mw-bestptalk.tex November 29, 2006

The Teaching of Fractions and its Discontents

Frederick Greenleaf, Professor of Mathematics, NYU/Courant Institute

November 28, 2006

Despite my background as a research mathematician, I have actually had quite a bit

of experience writing mathematics at or near the middle school level, and a substantial

amount of that e
ort was concerned with the teaching of fractions in the higher elementary

grades. It is an issue I take seriously because I am concerned with what it takes

in K-12 math education to produce students who are \math-ready" for the demands of

college level work. At some point, more than 50% of high school graduates will enter a

college level program, many of which have Calculus-level math requirements. No one can

tell in advance who will or will not seek such career paths, so college-readiness should be

the ultimate goal of any sound K-8 math program.

The topic of fractions is a major stumbling block for many students, otherwise we

would not be here tonight. Nevertheless, it is crucial that students become adept at

handling fractions { and by that I mean fractions as

into a calculator { because fractions and their arithmetical properties provide the bedrock

of intuition upon which all later concepts of algebra and Calculus rest.

What I will say is partly inspired by my reading of portions of a forthcoming book

directed toward the concerns of K-12 teachers by Prof. Hung-Hsi Wu, of Berkeley,

fractions, and not numbers punched1

the recent book

Baldridge, and from my own experience working with Stanley Ocken of CCNY creating

a mathematically coherent account of how fractions work (unpublished), used to train

tutors providing remedial math assistance to 9th graders at the University Neighborhood

High School in New York City.

Let me begin with what I see as the preconditions for any program that aspires to

the successful teaching of fractions. Although I may not have time to pass on to the next

topic { speci c pedagogical issues I see in the teaching of fractions { I have nevertheless

included a few thoughts on these at the end of the the written article I have prepared

for this convocation. I may have an opportunity to mention them in the open discussion

sessions.

Elementary Mathematics for Teachers by Thomas Parker and Scott

The Suppport Base for Teaching Fractions in Grades 5-8

In my opinion the following issues must be taken into account if we hope to resolve the

diculties commonly encountered in teaching fractions.

1. Teaching to mastery.

for example { owing to its intrinsically \vertical" structure. Its concepts build upon each

other, layer-by-layer, as we progress from counting to fractions, to algebra, to calculus,

and beyond. As you go higher you come to realize more and more ways in which the

external world is mirrored in mathematics. But, each level must be

can con dently use its concepts as they reach for the next level of sophistication. The

need for mastery before proceding to the next level is not given the respect it demands

in quite a few K-8 math curricula I have reviewed.

The need for step-by-step mastery is evident in microcosm in the teaching of fractions

in the K-8 grades, where the natural progression of concepts is pretty clear to any

Math is di
erent from many other subjects { the sciences,mastered so students

1

I refer to the chapter concerned with fractions which can be found on Prof. Wu's website:

http://math.berkeley.edu/

Appendix pp.117-123 of that article, which highlights some of the major practical pedagogical issues in

the teaching of fractions.

wu/EMI2a.pdf I especially recommend that you take a look at the brief

1

mathematician.

2. The need for logical progression and clear de nitions.

that \You cannot teach what you do not understand." In discussing fractions we are concerned

with its corollary: \You cannot really understand what has never been de ned."

I will have more to say about this, but for the moment let me say that mathematics is

founded on clear

There

a point on the number line whose position can easily be found from the symbol

n

represents it. By grade 5 most children understand the concept of number line { it is

visual and intuitive. The trouble begins when one fails to make a clear the distinction

between a fraction (a number) and the symbol that represents it. Students soon discover

that the same number (length of a line segment) can be represented by many symbols,

as with

2

12

2

on. It is not so confusing if you think of it this way: the family pet is often referred to

by various whimsical names, but whatever the name-of-the day, it is still the same pet.

Even so with fractions and their representations.

It has often been saidde nitions, and the logical relations between them.is a simple and easily understood de nition of a fraction as a (rational) number:mthat3= 48= 11, etc, and get confused if there is not one basic de nition to fall back

3. Sometimes less actually

instructional materials. I have in my oce a cubic foot of student booklets for Grade 6

in a well-know math program; the Singapore Math materials for Grade 6 consist of two

slim textbooks and two practice Workbooks [hold them up to view]; the entire set costs

about $45 online at

Part of the trouble stems from ill-considered State standards, which desperately need

pruning in the face of \topic ination," with every Committee member shoehorning in

his or her pet topic, whose coverage then becomes mandatory statewide. There is no

need for this, and it is counterproductive. The most successful countries in the world

{ Singapore, China, Japan, etc { have lean and clear-cut curricula which allow them to

make sure that students achieve real mastery of the topics that

is more. This seems to be the era of bloated sets ofsingaporemath.com.are covered.

4. The need for clear, concise, mathematically sound textbooks.

students, and even parents need an actual

that students can study at home to reinforce what they have learned in class. Without a

coherent textbook, parents cannot help their children learn and are disenfranchised from

the educational process. Without a textbook students cannot refer back to previous

topics relevant to the tasks at hand. On both counts the absence of a coherent textbook

seems to me indefensible.

