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Encyclopedia of Mathematics Education

Encyclopedia of Mathematics Education

Keywords: Mathematics Education

Visualization and Learning in Mathematics Education V637

1989; Zimmermann and Cunningham 1991) to Theoretical Lenses V
provide the following summary: Early research on visualization in mathematics
(e.g., Clements 1981, 1982) used a conceptual
Visualization is the ability, the process and lens that opposed analysis and visualization, an
the product of creation, interpretation, use of “ana-vis” scale, on which individuals could be
and reflection upon pictures, images, diagrams, placed according to the preponderance of logical
in our minds, on paper or with technological analysis or visualization in their mathematical
tools, with the purpose of depicting and thinking. However, Krutetskii (1976) argued, on
communicating information, thinking about the basis of his vast data pool, that without logical
and developing previously unknown ideas and analysis there is no mathematics, whereas the use
advancing understandings. of what he termed “visual supports” is optional.
Logic determines the strength of mathematical
Preference for Visualization in Mathematics processing, whereas visualization (or its absence)
Under the supervision of Ken Clements, early determines the type. One might consider
research on learners’ preferences for using these two aspects of mathematical thinking on
visualization in mathematics was carried out orthogonal axes: strength of logic on the x-axis
by Suwarsono (1982), who developed a and amount of visualization on the y-axis.
mathematical processing instrument (MPI) for Krutetskii (1976) worked with students who
use with seventh graders in Australia. This instru- were considered “capable” in mathematics. On
ment included word problems capable of solution the basis of their problem solving in task-based
by visual or by nonvisual mathematical means, interviews, he classified these students
and a questionnaire in which learners could into groups according to the type of their
identify the means they had used to solve the thinking, i.e., according to whether verbal-logical
problems, yielding a score of mathematical or visual-pictorial thinking predominated
visuality (MV). Presmeg (1985) followed (analytical and geometric types, respectively) or
Suwarsono’s methodology in constructing his whether these aspects were in equilibrium (two
instrument, but designed her MPI for use with types of harmonic thinkers – abstract-harmonic
learners in grades 11–12 (parts A and B) and and pictorial-harmonic). These types would all
their mathematics teachers (parts B and C, more lie in the right-hand quadrants of the orthogonal
difficult), thus enabling comparison of the MV model, because of the ubiquitous strength of logic
scores of teachers and students on part B, which demonstrated by Krutetskii’s learners. However,
was common. Nonparametric statistics revealed when Presmeg (1985) analyzed the mathematical
no significant difference between boys and girls achievements and type of thinking of grade
in her study, but a significant difference between 11 students according to her preference for
teachers and students: the learners needed visualization test, individuals could be classified
more visual means than did their mathematics in approximately equal numbers in all four of the
teachers. For most populations the preference quadrants. It is significant that not all students
for visuality (MV) scores follows a normal, with strong spatial ability, who are capable
Gaussian, frequency distribution. Factors that of using visualization in their mathematical
determine how a task will be approached include thinking, choose to do so. This aspect points to
the following: the task itself, instructions to do the interaction of visualization learning styles
the task in a certain way, individual preferences, with other aspects of the classroom, as
and, finally, the culture of the mathematical summarized in the next section.
learning environment including whether or not
visualization is valued. At the far ends of the Interaction of Visualization Styles in Learning
frequency distribution, some learners seldom and Teaching
resort to visualization, whereas there are others Dreyfus (1991) and Eisenberg (1994) suggested
who always do so. The latter form part of a group from their research that students are reluctant to
of learners who are called visualizers.

