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. This can be understood as one of characteristics of the textbook. Pre-Algebra gives the definition of rational number formally. It classifies rational numbers compared with a counting number, a whole number, and an integer, and has a section for explaining an equivalent fraction for a given fraction.
Definition of Rational Number
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. when each fraction is represented in a formal form, students are asked to place it in its right place.
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. Pre-Algebra should have made a distinction between a fraction and a rational number.
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, Pre-Algebra gives a formal approach by using the idea of least common multiple. Students are provided the rule for adding fractions.
Rule for Adding Fraction
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 the curriculum and evaluation standard for teaching mathematics (p.97), this problem was corrected by using estimation instead of computation. His answer should be greater than 1/2+1/2, or larger than 1 since 3/4 is greater than 1/2. Effective Teaching of Mathematics shows a different approach by using two concepts of number lines and equivalent fractions. I think this approach would have been included in Pre-Algebra because it introduced the idea of number lines and equivalent fractions in Example 1 and Example 2. 1/3+1/2=2/6+3/6=5/6 can be calculated by linking each fraction to its equivalent fraction in sixths as 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. In the time planning of teaching fractions, 29 days are assigned to cover "Rational Numbers" in Pre-Algebra.. The time that a teacher can use for teaching the multiplying and dividing of fractions is 3 days. The same amount of time is allotted to the addition and subtraction of fractions. I think that more time should have planned on the addition and the subtraction parts.
Pre-Algebra does not provide many mathematical modeling. Though it has class exercises, written exercises, and review parts in every chapter most of them are simple drills for a certain skill. It is also interesting that Pre-Algebra has just a few word problems where the concept of fraction can be addressed in its best. Word problems which require the applications of a number of mathematical ideas should have been included in Pre-Algebra . So it seems that teachers should find other materials during the teaching of fractions. Models for a fraction are usually a pizza-shape figure or a rectangle with several parts.
Pre-Algebra shows a fractions as part of a whole. If students are asked to solve Example 4 which came from Effective Teaching of Mathematics they will have another concept about a fraction. Many students might have a problem in understanding Example 4. When students solve these two problems, they should see the relationship between a numerator and a denominator. I think the value of solving Example 4 will be more than solving many exercise problems in Pre-Algebra.
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
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 Pre-Algebra. It was part of a whole, too. Therefore Pre-Algebra needs to be modified with some word-problems so that students can think about fractions differently. Another concept including the idea of division of one number by another can be taught.
re-Algebra does not recommend the use of calculators in solving fraction problems When it comes to an application of calculators in mathematics teaching, Pre-Algebra has just has a calculator option section where students can divide fractions and mixed numbers on a calculator by first changing the numbers to decimals and then dividing the decimals. But as we have seen in the standards, calculators can be a useful teaching tool in a mathematics classroom. In fact, it is possible that many students see a connection between 3 4 and 3/4 if teachers use calculators in teaching of fractions. In the Effective Teaching of Mathematics, the SESM project encouraged to determine the results of 1 2, 2 4, 3 4 etc. by using calculators. Students compared these decimal fraction results with their fraction equivalents 1/2, 1/4, 3/4 etc.
Inequality can be taught in Example 5 and Example 6. Pre-Algebra could have contained a conceptual idea about deciding on inequality of two fractions. Pre-Algebra has a separate section for inequality in fractions. The multiplcation and division section has next two examples. (P. 216)
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.
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-Algebra 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.
`It is not altogether impossible that even an average teacher using this kind of textbook (colorless) and implementing a theory of his own does better than a good teacher who is bound to a textbook of a character that does not match the character of his own teaching'. ... instruction is probably better than its textbook. (Freudenthal ,1973, pp. 159-160)
Clyde L. Corcoran, L. Carey Bolster, & Jewell G. Green. (1987) Pre-Algebra. Teacher's edition. Scott, Foresman.
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,
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.