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Problems In The History And Philosophy Of Education: Mathematics

March 19, 2008

[Warning: Long post.]

The issue of how to get youth to study math has been ongoing in my lifetime. As soon as I understood the endeavor of education as a problem unto itself, as an object of thought, I learned of math as an important subtopic. How can educators inspire students to a love of math study?

I concede that I haven’t ~fully~ explored the history of the problem of math education. I don’t know when getting students to study math, in the United States or elsewhere, became difficult. I know, like most others, that women have not traditionally been encouraged to study the subject. It was believed during the American Victorian era that math and science studies would lessen one’s femininity: many men and women believed that women couldn’t handle the stress. I’ll have to look more into this branch of the history of education.

I can, however, date one strain of concern—international competitiveness. This basically began on October 4, 1957. With the launching of the Russian Sputnik 1 satellite, many U.S. educators started bemoaning our status as a world leader in science development. This caused a public furor. Very soon thereafter Congress funded programs, via the 1958 National Defense Education Act, that encouraged students to study math and science in the public school system.

Wikipedia has brief entries on NDEA and the general “Sputnik Crisis” that ensued. The second entry contains a colorful quote from Clare Boothe Luce, where she metaphorically described the launch as “an intercontinental outer-space raspberry to a decade of American pretensions that the American way of life was a gilt-edged guarantee of our national superiority.” The larger social, political, and economic result was so-called Space Race. [Aside: One of the Wikipedia articles asserts that “New Math” came out of the Sputnik Crisis. I’ll have to look into that.]

A colleague of mine, Andrew Hartman, a fellow USIH enthusiast and assistant professor of history at Illinois State University, recently finished a book covering education reforms that surrounded the Sputnik Crisis. In Education and the Cold War, Hartman contextualizes the ideologies and politics of congressional and education leaders during the period. In fact, anti-Progressive education critics—such as Arthur Bestor, Admiral Hyman Rickover, and even President Dwight Eisenhower—seized on the Sputnik Crisis to enact long-hoped-for reforms.

Rickover’s desires in particular seemed to have found an audience. He explicitly advocated that math and science be given much more attention in schools. Rickover, moreover, thought that U.S. educators needed to focus more on educating the brightest. Hartman characterizes Rickover and likeminded reformers as advocating for a “national-security-style educational instrumentalism.” But you should read Hartman’s book for more; in fact, I’m in the middle of it. Suffice it to say that Sputnik provided an opening for anti-Progressive education (whether really derived from Dewey or from others acting in his “spirit”).

Returning to the present, this New York Times article by Tamar Lewin outlines some of today’s issues. Here are some excerpts:


