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OCR for page 121
309
Pipes,Tubes, and Beakers:
New Approaches to Teaching the
Rational-Number System
Joan Moss
4CdoT~A,~A~eE\ 'I (,˘-4 O calved ) THAT5 FRACTiONS!! YoU KNOtt' i LL NEVER
N - ~ i. r qou'RE TRYI~6 TO STAND Froths!
_4 W: :0~51 'at
e 1977 United Features Syndicare inc.
PEANUTS repan ad by permission of Uni ad Feature Syndicate, Inc
Poor Sally. Her anger and frustration with fractions are palpable. And
they no doubt reflect the feelings and experiences of many students. As
mathematics education researchers and teachers can attest, students are of-
ten vocal in their expression of dislike of fractions and other representations
of rational numbers (percents and decimals). In fact, the rational-number
system poses problems not only for youngsters, but for many adults as well.'
In a recent study, masters students enrolled in an elementary teacher-train-
ing program were interviewed to determine their knowledge and under-
standing of basic rational-number concepts. While some students were con-
fident and produced correa answers and explanations, the majority had
difficulty with the topic. On attempting to perform an operation involving
fractions, one student, whose sentiments were echoed by many, remarked,
"Oh fractions! I know there are lots of rules but I can't remember any of
them and I ne ver understood them to start with."Z
OCR for page 122
310 H OW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM
We know from extensive research that many people—adults, students,
even teachers—find the rational-number system to be very difficult.3 Intro-
duced in early elementary school, this number system requires that students
reformulate their concept of number in a major way. They must go beyond
whole-number ideas, in which a number expresses a fixed quantity, to un-
derstand numbers that are expressed in relationship to other numbers. These
new proportional relationships are grounded in multiplicative reasoning that
is quite different from the additive reasoning that characterizes whole num-
bers (see Box 7-1).4 While some students make the transition smoothly, the
majonty, like Sally, become frustrated and disenchanted with mathematics.5
Why is this transition so problematic?
A cursory look at some typical student misunderstandings illuminates
the kinds of problems students have with rational numbers. The culprit ap-
pears to be the continued use of whole-number reasoning in situations where
it does not apply. When asked which number is larger, 0.059 or 0.2, a major-
ity of middle school students assert that 0.059 is bigger, arguing that the
number 59 is bigger than the number 2 6 Similarly, faulty whole-number
reasoning causes students to maintain, for example, that the fraction 1/8 is
larger than 1/6 because, as they say, 8 is a bigger number than 6. 7 Not
surprisingly, students struggle with calculations as well. When asked to find
the sum of 1/2 and 1/3, the majority of fourth and sixth graders give the
answer 2/5 Even after a number of years working with fractions, some
eighth graders make the same error, illustrating that they still mistakenly
count the numerator and denominator as separate numbers to find a sum 8
Clearly whole-number reasoning is very resilient.
t)ecimal operations are also challenging.9 In a recent survey, research-
ers found that 68 percent of sixth graders and 51 percent of fifth and seventh
graders asserted that the answer to the addition problem 4 + .3 was .7.'° This
example also illustrates that students often treat decimal numbers as whole
numbers and, as in this case, do not recognize that the sum they propose as
a solution to the problem is smaller than one of the addends.
The introduction of rational numbers constitutes a major stumbling block
in children s mathematical development." It marks the time when many
students face the new and disheartening realization that they no longer un-
derstand what is going on in their mathematics classes.' This failure is a
cause for concern Rational-number concepts underpin many topics in ad-
vanced mathematics and car y significant academic consequences.'3 8tu-
dents cannot succeed in algebra if they do not understand rational numbers.
But rational numbers also pervade our daily lives.'4 We need to be able to
understand them to follow recipes, calculate discounts and miles per gallon,
exchange money, assess the most economical size of products, read maps,
interpret scale drawings, prepare budgets, invest our savings, read financial
statements, and examine campaign promises. Thus we need to be able to
OCR for page 123
NEW AFFFOACHES TO TEACH NO THE RAT ONAE NUMBEF SYSTEM 311
BOX7-] Additive and Multiplicative Reasoning
Lamon,'5 whose work on proportional reasoning and rational number has made a
great contribution to our understanding of students' learning, elucidates the dis-
tinction between relative and absolute reasoning. She asks the learner to con-
sider the growth of two fictitious snakes: String Bean, who is 4 feet long when
the story begins, and Slim, who is 5 feet long. She tells us that after 5 years, both
snakes have grown. String Bean has grown from 4 to 7 feet, and Slim has grown
from 5 to 8 feet (see the figure below). She asks us to compare the growth of
these two snakes and to answer the question, "Who grew more7"
String riean (4 [eetl
5 years from blow...
