# Some Remarks on the Teaching of Fractions in Elementary
School

It is widely recognized that there are at least two major
bottlenecks in

the mathematics education of grades K–8: the teaching of fractions and the

introduction of algebra. Both are in need of an overhaul. I hope to make

a contribution to the former problem by devising a new approach to elevate

teachers’ understanding of fractions. The need for a better knowledge of

fractions among teachers has no better illustration than the the following

story related by Herbert Clemens (1995):

Last August, I began a week of fractions classes at a
workshop

for elementary teachers with a graph paper explanation of why

The reaction of my audience astounded me.
Several

of the teachers present were simply terrified . None of my protestations

about this being a preview, none of my “Don’t worry”

statements had any effect.

This situation cries out for improvement.

Through the years, there has been no want of attempts from the mathematics

education community to improve on the teaching of fractions (Lamon

1999, Bezuk-Cramer 1989, Lappan Bouck 1989, among others), but much

work remains to be done. In analyzing these attempts and the existing

school texts on fractions, one detects certain persistent problematic areas in

both the theory and practice, and they can be briefly described as follows:

(1) The concept of a fraction is never clearly defined in
all of K–

12.

(2) The conceptual complexities associated with the common
usage

of fractions are emphasized from the beginning at the expense

of the underlying mathematical simplicity of the concept.

(3) The rules of the four arithmetic operations seem to be
made

up on an ad hoc basis, unrelated to the usual four operations on

positive integers with which students are familiar.

(4) In general, mathematical explanations of essentially
all aspects

of fractions are lacking.

These four problems are interrelated and are all
fundamentally mathematical

in nature. For example, if one never gives a clearcut definition of a

fraction, one is forced to “talk around” every possible interpretation of the

many guises of fractions in daily life in an effort to overcompensate. A good

example is the over-stretching of a common expression such as “a third of a

group of fifteen people” into a main theme in the teaching of fractions (Moynahan

1996). Or, instead of offering mathematical explanations to children

of why the usual algorithms are logically valid —a simple task if one starts

from a precise definition of a fraction,—algorithms are justified through
”connections

among real-world experiences, concrete models and diagrams, oral

language, and symbols (p. 181 of Huinker 1998; see also Lappan & Bouck

1998 and Sharp 1998). It is almost as if one makes the concession from the

start: “We will offer everything but the real thing”.

Let us look more closely at the way fractions are
introduced in the classroom.

Children are told that a fraction c/d , with positive integers c and d, is

simultaneously at least five different objects (cf. Lamon 1999 and Reys et al.

1998):

(a) parts of a whole: when an object is equally divided
into d

parts , then c/d

denotes c of those d parts.

(b) the size of a portion when an object of size c is
divided into

d equal portions.

(c) the quotient of the integer c divided by d.

(d) the ratio of c to d .

(e) an operator: an instruction that carries out a
process, such

as “2/3 of”.

It is quite mystifying to me how this glaring “crisis of
confidence” in fractions

among children could have been been consistently overlooked. Clearly, even

those children endowed with an overabundance of faith would find it hard to

believe that a concept could be so versatile as to fit all these descriptions.

More importantly, such an introduction to a new topic in mathematics is

contrary to every mode of mathematical exposition that is deemed acceptable

by modern standards. Yet, even Hans Freudenthal, a good mathematician

before he switched over to mathematics education, made no mention

of this central credibility problem in his Olympian ruminations on fractions

(Freudenthal 1983). Of the existence of such crisis of confidence there is no

doubt. In 1996, a newsletter for teachers from the mathematics department

of the University of Rhode Island devoted five pages of its January issue to

“Ratios and Rational Numbers” ([3]). The editor writes:

This is a collection of reactions and responses to the
following

note from a newly appointed teacher who wishes to remain anonymous:

“On the first day of my teaching career, I defined a
rational number

to my eighth grade class as a number that can be expressed

as a ratio of integers. A student asked me: What exactly are

ratios? How do ratios differ from fractions? I gave some answers

that I was not satisfied with. So I consulted some other teachers

and texts. The result was confusion . . . ”

