):1–16 (1992), Fuys et al. (1988), Burger and
Wright R (1994) A study of the numerical development of Shaunghnessy (1986), Hoffer (1981), Lesh
5-year-olds and 6-year-olds. Educ Stud Math 26:25–44 and Mierkiewicz (1978), Mayberry (1981), and
Wright R (1998) An overview of a research-based frame- Usiskin (1982).
work for assessing and teaching early number. In:
Kanes C, Goos M, Warren E (eds) Proceedings of the Characteristics
21st annual conference of the mathematics education
research group of Australasia, vol 2. Mathematics The Theory has two main aspects that combine to
Education Research Group of Australia, Brisbane, provide a philosophy of mathematics education
pp 701–708 (not only of Geometry). The two key aspects to
Wright R (2008) Mathematics recovery: an early interven- the theory are:
tion program focusing on intensive intervention. In: 1. Levels that students grow through in acquiring
Dowker A (ed) Mathematics difficulties: psychology
and intervention. Elsevier, San Diego, pp 203–223 competence and understanding
Wright R, Gould P (2002) Using a learning framework to 2. Teaching phases that assist students to move
document students’ progress in mathematics in a large
school system. In: Cockburn A and Nardi E (eds) Pro- through the levels
ceedings of the 26th annual conference of the interna- Van Hiele’s ideas have much in common with
tional group for the psychology of mathematics those of Piaget (Piaget et al. 1960) in that they
education, vol 1. PME, University of East Anglia ascribe student understanding to a series of levels
Wright R, Martland J, Stafford A (2006) Early numeracy: or stages. However, there are important differ-
assessment for teaching and intervention, 2nd edn. ences between the two theories. For example, the
Sage, London van Hiele Theory:
• Places explicit importance on the role of lan-
The van Hiele Theory guage in moving through the levels.
• Concentrates on learning rather than develop-
John Pegg ment; hence the focus is on how to help
Education, University of New England, develop student understanding.
Armidale, NSW, Australia • Postulates that ideas at a higher level result from
the study of the structure at the lower level.
Keywords Most of the research effort has been directed at
the van Hiele levels of thinking – a hierarchical
van Hiele theory; van Hiele levels; van Hiele series of categories that describe cognitive
phases; Geometry; Cognitive development growth in students. The second and equally
important aspect that has not received the same
Definition degree of scrutiny or acknowledgement is the
notion of five teaching phases to help guide activ-
The van Hiele Theory had its beginnings in the ities that lead students from one level to the next.
1950s in the companion doctoral work of

T 614 The van Hiele Theory

Van Hiele Levels Van Hiele Phases
Van Hiele envisaged five levels, and these are In terms of the van Hiele phases, the initial work
described below within the context of two- in this area appeared through the doctoral thesis
dimensional geometrical figures: of Dina van Hiele-Geldof. Her thesis was trans-
Level 1. Students recognize a figure by its lated from Dutch into English as part of the
investigation led by Geddes (see Fuys et al.
appearance (i.e., its form or shape). Prop- 1984) and provides a valuable insight into how
erties of a figure play no explicit role in its the phase concept emerged.
identification.
Level 2. Students identify a figure by its proper- The purpose of Dina’s study was to detail her
ties, which are seen as independent of one experiences and teaching procedures with 2 Year
another. 7 (12 years old) Geometry classes over a year.
Level 3. Students no longer see the properties of The students were studying Geometry for the first
figures as independent. They recognize that time, and the main question posed in the study
a property precedes or follows from other was to see if it was possible to follow a teaching
properties. Students also understand relation- approach that allowed students to develop from
ships between different figures. one level to the next in a continuous process.
Level 4. Students understand the place of deduc-
tion. They use the concept of necessary and As a result of this work, five phases were
sufficient conditions and can develop proofs identified that allowed students to move from
rather than learn them by rote. They can devise one level to the next. The descriptions of the
definitions. phases (see Pegg 2002) given below are adapted
Level 5. Students can make comparisons of var- from Dina’s last paper that was written just before
ious deductive systems and explore different her death and also translated into English by the
geometries based upon various systems of Geddes team.
postulates. Phase 1: Information (Inquiry). This part of the
Although these descriptions are content spe-
cific, van Hiele’s levels are actually stages of process allows students to discuss what the
cognitive development: “the levels are situated area to be investigated is about.
not in the subject matter but in the thinking of Phase 2: Directed Orientation. Out of the first
man” (van Hiele 1986, p. 41). Progression from phase and the resulting discussion, students
one level to the next is not the result of maturation begin to look at the area to be studied in
or natural development. It is the nature and qual- a certain way. This part of the process involves
ity of the experience in the teaching/learning the teacher in directing the class to explore the
program that influences a genuine advancement object of study by means of a number of sim-
from a lower to a higher level, as opposed to the ple tasks.
learning of routines as a substitute for Phase 3: Explicitation. As a result of the manip-
understanding. ulation of materials and the completion of
It is this focus on teaching that pervades the simple tasks set by the teacher, the need to
ideas inherent in the van Hiele’s writings – so talk and to converse about the subject matter
much so that the “theory” is perhaps better becomes important. During the early part of
described as pedagogical rather than psychologi- the process, the students are encouraged to
cal, as many (or most) of the problems identified use their own language. However, over time
in students’ learning have their basis in teaching the teacher gradually incorporates, where
practices rather than in the cognitive processes appropriate, correct technical terms.
that may underlie performance. Phase 4: Free Orientation. Here students are
It is important to state that the van Hiele levels given a variety of activities and are expected
are not without controversy. Some of these issues to find their own way to a solution. The
are discussed in Pegg and Davey (1998). teacher’s role is to encourage different solu-
tions to the problems as well as the inventive-
ness of the students.

Theories of Learning Mathematics T615

Phase 5: Integration. The students achieve an Pegg J, Davey G (1998) Interpreting student understand- T
overview of the area of study by themselves. ing in geometry: a synthesis of two models. In: Lehrer
They are now clear of the purpose of the study R, Chazan C (eds) Designing learning environments
and have reached the next level. for developing understanding of geometry and space.
As with the van Hiele levels, there is an intu- Lawrence Erlbaum, New Jersey, pp 109–135

itive appeal about the learning phases outlined Piaget J, Inhelder B, Szeminska A (1960) The child’s
above. conception of geometry. Basic Books, New York

Summary Usiskin Z (1982) Van Hiele levels and achievement in
The van Hiele theory is directed at improving secondary school geometry (final report of the cogni-
teaching by organizing instruction to take into tive development and achievement in secondary
account students’ thinking, which is described school geometry project). University of Chicago,
by a hierarchical series of levels. According to Department of Education, Chicago
the theory, if students’ levels of thinking are
addressed in the teaching process, students have Van Hiele PM (1986) Structure and insight: a theory of
ownership of the encountered material. As mathematics education. Academic, New York
a result, they have the potential to develop insight
(the ability to act adequately with intention in Theories of Learning Mathematics
a new situation). For the van Hieles, the main
purpose of instruction is the development of Richard A. Lesh1, Bharath Sriraman2 and
such insight. Lyn English2
1School of Education, Counseling and
References Educational Psychology, Indiana University,
Bloomington, IN, USA
Burger WF, Shaughnessy JM (1986) Characterizing the 2Department of Mathematical Sciences,
van Hiele levels of development in geometry. J Res The University of Montana, Missoula,
Math Educ 17:31–48 MT, USA

Clements D, Battista M (1992) Geometry and spatial Keywords
reasoning. In: Grouws D (ed) Handbook of research
on mathematics teaching and learning. Macmillan, Complexity; Learning theories; Models and
New York, pp 420–464 modeling; Models versus theories; Theories of
mathematics education
Fuys D, Geddes D, Tischler R (1984) English translation
of selected writings of Dina van Hiele-Geldof Definition
and Pierre M. van Hiele. Brooklyn College. ERIC
document reproduction service no ED 287 697 According to Karl Popper, widely regarded as
one of the greatest philosophers of science in
Fuys D, Geddes D, Tischler R (1988) The van Hiele model the twentieth century, falsifiability is the primary
of thinking in geometry among adolescents. J Res characteristic that distinguishes scientific theo-
Math Educ Monogr 3 ries from ideologies – or dogma. For example,
for people who argue that schools should treat
Hoffer A (1981) Geometry is more than proof. Math creationism as a scientific theory, comparable to
Teach 74:11–18 modern theories of evolution, advocates of crea-
tionism would need to become engaged in the
Lesh R, Mierkiewicz D (1978) Perception, imaging generation of falsifiable hypothesis and would
and conception in geometry. In: Mierkiewicz D, need to abandon the practice of discouraging
Lesh R (eds) Recent research concerning the questioning and inquiry. Ironically, scientific
development of spatial and geometric concepts.
ERIC, Columbus

