The New Math on Goodreads
[I apologize for skipping last week – I just came back from vacation and wasn’t quite back in the game.]
This book was suggested to me by a colleague when we were talking about different pedagogical philosophies/emphases in computer science. An op-ed version was published by the NYTimes, but I found the whole book engrossing. As the title suggests, the book is about the rise and fall of the New Math reforms in the late 1950s and early 1960s. Motivated by Russia’s successful launch of Sputnik, the NSF funded a reform of the K-12 mathematics curriculum, with the idea that Americans need to move away from rote memorization of calculations, and focus on what mathematicians consider core to mathematics: the structure of objects and the relations between them. For example, grade school textbooks were written that first introduced groups of objects (ie. sets), then cardinality, then unions, before finally getting to addition. The idea is that schools should teach students what mathematicians actually do, as opposed to the memorization and symbol manipulation that are normally presented as mathematics.
There are a lot of details that I won’t repeat here, but the punchline is that the fate of the reform rested entirely on political favor. It garnered attention because of the Cold War, and the belief that communist Russian children can only follow rules – much like how they merely memorize the multiplication table or rules for symbol manipulation – while the free American children would be better served by learning how to think. That is, the teaching and learning of mathematics was politicized such that it was no longer about the subject, but about “intellectual rigor and discipline” and the ability to determine the Truth. The nationalistic fervor was how the reform was funded by the NSF, but also how it was defunded later, when the Cold War lost its urgency in the late 1960s, and also when parents were aghast that their children could not rapidly provide the answer to (for example) 6*7. The radical textbooks thus fell out of favor, although the reform was not entirely erased: the presentation of connections between geometry and algebra in high school mathematics trace back to the New Math.
(Side note: I have had much of an interest in political history, but I would read a thousand pages about how governments influenced science and education.)
My own personal interest in the book is simply the substitution of computer science for mathematics in the major theme. What caught my attention first was the emphasis of the reformers that students should learning the mathematical way of thinking, and not merely mathematical “facts”. Implicitly, this is also a statement that mathematics is about the thinking and not about the calculation. Third, designing the curriculum that follows the standard mathematical approach (ie. by first providing definitions and axioms before proving some result) was a pedagogy that I had not considered before. Finally, I was hoping that I apply what I learned to the debates about K-12 computer science education. I will address these interests in reverse order, and talk about both what happened in the New Math and my thoughts on its relevant to computer science education.
Let me first talk about the New Math as a political reform. I must admit that I was shocked by the degree to which education reform was political. The author, Phillips, took great pains to omit any discussion of whether the reformers were correct in their view that students should be trained to think. This was frustrating for me at times (see below), but it highlighted how it wasn’t truly the education of children that was under discussion. The obvious parallel to computer science is the rhetoric around why earlier instruction should be required. Yes, computers are increasingly important, but just as frequently cited is the fact that other countries – especially China and India – are producing more STEM majors (proportionally). In some abstract way, this parallels the Cold War story in how the US is reacting to a diminished nationalistic pride, measured either by Sputnik or by standardized math tests.
If we take the New Math reform as a parable, this says that computer science education may fall out of favor for the same reason: that the government could either accept that other countries rank higher in STEM education, or decide that the ranks are not meaningful, and simply decide that Americans should be more (say) “well-rounded” than other students and defund computer science. The extreme version of this is unlikely, since mathematics education was already in place before the New Math; it is possible, however, for politicians to consider the use of email or other such “information technology” as computer science and only provide funding for that purpose. Regardless, it is likely trivially true that computer science education is at the mercy of political rhetoric of why it is necessary, and the history of the New Math (and of the space program) hints at what could happen.
As I mentioned above, Phillips’ book is more about the political history of the New Math than its pedagogical merits. Nonetheless, the book describes the New Math as a curriculum, which includes not only the placement of set theory as central to mathematics, but also the need to move between different perspectives (say, geometric and algebraic) of the same objects. The latter has remained in high school mathematics, so I want to say more about introducing set theory before addition. Even as I was reading, the idea struck me as either utterly fantastic or utterly idiotic.
