As you may or may not know, I ordered a copy of the book we used in high school Algebra I class many years ago (see here and here). Now indulge me, please. Imagine for a moment that you have been out of high school over three decades, that you have been teaching a substantial amount of time since, and you got a copy of the textbook you used in your favorite class. What would you do? What would you say?
There are many things I could say about this textbook, or by extension, New Math (New Math proper, space race math, not the current fuzzy math, which is sometimes confusingly called New Math). I could zoom in on exercises (I’ve done that), or the organization of the textbook. And do I take a student’s perspective or an educator’s perspective?
This article has gone through about fifteen false starts because I kept changing my mind about what I wanted it to be. At one point, I was going to use the textbook to critique New Math using the intent of the architects as my perspective, but decided against it–so there are no critiques of New Math linked in this article, and I purged them from my memory as best I could. I decided instead to take a unique perspective: Not what was New Math designed to do and how, but as both a student of New Math (with a textbook to refresh my memory) and an educator, what did I learn from New Math, what were its greatest strengths, and what were its greatest weaknesses?
The textbook is Pearson and Allen’s Modern Algebra: A Logical Approach, and I’d like to begin by giving the basic organization of the textbook, which I will use later.
- Sets
- The Number Line
- Expressions and Sentences
- Logic
- Operations with the Numbers of Arithmetic
- The Real Numbers
- Division of Real Numbers
- Factors and Exponents
- Polynomials and Rational Expressions
- The Real Number Plane
- Radicals
- Functions and Other Relations
Those who are unfamiliar with New Math will most likely be surprised by the appearance of sets, much less its place as the first chapter. But set theory was a fundamental component of New Math, and we started learning about sets in elementary school, then by the time we were in high school, Boolean algebra. New Math grounded nearly everything in set theory. For example, we never talked about the solution to a problem, but the solution set.
The second and third chapters are definitional, and along with the first chapter on sets, lead up to the fourth chapter: Logic. These four chapters form the foundation upon which the rest of the textbook (and course) rests.
This textbook rests on the premise that all mathematics is grounded in logic. This first section of the textbook ends with a chapter on logic, but logic problems continue in every chapter that follows. We did not just do proofs in geometry (though we certainly did many); we did proofs in all four years of high school math, using algebraic expressions, Boolean algebra, and set theory. Proofs–logic problems–were presented as an integral part of mathematics.
Here is an example proof from the functions chapter of the texbook (and I’m not doing this again, because this was an incredible pain in the ass):
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Prove:
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IF
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S
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=
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ax + by + c = 0
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and
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dx + ey + f = 0
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T
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=
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k1(ax + by + c) + k2(dx + ey + f ) = 0
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where
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k3(ax + by + c) + k4(dx + ey + f ) = 0
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k1k2k3k4 ≠ 0 and k1k4 ≠ k2k3, then the systems S and T are equivalent
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If you’re not a math geek, you’re shuddering in horror, I know. But logic problems were presented in many forms, not just formal proofs. Here is one from the beginning of the logic chapter:
What conclusion can be drawn from the following pair of statements?
John is a freshman
All freshmen study mathematics
It’s important to understand that mathematics didn’t take a back seat to logic. Only in the logic chapter do logic problems predominate. But logic is presented througout the book, both in examples and problems. Logic is the basis upon which mathematics rests, and logic is also used to lend further depth to explanations and definitions, as well as test learning (see the functions chapter logic problem above). New Math was "proofy" because of the fundmental importance of logic in mathematics.
New Math was also rigorous. It was, if you will, the antithesis of fuzzy math. We lived, breathed, ate, drank, slept, and dreamed expressions, inequalities, and equations. This isn’t to say that most of our problems, or math work, was of the "solve the equation" variety. At least half of the problems were so-called "story problems." But they rapidly became more complex as we advanced through the course. (I should probably also say that back then, we didn’t have AP or honors courses, but two course tracks, college and vocational, and though you could take courses from the other track–my parents forced me to take typing, and it’s one of the most valuable courses I’ve ever taken–most did not. So you were placed in one track or the other by the school advisor based on ability, and the students who would have struggled the most with this material were taking accounting instead of algebra–and that’s not meant as a sneer, by the way, just an explanation. Nobody then expected every student or most students to go to college; only a little over 14% of my graduating class went to college. College track courses were all rigorous, even English, though math was probably the most rigorous of all.)
If you glance through the text, you see that every example is presented using the same series of formalistic steps. We began learning this series–methodology, if you will–in elementary school. This formalistic approach to problem solving was highly detailed, often to the point of seeming silly to the math-inclined, but it had a purpose: We learned a coldly logical, step-by-step method for dissembling and solving a problem, and the same general methodology could be applied to any problem, not just 4th grade math or algebra or trigonometry. We were learning logic without realizing we were learning logic.
There was also a strong focus on process and understanding that went hand-in-hand with this formalism. In elementary school, for example, we learned long division by dividing into hundreds, then tens, then ones, instead of the traditional method our parents learned. It was the same long division and the same ultimate process–but the idea was to make sure we understood what we were doing.
