Thursday, November 30, 2017

Teaching Algebra To A Fourth Grader

I'm really excited about yesterday's algebra post and the worked the followed last night proving I was totally wrong.  I'd delete the post entirely if it wasn't chock full of good advice that sets the stage for today's topic, which is teaching Algebra to a 4th grader.  Which is going super well, thank you, and after I complain that this is insane and ruining our children, I'll show you how.

There is a branch of math curriculum that is not very mathy.  It's fuzzy and intuitive and wholelistic and verbal and it doesn't really care if the kid currently or will ever know anything about math.  I love this type of math.  It's used in our school, chosen by a group of teachers who have about 9,282 years of teaching experience between the 4 of them.  One is named Yoda, just so you know.

What I've found is math that works for average or below average children in the US will work for your slightly above average child much earlier.  I think the best math curriculum of all time, even surpassing Sylvan's Kindergarten math book is the middle school math Jo Boaler created.  It involves no math at all, and then she just walks kids into 'WHAM' real math.  Her problems show up in CMP math, which is what our school uses.

On the other end of the spectrum is Singapore math.  I've decided that Singapore math is now Public Enemy #1, replacing Kumon as the worst thing you can do to your child.  One of my early famous articles describes Anti-Kumon, a program that I felt so strongly about that it ended up being the Test Prep Math series.  I also hate Mathasium and Level One.  I'm going to hunt down Singapore math grades 5 and 6 ASAP and start using them.  I've already recommended Kumon pre-algebra grade 6, a good book once you rip out the part at the beginning that spoon feeds how to do each problem and the part at the end called Solutions.

What Singapore found out is that you can train your child on advanced math and they'll look pretty capable as a result.  The top high schools in the country are full of overstressed, overanxious kids spending long hours doing homework and beginning the teacher to just please please tell them exactly what to do to get an A because they've never been given the opportunity to learn  and don't know how.  The Singapore material itself isn't such a bad idea (which is why they're about to get an order for books from me), it's what parents do with the material that is detrimental to their child's future.

Recently I conducted a search for problem solving books.  I've developed my methodology based on Poyla's 1945 book and was wondering if anyone else came up anything else that was helpful.  (The short answer was The Art of Problem solving that came out 10 years later and is helpful for wealthy people who live in suburban Connecticut.)  For the 8 weeks prior, all of my Google searches were 'Algebra II problems', for the older brother, and Google's search engine is nothing if not intuitive, so it gave me an extensive list of books and websites devoted to detailing the step-by-step solution to every algebra problem ever devised.  I was horrified.  This is where math training leads.  A drug addiction to solution guides.

Anyway, I'm going to apply hard core Poyla to 2 problems.  The first problem is below, and the second problem is that you have to get your child to learn it on their own, no cheating.   In yesterday's post, there were two problems, and children naturally gravitate to the the second one.  I should have recognized this one as a necessary step for the first one but I blew it. 

Here we go with the harder question.

There were three times as many jelly beans in Jar A as in Jar B. After 2685 jelly beans in Jar A were sold, Jar B had twice as many jelly beans as Jar A. How many more jelly beans were there in Jar A than in Jar B at first? 

We followed exactly the problem steps from yesterday's article, including a diagram that was totally unhelpful, but there was no way to solve this equation without extensive backtracking and #7 the missing element.   I hate the missing element technique.  It is absolutely fundamental to geometry proofs, but it takes a lot of work for a child to derive the missing element in 1 hour that took brilliant mathematician's 250 years to derive.  The secret is #8 use everything you ever learned that points to the missing element, and if you haven't actually learned it yet, you need backtracking.

Step 1:  Backtracking
For backtracking, I used IXL and Khan Academy algebra problems with parenthesis.  I did this a few months ago. These are real powder puff exercises, like 4(x - 10) = 23.  I look for parenthesis because kids who grow up with wholelistic language-based thinking math take many months to remember how parenthesis work no matter how many ways you spoon-feed it to them.  I finally created Kumon style parenthesis worksheets and told them just to memorize it.

I'll tolerate estimate-iterate for a while (is x 3?  How about 30?  What about 12.2?) because it's good arithmetic practice and builds the type of number sense needed for statistics, but eventually I'll resort to something like 1/x(23 - x) = x1/2 so they quit guessing and ask for help.

The missing element is the equation x = 3(13 - 2).

What makes x = 3(13 - 2) a better problem than 1/x(23 - x) = x1/2 or  4(x - 10) = 23?  The answer might take multiple 30 minutes daily discussions.  The answer is that in the easy equation, x is on one side and all of the numbers are on the right.  In fact, the easiest equation of all is x = 33.  x on the left, a number on the right.  The goal of algebra is to get x on one side and the numbers on the other side, in the cheatiest least effort way possible.  The goal of prealgebra is to handles all types of numerical operations including exponents.  Algebra adds 'x' and mixes things up.

From that point forward, we had an 8 week battle to see whether or not the stubborn kid could solve the problem without resorting to algebra, no matter how long it took, or whether he had to learn the fundamental principle of algebra:  you can add/subtract/multiple/divide/power up/power down each side of the equation by the same factor (whether it's 5 or (x - 2)) and the equation will take a step in the direction of 'easier' if you didn't screw up the parenthesis again.

We spent so much time analyzing what was wrong with equations (the x is not on the opposite side as numbers) that it qualified as a principle on which to build.