~ S
%00
Stan 15 [eel}
String Bean (7 feet)
Slim (R feetl
Lao on suggests that there are two answers. First, if we consider absolute
growth, both snakes grew 3 feet, so both grew the same amount. The second
answer deals with relative growth: from this perspective, String Bean grew the
most because he grew 3/4 of his length, while Slim grew only 3/5 of his length. If
we compare the two fractions, 3/4 is greater than 3/5, and so we conclude that
String Bean has grown proportionally more than Slim.
Lamon asks us to note that while the first answer, about the absolute differ-
ence, involves addition, the second answer, about the relative difference, is solved
through multiplication. In this way she shows that absolute thinking is additive,
while relative thinking is multiplicative.
OCR for page 124
312 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM
understand rational numbers not only for academic success, but also in our
lives as family members, workers, and citizens.
Do The principles of learning highlighted in tills book help illuminate
The widespread problems observed as students grapple widh rational num-
ber? Can they point to more effective approaches to teaching rational num-
ber? We believe The answer to bodh These questions is "yes." In The first
section below we consider each of The dlree principles of How Students
Learn, beginning with principle 2—the organization of a knowledge net-
work That emphasizes core concepts, procedural knowledge, and their con-
nections. We then turn to principle 1—engaging student preconceptions and
building on existing understandings. Finally we consider metacognitive in-
struction as emphasized in principle 3.
The second section focuses on instruction in rational number It begins
with a description of frequendy used instructional approaches and The ways
in which they diverge Tom The above dlree pnnciples. We then describe an
experimental approach to teaching rational number That has proven to be
successful in helping students in fourth, ffdh, and srxdh grades understand
The interconnections of The number system and become adept at moving
among and operating with The various representations of rational number
Through a description of lessons in which The students engaged and proto-
cols taken from The research classrooms, we set out the salient features of
the instructional approach that played a role in shaping a learning-centered
classroom environment. We illustrate how in This environment, a focus on
The interconnections among decimals, fractions, and percents fosters stu-
dents' ability to make informed decisions on how to operate effectively widh
rational numbers. We also provide emerging evidence of The effectiveness of
The instructional approach. The intent is not to promote our particular cur-
nculum, but radher to illustrate The ways in which it incorporates The prin-
ciples of How People Learn, and The observed changes in student under-
standing and competence widh rational numbers that result.
RATIONAL-NUMBER LEARNING AND THE
PRINCIPLES OF HOW PEOPLE LEARN
The Knowledge Network: New Concepts of Numbers and
New Applications (Principle 2)
What are the core ideas that define The domain of rational numbers?
What are The new understandings that students will have to construct? How
does a beginning student come to understand rational numbers?
Let us look through The eyes of a young student who is just beginning to
learn about rational number Until tills point, all of her formal instruction in
OCR for page 125
NEW AFFFOACHES TO TEACH NG THE RAT ONAE NUMBEF SYSTEM 313
arithmetic has centered on learning the whole-number system. If her learn-
ing has gone well, she can solve andhmetic problems competendy and easily
makes connections between The madhematics she is learning and expen-
ences of her daily life. But in This next phase of her learning, The introduc-
tion of rational number, There will be many new and intertwined concepts,
new facts, new symbols that she will have to learn and understand a new
knowledge network, if you will. Because much of tills new learning is based
on multiplicative instead of whole-number relations, acquiring an under-
standing of dais new knowledge network may be challenging, despite her
success Thus far in mathematics. As with whole-number arithmetic, This do-
main connects to everyday life. But unlike whole numbers, in which the
operations for The most part appear straightforward, the operations involved
in The learning of rational numbers may appear to be less intuitive, at odds
widh earlier understandings (e.g., dhat multiplication always makes Things
bigger), and hence more diff cult to team,
New Symbols, New Meanings, New Representations
One of The first challenges facing our young student is that a particular
rational number can take many forms. Until now her experience widh sym-
bols and Their referents has been much simpler. A number—for example,
four—is represented exclusively by one numeral, 4. Now The student will
need to learn that a rational number can be expressed in different ways as
a decimal, fraction, and percent. To further complicate matters, she will have
to learn that a rational-number quantity can be represented by an infinite
number of equivalent common and decimal fractions. Thus a rational num-
ber such as one-fourth can be written as 1/4, 2/8, 3/12, 4 16, 0.25, 0.250, and
50 011.
