Of course, nothing against my fellow panelists, each of whom were accomplished scientists, mathematicians, and technology experts. Their only failing was making a singular assumption that is shared by many math educators--to assume that students don't WANT to do math, NEVER want to do math, and simply need to be encouraged to keep plugging away at the tedious, awful chore that is math so they can get to the fun stuff. Sadly, when you get a student who's run up against the same brick wall of math every year without any improvement, or worse, clarity, that's probably a safe assumption to make. But it really really really doesn't have to be this way.
Being someone who straddles the line between science and art, I'm surrounded by both math lovers and math loathers. Many of the math lovers would say they had a predilection for math from an early age (I'd put myself in this group) and just received the appropriate challenges and encouragement throughout their development. Their favorite areas of math fall all over the board (I'm big on algebra, myself). The math loathers are an interesting bunch, because many seem to all fall into a single category: people who only get geometry (and are often pretty dismissive of algebra). In other words, these are individuals who need math to be concrete.
It seems strange that algebra and geometry, two areas of math that cover the same types of functions, might generate such divisive reactions. But when you see this:
c2 = 32
+ 42
Solve for c.
But then see this:
I'd be willing to bet you'd see a lot more people who could solve that function after seeing the triangle, because suddenly they can make sense of what that variable c is supposed to stand in for. You've taken a completely abstract concept and paired it with something concrete that students can see and understand the reasoning behind (FYI: high school math teacher Dan Meyer has a wonderful TED talk on this very concept here: Dan Meyer: Math Class Needs a Makeover).
So my humble request of all you math (and STEM) educators out there is to PLEASE understand that there are very straightforward reasons why your students just don't get math, and forcing a single perspective on them is exclusionary and unfair, as is punishing them by having them do more homework because they aren't getting it. Your math curriculum shouldn't exclude concrete thinkers in favor of abstract thinkers for the same reason that you shouldn't exclude students of any race, color, or creed.
I mean, duh.
Thanks for writing this. I have a great fear that I'll instill my fear of math in my daughter, who thankfully seems to have quite an early head for it.
ReplyDeletePart of my problem was that for my early years, I wasn't at schools where they stressed math. I swear, I was still doing multiplication in 6th grade. I can multiple like a m*f*cker.
Also, my parents are not great at math either (not helped by my mom always talking about how she's bad at math). But they did invest for years in private tutors to help me, which is the only reason I didn't completely fail every math class I ever took.
And in college, I had to take a math class so I essentially took math for stupid people and, truly, only passed because the T.A. took pity on me. I attended several study seminars before the final and I couldn't have passed it because I couldn't even try to answer half the questions. He knew I would never pass and socially promoted me.
So, that leaves me with, why? I don't know how to describe it other than to say that it never made sense to me. And because it never made sense how any of it was related to any other part of it, or the "real world," it was even harder to remember by rote any of the equations or rules.
Now, I want to encourage Elyse as much as possible while being as silent as possible about my math disability so it doesn't hold her back, as a student and especially as a female student.
Thanks for the comment! You bring up some really important points, like lack of uniformity in math education across the U.S., lack of parental ability to help with math homework, and schools that require math courses but aren't really going to make the effort to ensure comprehension. These issues plague the U.S. educational system, and nobody has thought of a way to resolve them yet. It just seems like there is opportunity for failure every step of the way, and that is troubling.
ReplyDeleteI admit that I really benefited from having a father who was a mathematician and statistician, who really tried to impress on me the fact that the functions/equations/rules didn't matter as much as the process and the logic it took to solve math problems. Thanks to SOLs, American math education doesn't have the same emphasis.
Keep an eye out for a forthcoming post about how we can keep kids (girls, especially) interested in math. It's a bit of a personal issue for me too - my boyfriend's daughter has told me that her favorite subject is math, and I know I'm the one that needs to keep her engaged, as the rest of her family will be unable to once she gets to middle school.
I agree with your point about making problems more concrete. I think that's important. At one point I tutored a highschool remedial algebra class and the students could do the arithmetic, but were mystified by the leap to variables because of the abstraction level they introduced. So trying the abstract to concrete problems is an important step, but I don't think it goes far enough.
ReplyDeleteI really feel that one of the big problems in education in general and math education in particular, is that it's generally taught as a fait accompli rather than as part of a process of learning and discovery about the world. I think this is really true for primary education in just about all subjects but math may suffer from it more than most.
