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A proposal for the improvement of the mathematics training of elementary school teachersAuthor(s): HERBERT F. SPITZERSource: The Arithmetic Teacher, Vol. 16, No. 2 (FEBRUARY 1969), pp. 137-139Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185940 .Accessed: 17/06/2014 01:12

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Forum on Teacher Preparation Francis /. Mueller

A proposal for the improvement of the mathematics training of elementary school teachers

HERBERT F. SPITZER The University of Texas, Austin, Texas

Herbert Spitzer is, this year, visiting professor of mathematics education. He is also assisting with the Elementary Mathematics Migrant Program of the Southwest Educational Development Laboratory.

-Leaders in mathematics education have long recognized that the elementary school teacher's lack of mathematical background was a major deterrent to improvement of instruction in this curricular area. While a number of recommendations for improv- ing the mathematics part of teacher-educa- tion programs were made before the new mathematics era,1 very little change in the mathematics preparation of elementary school teachers took place until after new mathematics began to be widely used in the elementary schools. Then, when teachers began to meet elements of strange new content, the need for additional math-

iSee Foster E. Grossnickle, "The Training of Teachers of Arithmetic," and C. V. Newsom, "Math- ematical Background Needed by Teachers of Arith- metic," in The Teaching of Arithmetic, Fiftieth Year- book, Part II, of the National Society for the Study of Education (Chicago: University of Chicago Press, 1951); and Arden K. Ruddell, Wilbur Dutton, and John Reckzeh, "Background Mathematics for Ele- mentary Teachers," in Instruction in Arithmetic, Twenty-fifth Yearbook of the National Council of Teachers of Mathematics (Washington, D.C.: The Council, 1960), pp. 316-17.

February 1969

ematical knowledge became readily ap- parent.

The teacher-training panel of CUPM prepared a report published in 1961 rec- ommending that the minimum mathematics training of elementary teachers include twelve semester hours of mathematics, with major emphasis on the arithmetic of the real numbers, introductory algebra, and informal geometry. College courses based on such recommendations, and books to be used in the courses, quickly came into existence. In spite of the fact that these courses and materials were prepared in re- sponse to a recognized need, they have ap- parently not been very successful in pro- viding elementary school teachers with the mathematical knowledge needed to teach in many current elementary school pro- grams. The apparent inadequacy of current college mathematics courses for elementary school is frequently mentioned by teachers of mathematics methods, by supervisors, and by the teachers who have taken the courses. Further indications of the inade-

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quacy of the current mathematics programs for training elementary school teachers are found in such publications as "The 1967 Report of the Cambridge Conference on Teacher Training," "Improving Mathemat- ics Education For Elementary School Teachers, A Report of a 1967 Conference at Michigan State," and the January, Feb- ruary, and March 1968 issues of The Arithmetic Teacher.

This dissatisfaction with the college mathematics courses for elementary teach- ers is a problem that warrants careful consideration. It seems to me that a major reason for students' dissatisfaction is their failure to see much relationship between what they study in these courses and their image of what mathematics they will teach to children in the elementary school.

Because the content for these mathe- matics courses was selected for its rele- vance to the K-6 elementary school math- ematics program, it would seem that this criticism could hardly be valid. Yet it exists. Reasons for this are probably varied, but chief among them is the manner in which the content is presented - either by the college textbook or by the college in- structor or by both.

To illustrate, consider a typical introduc- tion to the study of geometry as presented in a representative mathematics book for elementary school teachers. First, there is a brief review of the concept of set, fol- lowed by a statement that study of sets of points is to be the next topic for considera- tion. Then this is followed by an allusion to the three infinite sets of points to be studied: the line, the plane, and all space. After representations of some subsets of planes and lines are presented, a statement is made that such representations are re- ferred to as geometrical figures. Following four pages of such expository material, some study exercises are offered. These require the student to draw representations of a line, a plane, and a point; to use the line symbol and letters to name a line, and so on.

That the college freshman or sophomore

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fails to see a connection between such mathematical exposition and the curricu- lum of the kindergarten or first grade, where the college student hopes to teach in three or four years, is not at all sur- prising. The content presented in the col- lege book is mathematically sound, and it is indeed relevant to kindergarten-primary mathematics. What, then, can be done to help college students recognize the re- lationship of this mathematics to their teaching, and what can be done to help them acquire an interest in the study of such content? One proven approach is illustrated by the following experience.

The study of geometry in a college class for elementary education majors began through consideration of the following dittoed material, describing a classroom activity.

Kindergarten Mathematics

In an elementary school, noted for its good mathematics program, a major kindergarten mathematics activity made use of three repre- sentations of intersecting geometric figures marked on the floor (Fig. 1). The side of the square was 3 feet long. Small wooden blocks in the shapes of squares, triangles, and circles were used in the activity. There were three colors (red, yellow, black), two thicknesses, and two area sizes of each block. At the beginning of the activity the blocks were randomly placed inside and (a few) outside the geometric figures on the floor.

