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Redesign from a
Research Perspective
Recent international come
parative studies indicate
that some of our basic
assumptions about the
structure and goals of
schooling, about children's
abilities, and about the
structure of curricula are
determined more by tracli-
tion than by fundamental
educational principles (Cross-
white et al,, 19861. For exam-
ple:
Mothers of American chilciren are far more likely than are
mothers of Japanese and Taiwanese children to believe
that innate ability underlies chilciren's success in mathe-
matics (Stevenson et al., 1 9861.
.
The curricula of other countries reflect very different
beliefs about what children are capable of learning.
American -textbooks tend to develop ideas very slowly by
progressing through a hierarchy of small, straightforward
learning tasks. Texts from Asian countries and from the
Soviet Union immerse students in much more demanding
problem situations from the beginning (Fuson et al., 1988~.
· Mathematics classrooms in Japan use instructional time in
quite different ways than American schools. For example,
group work and cooperative problem solving are stressed
throughout the earliest grades (Easiey and Easley, 1982;
Enloe and Lewin, 19871.

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28
Reshaping School Mathematics
These comparisons underscore that schooling is a reflection
of societal values, that we must think of "curriculum" in societal
as well as in classroom terms. The emphasis on mathematics in
Chinese and Japanese classrooms is a reflection of the impor-
tance that the Chinese and Japanese parents ascribe to
mathematics learning.
The findings of international comparisons are confirmed by
National Assessments of Eclucational Progress (Dossey et al.,
1988~. Ail studies consistently show disappointing levels of sTu-
dent performance in areas of mathematical power, uncler-
stancling, and relevant applications. Performance is most dis-
appointing in understanding and problem solving, the very
aspects of mathematics most important for working and living
in a technology-intensive society.
Learning Mathematics
These difficulties can be overcome only through new modes
of teaching (Resnick, 1987~. By synthesizing a broad range of
observational research, Romberg and Carpenter (1986) docu-
ment a curricular tradition in mathematics built on a massive
record of knowlecige divorced from science and other disci-
plines. Mathematics education, unlike mathematics itself, is
separated into subjects and subdivided into topics, stuclies,
lessons, facts, and skills. "This fragmentation of mathematics
has divorced the subject from reality and from inquiry. Such
essential characteristics of mathematics as abstracting, invent-
ing, proving, and applying are open lost."
Research on teaching for higher-orcler thinking lends sup-
port to the notion that instruction needs to change from the
traditional mode where the teacher presents material to a less
structured, more indirect style of teaching (Dessart and Suy-
clam, 1983; Grouws et al., 1988; Peterson, 1988; Peterson and
Carpenter, 1989), Because the development of higher-level
thinking in mathematics depends on autonomous, indepen-
clent learning behavior, teachers (and parents) must learn
how to encourage more self-reliance in students who are
learning mathematics.
Explicit teaching of cognitive and metacognitive strategies
can enhance students' learning (Schoenfeld, 1987), as can
small group cooperative learning (Shavelson et al., 1988;
Peterson and Carpenter, 1989), Cognitive research in other
content areas (Campione et al., 1988) shows the effectiveness
of reciprocal teaching in which children take turns playing
teacher, posing questions, summarizing, clarifying, and pre-
dicting. Reciprocal teaching is based on the premise that the

