Conceptual mathematical methodology for prospective elementary school teachers

Conceptual mathematical methodology for prospective elementary school teachersAuthor(s): ANDRÉ BROUSSEAUSource: The Arithmetic Teacher, Vol. 18, No. 4 (APRIL 1971), pp. 265-267Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186378 .

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Forum on teacher preparation

Edited by Francis J. Mueller

Conceptual mathematical

methodology for prospective elementary school teachers ANDRÉ BROUSSEAU

André Brousseau teaches mathematics and mathematics education at Centre College of Kentucky, Danville, Kentucky. He teaches both elementary and secondary education majors and assists students with practice teaching.

V^lne who questions a three- or four- year-old is often surprised at the child's understanding and knowledge of mathematical concepts. Many researchers, among them Davis (1962, p. 57) and Piaget (1953), have concluded that children at an early age, given the proper motivation, references, and catalysts, can grasp mathematical concepts normally delayed until later in their education. In recognition of this finding, and of the impact of television on a child's accumulated knowledge upon entrance into today's elementary schools, teachers must be better prepared. They face a challenge dreamed about but seldom approached before the coming of our pres- ent generation of preschoolers.

At the same time that we are faced with this enlightened generation of children, teachers are not being adequately prepared to exploit their accumulated reservoir of

knowledge. In a recent article in the Arithmetic Teacher, John LeBlanc (1970) points out that some elementary school teachers do not have a clear understanding of what they are attempting to accomplish in the overall elementary school mathematics program.

Recent studies have not only confirmed this assessment but also amplified it to show that many elementary school teachers are suffering further from a lack of apprecia- tion of the mathematical concepts they are required to teach. These studies, by Emma Garnett (1968) and Melvin Whitnell (1967), point up the contention that pres- ent methods of teaching mathematics to prospective elementary school teachers are far from satisfactory. Whitnell concluded that prospective elementary teachers are not being adequately prepared by currently recommended mathematics courses. Gar-

zi pro 1971 265



nett observed that the knowledge of mathematical concepts increases as the number of mathematics courses increases.

Following work begun in 1960, the Committee on the Undergraduate Program in Mathematics recommended courses for prospective Level I teachers. Many books have appeared purporting to accomplish the goals established for these courses. However, the two studies cited above amply illustrate that we are far from reaching desired goals and that newly designed courses have been less than successful in achieving them. Obviously something else is needed. I should like to propose a solu- tion to the dilemma in which we find our- selves - that is, too little comprehension in the time allotted to mathematics for elementary teachers yet insufficient time in four years to provide an abundance of courses to overcome this deficiency.

Every mathematician is aware that one learns concepts and becomes cognizant of the subtleties of mathematics much better when he or she is forced to answer the student's questioning "Why?" So the solu- tion naturally arises: Why not teach the concepts of elementary school mathematics from exactly that viewpoint, that is, through the eyes of the elementary student?

In my own experience I have found it difficult to motivate prospective elementary school teachers to really delve into the concepts of mathematics. Pointing out to them that they will someday teach these same concepts and must therefore understand them is analogous to telling an eleven-year-old boy that he should leârn to dance because one day he will want to dance with girls. In either case, they will hear none of it.

But placing the prospective teacher in the position of having to answer the whys of his students brings a motivation that we could not elicit before. There are two approaches that might be used to accomplish this. I offer them for your consideration. Both rely heavily on combining into integrated courses the methods of teaching elementary school mathematics and the

traditional mathematical concepts one would expect elementary teachers to mas- ter. Hence I have coined the phrase "conceptual mathematical methodology," which, simply stated, is the integration of mathematical concepts with the methods employed to teach mathematics in elementary schools.

