NACOME: Implications for Teacher EducationAuthor(s): HAROLD TRIMBLESource: The Mathematics Teacher, Vol. 69, No. 6 (OCTOBER 1976), pp. 465-468Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27960542 .

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NACOME: Implications for Teacher Education

By HAROLD TRIMBLE The Ohio State University

Columbus, OH 43210

MY FIRST reaction to NACOME was one of unqualified delight. I still feel grate ful to the committee. It was no small task to ferret out data to describe the "healthy but not untroubled teen-ager" (p. 149), the

phrase that the committee used to charac terize school mathematics in 1975. It took remarkable restraint to resist temptations to overgeneralize from biased samples; to become dogmatic about issues of special personal significance to committee mem

bers; to attack critics; and so on. NA COME has given our profession a bal

anced, thoughtful "benchmark 1975."

Clearly, the most important recommenda tion that I can make is to read the report. I assume that you will do this before you read further.

On second and third reading of the re

port I began to wonder whether, or when, professionals should take stronger posi tions. Is it now possible to identify some "truths" that we, like Thomas Jefferson, can declare to be "self-evident"? Are we,

perhaps, suffering from rigor? Do we really "know" some things without having what NACOME calls "hard data"?

The committee performs an important service when it warns against forming di chotomies (p. 136, policy recommendation

1). Do we favor "new math" or "old math"? Should we teach for skills or con

cepts? Let's adopt the inclusive use of the connective "or." Let's just say yes to ques tions of this sort. The "stronger positions" I seek must go deeper than fighting on one or the other side of such debate topics.

Beyond these simple either-or questions, there are others almost as unworthy of our

attention: Is work at laboratory stations in

the classroom more effective than dis cussions among clusters of students? Is it better to hold a class discussion prior to

making a reading assignment or after mak

ing a reading assignment? Perhaps these are

just either-or questions stated differently. But, again, the "stronger positions" I want us to take are of a different order.

These are different, too, from the "rec ommendations and perspectives" made in

chapter 6 of the NACOME report. Policy recommendation 4 on Teacher Education

(p. 139) has my unqualified support:

Colleges of education, professional mathematics or

ganizations, accrediting agencies of teacher certifica

tion, and the mathematics community m?st cooperate to produce mathematics teachers knowledgeable in

mathematics, aware of, oriented to, and practiced in a multitude of teaching styles and materials and philo sophically prepared to make decisions about the best

means to facilitate the contemporary, comprehensive mathematics education of their students. Further, the above bodies, together with local school boards and

organizations representative of teachers must contin

ually facilitate the maintenance of teachers' awareness of and input to current programs and issues.

But perhaps it is on the one hand too

general and on the other hand too specific to be implemented in schools and in col

leges of education. Perhaps its impact will be that of more words added to the already available plethora of words.

Could principles be established that would provide a solid basis for judging rec ommendations like the ones made by NA COME? For appraising such difficult mat ters as the effectiveness of a teacher or the

likely impact of a proposed teaching strat

egy? I believe that the answer is yes and that

formulating such principles is an urgent matter. What I am suggesting is a follow-up to the NACOME report.

In an article in the American Mathemati cal Monthly (vol. 70, June-July 1963, pp. 605-19), George Polya dares to address the

October 1976 465

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topic: "On Learning, Teaching, and Learn

ing Teaching." I think he exhibits exactly the kind of presumption that is now appro

priate for our profession, for people who want to provide leadership to teachers of school mathematics. Polya sets forth three

principles of learning. He then finds in them corresponding guides for teaching. Commenting on these principles, Polya says, "Their formulation and combination is of my choice, but the 'principles' them selves are by no means new; they have been stated and restated in various forms, they are derived from the experience of the ages, endorsed by the judgment of great men, and also suggested by the psychological study of learning''' (italics mine). Polya seems to be saying that he "knows" that these are correct principles; that they are

analogous to the "laws" that scientists as sert. The gas laws of chemistry are sup

posed by chemists to be true. Chemists claim that new data may force them to formulate new laws. But until such new data become available and stand the test of careful scrutiny by chemists, the gas laws will be used as if they were true. My ques tion is, simply, why can't we behave like the

most respectable of the scientists? Why can't we take some "stronger" positions and begin to identify recommendations like those of NACOM E as consequences of these "laws ?f learning and teaching"?