V 638 Visualization and Learning in Mathematics Education

visualize in mathematics. The evidence for their • A prototype image may induce inflexible
claim was largely from students learning college- thinking.
level mathematics. However, this phenomenon
could be the result of cultural environments in • An uncontrollable image may persist,
which visualization is not valued in mathematics. thereby preventing more fruitful avenues of
Presmeg’s (1985 and later) frequency distribu- thought.
tion graphs showed clearly that there is not Implicit in these difficulties is compartmental-
a shortage of visualizers in mathematics. She
explored the interactions between the teaching ization, whose damaging effect in learning
styles of 13 high school teachers and 54 visual- mathematics has been noted by several authors
izers in the mathematics classes of these teachers. (Duval 1999; Nardi et al. 2005; Presmeg 1992).
It was noteworthy that there was a correlation of There are two basic ways in which these
only 0.4 (Spearman’s rho) between the teachers’ difficulties can be overcome (Presmeg 1986,
mathematical visuality (MV) and teaching 1992). Firstly, a visual image or inscription of
visuality (TV) scores. Several teachers realized one concrete case can be the bearer of abstract
that their students required more visual supports information, that is, a sign for an abstract object.
than they did and taught accordingly. The TV Dynamic imagery and pattern imagery are types
scores enabled the teachers to be distributed into of imagery that are useful in this regard.
a visual group, a middle group, and a nonvisual Secondly, metaphor can link the domain of
group. Visualizers in the classes of teachers in the abstract mathematical objects with visual
nonvisual group attempted to follow the styles imagery or inscriptions in a different domain.
of their teachers, without visualization, and Visual images of all types have mnemonic
the result was lack of success, involving memo- advantages; pictures and spatial patterns are
rization without understanding. Surprisingly, often memorable.
visualizers with visual teachers also often expe-
rienced difficulty. It was the pedagogy of teachers Questions for Research on Visualization and
in the middle group that was optimal for these Learning in Mathematics
visual learners. These teachers used and Presmeg (2006, p. 227) put forward a list of
encouraged visual methods of working, but they 13 questions requiring further research, which
also stressed that abstraction and generalization she considered to be of significance for mathe-
are important in mathematics. matics education. Many of these questions have
received attention (e.g., Arcavi 2003; Nardi et al.
Difficulties and Affordances of Use of 2005; Owens 1999; Presmeg 1992, 2008;
Visualization in Mathematics Yerushalmy et al. 1999), but the list is still
Several research studies have emphasized that indicative of areas in which research is needed
visualization needs to link with rigorous logic in order to increase knowledge of the role of
and analytical thought processes to be effective visualization in effective learning of mathemat-
in mathematics (Arcavi 2003). Presmeg (1985, ics. Particularly in the computer age, the
1986) identified difficulties and strengths of affordances of technology inevitably change the
mathematical visualization in data from dynamics of the way in which mathematics is
task-based interviews with the 54 visualizers learned, including its visualization (Yerushalmy
in her study. All the difficulties related in one et al. 1999; Yu et al. 2009). Yu and colleagues
way or another to the abstraction and generaliza- found that the use of interactive dynamic
tion that are essential aspects of doing geometry software in learning geometry at
mathematics. middle school level inverted the order of the
• The one-case concreteness of an image may levels of learning geometry established by van
Hiele and van Hiele-Geldof in The Netherlands
be tied to irrelevant details or introduce false in the 1950s (Battista 2009).
information.
1. What aspects of pedagogy are significant in
promoting the strengths and obviating the

Visualization and Learning in Mathematics Education V639

difficulties of use of visualization in learning ▶ Epistemological Obstacles in Mathematics V
mathematics? Education
2. What aspects of classroom cultures promote
the active use of effective visual thinking in ▶ Information and Communication Technology
mathematics? (ICT) Affordances in Mathematics Education
3. What aspects of the use of different types of
imagery and visualization are effective in ▶ Mathematics Teachers and Curricula
mathematical problem solving at various ▶ Metaphors in Mathematics Education
levels? ▶ Problem Solving in Mathematics Education
4. What are the roles of gestures in mathemat- ▶ Semiotics in Mathematics Education
ical visualization? ▶ Shape and Space – Geometry Teaching and
5. What conversion processes are involved in
moving flexibly amongst various mathemat- Learning
ical registers, including those of a visual ▶ Visualization and Learning in Mathematics
nature, thus combating the phenomenon of
compartmentalization? Education
6. What is the role of metaphors in connecting
different registers of mathematical inscrip- References
tions, including those of a visual nature?
7. How can teachers help learners to make Arcavi A (2003) The role of visual representations
connections between visual and symbolic in the learning of mathematics. Educ Stud Math
inscriptions of the same mathematical notions? 52:215–241
8. How can teachers help learners to make
connections between idiosyncratic visual Battista MT (2009) Highlights of research on learning
imagery and inscriptions and conventional school geometry. In: Craine TV (ed) Understanding
mathematical processes and notations? geometry for a changing world. Seventy-first Year-
9. How may the use of imagery and visual book, National Council of Teachers of Mathematics.
inscriptions facilitate or hinder the reification Reston, Virginia, pp 91–108
of processes as mathematical objects?
10. How may visualization be harnessed to Bishop AJ (1973) The use of structural apparatus and
promote mathematical abstraction and spatial ability – a possible relationship. Res Educ
generalization? 9:43–49
11. How may the affect generated by personal
imagery be harnessed by teachers to increase Bishop AJ (1980) Spatial abilities and mathematics
the enjoyment of learning and doing education–a review. Educ Stud Math 11:257–269
mathematics?
12. How do visual aspects of computer technol- Clements MA (1981) Visual imagery and school
ogy change the dynamics of the learning of mathematics. Part 1. Learn Math 2(2):2–9
mathematics?
13. What is the structure and what are the Clements MA (1982) Visual imagery and school
components of an overarching theory of visu- mathematics. Part 2. Learn Math 2(3):33–38
alization for mathematics education?
Dreyfus T (1991) On the status of visual reasoning in
Cross-References mathematics and mathematics education. In:
Furinghetti F (ed) Proceedings of the 15th Conference
▶ Abstraction in Mathematics Education of the International Group for the Psychology of
▶ Creativity in Mathematics Education Mathematics Education (PME), vol 1, pp 33–48