Not only does The learning of rational number entail the mastery of
These fomms and of The new symbol systems that are implied, but The learner
is also required to move among These various forms flexibly and eff~ciendy.'6
Unfortunately, This flow between representations does not come easily.'7 In
fact, even mature students are often challenged when They try to understand
The relations among the representations.' To illustrate how diff alit translat-
ing between fractions and decimals can be, I offer two examples taken from
our research,
In a recent series of studies, we interviewed fourth, sixth, and eighth
graders on a number of items that probed for rational-number understand-
ing. One of the questions we asked was how The students would express the
quantity 1/8 as a decimal. This question proved to be very challenging for
many, and aldhough The students' ability increased with age and expenence,
more Than half of The sixth and eighdh graders we surveyed asserted dhat as
a decimal, 1/8 would be 0.8 (radher than The correct answer, 0.125).
OCR for page 126
314 H OW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM
In the next example, an excerpt taken from an interview conducted as
part of a pretest, Wyatt, a traditionally trained ffth-grade student, discussed
ordering a ser es of rational numbers presented to him in mixed representa-
tior s.
Interviewer Here are 3 numbers: 2/3, 0.5, and 3/4. Could
you please pat these numbers in orderfrom
smallestto largest7
Well, to start with, I think that the decimal 0.5
is bigger than the fractions because it's a
decimal, so it's just bigger, because fractions
are really small things.
Wyatt
The response that 1/8 would equal 0.8 should be familiar to many who
have taught decimals and fractions. As research points out, students have a
difficult time understanding the quantities involved in rational number and
thus do not appear to realize the unreasonableness of their assertion.'9 As
for Wyatt's assertion in the excerpt above that decimals and fractions cannot
be compared, this answer is representative of the reasoning of the majority
of the students in this class before instruction. Moreover, it reflects more
general research findings 20 Since most traditional instruction in rational num-
ber presents decimals, fractions, and percents separately and often as dis-
tinct topics, it is not surprising that students f rid this task confusing. Indeed,
the notion that a single quantity can have many representations is a major
departure from students' previous experience with whole numbers; it is a
difficult set of understandings for them to acquire and problem-laden for
many.:'
But this is not the only divergence from the familiar one-to-one corre-
spondence of symbol to referent that our new learner will encounter An-
other new and difficult idea that challenges the relatively simple referent-to-
symbol relation is that in the domain of rational number, a single rational
number can have several conceptually distinct meanings, referred to
as "subconstructs." Now our young student may well become completely
confused.
The Subconstructs or the Many Personalities of Rational Number
What is meant by conceptually distinct meanings? As an illustration,
consider the simple fraction 3/4. One meaning of this fraction is as a part-
whole relation in which 3/4 describes 3 of 4 equal-size shares. A second
interpretation of the fraction 3/4 is one that is referred to as the quotient
interpretation. Here the fraction implies division, as in 4 children sharing 3
OCR for page 127
NEW AFFFOACHES TO TEACH NG THE RAT ONAE NUMBEF SYSTEM 315
pies, As a ratio, 3/4 might mean There are, for example, 3 red cars for every
4 green cars (dhis is not to be confused widh The part-whole interpretation
that 3/7 of The cars are red). Rational numbers can also indicate a measure
Here rational number is a fixed quantity, most frequendy accompanied by a
number line, dhat identifies a situation in which the fraction 1/4 is used
repeatedly to determine a distance (e.g., 3/4 of an inch = 1/4, 1/4, 1/4).
Finally, the e is The interpretation of rational number as a mu/fip/icafive
operator behaving as an operation dhat reduces or enlarges the size of an-
odher quantity (e.g., the page has been reduced to 3/4 its original size).