Math is mostly taught as a set of rules that students must memorize in order to be able to crank through fairly mechanical problems. This partly has to do with making it concrete, but cloaking the banal arithmetic behind word problems really doesn't suffice. I'd like to see the teaching as more of an exploration of the problem and how we arrive at solutions. Math is largely an approach to problem solving more than it is any specific technique. The Pythagorean theorem is a great example here; the picture may help, but starting from a real problem (e.g., how tall does my ladder have to be), recognizing the step of abstraction to all right triangles, and finally working through some of the many proofs of the theorem would really cement not only what the theorem is, but also why it's true, and how theorems are created.
This might just be my personal bias as someone interested in the concept of formal reasoning systems, but I see this process of problem -> abstraction -> proof [-> unexpected application] as being one of the most important, beautiful, and difficult to teach parts of math.
I think I fall somewhere between abstract and concrete thinking, but I used to hate math for the reasons that you discuss. Then in early college, I started reading math theory and my interest level spiked. Finitism fascinated me and logicism made so many mathematical concepts make much more sense. I read books such as “Innumeracy,” “The Numerati,” “Why Beauty Is Truth: The History Of Symmetry,” and “The Beginner’s Guide To Constructing The Universe.” I bought a math dictionary that breaks down mathematical terms—this is an important tool for me in science and math because if I know the etymology of a term, I can usually glean much of its meaning and application instead of the term seeming arbitrarily chosen.
ReplyDeleteFor example, what is the identity function? Well, it’s this:
f(x) = x
But what does that even mean? Let’s break it down further: What is an identity? What is a function? In math, an identity is a set of two or more values that are identical.
Such as: 1 + 2 + x = 3 + 0 + x
meaning that, since 1+2 = 3 and 3+0 = 3, then whatever value we assign for x will result in the same sum on both sides of the equation, right? They are identical.
But, what is a function? In simple terms, it’s a factor that is dependent on another factor (e.g. My final take-home pay is dependent on my withholdings.). Now, take that concept and apply it mathematically. What is a function? It is a relation that takes an input (x) and produces a unique output (f(x)).
Example: f(x) = x + 1
Basically, this just means whatever value we input for x will result in a unique sum output, such as 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4. As you can see, all the outputs are unique.
Now, take these two definitions and combine them. What is the identity function? Well, we know that for it to be an identity, it must involve two or more values that are identical, and for it to be a function, whatever value we input for x must result in its own unique output. Therefore, if we took our identity function:
f(x) = x
and made x = 1, then our function’s output would also be 1. If that seems a little confusing, it might help to write it this way instead:
f(x) = x + 0
So, if we input 1 for x, what is our resulting sum output? It’s 1, the same value we input for x. And this would be true for any value in that whatever we input for x will result in x because when you add 0 to any number, the result is that same number again, right? Therefore, f(x) = x + 0 is also an identity function.
Hopefully the above example gets across what I mean about breaking down terms. I say all this as an example of someone who was convinced for the longest time that she would never be good at math and didn’t have much interest in it either. I think it had largely to do with most of my teachers who adopted the same solution of, “Practice more. Work harder.” While I do agree with your colleagues that practicing problems and fair effort made is essential to learning math, I think there should be at least a little time dedicated to math theory, especially at an early age. It always irritated me hearing my high school peers complain about learning math and ask, “Why am I learning this? I’m never going to use this again!”. Students need to learn why in a way that is meaningful to them. And be able to break down a mathematical problem. And yes, you really should take the time to show your work because not only can you account for the steps you’ve taken to solve the problem, if you make a mistake you will have an easier time looking back to see where it occurred. Wouldn’t it make sense to adopt this way of problem-solving in other areas of life? Absolutely. Math is a life skill, I think, and it makes for an informed consumer.
I look forward to your next post on this. To help inspire you, I have a woman friend who was very good at science -- and who is an artist. But her father kept telling her that she could be a scientist, kept helping her think that it was possible. She went to art school and while there realized that science was her true calling. Today, she is a research microbiologist. And an artist, because of course one doesn't preclude the other. But it is much easier to get permanently waylaid from a science career if you're a woman that from an art-rich life.
ReplyDelete@Angela, I really like what you said about breaking down the terms.
ReplyDeleteI also find the history of math to be a fascinating subject. Which sort of connects to my point about it being taught as too much of a set of arbitrary rules. Rarely do educators connect (e.g.,) the invention of calculus (first published in 1671) to the founding of the royal society (1660), the London plague (1665-1666), or the great fire of London (1666). But these were very much interrelated events. I feel like understanding the context in which math was originally invented/discovered (now there's a great philosophy of math debate stretching back thousands of years) helps to ground it.