American students’ math achievement is “at a mediocre level” compared with that of their peers worldwide, according to a new report by a federal panel. The panel said that math curriculums from preschool to eighth grade should be streamlined to focus on key skills — the handling of whole numbers and fractions, and certain aspects of geometry and measurement — to prepare students to learn algebra.
“The sharp falloff in mathematics achievement in the U.S. begins as students reach late middle school, where, for more and more students, algebra course work begins,” said the report of the National Mathematics Advisory Panel, appointed two years ago by President Bush. “Students who complete Algebra II are more than twice as likely to graduate from college, compared to students with less mathematical preparation.”
The report, to be released Thursday, spells out specific goals for students. For example, it says that by the end of the third grade, students should be proficient in adding and subtracting whole numbers; two years later, they should be proficient in multiplying and dividing them. By the end of sixth grade, it says, students should have mastered the multiplication and division of fractions and decimals.
The report tries to put to rest the long and heated debate over math teaching methods. Parents and teachers in school districts across the country have fought passionately over the relative merits of traditional, or teacher-directed, instruction, in which students are told how to solve problems and then are drilled on them, as opposed to reform or child-centered [Progressive] instruction, which emphasizes student exploration and conceptual understanding. The panel said both methods have a role .
“There is no basis in research for favoring teacher-based or student-centered instruction,” said Dr. Larry R. Faulkner, the chairman of the panel, at a briefing for reporters on Wednesday. “People may retain their strongly held philosophical inclinations, but the research does not show that either is better than the other.”
The president convened the panel to advise on how to improve math education for the nation’s children. Its members include math and psychology professors from leading universities, a middle-school math teacher and the president of the National Council of Teachers of Mathematics.
Closely tracking an influential 2006 report by the National Council of Teachers of Mathematics, the panel said that the math curriculum should include fewer topics, and then spend enough time on each of them to make it is learned in depth and need not be revisited in later grades. This is how top-performing nations approach the curriculum.
The new report does not call for a national math curriculum, or for new federal investment in math instruction. It does call for more research on successful math teaching, and recommends that the Secretary of Education convene an annual forum of leaders of the national associations concerned with math to develop an agenda for improving math instruction.
The report cites a number of troubling international comparisons, including a 2007 assessment finding that 15-year-olds in the United States ranked 25th among their peers in 30 developed nations in math literacy and problem solving.
The report says that Americans fell short, especially, in handling fractions. It pointed to the National Assessment of Educational Progress, standardized-test results that are known as the nation’s report card, which found that almost half the eighth graders tested could not solve a word problem that required dividing fractions.
After hearing testimony and comments from hundreds of organizations and individuals, and sifting through 16,000 research publications, the panelists shaped their report around recent research on how children learn.
For example, the panel found that it is important for students to master their basic math facts by heart. “For all content areas, practice allows students to achieve automaticity of basic skills — the fast, accurate, and effortless processing of content information — which frees up working memory for more complex aspects of problem solving,” the report said.
Dr. Faulkner, a former president of the University of Texas at Austin, said the panel “buys the notion from cognitive science that kids have to know the facts.” “In the language of cognitive science, working memory needs to be predominately dedicated to new material in order to have a learning progression, and previously addressed material needs to be in long-term memory,” he said.
The report also cites recent findings that students who depend on their native intelligence learn less than those who believe that success depends on how hard they work. Dr. Faulkner said the current “talent-driven approach to math, that either you can do it or you can’t, like playing the violin” needed to be changed.
“Experimental studies have demonstrated that changing children’s beliefs from a focus on ability to a focus on effort increases their engagement in mathematics learning, which in turn improves mathematics outcomes,” the report says “When children believe that their efforts to learn make them ‘smarter,’ they show greater persistence in mathematics learning.”


But what can we do to improve this belief? What can we do to combat this perennial problem in the history and philosophy of education?

I have a working hypothesis. My theory is based on two things: my own data set of experiences with regard to learning math, and a fledgling idea co-opted from Plato.

My math education began with a kind of regular, public school sequence. As a high schooler who had an idea that the sciences interested me, my college prep program consisted of taking math courses all four years. Since I transferred from a Missouri school district with one planned sequence to another with a different sequencing, I never made it to the summit in terms of high school math studies: calculus. This limited me when I got to college, but I didn’t mind, as a high schooler, that I only reached pre-Calculus. I was always one of the best in class, so it probably stroked my adolescent ego.

In college, as an engineering major, I was forced—via a math placement test—to re-take pre-calculus. This should’ve been my first clue that things weren’t going to go well. Anyway, to make a long and tumultuous story short, I made it through differential equations (via a two-semester stay in Calculus I. Sigh.). This gives me a pretty extensive data set from which to draw upon for sorting out my abovementioned working hypothesis.

The idea I’m co-opting from Plato arose during my reading of The Republic, particularly Part II, Book VII, “Education of the Philosophers” (circa 360 B.C.E, translated by Benjamin Jowett). Here is an extended excerpt (bolds mine) from conversations between Socrates, Glaucon, and Adeimantus (I think, here’s a Wikipedia link with more information):


Undoubtedly; and yet if music and gymnastic are excluded, and the arts are also excluded, what remains?

[Socrates] Well, I said, there may be nothing left of our special subjects; and then we shall have to take something which is not special, but of universal application.

What may that be?
A something which all arts and sciences and intelligences use in common, and which every one first has to learn among the elements of education.

What is that?
The little matter of distinguishing one, two, and three –in a word, number and calculation: –do not all arts and sciences necessarily partake of them?

Then the art of war partakes of them?
To be sure.
Then Palamedes, whenever he appears in tragedy, proves Agamemnon ridiculously unfit to be a general. Did you never remark how he declares that he had invented number, and had numbered the ships and set in array the ranks of the army at Troy; which implies that they had never been numbered before, and Agamemnon must be supposed literally to have been incapable of counting his own feet –how could he if he was ignorant of number? And if that is true, what sort of general must he have been?

I should say a very strange one, if this was as you say.
Can we deny that a warrior should have a knowledge of arithmetic?
Certainly he should, if he is to have the smallest understanding of military tactics, or indeed, I should rather say, if he is to be a man at all.

