on Rational Numbers

Jeon, Kyungsoon

Dept. of Mathematics Education

The University of Georgia

**Introduction**

I understand that the most common college preparatory mathematics program
in the secondary mathematics education in the United States is based on
the Algebra1-Geometry-Algebra2-Trigonometry-Calculus sequence. While talking
to other graduate students they expressed a great deal of doubts about the
mathematical ability of American students. This made me interested in looking
at what mathematical knowledge they should have before they enter the sequence
and what kind of teaching materials should be provided for teachers and
students. I think we need to go back to the basic question of what should
be taught and how the content should be presented in the prealgebra stage
of secondary mathematics education. I would like to talk about rational
numbers.

I have often seen a problem with the students in Korea since the curriculum
is fixed. So unfortunately, their knowledge is often delayed because they
should know a great deal of mathematics before they can integrate former
knowledge. To my point of view, United States has a flexible system to support
students who have a difficulty in conceptual understanding through the process
of their high school mathematics learning. If students can choose the level
of mathematics course depending on their aptitudes, interest, and ability
and can drop out at various points during their high school education, then
students and their teachers should be able to make a wise decision on the
quality of students' education. In that sense, prealgebra must be help.

This paper is not written just for comparing two textbooks. In fact, most
of the prealgebra textbooks that I investigated didn't show me any big difference
in their formation. In the first place, I wanted to give a special insight
into high school textbooks but it was hard for me to find any big difference
between two textbooks. So I took a textbook which has been used in actual
mathematics classrooms and a book which was written for teachers wishing
to improve their mathematics teaching skills. The former is *Pre-Algebra*
published by Scott, Foresman in 1987 and the latter is *Effective Teaching
of Mathematics* published by Longman in 1993. The second chapter of *Effective
Teaching of Mathematics* was "some early considerations when planning
lessons". It had several subtopics in the secondary mathematics curriculum.
I focused on the part of "fractions". In particular, "Fractions"
contains the results of two curriculum project, CSMS (the Concepts in Secondary
Mathematics and Science Project, 1974-1979) and SESM (the Strategies and
Errors in Secondary Mathematics Project, 1980-1983).

I will highlight the following points in this paper. First, I understood
that *Pre-algebra* is for students who don't feel confident in their
mathematics learning but have enough ability to understand a concept provided
by the teacher's careful preparation. Second, I chose the teacher's edition
of *Pre-Algebra* to see explicitly what objectives are pursued and
how content is addressed. I took into consideration that these materials
have distinct publication purposes. Third, I will analyze the chapter of
rational numbers from *Pre-Algebra.* Moreover, I will include my recommendation
in necessary parts. Fourth, I will introduce good approaches from CSMS and
SESM which investigated the types of difficulties that students had in this
content area. These difficulties were

(1) that many students could not think of a fraction as a number

(2) that a common presentation of a fraction, diagrammatically as part of
a whole, has severe

limitations on ideas about fractions and operations on fractions.

(3) that although students readily identify equivalent fractions they are
unable to use equivalence

to determine relative sizes or to perform operations on fractions.

**Analysis and suggestions
**Pre-Algebra stresses out the formal approach

A rational number is any number that can be written in the form a/b, where a and b are integers

and b is not zero.

Pre-Algebra uses a number line

Example 1

Through the process of filling a number line, students will be able to think that rational number is a number like integers. They can also see that the relative sizes of these numbers can be compared.

Integers are also rational numbers with denominator 1. Integers can be expressed in many different forms.

Example 2 Write each of the rational numbers 5, -3, 0.5, -11/8, and 4.20 in the form a/b and give

two other equivalent fractions.

Rules for operations with fractions are not enough. It is well-known fact that many students have a common error of adding numerators together and denominators together in the learning of the addition of fractions,

For fractions a/c and b/c, where c is not equal to zero, a/c+b/c=(a+b)/c

Then there are almost 60 exercise problems that students have to solve. They are asked to rename the fractions with their LCD. I don't think that Pre-Algebra tried to give students conceptual understanding of the rule. And there is no remedial way to correct the typical problem about the addition of fractions that students have shown for a long time in learning of fractions. For example, Hector had 4/6 when he added 3/4+1/2. He confused the rules for adding fractions with those for multiplying fractions and added the numerators and then the denominators. In

Example 3

Teachers will be able to explain why b/a+d/c is not equal to (b+d)/(a+c) and why it should be (bc+ad)/ac. I think that the degree of acceptance for formula provided by a textbook should be noticed by mathematics teachers in their classroom.

Time allotment is not proper

Pre-Algebra does not provide many mathematical modeling

Pre-Algebra shows a fractions as part of a whole. If students are asked to solve Example 4 which came from

Example 4

1. The 2 pints of milk are divided equally between the 3 cups. How much milk is there in each cup?

2. The 3 pints of water are divided equally between the 2 jugs. How much water is there in each

jug?

