# Some remarks on the teaching of fractions in elementary school

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1 Some remarks on the teaching of fractions in elementary school H. Wu Department of Mathematics #3840 University of California, Berkeley Berkeley, CA USA wu/ 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 2 1 = 2 4. 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 0 October 18,

2 2 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, with positive integers c and d, is d 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.

3 3 (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 ). 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

6 6 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 (i.e., a b + c d = ad+bc bd ).

7 7 (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

9 9 months. References [1] Principles and Standards for School Mathematics, Discussion Draft. (1999). Reston, VA: National Council of Teachers of Mathematics. [2] Mathematics Framework for California Public Schools, (1999). Sacramento, CA: California Department of Education. [3] Creative Math Teaching in Grades 7 12 (Newsletter of the Mathematics Department of the University of Rhode Island), (1996). Volume 2, Number 1, 1 5. Bezuk, N. & Cramer, K. (1989) Teaching about fractions: what, when and how? In: New Directions for Elementary School Mathematics. Trafton, P. R. & Schulte, A. P. (editors) Reston, VA: National Council of Teachers of Mathematics. Clements, Herb. (1995) Can university math people contribute significantly to precollege mathematics education (beyond giving future teachers a few preservice math courses)? In: Changing the Culture: Mathematics Education in the Research Community, Fisher, N D., Keynes, H. B. & Wagreich, P. D., (editors). Providence, RI: American Mathematical Society, Freudenthal, Hans. (1983) Didactic Phenomenology of Mathematical Structures. Boston, MA: Kluwer. Huinker, De Ann. (1998) Letting fraction algorithms emerge through problem solving. In: The Teaching and Learning of Algorithms in School Mathematics. Morow, L. J. & Kenny, M. J. (editors). Reston, VA: National Council of Teachers of Mathematics. Lamon, Susan J. (1999) Teaching Fractions and Ratios for Understanding. Mahwah, NJ: Lawrence Erlbaum Associates.

10 10 Lang, Serge (1988) Basic Mathematics. New York, NY: Springer-Velag. Lappan, Glenda & Bouck, Mary K. (1998) Developing algorithms for adding and subtracting fractions. In: The Teaching and Learning of Algorithms in School Mathematics. Morow, L. J. & Kenny, M. J. (editors). Reston, VA: National Council of Teachers of Mathematics. Moynahan, J. (1996) Of-ing fractions. In: What is happening in Math Class? Schifter, Deborah (editor). New York, NY: Teachers College Press. Reys, R. E. et al. (1998) Helping Children Learn Mathematics. 5th edition. Boston, MA: Allyn and Bacon. Sharp, Janet (1998) A constructed algorithm for the division of fractions. In: The Teaching and Learning of Algorithms in School Mathematics. Morow, L. J. & Kenny, M. J. (editors) Reston, VA: National Council of Teachers of Mathematics. Wu, H. (1998) Teaching fractions in elementary school: A manual for teachers. wu Department of Mathematics #3840, University of California, Berkeley, CA

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