Math isn't hard, although many people would disagree with that statement. In fact, basic calculus is something almost any high school graduate should be able to master (although almost none of them do).
It's a commonplace among folks who grok the calculus that the material's not difficult, it's just badly taught. I'd like to extend that statement to math in general, starting with first-grade arithmetic.
How can I say that arithmetic is badly taught? Because once students get out of elementary school -- once they get presented arithmetic problems that aren't already laid out in numbers, columns, operators and a nice neat line separating the problem from the space where you write the answer -- they often freeze, or at very least hesitate. Given the number of college graduates who "don't do math", K-12 education is clearly dropping some ball. Unfortunately, it's a ball that's critical to our society's ability ever to sustain itself.
Whether we're talking about the concentration of greenhouse gases in the atmosphere or the acidification of ocean waters or the erosion and desertification of productive soils, we're concerned with a process of which some elements can only be understood in mathematical terms. These problem aspects can be presented as simple arrays of numbers, but what portion of the populace will ever engage such arrays? They can be presented (somewhat less precisely, to be sure) as line charts, but how many folks will take the time to study those charts and draw the proper inferences? The sad fact is that sustainability challenges can only be fully described with a combination of qualitative and quantitative logic, while the way we teach math (separate from language) at the earliest grade levels establishes a pattern of thought in which words are words, numbers are numbers, and never the twain do meet.
If that all sounds a little abstruse, just talk to your favorite non-engineering undergrad. Or, perhaps better, your favorite high school student. Direct the conversation to the topic of math class, and then ask what type of problem (s)he really hates. I'll give you long odds that the answer is "word problems". Which is why, of course, many of the items on the SAT and ACT and similar tests are math problems set forth in words. Test designers know that it's the exceptional student who can reliably draw the math problem out of the verbal presentation. But all students -- all adults -- need the ability to do that. Any educational system which doesn't develop that ability is seriously deficient.
When my own children were in K-12 school, I repeatedly tried to get their teachers to understand the difference between mathematics and computation, with limited success. What's done with numbers is mere computation. Real-world math -- the application of mathematical concepts to real-world problems -- is as much about figuring out what the numbers are as it is about knowing how to turn the computational crank once you get them. The significant problems we face in the real world aren't exclusively qualitative or quantitative, they're a combination of the two. And if our students are going to able to grapple with those problems effectively, the silo wall between words and numbers needs to come down. Better, it needs never to be constructed in the first place.
Think about students in first grade (or, these days, kindergarten) when faced with their first arithmetic lesson. 1 + 1 = 2. If the kid's bright, (s)he'll ask "one what? two what?", and get rewarded with the first of many classroom exercises in the diminution of curiosity. The teaching of "arithmetic facts" or "plus/minus/times tables" is an exercise in rote pedagogy because, with numbers abstracted from any real-world application, nobody knows how else to do the job. Even when students are presented with "manipulables" as a way to bridge the gap between the abstract and the concrete (a practice which, I believe, has fallen somewhat out of favor), the direction in which the bridging is attempted is from the abstract to the concrete. In real life, understanding, logic and application all flow in the opposite direction. Constructivist pedagogy may make standardized testing more difficult, but it makes learning far easier. And far more profound.
Why are we teaching elementary students about the characteristics of numbers abstracted from any real-world application? Why are we creating an artificial boundary between "how" and "how many" or "how much"? Why are we, at such an early stage of education, forcing students' thought into disciplinary silos, so that they can spend the rest of their lives either struggling to overcome the artificial division or succumbing to it (e.g., "I just suck at math.")?
Would it be acceptable to graduate high school students who are convinced that they suck at their native language? Or that they suck at thinking and logic? That they suck at life? Were any of those conditions anywhere near as prevalent as the conviction that "I suck at math", I suspect that public schooling would enjoy far less popular support than is now the case.
In truth, virtually all the sustainability challenges we face as a society (and I'm thinking about the "developed world" here) got that way because a functional majority of society divorces its qualitative enjoyment of goods and services from any quantitative assessment of the real-world impacts of their production. I could certainly be wrong, but I suspect that lots of folks just aren't cognitively equipped -- haven't been appropriately prepared -- to consider qualitative and quantitative aspects of a process in any sort of comprehensive analysis (even an informal one).
It's true, as I mentioned last time, that many (most?) students don't come out of high school with the verbal skills -- clarity of thought and expression, breadth of vocabulary -- that would facilitate their qualitative analysis of any complex process or problem. But part of the reason for that seems to be the divorce between "language arts" and "math". Learning to grapple with and express mathematical concepts in verbal terms develops precision and concision of thought. If our graduates are to address sustainability challenges successfully, they don't just need to be adept in each of language and 'math', they need to be adept in their combination.
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