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217
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
5
Mathematical Understanding:
An Introduction
Karen C. Fuson, Mindy Kalchman, and John D. Bransford
For many people, free association with the word “mathematics” would
produce strong, negative images. Gary Larson published a cartoon entitled
“Hell’s Library” that consisted of nothing but book after book of math word
problems. Many students—and teachers—resonate strongly with this cartoon’s
message. It is not just funny to them; it is true.
Why are associations with mathematics so negative for so many people?
If we look through the lens of How People Learn, we see a subject that is
rarely taught in a way that makes use of the three principles that are the
focus of this volume. Instead of connecting with, building on, and refining
the mathematical understandings, intuitions, and resourcefulness that stu-
dents bring to the classroom (Principle 1), mathematics instruction often
overrides students’ reasoning processes, replacing them with a set of rules
and procedures that disconnects problem solving from meaning making.
Instead of organizing the skills and competences required to do mathemat-
ics fluently around a set of core mathematical concepts (Principle 2), those
skills and competencies are often themselves the center, and sometimes the
whole, of instruction. And precisely because the acquisition of procedural
knowledge is often divorced from meaning making, students do not use
metacognitive strategies (Principle 3) when they engage in solving math-
ematics problems. Box 5-1 provides a vignette involving a student who gives
an answer to a problem that is quite obviously impossible. When quizzed,
he can see that his answer does not make sense, but he does not consider it
wrong because he believes he followed the rule. Not only did he neglect to
use metacognitive strategies to monitor whether his answer made sense, but
he believes that sense making is irrelevant.

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218 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
BOX 5-1 Computation Without Comprehension: An Observation by
John Holt
One boy, quite a good student, was working on the problem, “If you have 6 jugs,
and you want to put 2/3 of a pint of lemonade into each jug, how much lemonade
will you need?” His answer was 18 pints. I said, “How much in each jug?” “Two-
thirds of a pint.” I said, “Is that more or less that a pint?” “Less.” I said, “How
many jugs are there?” “Six.” I said, “But that [the answer of 18 pints] doesn’t
make any sense.” He shrugged his shoulders and said, “Well, that’s the way the
system worked out.” Holt argues: “He has long since quit expecting school to
make sense. They tell you these facts and rules, and your job is to put them down
on paper the way they tell you. Never mind whether they mean anything or not.”1
A recent report of the National Research Council,2 Adding It Up, reviews
a broad research base on the teaching and learning of elementary school
mathematics. The report argues for an instructional goal of “mathematical
proficiency,” a much broader outcome than mastery of procedures. The
report argues that five intertwining strands constitute mathematical profi-
ciency:
1. Conceptual understanding—comprehension of mathematical con-
cepts, operations, and relations
2. Procedural fluency—skill in carrying out procedures flexibly, accu-
rately, efficiently, and appropriately
3. Strategic competence—ability to formulate, represent, and solve math-
ematical problems
4. Adaptive reasoning—capacity for logical thought, reflection, expla-
nation, and justification
5. Productive disposition—habitual inclination to see mathematics as
sensible, useful, and worthwhile, coupled with a belief in diligence and
one’s own efficacy
These strands map directly to the principles of How People Learn. Prin-
ciple 2 argues for a foundation of factual knowledge (procedural fluency),
tied to a conceptual framework (conceptual understanding), and organized
in a way to facilitate retrieval and problem solving (strategic competence).
Metacognition and adaptive reasoning both describe the phenomenon of
ongoing sense making, reflection, and explanation to oneself and others.
And, as we argue below, the preconceptions students bring to the study of
mathematics affect more than their understanding and problem solving; those
preconceptions also play a major role in whether students have a productive

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219
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
disposition toward mathematics, as do, of course, their experiences in learn-
ing mathematics.
The chapters that follow on whole number, rational number, and func-
tions look at the principles of How People Learn as they apply to those
specific domains. In this introduction, we explore how those principles ap-
ply to the subject of mathematics more generally. We draw on examples
from the Children’s Math World project, a decade-long research project in
urban and suburban English-speaking and Spanish-speaking classrooms.3
PRINCIPLE #1: TEACHERS MUST ENGAGE
STUDENTS’ PRECONCEPTIONS
At a very early age, children begin to demonstrate an awareness of
number.4 As with language, that awareness appears to be universal in nor-
mally developing children, though the rate of development varies at least in
part because of environmental influences.5
But it is not only the awareness of quantity that develops without formal
training. Both children and adults engage in mathematical problem solving,
developing untrained strategies to do so successfully when formal experi-
ences are not provided. For example, it was found that Brazilian street chil-
dren could perform mathematics when making sales in the street, but were
unable to answer similar problems presented in a school context.6 Likewise,
a study of housewives in California uncovered an ability to solve mathemati-
cal problems when comparison shopping, even though the women could
not solve problems presented abstractly in a classroom that required the
same mathematics.7 A similar result was found in a study of a group of
Weight Watchers, who used strategies for solving mathematical measure-
ment problems related to dieting that they could not solve when the prob-
lems were presented more abstractly.8 And men who successfully handi-
capped horse races could not apply the same skill to securities in the stock
market.9
These examples suggest that people possess resources in the form of
informal strategy development and mathematical reasoning that can serve as
a foundation for learning more abstract mathematics. But they also suggest
that the link is not automatic. If there is no bridge between informal and
formal mathematics, the two often remain disconnected.
The first principle of How People Learn emphasizes both the need to
build on existing knowledge and the need to engage students’ preconcep-
tions—particularly when they interfere with learning. In mathematics, cer-
tain preconceptions that are often fostered early on in school settings are in
fact counterproductive. Students who believe them can easily conclude that
the study of mathematics is “not for them” and should be avoided if at all
possible. We discuss these preconceptions below.

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220 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Some Common Preconceptions About Mathematics
Preconception #1: Mathematics is about learning to compute.
Many of us who attended school in the United States had mathematics
instruction that focused primarily on computation, with little attention to
learning with understanding. To illustrate, try to answer the following ques-
tion:
What, approximately, is the sum of 8/9 plus 12/13?
Many people immediately try to find the lowest common denominator
for the two sets of fractions and then add them because that is the procedure
they learned in school. Finding the lowest common denominator is not easy
in this instance, and the problem seems difficult. A few people take a con-
ceptual rather than a procedural (computational) approach and realize that
8/9 is almost 1, and so is 12/13, so the approximate answer is a little less
than 2.
The point of this example is not that computation should not be taught
or is unimportant; indeed, it is very often critical to efficient problem solv-
ing. But if one believes that mathematics is about problem solving and that
computation is a tool for use to that end when it is helpful, then the above
problem is viewed not as a “request for a computation,” but as a problem to
be solved that may or may not require computation—and in this case, it
does not.
If one needs to find the exact answer to the above problem, computa-
tion is the way to go. But even in this case, conceptual understanding of the
nature of the problem remains central, providing a way to estimate the cor-
rectness of a computation. If an answer is computed that is more than 2 or
less than 1, it is obvious that some aspect of problem solving has gone awry.
If one believes that mathematics is about computation, however, then sense
making may never take place.
Preconception #2: Mathematics is about “following rules” to
guarantee correct answers.
Related to the conception of mathematics as computation is that of math-
ematics as a cut-and-dried discipline that specifies rules for finding the right
answers. Rule following is more general than performing specific computa-
tions. When students learn procedures for keeping track of and canceling
units, for example, or learn algebraic procedures for solving equations, many