Duval R (1999) Representation, vision, and visualization:
cognitive functions in mathematical thinking. Basic
issues for learning. In: Hitt F, Santos M (eds) Proceed-
ings of the 21st North American PME conference,
vol 1, pp 3–26

Eisenberg T (1994) On understanding the reluctance to
visualize. Zentralbl Didakt Math 26(4):109–113

Goldin GA (1992) On the developing of a unified model
for the psychology of mathematics learning and
problem solving. In: Geeslin W, Graham K (eds)
Proceedings of the 16th PME international conference,
vol 3, pp 235–261

Hershkowitz R, Ben-Chaim D, Hoyles C, Lappan G,
Mitchelmore M, Vinner S (1989) Psychological
aspects of learning geometry. In: Nesher P, Kilpatrick
J (eds) Mathematics and cognition. Cambridge
University Press, Cambridge, pp 70–95

V 640 Visualization and Learning in Mathematics Education

Krutetskii VA (1976) The psychology of mathematical Presmeg NC (2008) An overarching theory for research on
abilities in schoolchildren. University of Chicago visualization in mathematics education. In: Plenary
Press, Chicago paper, proceedings of Topic Study Group 20:
visualization in the teaching and learning of mathe-
Nardi E, Jaworski B, Hegedus S (2005) A spectrum of matics. 11th International Congress on Mathematics
pedagogical awareness for undergraduate mathemat- Education (ICME-11), Monterrey, 6–13 July 2008.
ics: from “tricks” to “techniques”. J Res Math Educ Published on the ICME-11 web site: http://tsg.
36(4):284–316 icme11.org(TSG20)

Owens K (1999) The role of visualization in young Suwarsono S (1982) Visual imagery in the mathematical
students’ learning. In: Zaslavsky O (ed) Proceedings thinking of seventh grade students. Unpublished PhD
of the 23rd PME international conference, vol 1, dissertation, Monash University
pp 220–234
Yerushalmy M, Shternberg G, Gilead S (1999)
Piaget J, Inhelder B (1971) Mental imagery and the child. Visualization as a vehicle for meaningful problem
Routledge and Kegan Paul, New York solving in algebra. In: Zaslavsky O (ed) Proceedings
of the 23rd PME international conference, vol 1,
Presmeg NC (1985) The role of visually mediated processes pp 197–211
in high school mathematics: a classroom investigation.
Unpublished PhD dissertation, University of Cambridge Yu P, Barrett J, Presmeg N (2009) Prototypes and
categorical reasoning: a perspective to explain how
Presmeg NC (1986) Visualization in high school mathe- children learn about interactive geometry. In:
matics. Learn Math 6(3):42–46 Craine TV (ed) Understanding geometry for
a changing world. Seventy-first Yearbook, National
Presmeg NC (1992) Prototypes, metaphors, metonymies, Council of Teachers of Mathematics. Reston, Virginia,
and imaginative rationality in high school mathemat- pp 109–126
ics. Educ Stud Math 23:595–610
Zimmermann W, Cunningham S (1991) Visualization in
Presmeg NC (2006) Research on visualization in learning teaching and learning mathematics. Mathematical
and teaching mathematics: emergence from psychol- Association of America, Washington, DC
ogy. In: Gutierrez A, Boero P (eds) Handbook of
research on the psychology of mathematics education.
Sense, Rotterdam, pp 205–235

W

Wait Time in Mathematics Teaching Tobin (1986), as advantages are lost) has gained
further prominence by being seen as a key element
Stephen Lerman of formative assessment (Black et al. 2002). Future
Department of Education, Centre for research questions include research on gender- or
Mathematics Education, London South Bank other group-specific effects, the cognitive level of
University, London, UK teachers’ initial questions, and relationship to
social and socio-mathematical norms.

