The New Math as Politics

The New Math, by Phillips, is subtitled as a “Political History”. The New Math movement was an attempt to rejuvenate the teaching of mathematics in secondary schools and ultimately in the Primary schools. To the proponents of this movement there was staleness in the teaching of mathematics which was reflected in their belief that it was merely an exercise in memorization and lacked any true understanding of the elements of mathematical thinking.

For example, most elementary school students had memorized multiplication tables, learned fractions and division as a mechanical process, and dealt with “word problems” with abject terror. Mathematics was taught as if it were some mechanical set of processes that one set to memory and the students failed to have an understanding of what they were doing.

On pp 13-15 the author provides some baseline information on the creation of the New Math. The intent was to imbue understanding of what they student was doing not just a set of rote manipulations to produce an answer. There was an abject fear that memorization was futile and that the expression of what the processes were became the goal. Students must learn why they could multiple 3 X 2 and then 3 X 2+4 and find an answer. They must understand the processes of manipulations, at the risk of never memorizing that 3 X 4 are 12.

On p 27 the authors also discusses some of the political movements pressing for improvement. It mentions Rickover and Doolittle as applying their influence to promote improved education due to the need seen in WW II to “educate” many enlistees to be able to perform what were technical tasks. For example, to train an enlisted sailor in Fire Control Systems or in Radio or Radar, there was a prerequisite in understanding Geometry, Algebra and Trigonometry. Many schools never taught the skills to students, and thus the need to re-educate. Thus on one hand the need was to have a better baseline education and on the other hand to attempt to emphasize fundamentals as the core of that education.

What then were the principles? It all depended on whom one spoke with. What happened was that a group of “mathematicians” saw that they needed understanding of set theory, complex rules of algebra, base n systems of numbers and the like. This then changed the core of many of the courses.

The attack then also went to the teachers themselves. Teachers were all too often the product of teachers colleges, often state run institutions, to produce individuals to manage the utilization of the state mandated texts and managed by state mandated exams. The Regents of the State of New York was in many ways a classic example. Geometry was defined by them and each instructor taught that material.

The author on p 31 refers to Hofstadter and his book on “anti-intellectualism” and the argument that teachers had become “estranged” from academia. In reality the Hofstadter book is a polemic of a Columbia Professor against what he perceives is the “anti-intellectuals”, namely the Republicans, Catholics, and anyone opposed to his politics. In Hofstadter’s book on pp 138-141 is one of his many rants against Catholics and the Church, ironically because the intellectual were also strong supporters of Kennedy.

Thus the intellectualism at Columbia and of Hofstadter was at best problematic and use of the author of Hofstadter as a baseline is also problematic. Likewise, for example, on p 394 of the anti-intellectual treatise of Hofstadter, Hofstadter calls the Partisan Review the “house organ” of the “intellectuals”. It is in William Barrett’s writings of his time at the Partisan Review that he noted is strong Communist bent. Thus, using Hofstadter by the author is an attempt to set up the New Math as the “intellectual’s movement” and then subsequently to use this as the basis for arguing that its demise was the result of some right wing attempt to defeat it appears as a bit of a straw man strategy.

One of the problems I have is that the author fails to clearly identify what he means by New Math and what the New Math was. In a classic paper by Feynman in 1965 criticizing the New Math he states:

Many of the books go into considerable detail on subjects that are only of interest to pure mathematicians. Furthermore, the attitude toward many subjects is that of a pure mathematician. But we must not plan only to prepare pure mathematicians. In the first place, there are very few pure mathematicians and, in the second place, pure mathematicians have a point of view about the subject which is quite different from that of the users of mathematics. A pure mathematician is very impractical; he is not interested - in fact, he is purposely disinterested - in the meaning of the mathematical symbols and letters and ideas; he is only interested in logical interconnection of the axioms

This was the problem of the New Math. A radar technician does not need to understand set theory to understand the probability of a false alarm and the signal to noise ratio. Specifically Feynman states:

What is the best method to obtain the solution to a problem? The answer is, any way that works. So, what we want in arithmetic textbooks is not to teach a particular way of doing every problem but, rather, to teach what the original problem is, and to leave a much greater freedom in obtaining the answer - but, of course, no freedom as to what the right answer should be.