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221
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
view use of these procedures only as following the rules. But the “rules”
should not be confused with the game itself.
The authors of the chapters in this part of the book provide important
suggestions about the much broader nature of mathematical proficiency and
about ways to make the involving nature of mathematical inquiry visible to
students. Groups such as the National Council of Teachers of Mathematics10
and the National Research Council11 have provided important guidelines for
the kinds of mathematics instruction that accord with what is currently known
about the principles of How People Learn. The authors of the following
chapters have paid careful attention to this work and illustrate some of its
important aspects.
In reality, mathematics is a constantly evolving field that is far from cut
and dried. It involves systematic pattern finding and continuing invention.
As a simple example, consider the selection of units that are relevant to
quantify an idea such as the fuel efficiency of a vehicle. If we choose miles
per gallon, a two-seater sports car will be more efficient than a large bus. If
we choose passenger miles per gallon, the bus will be more fuel efficient
(assuming it carries large numbers of passengers). Many disciplines make
progress by inventing new units and metrics that provide insights into previ-
ously invisible relationships.
Attention to the history of mathematics illustrates that what is taught at
one point in time as a set of procedures really was a set of clever inventions
designed to solve pervasive problems of everyday life. In Europe in the
Middle Ages, for example, people used calculating cloths marked with ver-
tical columns and carried out procedures with counters to perform calcula-
tions. Other cultures fastened their counters on a rod to make an abacus.
Both of these physical means were at least partially replaced by written
methods of calculating with numerals and more recently by methods that
involve pushing buttons on a calculator. If mathematics procedures are un-
derstood as inventions designed to make common problems more easily
solvable, and to facilitate communications involving quantity, those proce-
dures take on a new meaning. Different procedures can be compared for
their advantages and disadvantages. Such discussions in the classroom can
deepen students’ understanding and skill.
Preconception #3: Some people have the ability to “do math”
and some don’t.
This is a serious preconception that is widespread in the United States,
but not necessarily in other countries. It can easily become a self-fulfilling
prophesy. In many countries, the ability to “do math” is assumed to be
attributable to the amount of effort people put into learning it.12 Of course,

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222 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
some people in these countries do progress further than others, and some
appear to have an easier time learning mathematics than others. But effort is
still considered to be the key variable in success. In contrast, in the United
States we are more likely to assume that ability is much more important than
effort, and it is socially acceptable, and often even desirable, not to put forth
effort in learning mathematics. This difference is also related to cultural
differences in the value attributed to struggle. Teachers in some countries
believe it is desirable for students to struggle for a while with problems,
whereas teachers in the United States simplify things so that students need
not struggle at all.13
This preconception likely shares a common root with the others. If
mathematics learning is not grounded in an understanding of the nature of
the problem to be solved and does not build on a student’s own reasoning
and strategy development, then solving problems successfully will depend
on the ability to recall memorized rules. If a student has not reviewed those
rules recently (as is the case when a summer has passed), they can easily be
forgotten. Without a conceptual understanding of the nature of problems
and strategies for solving them, failure to retrieve learned procedures can
leave a student completely at a loss.
Yet students can feel lost not only when they have forgotten, but also
when they fail to “get it” from the start. Many of the conventions of math-
ematics have been adopted for the convenience of communicating efficiently
in a shared language. If students learn to memorize procedures but do not
understand that the procedures are full of such conventions adopted for
efficiency, they can be baffled by things that are left unexplained. If students
never understand that x and y have no intrinsic meaning, but are conven-
tional notations for labeling unknowns, they will be baffled when a z ap-
pears. When an m precedes an x in the equation of a line, students may
wonder, Why m? Why not s for slope? If there is no m, then is there no slope?
To someone with a secure mathematics understanding, the missing m is
simply an unstated m = 1. But to a student who does not understand that the
point is to write the equation efficiently, the missing m can be baffling.
Unlike language learning, in which new expressions can often be figured
out because they are couched in meaningful contexts, there are few clues to
help a student who is lost in mathematics. Providing a secure conceptual
understanding of the mathematics enterprise that is linked to students’ sense-
making capacities is critical so that students can puzzle productively over
new material, identify the source of their confusion, and ask questions when
they do not understand.

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MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
Engaging Students’ Preconceptions and Building on
Existing Knowledge
Engaging and building on student preconceptions, then, poses two in-
structional challenges. First, how can we teach mathematics so students come
to appreciate that it is not about computation and following rules, but about
solving important and relevant quantitative problems? This perspective in-
cludes an understanding that the rules for computation and solution are a
set of clever human inventions that in many cases allow us to solve complex
problems more easily, and to communicate about those problems with each
other effectively and efficiently. Second, how can we link formal mathemat-
ics training with students’ informal knowledge and problem-solving capaci-
ties?
Many recent research and curriculum development efforts, including
those of the authors of the chapters that follow, have addressed these ques-
tions. While there is surely no single best instructional approach, it is pos-
sible to identify certain features of instruction that support the above goals:
• Allowing students to use their own informal problem-solving strate-
gies, at least initially, and then guiding their mathematical thinking toward
more effective strategies and advanced understandings.
• Encouraging math talk so that students can clarify their strategies to
themselves and others, and compare the benefits and limitations of alternate
approaches.
• Designing instructional activities that can effectively bridge commonly
held conceptions and targeted mathematical understandings.
Allowing Multiple Strategies
To illustrate how instruction can be connected to students’ existing knowl-
edge, consider three subtraction methods encountered frequently in urban
second-grade classrooms involved in the Children’s Math Worlds Project (see
Box 5-2). Maria, Peter, and Manuel’s teacher has invited them to share their
methods for solving a problem, and each of them has displayed a different
method. Two of the methods are correct, and one is mostly correct but has
one error. What the teacher does depends on her conception of what math-
ematics is.
One approach is to show the students the “right” way to subtract and
have them and everyone else practice that procedure. A very different ap-
proach is to help students explore their methods and see what is easy and
difficult about each. If students are taught that for each kind of math situa-
tion or problem, there is one correct method that needs to be taught and
learned, the seeds of the disconnection between their reasoning and strat-
egy development and “doing math” are sown. An answer is either wrong or