Keywords References

Achievement; Cognitive level Black P, Harrison C, Lee C, Marshall B, Wiliam D (2002)
Working inside the black box: assessment for learning
Definition in the classroom. King’s College London, Department
of Education and Professional Studies, London
Research on the effects of the amount of time
teachers wait for student responses after asking Rowe MB, Hurd P (1966) A study of small group dynam-
a question. ics in the BSCS laboratory block program. J Res Sci
Teach 4(2):67–73

Tobin K (1986) Effects of teacher wait time on discourse
characteristics in mathematics and language arts
classes. Am Educ Res J 23(2):191–200

Characteristics

Research in this area began in the 1960s in the Word Problems in Mathematics
science education field (Rowe and Hurd 1966) Education
and asked the question of what connections there
might be between the time teachers wait after ask- Lieven Verschaffel, Fien Depaepe and
ing a question and the cognitive level of students’ Wim Van Dooren
responses. Tobin (1986) found, in a scientific study Instructional Psychology and Technology,
set in a whole-class mathematics instructional set- Katholieke Universiteit Leuven, Leuven, Belgium
ting, that there were improvements in teachers’ and
students’ discourse, including length of student Keywords
responses, as well as higher mathematics achieve-
ment with increased wait time. More recently, the Affect; Algebra; Arithmetic; Cognitive
encouragement to teachers to wait longer than the psychology; Ethnomathematics; Metacognition;
typical 1s (though not more than 5, according to

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

W 642 Word Problems in Mathematics Education

Modeling; Problem solving; Sociocultural during the past decades writers have called for a
theories; Word problems reexamination of the rationale for this privileged
position (see, e.g., Lave 1992). It can be inferred
Definition and Function of Word that word problems have been included to accom-
Problems plish several goals, the most important one being
to offer practice for everyday situations in which
Word problems are typically defined as verbal learners will need what they have learned in
descriptions of problem situations wherein one or their arithmetic, geometry, or algebra lessons at
more questions are raised the answer to which can school (the so-called application function). Other
be obtained by the application of mathematical goals were and still are to motivate students to
operations to numerical data available in the prob- study mathematics, to train students to think cre-
lem statement (Verschaffel et al. 2000). As such atively and to develop their problem-solving abil-
they differ both from bare sums presented in writ- ities, and to develop new mathematical concepts
ten (e.g., 4 + 5 ¼ ?; 5x + 2 ¼ 22) or oral form (e.g., and skills.
How much is 40 divided by 5?; What is the mean
of the numbers 12, 17, 17, 18?), as well as from Characteristics
quantitative problems encountered in real life
(e.g., Which type of loan should we take? Can Research Perspectives on Word Problem
I drive home from here without filling the tank?). Solving
Word problems have already for a long time
Importantly, the term “word problem” does attracted the attention of researchers in psychol-
not necessarily imply that every task that meets ogy and (mathematics) education (see, e.g.,
the above definition represents a true problem, in Thorndike 1922). Before the emergence of the
the cognitive-psychological sense of the word, information-processing approach, research on
for a given student, i.e., a task for which no word problems focused mainly on the effects on
routine method of solution is available and performance of various kinds of linguistic,
which therefore requires the activation of (meta) computational, and/or presentational task
cognitive strategies (Schoenfeld 1992). Whether features (e.g., number of words, grammatical
a word problem that a student encounters complexity, presence of particular key words,
constitutes a genuine problem depends on his/her number and nature of the required operations,
familiarity with the problem, his/her mastery of nature and size of the given numbers) and
the various kinds of required knowledge and subject features (e.g., age, gender, general
skills, the available tools, etc. intelligence, linguistic, and mathematical ability
of the problem solver) (Goldin and McClintock
Word problems have always constituted an 1984).
important part of school mathematics worldwide.
Historically, their role in mathematics education With the rise of the information-processing
dates back even to antiquity. One can find word approach, researchers’ attention shifted from
problems already almost 4,000 years ago in learners’ externally observable performance to
Egyptian papyri. They also figure in, for instance, the underlying cognitive schemes and thinking
ancient Chinese and Indian manuscripts as well processes of students solving various kinds of
as in arithmetic textbooks from the early days of word problems, and, accordingly, their research
printing, such as the Treviso arithmetic of 1487, methods changed as well. Analyses of response
and they continue to fill current mathematics accuracies were complemented with analyses of
textbooks (Swetz 2009). thinking aloud or retrospective protocols, indi-
vidual interviews, reaction times, eye move-
Despite this continuity across time and ments, and, most recently, neuropsychological
cultures, there has been little explicit discussion measurements.
of why word problems should (continue to) be
such a prominent part of the curriculum, and