Mayberry J (1981) An investigation of the van Hiele levels
of geometric thought in undergraduate preservice
teachers. Unpublished doctoral Dissertation, Univer-
sity of Georgia. University Microfilms no DA 8123078

Pegg J (2002) Learning and teaching geometry.
In: Grimison L, Pegg J (eds) Teaching secondary
mathematics: theory into practice. Nelson Thomson,
Melbourne, pp 87–103

T 616 Theories of Learning Mathematics

theories themselves are accepted or rejected guide their research or development activities,
based on a principle that might be called survival these theories were mainly borrowed from edu-
of the fittest. So, for healthy theories on develop- cational psychology such as Bloom’s taxonomy
ment to occur, four Darwinian functions should of educational objectives, Gagne’s behavioral
function: (a) variation, avoid orthodoxy and objectives and learning hierarchies, Piaget’s
encourage divergent thinking; (b) selection, sub- stage theory, Ausabel’s advanced organizers
mit all assumptions and innovations to rigorous and meaningful verbal learning, and later
testing; (c) diffusion, encourage the shareability Vygotsky’s socially mediated learning, and
of new and/or viable ways of thinking; and (d) Simon’s artificial intelligence models for cogni-
accumulation, encourage the reusability of viable tion. However, the practitioners’ side of these
aspects of productive innovations. mathematics education researchers made it diffi-
cult for them to ignore the fact that very few of
Characteristics their most important day-to-day decision-making
issues were informed in any way by these
The History and Nature of Theory borrowed theories.
Development
To describe the nature of theories and theory In contrast to the preceding state of affairs,
development in mathematics education, it is use- Sriraman and English’s (2010) book clearly doc-
ful to keep in mind the preceding four functions uments a shift beyond theory borrowing
and to focus on two books that have been pro- toward theory building in mathematics educa-
duced as key points during the development of tion; the relevant theories draw on far more
mathematics education as a research community: than psychology, and the mathematics education
Critical Variables in Mathematics Education research community has become far more inter-
(Begle 1979) and Theories of Mathematics national – and far more multidisciplinary in
Education (Sriraman and English 2010). its membership. Furthermore, the field changed
significantly after the National Council of
Begle was one of the foremost founding Teachers of Mathematics (NCTM) published its
fathers of mathematics education as a field of nationally endorsed Curriculum and Evaluation
scientific inquiry; and his book reviews the liter- Standards for School Mathematics (NCTM
ature and characterizes the field when it was in its 1989), commonly referred to as the Standards in
infancy. For example, before 1978, the USA’s the USA. Since then, similar documents were
National Science Foundation had funding produced in many other countries throughout
programs to support curriculum development, the world. But, to what extent have these docu-
teacher development, and student development; ments been products of empirical research and
but, it had no comparable program to support theory development instead of dogma? The
knowledge development (i.e., research). Simi- NCTM Standards themselves were not based on
larly, before 1970, there was no professional any research per se, but simply an envisioning
organization focusing on mathematics education of what mathematics education in classrooms,
research or theory development; there was no i.e., in practice might look like and what the
journal for mathematics education research; and appropriate content might look like, keeping the
in the USA, just as in most other countries, there learner in mind.
existed no commonly recognized curriculum
standards for school mathematics. Furthermore, Curricular Standards and Mathematics
most mathematics educators thought of them- Education Research
selves as being curriculum developers, program Two decades later, consider the USA’s newest
developers, teacher developers, or student devel- Common Core State Curriculum Standards
opers (i.e., teachers) – and only secondarily as (CCSC 2012). In this case, there clearly exist
researchers. And, if any theories were invoked to some instances where these CCSC Standards
were informed by the work of a few researchers.

Theories of Learning Mathematics T617

But, it is equally clear that this document was (b) students’ conceptual understandings of T
produced using a process of political consensus these “big ideas” are a great deal more situated
building in which the views of some stakeholders and socially mediated than theories of 30 years
were given great attention (e.g., university-based ago led educators to believe.
mathematicians and teacher educators), whereas • Modeling continues to be characterized as the
others were ignored almost completely (e.g., engi- application of concepts (traditionally) taught in
neers, social scientists, and other heavy users of school. Yet, research in the learning sciences
mathematics outside of schools). Consequently, clearly is showing that, in modern societies, in
the CCSC Standards exhibit little recognition of students’ everyday lives outside of schools and
the fact that, outside of school in the twenty-first departments of mathematics, many of the situ-
century, many new kinds of problem-solving situ- ations that students need to mathematize
ations abound in which new types of mathematical involve (a) integrating ideas and procedures
thinking are needed. In fact, little is said in the drawn from more than a single textbook topic
CCSC Standards that could not have been said area and (b) using more than a single, solvable,
when Begle was in his prime. For example: and differentiable function. For example, in
• The mathematics education community still problem-solving situations that involve data
analysis and statistics, Bayesian and Fisherian
does not know how to operationally define computational models tend to be far more
measurable conceptions of almost any of the accessible and powerful than traditional
higher-level understandings or abilities that methods that depend on Calculus and the use
the CCSC Standards refers to as “mathemati- of traditional analytic methods. And, in situa-
cal practices.” So, the only goals that are tions that involve several interacting agents,
stated in ways that can be documented and issues often arise that involve feedback loops,
assessed tend to be the CCSC’s long lists of second-order effects, and issues such as maxi-
“things students should know and be able to mization, minimization, or stabilization. And
do” (i.e., declarative statements {facts} or again, graphics-oriented computational models
condition-action rules {skills}). make it possible for quite young children to
• In spite of the CCSC’s claim of being based on deal effectively with situations that no longer
research-based learning progressions, it still need to be postponed until after courses in
is unclear how the mastery of the CCSC’s lists Calculus.
of “things students should know and be able to Perhaps the most important general theme
do” interacts with the development of higher- that cuts across many of the chapters in
order “conceptual understandings of the type Sriraman and English’s book is that, from early
which are needed to conceptualize (i.e., math- number concepts through proportional reasoning
ematize by quantifying, dimensionalizing, and Calculus, the mathematics education com-
coordinatizing, systematizing) situations that munity in general has been quite na¨ıve about:
do not occur in a pre-mathematized form. In (a) what it means to “understand” nearly every
particular, it is unclear how (or whether) the “big idea” in the K-12 curriculum; (b) how these
CCSC’s lists of “things students should know understandings develop along dimensions such
and be able to do” should be treated as “pre- as concrete-abstract, intuition-formalization, or
requisites” which must be “mastered” before situated-decontextualized; (c) what it means for
students should be introduced to deeper and one concept or ability to be prerequisite to
higher-order conceptual understandings and another; and (d) how understandings of both
abilities. Furthermore, modern research in “big ideas” and basic “facts and skills” evolve
the learning sciences clearly has shown that as interconnections and distinctions develop.
(a) students’ and teachers’ conceptual under- Begle’s powerfully influential School Mathe-
standings of most “big ideas” in the K-12 matics Study Group (SMSG) projects provide
curriculum develop (in parallel and interac- clear instances of a curriculum development
tively) over time periods of many years and