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29
A Philosophy and Framework
opportunity to construct meanings communally internalizes
the process and gives these constructions permanence
(Resnick, 1988),
Constructed Knowledge
There is now wide agreement among researchers (Resnick,
1983; Linn, 1986) of the need to pay careful attention to stu-
clent-constructed knowledge (Piaget, 1954~. For example,
Resnick (1976), Carpenter et al. (1982), and Steffe et al. (1983)
have shown that when aciding two numbers, say 3 + 6, stu-
dents generally invent the method of counting on (that is, "7,
8, 9"~. Children come to school with a rich body of knowlecige
about the world around them, including well-developecl infor-
mal systems of mathematics (Ginsburg, 19771. Education fails
when children are treated as "blank slates" or "empty jugs,"
ignoring the fact that they have a great deal of mathematical
knowledge, some of which surpasses-and some of which
may contradict-what they are being taught in school (Erl-
wanger, 1974; Clement, 1977, Gelman and Gallistal, 1978;
Ginsburg, 1983~.
Children are active interpreters of the world around them,
including the mathematical aspects of that world. In the words
of Piaget (1948), "To understand is to invent." Topics in school
should be arranged to
exploit intuitions and infor-
mal numerical notions that
students bring with them
to school. Moreover,
teaching methocis must
adapt to the notion of the
child as interpreter and
constructor of (possibly
wrong) theories as opposed
to the child as absorber.
Thinking
Visually
Procedural
Knowledge
The recommendation
to reduce emphasis on
procedures used for
paper-and-pencil calcu
Here is a picture of a roller-coaster track:
Sketch a graph to show the speed of the roller coaster
versus its position on the track.
A
_
~ an,
. XL~ .
l X~<
1~< ,
'<
. ~ X
>< X
X ~
X
X ~
X X
X ~

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30
Reshaping School Mathematics
rations rests in part on extensive evidence that these proce-
dures, in themselves, do not aid conceptual understanding
(Mestre, 1987), The literature on arithmetic "bugs" (Brown and
Burton, 1978, Maurer, 1987) reveals that many mistakes stu-
dents make follow predictable patterns. Such consistent but
mistaken procedures learning bugs-have a natural origin in
the invention of the student: they are intelligent attempts to
modify memorized procedures that ore poorly understood.
Even when correctly learned, purely procedural knowl-
edge-the ability to implement mathematical algorithms with-
out underlying conceptual knowledge~can be extremely
fragile. Clement et al. (1979) have shown that even a solid
procedural knowledge of algebra, such as is held by universi-
ty-level engineering students, does not in most cases (over
80%) imply an ability to interpret the meaning of algebraic
symbols, Several recent studies show that mathematical learn-
ing is more robust when taught in a fashion that stresses under-
lying conceptual models (e.g., Carpenter et al., 1982; Davis,
1984; Hiebert, 1986; Romberg and Carpenter, 1986~.
Mastery of Subject Matter
Mastery of subject matter has for years been the predomi-
nant focus of mathematics education research. Yet
researchers have only just begun to construct a detailed map
of the phases children can go through as they build under-
standing of arithmetic (Steffe et al,, 19831. Even at early ages,
the picture is quite complex, Contrary to much present prac-
tice, it is generally most effective to engage students in mean-
ingful, complex activities focusing on conceptual issues rather
than to establish all building blocks at one level before going
on to the next level (Hatano, 1982; Romberg and Carpenter,
1986; Collins et al., 1989~.
In certain cases, the order of presentation appears critical.
Wearne and Hiebert (1988) showed that students who learn to
calculate too early may find it more difficult to reach an
understanding than students who have had no such experi-
ence. On the other hand, Steffe et al, (1983) showed that chil-
dren must be able to recite the number words in order before
they can develop a concept of counting or number.
There is some evidence to suggest that paper and pencil
calculation involving fractions, decimal long division, and pos-
sibly multiplication are introduced far too soon in the present
curriculum. Under currently prevalent teaching practice, a
very high percentage of high school students worldwide never