The necessity of coordinating These different interpretations requires a
deep understanding of the concepts and interrelationships among them. On
the one hand, a student must think of rational numbers as a division of two
whole numbers (quotient interpretation); on the other, she must also come
to know these two numbers as an entity, a single quantity (measure), often
to be used in another operation. These different interpretations, generally
referred to as The "subconstructs" of rational number, have been analyzed
extensively and are a very important part of The knowledge network dhat
the learner will construct for rational number,
Reconceptualizing the Unit and Operations
While acquiring a knowledge network for rational-number understand-
ing means dhat new forms of representation must be learned (e.g., decimals,
fractions) and different interpretations coordinated, The learner will encoun-
ter many other new ideas ideas dhat also depart from whole numbers. She
will have to come to understand dhat rational numbers are "dense"—mean-
ing dhat between any two rationals we can fmd an infinity of o h e- numbers.
In The whole-number domain, number is discrete radher than continuous,
and The main operation is counting. This is a very big change indeed.33
Another difficult new set of understandings concerns The fundamental
change that students will encounter in The nature of The unit. In whole num-
bers, The unit is always explicit (6 refers to 6 units). In rational numbers, on
The o he- hand, The unit is often implied. But it is the unstated unit that gives
meaning to the represented quantities, operations, and the solutions. Con-
sider The student trying to interpret what is meant by The task of multiplying,
for example, 1/2 times 1/8. If The student recognizes dhat The "1/8" in The
problem refers to 1/8 of one whole, she may reason COI I ectlv that half of The
quantity 1/8 is 1/16. However since the 1 is not stated but implied, our
young student may err and, Thinking The unit is 8, consider The answer to be
1/4 (since 4 is one-haU of 8)—a response given by 75 percent of traditionally
instructed fourth and sixth graders students in our research projects.
OCR for page 128
316 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM
New Conceptualizations: Understanding NumbersAs
Multiplicative Relations
Clearly the transition to learning rational numbers is challenging. Funda-
mentally, students must construct new meanings for numbers and opera-
tions Development of the network of understandings for rational numbers
requires a core conceptual shift: numbers must be understood in multiplica-
tive relationship,
As a final illustration, I offer one more example of this basic shift. Again,
consider the quantity 3/4 from our new learner's perspective. All of our
student's prior learning will lead her to conclude that the 3 and 4 in 3/4 are
two separate numbers that define separate quantities. Her knowledge of
whole numbers will provide an additive understanding. Thus she will know
that 3 and 4 are contiguous on the number line and have a difference of 1.
But to interpret 3/4 as a rational number instead of considering these two
numbers to be independent, as many students mistakenly continue to do,24
our student must come to understand this fraction as a new kind of quantity
that is defined multiplicatively by the relative amount conveyed by the sym-
bols. Suddenly numbers are no longer simple. When placed in the context
of a fraction, 3 and 4 become a quantity between 0 and 1. Obvious to adults,
this numerical metamorphosis can be confusing to children.
How can children learn to make the transition to the complex world of
rational numbers in which the numbers 3 and 4 exist in a relationship and
are less than 1? Clearly, instruction will need to support a major conceptual
change. Looking at students' prior conceptions and relevant understandings
can provide footholds to support that conceptual change.25
Students' Errors and Misconceptions Based on
Previous Learning (Principle 1)
As the above examples suggest, students come to the classroom with
conceptions of numbers grounded in their whole-number learning that lead
them astray in the world of rational numbers. If instruction is to change
those conceptions, it is important to understand thoroughly how students
reason as they puzzle through rational-number problems. Below I present
verbatim interviews that are representative of faulty understandings held by
many students.
In the following excerpt, we return to our fifth grader, Wyatt. His task
was to order a series of rational numbers in mixed representations Recall
his earlier comments that these representations could not be compared.
Now as the interview continues, he is trying to compare the fractions 2/3
and 3/4. The interview proceeds:
OCR for page 129
NEW AFFFOACHES TO TEACH NG THE RAT ONAE NUMBEF SYSTEM 317
Wyatt
Wyatt
Interviewer What about 2/3 and 3/47 Which of those is
bigger7
Well, I guess that they are both the same size
because they both have one piece missing.