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224 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Three Subtraction Methods
BOX 5-2
Maria’s add-equal- Peter’s ungrouping Manuel’s mixed
quantities method method method
11 14 11 14
1 2 14 12 4 12 4
– 15 6 – 15
–56 6
68 68 5 8
right, and one does not need to look at wrong answers more deeply—one
needs to look at how to get the right answer. The problem is not that stu-
dents will fail to solve the problem accurately with this instructional ap-
proach; indeed, they may solve it more accurately. But when the nature of
the problem changes slightly, or students have not used the taught approach
for a while, they may feel completely lost when confronting a novel prob-
lem because the approach of developing strategies to grapple with a prob-
lem situation has been short-circuited.
If, on the other hand, students believe that for each kind of math situa-
tion or problem there can be several correct methods, their engagement in
strategy development is kept alive. This does not mean that all strategies are
equally good. But students can learn to evaluate different strategies for their
advantages and disadvantages. What is more, a wrong answer is usually
partially correct and reflects some understanding; finding the part that is
wrong and understanding why it is wrong can be a powerful aid to under-
standing and promotes metacognitive competencies. A vignette of students
engaged in the kind of mathematical reasoning that supports active strategy
development and evaluation appears in Box 5-3.
It can be initially unsettling for a teacher to open up the classroom to
calculation methods that are new to the teacher. But a teacher does not have
to understand a new method immediately or alone, as indicated in the de-
scription in the vignette of how the class together figured out over time how
Maria’s method worked (this method is commonly taught in Latin America
and Europe). Understanding a new method can be a worthwhile mathemati-
cal project for the class, and others can be involved in trying to figure out
why a method works. This illustrates one way in which a classroom commu-
nity can function. If one relates a calculation method to the quantities in-
volved, one can usually puzzle out what the method is and why it works.
This also demonstrates that not all mathematical issues are solved or under-
stood immediately; sometimes sustained work is necessary.

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225
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
Engaging Students’ Problem-Solving Strategies
BOX 5-3
The following example of a classroom discussion shows how second-
grade students can explain their methods rather than simply performing
steps in a memorized procedure. It also shows how to make student
thinking visible. After several months of teaching and learning, the stu-
dents reached the point illustrated below. The students’ methods are
shown in Box 5-2.
Teacher Maria, can you please explain to your friends
in the class how you solved the problem?
Maria Six is bigger than 4, so I can’t subtract here
[pointing] in the ones.
So I have to get more ones. But I have to be
fair when I get more ones, so I add ten to both
my numbers. I add a ten here in the top of the
ones place [pointing] to change the 4 to a 14,
and I add a ten here in the bottom in the tens
place, so I write another ten by
my 5.
So now I count up from 6 to 14, and I get 8
ones [demonstrating by counting “6, 7, 8, 9,
10, 11, 12, 13, 14” while raising a finger for
each word from 7 to 14]. And I know my
doubles, so 6 plus 6 is 12, so I have 6 tens left.
[She thought, “1 + 5 = 6 tens and 6 + ? = 12
tens. Oh, I know 6 + 6 = 12, so my answer is 6
tens.”]
Jorge I don’t see the other 6 in your tens. I only see
one 6 in your answer.
Maria The other 6 is from adding my 1 ten to the 5
tens to get 6 tens. I didn’t write it down.
Andy But you’re changing the problem. How do you
get the right answer?
Maria If I make both numbers bigger by the same
amount, the difference will stay the same.
Remember we looked at that on drawings last
week and on the meter stick.
Michelle Why did you count up?
Maria Counting down is too hard, and my mother
taught me to count up to subtract in first
grade.

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226 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Continued
BOX 5-3
Teacher How many of you remember how confused we
were when we first saw Maria’s method last
week? Some of us could not figure out what
she was doing even though Elena and Juan
and Elba did it the same way. What did we do?
Rafael We made drawings with our ten-sticks and
dots to see what those numbers meant. And
we figured out they were both tens. Even
though the 5 looked like a 15, it was really just
6. And we went home to see if any of our
parents could explain it to us, but we had to
figure it out ourselves and it took us 2 days.
Teacher Yes, I was asking other teachers, too. We
worked on other methods too, but we kept
trying to understand what this method was
and why it worked. And Elena and Juan
decided it was clearer if they crossed out the 5
and wrote a 6, but Elba and Maria liked to do it
the way they learned at home. Any other
questions or comments for Maria? No? Ok,
Peter, can you explain your method?
Peter Yes, I like to ungroup my top number when I
don’t have enough to subtract everywhere. So
here I ungrouped 1 ten and gave it to the 4
ones to make 14 ones, so I had 1 ten left here.
So 6 up to 10 is 4 and 4 more up to 14 is 8, so
14 minus 6 is 8 ones. And 5 tens up to 11 tens
is 6 tens. So my answer is 68.
Carmen How did you know it was 11 tens?
Peter Because it is 1 hundred and 1 ten and that is 11
tens.
Carmen I don’t get it.
Peter Because 1 hundred is 10 tens.
Carmen Oh, so why didn’t you cross out the 1 hundred
and put it with the tens to make 11 tens like
Manuel?
Peter I don’t need to. I just know it is 11 tens by
looking at it.
Teacher Manuel, don’t erase your problem. I know you
think it is probably wrong because you got a
different answer, but remember how making a
mistake helps everyone learn—because other

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MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
students make that same mistake and you
helped us talk about it. Do you want to draw a
picture and think about your method while we
do the next problem, or do you want someone
to help you?
Manuel Can Rafael help me?
Teacher Yes, but what kind of helping should Rafael
do?
Manuel He should just help me with what I need help
on and not do it for me.
Teacher Ok, Rafael, go up and help Manuel that way
while we go on to the next problem. I think it
would help you to draw quick-tens and ones to
see what your numbers mean. [These draw-
ings are explained later.] But leave your first
solution so we can all see where the problem
is. That helps us all get good at debugging—
finding our mistakes. Do we all make mis-
takes?
Class Yes.
Teacher Can we all get help from each other?
Class Yes.
Teacher So mistakes are just a part of learning. We
learn from our mistakes. Manuel is going to
be brave and share his mistake with us so we
can all learn from it.
Manuel’s method combined Maria’s add-equal-quantities method,
which he had learned at home, and Peter’s ungrouping method, which he
had learned at school. It increases the ones once and decreases the tens
twice by subtracting a ten from the top number and adding a ten to the
bottom subtracted number. In the Children’s Math Worlds Project, we
rarely found children forming such a meaningless combination of meth-
ods if they understood tens and ones and had a method of drawing them
so they could think about the quantities in a problem (a point discussed
more later). Students who transferred into our classes did sometimes
initially use Manuel’s mixed approach. But students were eventually helped
to understand both the strengths and weaknesses of their existing meth-
ods and to find ways of improving their approaches.
SOURCE: Karen Fuson, Children’s Math Worlds Project.

