Wednesday, August 2, 2017

Why I Don't Check Solutions

Here is a math example of why solutions are so bad for learning.  Here is a multiple choice question for 3 x 5 = ____.
A)   30    B)    22   C)    9   D)   15

If your child is 5 or 6 years old and brand new to multiplication, this is a challenging problem.   At first glance, it appears that the goal of this problem is to figure out what 3 x 5 is and get the correct answer of 15.   If you want your child to do OK in school, then this is the goal.  

But if you want your child to crush the COGAT and get 99% on the math section of the ITBS/MAP/PARC and SAT, then the answer of 15 is irrelevant.  In fact, if my child replied "3 x 5 = 15" then we'd move on to something harder.  Here are the answers to this question, and they each relate to useful skills that are required for success:
  • 30 is too big.  That can't be an answer.
  • When I count by 5's, I always get a number that ends in 5 or 10, so 22 and 9 are out.  But 9 is a multiple by 3.
  • 22 is an even number.  I never get an even number when I count by 3's.
  • When I count by 3's, I get 14, and when I count by 5's, I get 15, so I did something wrong and have to start over.
  • 2 x 5 is 10, because I've done that before, so if I add another 5, I get 15.

Have you ever met the parent of a child who taught herself math?  The parent thinks the child is a genius.  In fact, the child is using skills like these.  Every child will one day learn that 3 x 5 = 15, but only some of the children will learn all of the subskills hidden in this problem.  The difference is whether or not the parent cares if the child knows what 3 x 5 is, or whether the parent cares if the child can think.

Without accelerated math, I don't see an opportunity to pick up these skills. Children doing grade level math can do all of their work simply using the mechanics of example-practice and not have to think at all.

On numerous problems between 3 x 5 (and (12 - 4f) = -2f - (4 + 2f), which is about where we are now on Khan Academy), I'll get one of 3 results:  Is it 15? or Is it 9?  or 'I get 16 but it's not on the list'.  We may not get to '15' today, but we'll get somewhere.  Here are my responses:
  1. I don't know.  How did you figure out the answer?
  2. Why isn't it 30?  I think it's 30.  What is 30?  
  3. Why didn't you pick the other answers?
  4. Do it again and tell me if you got it right.
  5. Can you figure out an easier way to do the problem.
This of course turns a 30 second problem into a 15 minute problem, which is our normal pace.  That's the pace of picking up all of the subskills need to get me out of the picture of helping with math.

But we're not working on arithmetic today.  We're working on this:


The first group of shapes is transformed into the second group.  Apply the transformation to the third group.  

I'm not looking for the correct answer.  I'm looking for this:
1. There's a cat shape (octagon) on it's side, with a little jug (no sides) in the middle, on top of a hexagon.
2.  It looks like the hexagon gains 2 sides and the jug gets taller by double.  The cat shape might flip vertically because it is symmetric in this way, or do nothing.

Here are the answer choices:

While determining the operations on the first group of shapes, it's not clear in the answers which shape is which.  In fact, the thumb shape doubles in 2 answer choices and the pentagon doubles in 2 shapes.  What about the arrow?  It looks like it doubled instead of just increasing it's height.  Maybe it got 2 sides.  Is that the same operation, or is it close enough in a situation where decisions have to be made based on partial information?  Did the jug flip?  I didn't think of that the first time.  Which shape gets 2 new sides?  That eliminates B.

I count somewhere between 10 and 15 steps of thinking to solve this problem. I'm not going to get them all.  Instead, I'll get some progress on thinking, a workout for working memory, and a child who learns to concentrate for longer and longer periods of time.  Magically, visualization skills, including resizing and rotating just magically appear.  In the end, picking a subpar answer from among 4 subpar choices is a hugely important skill.  I'm going to go with D for now.

Did you notice that A and D are the same?  They are not the same.  If the child noticed this, somewhere around the last step, that's a win.

By the way, this is problem #93.  It's not really designed as test prep in case you are wondering.  It's designed so that the test is a small, easy step on the way to something much bigger.








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