T 618 Theories of Learning Mathematics

project that attempted to make research and the- variables (p. 32) . . . . Many of our common
ory development important parts of their collec- beliefs about teachers are false, or at the very
tive agenda. For example, the National best rest on shaky foundations. For example,
Longitudinal Study of Mathematics Abilities the effects of a teacher’s subject matter knowl-
(NLSMA) was an important part of SMSG initia- edge and attitudes on students learning seem
tives. Nonetheless, in their introduction to to be far less powerful than most of us had
Begle’s book, Wilson and Kilpatrick reported realized (p. 54).
that “(Begle) tried to persuade the SMSG Most mathematics educators surely believe
advisory board to sponsor research as well as that teacher-level understandings of topics to be
curriculum development, but he was not successful taught should involve understanding both more
(p. x.).” Similarly, in his keynote address at and also differently than students. But, we still
the First International Congress of Mathematics know little about the nature of these teacher-level
Education, Begle stated: understandings.
• Concerning Problem Solving: A substantial
I see little hope for any further substantial improve- amount of effort has gone into attempts to find
ments in mathematics education until we turn out what strategies students use in attempting
mathematics education into an experimental to solve mathematical problems . . .. But no
science – until we abandon our reliance on philo- clear-cut directions for mathematics education
sophical discussions based on dubious assump- are provided by the findings of these studies. In
tions, and instead follow a carefully constructed fact, there are enough indications that prob-
pattern of observation and speculation, the pattern lem-solving strategies are both problem- and
so successfully employed by physical and natural student-specific to suggest that hopes of finding
scientists. one (or a few) strategies which should be
taught to all (or most) students are far too
Much of what goes on in mathematics simplistic (p. 145).
education is based on opinions that are so firmly In the NCTM’s most recent Handbook of
held that the thought of doubting them crosses very Research in Mathematics Education (Lester
few minds. Yet, most of these opinions have no 2007), the chapter on problem solving (Lesh
empirical substantiation, and in fact many of them and Zawojewski 2007) concludes that very little
are, if not wrong, at least in need of serious has changed since Begle’s time. New words (such
qualifications (p. xvi). as metacognition, or habits of mind) have been
introduced to replace previously discredited con-
In other words, Begle believed that a large structs (such as those reviewed by Begle), but the
share of what paraded as theory in mathematics following fundamental issues remain. (a) Strate-
education was (and continues to be) dogma. For gies, heuristics, or other meta-level procedures
example, in his reviews of the literature in topic which seem to provide useful after-the-fact
areas ranging from problem solving to teacher descriptions of what successful problem solvers’
development, Begle identified many examples behaviors seem to have done do not necessarily
of dubious opinions which continue to go provide prescriptions of what novice problem
unquestioned. solvers should do next during ongoing problem-
• Concerning Teacher-Level Knowledge: solving activities, and (b) if attention focuses on
a small number of larger or more general rules of
Despite all of our efforts, we still have no behavior, then these general rules tend to lack
way of deciding, in advance, which prescriptive power. But, if attention focuses on
teachers will be effective and which will not. a larger number of smaller or more specific rules
Nor do we know which training programs will of behavior, then knowing when to use such
turn out effective teachers and which ones will behaviors is a large part of what it means to
not (p. 29) . . . . The outcomes of teaching does understand them. And transfer of learning that
not depend just on the teacher (or the program
used) but rather is the result of complex inter-
actions among teachers, students, the subject
matter, the instructional materials available,
the instructional procedure used, the school
and community, and who knows what other

Theories of Learning Mathematics T619

was expected to occur in such studies has been shift attention to curriculum development and T
unimpressive. program development. Critics often accuse the
mathematics education research community of
Theories Versus Models failing to provide “scientifically sound” empirical
Sriraman and English’s book identifies a trend in evidence about curriculum materials that “work”
which theory development shifts toward model (ref needed). But, most mathematics education
development; and modeling perspectives are researchers are also practitioners – e.g., teachers,
being used to provide alternatives to traditional teacher educators, or developers of curriculum
theories related to topic areas ranging from materials. And it is precisely their practitioner
teacher development to problem solving; and side that makes them aware of the uselessly sim-
accompanying design research methodologies plistic nature of most studies claiming to show
are being used to supplement what can be inves- that some curriculum innovation “works” – using
tigated using more traditional methods. standardized and randomly assigned “treatment
groups” and “control groups” in situations where
The key assumption that underlies a model (a) the criteria for “working” tend to be poorly
and modeling perspective is that all relevant aligned with the most important goals of the
“subjects” – including not only students and curriculum that is used, (b) it is well known that
teachers but also researchers themselves – are “working” depends on far more than the curricu-
model developers. Students develop models to lum materials themselves, and (c) curriculum
make sense of mathematical problem-solving sit- innovations don’t simply act on students and
uations that do not occur in a pre-mathematized teachers – students and teachers also react (or
form. Teachers develop models to make sense of act back)! So, successful curriculum innovations
students’ model development activities. And usually involve continual adaptations – based on
researchers develop models of interactions the strengths and weaknesses of individual stu-
among students, teachers, and learning environ- dents and teachers and based on their reactions at
ments. For example, in the case of both teaching various stages of implementation.
and problem solving, it is widely recognized that
highly effective people not only do things differ- The Complexity of Models in Mathematics
ently than their less experienced or less effective Education
counterparts, but they also see (or interpret) To see why no two situations are never exactly
things differently. Furthermore, the interpretation alike and why the same thing never happens
systems that they develop are both learnable and twice, consider the following. During the 1980s
assessable – as well as being powerful, sharable, and 1990s, a number of learning theorists who
and reusable (i.e., transferrable). wanted to apply their learning theories to mathe-
matics education developed a methodology
Similarly, according to MMP, students’ con- called aptitude-treatment-interaction studies
ceptual understandings of “big ideas” are expected (ATI). These ATI studies recognized that, even
to involve conceptualizing (mathematizing or in very simple learning situations (e.g., one stu-
mathematically interpreting) situations; relevant dent and one teacher), different students reacted
models are expected to involve the gradual inte- differently to a given treatment. So, attempts
gration, differentiation, reorganization, and adap- were made to identify profiles of student attri-
tation of existing models. In other words, for a butes (A1, A2, . . ., An) which could be matched
given “big idea” in the K-12 curriculum, a large with alternative preplanned treatment attributes
part of “conceptual understanding is expected to (T1, T2, . . . Tm). But, the results of these ATI
involve the development of powerful, sharable, studies showed that progressively finer-grained
and reusable models. student and treatment profiles not only led to
unworkable combinatorial nightmares, but they
To highlight some other important ways that also involved feedback loops in which students
model development is expected to contribute to
theory development, while at the same time being
different than theory development, it is useful to

T 620 Theories of Learning Mathematics

Theories of Learning Mathematics, Fig. 1 Two identical starting points for a double pendulum system

acted on treatments as much as treatments acted No research methodology is “scientific” if it is
on students. So, what emerged in such situations based on assumptions that are inconsistent with
is similar to what happens when two identically those that are considered to be reasonable for the
configured double pendulums are set in motion at subjects and situations being investigated. So,
exactly the same time. Within a few cycles, iden- a fundamental dilemma that mathematics educa-
tical systems will function in ways that are quite tion researchers face is that (quite often) they are
different – and unpredictably so (as shown in the trying to understand subjects that they (as
Figs. 1 and 2 below). a community) also are trying to change, design,
or develop. This means that mathematics educa-
These kinds of systems are studied in a branch tion researchers tend to be more like engineers
of mathematics known as complexity theory. and other “design scientists” than they are like
And one thing complexity theory implies is that, “pure” scientists in fields such as physics or
even in situations that are as simple as a double chemistry. In a completely “pure” science,
pendulum, feedback loops tend to lead to a theory would tell which problems are priorities
unpredictable behaviors in only a few cycles. to solve; the theory also would determine the
So, simple input–output rules of the form correctness of permissible solution processes;
{Use treatment A and result B will occur.} are and the theory also would determine when the
not likely to work for situations involving problem is solved. Whereas, in design sciences,
student-teacher interactions, student-student problems arise in the “real world” (outside of any
interactions, teacher-treatment interactions, and theory); solution processes usually need to inte-
student-treatment interactions – all functioning grate ideas and procedures drawn from a variety
simultaneously.

