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.

Wednesday, November 29, 2017

Teaching 4th Grade Math to a 2nd Grade Child

I received this great question from Anonymous that deserves at least one post, if not a book.  Your child and skill level may vary, but from my stand point, it's the same question. 

I'm struggling with 4th grade math materials. What's the best way to teach my second grader how to solve these questions? 

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? 

Aileen and Barry had an equal number of postcards. After Barry had given Aileen 20 postcards, Aileen had five times as many postcards as Barry. Find their total number of postcards.

I'm going to provide a step-by-step guide, and this is going to be a long article.  Brace yourself.

There are many great reasons to teach a 2nd grade child 4th grade math.  Here they are ordered from most important to least important.
  1. You didn't think to teach your child 2nd grade math when they were in Kindergarten.  Frankly, it won't matter by middle school when you begin, but the earlier you start, the more time you have to block out all of the memories of frustration until you just remember what a great idea it was.  An earlier start imparts more technical skills and a later start imparts more grit, but grades are high in science and language arts, which is what you really want.
  2. You want to imbue your child with unmatched grit and generalized problem solving skills so that the rest of their academic career will be easy no matter what the challenge.
  3. You want your teacher to notice that your child is bored in math and recommends your child for an advanced or accelerated program.
  4. You are blatantly cheating your way to a high score on the MAP test.
This problem is from Singapore math.  Be very careful with Singapore math because like Kumon, it shows the kids how to do the math and undermines a host of more important skills like how to think.  There are problem solving guides that come with these types of math courses and they short circuit learning.  You can destroy your child's thinking ability in one shot and it's hard to undo the damage.

One more thing to keep in mind with Singapore math.  4th grade math compares with 5th grade or 6th grade math the way we normally refer to math curriculum in the US.  I've seen 2nd graders do 2nd and 3rd grade level Singapore and come out ahead.  You might want to think about switching to 3rd grade Singapore math or 4th grade lame standard US math.

Rule #1:  Don't, under any circumstances, teach math.  
You don't want your child to learn math.   If you focus on the more important skills, they will learn really advanced math on their own.  But if you try to teach them math concepts to solve these two problems, they are not going to learn math or anything else.  It's not about math.  The child is in charge of math, and you are in charge in an environment and experience where learning will explode.

Rule #2:  It's going to go painfully slow at the beginning.
It's really hard to watch a child tackle a problem that requires basic problem solving skills while they pick up basic problem solving skills.  It's painful.  If you want your child to learn how to learn, you can help by being confused, by being patient, by asking questions, but you can't just tell them how to do it.

It does not surprise me when a child takes 2 or 3 days to get past the first problem.  It does not surprise me that they forget something we did or said 10 minutes ago.  But I'm always totally shocked that in a few months they're zooming through 4th grade material like a slightly below average 4th grader, and I'm pleasantly surprised that test scores are now 100% across the board.

I'm always happy to receive an email from a parent that starts out with "I was doubtful at first because we got no where in the first 3 weeks..." because I know exactly where it's going.   If your child does ballet every day, they will probably become adept at ballet.  In the same way, success is inevitable on 4th grade math.  Give it time.

I like to say "of course your child can't do 4th grade math, because she is only a 2nd grader".  But she will.  These problems, however, are challenging for a 6th grader.  At Math House, we've worked through much more inappropriate problems, so I say go for it.

Rule #3:  Let's teach something besides math.
Language is probably the most important.  In the 2 problems above, there are at least a dozen words that your child could read and not understand, at least not in the context of the problem.  I'm going to provide some solution strategies that will help you in the first few weeks, but you need to get to a discussion of the problem as the primary way to work through it, not just because you want a high reading comp score as a bonus, but because understanding of math and language are linked.  I'm not sure math itself is linked to language, probably, but understanding math definitely is.

In the first few passes of each problem, invite your child to explain it to you, word-by-word and sentence-by-sentence.  For many parent-child teams, this will be total culture shock.  It takes changing gears and practice.  If your child can't articulate the question on the 7th try, word by word, you may ask for a picture or try again the next day.

Being confused, having to read a question 5 times, and getting it wrong are 3 important skills that have to be practiced and developed.   If your child doesn't become an expert at these 3 skills, and you as the primary academic coach aren't totally on board, more advanced work is going to be a real struggle. 