Word Problems in Mathematics Education W643

For instance, in the domain of one-step account the tactics and the affects that they have W
addition and subtraction word problems, a basic built up along with their participation in the prac-
distinction emerged between three classes of tice and culture of the mathematics classroom.
problem situations: change problems (involving
a change from an initial to a final state through the Phases and Components of Competent
application of a transformation), combine prob- Word Problem Solving
lems (involving the combination of two discrete Currently the competent solution of a word prob-
sets or splitting of one set into two discrete sets), lem is thought of as a complex multiphase
and compare problems (involving the quantified process the “heart” of which is formed by
comparison of two discrete sets of objects), each (1) the construction of an internal model of the
of which was further subdivided leading to a problem situation, reflecting an understanding of
classification scheme of 14 problem types. the elements and relations in the problem situa-
Numerous cognitive-psychological studies with tion, and (2) the transformation of this situation
lower elementary school children provided model into a mathematical model of the elements
evidence for the psychological validity of this and relations that are essential for the solution.
classification scheme (Fuson 1992; Verschaffel These two steps are then followed by (3) working
et al. 2007). through the mathematical model to derive math-
ematical result(s), (4) interpreting the outcome of
Researchers also analyzed pupils’ solution the computational work, (5) evaluating if the
strategies in the domain of one-step addition interpreted mathematical outcome is computa-
and subtraction word problems (Carpenter and tionally correct and reasonable, and (6) commu-
Moser 1984). These analyses first demonstrated nicating the obtained solution. This multiphase
that early in their development, children have a model is not considered to be purely sequential;
wide variety of successful material and verbal rather, individuals can go back and forth through
counting strategies, many of which are never the different phases of the model (Blum and Niss
taught explicitly and/or systematically at school. 1991; Verschaffel et al. 2000).
Gradually, these strategies develop into more
formal mental solution strategies based on Arguably, pupils’ actual problem-solving pro-
known and derived number facts. Second, it was cesses do not always fit with this theoretical
found that the situational structure of a word prob- model. To the contrary, the process of actually
lem significantly affects the nature of children’s solving word problems for many students is often
strategy choices. More specifically, children along the lines of a “truncated” model, wherein
tended to solve each word problem with the type the problem text immediately guides the mathe-
of strategy that corresponds most closely to its matical model – the choice of an arithmetic
situation model. Similar findings have been found operation, the selection of a geometric formula,
for the domain of multiplication and division or the composition of an algebraic expression –
word problems (Verschaffel et al. 2007). based on a quick and superficial analysis of the
problem statement (e.g., by relying on key words
Especially since the 1990s, insights from in the text, such as the word “more” in the prob-
ethnomathematics and sociocultural theories lem text automatically triggers an addition). The
have contributed to the insight that classical directly evoked mathematical operation, formula,
information-processing models are insufficient or expression is then worked through, and the
to grasp the full complexity of learners’ word result of the calculation is found and given as
problem-solving processes. They need to be the answer, typically without reference back to
enriched with the idea that word problem solving the problem text to verify whether the answer is
is a human activity situated in the particular meaningful in view of the original problem situ-
microcosm of a mathematics classroom (Lave ation (Verschaffel et al. 2000). Concerning the
1992; Verschaffel et al. 2000), and that, therefore, competencies that are required to solve word
students’ word problem-solving behavior can problems, there is nowadays a rather broad
only be understood by also seriously taking into