As a student and an educator, I would say that the integration of mathematics and logic was highly successsful. It almost had to be, as consistently as the two were presented together. This was one of the strongest attributes of New Math. The other was the methodology, or rather the way it could be applied to any problem. I took this with me when I graduated, and it has served me very well since.
New Math had its flaws. Our parents hated it, because they didn’t understand the focus on process (because they were never asked to understand or even see the process). This was probably the one thing that contributed more to the death of New Math than anything else (I suspect our parents finally went over the edge when the government tried to get us to adopt the metric system–this and New Math was just too much).
New Math took excellent ideas and, I think, pushed them too far, almost as if those who designed the curriculum were writing it for future mathematicians (which they were, but that was unrealistic, and ultimately, destructive). I don’t object to rigor and feel there is far too little in math as it is currently taught, but as someone who has a great deal of curriculum design experience, I can only say that there was too much information that had to presented in too little time. When I look over the textbook as an educator, I wonder how anyone could get through all that material. We did, of course, but if you fell behind, you stayed behind. Math was a sink or swim course–like the two-semester ten-credit calculus and analytic geometry for math majors when I was an undergraduate. In English class, at least, from time to time you could tread water for a couple of minutes, but in math class, you barely had time to breathe. Too much. Way too much.
Although I understand the importance of set theory in mathematics, even I, major math geek that I was, did not understand the heavy emphasis of set theory in the math curriculum. Set theory was poorly incorporated, and at times, seemed tangential, if not sometimes irrelevant. And it goes almost without saying that this was yet another thing our parents hated about New Math, given that they had never heard of a set or an operation or a union or an intersection or a member.
New Math could be retooled to fix the problems, certainly. But the most striking about perusing the text has also been the most depressing. You won’t see math textbooks this rigorous used for undergraduates, save for math major courses. There is no way this material with this level of rigor and logic could be taught in today’s schools without a revolution in the educational system–and I say revolution because everything would have to change. The purpose of education. Expectations of students (and teachers). Attitudes toward student work, in and out of class. Assessment and teaching methods. Everything. And the revolution would have to change the whole system, from the first grade up. Even with such a revolution, the only way we could go back to this level of material and rigor would be to begin in elementary school, like New Math did. Throwing something like this textbook at high school students today would be a disaster. They would sink like stones.
And that’s a sad thing, when you think about it.
Adrian says:
I have two comments, one short and serious and the other long and nutty:
1) You’re right about it being sink or swim. Any genuine education is. The problem is that we think we can educate the masses on some set time frame. The truth is that it doesn’t even take intelligence, in the end, so much as temperament (not that you can be dumb as rocks or anything). Only a small group of people in society ever develop that temperament, and even fewer develop it before 18. And so, this brings me to the next point…
2) I pay you to keep the math. Yes, I know, you probably thought that you just magically got paid for being smart or something. In reality, whether you work for a state university or a private one, you and everyone else there are only able to stick around because of the particular role you play in society. In your case, it is to be a good custodian of mathematics which includes collectively establishing standards of rigor, producing new mathematics, and preserving the mathematics that already exists, and so on. But, that also means being the last stop for the rest of us for finding out what that math is. In fact, that is the whole point — you are to be a good custodian of mathematics and then you are to deliver it to us — the ones that put you there. Needless to say, I was little pissed off when I found out the truth about calculus, for instance. I think everyone else would be, too, if they knew what I know.
Needless to say, I am being more than a little sarcastic here, but the point remains. The point is that it doesn’t matter how far we think enrollment will drop or what the patrons really want or if they can handle it or what scientists and engineers “need”. By doing what we are doing now, I really believe that in a very meaningfully significant way we are sort of “defrauding the consumer”. (Actually, by “we” I mean “you” since I don’t work in academia — but, I am just talking in general about how education works….)
The point just is that math ought to be taught the way it really exists and not just passed off as so many spurious treatments of various topics the way it is taught these days. I might even be willing to accept a “Math Methods for the Physical Sciences” here or there. But, at this point, almost every undergraduate math curriculum out there is one big “Math Methods for the Physical Sciences”. What little legitimacy a methods course has in the first place is predicated on it being taught in the context of an otherwise rigorous program — not just as a follow-up on yet another methods course that came before it.
There was a time in which calculus was considered far too advanced for college freshman. Thank god the positivists came and saved us from those misguided fools, eh? (Don’t tell me. You probably like calculus — the course not the subject. Oh well — some of my best friends like calculus….)
May 22, 2007, 7:36 pmbird dog says:
I had the New Math for two years in jr. high school, and it set back my basic math education by four years. The level of abstraction is too high for 12 year-old brains, and it takes you off the basic path to trig and calc.
May 23, 2007, 7:04 amHighlights From the 120th Carnival of Education at matthewktabor.com says:
[…] It took a while, but now we can see Right Wing Nation’s analysis of the New Math. […]
May 23, 2007, 10:22 pmBarry Garelick says:
Your disdain for anyone who doesn’t think like you is exceeded only by your arrogance.
May 24, 2007, 6:04 amAdrian says:
I know — ain’t it great. By the way, I can’t help but notice that this is one of those kinds of retorts that paradoxically expresses the very same disdain and arrogance it would seem to disapprove of.
May 24, 2007, 11:12 am