Rule #4:  You need solution strategies to survive.
You, the parent need the solution strategies.  My kids know all of them and are ready to tackle graduate study of Lie Groups, but if they use them, they use them behind my back.   I've never met a problem anywhere that can't be solved by these, so when they are stuck, I just shout out random solution strategies and we're back in business.

Now about that solution.

The challenge with the 2 problems above for a 2nd grade child is "2685" and "five times".  I don't care if my 2nd grader picks up an understanding of 4 digit numbers and multiplication/division.  That's his problem.  I want him to understand the essence of the logic and problem definition.

If the child understands the problem, in second grade, we're way ahead of the game.  Moving forward with strategy and solution will follow in time.   I prefer the child to get there when they get there, on their own.

By the way, you can just google these problems, tell your child the solution framework, and set your child up for failure down the road.   It's your choice.  

Here is the parent tool set:
  1. Draw a picture.  This doesn't work really well with 2,685.  Plus, this strategy is appropriate to geometry and should only be used as a fallback when your child is really frustrated.  Drawing is relaxing.   In this case, I would ask them to draw a diagram to show me the before and after (with colored bars instead of cards) just so I could see that they understand the problem.  Given the difficulty level of these problems, a drawing is inevitable, or acting them out with a stack of pennies.
  2. I tried algebra.  Total failure in 2nd grade.  The 4th grader is now starting to get it because I told him it's total cheating.  Yahoo answers recommend algebra for 4th graders, but if you are successful, by the time your child gets to 4th grade they will just look at the question, stare at it silently, and announce the right answer.  They will be using elements from the rest of the list.
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?

This problem needs #3: Make a simple problem.  In competitive math and math after calculus (like infinite series), a simple problem is followed by incrementally harder problems until we've developed a generalized algorithm.  In this case, we just want to understand the problem.

There were three times as many jelly beans in Jar A as in Jar B. After 25 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?

Now we've got a problem that a 2nd grader can work through, although it's going to take a few days at 30 minutes of concentration time per day.  I would recommend getting a bag of jelly beans after the first day.  Hopefully, you have lots of pennies, but now we've got a problem that deserves a picture.  Regardless, going back to 2,685 is going to add nothing to the problem for 2nd grade.

How did I pick 25?  I estimated and iterated (solution technique #4 which kids get really good at for problems like this after a few months of work).

Lay out the 25 sold jelly beans, and ask your child what we don't know.  (Many readings of the question later and some discussion) and we don't know how many beans are in Jar B and how many beans in Jar A were not sold.  You can do this on a 3 part diagram and place the sold beans in part 2 of Jar A.

Then invite your child to start putting down beans in the 2 missing places (#4 estimate) until we've got the beans left in Jar A to equal those in Jar B.  Finally, have your child read the question out loud and explain the answer to you.  Here's a tip.  Start with 1 bean in A for the part left after the sale (solution strategy #5 - start with 1) and ask how many need to go in B to establish twice.  Ask whether or not 1 in A and 2 in B satisfy the initial condition.  Your child is going to go "What does initial condition mean" so you have to read the problem again and write down the 2 conditions the beans have to satisfy.  As your child adds beans so that the part in A that is not sold is 1/2 of the part in B, see whether or not you got the solution.

In this way, a 2nd grader will build number sense, learn multiplication/division from the ground up, and have to concentrate really hard to get through it.  All great skills.  If you throw in discussion skills, your child is going to make a lot of progress.  It is unlikely that your child will get any where near competent on 4th grade Singapore math.  This has never been part of teaching current+2, but eventually it will happen.  The first year is mainly about grit.

On to the next question.  Solve these in order:
Aileen and Barry had an equal number of postcards. After Barry had given Aileen 1 postcard, Aileen had two times as many postcards as Barry. Find their total number of postcards.
Aileen and Barry had an equal number of postcards. After Barry had given Aileen 2 postcards, Aileen had three times as many postcards as Barry. Find their total number of postcards.
Aileen and Barry had an equal number of postcards. After Barry had given Aileen 4 postcards, Aileen had four times as many postcards as Barry. Find their total number of postcards.
Aileen and Barry had an equal number of postcards. After Barry had given Aileen 20 postcards, Aileen had five times as many postcards as Barry. Find their total number of postcards.

In addition to problem decomposition (inherent in these problems), estimating+iterating, and diagramming, I recommend solving these problems in reverse.  #5 Start with the end state and see if you can work your way backwards to the initial condition.  It's good practice on an important solution strategy.

