The elementary school curriculum
This article (like the talk itself!) is already inordinately long. Thus I will permit myself to be much briefer with my commentary than in the discussion of the secondary curriculum, believing that the rationale for my recommendations will be clear in the light of the preceding discussion and the reader's own experience. I will again organize the discussion on the basis of the "In" and "Out" format beginning with the "Out" list.
OUT (Elementary Level)
1. TEACHING STRATEGIES
Just as for the secondary level.
Emphasis on accuracy.
2. TOPICS
Emphasis on hand algorithms.
Emphasis on addition, subtraction, division and the order relation with fractions.
Improper work with decimals.
The remarks about teaching strategies are, if anything, even more important at the elementary level than the secondary level. For the damage done by the adoption of objectionable teaching strategies at the elementary level is usually ineradicable, and creates the mass phenomenon of "math avoidance" so conspicuous in present-day society. On the other hand, one might optimistically hope that the student who has received an enlightened elementary mathematical education and has an understanding and an experience of what mathematics can and should be like may be better able to survive the rigors of a traditional secondary instruction if unfortunate enough to be called upon to do so, and realize that it is not the bizarre nature of mathematics itself which is responsible for his, or her, alienation from the subject as taught.
With regard to the topics, I draw attention to the primacy of multiplication as the fundamental arithmetical operation with fractions. For the notion of fractions is embedded in our language and thus leads naturally to that of a fraction of a fraction. The arithmetical operation which we perform to calculate, say, 3/5 of 1/4 we define to be the product of the fractions concerned. Some work should be done with the addition of elementary fractions, but only with the beginning of a fairly systematic study of elementary probability theory should addition be given much prominence. Incidentally, it is worth remarking that in the latter context, we generally have to add fractions which have the same denominator - unless we have been conditioned by prior training mindlessly to reduce any fraction which comes into our hands.
Improper work with decimals is of two kinds. First, I deplore problems of the kind 13.7 + 6.83, which invite error by misalignment. Decimals represent measurements; if two measurements are to be added, they must be in the same units, and the two measurements would have been made to the same degree of accuracy. Thus the proper problem would have been 13.70 + 6.83, and no difficulty would have been encountered. Second, I deplore problems of the kind 16.1 x 3.7, where the intended answer is 59.57. In no reasonable circumstances can an answer to two places of decimals be justified; indeed all one can say is that the answer should be between 58.58 and 60.56. Such spurious accuracy is misleading and counterproductive. It is probably encouraged by the usual algorithm given for multiplying decimals (in particular, for locating the decimal point by counting digits to the right of the decimal point); it would be far better to place the decimal point by estimation.
Again, we return to the postiive side.
IN (Elementary Level)
1. TEACHING STRATEGIES
As for the secondary level.
Employment of confident, capable and enthusiastic teachers.
2. TOPICS
Numbers for counting and measurement- the two arithmetic's.
Division as a mathematical model in various contexts.
Approximation and estimation.
Averages and statistics.
Practical, informal geometry.
Geometry and mensuration;
geometry and probability (Monte Carlo method).
Geometry and simple equations and inequalities.
Negative numbers in measurement, vector addition.
Fractions and elementary probability theory.
Notion of finite algorithm and recursive definition (informal).
Some may object to our inclusion of the teacher requirement among the "teaching strategies" - others may perhaps object to its omission at the secondary level! We find it appropriate, indeed necessary, to include this desideratum, not only to stress how absolutely essential the good teacher is to success at the elementary level, but also to indicate our disagreement with the proposition, often propounded today, that it is possible, e.g. with computer aided instruction, to design a "teacher-proof" curriculum. The good, capable teacher can never be replaced; unfortunately, certain certification procedures in the United States do not reflect the prime importance of mathematical competence in the armory of the good elementary teacher. We close with a few brief remarks on the topics listed. It is an extraordinary triumph of human thought that the same system can be used for counting and measurement - but the two arithmetic's diverge in essential respects- of course, in many problems both arithmetic's a:e involved. Measurements are inherently imprecise, so that the arithmetic of measurement is the arithmetic of approximation. Yes, 2 + 2 = 4 in counting arithmetic; but 2 + 2 = 4 with probability of 3/4 if we are dealing with measurement.
The separation of division from its context is an appalling feature of traditional drill arithmetic. This topic has been discussed elsewhere ; here let it suffice that the solution to the division problem 100 + 12 should depend on the context of the problem and not the grade of the student. Geometry should be a thread running through the student's entire mathematical education - we have stressed this at the secondary level. Here we show how geometry and graphing can and should be linked with key parts of elementary mathematics. We recommend plenty of experience with actual materials (e.g., folding strips of paper to make regular polygons and polyhedra), but very little in the way of geometric proof. Hence we recommend practical, informal geometry, within an integrated curriculum. We claim it is easy and natural to introduce negative numbers, and to teach the addition and subtraction of integers - motivation abounds. The multiplication of negative numbers (like the addition of fractions) can and should be postponed.
As we have said, multiplication is the primary arithmetical operation on fractions. The other operations should be dealt with in context - and probability theory provides an excellent context for the addition of fractions. It is however, not legitimate to drag a context in to give apparent justification for the inclusion, already decided on, of a given topic. The idea of a finite algorithm, and that of a recursive definition, are central to computer programming. Such ideas will need to be clarified in the mathematics classroom, since nowhere else in the school will the responsibility be taken. However, it is reasonable to hope that to days's students will have become familiar with the conceptual aspects of the computer in their daily lives - unless commercial interests succeed in presenting the microcomputer as primarily the source of arcade games.
But this is just one aspect of the general malaise of our contemporary society, and deserves a much more thorough treatment than we can give it here. It is time to rest my case.
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