The SESM project analyzed the cause of students' problem and found out the reason of it. It is probably the result of many textbooks presenting fractions in typical mathematical modeling as the part of a whole. I have already mentioned about the mathematical model for a fraction in

re-Algebra does not recommend the use of calculators in solving fraction problems When it comes to an application of calculators in mathematics teaching,

Inequality can be taught in Example 5 and Example 6.

Example 5. Divide 3/4 by 2/5 Solutions Example 6. Find 6(2/3)8 Dividing by a number is the Use the rule for dividing same as multiplying by its fractions reciprocal 3/42/5=3/4*5/2 6(2/3)8=20/38/1 =(3*5)/(4*2) =15/8 =20(1/8) =20/24 =5/6

Since *Pre-Algebra* gave the rule for dividing fractions before
these examples, it is natural for teachers to expect that their students
can solve these problems. But if teachers intend more than getting right
answers from their students, these problems can be extended. Teachers can
explain the inequalities over the rational numbers. When the divisor is
less than the dividend, the quotient is greater than 1 as in Example 5.
When the divisor is greater than the dividend, the quotient is less than
1 as in Example 6. Ofcourse there is a separate section in *Pre-Algebra*
dealing with the inequalities over rational numbers. But the approach is
based on the same skills as those used to solve inequalities over the integers.
*Pre-Algebra* explains that if both sides of an inequality are multiplied
or divided by the same positive number, the order of the resulting inequality
is unchanged and if both sides of an inequality are multiplied or divided
by the same negative number, the order of the resulting inequality is reversed.

**Discussion**

I criticized *Pre-Algebra* and tried to supplement its content in
the previous section. But I feel that *Pre-Algebra* has many good things
to offer as a textbook. *Pre-Algebr*a is very different in many ways
from any of high school textbooks which I used to study in my country. First,
authors are all secondary education-related people. While in Korea, most
of the school textbooks are written by university professors. Second, *Pre-Algebra*
contains many special features other than certain mathematical topics. It
has statement of objectives, teaching examples, supplementary materials,
enrichment, chapter test, decision making in problem solving, questions
for investigation, maintenance, review, and testing. I think that these
features help to provide essentials that teachers need to prepare students
for success. If I were a teacher, I would accept most parts of this textbook
and mix in good points from *the Effective Teaching of Mathematics.*

I think that definitions or properties might not be applied indiscriminately
by students in solving fraction problems without a firm conceptual establishment.
Concepts should be taught through planned activities that focus on the understanding
of interrelationships among many mathematical ideas. Instead of devoting
large blocks of time to developing a mastery of paper-and-pencil manipulative
skills, more time and effort should be spent on developing a conceptual
understanding of key ideas and their applications. If teachers want their
students to have mathematical power, they first should have their pedagogical
power. Needless to say, textbooks and all other educational material should
be able to support both power.

I think about the role of teachers. If a teacher just tolerate errors and
doesn't use them as a feedback mechanism for real learning on the basis
of actual performance, he should be criticized. Though the teacher says
that he covered everything in the textbook, he has no excuse for ignoring
these errors. Teachers should make themselves open to a lot of materials
that can broaden their teaching. Many research finding will be a good choice
to build developmental works.

Finally, there is a good statement to which all curriculum developers should
pay attention. It came from "Learning How to Teach via Problem Solving"
from* Professional Development for Teachers of Mathematics. (1994 NCTM
Yearbook)
Instead of the expectation that skill in computation should proceed word
problems, experience with problems helps develop the ability to compute.
Thus, present strategies for teaching may need to be reversed...Students
need to experience genuine problems regularly.*

The next statement is for all mathematics teachers because I think they also have a responsibility in reversing teaching strategies.

Clyde L. Corcoran, L. Carey Bolster, & Jewell G. Green. (1987)

Glenview, Illinois

Frank K. Lester, Jr., Joanna O. Masingila, Sue Tinsley Mau, Diana V. Lambdin,
Vania Maria Pereira does Santos, &

Anne M. Raymond. (1994). Learning How to Teach via Problem Solving__,__
__Professional Development for __

__ Teachers of Mathematics__, (1994 Yearbook of the National Council
of Teachers of Mathematics, p. 152) Reston,

VA

Freudenthal H. (1973). __Mathematics as an Educational Task.__ Reidel

Geoffrey Howson, Christine Keitel, & Jeremy Kilpatrick. (1981). __Curriculum
development in mathematics.__

Cambridge University Press, New York, NY

Hirsch, C. R. & Zweng, M. J. (1985). The secondary school mathematics
curriculum. (1985 Yearbook of the National

Council of Teachers of Mathematics) Reston, VA

Malcolm Simmons. (1993). __The Effective Teaching of Mathematics.__ Longman,
London and New York, NY

National Council of Teachers of Mathematics. (1989). __Curriculum and evaluation
standards for school mathematics__. Reston, VA

National Council of Teachers of Mathematics. (1989). __Professional Standards
for Teaching of Mathematics.__

Reston, VA