Preparing Teachers to Teach Rational Numbers

Preparing Teachers to Teach Rational NumbersAuthor(s): Frank K. Lester, Jr.Source: The Arithmetic Teacher, Vol. 31, No. 6 (February 1984), pp. 54-56Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191013 .

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Teacher Education Preparing Teachers

to Teach Rational Numbers

By Frank K. Lester, Jr. Indiana University, Bloomington, IN 47401

Other articles in this issue present valuable ideas and points of view about learning and teaching concepts and skills involving rational numbers. In particular, these articles point out the trouble spots for children, offer suggestions for teaching these difficult topics, and offer some useful suggestions about sequencing instructional activities. An underlying theme of these articles is that rational-number topics are difficult to teach and difficult to learn. I propose that a major reason why elementary school children find rational numbers so trouble- some is that some of their teachers have an inadequate understanding of rational-number concepts and just as poor facility with rational-number skills. Because of this lack of understanding and skill, teachers often cannot decide how to present these topics to children and which ideas to empha- size. Too often, teachers resort to little more than rote-level instruction. In this article, I offer some suggestions for resolving this problem during the mathematics education training of prospective teachers.

Prospective Teachers' Knowledge of Rational Numbers During my career as a teacher educa- tor, I have taught mathematics methods courses to well over 1000 preservice elementary teachers. Anyone who has taught such a course is well aware that as a group, elementary teachers' understanding of mathematics is weak. Furthermore, their mathematics deficiencies are compounded

by generally negative attitudes about, or even anxiety toward, mathematics. The inadequacy of elementary teachers' mathematical knowledge is no- where more apparent than in their understanding of rational numbers. Let me document this statement.

For the past five years all preservice elementary teachers at Indi- ana University have been required to pass an arithmetic competency test to receive credit for the first of three three-credit-hour courses in mathematics for elementary teachers. Items on the test assess mastery of arithmetic concepts and skills at no higher

Rote learning leads to rote teaching.

than a seventh-grade level, and most items require no more than fifth- or sixth-grade knowledge of arithmetic. To pass this test a student must cor- rectly answer approximately 75 percent of the items in each of seven categories (numeration, arithmetic operations, fractions and decimals, percentages, signed numbers, ratio and proportion, and word problems). Of the more than 600 students who have taken the test to date, approximately 50 percent have failed on the first attempt to reach the 75 percent criterion for passing. Failure most often results from poor performance in the rational-number categories. To learn more about the nature of their difficulties, I have talked privately with students who have failed the test. A typical conversation usually pro- ceeds as follows:

Teacher: You failed because you missed too many of the items on fractions and decimals. Did you just have a bad day, or don't you understand how to do these problems?

Student: Well, I think I understand them pretty well, but I just don't do very well on math tests. Yeah, I had a bad day.

T: That's possible, we all have off days from time to time. I want to help you prepare for the next test so you will do much better on the sections you failed. To help you I need to know more about why you got the problems wrong. For example, you wrote

56 ̂ 2_= _3_ 21 * 7 28 '

How did you do this one? S: Let's see. Well, I remembered that

you have to ' 'invert and multiply" when you divide fractions, so I inverted Щ and got

56 X 7 *

Then I did some canceling and finally multiplied to get ¿.

T: Do you see why that's wrong? S: I guess I inverted the wrong part.

It must be

56 x7 2íxř

Right? T: That would be one correct way to

do it. But what I'd like to know is why did you think to "invert and multiply"?

54 Arithmetic Teacher



S: That's what I was told in fifth grade or whenever it was. I guess I just forgot which one to invert.

T: So you think that if you brush up on rules such as ' 'invert and multiply" you'll pass next time?

S: I think so, because I really didn't take this test seriously. I used to know this stuff, I just forgot.

T: But you didn't forget how to do

347 x86

Why is that? S: It's easier. Anyway, it seems like

there are so many rules to remem- ber when you work with fractions. It's hard to keep them straight.

T: OK! Let me ask you another question. Do you think you could teach fractions to fourth or fifth graders?

S: Well, I do need to brush up, but I think I could.

T: Would you enjoy it? S: I really like working with kids. I

wouldn't really enjoy teaching fractions, but since I like teaching, it would be OK. My biggest problem would be not to make it boring. Fractions were boring for me in school. In fact, that's one reason I started hating math.

T: Because math is boring or because fractions are boring?

S: Both! By the time you're in fifth or sixth grade it seems like all you do in math is fractions and decimals and things like that. They are boring, so math was boring.

T: Was it boring because it was hard or because it wasn't interesting?

S: I'm not sure, but probably both. T: I wouldn't think I would enjoy

teaching a topic I thought was boring. I'll bet the kids would think it was boring, too.

S: Probably so! To be honest, I don't plan to teach the upper grades. I want to teach grade one or two.

T: What if you don't have any choice and you are offered a job in a good location, but it's for sixth grade?

The point of this simulated conversation is to show that students' fail- ures on rational-number exercises often stem from their having learned by

rote a collection of disconnected rules and procedures. Furthermore, because these rules and procedures were not practiced regularly, they were easily forgotten - a striking illus- tration of the limitations of rote learning. In addition, instruction that leads to mechanical learning will probably be repeated when these students be- come teachers. As undesirable as rote learning is, it is especially so for a prospective teacher. An individual who has only rote-level mastery of a topic cannot be expected to guide others to any more than rote-level mastery of that topic.