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246 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
this work indicates that we have begun the crucial journey into mathemati-
cal proficiency for all and that the principles of How People Learn can guide
us on this journey.
NOTES
1. Holt, 1964, pp. 143-144.
2. National Research Council, 2001.
3. See Fuson, 1986a, 1986b, 1990; Fuson and Briars, 1990; Fuson and Burghardt,
1993, 1997; Fuson et al., 1994, 2000; Fuson and Smith, 1997; Fuson, Smith, and
Lott, 1977; Fuson, Wearne et al., 1997; Fuson, Lo Cicero et al., 1997; Lo Cicero
et al., 1999; Fuson et al., 2000; Ron, 1998.
4. Carey, 2001; Gelman, 1990; Starkey et al., 1990; Wynn, 1996; Canfield and
Smith, 1996.
5. Case et al., 1999; Ginsburg, 1984; Saxe, 1982.
6. Carraher, 1986; Carraher et al., 1985.
7. Lave, 1988; Sternberg, 1999.
8. De la Rocha, 1986.
9. Ceci and Liker, 1986; Ceci, 1996.
10. National Council of Teachers of Mathematics, 2000.
11. National Research Council, 2001.
12. See, e.g., Hatano and Inagaki, 1996; Resnick, 1987; Stigler and Heibert, 1997.
13. Stigler and Heibert, 1999.
14. National Research Council, 2004.
15. See, e.g., Tobias, 1978.
16. Hufferd-Ackles et al., 2004.
17. Sherin, 2000a, 2002.
18. See, e.g., Bransford et al., 1989.
19. See, e.g., Schwartz and Moore, 1998.
20. Sherin, 2000b, 2001.
21. Lewis, 2002, p. 1.
22. Fernandez, 2002; Lewis, 2002; Stigler and Heibert, 1999.
23. Remillard, 1999, 2000.
24. Remillard and Geist, 2002.
25. Remillard, 2000.
REFERENCES
Anghileri, J. (1989). An investigation of young children’s understanding of multiplica-
tion. Educational Studies in Mathematics, 20, 367-385.
Ashlock, R.B. (1998). Error patterns in computation. Upper Saddle River, NJ: Prentice-
Hall.
Baek, J.-M. (1998). Children’s invented algorithms for multidigit multiplication prob-
lems. In L.J. Morrow and M.J. Kenney (Eds.), The teaching and learning of
algorithms in school mathematics. Reston, VA: National Council of Teachers of
Mathematics.

OCR for page 29

247
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
Baroody, A.J., and Coslick, R.T. (1998). Fostering children’s mathematical power: An
investigative approach to k-8 mathematics instruction. Mahwah, NJ: Lawrence
Erlbaum Associates.
Baroody, A.J., and Ginsburg, H.P. (1986). The relationship between initial meaning-
ful and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and
procedural knowledge: The case of mathematics (pp. 75-112). Mahwah, NJ:
Lawrence Erlbaum Associates.
Beishuizen, M. (1993). Mental strategies and materials or models for addition and
subtraction up to 100 in Dutch second grades. Journal for Research in Math-
ematics Education, 24, 294-323.
Beishuizen, M., Gravemeijer, K.P.E., and van Lieshout, E.C.D.M. (Eds.). (1997). The
role of contexts and models in the development of mathematical strategies and
procedures. Utretch, The Netherlands: CD-B Press/The Freudenthal Institute.
Bergeron, J.C., and Herscovics, N. (1990). Psychological aspects of learning early
arithmetic. In P. Nesher and J. Kilpatrick (Eds.), Mathematics and cognition: A
research synthesis by the International Group for the Psychology of Mathematics
Education. Cambridge, England: Cambridge University Press.
Bransford, J.D., Franks, J.J., Vye, N.J., and Sherwood, R.D. (1989). New approaches
to instruction: Because wisdom can’t be told. In S. Vasniadou and A. Ortony
(Eds.), Similarity and analogical reasoning (pp. 470-497). New York: Cambridge
University Press.
Brophy, J. (1997). Effective instruction. In H.J. Walberg and G.D. Haertel (Eds.),
Psychology and educational practice (pp. 212-232). Berkeley, CA: McCutchan.
Brownell, W.A. (1987). AT Classic: Meaning and skill—maintaining the balance. Arith-
metic Teacher, 34(8), 18-25.
Canfield, R.L., and Smith, E.G. (1996). Number-based expectations and sequential
enumeration by 5-month-old infants. Developmental Psychology, 32, 269-279.
Carey, S. (2001). Evolutionary and ontogenetic foundations of arithmetic. Mind and
Language, 16(1), 37-55.
Carpenter, T.P., and Moser, J.M. (1984). The acquisition of addition and subtraction
concepts in grades one through three. Journal for Research in Mathematics
Education, 15(3), 179-202.
Carpenter, T.P., Fennema, E., Peterson, P.L., Chiang, C.P., and Loef, M. (1989). Using
knowledge of children’s mathematics thinking in classroom teaching: An experi-
mental study. American Educational Research Journal, 26(4), 499-531.
Carpenter, T.P., Franke, M.L., Jacobs, V., and Fennema, E. (1998). A longitudinal
study of invention and understanding in children’s multidigit addition and sub-
traction. Journal for Research in Mathematics Education, 29, 3-20.
Carraher, T.N. (1986). From drawings to buildings: Mathematical scales at work. In-
ternational Journal of Behavioural Development, 9, 527-544.
Carraher, T.N., Carraher, D.W., and Schliemann, A.D. (1985). Mathematics in the
streets and in schools. British Journal of Developmental Psychology, 3, 21-29.
Carroll, W.M. (2001). A longitudinal study of children using the reform curriculum
everyday mathematics. Available: http://everydaymath.uchicago.edu/educators/
references.shtml [accessed September 2004].

