Goals for mathematical education of elementary school teachersAuthor(s): ARTHUR MORLEYSource: The Arithmetic Teacher, Vol. 16, No. 1 (JANUARY 1969), pp. 59-62Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187462 .

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Forum on teacher preparation Francis J. Mueller

Goals for mathematical education of

elementary school teachers

ARTHUR MORLEY Nottingham College of Education, Clifton, Nottingham, England

Arthur Morley is principal lecturer in mathematics at Nottingham College of Education.

L he Cambridge Conference Teacher Training Report1 is about matters of vital concern, to most countries, and so perhaps some comment from England is allowable.

I am doubtful of the report's wisdom in proposing such substantial mathematics courses for all generalists. The courses sug- gested are more like those we give in one of the two main strands of our three-year programs for teachers who, in the terms of the report, will become elementary mathematics specialists and have to some extent specialized in mathematics during their later years at school.

For many generalists in preservice train- ing, the paramount questions are interest in mathematics (as opposed to legitimate main interests in other subjects which have developed by this time) and the effect of past experience in school. An interest in the teaching of mathematics for those whose main academic preferences lie else- where comes from an acceptance of their professional task and preparation for it, including work with children. Those who

i Goals for Mathematical Education of Elementary School Teachers, a Report of the Cambridge Con- ference on Teacher Training (Boston: Houghton- Mifflin, 1967).

fear mathematics and have shown little ability in it (page 16 of the report) often find a renewal of confidence and interest through work on limited topics with a small group of children. For these reasons, in my experience, to completely dissociate discussion of aspects of teaching mathe- matics and, vitally, work with children, from teaching the mathematical content is to invite trouble, because it cuts these students off from their main source of moti- vation. However good the mathematics course may be, it seems irrelevant, and they switch off.

I am not here arguing against the need for teachers to know and understand more mathematics, but about the tactics of the college approach. The report puts too much faith in what can be done by improved "content" courses in isolation. Initially, mathematics workshop or laboratory-type courses are much more successful for many prospective elementary teachers. Experi- ences of this sort are closer to school work, involve the students individually, and allow their background to be reinforced in a more indirect way. An important aim of this work is to give the students experience in using materials to set up problem situa- tions.

January IVOV oy

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Some sample assignments used in units of this type of course in England are these:

1. (Number) Using only two squares, put them together to form as many different squares or rectangles as you can. Now try with three squares; four squares; and so on. Can you classify numbers (e.g., primes, composites, squares, etc.) using any patterns in your results?

2. (Number) Take a yellow Cuisenaire rod. Find all the different ways to form the same length by putting other rods end-to-end. Repeat for other starting rods. Can you find any general patterns in your results? Repeat using only two rods each time to form the same length as the chosen rod.

3. (Number) Investigate the patterns of multiples of different numbers on a grid using different numbers of columns.

l|2|3|4|5J6|7|8|9 10 11 12 13 14 15 16 17 18

Jl-J*LJL2!L2íL.- H ü H_ 4. (Geometry) Find as many different tri-

angles as you can on the 9-pin geo- board, and classify them in as many ways as you can.

5. (Geometry) Take a collection of equi- lateral triangles and see if these can be used as tiles to cover a surface without overlapping or leaving a gap anywhere. Would it work with triangles (all the same) of any shape? Parallelograms? Irregular quadrilaterals (number the corners 1, 2, 3, 4)? Regular polygons with 4, 5, 6, 8 sides? Try to explain your results.

6. (Graphs) Move the brick to different positions on the plank and each time take the readings on the two supporting balances. Draw a graph to show the variation in the load on the two balances and comment on your results.

The recent book, prepared by the Asso- ciation of Teachers of Mathematics, Notes

on Mathematics in Primary Schools (Cam- bridge University Press), and Appendix 3 to Curriculum Bulletin No. 1, Mathematics in Primary Schools (Her Majesty's Station- ery Office, High Holbern, London, $1.70), give further details of this kind of material.

Applying the content

A second difficulty I find with the report is the assumption that extra insights into mathematics gained by the treatment of a topic at a more sophisticated level will help a student teacher to devise an appropriate level of treatment in the classroom, or to teach better because she has this back- ground. This is rarely so for generalists, and I find it a basic fault in other texts for elementary teachers, such as SMSG Vol. IX2 and the ESI Entebbe books,3 both of which I saw in use in teacher training col- leges in Africa. The SMSG book (pages 55-61 ), for example, gives three definitions of subtraction:

1. In terms of the remainder set, A ~ Bt consisting of those elements in A which are not in B, where B is a sub- set of A. Thus n(A) - n(B) = n(A ~ B).

2. In terms of a set C, disjoint from A and B (which are disjoint sets) chosen so that A and (B U C) are in one-to-one correspondence, so that n(A) - n(B) = n(C).

3. As the inverse of addition, so that a - b = n where b + n = a.

Behind these definitions, as the book notes, lie what might be called the take- away and comparison situations and the complementary addition method of finding a difference. But I cannot see (1) and (2) being used at all as approaches to the idea of subtraction in the primary school (years 1 through 6 in England), or (3) used ex-

2 A Brief Course in Mathematics for Elementary Teachers, School Mathematics Study Group, Vol. IX (Pasadena, Calif.: A. C. Vroman, 1963).

3 Basic Concepts of Mathematics - An Introductory Text for Teachers (Watertown, Mass.: Educational Development Center, 1965).

