Sunday, April 26, 2015

Where I Check the Condition of My Cognition by Doing some Algebra

As Cognition Slips, Financial Skills Are Often the First to Go... NY Times

Early today I read the scary piece above from Saturday's business section of the NY Times about how age begins to affect your basic financial - and math skills. I went through the checkpoints and - phew - I passed. But having that big 70 in front of my age, I am nervous about slippage.

So this morning, glancing through yet another column on education no-nothing Nicholas Kristof, whose column I try assiduously to avoid, I noticed some math problems embedded in the column, Are You Smarter Than an 8th Grader?

So I tried to do this one without looking at the answer:

A piece of wood was 40 centimeters long. It was cut into 3 pieces. The lengths in centimeters are 2x -5, x +7 and x +6. What is the length of the longest piece?

I went back to 8th and 9th grade algebra -- that's 1958-9, folks -- about 60 years before common core. And like they ask, I am going to show my work.

First step: I have 2 options here. Either trial and error - take a guess and see where it lands you. This is often a process I follow and do pretty well.

But I decide on option 2 -- dredge up my old algebra.

I must find the value of x.

[NOTE - I taught 4-6 grade self-contained classes and math was easiest for me to teach --- and even without common core -- taught my kids they must first identify the crucial thing they must do first -- solving these things has an order --- not always a rigid order -- but you must work from a plan. I did teach them option 1 - just make an educated guess.]

Let's solve for x in the equation: 2x -5 + x + 7 + x + 6 = 40.
Got that so far? If the total length is 40, then the 3 pieces must add up to 40.

My pre-common core teachers taught us that an equation was like a see saw that always must have both ends in the same position -- you can't let one side tilt. Did I ever take kids to a park and try it? Can't remember but I think I did use a balance scale -- the law says you if you put stuff in one side you must put other stuff -- not the same necessarily - in the other side so they are balanced. A great way to teach the concepts of what an equation is. And I would teach this in 2nd - and maybe even 1st grade just by letting kids play with these as toys.

Now let's solve for x.

My pre-common core teachers taught us that in solving for x you must isolate x on one side of the equation (see-saw) -- use do unto one side of the equation as you would do unto the other -- as basic and important a rule anyone needs to know.
So -- I do +5 and -7 and - 6 to both sides of the = sign.

Thus -- we have the numbers on the left side all cancel each other out and end up with 2x + x + x (4x)

while on the right side we get 40 +5 -7 -6  =  45 - 13 = 32.

4x = 32. Follow our golden rule and divide both sides by 4 to isolate the x and we get x= 8

Now that we know x we are ready to solve the original question:
The lengths in centimeters are 2x -5, x +7 and x +6. What is the length of the longest piece?

Did I have to know the lengths were in centimeters instead of inches? A bit of a red herring but maybe kids need to see through the red herrings.
And also - why use "x" as a variable when X is also the multiply symbol? A bit of confusion I would say. I say that because I want to notate this as 2x4 =8 but using the x as a multiplier here is confusing.
So:
1st piece: 2 times 8 - 5 = 16 - 5 = 11
2nd piece: 8 + 7 = 15
3rd piece: 8 + 6 = 14

The longest piece = 15.
But let's always check out work - as those pre-common core teachers taught me using the old math in the 1950s - by adding up all the lengths to be sure they = 40.

11 + 15 + 14 = 40.

Voila.

And I want to make the point that through the 8th grade I struggled with math -- or arithmetic. It was the simple logic of algebra that opened my eyes mathematically in the 9th grade. [Just a message to the debunkers of the value of math - who argue kids should only be taught consumer math.]

In the 10th grade , geometry really shook up my brain -- a 98 on the regents.
11th grade Intermediate algebra and trig are less memorable.
12th grade Advanced algebra--- 100 on the regents -- the only one in an exceptionally bright senior class at the now closed Thomas Jefferson HS  - where I did not feel very bright compared to the competition. That same year, calculus for me was a disaster - so I reached my limit. I tried to avoid higher math in my masters in computer science and then found I had an interest in artificial vision which required differential equations and beyond  -- did you know your eye was doing calculus?

Hey, any teachers out there looking for a guest lecturer for their class? Just don't ask me about quadratic equations.

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