Those 4 versions of the problem are not just a variant of start easy and work your way up, but have an element of what I call 'Backtracking'.  When we do 'work ahead' these days, we'll come across something like arithmetic in the complex plane and have to take off time from the problem to practice adding etc complex numbers.   It can happen on any problem.  In your case, it could be arithmetic with multiple digits or decimals.  Be prepared.

On that note:

Rule #5:  Get a Fallback Book for Bad Days
I've used boring current+1 workbooks which just have pages of fill in the blank when we're having a bad day because at least I want daily math to be an established pattern during the current+2 year.  In your case, I highly recommend Singapore Math Grade 3, or grade 3 if some publisher stole this question, because you may find that the grade 3 book is already 2 years advanced over 2nd grade and end up switching to it.  Then get a boring 3rd grade fill in the blank book for bad days.

Plus, I can't help every day, and it's nice to have a worksheet that I don't have to grade.

Plus, we may need it to backtrack on missing math topics and a 3rd grade book would do it.

Rule #6:  You'll Never Succeed
You'll never succeed in a 2nd grader doing 4th grade math like a top notch 4th grader.   You don't want to, so don't set out with this goal in mind.  You want your 2nd grader to be an amazing kid in all subjects, prepared to take on the best of the best.  But a great 4th grade mathematician will crush him.  If you want a child to work quickly and accurately 2 years ahead at the end of 6 months (which may happen a few years in the future on its own), you'd have to spoon feed, memorize, and train, and you'd end up with a dummy who hates math.

Instead, after you get to about the 75% mark of the book (or the 3rd grade book in this super hard series once you come to your senses), when your child is only misses half of the problems and takes forever, look for amazing things in all subjects.   Take a year off of math and do other things if you can.  Then be prepared to spend the rest of grade school feeding your child advanced math so they aren't bored.

The original experiment for current+2 never got beyond adequate, although he works nicely on his own.  Sometimes he does really well with current+3 or current +5, and sometimes it's 100% wrong.  Recently, I created a new website for our Boy Scout troop.  He sat at the computer next to me because he wanted his own website. [Insert eye rolling here, because that's what I was doing.]  I sat there stunned when he typed html from scratch.  Who types html from scratch?  He certainly didn't learn this in school.  Then started adding detailed styling and animation like he has a programming gene.  The level of learning skills when he's motivated is at about current+7.  That's what I'm talking about.  I didn't give him a fish when he was hungry.  I didn't give him a fishing pole or a net.  Apparently by focusing on problem solving skills and not helping or caring about the answer to a math problem, I gave him a whole fleet of fishing trawlers.  That's what I'm talking about.

Monday, November 27, 2017

Add 20% To Your Child's Score

Here is a thorough paper summarizing early childhood studies.  It's slightly dry if you're not in to this sort of thing, but it's very inspiring how much success has followed investment in at risk children living in poverty.

The general conclusion is that taking kids who live in a home devoid of eduction and putting them in a top notch academic program is going to have a big impact.  Early studies found that when you send a kid back into the original environment, the scores and grades plummet back to where they came from.  It's nice to see later studies address this issue.

These authors ask an open question that I have already answered.  It's a really big question and has a big answer.
...many early child interventions are conducted with at-risk children living in poverty. There are many reasons to suspect that the same results may not occur if the same intervention were conducted with affluent children. 

"Affluent" in this case means a home with education and stability.  I would agree a child from an "affluent" home may not see much benefit from a program designed for an inner city child with a single parent who didn't finish high school, even though some of the at risk kids in these programs saw IQ leaps from IQ = 92 to IQ = 130.   That is one friggin' big leap.

Since I don't have any at risk kids in my home, I asked a different question "What type of radical dramatic change would I need to do around here to go from 110 to 125, or 125 to 135, or to 160 just for the day of the big test?"

Step 1:  I'm going to give myself 18 months.  It turned out that it took 14 months just for me to get my act together as a parent, followed by 2 months for my child to get past radical core skill therapy (in one case barely in time for the test) and then 2 months to ramp up to a new level.  You can get your act together on day 1, and I'd be happy to provide a list of mistakes not to make, but if you've been reading my blog I think you're past that.

Step 2:  I want my kids to experience the same shock that these at risk kids experienced walking out of poverty into an advanced academic program run by a bunch of PhD's and taught by their graduate students. 