Theories of Learning Mathematics T621

Theories of Learning Mathematics, Fig. 2 Stopping the two systems after 10 s

of disciplines (or textbook topic areas); and prob- portions of their time and energy to the develop- T
lems are not solved until the relevant real-life ment of tools to provide infrastructure for their
issue is resolved. own use. So, it is revealing that the mathematics
education research community still does not have
Why do realistically complex problems tend tools to document and assess the most important
to require solutions which draw on more than a achievements that are expected of students,
single theory? One reason is because “real-life” teachers, or programs.
problems often involve partly conflicting con-
straints – such as high quality and low costs, low To recognize why lack of accumulation has
risk and high gain, simple and complete. Is a Jeep been such a problem in mathematics education,
Cherokee a better buy than a Ford Taurus or consider the following facts. If it were possible (It
a Toyota Prius? Answers depend on preferences of isn’t!) to inspect the archives of all past curricu-
relevant decision makers. So, “one size fits all” is lum innovation projects which have been
seldom a principle that decision makers will accept. supported by agencies such as the US National
Science Foundation, then (beginning with early
The central shortcoming of mathematics edu- projects such as School Mathematics Study
cation research is not a lack of success in produc- Group, The Madison Project, and MiniMast and
ing effective programs and materials. The central continuing up to current times) inspectors of
problem is lack of accumulation – coupled with these archives would have no difficulty produc-
the repeated recycling of previously discredited ing convincing evidence that important parts of
ways of thinking. And for accumulation to occur, most of these projects would be highly likely to
it is important to notice that, in mature sciences, be useful and effective today (under some
research communities tend to devote large

T 622 Theories of Learning Mathematics

conditions and for some students, some teachers, 2. Research on models and modeling has shown
some schools, and some communities). On the that thinking is far more situated than tradi-
other hand, other parts clearly would be missing tional perspectives have suggested – because
or in need of significant revision. For example, thinking tends to be organized around experi-
most projects that focused on the development of ence as much as it is organized around
innovative learning materials for children were abstractions.
not accompanied by adequate teacher develop-
ment materials or implementation plans to help 3. For a given concept, understandings develop
projects evolve from entry-level implementations along a variety of interacting dimensions: con-
(during the first year) to more complex and com- crete-abstract, situated-decontextualized, spe-
prehensive implementations (during the Nth cific-general, intuition-formalization, etc.
year). Furthermore, most of these projects did
not provide assessment tools to document the 4. In each of the preceding dimensions, there
achievements of higher-level achievements of exist “zones of proximal development”
students, teachers, or programs. (ZPD) similar to those described by Vygotsky.
Can these ZPDs be unpacked?
If mathematics education researchers pointed
to one topic area where they believe theory 5. The development of “big ideas” interacts – so
development to be strongest, they’d likely point that understandings of any one of them depend
to either (a) early number concepts or (b) early partly on the development of others.
algebraic reasoning (or rational numbers and pro- We conclude this encyclopedic entry with
portional reasoning). Evidence of this theory
development in learning is found in the literature more questions than answers per se, with the
related to Piaget-like cognitive structures (Steffe hope of the community becoming interested in
1995; Steffe et al. 1996), cognitively guided answering these fundamental questions in their
instruction which focuses on task variables quest for developing theories of mathematical
which are not at all like Piagetian cognitive struc- learning.
tures, the focus on counting strategies and • How do understandings of various “big ideas”
Vygotsky’s socially mediated views of develop-
ment, and focus on computer-based embodiments interact?
which are in some ways similar to those used by • How does the development of “big ideas”
Zoltan Dienes (Sriraman 2008) but which also
emphasize constructs similar to those empha- interact with the development of “basic
sized by Steffe. skills”?
• How does the development of “big ideas”
Yet, each of the preceding perspectives are interact with the ability to use these ideas
based on significantly different (and in some in situations that are not pre-mathematized
ways incompatible) ways of thinking about math- (outside of mathematics classrooms)?
ematics concept development. One place where
differences can be seen where the preceding per- Cross-References
spectives differ has do with “learning trajecto-
ries” (or “learning progressions”) through which ▶ History of Mathematics Teaching and
development occurs. The notion of “learning Learning
trajectories” generally describes development
(in both learning and problem-solving situations) ▶ Policy Debates in Mathematics Education
as if it were like a point moving along a path. Yet, ▶ Zone of Proximal Development in
the following facts are well known:
1. It is easy to change the difficulty of a given Mathematics Education

task by several years by varying mathemati- References
cally insignificant aspects of the task.
Begle EG (1979) Critical variables in mathematics educa-
tion: findings from a survey of the empirical literature.
Mathematical Association of America, Washington, DC

Types of Technology in Mathematics Education T623

Common Core State Curriculum Standards with terms “computers,” “computer software,” T
(2012) Retrieved from http://www.corestandards.org/ and “communication technology,” according to
about-the-standards/key-points-in-mathematics Laborde and Str€aßer (2010), p. 122. Another term
“digital technology” which denotes a wide range
Lesh R, Zawojewski JS (2007) Problem solving and of devices including a hardware (such as proces-
modeling. In: Lester F (ed) Second handbook of sor, memory, input–output, and peripheral
research on mathematics teaching and learning. Infor- devices) and software (applications of all kinds:
mation Age Publishing, Greenwich technical, communicational, consuming, and
educational) is used by Clark-Wilson, Oldknow,
Lester F (2007) Second handbook of research on mathe- and Sutherland (2011). This is contrasted with yet
matics teaching and learning. Information Age Pub- another term Information and Communication
lishing, Reston/Greenwich Technology (ICT) widely used in a variety of
educational contexts and describes the use of
National Council of Teachers of Mathematics (1989) Cur- so-called “generic software” which means word
riculum and evaluation standards for school mathe- processing, spreadsheets, along with presenta-
matics. NCTM, Reston tional and communicational tools (such as
e-mail and the Internet) (2011).
Popper K (1963) Conjectures and refutations: the growth
of scientific knowledge. Routledge, London Historical Background

Sriraman B (2008) Mathematics education and the legacy Historically, technology and mathematics go
of Zoltan Paul dienes. Information Age Publishing, alongside by mutually influencing each other’s
Charlotte development (Moreno and Sriraman 2005). His-
tory does provide us with many technologies that
Sriraman B, English LD (eds) (2010) Theories of mathe- enhance people to count (stones, pebbles, bones,
matics education: seeking new frontiers, Advances in fingers), to calculate (abacus, mechanic devoices,
mathematics education series. Springer, Heidelberg electronic devices), to measure (ruler, weights,
calendar, clock), to construct (compass, ruler),
Steffe LP (1995) Alternative epistemologies: an educator’s and to record statistical data (cards with holes,
perspective. In: Steffe LP, Gale J (eds) Constructivism spreadsheets) (Fig. 1).
in education. Lawrence Erlbaum, Hillsdale, pp 489–523
As example of such devices, we can name the
Steffe LP et al (1996) Theories of mathematical learning. famous Ishango bone, an artifact of ingenious
Lawrence Erlbaum, Hillsdale mind of our ancestors recently analyzed by
Pletser and D. Huylebrouck (1999) who point at
Vygotsky L (1986) Thought and language. The MIT press, its possible function as one of the oldest known
Cambridge computational tool along with its other possible
uses (calendar, number system, etc.). The inven-
Types of Technology in Mathematics tion of mechanical counting devices takes its ori-
Education gins from different kinds of abacus, such as Greek
abax, meaning reckoning table covered with the
Viktor Freiman dust or later version with disks moving along some
Faculte´ des sciences de l’e´ducation, Universite´ de lines (strings) (Kojima 1954). It is interesting that
Moncton, Moncton, NB, Canada in some cultures, abacus was used till very recent
times, as in Russia, in the everyday commerce to
Keywords do calculations with moneys. Today, they may
appear as educational support to enhance reasoning
Computers; Computer software; Communication about quantities, such as Rekenrek (Blanke 2008).
technology; Handheld; Mobile; E-learning

Terms and Definitions

Many of today’s mathematics classrooms around
the world are nowadays equipped with a variety
of technologies. By using the term “technology,”
we mainly mean “new technology,” as we refer to
the “most prominent,” recent, and “modern tool”
in the teaching of mathematics that is labeled