. . why can't we behave like the most respectable of the scientists?"

Personally, I like Polya's three principles. I shall not try to describe them in detail.

Polya does this better than I could. And in

any event, I would want to assign to NA COM E, or to a successor to this excellent

committee, the task of drafting the laws of

learning and teaching. Briefly, Polya's three "laws" concern the

following:

1. Active learning: "For efficient learning,

the learner should discover1 by himself as

large a fraction of the material to be learnt as feasible under the given circumstances."

2. Best motivation: "For efficient learn

ing, the learner should be interested in the material to be learnt and find pleasure in the

activity of learning. Yet, besides these best motives for learning, there are other motives

too, some of them desirable. (Punishment for not learning may be the least desirable

motive.)" 3. Consecutive phases: "Learning begins

with action and perception, proceeds from thence to words and concepts, and should end in desirable mental habits." Polya calls the three phases exploration, formalization, and assimilation.

"Could not NACOME be more dogmatic?"

Readers of the writings of great teachers of mathematics will find it easy to fill in the details for themselves. A follow-up com mittee charged with drafting laws of learn

ing and teaching should study the writings of great teachers like Socrates, Felix Klein, Max Beberman, and Thomas Butts. The committee members should, like George Polya, interpret as hard data the experience of the ages, the endorsement of great men, and the suggestions beginning to come from the psychological study of learning.

Permit me to dream a bit about the ana

lytical powers that adopting Polya-like laws of learning and teaching would give us.

Take, for example, the preservice and in service education of teachers. This is a good time for the reader to study policy recom

mendation 4: Teacher Education (p. 139)

1. Polya does not in his discussion equate active

learning with what has, in recent years, been called

discovery learning. He seems to mean "active" in the

usual sense of becoming involved, of working on a

homework problem, trying to build a winning strategy for the solution.

466 Mathematics Teacher

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quoted earlier. Is it not clear that NA COM E agrees with Polya that active learn

ing, best motivation, and consecutive

phases are necessary conditions for efficient

learning and hence for efficient teaching? Could not NACOME, then, be more dog matic? Could it not say, for example, that

teaching is a very personal act and, hence, that competency-based teacher education will not be fully effective in the foreseeable future? That trying to "cover" algebra, geometry, analysis, probability and statis

tics, and computer science during the pre service years cannot be accomplished while

paying the needed attention to the con

secutive phases of learning? That it is better to do a few things well than many things badly? That teachers need to participate in the creation of mathematics before they can

lead boys and girls to become active as

learners? That teacher education must be come a lifelong enterprise? That, even so, each teacher will be limited to do his or her own thing? That staffing a school program calls for forming a team of teachers that

have, as a collection, the variety of mathe matical interests and competencies that rea

sonably represents the spectrum from pure to applied mathematics, from historical

perspective to involvement in the latest technical advances, and so on?

We should adopt Polya's laws of learning and his correspond ing guides for teaching.

It will be hard for me to go on at greater length without revealing biases of my own.

Biases are, I believe, exactly what we need. All 1 ask is that a responsible, representa tive group, like the decision groups NA COME describes in their recommendation

4, test proposed positions against laws of

learning and teaching. Here are a few that I

believe are consequences of Polya's laws:

1. Courses in mathematics for preservice and in-service teachers must pass the test of

Polya's laws.

2. Methods courses pursued apart from contact with boys and girls in schools or other intended learning situations are, at

best, only mildly effective.

3. Mathematics learning in schools would be enhanced by providing at least one day a week for teachers to become learners.

"Mathematics learning in schools would be enhanced by providing at least one day a week for teachers to become learners."

