Common Core math thing, round two ... a mathematician.

I was floating around the Internet and came across The Mindful Mathematician's A Letter to Frustrated Parents

I was never taught to make sense of numbers, I was taught one way to solve every problem, every problem had ONE way, memorize these steps and you will be able to solve this problem.  Sorry if you can't remember the steps.  This is how we do it.  I was robbed.  I was not taught to persevere and try to make sense of the problem... who cares what it means, here's how you do it, just do this.  

I would feel sorry for you, but I cannot accept that this is truly what happened to you in school. As much as I get irritated by some of what I hear about elementary school teachers, I cannot believe that anyone took this approach.

You are flat-out misrepresenting what the "traditional" approach was in order to bolster support for the New way of doing things - to the exclusion of the algorithm. "I call Shenanigans."

Hold on, why am I crossing out this number and changing that one? 

Because you don't have enough to take away.  Just do it!  

But wait, I have 453 and I'm just trying to take away 17. I think there is more than enough to take away.  

No you can't take 7 away from 3... Just cross out the 5. Just do it!

But wouldn't 3 take away 7 be negative...

NO! You can't take a bigger number from a smaller number, sit down, JUST DO IT MY WAY!

Bullshit. Or, to put it more kindly, IF this is a true and accurate transcript of the conversation, this teacher is not very good and probably would teach everything in the same tyrannical fashion. Rather, it sounds like the kid with a poor understanding and this is the "excuse" for why.

"Hold on, why do I have to have common denominators? "Because you can't add apples and oranges! Just do it"

"I was wondering... why do I have to flip the fraction upside down if I'm dividing?" "It's not your place to reason why, just invert and multiply!  JUST DO IT!"

Elementary teachers have enough problems teaching math, without your strawman argument and fairly obvious projection.

Second, isn't it interesting that you claim this fictional teaching method is the reason that all of our students hate math, yet you are the first counterexample among many ... most math teachers included. If you are looking for a cause-effect relationship, you've disproved it.

The probable causes for the lowered "love of mathematics" are the lack of enjoyment of the subject by those teachers, the nervousness and trepidation with which they approach math, and the over-reliance on discovery methods of teaching and the "Guide on the Side, not a Sage on the Stage."

The "new" methods you're seeing are not being taught.  They are methods that students naturally invent.  Just the way that mathematicians invented them before our formal mathematics system existed.  

That is the crux of the problem. Too much of teaching now is centered around letting the kids "discover" a way of their own. Having the kids to "discover" their own way does not create "better understanding", it merely forces them to re-invent the wheel ... and then go through the trouble of learning. (I guess this is the next post.)

But I digress.

Students find comfort in tried and true methods that work without major thinking. They'll accept the  struggle with any method at first, because it is new. Once learned, there is a sense of pride of ownership, of knowing that they've got something to call their own, something that eliminates the need to count by ones on their fingers. They could subtract 37 from 63 by counting but it's slow, so we give them new methods. One in which you count up like a shopkeeper making change and the other, the "algorithm."

The algorithm, originally developed for simplicity, required the student subtracting 37 from 63 to "borrow 1 ten from the tens place to make 13, and 13-7 is 6. Then 5-3 is 2. Ah, 26." That requires understanding of place-value. That's very important.

The "new method" takes a different kind of thinking. "Add 3 to get to 40. Add 23 to get to 63. Ah, 26." This thinking is also very important.

These two methods are not mutually exclusive in a student.

For 63-37, method 2 works better. For 8569 - 6325, the first one is superior because there's no cancelling and because the numbers are larger.

When people say that "borrowing" is unnatural, I present the addition to the right. Go.

Did anyone add 3 to one of the 27s to get to 30, then add 24 to get to 54? Probably not. That's the method for subtraction, done in reverse.

Did you add the twenties and then add the sevens? 20+20 = 40 and 7+7=14. Ah, 54." Maybe. Depends on how old you are. For 27+27, it's the most efficient.

More likely, you added 7+7 = 14, carry the 1; 1+2+2 = 5; Ah, 54.

If we're okay with "carry the one" why are we so all-fired-up about "borrow a 1" when subtracting?

Should kids be able to do it all three ways? Yes. It's math, and math is fun.

I'll leave you with this final example, from our intrepid mathematician -- right at the top of the page -- that exemplifies the best time to use the old algorithm.

Students must be able to do both. (or all three, or four).
Which they choose is up to them.

Why do we have so much trouble with that?