T 624 Types of Technology in Mathematics Education

Types of Technology in
Mathematics Education,
Fig. 1 http://nrich.maths.
org/6013

Types of Technology in Mathematics Education, the Pascaline, a mechanical calculator invented in
Fig. 2 http://www.sciencemuseum.org.uk/objects/math- 1642 by Pascal. Leibnitz (1673) and Babbage
ematics/1927-912.aspx (1822) were among others who significantly con-
tributed to the advancement in creation of auto-
Punch cards were invented by Hollerith, and his matic calculators which led, in the first half of the
machine was used by the US Census Bureau to 1920s century, to the construction of the first
process data from 1890 till 1950s when it was computers, such as ENIAC (Electrical Numerical
replaced by computers (http://www.census.gov/ Integrator and Calculator), by Mauchly and
history/www/innovations/technology/the_hollerith_ Eckert, in 1946, mainly for military purposes.
tabulator.html) (Fig. 2). The second half of the twentieth century was
marked by the rise of the IBM (International
First Computers and Their Use in Business Machines); one of its models was used
Education to prove the famous Four-Color Theorem (Appel
and Haken 1976) (Fig. 3).
Computers themselves can be seen as “mathe-
matical devices,” and their timeline goes back The time period after 1950 and till early 1980s
to abacus and is further marked by names of was marked by as rather slow but sure penetration
Leonardo da Vinci who conceived the first of mainframe and minicomputers in education,
mechanical calculator (1500), followed by including mathematics education. With the main
“Napier’s bones” invented by Napier for multi- focus on accessibility of such devices for schools
plication (1600), based on the ancient numerical (question of costs and space), other questions
scheme known as the Arabian lattice; then comes arose by mathematics educators at that time
regarding the purposes of its use and impact on
learning. Zoet (1969) pointed at several
dilemmas, namely, (1) about the capacity of com-
puters to process data, like in business manage-
ment to produce bills for millions of customers,
on the one side, and to compute data, like in
mathematical modelling where scientists need
to do large amount of calculations in a short
period of time; (2) about the time needed to
master a particular part of technology (to solve
mathematical problems) . . . which will soon be
replaced with a new one; and (3) about the pos-
sibility of computer to assist a greater number of
students to grasp principles of mathematics, as
well as strengthen and broaden students’ under-
standing, about whether mathematics learned by

Types of Technology in Mathematics Education T625

Types of Technology in
Mathematics Education,
Fig. 3 http://www-03.ibm.
com/ibm/history/exhibits/
mainframe/mainframe
_PP3168.html

Types of Technology in Mathematics Education, Mathematics Education (Hansen and Zweng T
Fig. 4 http://el.media.mit.edu/logo-foundation/logo/turtle. 1984) portrays newest types of technologies
html, repeat 3 [forward 50 right 60] called microcomputers as having endless list of
applications available for mathematics teachers
the students will be more functional, once they and learners which are becoming widely accessi-
see how it is used in computers, or if small com- ble for schools at low cost; it also adds graphics
puters can be integrated into mathematics pro- capabilities to support mostly two-dimensional
grams as the slide rule in the training of representations (Fey and Heid 1984). Again this
engineering students. technology development interacts with pedagogi-
cal use as tutor, tool, and tutee (Fey and Heid
In 1970s–1980s, special languages, like FOR- 1984, referred to Taylor 1980) with questioning
TRAN, PASCAL, BASIC, were used as the first whether “traditional collection of mathematical
software, and their mastery was necessary to skills and ideas needs” to be acquired by students
use computers effectively including mathematics to enable them “to operate intelligently in the
calculations and modelling of mathematical computer-enhanced environment for scientific
processes and thus enhancing learning. One of work,” or one must have “new skills or under-
such languages (LISP) was used to create the standings” to get prepared “for mathematical
LOGO, a programming language designed by demands that lie in the twenty-first century”
Papert (1980) specifically for educational pur- (Taylor 1980, p. 21). Regarding the format of
poses. According to Pimm and Johnston-Wilder integration of such technology in the process of
(2005), a common starting point in creating teaching, educational institutions usually put
LOGO programs was writing commands computers in one classroom (computer lab)
allowing for directing and controlling a “turtle” shared by several groups of students, or they can
on the screen. This idea led to construction put a number of desktop computers (1–4) in
of specific mathematically rich learning environ- a regular classroom, so teachers and students can
ments called microworlds (Pimm and Johnston- work with them individually or in small groups.
Wilder (2005)) (Fig. 4).
On those computers, teachers could find gen-
In 1984, the NCTM (National Council of the eral software, including spreadsheets (like
Teachers of Mathematics) produced a yearbook Supercalc, Lotus, or Excel) that could be used in
entirely devoted to the topic on Computers in multiple teaching and learning purposes, for
example, to conduct probabilistic experiments

T 626 Types of Technology in Mathematics Education

and simulations (Anand et al. 2012). 1980s and creating and sharing musical fragments. In 2004,
1990s were also marked by widely spread use of the Internet bookstore Amazon.com allowed
educational games, on small floppy disks, and buying books entirely online. In 2005, the
later multimedia on CD-ROMs and DVDs, help- video-sharing site Youtube.com appeared,
ing even the very young students to learn basics allowing producing and sharing short video
about numbers and shapes and develop sequences. The authors state that by the year
mathematical thinking while playing with pat- 2005, the Internet had grown more in 1 year
terns. Another kind of the software specifically than in all the years before 2000, reaching
designed for mathematics classroom based on a 1,000,000,000 sites by 2006.
constructionist’s ideas leads to the development
of dynamic and interactive computer environ- The result of this tremendous growth of Inter-
ments in geometry (dynamic geometry systems) net-based environments and the educational
and algebra (computer algebra systems). Different resources generated by them is a transformation
types of virtual manipulatives thus become avail- of e-learning itself. According to O’Hear (2006),
able to teachers to make learning more visual, the traditional approach to e-learning was based
dynamic, and interactive (Moyer et al. 2002). on the use of a Virtual Learning Environment
(VLE) which tended to be structured around
Computer networks – systems of courses, timetables, and testing. That is an
interconnected computers and systems of their approach that is too often driven by the needs of
support called Intranet and Internet that emerge the institution rather than the individual learner.
and spread out in 1990s and 2000s. The first In contrast, the approach used by e-learning 2.0
(Intranet) allows to connect computers with (a term introduced by Stephen Downes) is “small
a restraint number of people having access to it; pieces, loosely joined,” as it combines the use of
often it is used within an organization, like school discrete but complementary tools and web ser-
or school board, or university. The second (Inter- vices – such as blogs, wikis, and other social
net) is open to a much wider audience, in many software – to support the creation of ad hoc learn-
cases worldwide, although it can serve closed ing communities. Let us look at several features
groups/communities built with different pur- of these tools as we analyze a few examples of
poses. This technology, with the time becoming mathematical opportunities they create (adapted
more rapid (high-speed), wireless, and handheld, from Freiman 2008).
enhances communication of people or machines
with other people or machines to share informa- Wiki is an Internet tool allowing a collective
tion and resources in all areas including mathe- writing of different texts as well as sharing
matics. As example of such kind of technology, a variety of information. Everybody can eventu-
we will analyze web 2.0 tools. ally be a contributor to the creation of a web site
on a certain topic (or several topics, as it is in the
E-learning: Web 2.0 Tools and Their Use case of the Wikipedia, www.wikipedia.org/).
in Mathematics
Podcasts can be used to audio-share mathe-
Solomon and Schrum (2007) use the year 2000 as matical knowledge among a larger auditorium
a turning point in the development of a new Inter- than one with people sitting in a traditional class-
net-based technology called Web 2.0. They begin room. It can be used as a method of delivering
their timeline with year 2000 when the number of mathematical lectures online as well as for the
web sites reached 20,000,000. The year 2001 was promotion of mathematics.
marked by the creation of Wikipedia, the first
online encyclopedia written by everyone who Video-casting opportunities are provided by
wanted to contribute to the creation of the shared multiple Internet sites, allowing the creation and
knowledge. In 2003, the site iTunes allowed sharing of video sequences produced by the users.
For example, an article published in one local
newspaper informs the readers about one univer-
sity professor who put a 2-min video about
a Mobius strip on the Youtube.com site.