OCR for page 29

248 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Carroll, W.M., and Fuson, K.C. (1999). Achievement results for fourth graders using
the standards-based curriculum everyday mathematics. Unpublished document,
University of Chicago, Illinois.
Carroll, W.M., and Porter, D. (1998). Alternative algorithms for whole-number opera-
tions. In L.J. Morrow and M.J. Kenney (Eds.), The teaching and learning of
algorithms in school mathematics (pp. 106-114). Reston, VA: National Council of
Teachers of Mathematics.
Case, R. (1985). Intellectual development: Birth to adulthood. New York: Academic
Press.
Case, R. (1992). The mind’s staircase: Exploring the conceptual underpinnings of
children’s thought and knowledge. Mahwah, NJ: Lawrence Erlbaum Associates.
Case, R. (1998). A psychological model of number sense and its development. Paper
presented at the annual meeting of the American Educational Research Associa-
tion, April, San Diego, CA.
Case, R., and Sandieson, R. (1988). A developmental approach to the identification
and teaching of central conceptual structures in mathematics and science in the
middle grades. In M. Behr and J. Hiebert (Eds.), Research agenda in mathemat-
ics education: Number concepts and in the middle grades (pp. 136-270). Mahwah,
NJ: Lawrence Erlbaum Associates.
Case, R., Griffin, S., and Kelly, W.M. (1999). Socioeconomic gradients in mathemati-
cal ability and their responsiveness to intervention during early childhood. In
D.P. Keating and C. Hertzman (Eds.), Developmental health and the wealth of
nations: Social, biological, and educational dynamics (pp. 125-149). New York:
Guilford Press.
Ceci, S.J. (1996). On intelligence: A bioecological treatise on intellectual development.
Cambridge, MA: Harvard University Press.
Ceci, S.J., and Liker, J.K. (1986). A day at the races: A study of IQ, expertise, and
cognitive complexity. Journal of Experimental Psychology, 115(3), 255-266.
Cotton, K. (1995). Effective schooling practices: A research synthesis. Portland, OR:
Northwest Regional Lab.
Davis, R.B. (1984). Learning mathematics: The cognitive science approach to math-
ematics education. Norwood, NJ: Ablex.
De la Rocha, O.L. (1986). The reorganization of arithmetic practice in the kitchen.
Anthropology and Education Quarterly, 16(3), 193-198.
Dixon, R.C., Carnine, S.W., Kameenui, E.J., Simmons, D.C., Lee, D.S., Wallin, J., and
Chard, D. (1998). Executive summary. Report to the California State Board of
Education, review of high-quality experimental research. Eugene, OR: National
Center to Improve the Tools of Educators.
Dossey, J.A., Swafford, J.O., Parmantie, M., and Dossey, A.E. (Eds.). (2003). Multidigit
addition and subtraction methods invented in small groups and teacher support
of problem solving and reflection. In A. Baroody and A. Dowker (Eds.), The
development of arithmetic concepts and skills: Constructing adaptive expertise.
Mahwah, NJ: Lawrence Erlbaum Associates.
Fernandez, C. (2002). Learning from Japanese approaches to professional develop-
ment. The case of lesson study. Journal of Teacher Education, 53(5), 393-405.

OCR for page 29

249
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
Fraivillig, J.L., Murphy, L.A., and Fuson, K.C. (1999). Advancing children’s math-
ematical thinking in everyday mathematics reform classrooms. Journal for Re-
search in Mathematics Education, 30, 148-170.
Fuson, K.C. (1986a). Roles of representation and verbalization in the teaching of
multidigit addition and subtraction. European Journal of Psychology of Educa-
tion, 1, 35-56.
Fuson, K.C. (1986b). Teaching children to subtract by counting up. Journal for Re-
search in Mathematics Education, 17, 172-189.
Fuson, K.C. (1990). Conceptual structures for multiunit numbers: Implications for
learning and teaching multidigit addition, subtraction, and place value. Cogni-
tion and Instruction, 7, 343-403.
Fuson, K.C. (1992a). Research on learning and teaching addition and subtraction of
whole numbers. In G. Leinhardt, R.T. Putnam, and R.A. Hattrup (Eds.), The
analysis of arithmetic for mathematics teaching (pp. 53-187). Mahwah, NJ:
Lawrence Erlbaum Associates.
Fuson, K.C. (1992b). Research on whole number addition and subtraction. In D.
Grouws (Ed.), Handbook of research on mathematics teaching and learning
(pp. 243-275). New York: Macmillan.
Fuson, K.C. (2003). Developing mathematical power in whole number operations. In
J. Kilpatrick, W.G. Martin, and D. Schifter (Eds.), A r esearch companion to prin-
ciples and standards for school mathematics (pp. 68-94). Reston, VA: National
Council of Teachers of Mathematics.
Fuson, K.C., and Briars, D.J. (1990). Base-ten blocks as a first- and second-grade
learning/teaching approach for multidigit addition and subtraction and place-
value concepts. Journal for Research in Mathematics Education, 21, 180-206.
Fuson, K.C., and Burghardt, B.H. (1993). Group case studies of second graders in-
venting multidigit addition procedures for base-ten blocks and written marks. In
J.R. Becker and B.J. Pence (Eds.), Proceedings of the fifteenth annual meeting of
the North American chapter of the international group for the psychology of
mathematics education (pp. 240-246). San Jose, CA: The Center for Mathematics
and Computer Science Education, San Jose State University.
Fuson, K.C., and Burghardt, B.H. (1997). Group case studies of second graders in-
venting multidigit subtraction methods. In Proceedings of the 19th annual meet-
ing of the North American chapter of the international group for the psychology
of mathematics education (pp. 291-298). San Jose, CA: The Center for Math-
ematics and Computer Science Education, San Jose State University.
Fuson, K.C., and Fuson, A.M. (1992). Instruction to support children’s counting on
for addition and counting up for subtraction. Journal for Research in Mathemat-
ics Education, 23, 72-78.
Fuson, K.C., and Kwon, Y. (1992). Korean children’s understanding of multidigit
addition and subtraction. Child Development, 63(2), 491-506.
Fuson, K.C., and Secada, W.G. (1986). Teaching children to add by counting with
finger patterns. Cognition and Instruction, 3, 229-260.