This is followed by the input of many teachers as well as
the editor on this

topic, each detailing his or her inconclusive findings after consulting existing

texts and dictionaries (!). In a similar vein, Lamon (1999) writes: “As one

moves from whole number into fraction, the variety and complexity of the

siutation that give meaning to the symbols increases dramatically. Understanding

of rational numbers involves the coordination of many different but

interconnected ideas and interpretations. There are many different meanings

that end up looking alike when they are written in fraction symbol” (pp. 30–

31). All the while, students are told that no one single idea or interpretation

is sufficiently clear to explain the “meaning” of a fraction. This is akin to

telling someone how to get to a small town by car by offering fifty suggestions

on what to watch for each time a fork in the road comes up and how

to interpret the road signs along the way, when a single clearly drawn road

map would have done a much better job. Given these facts, is it any wonder

that Lappan-Bouck (1998) and Lamon (1999) would lament that students

“do” fractions without any idea of what they are doing? For example, it is

certainly difficult to learn how to add two “operators” in the sense of (e).

Sometimes one could “get by” a mathematical concept
without a precise

definition if its rules of operation are clearly explained. Conjecturally,

that was how Europeans in the 14th and 15th centuries dealt with negative

numbers. In the case of fractions, however, this is not true even when
interpretation

(b) of fractions is used. The worst case is the rule of adding two

fractions. In book after book (with very few exceptions, such as Lang (1988)),

is defined as (pa+cq)/m, where m = lcm{b, d}
and m = bp = cq. Now

at least two things are wrong with this definition. First, it turns off many

students because they cannot differentiate between lcm and gcd . This definition

therefore sets up an entirely unnecesary roadblock in students’ path

of learning. Second, from a mathematical point of view, this definition is
seriously

flawed because it tacitly implies that without the concept of the lcm

of two integers, fractions cannot be added. If we push this reasoning another

step, we would arrive at the absurd conclusion that unless an integral domain

has the unique factorization property, its quotient field cannot be defined.

Informal surveys among teachers consistently reveal that
many of their

students simply give up learning fractions at the point of the introduction of

addition. It is probably not just a matter of being confused by gcd and lcm,

but more likely a feeling of bewilderment and disgust at being forced to learn

a new way of doing addition that seems to bear no relation to the addition of

whole numbers. This then brings us to the problem area (3) at the beginning

of this article. We see, for example, that Bezuk and Cramer (1989) willingly

concede that “Children must adopt new rules for fractions that often conflict

with well-established ideas about whole number” (p.156). In mathematics,

one of the ultimate goals is to achieve simplicity. In the context of learning,

it is highly desirable, perhaps even mandatory, that we convey this message

of simplicity to students. However, when we tell students that something

as simple as the addition of two numbers is different for whole numbers and

fractions, we are doing them a great disservice. Even when students are

willing to suspend disbelief and go along on such a weird journey, they pay a

dear price. Indeed, there are recurrent reports of students at the University of

California at Berkeley and at Stanford University claiming in their homework

and exam papers that and

All in all, a mathematician approaching the subject of
fractions in school

mathematics cannot help but be struck by the total absence of the characteristic

features of mathematics: precise definitions as starting point, logical

progression from topic to topic, and most importantly, explanations that

accompany each step. This is not to say that the teaching of fractions in

elementary school should be rigidly formal from the beginning. Fractions

should be informally introduced as early as the second grade (because even

second graders need to worry about drinking “half a glass” of orange juice!),

and there is no harm done in allowing children to get acquainted with fractions

in an intuitive manner up to, say, the fourth grade. An analogy may be

helpful here. The initial exploration of fractions may be taken as the
“datacollecting

phase” of a working scientist: just take it all in and worry about

the meaning later. In time, however, the point will be reached when the said

scientist must sit down to organize and theorize about his or her data. So

it is that when students reach the fifth grade ([2]) or the sixth grade ([1]),

their mathematical development cannot go forward unless “miracles” such

as having one object c/d enjoying the five different properties of (a)–(e) above

are fully explained, and rules such as
justified. And it at this

critical juncture of students’ mathematical education that I hope to make a

contribution.