Phillips does not offer a definitive opinion either way, remaining neutral and only reporting what contemporary mathematicians, politicians, teachers, and parents said. Although some arguments were provided for why the historical progression of mathematics should not be followed – namely, that modern mathematical development provides much better grounded – I found the explanation a little shallow. Set theory was developed until the late 1800s because not directly connected to everyday experiences, and this abstractness made it difficult to teach even to college students, never mind grad school children. The equivalent in computer science would to be first talk about Turing machines and computability, before developing programming languages as symbol systems that encapsulate some of those ideas. Students would not even know what the “computability” means without prior experience with programming, never mind the abstract idea of a universal Turing machine (which to me a while to understand it was only a mathematical model, not a physical object). It’s possible that this curriculum could work – I have no data either way – but I was astonished that the reformers, even with the help of grade school teachers, had not planned for this obstacle.
The third aspect of the New Math (for me) is the discussion of what is the nature of mathematics. Phillips writes that the mathematicians have “neatly divided the mathematical world into two camps: one that envisioned the practice of mathematics as fundamentally divorced from the subject’s usefulness, and another that believed physical applications determined the course of mathematics.” Implicit here is that mathematicians will push whatever agenda they believe in onto their students. This particularly affected me, since I am still thinking about computer science curricula and, in a broader sense, what computer science is about. I think the divides in computer science – being both a science and an engineering discipline – are not quite the same as those in math. If I had to draw the lines, I would say that there are three things:
- Computational thinking, the ability to consider the structure and processing of information. The mathematical equivalent of this is what the New Math reformers focused on.
- Computational skill, the ability to write correct and efficient programs. The mathematical equivalent of this, at least at the grade school level, is to be able to do mental math, although I think it would extend to the ability to prove known theorems as well.
- Computational knowledge, having facts about how computers and specialized computer systems work. A mathematical equivalent might be the ability to explain what (say) Fermat’s last theorem is, without necessarily proving it or relating it to other contexts.
It’s uncontroversial that computer science is really all three, but if resources are limited to focus on only one of these, which one should be chosen? I am personally predisposed towards computational thinking as key, which is why I am sympathetic to the New Math reform. The issue – as occurred in the reform – is that people disagree on which aspects of computer science is most important. Their ordering in a computer science curriculum aside, it’s important to keep in mind that (contrary to the quote) that this is not a debate about the “core” of computer science, but about what should be taught to students.
The last perspective I want to take on the New Math is its philosophy of thinking as the goal of education, which is obviously related to what each instructor thinks is the core of computer science. Biased as I am towards computational thinking, the story of the New Math reform made me wonder if such a goal is attainable. It should noted that the New Math never promised to raise test scores (which made me wonder if a similar reform would succeed in today’s data-driven political culture), but that a new curriculum was necessary to develop students’ “intellectual rigor and discipline” – which is, of course, hard to test for and measure.
The question I don’t have an answer to is whether computational and mathematical thinking should be an educational goal at all, or if it develops naturally with increase in skill, regardless of what instruction is used in the classroom. My fear – despite explaining away the political causes of the New Math’s failure – is that there is no way to cultivate cross-domain transfer. I don’t have a good idea what this looks like – much as the New Math reformers can’t say what intellectual rigor looks like – and it’s possible that interdisciplinary computational thinking is not a thing at all. Still, I think there is a place for computer science in the liberal arts (that is, in the development of someone who might be considered a well-rounded human being) beyond learning about the objects capable of computation (ie. computers), and I think that is something worth setting as the goal of introductory computer science education. What the political history of the New Math reform cannot tell us is whether there is a way to confer these abstract benefits to students.
I will end with a short paragraph from the book about the theoretical and applied side of mathematics, and its ties to education and (through that) to society. I find it presents many of the concerns around the debate of mathematics and compter science education:
The issue was not reducible to a simplistic dichotomy between the backers of rote learning and those of conceptual learning. Mathematicians, teachers, and citizens have never agreed completely on the proper balance between those two poles, but few, if any, have thought that it was a matter of only one or the other. That sort of easy distinction is a red herring, drawing attention away from the more important assumption that learning mathematics counts as learning to think. One’s view of the new math was understood to be a view about the desirability of elite, technical, structural knowledge in the shaping of American minds.