OCR for page 29

250 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Fuson, K.C., and Smith, T. (1997). Supporting multiple 2-digit conceptual structures
and calculation methods in the classroom: Issues of conceptual supports, in-
structional design, and language. In M. Beishuizen, K.P.E. Gravemeijer, and
E.C.D.M. van Lieshout (Eds.), The role of contexts and models in the development
of mathematical strategies and procedures (pp. 163-198). Utrecht, The Nether-
lands: CD-B Press/The Freudenthal Institute.
Fuson, K.C., Stigler, J., and Bartsch, K. (1988). Grade placement of addition and
subtraction topics in Japan, mainland China, the Soviet Union, Taiwan, and the
United States. Journal for Research in Mathematics Education, 19(5), 449-456.
Fuson, K.C., Perry, T., and Kwon, Y. (1994). Latino, Anglo, and Korean children’s
finger addition methods. In J.E.H. van Luit (Ed.), Research on learning and
instruction of mathematics in kindergarten and primary school, (pp. 220-228).
Doetinchem/Rapallo, The Netherlands: Graviant.
Fuson, K.C., Perry, T., and Ron, P. (1996). Developmental levels in culturally different
finger methods: Anglo and Latino children’s finger methods of addition. In E.
Jakubowski, D. Watkins, and H. Biske (Eds.), Proceedings of the 18th annual
meeting of the North American chapter for the psychology of mathematics educa-
tion (2nd edition, pp. 347-352). Columbus, OH: ERIC Clearinghouse for Science,
Mathematics, and Environmental Education.
Fuson, K.C., Lo Cicero, A., Hudson, K., and Smith, S.T. (1997). Snapshots across two
years in the life of an urban Latino classroom. In J. Hiebert, T. Carpenter, E.
Fennema, K.C. Fuson, D. Wearne, H. Murray, A. Olivier, and P. Human (Eds.),
Making sense: Teaching and learning mathematics with understanding (pp.
129-159). Portsmouth, NH: Heinemann.
Fuson, K.C., Smith, T., and Lo Cicero, A. (1997). Supporting Latino first graders’ ten-
structured thinking in urban classrooms. Journal for Research in Mathematics
Education, 28, 738-760.
Fuson, K.C., Wearne, D., Hiebert, J., Murray, H., Human, P., Olivier, A., Carpenter, T.,
and Fennema, E. (1997). Children’s conceptual structures for multidigit numbers
and methods of multidigit addition and subtraction. Journal for Research in
Mathematics Education, 28, 130-162.
Fuson, K.C., De La Cruz, Y., Smith, S., Lo Cicero, A., Hudson, K., Ron, P., and Steeby,
R. (2000). Blending the best of the 20th century to achieve a mathematics equity
pedagogy in the 21st century. In M.J. Burke and F.R. Curcio (Eds.), Learning
mathematics for a new century (pp. 197-212). Reston, VA: National Council of
Teachers of Mathematics.
Geary, D.C. (1994). Children’s mathematical development: Research and practical
applications. Washington, DC: American Psychological Association.
Gelman, R. (1990). First principles organize attention to and learning about relevant
data: Number and the animate-inanimate distinction as examples. Cognitive Sci-
ence, 14, 79-106.
Ginsburg, H.P. (1984). Children’s arithmetic: The learning process. New York: Van
Nostrand.
Ginsburg, H.P., and Allardice, B.S. (1984). Children’s difficulties with school math-
ematics. In B. Rogoff and J. Lave (Eds.), Everyday cognition: Its development in
social contexts (pp. 194-219). Cambridge, MA: Harvard University Press.

OCR for page 29

251
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
Ginsburg, H.P., and Russell, R.L. (1981). Social class and racial influences on early
mathematical thinking. Monographs of the Society for Research in Child Develop-
ment 44(6, serial #193). Malden, MA: Blackwell.
Goldman, S.R., Pellegrino, J.W., and Mertz, D.L. (1988). Extended practice of basic
addition facts: Strategy changes in learning-disabled students. Cognition and
Instruction, 5(3), 223-265.
Goldman, S.R., Hasselbring, T.S., and the Cognition and Technology Group at
Vanderbilt (1997). Achieving meaningful mathematics literacy for students with
learning disabilities. Journal of Learning Disabilities, March 1(2), 198-208.
Greer, B. (1992). Multiplication and division as models of situation. In D. Grouws
(Ed.), Handbook of research on mathematics teaching and learning (pp. 276-
295). New York: Macmillan.
Griffin, S., and Case, R. (1997). Re-thinking the primary school math curriculum: An
approach based on cognitive science. Issues in Education, 3(1), 1-49.
Griffin, S., Case, R., and Siegler, R.S. (1994 ). Rightstart: Providing the central concep-
tual structures for children at risk of school failure. In K. McGilly (Ed.), Class-
room lessons: Integrating cognitive theory and classroom practice (pp. 13-48).
Mahwah, NJ: Lawrence Erlbaum Associates.
Grouws, D. (1992). Handbook of research on mathematics teaching and learning.
New York: Teachers College Press.
Hamann, M.S., and Ashcraft, M.H. (1986). Textbook presentations of the basic addi-
tion facts. Cognition and Instruction, 3, 173-192.
Hart, K.M. (1987). Practical work and formalisation, too great a gap. In J.C. Bergeron,
N. Hersovics, and C. Kieren (Eds.), Proceedings from the eleventh international
conference for the psychology of mathematics education (vol. 2, pp. 408-415).
Montreal, Canada: University of Montreal.
Hatano, G., and Inagaki, K. (1996). Cultural contexts of schooling revisited. A review
of the learning gap from a cultural psychology perspective. Paper presented at
the Conference on Global Prospects for Education: Development, Culture, and
Schooling, University of Michigan.
Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics.
Mahwah, NJ: Lawrence Erlbaum Associates.
Hiebert, J. (1992). Mathematical, cognitive, and instructional analyses of decimal
fractions. In G. Leinhardt, R. Putnam, and R.A. Hattrup (Eds.), The analysis of
arithmetic for mathematics teaching (pp. 283-322). Mahwah, NJ: Lawrence
Erlbaum Associates.
Hiebert, J., and Carpenter, T.P. (1992). Learning and teaching with understanding. In
D. Grouws (Ed.), Handbook of research on mathematics teaching and learning
(pp. 65-97). New York: Macmillan.
Hiebert, J., and Wearne, D. (1986). Procedures over concepts: The acquisition of
decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural
knowledge: The case of mathematics (pp. 199-223). Mahwah, NJ: Lawrence
Erlbaum Associates.
Hiebert, J., Carpenter, T., Fennema, E., Fuson, K.C., Murray, H., Olivier, A., Human,
P., and Wearne, D. (1996). Problem solving as a basis for reform in curriculum
and instruction: The case of mathematics. Educational Researcher, 25(4), 12-21.