Step 2: Derive the Equations
In the problem above, this wasn't an issue because our math program is founded on convoluted complex word problems with double reverse logic.   We lost a few minutes because one of the unfortunate side effects of this approach is a kid smart enough to point out how stupid the problem is.  "Who buy's 2,685 jelly beans?  Like they're going to sit there and count them.  This is a dumb problem."

For some kids, backtracking might include writing equations from word problems.

So we got 3B = A and A - 2685 = 1/2B. 

The second equation was rewritten as B = 2A -5310.  The reason is that at one point in our backtracking, I told him if he see's x in an equation (aka a variable), then there is a 100% probability he'll have to work the equation with transformations to derive the answer, so stop wasting time trying to solve it in your head. 

The three important principles for this step that we haven't come to terms with fully are:
a) The best way to determine the correct equation is to write down the crap you know is wrong and fix it
b) don't write the two equations buried in a bunch of pictures
c) if the older brother wants to interrupt math with the new Avengers trailer, you're going to lose 20 minutes

No matter how many times I encourage mistakes and do overs, each new step up the math ladder is greeted with this expectation of getting things right the first time.  Mistakes are the fastest way to the goal.  Perfection is a hard stop on the road to learning.  We would save a lot of time if he would just write 2,685 - B = 2A, realize it's wrong, and fix it.

Step 3: Wait for the Leap
At some point during this problem I started cursing Anonymous for putting me in this position.  This would be a great problem for a long weekend.  To solve it, my kid has to figure out how to solve simultaneous equations, on his own, and all we've got are our problem solving techniques.

I am 100% sure that 100% of Singapore kids are told what simultaneous equations are and shown how to solve them, then they can practice this technique, get high test scores and great grades, without ever have experienced true learning.  It's like taking a Grade A steak and grinding it into dog food.  For my buddies from Southern India, I don't have a good analogy.  I once made Indian food and proudly brought it to work.  My coworkers told me it was 'bachelor food'.  All those great spices mixed into a tasteless mess.  That's what happens to Singapore math when it's trained and not learned.

My son pointed out that he can't solve the equation, and then complained and glared at me.

Why?  "Because it's got a B and an A.  It could be anything."

I asked him to specifically point to what is wrong with the equation.  After about 5 minutes, he pointed to "2A" in the equation B = 2A -5310.  So I asked him to fix it.

We had already established algebra is about fixing equations.  He knew the way to do this was transformations.  In the first 7 or 8 minutes, he just stared trying to determine how to transform the equation.  No luck.  Then he got really intense because somewhere in the pictures of his bear and a girl named 'Amy', he could sense 3B = A plays a role.

In Poyla, one of the foundations of understanding the question is 'use ALL available elements of the problem'.  This becomes really important in geometry.  We haven't spent much time on it.  I asked him if anything else could help.  Since 3B = A was buried in doodles, I asked him to show me all of the pieces of this problem.  I'm not sure this was necessary, but it was getting late and he had science homework and my spouse was yelling at me.  (Solution strategy #9, when your spouse is yelling about how late it is, start asking questions that direct your child.)

We had 7 or 8 minutes of silence and I could see he was becoming really excited in an intense concentrating way.  He said "2A is 6B" and wrote 5B = 5310.  When you're excited about learning, you can do 3 transformation in one step and I'm not going to complain.  This is how brainiacs get to the point where they solve things mentally to the consternation of their teachers.

What did I do?  I did three things.
1.  I didn't look up the solution and explain it to him.
2.  I didn't help other than ask questions and suggest one of the 8 problem solving techniques.  In this case, I suggested all 8 and we used all 8.  I will continue to do so until I'm banned from helping by my son, which is scheduled for middle school, at which point I will solidify my role as the dumbest, lamest parent on the planet and my child will reach self sufficiency.
3.  I waited, and waited, and was prepared to wait for the next 3 weeks if that's what it takes.

I was rewarded in 3 big ways.
1.  I concluded the whole session by mentioning that 2 equations with 2 variables is called 'simultaneous' equations.  I pointed how that he taught himself how to solve simultaneous equations and this is a big deal.  He already knew at this point that he taught himself and it was a big deal to him.
2.  3 months ago, it was horribly painful for him to transform x - 3 = 6 by adding 3 to both sides.  Now he was doing 4 steps in once (multiplying 3B = A times 2, substituting 6B for 2 A, subtracting 6B from each side of the equation and multiplying each side by -1).  I have repeatedly told parents to look for this effect, starting with phonics and first math when you get 3 weeks of zero and want to quit.  It's nice to see anyway.
3.  As a parent, I took a big leap myself in problem solving skills under the problem of how NOT to teach my child how to solve simultaneous equations even though Anonymous put me into this awful spot.

We are not going to have to practice simultaneous questions to perfect it. It's been earned, not trained. I don't like perfection, it removes the problem solving aspect that will be gained the next time the topic comes up, which will probably be this weekend with 8th grade simultaneous linear functions because I'm totally psyched.

In the last 6 weeks, I've come to the realization that the approach behind Test Prep Math is not at all compatible with Singapore math before grade 4.  Test Prep Math tries to avoid math at all costs while building up the skills underlying math, logic, and reading convoluted problems to earn the first 3 foundation problem solving skills that I covered in yesterday's article.  These are 2 wholly different world views.  I'm betting the farm that by middle school and then again in high school I will inevitably be proven correct.  I'm standing on a mountain of research, logic, and common sense from qualified teachers that I stole (problems solving technique #10).  By why wait until then?  4th grade is a great time to crush a few years of Singapore math.

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