It finally happened, WorldTeach put up a math focused module for me to do this month! Yeah, I know that's not actually exciting for anyone except me, but let me have my moment, ok? Modules are things that I have to do once a month for WT. Basically they're like the education assignments I used to do in college. They consist of a case study, an accompanying article, and reflection questions. Since all of the other volunteers teach English, most of the modules are very English focused. While they're not completely irrelevant to my classes (a lot of the broad themes and issues apply to all subjects), they're also not particularly helpful to my teaching experience (especially since my ed background means I've learned about/discussed a lot of the topics before). But, this month, WT finally released a math focused module, and I was excited to see what it would be about. The topic was "Balancing Factual, Procedural, and Conceptual Knowledge in Math" and the accompanying article basically put into words everything that's been rolling around in my brain about math education for the past four years. What it boils down to is that America (and many other countries) are not doing a good job at the balance of the three knowledge types. (Quick break down: factual knowledge--the quick recall of basic facts, i.e. multiplication tables, squares have four sides, 5 is bigger than 2 etc, the 'what'; procedural-- the steps you follow to solve a problem, the 'how'; conceptual-- the reasoning behind the procedure and the facts, the 'why'"). Reading the article helped me to either explain, or justify many of the frustrations I've encountered while teaching math both here in Namibia and back home.
The statement "I hate math, I'm not good at it" is one that has permeated our society. I hear it all the time, from people of all ages, all background, and all over the world. It is a statement that is said with a weird sort of pride, asserting that the speaker was a member of the every growing club of people who couldn't wait to graduate high school and leave math behind. It has gone so far that the New York Times published an op-ed where the author claimed that we should no longer require students to learn algebra, or really much math at all, since most people don't use those skills post-graduation. I hear this argument a lot, and it's one that actually causes my blood to boil, until I calm down and try to look at math education from an outsider's perspective. When I do, I start to see their point, and will conceded that math education the way it stands today is really not benefiting kids as much as it is hurting them. The fact is, we are completely failing at actually teaching kids how to do math. Instead, we teach them how to blindly follow steps. Not exactly the critical thinking skills we like to see in our citizens.
Math education has gone through a lot of phases throughout history. The ancient Greeks (think Euclid, Pythagoras, etc) focused mostly on concepts. Euclid's Elements- the standard for all geometry- hardly uses any numbers at all, it's all about the concepts and the proofs. The majority of math education in early America was factual, (the 'arithmetic' in the 'three R's'). Children learned how to count and how to do the basic operations. Slowly, we have moved to a system where the emphasis is on procedure-- how to follow steps to get a desired answer. The factual knowledge has still stuck around for the most part, but the focus has moved. Concepts, while still present occasionally, are often not really delved into.
The driving force for this push to procedure is, in my opinion, standardized tests. When a year's worth of learning comes down to the results of one test, the incentive is to teach procedure. Make sure the kids know the steps to follow to get the right answer, we don't care if they know why they're doing it. Because the standards for each year are so full of content, there is no time for exploration of concepts-- everything is watered down to "here are the steps: go." Students are so singularly focused on what the right answer is, that they stop caring about the why and the how, which are the most important parts of math.
Now, I'm not saying that we should go back to a Greek way of thinking and teach only concepts, that would be extremely impractical and ineffective for the world we live in today. However, we need to make conceptual knowledge a bigger part of math again. True understanding of math cannot exist without factual, procedural, and conceptual knowledge.
In the past few decades, factual knowledge has been on the decline. It's something I've noticed even since my days in elementary school. Raise your hand if you remember doing Mad Minutes in 4th grade to memorize your multiplication tables. My guess is that most people have their hand up. Would it surprise you then to learn that the majority of grade 8 and 9 students that I've taught needed a calculator to do 6*7? The memorization of facts has become demonized in a lot of education circles recently- earning the title "drill and kill". While I agree that memorizing facts is not the best way to learn, people always forget the old nutrition motto "everything in moderation". This is true of education as well. Sure, giving kids a history text book to memorize is not going to help them learn about the Revolutionary War, but there are some things that have to be memorized. Parts of speech, math facts, the date we signed the Declaration, the chemical make-up of water (H2O for anyone panicking). Here, memorization leads to instant recall which becomes very important when dealing with bigger problems. For example when our brains don't have to spend precious time and energy solving the basic problem, like 6*7, it can devote more time and energy to solving the equation. This not only makes math easier, but also less frustrating and therefore more enjoyable. So even though we live in a world where we have the ability to look up or calculate basic facts at our fingertips, a base of factual knowledge is still beneficial, and actual crucial, to deeper understand of more complex topics.