Step 3:  We're not going back.  I am on constant watch against video games, surfing, online chatting, and fun of any kind as my kids try their best to have a normal life.  It turns out that we only need about 20 or 30 minutes a day of heads down concentration on something inappropriately hard, but I've made those 20 to 30 minutes a prerequisite of fun.

Step 2 is formalized into Test Prep Math.  I want a single shocking 25 minute problem a day at first.  (Yes, I ramp up slowly because some kids cry and more adept kids can just zoom ahead feeling confident before the 'wham'.)   I want mistakes and confusion.  This is the birth place of problem solving skills.  If you present a child with a doable problem, there is no need for problem solving skills.  How about just easing your child along with some step-by-step and scaffolding?  You're not going to get a leap of 20+ points like these studies have found taking baby steps.

But the work is not done. There are two problems I'm dealing with in my own research. 

On one end, I just got 1 started on Amazon.  The reviewer complained "the book has so many errors".  Those "errors" are alternate solutions.  I stole this directly from the COGAT and love it.  You do a problem, get it wrong, don't understand the solution, and then dig in for 20 minutes to figure out that you assumed adding but the only available solution uses multiplication.  If you don't like confusion, don't by the book, because this is the most important skill and the base of the whole GAT skill pyramid.  I'm always worried about printing issues, so I'm getting a new copy just to check the solutions for the 10th time.  My other copies keep getting 'borrowed' by neighbors.

On the other end of the spectrum are kids who have really great math training and skip right past the confusion and problem solving steps because they already know how to do the problems.  The COGAT is a big stumbling block because it demands problem solving ingenuity.   A child never learns to solve problems if they are formally taught math, and if your child goes to a great math program like Mathasium, Level One, or Singapore, they have completely different academic world view than the COGAT.  It's not a bad thing, and could be a good thing, but it's the opposite of what I want for my children.  I'm laying awake at night wondering how to fix these kids.  It's one thing to lead a horse to water and they refuse to drink.  It's another thing if the horse is drinking gallons of water and is still thirsty. 

I'm thinking of just adding more bonus question to Level 2.  I am the master of giving a child a question that he can't answer without 20 minutes of logic and solution strategies, but it would just make people like the 1 star guy more baffled.  I could spoon feed everything in the solution, but this will just help others avoid the learning process.  I hate solutions.  Too many parents think that the whole purpose of test prep and math is to have your child know something.  It's not.  Think more radically, like 20 points radically, whether this is from 79 to 99 or from 99.1 to 99.7.  Step 0 is big goals.

Saturday, November 25, 2017

Math Over 20 Minutes a Day

I'm back in the daily math business thanks to the break.  It beats listening to 2 boys make up video games and verbally describe the action because they are not allowed to play real video games unless they do some really serious math, read, practice their instruments (together without fighting) and clean the kitchen and vacuum the place.

Each child began their official math career with 6 months of math at a level of current plus 2.  (This was Every Day Math, hard but not too hard.)  It's a right of passage and a way to impart senior executive functioning skills, problem solving skills, core learning skills, and grit.  The exercise below is exactly the exact identical same situation, just with different math.  Since then, I haven't worried about an organized program.

When the video game talk became pushing and shoving, I upped the ante.  The fourth grader would get a 6th grade math test from Virginia and the 7th grader got a rigorous high school Algebra II final.

For math at an inappropriate level, we follow these rules.  First, the kids do it.  Since they are missing vocabulary and concepts, because I haven't done current+2 in a long time, we do it again together the second time.  (We've been skimming current+4 and current+6).

I'm going to describe how we do some of these problems so that you'll see that it really isn't about math at all.  It's about a high degree of analysis and problem solving because a) the MAP is a big deal this year for us and b) multi-digit multiplication is 100% useless and distracting from important skills.

Here are the rules for doing math at home:
  1. Let the child do the work on their own first with no help.
  2. When you go through it, pause to take the material on it's own terms because the child hasn't seen it yet and he started with a final and not the actual text book.
  3. Figure out a way to cheat.
#3 is the key to everything including academic and life success.  It's why I expect my kids to open the book the first time the night before the Chemical Bio Organic Genetic Engineering Chemistry final and ace it the next morning.

Here's how it works in action.  As you can see from the picture, we got off to a slow start on the very first question.

The way to cheat on this question is to note a) the answer is in the vicinity of 25 or 26, and since no addend ends with 9 in the 10,000th place, the answer has to be C.  Much more gratifying than multi-digit addition which is used no where in life or in any other class or in college.