60 The Arithmetic Teacher

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plicitly until a late stage if operations with the integers are being dealt with. The only thing they do for the teacher of these grades is to cause needless anxiety.

Or take the seemingly endless defini- tions in the geometry sections of these books: point; line, ray, segment; plane and half-plane, spaces and half-spaces; curves, simple, plane, open or closed - and not a problem in sight out of which they might emerge! Are these definitions really the ultimate of our geometry classes with young children, and if not, where are the situa- tions and materials that the generalist can use himself to gain ideas for the classroom? Such texts lead to the teacher trying to give a watered-down version of the treatment in the teacher's text (whatever is in the chil- dren's texts) and create a school-teaching situation of long explanations centered on clarifying definitions which are imposed by the teacher with the pupils as spectators.

The discussion of the two proposals for generalists in the report (pages 15-21) shows some awareness of this problem of the relation of background mathematics courses and their effect on the teacher's classroom performance. The report does not provide fully developed courses, but the outlines given still seem to dwell on the merits of systematic exposition. The ex- amples I have cited above ought to warn us of what tends to happen when such a pro- gram is developed in detail.

Further, more emphasis needs to be placed on the involvement of the college student in mathematical activity. The re- port recognizes this (page 10) for school children in discussing" the 1963 Cambridge Report4 when it says, "Ideally such active exploration should convey to the student (in school) the important fact that mathe- matics is something one does, not some- thing one absorbs passively. One would hope to strengthen the impression that a mathematical idea appears first as some-

one's solution of a problem. The problems thus become a matter of importance equal to, or even greater than, that of the textual material itself." (Sadly, on the previous page we find the old false distinction be- tween "intuitive pre-mathematics" and "rigorous" mathematics. When does the rigorous mathematics begin in problem solving? Does not the truth rather lie with Polya's "Let us teach proving, but let us also teach guessing"?)

However, the outline of courses in the 1967 report and discussion of the pro- fessor's style seem to envisage that lectures to large groups of students are the basic teaching situation in college. Yet the limi- tations of such courses where no verbal ex- change can take place between student and lecturer, or student and student, are very real, though something can be done by ex- ploratory exercises worked prior to the lecture as well as follow-up ones after- wards.

Even if large groups do make such a situation inevitable for "content" courses, a much more radically problem-based ap- proach is needed. In particular, Niven's "Mathematics of Choice,"5 and others in the SMSG New Mathematics Library se- ries, are texts which in style indicate what I have in mind.

Of great value for all generalists and ele- mentary school specialists is, I believe, par- ticipation in problem-formulating-and-solv- ing in seminars of not more than twenty students, followed by individual investiga- tions by the students. The aim in these sessions is not to teach any specific content of ideas, but to provide a firsthand experi- ence of mathematical argument - classify- ing, generalizing, proving, symbolizing. This kind of understanding of mathematics as a creative activity is crucial for the ele- mentary teacher.

Examples of problems and situations that we have found useful follow, along with brief comment on their purpose and

4 Goals for School Mathematics, the Report of the Cambridge Conference on School Mathematics (Boston: Houghton-Mifflin, 1963).

s Ivan Niven, Mathematics of Choice (New York: L. W. Singer Co., 1965).

January 1969 61

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different related problems that may be sug- gested by the group. 1. Routes on a square grid. Number of

paths between two points. "Shortest" paths. Number of shortest paths to dif- ferent points from a given point. Points equidistant from a given point. Longest routes on a limited grid. Notation for paths and return paths along the same route.

2. On a square grid take any two intersec- tion points. Any move along the lines from one point is followed by the same move in the opposite direction from the other point. Do the paths meet, and if so where? (Conditions for existence of a solution.) Variations: a) The second move is some multiple

of the first in the opposite direc- tion.

b) The second move rotates through 90° (180°, 270°) before moving.

c) Play on plain paper, and turn through any angle before moving.

d) Play on a triangular grid. 3. The number of squares passed through

by a diagonal of a rectangle drawn on a square grid. (Generalization, counter^ example, new conjecture, and proof are raised here.)

4. Five heavy boxes, marked as shown, and which can only be moved by rolling about an edge, are placed as in the diagram.

The problem is to get them in a single line

ITITITITITI. (To show that something is impossible is worth attention; other initial positions can be tried.)

5. Areas on a geoboard and the relation between the area and the number of pins on the boundary and the number inside. (Holding one variable constant and altering the other is raised here. )

6. Coloring a tetrahedron or cube, with different numbers of colors allowed and restrictions on the positions of the colors on the faces. (Questions of "same" and "different" loom large in the discus- sion.)

A detailed discussion of the purpose of such work, its implementation in college, and a description of further problems and examples of students' work are to be found in a new report, Teaching Mathe- matics - Main Courses in Mathematics in Colleges of Education (Association of Teachers in Colleges and Departments of Education, 151 Gower Street, London, W. I., $1.50). Also, the reader may find considerable interest in the evidence sub- mitted to the 1966 International Congress, printed in The Development of Mathe- matical Activity in Children - The Place of the Problem in this Development (Asso- ciation of Teachers of Mathematics, Vine Street Chambers, Nelson, Lanes., $.72) in which the "problem" in mathematics and mathematics teaching is discussed by a number of contributors from different sec- tions of the mathematical community.

62 The Arithmetic Teacher

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