W 644 Word Problems in Mathematics Education

consensus that they involve (Schoenfeld 1992; studies differ widely in terms of the age and
De Corte et al. 1996): mathematical background of the learners (from
• A well-organized and flexibly accessible first graders up to university students) and the
aspect(s) of word problem-solving expertise
knowledge base involving the relevant con- they are primarily aiming at (i.e., schematic
ceptual knowledge (e.g., a schematic knowl- knowledge, problem-solving skills, attitudes,
edge of the different problem types) and and beliefs), common characteristics are:
procedural knowledge (i.e., informal and for- • The use of varied, cognitively challenging,
mal solution strategies) that is relevant for
solving word problems and/or realistic tasks, which lower the chance
• Heuristic methods, i.e., search strategies for of developing superficial coping strategies
problem analysis and transformation which (such as the key word strategy) and which
increase the probability of finding a solution involve the complexities of genuine mathe-
(e.g., making a drawing or a scheme) and matical application and modeling tasks (such
metacognition, involving both metacognitive as the necessity to seek and apply aspects of
knowledge and metacognitive skills the real context to proceed, to discuss alterna-
• Positive task-related affects, involving posi- tive models, to decide upon the required level
tive beliefs, attitudes, and emotions, as well of precision of the outcome).
as meta-affect, involving knowledge about • A variety of teaching methods and learner
one’s affects and skills for regulating one’s activities, including expert modeling of the
affective processes strategic aspects of the problem-solving pro-
While there is evidence for the role of each of cess, appropriate forms of scaffolding, small-
these aspects in students’ word problem-solving group work, and whole-class discussions;
processes and skills, it should be clear that they typically, the focus is not on presenting and
are strongly interrelated and interdependent. practicing well-established methods for solv-
ing well-defined types of problems, but rather
Solving Word Problems Versus Problems in on demonstrating, experiencing, articulating,
the Real World and discussing what applied problem solving
An issue that has received quite some attention and modeling is all about.
during the past decades is the complex relation • The creation of a classroom climate that is
between word problems and reality. For a very conducive to the development of a proper
long time, word problems have played their role view of applied problem solving and mathe-
as an unproblematic and transparent bridge matical modeling and of the accompanying
between the world of mathematics and the real skills and affects (Lesh and Doerr 2003;
world. However, during the last 10–15 years, Verschaffel et al. 2000).
more and more researchers have questioned this In most of these design experiments, (moder-
role, partly on the basis of increasing empirical ately) positive outcomes have been obtained in
evidence of students’ “suspension of terms of performance, underlying (meta)cogni-
sense-making” (Schoenfeld 1991) when doing tive processes, and affective aspects of learning.
school word problems and of aspects of the cur- A final issue that has not yet elicited a lot of
rent practice and culture of word problem solving research, but that will become more important
that seem directly responsible for this phenome- in the future, is whether word problems, which
non (Verschaffel et al. 2000). rely after all on an “old” vehicle for creating an
applied problem situation (namely, printed text),
Teaching Word Problem Solving will continue to keep their prominent position in
Besides ascertaining studies, researchers have the mathematics curriculum or whether they will
also done numerous intervention studies – both be replaced or at least complemented by new and
design experiments and (quasi)experimental potentially more effective ways of bringing rich
teaching experiments. While these intervention and real problems into the mathematics

Word Problems in Mathematics Education W645

classroom, based on new information and com- Lesh R, Doerr HM (eds) (2003) Beyond constructivism.
munication technologies, such as video, com- Models and modeling perspectives on mathematical
puter graphics, and the Internet. problem solving, learning and teaching. Erlbaum,
Mahwah
References
Schoenfeld AH (1991) On mathematics as sense-making:
Blum W, Niss M (1991) Applied mathematical problem an informal attack on the unfortunate divorce of formal
solving, modeling, applications, and links to other and informal mathematics. In: Voss JF, Perkins DN,
subjects: state, trends and issues in mathematics Segal JW (eds) Informal reasoning and education.
instruction. Educ Stud Math 22:37–68 Erlbaum, Hillsdale, pp 311–343

Carpenter TP, Moser JM (1984) The acquisition of Schoenfeld AH (1992) Learning to think mathemat-
addition and subtraction concepts in grades one ically. Problem solving, metacognition and
through three. J Res Math Educ 15:179–202 sense-making in mathematics. In: Grouws DA
(ed) Handbook of research on mathematics
De Corte E, Greer B, Verschaffel L (1996) Mathematics teaching and learning. Macmillan, New York,
teaching and learning. In: Berliner D, Calfee R (eds) pp 334–370
Handbook of educational psychology. Macmillan,
New York, Chapter 16 Swetz F (2009) Culture and the development of mathe-
matics: a historical perspective. In: Greer B,
Fuson KC (1992) Research on whole number addition and Mukhupadhyay S, Powell AB, Nelson-Barber S (eds)
subtraction. In: Grouws DA (ed) Handbook of research Culturally responsive mathematics education. Taylor
on mathematics teaching and learning. Macmillan, and Francis, Routledge, pp 11–42
New York, pp 243–275
Thorndike E (1922) The psychology of arithmetic.
Goldin GA, McClintock E (eds) (1984) Task variables in Macmillan, New York
mathematical problem solving. Franklin, Philadelphia
Verschaffel L, Greer B, De Corte E (2000) Making
Lave J (1992) Word problems: a microcosm of theories of sense of word problems. Swets & Zeitlinger, Lisse
learning. In: Light P, Butterworth G (eds) Context and
cognition: ways of learning and knowing. Harvester Verschaffel L, Greer B, De Corte E (2007) Whole number
Wheatsheaf, New York, pp 74–92 concepts and operations. In: Lester FK (ed)
Second handbook of research on mathematics
teaching and learning. Information Age, Greenwich,
pp 557–628