Question 3:  6x + 3 = 3(2x + 3)
Here are my comments:  Every time you see x in an equation, be prepared to rearrange and transform the equation.  This is a good place to learn how parentheses work without the spoon feeding and repetition of Pre-Algebra. "x" requires a few long discussions under the heading of Power Bucketing (aka setting up future math) but we already had those discussions.   The equation becomes 3 = 9.  Now pick the answer.  On the SAT, we'll switch to looking for a subset relationship to find the answer and other cheatiness, but for now they need to learn transformations.

Question 5 was even more fun.

It took a while, but we settled on putting 6.23 x 9.3 within the bounds of 6 x 9 and 7 x 10.  You can see this work in the middle of the page, and you can see on the right the framework my son used to actually calculate 6.23 x 9.3 using successive digits.  The cheaty-est way requires the most work, higher order problem solving skills, more creativity and more time.  I love it when a student is excited that cheating turned a 5 minute problem into a 20 second problem, never stopping to think that it took us 20 minutes to get there.  

By the way, one reason to let the child do the work first is because children will most likely resort to calculations and they all need practice in arithmetic.   The main reason is that whatever they answer is expected to be wrong, and Math House loves mistakes.  

Question 6 in the picture is awesome.  It involved a wiki definition of "statistical" followed by an evaluation of each statement on those terms.  'Statistical' is a summary or characterization of the data, and 3 of those answers ask for a single number.  The last one is seeking an average.  Also, statistics usually follows the rest of the material in a math course and in terms of timing usually ends up being taught after the you-know-what test.  I'm not sure how the little one ended up with the correct answer the first time.

The Algebra II final was loaded with solution strategies, most of which were not discussed yet and ended up being more boring to re-work than the 6th grade test.  The breadth of the Algebra II test was good so it was worth it.  On both tests there was this little gem:

Alegra II
Evaluate 3n/(n + 3) + 5/(n - 4) 
Order 1/4, 1/5, and 11/40 

In both cases, you can't compare apples and oranges, or 1/5 and 11/40.  And you can't directly add things with different denominators.  What happens if you multiple something by 1?  Does 100/100 qualify as 1?  What about (n + 4)/(n + 4)?  These are my comments as we meandered to comprehension.

Many parents complain that they don't feel qualified to understand math in a child's terms, let alone teach it.  Here is my response.  What they are really saying is that they don't want to take the time to understand and solve the problems of a) the math on it's own simple terms (the 'simple' takes time to get there) and b) the problem of working slowly with the child until the child has learned some skill.  And yet they expect their child to magically acquire patience in analysis and problem solving skills?

You've got a third problem, which is that at school the child is learning boring spoon fed repetitive work that values memorization, speed, and 100% accuracy.  So if they don't learn the important skills at home from you, they won't learn it.

Learning starts with unknown, slow, and 0% correct.  I'm happy if you feel like you're at this point too.

First, acknowledge that your baffled.  You can be baffled on how to teach this while the child is baffled on what to do.  This sets the right tone and makes everyone comfortable.

Next, look at what you know about the problem.  a) Your child stinks at parentheses, isn't 100% conversant with variables that span R, doesn't see the need for common factors and b) you don't know how to teach them.

Third, come up with a strategy.  It will have the following components:  a) it will take a long time, b) it will be step-by-step, c) you have to back track on something simpler, like having the older child do the fraction problem first and having the younger child compare 1/2 and 1/3 first, d) you will have to reevaluate and try again.

The preceding paragraph is going to be the child's take away.  Someday they'll understand "n", common denominators, how to use parentheses, algebraic transformations and the rest of it.  At least they'll be comfortable when they get to it in school or on a test.  But those problem solving skills starting with 'Baffled' and ending with 'Try Again' are so powerful both of you are on a path to 99%.

I'm going to conclude with a warning about accelerated math in school or after school.  It is now common to teach Algebra II in 7th grade and Geometry in 8th grade, or to skip to Geometry in 8th grade, despite the compelling evidence that the result is many kids quit math early in high school and reporting hating it.  At this young age, speeding along, the kids memorize and learn to use the concepts.  There is little if any time devoted to the key skills that are taught in high school, especially in Geometry, learning adult level problem solving skills doing Geometry proofs.  I would be surprised if any 8th grader could prove to me that a straight line is straight, let alone prove each building block of Euclidean Geometry all the way up to trig, but that's what a rigorous high school class teaches, because it is essential to Calculus.  It's essential to thinking.

