I have been mulling this question over for a while now, since last summer at least. It is a offshoot of the time I spent working with Exeter materials at an Exeter summer institute, and if anything the question has grown in my mind to the point where I must answer it for myself and act on it.
Here is the newest version of my question: If much of what and why we teach math the way we do is arbitrary, then why not change to make it easier to learn?
Now of course, there is a HUGE set of presuppositions / assumptions just in asking the question. First, I assume that much of what and why we learn math is arbitrary. Well, I don’t think I am that far off the mark. Let’s look at Algebra 1 as a course first. Honestly I am in good company with this thinking.
I agree. Algebra 1 has a huge failure rate because it is very abstract, meaningless content. We don’t really ever see why we are doing it, we are just learning to manipulate variables and constants around. Grant gives a small part of Lockhart’s Lament (pdf), and it is worth linking to (and reading) completely. Again, the ‘why’ of ‘why do we teach it this way’ is completely arbitrary. Which is why we get political science professors arguing that Algebra is unnecessary because it is hard in the NYTimes. That Algebra is a gateway topic is not in question. It is. The content is essential for future jobs and future success.
If we look a the content, we see arbitrary stuff all over. Heck, just look at the old y=mx + b. Why “m”, why “b”? There is no good answer. Do a google search and get 33 Million hits, none of which can definitively tell us why. I find the answer from the Drexel University Ask Dr. Math to be the most grounded answers, which you can find here for m, and here for b. And I LOVE the answer given here by math historian Howard W. Eves in Mathematical Circles Revisited (2003), where he suggests that it doesn’t matter why “m” has come to represent slope.
“When lecturing before an analytic geometry class during the early part of the course,” he writes, “one may say: ‘We designate the slope of a line by m, because the word slope starts with the letter m; I know of no better reason.’ ” via
I totally agree. In other countries they use other variables for the same meaning (scroll down), so clearly the “agreement” that we all must use the same convention is not universal. There are so many conventions in math that are purely arbitrary. Since they are arbitrary, we must feel free to throw them away when they interfere with good learning and teaching.
So the second question, and a very important one, is: How could we teach math differently to make it more understandable?
One thing I think is important is to connect the vocabulary / language / processes of linear functions with other polynomials / transcendental functions. After all, look at the amazing similarity and simplicity of understanding the transformation processes.
Don’t believe me that every single function listed has exactly the same transformation rules? Try this little GeoGebra applet I whipped up. Think about that for a second. When I have shown this to math teachers I get two reactions, “Well duh” and “OMG, I never thought about these like that.”
The teachers who see this as obvious are the teachers who are much more experienced and have taught for many years and have spent the time looking at the math. The crazy thing is that very few teachers have told me they teach this. Why not? Because it isn’t how they were taught, it isn’t how the books phrase it, it violates the conventions of math teaching. So they know it, but ignore it.
And don’t get me wrong. I am not suggesting this is where we stop teaching, we use the exact similarities as a springboard to bounce into the other types of functions. If the learner of math knew this with strong understanding, then the rest of algebra becomes a close examination of each type of function (which is all the different algebra courses are anyway.)
The Common Core Curriculum has mixed up the order of teaching these functions, but the fact that all algebra is just an examination of the skills (which are essentially the same, find ones by multiplying by the inverse, find zeros by adding the inverse) needed to solve, graph, and understand how each function is used.
The last question I have is: Why do we, as teachers put up with this, and what are we going to do about it?
I think the CCSS gives us the perfect opportunity to demand better from textbook publishers as well as our professional development opportunities. We, as teachers, need to be willing to throw the ‘conventions’ away and teach better.
Will it make a difference to the failure rates of algebra 1? I don’t know. but how can it hurt? How can it hurt to strongly connect all of algebra through trigonometry with an unbreakable thread so learners know that what works for one type of functions will work for every other type of function too. It shatters the concept that Chapter 3 doesn’t relate or have anything to do with Chapter 7. That is what learners think now.