Interviewer I am not sure I understand what you mean
when you say that there is one piece missing.
I'll show you. I Wyatt draws two u neven ci ~~ es,
roughly pa rtitions the fi rst in fou r pa rts, and
then proceeds to shade three parts. Next he
divides the second circle into three parts and
shades two of them (see Figu re 7-1). O.K., here
is 3/4 and 2/3. You see they both have one part
missing. I He points to the unshaded sections
in both circular regions.l You see one part is
left out, so they are both the same.
FIGURE 7 1
Wyatt's response is typical in asserting that 2/3 and 3/4 must be the
same size. Clearly he has not grasped the multiplicative relations involved in
rational numbers, but makes his comparisons based on operations from his
whole-number knowledge. When he asserts that 2/3 and 3/4 are the same
size because there is "one piece missing," Wyatt is considering the differ-
ence of 1 in additive terms rather than considering the multiplicative rela-
tions that underlie these numbers.
Additive reasoning is also at the basis of students' incorrect answers on
many other kinds of rational-number tasks. Mark, a sixth grader, is working
on a scaling problem in which he is attempting to figure out how the length
and width of an enlarged rectangle are related to the measurements of a
smaller, original rectangle. His challenge is to come up with a proportional
relation and, in effect, solve a "missing-term problem" with the following
relations: 8 is to 6 as 12 is to what number?
Interviewer I have two pictures of rectangles here (see
Figu re 7-2). They a re exactly the same shape,
but one of them is bigger than the other I
OCR for page 130
318 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM
Ma rk
FIGURE 72
made this second one bigger by taking a
picture of the first one and then enlarging it
just a bit. As you can see, the length of the first
rectangle is 8 cm and the width is 6 cm.
Unfortunately, we know only the length of the
second one. That is 12 cm. Can you please tell
me what you think the width is7
Well, if the first one (rectangle) is 8 cm and 6
cm, then the next one is 12 cm and 10 cm.
Because in the 8 and 6 one (rectangle) you
subtract 2 from the 8 (to get the difference of
the width and the length). So in the bigger
rectangle you have to subtract 2 from the 12.
So that's 10. So the width of the big rectangle
is 10.
I 12cm 1
Mark's error in choosing IO instead of the correa answer of 9 is cer-
tainly representative of students in his age group—in fact, many adults use
the same kind of faulty reasoning. Mark clearly attempts to assess the
relations, but he uses an additive strategy to come up with a difference of 2.
To answer this problem correctly, Mark must consider the multiplicative
relations involved (the rectangle was enlarged so that the proportional rela-
tionship between the dimensions remains constant challenge that eludes
many.
It is this multiplicative perspective that is difficult for students to adopt
in working with rational numbers. The misconception that Mark, the sixth
grader, displays in asserting that the height of the newly sized rectangle is IO
cm instead of the correct answer of 9 cm shows this failure clearly. Wyatt
OCR for page 131
NEW AFFFOACHES TO TEACH NG THE RAT ONAE NUMBEF SYSTEM 319
certainly was not able to look at the relative amount in trying to distinguish
between The quantities 2/3 and 3/4 Rather, he reasoned in absolute terms
about the circles, that ". . . bodh have one piece missing.''
Metacognition and Rational Number (Principle 3)
A metacognitive approach to instruction helps students monitor their
understanding and take control of Their own learning.27 The complexity of
rational number—The different meanings and representations, the challenges
of comparing quantities across The very different representations, The un-
stated unit—all mean that students must be actively engaged in sense mak-
ing to solve problems competendy.r We know, however, that most middle
school children do not create appropriate meanings for fractions, decimals,
and percents; radher, they rely on memorized rules for symbol manipulation,
The student errors cited at the beginning of tills chapter indicate not
only The students' lack of understanding of rational number, but also their
failure to monitor Their operations and judge the reasonableness of Their
responses.29 If classroom teaching does not support students in developing
metacognitive skills for example, by encouraging Them to explain their
reasoning to Their classmates and to compare interpretations, strategies, and
solutions The consequences can be senous. Student can stop expecting
madh to make sense. Indeed for many students, rational number marks The
point at which they draw this conclusion.