A good text should strike a balance between explanation of concepts, worked examples,

exploratory projects illuminating the meaning of mathematical concepts, and plenty

of practice problems (perhaps in a separate workbook); contrary to popular belief, it is

not mandatory that practice problems be dull and boring { see the Singapore texts and

Workbooks for example. The text should also adhere to the principles set forth in

Teachers,textbook, with supporting practice workbooks,3.

The following quote from Wu is relevant here:

A mathematician approaching the subject of fractions in school cannot help

but be struck by the total absence of the characteristic features of mathematics:

precise de nitions as a starting point, logical progression from topic to topic,

and most importantly, explanations that accompany each step. This is not to

say that teaching of fractions ... should be rigidly formal from the beginning.

Fractions should be informally introduced as early as second grade

even second graders need to worry about drinking \half a glass" of orange

juice

(because):

5. Teachers must know their subject.

needs to know fractions and their applications cold. A good check (borrowed from Herb

2

Clemens via Prof. Wu): can the instructor explain why

7

To teach fractions e
ectively, a teacher2



9

7

sympathize with the panic response that often ensues { fractions is a subtle and many

faceted topic, and it often takes a lot of experience to gain command of it. I applaud

e
orts to enhance the math content training of K-8 teachers who face these challenges.

But until the millennium arrives, perhaps the time has come for us in the U.S. to consider

the turning math teaching in the upper middle grades to cadres of well-trained and

experiences

in China. This seems to be standard practice in K-8 math (and science) teaching in China,

Japan, and other highly successful countries.

1= 24? I can fullymath specialists, of the sort described in Li-Ping Ma's book on her experiences

Some Pedagogical Issues in the Teaching of Fractions

I list items as they arise in the natural sequence of fraction concepts.

1. The very de nition of \fraction" is a source of diculties.

lists 5 di
erent commonly used \explanations" of what a fraction \is." Is it any wonder

kids get confused by the concept? In my view there is just one basic de nition: a fraction

In his book Wu

m

n

you are given the symbol

n

is a rational number, a point on the number line whose location is easily found oncemusing a de nite algorithm: divide the unit interval into n

equal pieces, then chain together

m of them.

2. A fraction vs the symbols

n

fraction (a number) and the symbols used to represent it is not always made. The symbol

is

locate the number. But the encoding process is a bit redundant, with the result that a

single number can have many di
erent

3

18

3

The foundational concept of \equivalent fractions" (which should perhaps be referred to

as \equivalent representations") cannot be understood until students understand that

n

mthat represent it. The distinction between anot \the number"; it is a mnemonic device that encodes the information we need tosymbolic representations such as 4= 24= 11.m

is just a label for the actual object. They should be shown many examples illustrating

why many di
erent labels can be applied to the same object, until they are comfortable

with the idea.

3. Dealing with the ambiguity in the symbol

n

fractions should culminate in a clearly stated algorithm for \building up" and \reducing"

fractions, namely the identity

m. This discussion of equivalent

m

n

=

mk

nk

for any whole number

k

I have seem many rambling discussions and interpretations of equivalent fractions that

never get around to stating any such clear-cut nal result, which students can then use

at the next level in their study of fractions.

4. Confusing a fraction with its many real world manifestations.

many useful

concept with its interpretations. The fraction

8

situations that number might be regarded as describing: a ratio, a rate, a percentage,

or a collection of pizza slices. There is just

are

single

a number subsumes all its applied interpretations. In short,

Fraction haveinterpretations in real life, and students often confuse the mathematical5is a number, but in speci c real-worldone de nition of fraction; all other aspectsinterpretations of this mathematical concept in speci c real world situations. Nointerpretation can encompass the others; the single basic de nition of fraction as

If you know the one big thing, you can do many little things.

This is the whole point of mathematical abstraction, and the source of its power.

6. The addition algorithm made confusing.

addition is to rst deal with sums involving fractions with the same denominator such as

3

The natural way to explain fraction

2

7

7

7

7

just like addition of whole numbers. Then one employs the rules for handling equivalent

fractions to get the following algortihm for adding unlike fractions

+ 6= 8= 11. For these, addition of fractions interpreted as lengths of line segments is

m

n

+

p

q

=

mq

nq

+

np

nq

=

mq

nq

+ np

Done!

I have been told by various people that this \cross-multiply" algorithm is not allowed.

(By whom?) To the contrary, what is unnatural is the usual de niton in terms of

common divisor

greatestand all that. This concept only comes into play when you go to simplify

the output of this algorithm, and even then is often completely unnecessary if you are

comfortable working with equivalent fractions. The traditional de nition, by mashing

together two completely di
erent ideas, needlessly confuses the issue.

7. Distinguishing the (+) operation from the algorithm for computing it.

The (+) operation can be explained geometrically without invoking anything like the

cross-multiply formula. But to get beyond gluing together strips of paper and measuring

their lengths, we need an

output

Input: The symbols

algorithm { a procedure that tells us how to get from input to

m

n

;

p

q