We've been doing high school Geometry proofs for the last 6 months with both kids.  Algebra II is a belated detour.  You might imagine that the approach is dramatically different than an accelerated school math program.   In 14 months I'll tell you why this is so important.

Thursday, November 23, 2017


It's time to take a closer look at the WISC V.

I haven't looked at this test since I began my early research.  It is a combination of the Word Board, the COGAT, and an IQ test.   I studied the concept of IQ extensively, and once I realized it was a myth I turned my attention to cognitive skills.  IQ is not just a dumb concept, it is detrimental to cognitive growth.  I believe that studying for the COGAT when done properly has an immediate impact on the child's academic, problem solving, and cognitive abilities.  If the WISC preparation is done properly, growth will occur in the parts that are not designed like an IQ test.  The IQ part should be learned elsewhere.  It takes a bit more time, but it's doable.

Some school districts include the  WISC in their GAT screening, and a seat in a gifted and talented program can is worth it so let's beat the WISC IV and V.  The WISC is often paired with the MAP and the COGAT, and these 3 tests have some overlap, not to mention the fundamentals of thinking skills.

If you want to know what is on the WISC, take a look at critical thinking's page.  I like some of the material, but it's way too easy to get to the cutoff scores in almost all school districts.  Even worse, there is a chart showing indicating how fast a pace you would keep to get through BTS Level 1 for a first grader.  Working slowly through this book in K develops deep skills.  Going through this whole book quickly leads to what I don't know.  Nonetheless, I noticed their "Suggested Test Prep Plan" is chock full of best practices, as is the concept of a timing chart, but this content could use about 50 pages of explanation.  I'm a big fan of their website and how they've improved it over the years and if you're really facing 4 weeks to the test, get a practice test and do what they say.  (Someone once accused me of making money off my recommendations but it should be pretty obvious that companies wouldn't pay for this type of analysis.)

My starting point for a WISC refresh was the authors' books on comparing the WISC 4 to the WISC 5 and the new practitioner's guides.  Then I looked at adult level IQ tests and asked how I could present permutations of fundamental concepts to little kids so that they could understand the question but not get it right, apply the underlying skills, and thereby learn something.

The list of question types doesn't hint at the grilling your child is going to get on the verbal section of the WISC.  It's all Word Board.  Get a vocabulary workshop book, post the words on the fridge, and ask your child about similarities, synonyms, used by, part of and the rest.  If you have a 4 year old, get Pre-K Phonics Conceptual Vocabulary and Thinking and then you won't have to worry about vocabulary, reading, or grilling until the 7th grade MAP.  

Matrix reasoning, picture concepts, symbol search, perceptual reasoning and arithmetic are question types that would benefit from COGAT practice.

That leaves the IQ portion.  Sequencing, Cancellation and Symbol Search stand out as blatant attempts to measure IQ, but elements of IQ skills can be found in other question types as well by design when this test is properly administered.

The difference between measuring cognitive skills (COGAT) and IQ (WISC) is the difference between measuring the student's ability to solve a novel problem given enough time, and the student's ability to quickly solve a novel problem because they have had so much practice doing it in the past.  Some school districts want to find children that have the potential for a strong academic performance and use the COGAT.  Others want to see evidence that the child loves school so they add the MAP.  The most short sighted school districts only want kids who's parents have PhD's so they add the WISC.

The student develop the additional IQ related skills from an early age because they love to read and solve puzzles and do crafts and play with Legos and other activities that would give school districts confidence that this student lives in a primarily academic household and will perform in an accelerated program in the long term because the parent is actively involved in making the house a learning environment with no TV or screens.

School districts that use the WISC wasting a lot of academic potential in their student population.  They are also avoiding the problem of program full of kids who prepped for the test and then ended up with a subpar academic record by middle school.  The latter case can be solved with a better program.  The former case is called a city where you don't want to raise your children.

There are 2 challenges with teaching IQ skills.  First, turning your house into a top notch learning environment requires a lot of time and effort on the part of the parent.  You have to do it, but it takes time to make up for 20,000 hours that you spent with your child not acting like you have a joint PhD in literature and biology.  The second challenge is that training your child on sequencing and memory skills is counter-productive.  You are taking the thinking out of thinking, and giving your child a time limit on anything also takes learning out of learning.

