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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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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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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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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.
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SUGGESTED READING LIST FOR TEACHERS
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