Well, I ate far too much pie and sat stalled in traffic far too long waiting to get back into New York City via the Holland Tunnel. Happy belated Thanksgiving to my blog readers!
Now that I have Internet access again, I’ve been checking my various education sources and came across this interesting follow-up on the common core math standards. The author is a secondary school math teacher who writes frequently on education policy (and like me is a second career teacher).
I teach math only as needed for economics – which means I actually end up teaching a fair amount of math – but I don’t consider myself remotely expert in this area.
Still, one comment from the article resonated with me:
Let’s look first at the 97 pages of what are called “Content Standards.” Many of these standards require that students to be able to explain why a particular procedure works. It’s not enough for a student to be able to divide one fraction by another. He or she must also “use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9, because 3/4 of 8/9 is 2/3.”
It’s an odd pedagogical agenda, based on a belief that conceptual understanding must come before practical skills can be mastered. As this thinking goes, students must be able to explain the “why” of a procedure. Otherwise, solving a math problem becomes a “mere calculation” and the student is viewed as not having true understanding.
This reminded me of a ditty that helped me learn how to divide fractions back in the Triassic Era. I confess that I passed this advice along to my own children when they were struggling with elementary school math:
“When dividing fractions don’t ask why; just flip the sucker and multiply.”
Do I really agree with my own advice? Yes and no. It’s helpful to understand why dividing by a fraction yields a larger number – not least as a reality check against the answer that the calculator throws out. I also think that we can teach this concept with some of the manipulatives traditionally employed in math education. Grandma divided the cherry pie into 8 slices, but there are 12 people at the table. So each person gets 2/3 of a slice. 8 divided by 2/3 = 12, right? Try it with cookies you’ve baked for the class, and any teacher has a winner.
But 3/4 of 8/9s? That’s both difficult to demonstrate visually and difficult to grasp conceptually. I’m all for cutting up the pie or cookies once . . . and then flipping the sucker over and over again until the algorithm becomes second nature.
Teachers don’t have unlimited time. One colleague, who really is an expert math teacher, tells me that her biggest problem with the standards is not so much their content as their requirement that she teach twice as much material to students who are struggling with the material she’s already trying to present. If conceptualizing every problem really helped students learn more math more quickly or more thoroughly, I’d applaud this approach. But I worry that in practice “discovery” will drive out drill without producing much more conceptual understanding.
The seniors I’ve taught concurrent enrollment (college credit) economics would almost all have benefited from a lot more drill. Almost none of them, for example, can calculate percent change until I’ve retaught the algorithm . . . er, concept. On the other hand, my students would also benefit from a more solid conceptual understanding. I’m always startled at how few students can look at a linear graph and identify a direct or inverse relationship, based on the positive or negative slope – or explain what difference direct or inverse makes to the mathematical relationship they’re exploring. That should be second nature for students who all have a couple of years of algebra under their belts, right? But it’s not.
So what do you think?