INSTRUCTION IN RATIONS NAMER
Why does instruction so often fail to change students' whole-number
conceptions? Analyses of commonly used textbooks suggest That The prin-
ciples of How People Learn are routinely violated. First, it has been noted
That—in contrast to units on whole-number learning topics in rational num-
ber are typically covered quickly and superficially. Yet the major conceptual
shift required will take time for students to master Thoroughly. Within The
allotted time, too little is devoted to teaching The conceptual meaning of
rational number, while procedures for manipulating rational numbers re-
ceive greater emphasis.30 While procedural competence is certainly impor-
tant, it must be anchored by conceptual understanding. For a great many
students, it is not.
Odher aspects of The knowledge network are shortchanged as well, in-
cluding The presentation and teaching of The notation system for decimals,
fractions, and percents. Textbooks typically treat The notation system as some-
dling Flat is obvious and transparent and can simply be given by definition
at a lesson's outset. Further, operations tend to be taught in isolation and
OCR for page 152
340 HO
W STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM
Then I did 10 percent of 160, which is 16. Then
I did 5 percent, which was 8.1 added them 116
+ 81 to get 24, and added that to 80 to get 104.
For anyone who has seen a colleague pause when asked to compute a
percentage, as one must, say, to calculate a tip, the ease with which these
students worked through these problems is striking.
Knowledge Network
These are only a few examples from the posttest interviews that illus-
trate the kinds of new understandings and interconnections students had
been able to develop through their participation in the curnculum. Overall,
our analyses of the children's thinking revealed that students had gained (1)
an overall understanding of the number system, as illustrated by their ability
to use the representations of decimals, fractions, and percents interchange-
ably; (2) an appreciation of the magnitude of rational numbers, as seen in
their ability to compare and order numbers within this system; (3) an ulldel
standing of the proportional- and ratio-based constructs of this domain, which
underpins their facility with equivalencies; (4) an understanding of percent
as an operator, as is evident in their ability to invent a variety of solution
strategies for calculating with these numbers; and (5) general confidence
and fluency in their ability to think about the domain using the benchmark
values they had teamed, which is a hallmark of number sense.
Our research is still in an early stage. We will continue to pursue many
questions, including the potential limitations of successive halving as a way
of operating with rational numbers, downplaying of the important under-
standings associated with the quotient subconstruct, as well as a limited
view of fractions. Furthermore, we need to team more about how students
who have been introduced to rational numbers in this way will proceed with
their ongoing learning of mathematics.
While we acknowledge that these questions have not yet been answered,
we believe certain elements of our program contributed to the students'
teaming, elements that may have implications for other rational-number
curncula. First, our program began with percents, thus pemmitting children
to take advantage of their qualitative understanding of proportions and com-
bine that understanding with their knowledge of the numbers from 1 to 100,
while avoiding (or at least postponing) the problems presented by fractions.
Second, we used linear measurement as a way of promoting the multiplica-
tive ideas of relative quantities and fullness. Finally, our program empha-
sized benchmark values of halves, quarters, eighths, etc.—for moving among
equivalencies of percents, decimals, and fractions, which allowed students
to be flexible and develop confidence in relying on their own procedures
for problem solving.
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NEW AFFFOACHES TO TEACH NG THE RAT ONAE NUMBEF SYSTEM 341
CONCLUSION: HOW STUI)ENTS LEARN
RATIONAL NUMBER
Principle #1: Prior Understandings
For years madhematics researchers have focused their attention on un-
derstanding dhe complexities of His number system and how to facilitate
students' learning of the system. One well-established insight is dhat rational-
number teaching focused on pie charts and part-whole understandings rein-
forces dhe primary problem students confront in learning rational number:
dhe dominance of whole-number reasoning. One response is to place the
multiplicative ideas of relative quantity, ratio, and proportion at the center of
instruction.
However, our curriculum also builds on our Theory and research find-
ings pointing to The knowledge students typically bring to the study of ratio-
nal number dhat can serve as a foundation for conceptual change. Two
separate kinds of understandings dhat 10-year-olds typically possess have a
multiplicative orientation. One of These is visual proportional estimation; for
children, This understanding usually functions independency of numbers, at
least initially. The second important kind of understanding is the numerical
procedure for repeated halving By strengthening and merging These two
understandings, students can build a solid foundation for working flexibly
widh rational numbers.