I should like to know whether you have the same notion which I have of this study?

What is your notion?
It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the soul towards being.

Will you explain your meaning? he said.
I will try, I said; and I wish you would share the enquiry with me, and say ‘yes’ or ‘no’ when I attempt to distinguish in my own mind what branches of knowledge have this attracting power, in order that we may have clearer proof that arithmetic is, as I suspect, one of them.

Explain, he said.
I mean to say that objects of sense are of two kinds; some of them do not invite thought because the sense is an adequate judge of them; while in the case of other objects sense is so untrustworthy that further enquiry is imperatively demanded.

You are clearly referring, he said, to the manner in which the senses are imposed upon by distance, and by painting in light and shade.

No, I said, that is not at all my meaning.
Then what is your meaning?
When speaking of uninviting objects, I mean those which do not pass from one sensation to the opposite; inviting objects are those which do; in this latter case the sense coming upon the object, whether at a distance or near, gives no more vivid idea of anything in particular than of its opposite. An illustration will make my meaning clearer: –here are three fingers –a little finger, a second finger, and a middle finger.

Very good.
You may suppose that they are seen quite close: And here comes the point.

What is it?
Each of them equally appears a finger, whether seen in the middle or at the extremity, whether white or black, or thick or thin –it makes no difference; a finger is a finger all the same. In these cases a man is not compelled to ask of thought the question, what is a finger? for the sight never intimates to the mind that a finger is other than a finger.

And therefore, I said, as we might expect, there is nothing here which invites or excites intelligence.

There is not, he said.
But is this equally true of the greatness and smallness of the fingers? Can sight adequately perceive them? and is no difference made by the circumstance that one of the fingers is in the middle and another at the extremity? And in like manner does the touch adequately perceive the qualities of thickness or thinness, or softness or hardness? And so of the other senses; do they give perfect intimations of such matters? Is not their mode of operation on this wise –the sense which is concerned with the quality of hardness is necessarily concerned also with the quality of softness, and only intimates to the soul that the same thing is felt to be both hard and soft?

You are quite right, he said.
And must not the soul be perplexed at this intimation which the sense gives of a hard which is also soft? What, again, is the meaning of light and heavy, if that which is light is also heavy, and that which is heavy, light?

Yes, he said, these intimations which the soul receives are very curious and require to be explained.

Yes, I said, and in these perplexities the soul naturally summons to her aid calculation and intelligence, that she may see whether the several objects announced to her are one or two.

And if they turn out to be two, is not each of them one and different?

And if each is one, and both are two, she will conceive the two as in a state of division, for if there were undivided they could only be conceived of as one?

The eye certainly did see both small and great, but only in a confused manner; they were not distinguished.

Whereas the thinking mind, intending to light up the chaos, was compelled to reverse the process, and look at small and great as separate and not confused.

Very true.
Was not this the beginning of the enquiry ‘What is great?’ and ‘What is small?’

Exactly so.
And thus arose the distinction of the visible and the intelligible.
Most true.
This was what I meant when I spoke of impressions which invited the intellect, or the reverse –those which are simultaneous with opposite impressions, invite thought; those which are not simultaneous do not.

I understand, he said, and agree with you.
And to which class do unity and number belong?
I do not know, he replied.
Think a little and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks ‘What is absolute unity?’ This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.

And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude?

Yes, I said; and this being true of one must be equally true of all number?

And all arithmetic and calculation have to do with number?
And they appear to lead the mind towards truth?
Yes, in a very remarkable manner.
Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician

That is true.
And our guardian is both warrior and philosopher?
Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavour to persuade those who are prescribe to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being.

That is excellent, he said.
Yes, I said, and now having spoken of it, I must add how charming the science is! and in how many ways it conduces to our desired end, if pursued in the spirit of a philosopher, and not of a shopkeeper!

How do you mean?
I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. You know how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply, taking care that one shall continue one and not become lost in fractions.

That is very true.
Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, –what would they answer?

They would answer, as I should conceive, that they were speaking of those numbers which can only be realised in thought.

Then you see that this knowledge may be truly called necessary, necessitating as it clearly does the use of the pure intelligence in the attainment of pure truth?

Yes; that is a marked characteristic of it.
And have you further observed, that those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been


So what can we bring forward from The Republic?