Procedure is important too, as much as it may seem that I'm hating on it. Having intimate conceptual knowledge won't do you any good if you don't know how to use it. Procedure is how math can be made accessible. It's human nature to want to have steps to follow to reach an answer. The problem comes with complete dependence on procedure- when procedure is taught with no explanation as to why the steps are followed the way they are. Teaching students to blindly follow steps is detrimental. What happens when they encounter a problem that requires a slight deviation from the steps? If students understand what the steps are trying to achieve, they can use their knowledge of the procedure to figure out what to do next. If they have just memorized the steps, they will get stuck and likely give up.
I saw on the internet a while back a picture with a statement "I forgot the Law of Cosines on a test, so I used right triangles to solve the problem another way". The picture was of her notebook, showing her work. It turned out that she didn't solve it 'another way', she actually derived the Law of Cosines for the particular problem she was doing. Replace all her numbers with variables and there's the proof. This girl used her intuition to get her out of a jam. Instead of giving up when she forgot a formula, she said, 'let's see what I can do with what I know'. I was sad to realize how happy seeing this made me because, unfortunately, most students don't think like that-- if they forget a formula, they skip the problem. Most people have this idea of math being all about following a precise set of rules to arrive at a correct answer. Math is seen as rigid, calculated, and lacking all creativity, when in fact, creativity and imagination is crucial to math. There are a multitude of ways to arrive at every answer. Exploring these different ways, how they're different, why they all lead to the same solution, that's what math is. This is lost in today's education system. This is what happens when you teach the how and not the why.
This lack of conceptual learning also takes the excitement out of math. Math is a beautiful study of the patterns in nature. In math, things are even when it seems there is not real reason for them to be. When we're just told that they are- when concepts are just handed down as Mathematical Truths with no exploration- we take them for granted and we loose the beauty in their being. Let me give you an example from my own education. In Calculus, there's something called the Fundamental Theorem of Calculus. Sounds like a big deal, right? It is. What it says is that the integration of a function can be reversed by finding the derivative. Without getting too technical, they're sort of like opposites. This is a HUGE deal. Two things that seemingly shouldn't be related are actually very closely related. Now before learning about the FTC, we all anticipated it to be a big deal, but were very disappointed when we actually did learn it. Here's why: we were taught how to integrate first, and the way you do that is by finding the antiderivative (the opposite of the derivative). So in our heads, an antiderivative and an integral were the same thing by definition. So when we got the FTC that told us that the integral is actually equal to the antiderivative, we were all like "well yeah, duh". It wasn't until I was a TA for Calculus in college that I realized how important, amazing, beautiful, and well, cool, it is that the area under a curve (the integral) is the 'opposite' as the slope (the derivative) of the curve. Had we learned about the concepts and been told about the FTC first, and then discovered how to find an integral using the anti-derivative, I would have been fascinated, and it would have made a lot more sense.
Another thing that happens when concepts aren't fully understood is that things get misused. The most common place I see this is with the equals sign. The concept of equality- things being equivalent in value- has gotten lost with most students today. The equals sign has come to mean "put answer here". (In Namibia, this is exceptionally true. Learners use equals signs before writing the answer in any subject. If the directions say "Write 'I am eating breakfast' in the past tense." they will write " = 'I ate breakfast.'") This means that kids are constantly misusing equals signs, claiming things are equal when they are not. It also means that solving equations, when there is already something on the other side of the equals sign, gets them very confused. The typical way to teach the concept of equations is with the analogy of a balancing scale. I had another math teacher here tell me not to bother with the analogy because they just don't get it. When I asked what to use instead, he told me just to give them the steps. "THIS IS WHY THEY DON'T GET IT!" I wanted to scream, "BECAUSE NO ONE TEACHES CONCEPTS!" Instead I bit my tongue, but I sort of wish I hadn't. Now I am subjected to paper after paper where equals signs are grossly misused and I'm not sure how to fix it because, once again, exams are in full swing and I am simply out of time. "Maybe next term" I think, and it gets added to the mile high pile of topics to address in the two months we have before the end of year exams.
I am scared for the mathematical futures of this generation. Already college students are fleeing the math and science fields in droves. (A recent headline I found grimly amusing: "Math, Science Popular Until Student's Realize They're Hard" (and no, that's not an Onion article...)) If we keep teaching math by drilling procedure and eschewing concepts, we will be doing a dangerous disservice to the future of our nation. We need to bring conceptual knowledge back into the classroom, but to do that, we need actual math teachers driving math education policy. We need to tell those writing the standards that we've had enough of mile-wide-but-inch-deep curricula and should instead be teaching fewer topics more in depth. This is the only way we will get back to teaching actual math.
For anyone interested in this topic check out the article "Is It True That Some People Just Can't Do Math?"