Saturday, September 23, 2017

Every Math Problem Has Three Steps

Here is an experiment for a child between 2nd and 6th grade.  I'm assuming this child does a little extra math outside of school, like daily math or math on the weekends to earn computer time.

Ask this child to solve this problem:  8 + 9 = ?

One of two things is going to happen.

Either the child says 17 because the child has memorized his math facts, which you hope happens by 6th grade, or the child will look at this problem trying to remember how they did it so as to do the minimum work possible.  For older kids, you can substitute 85 + 78 if you want.

Many children are either Solvers or a Memorizers.  Memorizers tend to hurry through things and do things fast and accurately, and Solvers are always looking for a way to cheat because they are lazy.  Solvers take a long time.  If your child is in 7th grade and takes 20 seconds to solve this problem, she is an Concentrator, which is a Solver Plus Plus.  With 85 + 78, Memorizers have memorized the 'how to mechanism' complete with carry the one and apply it quickly.  Solvers are too lazy to carry the one or and they vaguely recall either 'carry the something' or 'drop the something'.

The correct answer is not 17.  This article explains why.

Some day this child will be asked to decompose x2 + 7x + 6 into it's component zeros (x+1)(x+6) and back.   If the child is new to 2nd order polynomials, I'll ask how this helps us graph this function and what else we need to cheat so we don't have to calculate a bunch of values. The Memorizers do not have the skills to do any of this.  Memorizers usually cry when presented with new topics. If I'm involved, it takes Solvers about 30 minutes to crush 2nd order polynomials and be ready to move on to asymptotic functions or the complex plane.  Solvers will have forgotten everything we worked on so it will take them 3 weeks for any of it to sink in, but they are way ahead of Memorizers.  Concentrators will solve the whole thing on their own and I'll have a hard time explaining why they need algebraic transformations in the first place.

This is why 8 + 9 does not equal 17.

Some time after Kindergarten, after the finger counting and mental counting stage, your child will see 5 + 6 or 7 + 9 and I'm telling you right now, any number that the child announces is wrong, especially one that comes quickly from memory.   The answer to 5 + 6 is "both hands plus my invisible finger", or "5 + 5 + 1" and the answer to 7 + 9 is "8 + 1 + 8 - 1", which becomes "8 + 8", but with a few more steps could end up being 4 x 4 which is equivalent to the area of a square 4 on a side, depending on the age of the child.

At the right age, 8 + 9 is a ten minute problem.  Step 1:  Look at this problem and decide that there's an easier cheaty way to solve it.  Step 2: rearrange, decompose, etc.  Step 3:  Get the wrong answer, like 14, and try again, because you stink.

Solvers have a ton of grit and go much farther and do much harder things than Memorizers.  All of the mistakes make for one persistent, durable, cool child.

For the last 6 years, I've been warning readers that a single hard problem that takes 20 minutes is 10 times the learning of a worksheet with 30 easy Kumon style math facts.  With that hard one problem your child can acquire the tools to break apart a problem, refactor it, substitute, get it wrong, try something different, and come up with something clever on their own.   This is how Solvers earn their name.  If this problem happens to be 3 problems in one with some ambiguity and other craziness, and I'm talking about Test Prep Math, then you can get a Solver out of a Memorizer.  TPM is a Solver Training Course.

Solvers don't seem to have a problem doing worksheets, but they do it much differently than Memorizers.  They score 100% because they check the answers out of habit because they never bothered to memorize anything and are accustomed to getting things wrong.  Solvers can take on new topics and thrive on tests like the MAP.  Memorizers are stumped by the COGAT.

One of the biggest mistakes a parent can make with the child's education is to teach the child math.  A similar mistake is to hire a tutor to teach the child math.   It seems like a good idea, and appears to solve the problem by increasing grades and test scores, but you are only solving the short term problem and neglecting the impending long term disaster.  The clock is ticking and the disaster approaches.  Instead, you need to teach solving, not math.  Or, if you really want to get fancy, teach cognitive skills and let the child learn solving on their own, and you will get a Concentrator.

How is a Memorizer made?  A child solves a challenging problem for their age, like '5 + 6' at age 4 or early 5, and the parent is totally overwhelmed with pride and praise.  So the child determines that 7 + 8 would garner more love and it does, so he just keeps going.  Stop it.   Instead, ask the child how he solved the problem and then be proud.  Even better, ask if there's a better, cheaty way to solve it, don't be impressed with the first few attempts, and you might get a Concentrator.

1 comment:

  1. This is right on. My 4 year old goes to a Montessori school where they're teaching addition like 7+2 using a number line...basically a process...no thinking needed...step 1 step 2...
    Meanwhile my 6 year old is learning addition following your recommended method at home.
    So I can understand the difference.
    And will have to get my son on the decomposition ball at some point!

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