# Some Remarks on the Teaching of Fractions in Elementary School

# Some Remarks on the Teaching of Fractions in Elementary School

The work done on the teaching of fractions thus far has
come mainly

from the education community. Perhaps because of the recent emphasis on

situated learning, fractions tend to be discussed at the source, in the sense

that attention is invariably focussed on the interpretation of fractions in a

“ real world ” setting. Since fractions are used in many contexts in many

ways, students are led through myriad interpretations of a fraction from the

beginning in order to get some idea of what a fraction is. At the end, a

fraction is never defined and so the complexities tend to confuse rather than

clarify (cf. (2) at the beginning of the article). More to the point, such an

approach deprives students the opportunity to learn about an essential aspect

of doing mathematics: when confronted with complications, try to abstract

in order to achieve understanding. Students’ first serious encounter with the

computation of fractions may be the right moment in the school curriculum

to turn things around by emphasizing its abstract, simple component and

make the abstraction the center of classroom instruction. By so doing, one

would also be giving students a substantial boost in their quest for learning

algebra. The ability to abstract, so essential in algebra, should be taught

as early as possible in the school curriculum, which would mean during the

teaching of fractions. By giving abstraction its due in teaching fractions, we

would be easing students’ passage to algebra as well.

It takes no insight to conclude that two things have to
happen if mathematics

education in K-8 is to improve: there must be textbooks that treats

fractions logically, and teachers must have the requisite mathematical knowledge

to guide their students through this rather sophisticated subject. I

propose to take up the latter problem by writing a monograph to improve

teachers’ understanding of fractions.

The first and main objective of this monograph is to give
a treatment

of fractions and decimals for teachers of grades 5–8 which is mathematically

correct in the sense that everything is explained and the explanations are

sufficiently elementary to be understood by elementary school teachers. In

view of what has already been said above, an analogy may further explain

what this monograph hopes to accomplish. Imagine that we are mounting an

exhibit of Rembrandt’s paintings, and a vigorous discussion is taking place

about the proper lighting to use and the kind of frames that would show

off the paintings to best advantage. Good ideas are also being offered on

the printing of a handsome catalogue for the exhibit and the proper way

to publicize the exhibit in order to attract a wider audience. Then someone

takes a closer look at the paintings and realizes that all these good ideas
might

go to waste because some of the paintings are fakes. So finally people see the

need to focus on the most basic part of the exhibit—the paintings—before

allowing the exhibit to go public. In like manner , what this monograph tries

to do is to call attention to the need of putting the mathematics of fractions

in order before lavishing the pedagogical strategies and classroom activities

on the actual teaching.

An abbreviated draft of the part of the monograph on
fractions is already

in existence (Wu 1998). The main point of the latter can be summarized as

follows .

(i) It gives a complete, self-contained mathematical
treatment of

fractions that explains every step logically.

(ii) It starts with the definition of a fraction as a
number (a point

on the number line, to be exact), and deduces all other common

properies ascribed to fractions (cf. (a)–(e) above) from this definition

alone.

(iii) It explicitly and emphatically restores the simple
and correct

definition of the addition of two fractions

(iv) The four arithmetic operations of fractions are
treated as extensions

of what is already known about whole numbers. This

insures that learning fractions is similar to learning the mathematics

of whole numbers.

(v) On the basis of this solid mathematical foundation for
fractions,

precise explanations of the commonly used terms such as

“17 percent of”, “three-fifths of”, “ratio”, and “ proportional to ”

are now given.

(vi) The whole treatment is elementary and, in particular,
is appropriate

for grades 5–8. In other words, it eschews any gratuitous

abstractions.

In the eighteen months since Wu (1998) was written, I have
gotten to

know more about the culture of elementary school teachers and have come to

understand better their needs. I have also gotten to know, quite surprisingly,

that there are objections to a logically coherent mathematical treatment of

fractions in grades 5–8 by a sizable number of educators. This objection

would seem to be grounded on a misunderstanding of the basic structure

of mathematics. There are also some gaps in the treatment of Wu (1998).

All this new information must be fully incorporated into the forthcoming

monograph. More specifically, the envisioned expansion will address the

following areas:

(a) Discuss in detail from the beginning the pros and cons
of the

usual “discrete” models of fractions, such as pies and rectangles,

versus the number line. Special emphasis will be placed on the

pedagogical importance of point (iv) above. Such a discussion

would address the concerns of many school teachers and educators

who are used to having “models” for an abstract concept

and have difficulty distinguishing between the number line as a

model for fraction and its use as a definition which underlies the

complete logical development of fractions.

(b) Explain carefully that at a certain point of
elementary education,

a mathematical concept should be given one definition

and then have all other properties must be deduced from this

definition by logical deductions. This need is not generally recognized

in the education community. For fractions, the point in

question would seem to be reached in the fifth grade ([2]) or sixth

grade ([1]). Whether such logical deductions are properly taught

is likely to determine whether the learning of fractions is an immense

aid or obstacle to the learning of algebra later on.

(c) More clearly delineate which part of the exposition
primarily

addresses teachers, and which part could be directly used in the

classroom. Early readers of Wu (1998) have been to known to

complain that “no student in the fifth grade could understand

the algebraic notation”, without realizing it was a document for

teachers.

(d) Add a treatment of negative fractions and complete the
discussion

of the rational number system .

(e) Add a treatment of decimals and the relationship
between

decimals and fractions. Emphasis will be placed on a precise

definition of decimals and the logical explanations of the common

properties of decimals, viz., why fractions are the same as

repeating decimals, and which fractions have finite decimal representations.

(f) Add a discussion of the role of calculators. Although
calculators

already appear in the exercises at the end of §3 and §4 in

Wu (1998), there is need of a discursive and direct discussion of

this important issue.

(g) Amplify on the brief discussion of ratio and
proportion in Wu

(1998). Carefully explain the unfortunate historical origin of the

concept of “ratio” in Euclid’s Elements which has led to immense

and unnecessary confusion about what it means. Clarify the concept

of so-called “proportional thinking” which has been treated

with unwarranted reverence in the literature.

(h) Add generous pedagogical comments on how to deliver
such

an approach to fractions in the classroom.

(j) Expand on the rather terse expository style of Wu
(1998), everywhere.

In particular more exercises and more examples are

needed.

Thus Wu (1998) will have to be fleshed out in a
substantial way, and

its vision has to be sharpened. More importantly, to ensure its usefulness

as a resource for professional development, it is essential that it be tested

on teachers before it is finalized. Perhaps a draft of this mongraph can be

put into service in some professional development activities in the coming

months.

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