OCR for page 29

252 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Hiebert, J., Carpenter, T., Fennema, E., Fuson, K.C., Wearne, D., Murray, H., Olivier,
A., and Human, P. (1997). Making sense: Teaching and learning mathematics
with understanding. Portsmouth, NH: Heinemann.
Holt, J. (1964). How children fail. New York: Dell.
Hufferd-Ackles, K., Fuson, K., and Sherin, M.G. (2004). Describing levels and com-
ponents of a math-talk community. Journal for Research in Mathematics Educa-
tion, 35(2), 81-116.
Isaacs, A.C., and Carroll, W.M. (1999). Strategies for basic-facts instruction. Teaching
Children Mathematics, 5(9), 508-515.
Kalchman, M., and Case, R. (1999). Diversifying the curriculum in a mathematics
classroom streamed for high-ability learners: A necessity unassumed. School
Science and Mathematics, 99(6), 320-329.
Kameenui, E.J., and Carnine, D.W. (Eds.). (1998). Effective teaching strategies that
accommodate diverse learners. Upper Saddle River, NJ: Prentice-Hall.
Kerkman, D.D., and Siegler, R.S. (1993). Individual differences and adaptive flexibil-
ity in lower-income children’s strategy choices. Learning and Individual Differ-
ences, 5(2), 113-136.
Kilpatrick, J., Martin, W.G., and Schifter, D. (Eds.). (2003). A research companion to
principles and standards for school mathematics. Reston, VA: National Council
of Teachers of Mathematics.
Knapp, M.S. (1995). Teaching for meaning in high-poverty classrooms. New York:
Teachers College Press.
Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and
Instruction, 3, 305-342.
Lampert, M. (1992). Teaching and learning long division for understanding in school.
In G. Leinhardt, R.T. Putnam, and R.A. Hattrup (Eds.), The analysis of arithmetic
for mathematics teaching (pp. 221-282). Mahwah, NJ: Lawrence Erlbaum Asso-
ciates.
Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday
life. London, England: Cambridge University Press.
LeFevre, J., and Liu, J. (1997). The role of experience in numerical skill: Multiplica-
tion performance in adults from Canada and China. Mathematical Cognition,
3(1), 31-62.
LeFevre, J., Kulak, A.G., and Bisantz, J. (1991). Individual differences and develop-
mental change in the associative relations among numbers. Journal of Experi-
mental Child Psychology, 52, 256-274.
Leinhardt, G., Putnam, R.T., and Hattrup, R.A. (Eds.). (1992). The analysis of arith-
metic for mathematics teaching. Mahwah, NJ: Lawrence Erlbaum Associates.
Lemaire, P., and Siegler, R.S. (1995). Four aspects of strategic change: Contributions
to children’s learning of multiplication. Journal of Experimental Psychology:
General, 124(1), 83-97.
Lemaire, P., Barrett, S.E., Fayol, M., and Abdi, H. (1994). Automatic activation of
addition and multiplication facts in elementary school children. Journal of Ex-
perimental Child Psychology, 57, 224-258.
Lewis, C. (2002). Lesson study: A handbook of teacher-led instructional change. Phila-
delphia, PA: Research for Better Schools.

OCR for page 29

253
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
Lo Cicero, A., Fuson, K.C., and Allexaht-Snider, M. (1999). Making a difference in
Latino children’s math learning: Listening to children, mathematizing their sto-
ries, and supporting parents to help children. In L. Ortiz-Franco, N.G. Hernendez,
and Y. De La Cruz (Eds.), Changing the faces of mathematics: Perspectives on
Latinos (pp. 59-70). Reston, VA: National Council of Teachers of Mathematics.
McClain, K., Cobb, P., and Bowers, J. (1998). A contextual investigation of three-digit
addition and subtraction. In L. Morrow (Ed.), Teaching and learning of algo-
rithms in school mathematics (pp. 141-150). Reston, VA: National Council of
Teachers of Mathematics.
McKnight, C.C., and Schmidt, W.H. (1998). Facing facts in U.S. science and math-
ematics education: Where we stand, where we want to go. Journal of Science
Education and Technology, 7(1), 57-76.
McKnight, C.C., Crosswhite, F.J., Dossey, J.A., Kifer, E., Swafford, J.O., Travers, K.T.,
and Cooney, T.J. (1989). The underachieving curriculum: Assessing U.S. school
mathematics from an international perspective. Champaign, IL: Stipes.
Miller, K.F., and Paredes, D.R. (1990). Starting to add worse: Effects of learning to
multiply on children’s addition. Cognition, 37, 213-242.
Moss, J., and Case, R. (1999). Developing children’s understanding of rational num-
bers: A new model and experimental curriculum. Journal for Research in Math-
ematics Education, 30(2), 122-147.
Mulligan, J., and Mitchelmore, M. (1997). Young children’s intuitive models of multi-
plication and division. Journal for Research in Mathematics Education, 28(3),
309-330.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation
standards for school mathematics. Reston, VA: National Council of Teachers of
Mathematics.
National Council of Teachers of Mathematics. (1991). Professional standards for teach-
ing mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (2000). Principles and standards for
school mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Research Council. (2001). Adding it up: Helping children learn mathemat-
ics. Mathematics Learning Study Committee, J. Kilpatrick, J. Swafford, and B.
Findell (Eds.). Center for Education, Division of Behavioral and Social Sciences
and Education. Washington, DC: National Academy Press.
National Research Council. (2002). Helping children learn mathematics. Mathemat-
ics Learning Study Committee, J. Kilpatrick, J. Swafford, and B. Findell (Eds.).
Center for Education, Division of Behavioral and Social Sciences and Education.
Washington, DC: The National Academies Press.
National Research Council. (2004). Learning and instruction: A SERP research agenda.
Panel on Learning and Instruction. M.S. Donovan and J.W. Pellegrino (Eds.).
Division of Behavioral and Social Sciences and Education. Washington, DC: The
National Academies Press.
Nesh, P., and Kilpatrick, J. (Eds.). (1990). Mathematics and cognition: A research
synthesis by the International Group for the Psychology of Mathematics Educa-
tion. Cambridge, MA: Cambridge University Press.

OCR for page 29

254 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Nesher, P. (1992). Solving multiplication word problems. In G. Leinhardt, R.T. Putnam,
and R.A. Hattrup (Eds.), The analysis of arithmetic for mathematics teaching
(pp. 189-220). Mahwah, NJ: Lawrence Erlbaum Associates.
Peak, L. (1996). Pursuing excellence: A study of the U.S. eighth-grade mathematics
and science teaching, learning, curriculum, and achievement in an interna-
tional context. Washington, DC: National Center for Education Statistics.
Remillard, J.T. (1999). Curriculum materials in mathematics education reform: A frame-
work for examining teachers’ curriculum development. Curriculum Inquiry, 29(3),
315-342.
Remillard, J.T. (2000). Can curriculum materials support teachers’ learning? Elemen-
tary School Journal, 100(4), 331-350.
Remillard, J.T., and Geist, P. (2002). Supporting teachers’ professional learning though
navigating openings in the curriculum. Journal of Mathematics Teacher Educa-
tion, 5(1), 7-34.
Resnick, L.B. (1987). Education and learning to think. Committee on Mathematics,
Science, and Technology Education, Commission on Behavioral and Social Sci-
ences and Education. Washington, DC: National Academy Press.
Resnick, L.B. (1992). From protoquantities to operators: Building mathematical com-
petence on a foundation of everyday knowledge. In G. Leinhardt, R.T. Putnam,
and R.A. Hattrup (Eds.), The analysis of arithmetic for mathematics teaching
(pp. 373-429). Mahwah, NJ: Lawrence Erlbaum Associates.
Resnick, L.B., and Omanson, S.F. (1987). Learning to understand arithmetic. In R.
Glaser (Ed.), Advances in instructional psychology (vol. 3, pp. 41-95). Mahwah,
NJ: Lawrence Erlbaum Associates.
Resnick, L.B., Nesher, P., Leonard, F., Magone, M., Omanson, S., and Peled, I. (1989).
Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for
Research in Mathematics Education, 20(1), 8-27.
Ron, P. (1998). My family taught me this way. In L.J. Morrow and M.J. Kenney (Eds.),
The teaching and learning of algorithms in school mathematics (pp. 115-119).
Reston, VA: National Council of Teachers of Mathematics.
Saxe, G.B. (1982). Culture and the development of numerical cognition: Studies
among the Oksapmin of Papua New Guinea. In C.J. Brainerd (Ed.), Progress in
cognitive development research: Children’s logical and mathematical cognition
(vol. 1, pp. 157- 176). New York: Springer-Verlag.
Schmidt, W., McKnight, C.C., and Raizen, S.A. (1997). A splintered vision: An investi-
gation of U.S. science and mathematics education. Dordrecht, The Netherlands:
Kluwer.
Schwartz, D.L., and Moore, J.L. (1998). The role of mathematics in explaining the
material world: Mental models for proportional reasoning. Cognitive Science,
22, 471-516.
Secada, W.G. (1992). Race, ethnicity, social class, language, and achievement in math-
ematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching
and learning (pp. 623-660). New York: Macmillan.
Sherin, M.G. (2000a). Facilitating meaningful discussions about mathematics. Math-
ematics Teaching in the Middle School, 6(2), 186-190.
Sherin, M.G. (2000b). Taking a fresh look at teaching through video clubs. Educa-
tional Leadership, 57(8), 36-38.

