Right Wing Nation

Looking through my high school algebra text is really quite fascinating. The last time I saw this book, I was a high school freshman, and I don’t recall having any reaction to the textbook, other than reading it and working through the problems. My perspective now is very different, after having taught for over two decades and reviewed God knows how many textbooks.

This is New Math. Not New New Math, but Cold War space race New Math, the only math pedagogy that has ever been designed by mathematicians. Yes, it had its weaknesses, which I will get to sometime in the future, but none in common with the weaknesses of today’s math pedagogy. Whereas today, students cover far too many topics and only shallowly, New Math covered fewer topics to great depth (perhaps in some cases, too much depth, but that’s for another article). Whereas math today is self-consciously non-linear and non-logical, New Math was coldly linear and logical. Whereas math today has no methodology and allows Janie to figure out her own way to solve the problem, New Math drilled a strict, step-by-step methodology. But let’s turn to the textbook.

First, the text is accurately named: Modern Algebra: A Logical Approach. This is not just a math textbook. Formal logic is built into each chapter. I’ll address the logic-specific problems in the future. Now, I want to address the problems, and the method we were taught to use when solving them. By the way, not only did we not use calculators, but they didn’t exist (we learned to use slide rules in high school).

Let’s look at that example problem I scanned yesterday. Here it is:

Let’s look at the methodology, which is the same, and was in general the same as what we were taught to do all the way back in elementary school. One of the characteristics of New Math is that even in elementary school, we learned the basic principles of algebra (even though that’s not what they were called) because we tackled each problem with the same step-by-step methodology. By the time we got to freshman algebra, we knew what you did to one side of the equation you had to do to the other, and so forth. We just didn’t know what “algebra” was until we got to high school.

First, we declared the variable. This may seem needlessly formalistic, but it isn’t. What, say, if you have a problem with two variables? You have to solve one in terms of the other, then substitute. If you declare the variables, it’s much easier then not to get confused about which one represents which value in the problem. In other words, declaring the variable clarifies the goal of our solution, and focuses us on that goal.

Next, we set up the problem directly from the text.

Next, we multiply both sides by 35.

Let’s stop here for a minute. Why did we multiply both sides by 35–or more to the point, how would we know to multiply both sides by 35? Because we had thoroughly learned fractions and the principle of the LCD long before we got to freshman algebra. I mention this because both fractions and LCD are topics that have fallen out of favor in current math curricula. Our teacher would have, if working through this, asked us how we would have gotten rid of the fractions. We would have known that the way to do that was multiply both sides by 35, because we had already worked with fractions a great deal, and to a relatively high level of complexity. So to us, this step would not have presented a cognitive leap.

Again, the remaining steps were the same general methodology we had been using all through elementary school. There were no surprises. The only thing “new” about methodology in algebra was the use of variables. We had already learned the basic principles–without calling them algebra.

Focus on Part 2. From what I can determine, students today “check” problems by guestimating the answer. We didn’t. This problem exemplifies one of the two ways in which we had been taught to check our solution: We substituted the solution and worked it to see if we got the correct answer. Again, we had been doing this since elementary school. It wasn’t new.

The other way we checked problems was to work it backward, but in a different way (sometimes, there is more than one way to work a problem). For example, here is problem 12 from the problems I scanned:

The music department of Eastern High School bought 12 band uniforms and 3 hats at a total cost of $615. Later the department bought 3 uniforms and 2 hats at a total cost of $165. If the two orders were for identical items, what was the cost of each uniform and each hat?

To solve this problem, we would have first solved one variable (the cost of a hat, say) in terms of the other (the cost of a uniform), substituted it, solved for the cost of a uniform, and then repeated the process to solve for the other variable. This reinforces the cold, linear, logical methodology and thought process.

Note, however, that if you know the value of one variable and the total cost, you can calculate the value of the other variable without substitution (in other words, if you know how much you spent total and how much you spent on hats, then you can use simple subtraction to figure out how much you spent on uniforms). This is an example of a problem that has more than one route to a solution. When checking a problem like this, we would have checked it by taking another route backwards, instead of the same route we used when solving it.

Why? Because it reinforces that there is more than one logical progression of steps to the answer. Note that there is nothing “fuzzy” about this. Many problems can be solved in more than one way. New Math focused on the logical process and the mathematics behind the problem. Our teachers wanted us not only to understand the mathematics, but they also wanted us to see the different routes to the solution–because that’s when you really understand the mathematics.

My point here is the methodology and its consistency throughout the math curriculum. Going from elementary to junior high math, then junior high math to algebra was a smooth, continuous process because we focused on the linear, logical progression of steps, and we did this all the way throughout the curriculum. New Math was designed as a linear continuum, where new material was not some kind of a leap, but logically progressed from what we had already learned. We knew all about fractions and finding the LCD, so that was nothing new. We knew that what you did to one side of the “equals sign” you had to do to the other, so that was nothing new. We had already spent eight years of math classes in school extracting information from text and turning it into numbers (equations), so that was nothing new.

We probably covered more in less time in freshman algebra than we had before, though here, I’m working on very old memories, and have no elementary New Math textbook from the same era to check. But we didn’t do calculus in high school then. The college math track was algebra I, geometry, algebra II, and “senior math,” which was trigonometry and pre-calculus math. The closest we got to statistics was calculating averages, and the first I learned about probability was as an undergraduate.

Also, the cognitive, logical process we had been using since elementary school immediately transferred to other courses without our having to figure it out, even in high school. When I took my first year of chemistry, nobody had to explain the step-by-step logical process of balancing equations. The variables, constants, and symbols were different, but the process was exactly identical. The logic transferred. The same was true with physics. Nobody had to make that connection for us, because we had learned it so thoroughly.

And this leads me to what is, perhaps, my greatest objection to those who sneer at “drill and kill.” Forget being able to recall what 12*11 is, though that is important. It was precisely this “drill and kill” that taught us this logical, step-by-step process which we could take from math to chemistry and physics and yes, even writing. We learned an analytical way of looking at the world, dissembling a problem to its component parts, and working our way to a solution. And so-called “fuzzy math” can never accomplish that.