Figure 1

In this activity one child who is "It" selects a block mentally, tells the teacher, and then other children attempt to identify the selected block by asking "yes-or-no" questions (see dialogue below). A "yes" answer permits the questioner to continue. A "no" answer results in the right to ask questions being passed to another child. The child who identifies the block earns the privilege of being "It" for the next round of

The Arithmetic Teacher

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the activity. A report of the complete dialogue of one round of the activity in a class follows.

Teacher: Janet has told me which block she has selected. Bill, you may ask questions first. Remember that your turn to ask questions stops when you get a "no" answer.

Bill: Is it in the round figure? (Yes.) Is it in the figure with four sides? (No.)

Teacher: Your turn, Carl. Carl: Is it a big block? (Yes.) Is it a thin

block? (Yes.) Is it a red block? (No.) Teacher: Alice, it's your turn. Alice: Is it black? (No.) Teacher: Henry, it's your turn. Henry: Is it inside this space (pointing to the

space inside the triangle and circle) ? (No.) Teacher: Bill, it's your turn again. Bill: Does it have sides? (Yes.) Is it in the

three-sided figure? (No.) Teacher: Carl. Carl: Is it yellow? (Yes.) Is it this one (point-

ing to a large, thin, yellow, triangular block inside the circle, outside the square and the triangle)'? (Yes.)

Teacher: Good, Carl. You are the winner. If the answer had been "No," Janet would

have been the winner and could have selected another block to start another round.

The teacher who directed this activity believes that pupils develop some important mathematical knowledge through the activity. What knowledge that could be useful to a child's mathematical growth might be developed in this way?

After a time for individual consideration of the question posed in the dittoed ma- terial, the students in the college class reacted. That at least some members of the class thought the idea portrayed in the material had merit is reflected in the fol- lowing seven points recorded during the discussion:

1. Any location (in this activity, an ob- ject) is either inside or outside of a given closed curve.

2. Some figures have round edges; others have line segments for edges.

3. Numbers may be helpful in identifying figures.

4. Given information is most useful if a logical pattern of thinking is followed. For example, if the object is within the circular region, then parts of the square region and the triangular region are excluded.

February 1969

5. There are different kinds of closed curves.

6. There is a need for accurate descrip- tions. For example: "Are the sides of the figure the same length?"

7. The longer accurate descriptions can be discarded when the shorter name (for example, "square") is recognized by all. In the discussion of the above seven

statements, and of other aspects of the kin- dergarten mathematics activity, some ques- tions that led to further study were asked. Among them:

a) What is a closed curve? This ques- tion was formulated after a student, in re- ferring to statement (1) above, asked, "Why do you say 'closed curve'? Wouldn't 'figure' be better?"

b) What is a curve? This question arose during the consideration of the preceding question.

c) Couldn't the location (see statement a) be on the closed curve?

d) Why use circular region, square re- gion, and triangular region? (See point 4 above. )

e) What are the really distinguishing characteristics of each of the geometric figures used in this activity?

It is my firm belief that the study of geometry, guided by such questions as those given in the preceding sections, proves to be no less fruitful than study guided by questions and suggestions based on material in current college textbooks. Such an appoach to the study of geometry makes it easier for college students to recognize that college mathematics is in- deed relevant to elementary school mathe- matics.

It also seems to me that the time has arrived when "elementary school teachers- to-be" should expect that the required col- lege mathematics classes, and the materials of instruction used in those classes, reflect the kind of a mathematics program that is envisioned for the elementary schools where these students are to teach.

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Article Contentsp. 137p. 138p. 139

Issue Table of ContentsThe Arithmetic Teacher, Vol. 16, No. 2 (FEBRUARY 1969), pp. 82-160Front MatterEditorial commentAs we read [pp. 82-83]

From the editor's desk [pp. 84-85]Mathematical puzzles and games [pp. 85, 114]We all teach mathematics [pp. 86-87]Introduction to ratio and proportion [pp. 89-90]A natural way to teach division of rational numbers [pp. 91-93]Letters to the editor [pp. 93, 135]A different look at decimal fractions [pp. 95-98]On planting and learning [pp. 98-98]Division of fractional numbers [pp. 99-103]Percent: a rational number or a ratio [pp. 105-109]Why teach division of common fractions? [pp. 111-112]Secret number sentence [pp. 113-114]A deck of cards, a bunch of kids, and thou [pp. 115-117]Focus on researchSelected geometric concepts taught to children ages seven to eleven [pp. 119-128]Suggestions from researchfractions [pp. 131-135]

Forum on Teacher PreparationA proposal for the improvement of the mathematics training of elementary school teachers [pp. 137-139]

In the ClassroomAddition and subtraction fraction algorisms [pp. 141-142]Pegboard multiplication of a fraction by a fraction [pp. 142-144]Graph paper: a versatile visual aid [pp. 144-148]Subtraction of integersdiscovered through a game [pp. 148-149]Prime: a drill in the recognition of prime and composite numbers [pp. 149-151]

ReviewsBooks and materialsReview: untitled [pp. 153-153]

Books received [pp. 153-153]

Message from the President [pp. 155-155]Proceedings of the Nineteenth Annual Delegate Assembly [pp. 156-157]Committees and representatives, 1968/69supplemental list [pp. 157-158]Your professional dates [pp. 158-160]Back Matter