W

Z

Zone of Proximal Development in between the actual developmental level as deter-
Mathematics Education mined by independent problem solving and the
level of potential development as determined
Wolff-Michael Roth through problem solving under adult guidance
Applied Cognitive Science, University of or in collaboration with more capable peers”
Victoria, Victoria, BC, Canada (Vygotsky 1978, p. 86).

Keywords Zone of Proximal Development:
An Example
Vygotsky; Learning; Development; Expansion of
agency; Collective agency; Activity theory The following example from a 3-D geometry
lesson in a second grade class illustrates how
Definition the zone of proximal development tends to be
used in education. On this day, the second graders
The zone of proximal development is a category each grabs a mystery object from a bag and places
that emerged from the work of L. S. Vygotsky, it with an existing group of objects or starts a new
the father of activity theory. Inspired by K. Marx, group. Connor has just completed placing his
Vygotsky came to understand the specifically “mystery object” with a group of objects next to
human characteristics in terms of society (Roth which one of the two teachers present has placed
and Lee 2007). Explicitly referring to Marx, a label with the words “squares” and “cubes.”
Vygotsky states that “any higher psychological The following fragment from the lesson begins
function was external; this means it was social... when Mrs. Winter asks Connor what the group
the relation between higher psychological func- was about.
tions was at one time a physical relation between 1. W: Em an’ what did we say that group was
people” (Vygotsky 1989, p. 56). As a result of
this perspective, our personalities are shaped by about?
society: “the psychological nature of man is the 2. C: What do you mean like?
totality of societal relations shifted to the inner 3. W: What was the– What did we put for the
sphere” (p. 59). Based on this understanding, he
created a definition of the zone of proximal devel- name of that group? What’s written on the
opment that now has aphoristic qualities in edu- card?
cational circles. Thus, it denotes “the distance 4. C: Squares.
5. W: Square and...?
6. J: Cube.
7. W: Cube.

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

Z 648 Zone of Proximal Development in Mathematics Education

At first, there is a counter-question rather Researchers have drawn on the asymmetry
than a reply: What does she mean? (turn 2). between learner and the social other, because it
Mrs. Winter begins to rephrase: There is one orients to “the ways in which more capable par-
abandoned question and then there are two full ticipants structure interactions so that novices
questions (turn 3). Now Connor replies providing (children) can participate in activities that they
one of the two words: Squares (turn 4). Mrs. are not themselves capable of” and to the fact that
Winter acknowledges his contribution by “with repeated practice, children gradually
restating the word with a constative statement: increase their relative responsibility until they
Square (turn 5). She then says “and” with the can manage the adult role” (Cole 1984, p. 155).
rising intonation of a question. Jane says “cube” It thereby leads us to think about the learning
(turn 6), and Mrs. Winter acknowledges by process through the lens of the teacher who,
repeating the word as she had done in the first because she/he is responsible for structuring the
instance (turn 7). learning situation, becomes the “real subject in/of
the child’s learning” (Holzkamp 1993, p. 418).
In the (dialogical) relation, Connor and Jane
arrive at providing the sought-for answer because Recent Critical Reworking of the Notion
they do not do the entire task on their own. Here,
they are part of a dialogical relation where the Recent work in mathematics education shows that
teacher takes one part of the task and the students the relations between teachers and students, such
the other. Now the task is spread across all par- as Mrs. Winter and Connor and Jane, are much
ticipants. Later, once they are able to state the more symmetrical (Roth and Radford 2010). This
name and properties of the group without being is so because each has to understand the other for
asked, the children are said to have internalized. the episode to unfold as it does. For example,
But it is evident that this description does not Connor already has to understand that Mrs. Win-
entirely match the situation. For the children to ter is asking him a question, and he has to under-
take their part in the relation, they already have to stand that he has trouble with her question.
mobilize their understanding so that they can take Similarly, Mrs. Winter has to understand that
their position in the question answer game that Connor is asking her to restate the question.
produces the result. The fact is, as Vygotsky’s Thus, the relation is more symmetrical than
other way of framing says much more clearly, researchers have led on in the past. For example,
that the higher psychological functions exist precisely because her first question (turn 1) was
in and as external relations between people. not intelligible, Mrs. Winter has to rephrase it. She
Thus, “the relation of psychological functions gives it several tries and eventually finds one that
is genetically linked to real relations between allows the children to provide first one and then
people: regulation of the word, verbalized the other expected word. That is, it is precisely in
behavior ¼ power–submission” (Vygotsky such relations that teachers such as Mrs. Winter
1989, p. 57). In the exchange, under the tutelage become better and better at asking appropriate
of the teacher, they learn to provide the right questions. The zone of proximal development,
words; she regulates the production of the words therefore, works both ways. Connor and Jane
and regulated verbal behavior. As a result of the learn to talk about, name, and characterize math-
exchange, when they no longer need the external ematical objects; and Mrs. Winter develops as
relation to name and characterize the group a teacher by learning to ask age-appropriate ques-
of cubes, Connor and Jane are in a position to tions in a unit of 3-D geometry (it is her first time
individualize the social relation – they develop. to teach such a unit at that grade level).