The solution is working memory, as in working + memory.  In Test Prep Math, the word problems, quantitative and visual spatial sections all have multiple problems superimposed.  The original intent was to slow the child down to the pace of learning by replacing one-step problems with multi-step problems, and cheat the cognitive skills tests by getting the child used to something 2 or 3 times as comlicated.  The magic number for working memory is 3 but substeps tend to push the ceiling.  For the word problems, I add confusion to the question (why ask how many they have altogether when you can ask it in 100 other ways?) and the result is that work to identify the equations to be solved, putting them in memory while uncovering the rest, and then keep them in memory while being doing the work.  Kids who have had Level One or Mathasium are forbidden from using a pencil because they've already had training in problem solving (instead of learning) and lose the 'working' part of working memory.

The result is the natural tendency of children to form their own algorithms and techniques to manage this process and not have to sit their for 25 minutes solving a problem 5 times to get the correct answer.  Those internal algorithms evolve naturally on the student's own terms, and once they are there, we step beyond solving novel problems to applying solution techniques.  When a child takes less time and makes less mistakes on a series of problems that continues to grow in complexity, we've stepped from cognitive skills into the realm of IQ skills.  What I don't like about after school math programs is that they skip the internal process and explicitly the methods to the child.  We get a child who appears to be great at math, but didn't make the internal effort and is not going to gain the long term benefit.  It's like a performance enhancing drug instead of a fundamental long term improvement; long term research is ongoing.

But Test Prep Math is for 2nd and 3rd grade.  I've gotten complaints from the very beginning that there is no Test Prep Math for 1st grade.  Almost there.  In the mean time, with what ever material you use, give your child space to grow their working memory.  If you don't help, at least for the first 10 minutes, or help by just asking the child to walk you through each micro bit of the problem in excruciating detail, you're letting memory work, as in holding things longer and letting those little analysis and problem solving skills develop on their own. 


Tuesday, November 14, 2017

Blanket Parenting Advice

I often get questions about different situations under the heading of 'Is This A Problem?' and while I'm struggling for an intelligent and informative response, I'm thinking Your child is awesome.  You should help me with my parenting issues.

The spouse of one of my Power Dads told me her mother directed her to add " by Kindergarten" to each of her questions to diffuse the anxiety of younger parents.  "Is my child going to learn to use the potty?" becomes "Is my child going to learn to use the potty by Kindergarten?" and an issue magically becomes a non-issue.

Our pediatric physician was more direct.  This person will easily have 90,000 people at his funeral because he was so awesome.  Throughout the entire time we qualified for his practice, no matter how serious our worries were, his answer as always "this is not a problem."  Looking back, I can see that he meant either a) this is not a problem, or b) this will take care of itself in a few years so stop worrying about it.

In the academic space, a bit more advice is needed because there are deadlines.  In most school districts, these deadlines reappear every year to provide second and third chances, although in some school districts there are far fewer seats after the first test.  Middle school provides a whole new round of opportunities, as does high school.  Frankly, the later opportunities are the only important ones in the long run, but this is not helpful if you have a shot at a great elementary school.

Here is my advice for academic excellence.  If you want a child who is far above average, you as the parent need to act far above average.  Average approaches are going to produce average results, except in cases where you've been doing something unusual for a long time and neglect to mention it to me.  When I ask  parents how their child scored a 99% on the COGAT and they respond "We didn't do anything" or "the kid just taught himself how to read" there has been much more going on than the parent realizes.  It's usually a case of the parent doing all the right things in a hand-on/hands-off environment without knowing it.

The material to grow cognitive skills is well known at this point.  I'm looking for a change in testing to respond to the routine high scores these days and I'm seeing early evidence.  There are programs, classes and books behind the rise in test scores.  The average outcome of this approach is great but no guarantee of 99%.  But there are thousands of parent-child interaction hours and when research takes the time to observe and measure the parent-child hours, the statistical importance of everything else diminishes. The what is important, obviously; a child can't exercise cognitive skills if the child is not exposed to anything that requires these skills.  The how is the deciding factor.

Ask yourself this question.  What is a GAT parent?  Am I measuring up?  How would a GAT parent spend their time?  How would a GAT parent show interest in things?  How would this person answer questions?  What books should I bring home?  Does a GAT parent use scaffolding, and if so, how much?  Should I push my child into something over their head or let them walk their step-by-step?  If you're looking to send your child somewhere to become GAT you might be looking in the wrong direction.

