Ereyesterday, I had a fun chat with my dear friend Dr. Hmnahmna regarding Common Core mathematics. Here’s a transcript, edited slightly for clarity and national security:
DR. HMNAHMNA: From what I’ve seen of “Common Core” math (the quotes are deliberate), it’s not nearly as stupid as people think. I say “Common Core” because all Common Core does is say that you should have certain skills at a certain point.
VDV: I don’t think it’s stupid, but I do think it’s needlessly complicated. They’re giving as much emphasis to shortcuts and tricks as they are to basic algorithms and tables. I think that’s a mistake.
DR. HMNAHMNA: What most people call “Common Core” is various curricula that implement those standards.
DR. HMNAHMNA: There’s actually deep concepts buried in those “shortcuts” and “tricks”.
VDV: I don’t deny that one bit. I question the utility of the approach, and I fully expect that as these students reach high school they’ll be worse at math than current high schoolers.
DR. HMNAHMNA: Here’s an example of chunking:
Basically, this breaks the problem down into easier to manage chunks. If you’ve ever made change running a cash register, this is the way it’s done. It also demonstrates the commutative and associative properties of addition and subtraction in a concrete way. And if you’re trying to quickly subtract 712 – 648 in your head, it works much better than trying to remember the borrowing algorithm that we learned.
VDV: Again, I’m not denying that the tricks/shortcuts/alternate methods aren’t useful. I think the traditional method has far greater utility on paper. And I don’t mean “on paper” to mean “theoretically”, I mean actually on paper. The larger the numbers get, the less I’d want Joe Average to rely on a mental calculation, and the more I’d like him to rely on tried-and-true. I’ve seen a lot of students try to guess their ways through simple arithmetic work, and if they don’t guess right, the attitude is “oh well”. I shudder to think what’s going to happen when the CC wave comes through. Maybe I’m wrong, though.
DR. HMNAHMNA: Is your students’ approach a matter of not knowing the method? Or is it just trying to blow through and not caring? Because I haven’t figured out a teaching method that can overcome cockiness/laziness.
VDV: [Redacted list of SMERSH-approved torture techniques] overcome cockiness and laziness. Unfortunately they aren’t permitted by the curriculum. Seriously, though, what I meant was that training the masses to do it in their heads will lead to less patience with pencil-and-paper, and less willingness to use it.
DR. HMNAHMNA: The other big advantage I’ve seen from these other methods is that they are designed to promote a deeper understanding of what you’re doing when you subtract 712-648. And demonstrating it using teaching methods that are applied again in higher math.
VDV: When you put it like that, I think of the analogy of plumbing vs. fluid dynamics, or history vs. historiography. Or of the argument over the correct “first ten numbers” (0-9 vs. 1-10). Do they need the deeper mathematical understanding, or do they need extensive practice in efficient arithmetic calculation?
DR. HMNAHMNA: I happen to think that innumeracy is a big problem and that deeper mathematical understanding is important, though practice for efficiency is also important.
VDV: I agree. I see no reason not to introduce algebra/geometry earlier.
DR. HMNAHMNA: If I had to choose, I’d rather have a deeper understanding and slightly less efficient.
VDV: If I had to choose that for the Hmnahmnas and the [our friend who majored in mechanical engineering and is an industrial manager]s and the [our friend in military intelligence who majored in history and physics]s of the world, yes. But we’re also talking about the people who mess up your change at McDonald’s.
DR. HMNAHMNA: I think the example above will actually help the people at McDonalds not mess up your change. Note I said that it is the exact method you use to count change– count up from the total to the amount of cash handed over.
VDV: That’s why I used that example.
DR. HMNAHMNA: And hopefully, the deeper understanding will remove the blank stares when the total is $6.03 and you hand them $10.10.
VDV: May I propose a minor flaw with the analogy? If chunking, cashiers don’t have to keep track of how much they’re handing back (i.e., the difference). They just have to keep adding bills and coins until (price + change) = (initial payment).
DR. HMNAHMNA: Maybe I don’t understand people that are stupid at math, which is entirely possible.
Dr. Hmnahmna has a doctorate in mechanical engineering. It’s entirely possible.
Byzantine arithmetic is not unique to our era. About a month back, I picked up an 1877 edition of Ray’s New Practical Arithmetic: A Revised Edition of the Practical Arithmetic. Four bucks at a flea market. Here’s some Core-esque material from page 29:
I shouldn’t have posted this. Someone, somewhere, might get some ideas.
Yesterday I was told by another friend, who has operated cash registers far more recently than either Dr. Hmnahmna or I have, that many modern cash registers have dedicated buttons for each denomination of currency. That means that a present-day cashier can get away with not knowing how much cash the customer handed over. If the customer gives the cashier two twenties, the cashier can just press the $20 button twice.
I replied that even that might prove too complicated one day. What if the cashier doesn’t recognize the digits on the bills or coins, or can’t tell the difference between Grant and Franklin, or doesn’t understand that a $20 is worth more than a $10? I mean, looking at two digits is so much tougher than looking at one. We need to make counting bills and coins as simple as humanly possible. Therefore I recommend that henceforth, all portraits and other decorative images on American currency be replaced with symbols from video game controllers. Problem solved, and we’re one step closer to pure idiocracy.
My proposal also eliminates any possible debate about controversial figures appearing on our currency. Honoring Andrew Jackson by putting him on the twenty is questionable for several reasons, but what did ✕ or △ ever do to anyone?
2 thoughts on “Ramblings on innumeracy.”
Just a few comments and clarifications:
1-10 are the first ten natural numbers. 0-9 are the first ten non-negative integers. Precision counts when defining mathematical concepts.
When counting change, the chunks are predefined by the denominations of currency. Adding price plus change to get the original amount is essentially how the chunking technique works for subtraction. Dom is correct in stating that you don’t really need the final step of adding up the chunks when making change, but it’s a nice thing to do out loud for the customer.
And if anyone really wants to discuss the Navier-Stokes equations for fluid mechanics, I have a passing understanding of them.
To clarify: the “argument over the correct ‘first ten numbers'” was not over what the first ten numbers were, it was an argument over which ten numbers to teach your kids first. One old friend of mine said you should teach your kids 0-9 first because it lays a better foundation for teaching more advanced mathematical principles later. The other old friend of mine said you should teach your kids 1-10 first because each of those numbers has a specific Biblical/Scriptural meaning.
Comments are closed.