Overcoming the Difficulties Teachers of elementary mathematics methods courses are faced with a difficult question: How does one instruct students how to teach topics that they neither like nor understand? The answer would seem to have at least three parts. First, students' competence in working with rational numbers must be improved. Second,

Teaching preservice teachers about rational numbers should follow the same approach as their teaching with their students.

teacher educators must provide their students with conceptually sound ideas about the sequencing of topics, use of instructional aids, introduction and development of new topics, appli- cations of concepts and skills, and other aspects of instruction in rational numbers. Finally, students' attitudes must be changed by reducing their anxieties about rational numbers and convincing them of the value of meaningful instruction.

Unfortunately, most programs for teacher education are designed so that these changes must occur in a few class meetings. To expect any signi- ficant improvements, prospective teachers should be taught the content of rational numbers in a manner simi-

lar in development and emphasis to the way they should teach elementary schoolchildren. The advantage of this approach is that the issues of content competence and knowledge of peda- gogy can be addressed simultaneous- ly. Furthermore, the study of mathematics content in relation to methods of teaching it to children should increase its relevance to the prospective teacher. As the content becomes more relevant, motivations to learn and attitudes improve. Let us now consider the appropriate development and emphases of this approach.

Development of Rational-Number Concepts Preservice teachers must recognize the importance of the development of rational-number ideas. A major reason for children's difficulties with rational numbers is that insufficient attention is paid in the primary grades to the establishment of a foundation on which to build further understanding and skills. Instruction in rational numbers for preservice teachers should begin with fundamental conceptuali- zations and proceed systematically to computational skills in, and their ap- plication to, various problems. Both preservice teachers and children should have enough exposure to models of rational numbers to allow them to form meaningful mental representa- tions. If this approach is taken, the symbolic representation of rational numbers becomes linked to real situations and concrete models.

Emphases in Teaching Rational Numbers To be effective, teachers must know what topics should be emphasized during instruction. Four basic categories of emphasis include (1) the development of rational-number ideas from real-world situations to symbolic forms; (2) the modeling of rational numbers; (3) the use of big ideas in developing comprehension of, and skill with, rational numbers; and (4) the importance of systematic analyses of children's error patterns. To illus-

February 1984 55



trate each category, let us discuss them with respect to fractions.

Development of rational-number ideas. The study of fractions should begin with examples from the children's lives. Models of fractional ideas, both concrete and pictorial, should be developed from these examples to aid the children in the for- mation of meaningful mental repre- sentations of fractional concepts. Real-world examples and models give meaning to both the symbolic form and the oral name for fractional numbers. If too little attention is paid to the establishment of solid connections among models, symbols, and names, children will have difficulty when more abstract work begins.

Models of rational numbers. Sever- al models should be used to introduce the concept of fractions. The most common model, and one that should be introduced first, is the region model. Various examples of regions should be used - rectangular, circular, and so on. Region models are effective in giving children a grasp of fundamental concepts about fractions. A second model is the set model. Whereas the region model depicts a whole unit divided into parts, the set model illustrates a part of a set of discrete objects. (This model is often more difficult to understand.) A third model is a measurement model, an example of which is a number line. Number lines are often effective in introducing equivalent fractions. Models are very important in establishing a base on which to build later concepts; it is important that fractions be modeled in different ways - with regions, sets, and number lines. The region and set models are discussed elsewhere in this issue. Some refer- ences on measurement models are in- cluded at the end of this article.

Big ideas. At least four big ideas must be stressed to develop children's competence with fractions. The first has already been discussed - the importance of establishing stable connections among models, symbols, and oral names. The notion of equivalent fractions is another big idea because it is critical to the further development

of ideas about rational numbers. The third big idea involves the development of techniques for comparing and ordering, performing operations with fractions, and changing fractions to equivalent forms. This idea receives the most attention, often at the ex- pense of the other big ideas. The fourth big idea involves the relation- ships among fractions, decimals, and percentages. Unfortunately, these re- lationships often are not emphasized.

Analysis of error patterns. Provid- ing prospective teachers with experience in analyzing samples of children's work to identify common errors is effective in sensitizing teachers to common difficulties that may arise. Once these trouble spots are identified, teachers can see why cer- tain ideas and skills must be presented carefully.

Summary The premise of this article is that teachers' lack of knowledge about rational numbers is a major reason for the difficulties children experience in learning rational-number concepts

and skills. Often, teachers' inadequate knowledge results in the presen- tation of rational-number topics as a series of disconnected ideas and procedures with little or no meaning. My solution to this problem is to teach rational numbers to prospective teachers in the same way as the concepts should be taught to children. Not only must teachers be taught how to teach rational numbers but also they must understand what they are teaching; they need to be confident that the instruction they are providing is promoting solid understanding of concepts, reasonable facility with skills, and attitudes conducive to sub- sequent learning.

Bibliography Ashlock, Robert B. Error Patterns in Computa-

tion , 2d ed. Columbus, Ohio: Charles E. Merrill Publishing Co., 1976.

Kennedy, Leonard M. Guiding Children to Mathematical Discovery, 3d ed. Belmont, Calif.: Wadsworth Publishing Co., 1980.

LeBlanc, John F., Donald R. Kerr, G. R. Croke, K. M. Hart, С J. Irons, and Thomas L. Schroeder. Rational Numbers with Inte- gers and Reals. Reading, Mass.: Addison- Wesley Publishing Co., 1976. W

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Preparing Teachers to Teach Rational Numbers