You can add other "positions" of your own. And I think that you will agree with me that many current programs are incon sistent with Polya's laws. For example:

1. Does programmed instruction in any of its various forms "end in desirable men tal habits"?

2. Do commonly used schemes for test

ing and grading provide "best motivation"?

3. Does following a textbook page by page promote active learning?

The outlook for teacher education need not be a dreary one. The variety of Patterns of Instruction described in chapter 3 of the NACOME report; research supported by federal agencies and research by doctoral students and classroom teachers; and pub lic concern as shown in such broad-scale

movement as formulating behavioral objec tives or trying to express teacher prepara tion in terms of demonstrable com

petencies?these and other current crusades surely indicate a high level of con cern. Periods of furious activity tend to

spawn fanatics, people who lose track of all but one objective and redouble their efforts toward that objective. To control the fanat ics with their oversimplified, immediate cures for the ills of school mathematics will, as I see it, call for a solid set of professional canons. Laws of teaching and learning

October 1976 467

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would, I believe, serve this purpose. They could also improve our image as profes sionals. I think the public has a right to ask, "What, exactly, do you stand for?" (The public doesn't worry about using a preposi tion to end a sentence with.) I believe we need ready, understandable, well-consid ered answers. If we have them, I think that

people will listen; that we shall be able to

compete for resources; that we shall be able to show results. To get a then that we want, I believe we must make the //true. All my hopes are pinned on the ij of having profes

sionals agree, rather dogmatically, on laws of learning and teaching.

When you think of the importance of active learning, best motivation, and con secutive phases, it may help you, as it has

helped me, to recite the following para phrase of Richard Lovelace:

Would you choose yourself a knight From those who never fought a fight? Or, would you trust a plumber more Who swore he never plumbed before? How could I love you half so well Had I not first loved Isabelle?

A Property of Even Perfect Numbers For several thousand years mankind has been en

chanted and bewitched by various number curiosities. One such aspect of number theory, presumably forma lized by the Pythagoreans, is that of a perfect num ber?defined as a natural number that is the sum of its

proper divisors. Here proper divisors refers to all the divisors of a number exclusive of the number itself. Hence the proper divisors of six are one, two, and three. Furthermore, 1+2 + 3 = 6; therefore, six is a

perfect number. Again, the proper divisors of 28 are 1, 2, 4, 7, and 14; and 1 + 2 + 4 + 7 + 14 = 28, making the latter another perfect number.

Euclid and Euler showed that number =

2n_1 (2n -

1) is perfect if and only if the factor 2n - 1 is prime. The burden of this note is to demonstrate that the product of all of the divisors of the Euclid-Euler

perfect number, = 2n~l (2n

- 1), is equal to Pn. For

example, when = 3, =

28, and the product of all of the divisors of 28 is 1?2?4?7?14?28 = 283.

If we display all the divisors of P, namely, 1, 2, 22, 23_, 2n'\ (2n

- 1), 2(2n

- 1), . . . , 2"-2 (2n

- 1),

2n~l (2n ?

1), then two methods of proof suggest themselves. The first treatment is reminiscent of the

legend of young Gauss's handling of the arithmetic

progression whose sum was required. He paired terms

equidistant from each end of the progression and ob served that their sum was constant. We note in our

array that divisors equidistant from each end of the

lineup have a common product of P. Furthermore, there are such pairs of divisors, from which the result follows immediately.

For a straightforward, but less inspired proof, let

signify the product of all of the divisors of P. Then,

== [20+1

+ 2+ ? ? ? +<n-1']2

(2n-l)n = 2 ( "1) ?

(2n -

l)n, or

= [2n~1(2n

- l)]n

= Pn.

Depending on individual taste, one might prefer to see this theorem stated instead as: The (

- 1 )th root of the product of all the proper divisors of the perfect number,

= 2n~l (2n -

1), is equal to P. Using = 3

and = 28 again for an example, we get (1?2?4?7?14)1/2

= 28.

Richard W. Shoemaker

University of Toledo

Toledo, OH 43606

468 Mathematics Teacher

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