In the past few years, I've answered all of these questions and the answer always involves both options (if there are 2 options) or all options (if there are more than 2).  In our case, we forgo anything that involves the car or a building that is not our house, but that's our choice.  The one thing I learned from home schoolers is that you can teach your child in 20 minutes what would take 3 hours in a classroom setting.   The key is that we do a lot less and we stretch things out a lot more.   It's not the only key, but it's a good start.

In the face of little available information, my starting point was the simple observation that if lots of people do something, we don't want to do it.  I don't want that outcome.  I don't expect this road to be easy on anyone.  In fact, we've become quite adept at doing things that aren't easy because we've done so much of it.  But we'll make it easy anyway by following the Zero Expectations and Lots of Mistakes Are Good rules.  There's the "both" again.

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.

Saturday, November 4, 2017

Practice, Training, and Learning

A few weeks ago, I re-reviewed the COGAT practice tests.  Then I pondered how different practice is from cognitive skills development.   I've been pondering ever since.

My goal for the period leading up to the test is to increase a child academic, cognitive, and problem solving skills.  There's very little difference between these skill sets.  The only academic skill that is not on the list of cognitive skills and problem solving skills is how to survive a 50 minute math class when you figured out next month's chapter in the first 60 seconds of class.

The first rule of cognitive skills development is that it's not going to happen if the child sits down and does more than 1 problem.  This is impossible for most situations, so the next best thing is a few problems for decoration surrounding 1 problem that will result in cognitive development.

The second rule is that if the child knows the rules, recognizes the problem format and knows what to do, then no cognitive development is going to take place.  Think about that.  I'm going to great lengths to make a problem confusing and hide that it's really just a plain old boring math problem, except when it's not a plain old boring math problem.

My absolute favorite conversation to have with kids goes like this:  "Mr. Math Guy, what does this mean?"  You tell me.  "I don't understand it."  Then explain it to me.  "How can I explain it to you?"  Read it again.  "Read it and still don't get it."  Then tell me what the first word means.  "Really?"  I've got all day.

The conversation works better if you use a 1970's Clint Eastwood western accent.  When this conversation happens, we're on the verge of a cognitive explosion and this kid is not just going to learn math, they are going to learn 5 or 6 really powerful life changing skills.  I'm currently working on suggested content for these kids when I get mentioned in Nobel Prize speeches.

The third rule is that you can't tell the child anything.  Telling is the opposite of learning.  This is really hard to do when the kid starts at square one so feel free to cheat a little.

The fourth through nth rule (for lack of time) is that you can't hurry, mistakes are required, you're going to end up on tangents, and any time your child gets anywhere near the Big Five problem solving skills, clear the decks and do not, under any conditions, do another problem after that because you may ruin the whole experience. You can't use a Big Five if you have a whole bunch of work to do, and I don't trust either child or parent to live in the moment and make it a life lesson if you follow up using one of these skills with more problems.

Of course, you can't do any of this 2 weeks before the test by zooming through practice tests.  Plus, practice tests are nowhere near as hard as the convoluted knot that is in Test Prep Math.  (Reading comp skills are an added bonus.)

So far, I've directly addressed the problem wherein the child does not have skills, by prepending easier problems.  I've addressed the problem wherein the parent does not have any skills by adding direction.  I've addressed the problem of what do you do if you end up with a braniac (section 2) I've addressed the problem with children who have unusual skill gaps but are competent in other areas (section 3).

The problem I'm working on now is children who are highly trained in math, like after school or weekend managed math program, and don't need the problem solving skills.

This is a pending disaster.  It's my worst nightmare and keeping me up at night thinking new problems.  I put my contact information right in Test Prep Math.  If anyone has any questions or problems, I get a call and the clock starts ticking on me fixing it.  I didn't get any calls for 12 months, other than 'great book, my son thinks your book is not lame' or 'great book, my daughter loves math now' but in the last 2 months, I've a few calls like "my son is already in the 99th percentile in math but can't pass the COGAT".  The culprit appears to be Level One and Mathasium.  I don't think it's the extra math.  These aren't bad programs.  The culprit is that being really great at math precludes the need for high level cognitive skills, the skills that are so important we should probably form a cult around them.  The COGAT, on the other hand is remedial math heavy on extra problem solving skills.  There's a disconnect.  It looks like managed math programs take a short cut and just teach and practicing math, then hard math, then really advanced hard math.  No skills required.   Then there is an add-on program for test prep skills, but it's too late.  The damage is done.  This is not going to end well.

I'm on the case.