Tuesday, November 7, 2017

Simple Down and Estimate

I'm amazed at how 2 solutions strategies grow and develop in children year over year.

The estimation strategy started as more of a graduate level proof strategies before I turned it in to a learning experience for 5 year olds.  It's related to start with a simpler version of the problem which I refer to as 'Simple Down' in the title to this article for lack of space.

Here's how the estimation strategy works.  A child is looking at a super hard problem that they know how to do, but they can't quite work out a strategy for decomposition (2 easy steps are better than 1 hard step) so I ask the question:  Is it 1?  Hopefully, I'll get the half eyelid you dummy look, but it usually takes a few times until the kid is on to me.  "No, it's not 1."  Is it 2?  "No, that one doesn't work either."  What about 3?  Eye rolling.  How about 4?  Sighing.

What I'm looking for is a whole bunch of more advanced super skills to emerge.  It takes restraint for me to explain how to do the problem, but this will destroy a tower of synaptic connections, not to mention their academic future.  So I'll just ask about 5 and maybe 6.

Inevitably, the child will be totally frustrated and bored by my useless guidance and will try 10.  A few more times, and the child might look at the picklist and see "9  12  15  19".  I hope this doesn't happen right away, because iterating through 1, 2, 3 and 4 might produce answers like 5, 7, 9, 11 and suddenly we're developing causal function sense, the Estimate Iterate skill.  Once they see that the answer set only requires 4 solution calculations, we've got head down the path of efficient time management.

These variants are just the tip of the skill ice berg, and to get deeper I believe very strongly that describing and explaining the solution strategy prevents the child from an internal mechanism that opens the door on more sophisticated methods that they will invent on their own in a few years.   Plus I tried just describing and recommending this approach and was ignored.  I'm disappointed when I meet a child who knows how to solve problems of all types.  It's unlikely they figured this out on their own.  There are many popular books describing solution methods for algebra problems of all types. You will need one of these if you teach your child how to do a math problem at a younger age.

The second strategy involves a simpler variant of the problem looks a bit like estimation but is used when your child is in over his head.  A 6 or 7 year old seeing multiplication for the first time, a 5 year old struggling with double digit addition, an 8 year old dealing with complex numbers, or a 9 year old graphing 2x over the positive and negative rational numbers*.   23 + 89 too hard?  Start with 10 + 10 and work your way up from there.  When the child gets back to 23 + 89, he will own it.  You are one step closer to never seeing a math grade again or caring.

These 2 solution methods are why I prefer one hard problem to six medium problems.  There is a much bigger payoff and it just builds on itself.  Suppose we're doing a few problems on a worksheet and get to a problem needing one of these methods.  I'll stretch out the time just on this one problem, so that the lesson stays with the kid and is not drowned out by more problems.  I don't see strong readers iterating through worksheets like "Jane eats corn.  Jane eats pie.  Jane eats broccoli.  Jane makes pie.  Bill makes pie..."  This is what most math worksheets look like to me.  The math version of a book is a single worthy problem that unfolds a whole new world of awesomeness.  (Readers read books to become strong readers not worksheets.)

I've seen these skills grow over the last 8 years.  What is amazing is all of the different manifestations of the subskills I didn't know exist until they happened and how these skills are applied elsewhere. 

*No one actually appreciates an exponential graph but me.  That was a fail.  Buy the square root of -1 is pretty cool.

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