– Math has universal application;
– Math attracts one toward reflection in general;
– Math draws the soul toward understanding being (metaphysics);
– Understanding mathematical calculation aids in the solving of various perplexities and curiosities (reconciling conflicting information);
– Math helps one in conceiving of unity and plurality;
– Understanding arithmetic helps one deal with the concept of truth;
– Math helps one deal with change;
– Learning general mathematical concepts helps one understand the subject outside of the realm of business and shopkeeping;
– Thinking about math fosters abstract thinking (or philosophizing);
– Math compels the soul to reason (for math is about logical order);
– Understanding math thoroughly helps one not get lost when dealing with partials or fractions (how very contemporary this sounds in light of the NYT article);
– Understanding math well has a trickle down effect of setting up quicker thinking about other kinds of knowledge.

I can imagine the skeptical reader, a math educator, saying that this is all known. So what?

My point is not that this isn’t known, but rather that these philosophical motivations are not translated to youth. In all my years in math classes, I don’t recall a ~consistent reinforcement~ of high-minded motivations for learning math. It was always about the practical: business, engineering, heat loss, friction, taxes, interest, etc. Math was never, in my experience, linked to humanistic thinking.

Is it not true that teaching is also coaching? Isn’t coaching about motivation? Is motivation only about loud or bright scientific demonstrations? Is motivation also about moving your deepest being, your soul, toward an end?

Perhaps if more students understand the high-minded, humanities-oriented reasons for studying math, there would be less need for the grind of practice. Practice is always necessary, but a rightly-motivated student would likely need less practice to get the movements down.

I want to wrap up this post with thoughts from this year’s Templeton Prize Winner, Michael Heller, a Polish cosmologist and Catholic priest. Here’s what he said in <a href="
“>an interview (bolds mine):


Question: In your statement today you said: “Things thought through by God should be identified with mathematical structures interpreted as structures of the world.” Does that mean that you see mathematics as the language of God?

Heller: In a word, yes. One of my heroes is [Gottfried] Leibniz, the great philosopher of the 17th century. In the margin of his work entitled Dialogus [can’t find] there is a short handwritten remark in Latin that says, “When God calculates and thinks things through, the world is made.” My philosophy is encapsulated in that.


Perhaps more students would be motivated to study math if they understood, in the depths of their soul, that math is more than mere calculation (i.e. skills) to be applied toward practical ends? Is it perhaps the case that, by focusing on drilling and methodology, that we’re putting the cart before the horse? Perhaps attempts to make math more prominent in education, American or otherwise, should be founded on philosophical and broad-ranging thought?

I realize that what I’m saying is counterintuitive: to gain better skills, we should stop focusing on teaching those same skills.

Maybe this is being done somewhere already? I expect so. But I’m reasonably sure, based on our perennial concerns, that it’s not being done everywhere. I look forward to your comments. – TL


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  1. TL: I've decided that students are like the American “patriots” in the 1760s–their objections are not precise, and they cannot be taken at face value. Tax their paper, and they'll complain that's internal taxation; tax their trade, and they'll say only the king can do that; demonstrate royal support, and they'll call the king a brute. It is my experience over the last year that has led me to conclude that people simply don't want to learn, and all this complaining about methods and boredom is so much nonsense. If teachers assigned video games, movies, soda, and candy, students would object.

    Personally, I concur with your mathematical experiences. I didn't need the practical applications of math, but I found trig and calculus to be very interesting–the idea that there could be a relationship between equations and shapes was profoundly interesting to me. And yet I became an historian!

  2. Ditto on the becoming-an-historian part.

    I'm so bummed—for you and the profession—over your experiences as an instructor this past year. – TL

  3. Nice post Tim. And thanks for the free publicity on my book. I agree with you that math education is best when placed in a larger humanistic framework. Alas, that is not how I learned math, even though I always enjoyed the problem-solving aspects of it.

    With regards to the 1950s debates surrounding Sputnik (all of this is in the book), Rickover was an elitist who wanted the best and the brightest to become more highly trained technicians. Bestor, on the other hand, wanted to extend humanistic learning to the democratic masses, an admirable goal. Bestor's problems were not in his intentions, but rather in his style, which was intemperate (he alienated many of his fellow educators). Also, Bestor was too vague: he was never very clear about what he meant by humanistic education, except that history was the core discipline os a liberal education.

    Cheers. Andrew

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