Types of Technology in Mathematics Education T627

The sequence was viewed by more than one trends related to the web 2.0 technology: knowl- T
million users within 2 weeks. The environment edge building/co-constructing, knowledge shar-
offers not only an opportunity to view the video ing, and socialization by interaction with other
but also to assess it (using a 5-star system) and to people. Moreover, further development towards
share it with others, as well as publish a comment. semantic web (web 3.0) technology has
a potential to enhance self-learning, critical
Photo sharing is yet another form of creating thinking, and collaborative and exploratory
and sharing knowledge, available on several learning.
dynamic sites with photo galleries like Flickr.
Regrouped by categories that can be found by M-Learning: Anytime, Anywhere with
an easy-to-use search engine, the photos can be Laptops and Other Handheld Devices
published and discussed by the members of
a community, as for example, the community Another recent trend is related to the rapid
that discusses geometric beauty which numbers changes brought by so-called mobile technology
almost 5,000 members. Each photo is provided that enhances anytime anywhere learning. Tak-
with a kind of ID card that documents useful ing its roots from different types of calculators, it
information such as the date of its publication, provides today’s mathematics classrooms with
the author’s (or publisher’s) username, as well as several types of portable devices, such as laptop
the list of all other categories to which the photo computers, iPads, iPhones, and other types of
belongs, the date when the photo was taken, and mobile technology (Fig. 5) (Jones et al. 2012).
how many other users added it to their albums.
According to Burrill et al. (2002), the first type
Discussion forums allow building online com- of handheld technology mentioned as a part of the
munities that talk to each other by posting ques- secondary school curriculum in 1986 was a Casio
tions and giving answers. This collective work fx-7000G model. Even if the appropriate role of it
may enable a student who is struggling with in mathematics classroom was at that time (and
mathematical homework to address other people still remains) debatable, it supported the creation
and ask for help, as illustrated by the following of new visions for mathematics education while
example from the math forum site (mathforum. calling for broader access to deeper mathematics
org). The message posted by one user says that for all students (Burrill et al. 2002). Regarding
“after having asked a teacher and having read the newest development of this type of technol-
a book,” she “still had a feeling” that she needed ogy, Burrill (2008) sees its potential to combine
more explanation, so she appealed to the whole various learning environments like computer
virtual community asking for help. The discus- algebra systems (CAS) and Dynamic Geometry
sion on some questions can take the form of computer software, such as Dynamic Geometry
multiple exchanges between members. Sketchpad or Cabri: “new technologies such as
TI-Nspire bring together both of these
Blogs may provide multiple educational environments in one handheld, providing the
opportunities as they are built by means of easy- opportunity to create an even wider variety of
to-use software that removes the technical dynamic linked representations, where
barriers to writing and publishing online. The a change in one representation is immediately
“journal” format encourages students to keep and visibly reflected in another” (http://tsg.
a record of their thinking over time facilitating icme11.org/document/get/218).
critical feedback by letting readers add
comments – which could be from teachers, peers, Several laptop studies report about a variety of
or a wider audience. Students may use blogs for teaching and learning opportunities to use 1–1
different purposes: to provide a personal space portable technology for several subjects includ-
online, pose questions, publish work in progress, ing mathematics. Freiman et al. (2011) developed
and link to and comment on other web sources. and implemented problem-based learning (PBL)

The learning model that can be extracted from
our examples features three major educational

T 628 Types of Technology in Mathematics Education

Bracket Basics ) the individual base, or working in small groups,
as the technology becomes less costly, more
Challenge. Make this number: 30 flexible in terms of usability, with better feedback
options, allows for better merging with other
×( + mathematical learning environments, such as
dynamic geometry (G€uc¸ler et al. 2012).

364 Cross-References

Types of Technology in Mathematics Education, ▶ Information and Communication Technology
Fig. 5 http://www.k12mathapps.com/ (ICT) Affordances in Mathematics Education

interdisciplinary scenarios (math, science, lan- ▶ Learning Environments in Mathematics
guage arts) to measure and document students’ Education
actual learning process, particularly in terms of
their ability to scientifically investigate authentic ▶ Learning Practices in Digital Environments
problems, to reason mathematically, and to com- ▶ Technology and Curricula in Mathematics
municate. In a rapidly changing world of technol-
ogy and infinity of educational applications, Education
mathematics teachers can now try to integrate ▶ Technology Design in Mathematics Education
newest technology, like iPads, in mathematics
lessons. While only few research available, first References
pilot studies, like one reported by HMH
(2010–2011, http://www.hmheducation.com/fuse/ Anand R, Manju M, Anju M, Kaimal V, Deeve NV,
pdf/hmh-fuse-riverside-whitepaper.pdf) seems to Chithra R (2012) Teaching computational thinking in
have a positive impacts on students’ performance. probability using spread sheet simulation. Int J Sci Res
In this study, individual iPads were used along Publ 2(12)
with the HMH Fuse: Agebra 1 programs. The
application helped students use its multimedia Appel K, Haken V (1976) Every map is four colourable.
components whenever and wherever they saw fit, Bull Am Math Soc 82:711–712
regardless of Internet availability. In addition,
students could take device home and “customize Blanke B (2008) Using the rekenrek as a visual model
them,” adding their own music, videos, and addi- for strategic reasoning in mathematics. The Math
tional applications (Freiman et al. 2011). Learning Center, Salem

Among other types of technologies to be Burrill G (2008) The role of handheld technology in
mentioned are interactive whiteboards which, teaching and learning secondary school mathematics.
according to Jones (2004), might encourage Paper presented at ICME-11, Monterrey, 2008
more varied, creative, and seamless use of teach-
ing materials; increase student’ enjoyment and Burrill G, Breaux G, Kastberg S, Leatham K, Sanchez W
motivation; and facilitate their participation (2002) Handheld graphing technology at the second-
through the ability to interact with materials. ary level: research findings and implications for class-
While the whiteboards support and extend room practice. Texas Instruments Corp, Dallas. http://
whole-class teaching in a more interactive way, education.ti.com/research
haptic (in-touch) devices have a potential to
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technologies and mathematics education: executive
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Kingdom, London

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T

U

Urban Mathematics Education and classrooms with majority Black, Brown,
and/or recent immigrant students [i.e., bilingual
David Wayne Stinson and multilingual students]).
Middle and Secondary Education Department,
Georgia State University, Atlanta, GA, USA Suburban School/Urban School Binary
Over the past 40 years or so, a discursive binary
Keywords between suburban school and urban school has
emerged that privileges the suburban. This
Culture; Diversity; Ethnicity; Race; “High- privileging has resulted in (re)segregated urban
needs” schools; Socioeconomic class; Urban schools being further defined by euphemisms
education; Urban schools such as “hard-to-staff schools,” “high-needs
schools,” and/or “low-performing schools”
Definition (Lipman 2011). Such euphemisms are used to
gloss over challenges that too often plague
Urban Mathematics Education (and/or Urban urban schools such as ageing and inadequate
Mathematics Education Research) is character- infrastructures; dense and disconnected bureau-
ized as a specific focus on the multilayered com- cracies; uncertified or inadequately trained
plexities as well as the challenges and promises teachers; limiting and misdirected funds; and
of mathematics teaching and learning in high- the ever-lingering damaging effects of race
density populated geographic areas. These and racism, and xenophobia in general (Darling-
“urban” areas more times than not contain greater Hammond 2010). These challenges, which
human and cultural diversity in terms of “race,” typically have been found within what was
ethnicity, socioeconomic class, language, reli- commonly known as the urban “inner-city”
gion, disabilities/abilities, and sexual orientation school, are increasingly found in suburban and
and gender expression. Often times the phrase even rural schools (i.e., metropolitan suburban/
urban schools and, in turn, urban mathematics rural sprawl) as schools in these geographic areas
education are used as euphemistic proxies for face similar challenges with the ever-changing
(re)segregated schools and mathematics class- racial, socioeconomic class, and citizenship
rooms with high concentrations of poor and his- status student demographic of suburban and
torical marginalized racial and/or ethnic student rural communities and schools. These
populations (e.g., in the United States, schools ever-changing demographics are motivated, in
part, by gentrification of inner-city urban spaces
(Lipman 2011).