As mentioned, there are two approaches to this, although both methods may be employed at once. In the first instance, one could stress the mathematical concepts as we have been doing, while introducing various methods and techniques for teaching these concepts. The difference in this method is that one would provide each elementary education major with the opportunity to actually teach the material to the level of his own choosing, K-6. This laboratory practice may be before peers who serve as students or in actual practice with elementary school children. In effect, then, one would provide a teaching laboratory in which the student realizes immedi- ately the need to understand the concepts of elementary school mathematics, at the same time providing him with the necessary guidance in both mathematical content and methods to be employed.

In the second approach, the course is designed in such a manner that prospective teachers are continually being asked "Why?" Most students already know many of the computational techniques they are required to teach. As pointed out, failure to appreciate the underlying concepts is the weakness we are attempting to correct. Hence, the content of the course must be designed to continually challenge the student teacher to understand the concepts, which are the foundation upon which the computational techniques are based. In this regard, questions should be designed and assignments given in such a manner that they stress understanding of the concept involved and not merely the technique. Some examples are:

How would you teach a first grader the concept of round?

266 The Arithmetic Teacher



Why do we teach "equivalence relations" to children in kindergarten relative to number concepts as opposed to simply teaching them the numerals?

Devise a method of teaching division to third graders that is based on their previ- ous knowledge. Preferable to either of the courses de-

scribed above, however, is a course designed to implement both of the above proposals in one course - that is, a course that stresses mathematical concepts combined with methods and techniques of instruction plus the opportunity to be challenged in a teaching environment. Such a course would begin with the underlying concepts of mathematics, integrated with methods and techniques of teaching. Each student would be expected, however, within a short period of time following the commencement of the course, to teach his chosen class. He would be required to repeat this exercise several times during the mathematics-education courses required for the major. I have found that relatively short periods of time - from fifteen to twenty minutes - are sufficient to judge a student's progress in the various courses. This allows several students to participate during a one-hour period (and could be useful in learning the techniques of team teaching).

Naturally, a critique of the student's per- formance would serve to point out his shortcomings. In addition, such a critique could be used to point up his strengths. Such a critique should consider, and grade the student on, both the techniques of instruction and the mathematical concepts involved therein.

It is hoped that, through one of the proc- esses described, we shall realize the goal of preparing elementary-education majors to face the challenge of the enlightened children of this and future generations. Along with others, I call for a new look at our course offerings. It is much too easy for us, as mathematicians, to teach as we have done in the past - when what is really needed is a new approach that will motivate our students.

References

Davis, Robert. "Algebra in Grades Four, Five, and Six." Grade Teacher 79 (April 1962) :57.

Garnett, Emma W. "A Study of the Relationship between the Mathematics Knowledge and the Mathematics Preparation of Undergraduate Elementary Education Majors." Ph.D. dissertation, George Peabody College, 1968.

LeBlanc, John F. "Pedagogy in Elementary Math- ematics Education - Time for a Change." Arithmetic Teacher 17 (November 1970): 605-9.

Piaget, Jean. "How Children Form Mathematical Concepts." Scientific American 189 (1953): 74-79.

Whitnell, Melvin C. "A Comparison of the Math- ematical Understandings of Prospective Ele- mentary Teachers in Colleges Having Different Mathematics Requirements." Ph.D. dissertation, The University of Michigan, 1967.

Remainder multiplying

[Continued from p. 249.] Step 3. Next you would multiply 6 by 2 and

add 5 (a combination of 1 plus 4) to get 17. Put down the 7; carry the 1.

G>4(15) ®227 ®227 X16

Step 4. You next would multiply 6 by 2 and add your 1 to get 13. Put it down.

227 X 16

Step 5. When it comes to multiplying the 10 (from the 16) by 227, you simply ignore the 15.

227 X 16

1377

Step 6. You finish your problem like this: 227

X 16

1377 +2270

[36471 And this is how to remainder multiply. - Betsy Benner, Sixth-Grade Student, Bulman Elemen- tary School, Redford Union School, Redford, Michigan.

April 1971 267



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Conceptual mathematical methodology for prospective elementary school teachers