OCR for page 29

255
MATHEMATICAL UNDERSTANDING: AN INTRODUCTION
Sherin, M.G. (2001). Developing a professional vision of classroom events. In T.
Wood, B.S. Nelson, and J. Warfield (Eds.), Beyond classical pedagogy: Teaching
elementary school mathematics (pp. 75-93). Mahwah, NJ: Lawrence Erlbaum
Associates.
Sherin, M.G. (2002). A balancing act: Developing a discourse community in a math-
ematics classroom. Journal of Mathematics Teacher Education, 5, 205-233.
Shuell, T.J. (2001). Teaching and learning in a classroom context. In N.J. Smelser and
P.B. Baltes (Eds.), International encyclopedia of the social and behavioral sci-
ences (pp. 15468-15472). Amsterdam: Elsevier.
Siegler, R.S. (1988). Individual differences in strategy choices: Good students, not-so-
good students, and perfectionists. Child Development, 59(4), 833-851.
Siegler, R.S. (2003). Implications of cognitive science research for mathematics edu-
cation. In J. Kilpatrick, W.G. Martin, and D.E. Schifter (Eds.), A research com-
panion to principles and standards for school mathematics (pp. 1289-1303).
Reston, VA: National Council of Teachers of Mathematics.
Simon, M.A. (1995). Reconstructing mathematics pedagogy from a constructivist per-
spective. Journal for Research in Mathematics Education, 26, 114-145.
Starkey, P., Spelke, E.S., and Gelman, R. (1990). Numerical abstraction by human
infants. Cognition, 36, 97-127.
Steffe, L.P. (1994). Children’s multiplying schemes. In G. Harel and J. Confrey (Eds.),
The development of multiplicative reasoning in the learning of mathematics (pp.
3-39). New York: State University of New York Press.
Steffe, L.P., Cobb, P., and Von Glasersfeld, E. (1988). Construction of arithmetical
meanings and strategies. New York: Springer-Verlag.
Sternberg, R.J. (1999). The theory of successful intelligence. Review of General Psy-
chology, 3(4), 292-316.
Stigler, J.W., and Hiebert, J. (1999). Teaching gap. New York: Free Press.
Stigler, J.W., Fuson, K.C., Ham, M., and Kim, M.S. (1986). An analysis of addition and
subtraction word problems in American and Soviet elementary mathematics text-
books. Cognition and Instruction, 3(3), 153-171.
Stipek, D., Salmon, J.M., Givvin, K.B., Kazemi, E., Saxe, G., and MacGyvers, V.L.
(1998). The value (and convergence) of practices suggested by motivation re-
search and promoted by mathematics education reformers. Journal for Research
in Mathematics Education, 29, 465-488.
Thornton, C.A. (1978). Emphasizing thinking in basic fact instruction. Journal for
Research in Mathematics Education, 9, 214-227.
Thornton, C.A., Jones, G.A., and Toohey, M.A. (1983). A multisensory approach to
thinking strategies for remedial instruction in basic addition facts. Journal for
Research in Mathematics Education, 14(3), 198-203.
Tobias, S. (1978). Overcoming math anxiety. New York: W.W. Norton.
Van de Walle, J.A. (1998). Elementary and middle school mathematics: Teaching
developmentally, third edition. New York: Longman.
Van de Walle, J.A. (2000). Elementary school mathematics: Teaching developmen-
tally, fourth edition. New York: Longman.
Wynn, K. (1996). Infants’ individuation and enumeration of actions. Psychological
Science, 7, 164-169.

OCR for page 29

256 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Zucker, A.A. (1995). Emphasizing conceptual understanding and breadth of study in
mathematics instruction. In M.S. Knapp (Ed.), Teaching for meaning in high-
poverty classrooms. New York: Teachers College Press.
SUGGESTED READING LIST FOR TEACHERS
Carpenter, T.P. Fennema, E., Franke, M.L., Empson, S.B., and Levi, L.W. (1999).
Children’s mathematics: Cognitively guided instruction. Portsmouth, NH:
Heinemann.
Fuson, K.C. (1988). Subtracting by counting up with finger patterns. (Invited paper
for the Research into Practice Series.) Arithmetic Teacher, 35(5), 29-31.
Hiebert, J., Carpenter, T., Fennema, E., Fuson, K.C., Wearne, D., Murray, H., Olivier,
A., and Human, P. (1997). Making sense: Teaching and learning mathematics
with understanding. Portsmouth, NH: Heinemann.
Jensen, R.J. (Ed.). (1993). Research ideas for the classroom: Early childhood math-
ematics. New York: Macmillan.
Knapp, M.S. (1995). Teaching for meaning in high-poverty classrooms. New York:
Teachers College Press.
Leinhardt, G., Putnam, R.T., and Hattrup, R.A. (Eds.). (1992). The analysis of arith-
metic for mathematics teaching. Mahwah, NJ: Lawrence Erlbaum Associates.
Lo Cicero, A., De La Cruz, Y., and Fuson, K.C. (1999). Teaching and learning cre-
atively with the Children’s Math Worlds Curriculum: Using children’s narratives
and explanations to co-create understandings. Teaching Children Mathematics,
5(9), 544-547.
Owens, D.T. (Ed.). (1993). Research ideas for the classroom: Middle grades math-
ematics. New York: Macmillan.
Schifter, D. (Ed.). (1996). What’s happening in math class? Envisioning new practices
through teacher narratives. New York: Teachers College Press.
Wagner, S. (Ed.). (1993). Research ideas for the classroom: High school mathematics.
New York: Macmillan.