Our initial instructional activities are designed to elicit These informal
understandings and to provide instructional contexts that bring Them to-
gether We believe this coordination produces a new interlinked structure
that serves bodh as foundation for The initial learning of rational number and
subsequently as The basis on which to build a networked understanding of
dais domain.
Principle #2: Network of Concepts
At He beginning of tills chapter, I outlined The complex set of core
concepts, representations, and operations students need to acquire to gain
an initial grounding in the rational-number system As indicated above, The
central conceptual challenge for students is to master proportion, a concept
grounded in multiplicative reasoning. Our instructional strategy was to de-
sign a learning sequence that allowed students to fret work with percents
and proportion in linear measurement and next work with decimals and
fractions Extensive practice is incorporated to assure that students become
fluent in translating between different fomms of rational number. Our inten-
tion was to create a percent measurement structure that would become a
central network to which all subsequent mathematical learning could be
OCR for page 154
342 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM
linked. This design is sign ficandy different from traditional instruction in
rational number, in which topics are taught separately.
Principle #3: Metacognition
In dais chapter, I have not made detailed reference to students' develop-
ing metacognition. Yet the fostering of metacognition is in fact central to our
curnculum. First, as The reader may have noted, we regularly engaged the
students in whole-group discussions in which they were asked to explain
their reasoning and share invented procedures with Their classmates. We
also designed The lessons so That students worked in small groups to col-
laborate in solving problems and constructing matenals; we Thereby pro-
vided students widh a forum to express and refine their developing under-
standings. There were also many opportunities for students to consider how
they would teach rational number to odhers, eidher younger students or their
own classmates, by designing Their own games and producing teaching plans
for how These new concepts could be taught. In all These ways, we allowed
students to redect on their own learning and to consider what it meant for
Them and odhers to develop an understanding of rational number Finally,
we fostered metacognition in our program Through the overall design and
goals of The experimental curnculum, with its focus on interconnections and
multiple representations. This focus, I believe, provided students with an
overview of The number system as a whole and Thus allowed Them to make
informed decisions on how best to operate widh rational numbers.
Final Words
I conclude dais chapter with an interchange, recorded verbatim, be-
tween a fourth-grade student and a researcher Zach, The fourth grader, was
being interviewed by The researcher as part of a posttest assessment. The
conversation began when Zach had completed two pages of the srx-page
posttest and remarked to The interviewer, "I have just done 1/3 of the test;...dhat
is 33.3 percent." When he finished the Third page, he noted, "Now I have
finished 1/2 or 50 percent of The test." On completing The fourth page he
remarked, "Okay, so l have now done 2/3 of The test, which is The same as
66 percent." When he had completed The penultimate page, he wondered
out loud what The equivalent percentage was for 5/6: "Okay, let's see; it has
got to be over 66.6 percent and it is also more than 75 percent. I'd say That
it is about 80 percent....No, wait; it can't be 80 percent because That is 4/5
and This [5/6] is more Than 4/5. It is 1/2 plus 1/3.. .so it is 50 percent plus 33.3
percent, 83.3 percent so I am 83.3 percent finished."
This exchange illustrates The kind of metacognitive capability that our
cl ~~icLIL~T~ is intended to develop. First, Zach posed his own questions,
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NEW AFFFOACHES TO TEACH NG THE RAT ONAE NUMBEF SYSTEM 343
unprompted. Further, he did not expect that The question had to be an-
swered by the teacher Radher, he was confident That he had The tools, ideas,
and concepts that would help him navigate his way to The answer We also
see Tat Bach tigot ously assessed The reasonableness of his answers and That
he used his knowledge of translating among The various representations to
help him solve The problem. I conclude with this charming vignette as an
illustration of The potential support our curriculum appears to offer to stu-
dents beginning their learning of rational number
Students Then go on to learn algoridlms Tat allow Them to calculate a
number like 83.3 percent from 5/6 efficiently. But The foundation in madh-
ematical reasoning That students like Bach possess allow Them to use Those
algondhms with understanding to solve problems when an algoridlm has
been forgotten and to double check their answers using multiple methods.
