The good news
As it turns out, I have managed to spend a good bit of time reflecting on the educational process over the last semester-- just not blogging about it until today. Rather than keep you in suspense-- because I know you could be watching YouTube videos of cats in amusing situations if you weren't reading this-- I will cut to the chase.
Fortunately, I have figured out what the problem with the American mathematics education system is. In fairness, I should point out that I am not unique in this. Or even ahead of the curve, actually. To whit, have a look at the following TEDx talk of Gary Stager.
Additionally, hop on over to Generation YES and have a gander at Sylvia's thoughts on the Khan Academy.
So, what's the deal?
Simply put, the problem that we have with American mathematics is this. From my bridge, I see basically two reasons that motivate people to learn about any particular thing. The first, and most natural, is that the learning process is exciting and enjoyable. Ever wonder why teenagers who won't learn to factor trinomials can effortlessly survive more than 20 waves of a zombie attack?
I have played Nazi Zombies into the 30s, and I can tell you that factoring trinomials is much easier. So how do guys like Syndicate get good at it? The same way you get to Carnegie Hall. They practice. A lot. Jazz musicians call it the woodshed-- supposedly named after the location of Charlie Parker's intensive practice sessions. So, here's the big question. Why do they practice so much?
The obvious, yet often overlooked answer is because it's fun. People who obsessively practice their craft do so partially because they want to get better, but also because the process of getting better is fun and exciting to them. You tell me, is the following fun and exciting?
Notice a few things about the preceding video. First, Mr. Jones does an excellent job of presenting his example in a clear, concise, and easily understood manner. Even though voluntary response sampling is rather suspect, the comments below the video certainly don't contradict this conclusion.
Second, notice that the example is presented almost entirely out of context. We can reasonably assume that Mr. Jones has provided some of that either in class or other videos, but there really isn't anything motivating the mechanics of this factoring technique other than an amorphous desire to find the monomial factors of the indicated polynomial.
And that brings me to the second thing that motivates people to learn something. What will I be able to do with this? Here's one of my favorite examples.
One of the nice things about this presentation is that Sean purposely avoids the gory details of data mining and exponential/logarithmic regression. This is a good decision on his part for multiple reasons. First, it would take way too long to provide enough detail to mean anything to the audience. Second, these kind of details are really boring. They would completely cut the legs out from under his presentation. The only purpose those details would serve would be to convince the audience that it's a complicated process. Guess what. They can already tell.
The problem with mathematics education
The problem I'm dancing around is that our educational system completely ignores the two things that would motivate students to learn mathematics. Our American teacher-centered, didactic-ueber-alles approach designed for the sole purpose of appeasing the Gods of Standardized Testing is not only educationally unsound, it also gives mathematics a bad name.
But how did this happen? Simple. We allowed it to happen. The more proper question is why did this happen? Unfortunately, the answer to that question is also simple. We are lazy. It is easier to assess whether students can factor trinomials than to assess how well they can creatively solve real world problems that involve the solution of a second-degree polynomial equation. Perhaps more importantly, it is much easier to assess such things in a standardized testing environment.
Our misplaced focus
According to Don Small, writer of one of the more highly regarded reform textbooks for college algebra, the Problem-Solving/Modeling Process consists of three steps illustrated like so.
The three steps are model creation, analysis, and interpretation. In Sean Gourley's TED talk, he focused on model creation and interpretation. And these are the exciting steps. This is the sort of thing that causes people to say, "That's why we learned this? Awesome!"
But we have chosen to focus on the analysis step in our classrooms. While this is an important step, we are ignoring both steps that involve interfacing with the real world. The result is all over our contemporary mathematics classrooms. Ask any high school math student to come up with a real world example where he or she would need to use algebra to solve a problem.
And the worst part is that the analysis step is the part that people are worst at. Consider the following Wolfram Alpha widget created by yours truly.
If you have a job factoring polynomials, guess what. You've just been replaced by a computer. But guess what else. Nobody has a job factoring polynomials. In reality, factoring polynomials is a small part even of an algebra teacher's job. Similarly, it's a small part of solving the analytical step of any real life problem involving a polynomial function. And yet, we act as though this skill is the lynch pin of higher mathematics.
Imagine if we took the same approach with teaching English. A person who still muddles up the difference between the nominative and objective cases would never be allowed to write her memoirs. But then the world would be a safe place for people who say, "It is only I."
And now for the bad news
Of course, the natural question is "What do we do to fix this problem?" The bad news is that it won't be easy. We have allowed this twisted view of mathematics grandfather itself into our culture as well as our educational hierarchy. Being an industrial engineer, I am a strong advocate of Deming's principles for trans-formative change. Rather than summarize all 14 points, here are three that make for a good conversation starter.
Point 1: Create constancy of purpose towards improvement. This is the big one. "As they say on TV, the mere fact that you realize you need help indicates that you are not too far gone."
You see, it will be hard to convince the powers-that-be that our system is a) broken and b) can't be fixed by more standardized testing. But we as educators have to accept that we can't simply put our heads down, change our own classrooms, and expect the system to fix itself.
Point 3: Cease dependence on inspection. The goal of total quality management (TQM) is to reduce variation and constantly improve the product. If we are honest with ourselves, we will accept that the purpose of standardized testing is to identify "defective products"-- that is, deficient students. There will be no need to waste countless hours of instructional time on standardized testing if we build quality into the product. Identifying and quantifying mistakes is not a way of life in a system that is designed to produce a quality product. Of course, a certain amount of testing is needed for assessment purposes, but it shouldn't be the tail that wags the dog.
Point 10: Eliminate slogans. Deming had a rather cynical view of management, it would seem. In the world of sub-par management, people make mistakes and need to be hassled until they straighten up and fly right-- or replaced with different people. In Deming's worldview, most mistakes are results of a poorly designed system. But hassling employees is easier than improving the system. The short version is, "Blame the system, not the people."
Rather than wallow in helplessness, it is my intention to contribute to the current dialogue in reform of the US mathematics education system. It will be a long, slow road to modernizing our system, but it is incumbent on us as educators, parents, and citizens to insist on nothing less than a first-class education system.
What we have in our education system-- particularly in K-12, but colleges are not exempt-- is a culture where "bad at math" is not only socially acceptable but also the cultural norm. Simply put, we have decided as a society that we are willing to accept our substandard results. And the reason is that we are unwilling to agree that the system is fundamentally flawed.