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?

I hate it when smart people are wrong in public.

Ordinarily, this guy is correct, but this time he screws up:

"If you went to elementary school in the US, you almost certainly learned about the order of operations."

No, you didn't.

What he should have said was, "If your elementary school teacher was incompetent in math, she taught you this."Which is why we need to seriously rethink the training of teachers in this country.

"If you went to elementary school in the United States (or much of the rest of the world), you almost certainly learned about something boringly called the "order of operations"- a set of rules for whether or not you should do multiplication before addition or addition before subtraction to get the right answer on your math test.

Except, you don't always get the right answer, or even, one answer - I mean, is 8-2+1 equal to 5 or 7? and is 6/3/3 equal to two thirds or six? - the problem is, focusing on the order of operations can lead to ambiguity and obscures the real, underlying, and often beautiful mathematics."

 

FAIL.

http://www.youtube.com/watch?v=y9h1oqv21Vs#

It's just an arbitrary rule ...

Okay, so there's an easy one for your students, right? I thought I'd try it with mine and let them argue a bit over what the answer is and why they came up with it.  When someone asks, I'll tell them that this is why we developed the arbitrary rule of PEMDAS - so that there would be no confusion, so that anyone who approached it came back with something we could all agree on.

But that's too easy, you say?

Well, following the break (Sorry to those reading in Google Reader) is a small portion of the Facebook feed. This is 100 or so of 254,542 ! According to the poster, only 30% got it right.

AnyQs - Cleaners

Two seemingly identical bottles. 28 and 32 oz.

If you can't read it, it says
33% more 
than other leading national brands **
**Compared to 24 fl.oz. of other leading national brands.

Algebra in the Third Grade?

I got the NASCO 2011 catalog in my mailbox this morning. If you haven't dealt with them, they are a pretty typical school supply catalog - prices are double what you can find if you search diligently or ten times the scrounger's price. What it is really good for is as a wish list, a source for ideas. "How does that toy work? I could do that in the garage for $2 in materials."

My problem with them is that sometimes they really miss the mark. Here's the cover:

Really? Third grade and up? I rail all the time about my students arriving in 9th grade without a clear grasp of fractions and clueless about operations with fractions (sans calculator), limited fluency with percents and decimals, and an unsteady grasp of their multiplication tables and other basic stuff. I had usually blamed the 3rd through 6th grade teachers for not really understanding arithmetic and passing on any math phobias, but this seems like a major problem right here.

If the expectation is that this kind of thing is possible for 3rd grade and up, maybe it shouldn't be a surprise that fractions and arithmetic aren't getting as much attention.

My questions:

• Is this typical? Do third grade teachers really do this?
• Is it just a stupid cover by a catalog desperate to sell overpriced AlgeBlocks to a gullible school system?
• Shouldn't we be solidifying knowledge to the point of automaticity instead of spreading algebraic materials ever lower?
• I'm pretty sure that a few third graders could get this but is it appropriate for that level? 
• Is it possible without manipulatives at this age?

 Then, there's the other debate:

• Are manipulatives appropriate? 
• Does the use of something tangible and obviously fixed in size get in the way of learning an abstract idea about a variable?

The Trader's Puzzle - Balance Weights

What are the weights of the four rings to as to measure any desired weight from a quarter-pound up to ten, in quarter-pound increments?

Archery Puzzle

How close can the young archer come to scoring a total of 100 - using as many arrows as she pleases.
The rings are numbered 16, 17, 23, 24, 39, 40

The Milkman's Puzzle

Honest John says that what he "don't know about milk is scarcely worth mentioning," but he was flabbergasted the other day when he had nothing but two ten-gallon cans of milk, and two customers with a five and a four quart measure wanted two quarts put into each measure.

It is a juggling trick, pure and simple, devoid of trick or device, but it calls for much cleverness to get two exact quarts of milk into those measures employing no other receptacles of any kind except the two measures and the two full cans. You can try the problem with the fullest assurance that it is a legitimate and not a silly catch.

Sam Loyd, Cyclopedia of Puzzles, 1914

see the answer.

The Weight of a Brick

If a brick balances with three-quarters of a brick and three-quarters of a pound, how much does a brick weigh?
~Sam Loyd, Cyclopedia of Puzzles, 1914

I won't bother with a separate post: answer

Cats and Kittens Puzzle

Seeing that four cats and three kittens weigh thirty-seven pounds, while three cats and four kittens weigh but thirty three pounds, we are asked to tell the respective weights of cats and kittens.

- Sam Loyd, Cyclopedia of Puzzles, 1914

Koi Pond Puzzle

This geometric city park is perfectly square with a square koi pond in the middle. There's a circular walking path that is tangent to both squares at the indicated points. The pond is 3 feet deep. The area of the park is 4 acres.

What is the volume of the koi pond in gallons? (You’ll need to Google a few conversions.)

Answer in this backdated post.