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Some Reflections on Teaching Mathematics for Elementary School TeachersAuthor(s): Lowell LeakeSource: The Arithmetic Teacher, Vol. 28, No. 3 (November 1980), pp. 42-44Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191833 .Accessed: 14/06/2014 07:37

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Some Reflections on Teaching Mathematics for Elementary School Teachers By Lowell Leake

Virtually every college or university training elementary school teachers of- fers a course in mathematics content with a title similar to "Mathematics for the Elementary School Teacher", the title we use at the University of Cincin- nati. The course is usually a year long and in most states it is a certification requirement for grades kindergarten through eight; it evolved from the rec- ommendations of the Committee on the Undergraduate Program in Mathe- matics (CUPM) of the Mathematical Association of America in the early 1960s. Little, however, has appeared in journals such as the Arithmetic Teacher about the experiences various universi- ties or instructors have had with this course. This paper describes some as- pects of such a course at the University of Cincinnati, and 1 have had two ma- jor purposes in mind in writing it. One is to communicate some successful ideas to others who teach similar courses, and the other is to encourage others to submit their ideas, experi- ences, and suggestions so that all of us can share and profit from each other's successes and failures. A third possi- bility is to generate some comments from teachers in elementary and middle schools who once took such a course and now, in retrospect, have suggestions to make to those of us who teach it.

Lowell Leake is a professor in the Department of Mathematical Sciences at the University of Cin- cinnati in Cincinnati, Ohio. Prior to his university teaching, he taught in the junior and senior high school. He was one of the translators of The Ori- gin of the Idea of Chance in Children by Piaget and Inhelder and recently served a three-year term as a review editor for the Mathematics Teacher.

One of the most successful ideas I have used is to require that my stu- dents read articles in the Arithmetic Teacher related to the content we are studying that quarter. In each of the three quarters of my section of our course, students who want to receive an A must, in addition to meeting the standards required on tests, and so on, read three articles from recent issues of the Arithmetic Teacher. For each article I require a one- or two-page "reaction paper" to be handed in any time prior to the last day of class for that quarter. I ask students to include both a short summary of the article and their own personal reaction to it in terms of their perception of the article's quality and the reasons for their assessment. I do not grade the papers but I read all of them carefully. To get a B two such re- ports must be handed in, and C and D students must hand in one paper.

For several years I superimposed on this requirement the option of reading up to five articles each quarter (with reaction papers) for anyone, and I added one point per paper to the over- all numerical average of the student. I abandoned this approach as too gener- ous since it inflated a number of grades that I felt should not have been raised.

The students' response to this read- ing requirement, once they get over the initial shock on hearing about it, has been totally positive. Not only do their comments in the reaction papers in- dicate curiosity and enthusiasm, but they also indicate clearly that the read- ings help them relate the course con- tent to their future careers in the class- room. It has done a great deal to relate content to methodology without taking

any time from classroom instruction. The other clear benefit is that when these students graduate they have a personal acquaintance with a journal that may turn out to be their best friend in the classroom.

In Cincinnati we have an active group, the Mathematics Club of Greater Cincinnati, which holds fre- quent meetings. We are also, from time to time, the host city for an NCTM convention. When these meetings oc- cur I encourage my students to attend, and I allow them to substitute a report on the attendance at a section meeting for elementary teachers for one of the reaction papers. A few students have done this, but not very many.

An alternative to assigning readings in the Arithmetic Teacher itself is to re- quire that the students purchase a good book of readings, which would of course include articles from many other journals. An excellent choice for this would be a book such as Activity Oriented Mathematics, edited by Schall (1976). The advantage of this approach is that the students become exposed to the idea that many good journals exist that can help the elementary classroom teacher become a better mathematics teacher. The disadvantages are the ex- pense to the student and the fact that students do not actually see any partic- ular journal and what it has to offer on an issue-by-issue basis.

Another aspect of my experience that has been quite successful, but for relatively few students, has been an ar- rangement to match bright, able stu- dents on a one-to-one tutoring basis with students who are struggling to pass. This is strictly an optional ar-

42 Arithmetic Teacher

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rangement. After the first hour-long examination, students who received an A may volunteer to be a tutor for any students who want help. Both those who want to tutor and those who want to be tutored let me know about this in writing and I try to match them up. Students who tutor are excused from the final examination, but they must take the other hour-long examinations (a total of three each quarter) and they must do the readings described earlier. They also are required to attend class, to meet with the student or students they tutor at least twice a week, and to turn in a written summary (3-5 pages) of the tutoring experience. I explain at the outset that this is a very difficult way to earn an A; it would be far easier to take the final. On the other hand, I point out that they will learn much more if they have to teach the material, both about the content and about the learning problems others face, and that they will gain a good deal of satisfac- tion from helping fellow students. On the average, about 10 percent of the A students volunteer for this experience. Sometimes they have to tutor two stu- dents (if they agree) since I can't guar- antee a one-to-one correspondence be- tween those helping and those needing help. The students who get the help have the advantage of free tutoring from someone who actually sits in on the lectures and takes the tests. I am toying with the idea of requiring this tutoring experience for anyone who wants an A, but I am not sure this would be a wise step.

A third extremely successful experi- ment of the things I have tried has been to have each student buy a calcu- lator. Most buy inexpensive but ade- quate four-function calculators. Some have better machines, with keys for square roots, squares, reciprocals, and so on, and some have very good scien- tific calculators. Before they buy - only about 25 percent do not already own one - I suggest what they should look for.

The students use the calculators for homework, classwork, and on most tests. When I first announce this re- quirement a few students grumble, but these expressions of dissatisfaction evaporate almost immediately. To sat- isfy my own curiosity I always ask the

class, when I first announce this re- quirement, how many of them dis- approve the use of calculators in ele- mentary school. The vast majority do not approve, but at the end of the year, when I repeat the same question, al- most no one disapproves. Of course, during the year I make sure that in- struction on when to use calculators, how to use them, and the pros and cons of their use are an integral part of the course. As most readers realize, calcu- lators not only help with complicated calculations, but they also allow the use of more realistic data in problems. Furthermore, they definitely help to teach concepts. For example, in study- ing square roots, my students are asked to find square roots without using the square root key - by guessing and ad- justing the guess. The same idea works for cube roots. If you want to discuss this method more formally, using what is known as Newton's method, the calculator makes it possible to do the it- erations for the third and fourth approx- imations (and beyond), whereas paper- and-pencil techniques bog down