In the way the zone of proximal development From a systemic perspective on cultural-
is defined, there is an asymmetrical relation historical activity theory, the concept of a zone
between those who know (teachers, more
advanced peers) and those who do not (students).

Zone of Proximal Development in Mathematics Education Z649

of proximal development can be reformulated as which, in the Vygotskian framing, are part of Z
the “distance between the present everyday one and the same process. In the relation, the
actions of the individuals and the historically students expand their agency and control over
new form of the societal activity that can be the mathematical task conditions: They learn.
collectively generated as a solution to the double But when they no longer need the relation with
bind potentially embedded in the everyday the teacher or peers, they have reached the new
actions” (Engestro¨m 1987, p. 174). We may developmental level. Learning and development,
exemplify the core idea in this revised definition very different concepts in the constructivist par-
in the following way. If Connor and Jane had adigm of J. Piaget, are now two different sides of
been in a class based on discovery learning, the same movement (Roth and Radford 2011).
they would have been left on their own to make Similarly, by working with the children, in the
mathematical discoveries. They would have interest of allowing them to learn, the teacher
arrived at certain results, which, in all likelihood, expands her own agency and control over the
would have been less advanced than the results conditions: She develops.
that they contribute to producing in the presence
of the teacher. Because there are now more Summary
people working together, but with clearly
different role in the division of labor, a new In summary, the zone of proximal development is
form of activity has emerged. This new form of a powerful category for understanding learning
societal activity gives rise to higher-level that arises when people enter relations with
actions on the part of the children then in the others. Aphoristically we may state: What these
hypothetical discovery learning context; it also relations are today will be psychological func-
gives the teacher new opportunities to learn tions of the participants tomorrow.
to teach.
Cross-References
A third way of defining the zone of proximal
development takes the perspective of individuals ▶ Activity Theory in Mathematics Education
who are integral and irreducible parts of society ▶ Concept Development in Mathematics
(Holzkamp 1993). Individuals can expand their
individual agency and control over life conditions Education
by contributing to collective agency and collec- ▶ Dialogic Teaching and Learning in
tive control over conditions. In mathematics
classrooms, this means that students engage in Mathematics Education
collaborations with others, because they increase ▶ Learner-Centered Teaching in Mathematics
their individual agency and task control when
they contribute to the expansion of collective Education
agency and control by active participation. ▶ Scaffolding in Mathematics Education
Thus, Connor and Jane already participate with
the teacher; and it is because of their participation References
that their agency expands. If one or the other had
said to the teacher, “I want to do this on my own” Cole M (1984) The zone of proximal development:
or “I don’t need help,” then they would have where culture and cognition create each other. In:
actively rejected contributing to the collective Wertsch JV (ed) Culture, communication, and cogni-
agency and control and, perhaps, never arrived tion. Cambridge University Press, Cambridge,
at the point where they did. pp 146–161

This final definition allows us more easily than Engestro¨m Y (1987) Learning by expanding: an activity-
the other two to conceptualize the important dis- theoretical approach to developmental research.
tinction between learning and development, Orienta-Konsultit, Helsinki

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Holzkamp K (1993) Lernen: Subjektwissenschaftliche Roth W-M, Radford L (2011) A cultural-historical per-
Grundlegung. Campus, Frankfurt/M spective on mathematics teaching and learning. Sense,
Rotterdam
Roth W-M, Lee YJ (2007) “Vygotsky’s neglected leg-
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Roth W-M, Radford L (2010) Re/thinking the zone of
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