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31
A Philosophy and Framework
masters these topics just what one would expect in a case
where routinized skills are blocking semantic learning (e.g.,
Benezet, 1935~. The challenge for curriculum development
(and research) is to determine when routinized rules should
come first and when they should not, as well as to investigate
newer whole-language strategies for teaching that may be
more effective than traditional methods. This is an area where
far more research needs to be done.
Problem Solving
Problem solving is a central focus of the mathematics cur-
riculum. There is by now an extensive body of literature (Krulik,
1980; Mason et al., 1982; Schoenfeld, 1985; Silver, 1985; Charles
and Silver, 1988; Nodclings, 1988) indicating that strategies for
problem solving can be taught effectively. The main warning
from the research literature is that one should be careful not to
trivialize problem-solving strategies, teaching a collection of
isolated tricks (e.g., "of" means multiply, or cross-cancelling
factors). Problem-solving strategies, in the spirit of Polya (1945),
are subtle and complex. Important strategies such as "look for
a pattern by plugging in values for n = 1, 2, 3, 4,, . ." cannot be
taught effectively apart from the situational clues that indicate
when such strategies are appropriate. For a nice collection of
"starter" problems and a discussion of mathematical thinking,
see Mason et al. (19821. For a discussion of problem posing (as
opposed to mere problem solving), see Brown and Walter
(1 983~.
An effective approach to solving problems is provided by
metacognition, the self-conscious ability to know when and
why to use a procedure. There is ample evidence (Schoen-
feld, 1985; Silver, 1985; Campione et al!, 1988; Collins et al.,
1989) that students who know more than enough subject mat-
ter often fail to solve problems because they do not use their
knowlecig0 wisely. They may jump info problems, doggedly
pursuing a particular ill-chosen approach to the exclusion of
anything else; they may raise profitable alternatives, but fail to
pursue them; they may get side-trackecl into focusing on trivia
while ignoring the "big picture."
Research indicates that such "executive" skills can be
learnecO, resulting in significant improvements in problem-
solving performance. Effects can be obtained with interven-
tions as simple as holding class discussions that focus on exec-
utive behaviors, and by explicitly and frequently posing

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32
Reshaping School Mathematics
questions such as: "What are you doing?" "Why are you doing
it?" "How will it help you?"
Making Sense of Mathematics
The current curriculum fails badly in teaching genuine appli-
cations of mathematics (Dossey et al,, 1988~. In a report of the
Thircl National Assessment of Educational Progress (Carpenter
et al., 1983), nearly 30% of children reported the answer of "31
remainder 12" to the problem:
An Army bus holds 36 solcliers. If 1 128 soldiers are
being bused to their training site, how many buses
are neecled?
Fewer than one in four gave the correct answer to the prob-
lem. Approximately 70 percent of the students who took the
examination performed the right operation (1 128 divided by
36 yields "31 remainder 12"), However, fewer than one-third of
those students checked "32 buses" as their answer. Many stu-
dents extract from their school experience the general view
that the result of an algorithm (e.g., "31 remainder 12") is the
correct and complete answer Very little in their experience
would suggest the need to interpret the result of a mathemati-
cal procedure.
For most stuclents, school mathematics is a habit of prob-
lem-solving without sense-making: one learns to read the
problem, to extract the relevant numbers and the operation
to be used, to perform the operation, and to write down the
result-without ever thinking about what it all means.
Reusser (1986) reports that three school children in four will
produce a numerical answer to nonsense problems such as:
There are 125 sheep and 5 dogs in a flock. How old is
the shepherd?
Typical responses, produced by a student solving the prob-
lem out loud, exemplify the mindless world of school mathe-
matics:
125 + 5 = 130 . . . this is too big, and 125 - 5 = 120 is still
too big . . . while 125/5 = 25. That works. I think the
shepherd is 25 years old.
Students constantly strive to make sense of the rules that
govern the world around them, including the world of their
mathematics classrooms. If the classroom patterns are per

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33
A Philosophy and Framework
ceived to be arbitrary and the mathematical operations
meaningless-no matter how well "masterecl" as proce-
dures-students will emerge from the classroom with a sense of
mathematics as being arbitrary, useless, and meaningless.
The classroom culture in which students learn mathematics
shapes their understanding of the nature of mathematics as
well as the ways they will use the mathematics they have
learned. Many studies (Fawcett, 1938; Bruner, 1964; Mason et
al., 1982; Burton, 1984; Love et al., 1988; Schoenfelcl, 1988b)
indicate that it is indeed possible to create classroom environ-
ments that are, in essence, cultures of sense-making-cultures
from which students emerge with an understanding of mathe-
matics as a discipline that helps to make sense of things. The
goal of teaching sense-making via mathematics should be a
central concern of all curricular reform.
To "see" abstract ideas such as limits, it is helpful to use pictures. For example, the sum
1/2+ 1/4 + 1/8 + 1/16 + 1/32+ . . . can tee seen as totaling 1 either by cone-dimentionalline:
1 1 1 1 1 11
1/4 1/8 1/16
1/2
or by a two-dimensional square:
1/8
1/2
. =
32 _
1/16
1/4
Proofs Without
Words

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