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

U 632 Urban Mathematics Education

Research in Urban Mathematics Education education research, see the Journal of
Many researchers who work within the urban Urban Mathematics Education (JUME), a
mathematics education domain deconstruct the peer-reviewed, open-access, academic journal
euphemisms of urban schools as they make published twice a year: http://education.gsu.edu/
the social (Lerman 2000) and sociopolitical JUME.
(Gutie´rrez 2013) turns in mathematics education
research. These researchers most often place an Cross-References
emphasis on contextualizing not only the mathe-
matics classroom but also the concentric circles ▶ Bilingual/Multilingual Issues in Learning
of school, district, community, and society at Mathematics
large in which the urban mathematics classroom
is embedded (Weissglass 2002). Such contextu- ▶ Cultural Diversity in Mathematics Education
alization makes possible a more complete ▶ Cultural Influences in Mathematics Education
analysis of the effects of the neoliberal and ▶ Equity and Access in Mathematics Education
neoconservative agenda of urban (mathematics) ▶ Immigrant Students in Mathematics Education
education (Lipman 2011). Here, the mathematics ▶ Rural and Remote Mathematics Education
teaching and learning dynamic of the classroom ▶ Socioeconomic Class in Mathematics
is not stripped of the sociocultural and sociopo-
litical power relations that exist within the Education
multiplicity of interactions that occur in the
mathematics classroom among teachers and stu- References
dents and the mathematics being taught and
learned. Analyses of such power relations bring Darling-Hammond L (2010) The flat world and education:
to the fore issues of equity and access, identity, how America’s commitment to equity will determine
and race, class, gender, language, and other our future. Teachers College Press, New York
sociocultural and sociopolitical discourses and
practices that marginalize or silence groups of Gutie´rrez R (2013) The sociopolitical turn in mathematics
students, which, in turn, limit mathematics education. J Res Math Educ 44(1):37–68. Retrieved
access, participation, and contribution of large from http://www.nctm.org/publications/toc.aspx?jrn
groups of students. l¼JRME&mn¼6&y¼2010

Since the early to mid 2000s, a new trend in Leonard J, Marin DB (eds) (2013) The brilliance of Black
urban mathematics education research has children in mathematics: beyond the numbers and
emerged that highlights and examines the math- toward new discourse. Information Age, Charlotte
ematics achievement and persistence of Black
and Brown children within urban contexts and Lerman S (2000) The social turn in mathematics
the effectiveness of urban teachers, schools, and education research. In: Boaler J (ed) International per-
districts (see, e.g., the edited volumes Leonard spectives on mathematics education. Ablex, Westport,
and Marin 2013; Martin 2009; Te´llez et al. 2011). pp 19–44
Much of this emerging research provides
counter-stories or -narratives to the discourses Lipman P (2011) The new political economy of urban
of deficiency and ineffectiveness that too often education: neoliberalism, race, and the right to the
frame urban mathematics students, teachers, and city. Routledge, New York
classrooms. For exemplars of urban mathematics
Martin DB (ed) (2009) Mathematics teaching, learning,
and liberation in the lives of Black children.
Routledge, New York

Te´llez K, Moschkovich J, Civil M (eds) (2011) Latinos/as
and mathematics education: research on learning and
teaching in classrooms and communities. Information
Age, Charlotte

Weissglass J (2002) Inequity in mathematics education:
questions for educators. Math Educ 12(2):34–39

V

Values in Mathematics Education education and how they seem to develop in both
the individual and society, it finally introduces
Alan Bishop the challenging issues of whether desirable
Faculty of Education, Monash University, values can be developed in students through
Melbourne, VIC, Australia mathematics education and how values in
mathematics education should be developed.

Introduction Values as Personal Constructs
Krathwohl et al. (1964) significant book on
Values are a significant feature of education in educational goals gives us a useful starting
any field, but it is only recently that values in point. Their work was based on a behaviorist
mathematics education have been considered sig- approach and was hierarchical in structure. Thus
nificant, or even recognized. This entry provides at their levels 3 and 4 (from 5), one finds the
a historical perspective to the growing relevance following categories of goals:
of values in mathematics education. It also 3. Valuing: 3.1 acceptance of a value, 3.2 prefer-
illustrates how different researchers have
addressed different aspects of values depending ence for a value, and 3.3 commitment
on their theoretical and educational foci. 4. Organization: 4.1 conceptualization of a value

One focus has values being addressed as and 4.2 organization of a value system.
a characteristic of the person, which approaches Of particular interest is their behaviorist
values as a psychological construct. It builds on background theory which gives us the distinction,
the research in mathematics education which and relationship, between values and valuing.
explores values as related to learners’ and From an educational viewpoint, this distinction is
teachers’ attitudes, beliefs, and affect generally. highly significant. “Valuing” is clearly a behavior
but with no specification of what is to be valued.
The other main research focus conceptualizes “Values” on the other hand represent what is to be
values as a sociocultural construct and is more valued, a totally different educational objective.
concerned with the sociocultural context of The research of Raths et al. (1987) offers
mathematics education in which values are a related perspective. They describe seven gen-
observed and negotiated. This approach builds eral criteria for calling something a value. Their
on the relevant historical, cultural, and philosoph- criteria are (1) choosing freely, (2) choosing from
ical literatures at the intersection of mathematics alternatives, (3) choosing after thoughtful consid-
and education. eration of the consequences of each alternative,
(4) prizing and cherishing; (5) affirming,
While this entry describes generally the (6) acting upon choices, and (7) repeating.
different meanings of values in mathematics

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

V 634 Values in Mathematics Education

They say “unless something satisfies all seven culture of the context in which the learner
of the criteria, we do not call it a value, but rather (in this case) is operating. Lancy (1983) was an
a ‘belief’ or ‘attitude’ or something other than early researcher in this area, and he updated
a value” (Raths et al. 1987, p. 199). They add Piaget’s work with his research from Papua
“those processes collectively define valuing. New Guinea. He proposed that three stages
Results of this valuing process are called values” were/are significant in a learner’s development
(p. 201). Their emphasis on choices and choosing where cultural influence is paramount:
is also important in separating values from Stage 1, where genetic programming has its
beliefs. One may hold several different beliefs
but values are most likely to appear when the major influence and where socialization is
individual makes specific choices. This point is the key focus of communication.
important for both research and practice. Stage 2, where enculturation takes over from
socialization and, for example, where
Values in mathematics education have how- ethnomathematics becomes relevant.
ever generally been couched in terms of affect Stage 3, which concerns the metacognitive level
and attitudes. As a leading proponent of this and where different cultural groups emphasize
research, McLeod (1989, 1992) separates beliefs, different theories of knowledge. These theo-
attitudes, and emotions, where beliefs can be ries of knowledge represent the ideals and
about mathematics (e.g., mathematics is based values lying behind the actual language and
on rules), about self (e.g., I am able to solve symbols developed by a cultural group. Thus
problems), about mathematics teaching (e.g., in relation to the previous section, it is in
teaching is telling), and about the social context Stages 2 and 3 that values are inculcated in
(e.g., learning is competitive). Attitudes can be the individual learners, and Stage 3 is where
exemplified by a dislike of geometric proof, the the value system is developed.
enjoyment of problem solving, or a preference for In the classic work by Kroeber and Kluckhohn
discovery learning. Emotions appear through, (1952), they strongly support this general idea:
for example, joy (or frustration) in solving “Values provide the only basis for the fully intel-
nonroutine problems or an aesthetic response to ligible comprehension of culture because the
mathematics. actual organisation of all cultures is primarily in
terms of their values” (p. 340). Moreover culture
However he like others at that time made no has been defined as an organized system of values
reference to values, but one senses from his writ- which are transmitted to its members both for-
ing that he would see values as linking strongly mally and informally (McConatha and Schnell
with both beliefs and attitudes. Krathwohl et al. 1995, p. 81).
(1964) support this view: “Behaviour categorized Thus from the perspective of mathematics
at this level (3) is sufficiently consistent and sta- education, the idea of mathematical thinking as
ble to have taken on the characteristics of a belief a form of metacognition affected by the norms
or an attitude. The learner displays this behaviour and values of the learner’s society and culture is
with sufficient consistency in appropriate helpful. But where do these norms, values, and
situations that he (sic) comes to be perceived as knowledge come from, and how are they framed
holding a value” (p. 180). So from this perspec- in educational contexts?
tive, values grow out of beliefs and attitudes. Two points must be made here – firstly as
Bishop (1988) has explained, it was the values
Values as Sociocultural Constructs which have been held by previous mathemati-
The seminal work of Kroeber and Kluckhohn cians which have shaped the field we know as
(1952) and Kluckhohn (1962) gives us an entre´e mathematics today. Secondly, the research field
into this other historical, and related, dimension of ethnomathematics has demonstrated that all
of research on values. This is best summed up by cultures develop their own mathematical ideas
the construct “cultural psychology,” a branch of and practices. This has not only generated
psychology which takes into consideration the

