Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Commentary: Algebra – yes or no?

Posted by mark schwartz on August 17, 2016

Commentary: Algebra – yes or no?

Andrew Hacker is proposing not teaching algebra but rather teaching math in a real-world context. He has created a conversation about the utility of algebra and proposes in his book “the math myth” that algebra is a cause of the loss of talent because many students can’t get past the algebra filter. I propose that it’s not the algebra content but rather the bad teaching of algebra. Consider this.

What does a student see when presented with 2(3x ̶ 1) = 10? As they have been taught, they will start rummaging through the rules for solving equations – order of operations, distributive law, operations with signed numbers, basic addition and subtraction – and anything else needed to solve for “x”.

This is sad. Algebra is an aid to help us see patterns and relationships and the equation presented above contains those things, yet that’s not what students have been taught to see. Poor teaching of algebra is the issue, not the algebra content itself. The argument about when will students use algebra in the future or in their daily lives can be asked also of history and poetry. If the premise is utility in the future, then many topics need not be taught at all. I’m going to shy away from the philosophical framework of the purpose of education, outside of stating it’s a way of introducing people to the social, economic, religious, literate and cultural concepts in which they will live their lives.

As humans, we tend to learn not individual facts but rather how these facts aggregate to patterns and relationships. Using math as the example, students in elementary school learn arithmetic – the four basic operations and maybe a few extended procedures, for example what is called long division. But what is really being learned? Are there patterns and relationships from which the four basic operations emerge?

Consider addition. The “new” concepts being presented in the common core curriculum have students thinking about numbers differently. In essence, they are taught that it is ok to rephrase the problem. They are presented with a specific procedure on how to do this but what I suspect is that they aren’t taught that they have some alternatives which also work. For example, 14 ̶ 9 can be rephrased as 5 + 9 ̶ 9, giving the answer 5. There are alternatives, which I have seen students create and use. One example is a student who realized that in the problem 14 ̶ 9 if 1 is added to both numbers, you see 15 ̶ 10, giving 5. This is a sound “theory” and if extended leads to a different conception of subtraction. This students can’t “prove” (or demonstrate) why this procedure works, but he knows it does consistently.

This type of thinking happens at the algebraic level as well. Hacker spends a lot time focusing on the idea of “rigor” and provides examples how rigor is factor which keeps students from passing algebra. Hacker misses what this student has done; Immanuel Kant stated that Math requires two things: imagination and rigorous logic and this student has employed both of these aspects, while hacker focused on only one.

I’ve strayed from the basic theme here. To repeat, it’s not algebra it’s the way it is taught. What’s the purpose of teaching math beyond the basics of arithmetic? Simplistically, our society needs to identify students with math capability who really will be needed as our society develops and grows. Mathematical modeling has become a core element across a wide spectrum of our lives.

Beyond this identification purpose, math allows us to examine a cluster of information and toss out the distractors, those elements that have no serious relationship to the core relationships in that set of information. This is what happens in what are called application problems. This idea alone is a very essential life value. Again, given a flurry of information, we tend to look for the essential pattern and relationship. This happens when driving. There’s a heap of stuff we see and hear but we pay attention only to those things that help us drive safely. Is this algebra? I believe some of it is.

Allow me the license of calling algebraic math something that we do automatically at incredible speed. The stuff we see in texts and the formulae and procedures are nothing more than slowed-down and recorded versions of what we do automatically. For example, I will start a math class by asking if there are any softball, baseball or basketball players and usually there’s more than one in the class. I ask one of them to stand and I toss them an eraser. They catch it and throw it back. I then ask if there was any math involved. As the discussion develops, it turns out that the class identifies a bunch of quantitative judgements involved. For example, how much speed must be used to get the eraser from the thrower to the catcher; how is the trajectory of the eraser tracked; how does the catcher determine where to place a hand to intercept the eraser’s path; how much energy must be applied to grip and hold the incoming eraser? There are other issues related to these core activities but the overall effect is that students start realizing that they employ “math” all the time. What’s more, they offer many more examples of activities that involve making quantitative judgements – driving, even walking up and down steps.

So, in my judgement, algebra is a tool we use to help us identify, connect and summarize quantifiable relationships and if taught this way, I tend to believe people would actually enjoy it.

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