Feynman, a product of the New York City School System, and then MIT and Princeton, is correct. His own technique was to intuit the answer and then find the framework to support it. I doubt he ever used a single element of set theory. The conclusion even in 1965 was the core of the New Math was flawed as a pedagogical approach. It in fact was intellectualism gone astray.

On p 103 the author describes some of the texts which resulted from this effort. Take the Moise and Downs text on Geometry, and compare it to text by Wells in 1908. Wells was brief and to the point and one walked away understanding enough geometry to measure angles, understands triangles and the like. The Moise and Downs book makes the development of Proofs impossible. The simple example on pp 190-191 (of the 1982 edition) is a classic obfuscation of the obvious, a proof of the existence of a perpendicular line.

Kline also discussed the shortcomings in his superb book “Why Jonny Can’t Add” Mathematics is a tool for almost all of its users. It is “learned” by application. No user of an Excel spreadsheet would benefit from the New Math.

The author then proceeds to discuss the political opposition from the right to New Math and the Back to Basics movement. On p 145 he opens the Epilogue with the statement “Opponents of the new math won.” In reality the weaknesses of the New Math caused its own demise. With the like of Kline and Feynman against it than what chance would it have? It just did not work.

The last sentence is also worthy of comment:

“Yet math classroom will remain a political venue as long as learning math counts as learning to think. Debates about American math curriculum are debates about the nature of the American subject.”

It is not clear to me what he means by the phrase, “nature of the American subject”. Is subject the material taught, the individual, some broader idea not explained?

Overall this book has two tales. One is the intended one of the birth and death of the New Math. It has not totally died but still is found floating around a bit. It is also a tale of pedagogy in the state run schools and who decides what students must know and why they must know it. For most of us, mathematics is a tool, it is a way to express facts and explore reality. My day is often driven by mathematical realities, albeit those of an engineer, pedantic, utilitarian, and lacking in questioning principles. I assume solutions exist; I do no really pay attention to uniqueness theorems, and use them as a tool kit to gain knowledge. Almost all who rely on mathematics do so. The pure mathematician asks fundamental questions, questions about fields, convergence, existence, measurability and the like. They do affect reality from time to time. But rarely, yet when the do the impact is significant, just look at the analysis of the Wiener Process in dynamic systems, and the Ito integral.

However, for the most part we want students to understand technique, to a point. Out of the mix will come the engineers, physicists, and yes the mathematicians, the very few mathematicians who have the unique capabilities to abstract thinking.

Overall the book is a reflection of the political processes surrounding education. This has been all too common especially since the advent of Dewey and the education movement he was so prominent in. This book is a useful exercise in grasping with the tendencies to make material relevant on the one hand and a facilitator to understand society for good citizens on the other. The book has certain weaknesses but it also has certain positive points. It allows one to see how the arguments can be made. One may then ask in a similar fashion; if these same arguments and this same process will follow through with Common Core?

The book also shows the break between the academic practitioners and the practitioners who teach the subjects. At the University level we still see a great deal of freedom. At MIT for example courses change on almost an annual basis as the technology and science progress. The need for “standards” is non-existent. At the secondary level this is hardly the case, due to the size and complexity. That perhaps is worthy of a similar study.

But one of the important observations here is the movement of "protected" groups like the "mathematicians" who may very well have been used by political operatives to gain deeper control in schools. As indicated colleges and universities are somewhat protected. But if the Government extends its control to Community Colleges we can easily see the movement of Washington control to move there. That may very well be the unintended less of this book. Namely, beware the Politician, they ultimately want to control everything,