Values in Mathematics Education V635

a great deal of interesting evidence, but it has any of the value goals and objectives outlined V
fundamentally changed many of our research there, apart from the idea that values education
ideas and constructs. The most significant influ- should involve the existence of alternatives,
ences have been in relation to: choices and choosing, preferences, and consis-
• Human interactions. Ethnomathematics tency. Bishop et al. (2001) set out to investigate
this in practice. The main conclusion was that
research concerns mathematical activities values did not seem to mean much to the mathe-
and practices in society, which take place out- matics teachers in the study, while much harder
side school, and it thereby draws attention to still for them was the idea of trying to “teach”
the roles which people other than teachers and different values from the ones they normally
learners play in mathematics education. “taught.” A further study focused on understand-
• Values and beliefs. Ethnomathematics ing the values that the students were learning.
research makes us realize that any mathemat- The idea that values are revealed at choice points
ical activity involves values, beliefs, and is only helpful when people have the opportunity
personal choices. to make valid and consistent choices. If one
• Interactions between mathematics and considers a “normal” mathematics classroom,
languages. Ethnomathematics research however, students rarely have the opportunity to
makes us aware that languages act as the exercise any choices.
principal carriers of mathematical ideas and
values in different cultures. There are many connections between values in
• Cultural roots. Ethnomathematics research mathematics and in science (Bishop et al. 2006).
is making us more aware of the cultural Their study showed that useful research on values,
starting points and histories of mathematical and its associated data collection, should stay close
development. to the experienced situation of the subjects, empha-
One example of an educational approach was sizing as Raths et al. (1987) argued, that values are
derived from the cultural perspective of White thoroughly personal attributes, and not easily
(1959), an anthropologist interested in the ways developed within the social context of a classroom.
cultures develop. Based on his research he
argued that for all cultures to develop, they need Not only are values personal attributes, they
cultural components which are technological, have a strong emotional characteristic, as
sentimental, societal, and ideological. McLeod (1992) also suggested. Future research
Translating this into mathematics, Bishop can potentially increase our understanding of the
(1988) argued that the value dimensions could relationship of values with the positive emotional
be formed of complementary pairs, using White’s side of mathematics learning.
categories, producing six values: rationalism and
objectivism (ideological), control and progress Cross-References
(sentimental), and openness and mystery
(sociological). The technological component is ▶ Anthropological Approaches in Mathematics
given by the symbolic technology of mathemat- Education, French Perspectives
ics. Using these categories research has explored
teachers’ values, students’ values, and values ▶ Ethnomathematics
in the mathematics curriculum and in teacher ▶ Teacher Beliefs, Attitudes, and Self-Efficacy
education.
in Mathematics Education
▶ Students’ Attitude in Mathematics Education

Educating Values and Developing References
Mathematics Education
One interesting fact is that there is little or no Bishop AJ (1988) Mathematical enculturation. Springer/
indications in the research literature above Kluwer, Dordrecht/Holland
concerning the educational means of attaining

V 636 Visualization and Learning in Mathematics Education

Bishop AJ, FitzSimons GE, Seah WT, Clarkson PC learners; Spatial abilities; Visual mental imagery;
(2001) Do teachers implement their intended values Inscriptions; Visual image; Ana-vis scale; Logic;
in mathematics classrooms? In: Heuvel-Panhuizen M Strength of mathematical processing; Type;
(ed) Proceedings of the 25th conference of the Verbal-logical; Visual-pictorial; Analytic, geo-
international group for the psychology of mathematics metric, and harmonic types; Reluctance to visu-
education. Freudenthal Institute, Utrecht, pp 169–176 alize; Pedagogy; Abstraction; Generalization;
One-case concreteness; Prototype; Uncontrollable
Bishop AJ, Gunstone D, Clarke B, Corrigan D (2006) image; Compartmentalization; Dynamic
Values in mathematics and science education: imagery; Pattern imagery; Metaphor; Mnemonic
researchers’ and teachers’ views on the similarities advantages; Interactive dynamic geometry
and differences. Learn Math 26:7–11 software; Gestures; Conversion processes;
Registers; Connections; Idiosyncratic visual
Kluckhohn C (1962) Values and value-orientations in the imagery; Reification; Computer technology;
theory of action: an exploration in definition and clas- Overarching theory of visualization
sification. In: Parsons T, Shils EA (eds) Toward
a general theory of action. Harper & Row Publishers, Definitions and Background
New York, pp 388–433
Visualization in mathematics learning is not new.
Krathwohl DR, Bloom BS, Masia BB (1964) Taxonomy Because mathematics involves the use of signs
of educational objectives, the classification of educa- such as symbols and diagrams to represent
tional goals: Handbook 2: affective domain. abstract notions, there is a spatial aspect involved,
Longmans, New York that is, visualization is implicated in its represen-
tation. However, in contrast with the millennia in
Kroeber AL, Kluckhohn C (1952) Culture: a critical review which mathematics has existed as a discipline,
of concepts and definitions. Peabody Museum Papers, research on the use of visual thinking in learning
vol 47 (1). Peabody museum of american archaeology mathematics is relatively new. Such research has
and ethnology, Harvard University, Cambridge, MA been growing in volume and depth since the
1970s, initiated by Bishop (1973, 1980) and later
Lancy DF (1983) Cross-cultural studies in cognition and Clements (1981, 1982), who investigated prefer-
mathematics. Academic, New York ences of individual learners with regard to visual-
ization in mathematics and how spatial abilities
McConatha JT, Schnell F (1995) The confluence of interacted with these preferences. Visualization
values: implications for educational research and has internal and external forms (Goldin 1992),
policy. Educ Pract Theory 17(2):79–83 which may be designated as visual mental imag-
ery and inscriptions, respectively (Presmeg 2006).
McLeod DB (1989) Beliefs, attitudes, and emotions: new Presmeg defined a visual image as a mental sign
views of affect in mathematics education. In: McLeod depicting visual or spatial information and
DB, Adams VM (eds) Affect and mathematical prob- inscriptions as symbols, diagrams, information
lem solving: a new perspective. Springer, New York, on computer screens, or any external representa-
pp 245–258 tion with a visual component. Following Piaget
and Inhelder’s (1971) claim that visual imagery
McLeod DB (1992) Research on affect in mathematics underlies the creation of a drawing or a
education: a reconceptualization. In: Grouws DA (ed) spatial arrangement, Presmeg did not pursue the
Handbook of research on mathematics teaching and distinction between external and internal visual
learning. Macmillan, New York, pp 575–596 images.

Raths LE, Harmin M, Simon SB (1987) Selections from Arcavi (2003, p. 217) blended definitions
values and teaching. In: Carbone PF (ed) Value theory given by previous authors (Hershkowitz et al.
and education. Krieger, Malabar, pp 198–214

White LA (1959) The evolution of culture. McGraw-Hill,
New York

Visualization and Learning in
Mathematics Education

Norma Presmeg
Department of Mathematics, Illinois State
University, Maryville, TN, USA

Keywords

Signs; Symbols; Diagrams; Spatial aspect;
Representation; Preferences of individual