The confidence created when a student's mathematical reasoning is secure
bodes well for future mathematics teaming,
NOTES
I ArrcLstrong and Bezuk, 1995; sall' 1990; Post et al., 1991.
2. Moss, 2000.
3 carpenter et al 1980
4. Ball, 1993; Hieuert and sehr, 1988; Kieren, 1993
5. Lamon, 1999.
6. Hieuert and Weame, 1986; Wearne and Hieuert, 1988.
7. Hieuert and sehr' 1988.
8. Ketslake, 1986.
9. Heroert, 1992.
10. National Research Council, 2001
I I carpenter et al 1993
12. Lamon, 1999.
13. Lesh et a ., 1988.
14. Be oody, 1999.
15. i amon, 1999.
16. National Council of Teachers of Mathematics, 1989, 2000.
17. Markovits and Powder, 1991, 1994; Powder, 1995.
18. Gamer et al., 1989.
19. Powder, 1995.
20. Powder, 1992.
21 Hieuert and Beer, 1988.
22. sehr et a ., 1983, 1984, 1992, 1993; Kieren, 1994, 1995; oh sson, 1988.
23. Hieuert and sehr, 1988.
24. Kerslake, 1986.
25. sehr et al., 1984; Case, 1998; Hieuert and sehr' 1988; Lamon, 1995; Mack,
1990, 1993, 1995; Hesnick and Singer, 1993.
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344 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM
26. Hart, 1988; Karplus and Peterson, 1970; Karplus et al., 1981, 1983; Gamer et
al., 1993; Noelting, 1980a, 1980b.
27. National Counci of Teachers of Mathematics, 1989, 2000; National Research
Counci, 2001.
28. Ba, 1993.
29. Sowder, 1988.
30. Ba oody, 1999; Heibert, 1992; Hiebert and Wea ne, 1986; Moss and Case, 1999;
Post et a ., 1993.
31. Armst ong and Bezuk, 1995; Ba 1, 1993; Hiebert and Wea ne, 1986; Mack, 1990,
1993; Markovits and Sowder, 1991, 1994; Sowder, 1995.
32. Conf˘y, 1994, 1995; Kieren, 1994, 1995; Post et al., 1993; 5treef and, 1991,
1993.
33. Kieren, 1994, 1995; Mack, 1993, 1995; Sowder, 1995; Streef and, 1993.
34. Kieren, 1994, p. 389.
35. Iiieren, 1992, 1995.
36. Confrey, 1995.
37. Lachamce and Confrey, 1995.
38. Streef and, 1991, 1993.
39. Mack, 1990, 1993.
40. Lamon, 1993, 1994, 1999.
41. As of this writing, this curriculum is being implemented with students of low
socioeconomic status in a grade 7 and 8 c ass P˘lim nary analyses have shown
that it is highly effective in helping struggling students relearn this number
system and gain a st onger conceptual understanding.
42. Kalchman et al., 2000; Moss, 1997, 2000, 2001, 2003; Moss and Case, 1999.
43. National Research Council, 2001
44. Parker and Leinhardt, 1995.
45. Case, 1985; Noelting, 1980a; Nunes and Bryant, 1996; Spinilo and Bryant,
1991.
46. Confrey, 1994; Kie˘n, 1994.
47. Resnick and Singer, 1993.
48. Case, 1985.
49. Conf˘y, 1994; Kie˘n, 1993.
50. Case and Okomoto, 1996.
51. Case, 1998; Kalchman et al., 2000.
52. Parker and Leinha dt, 1995.
53. LembLe and Reys, 1994.
54. Whi e the activities and lessons we designed am organized in three phases, the
actual order of the lessons and the pacing of the teaching, as well as the
particular content of the activities described below, vaned in different class-
tooms depending on the needs, capabi ities, and interests of the participatmg
students.
55. These materials are available at any building supply store
56. Parker and Leinha dt, 1995.
57. Hiebert et al., 1991.
58. Resnick et al., 1989.
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NEW AFFFOACHES TO TEACH NG THE RAT ONAE NUMBEF SYSTEM 345
59. Kalchman et al., 2000; Moss, 1997, 2000, 2001; Moss and Case, 1999.
60. From p e- to posttest, achieving effect sizes between and I and 2 standard
deviations.
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Representative terms from entire chapter:
onae numbef
