Wednesday, February 28, 2018

Skills In Perspective

In the last article, I went a little overboard on the technical detail with some middle school competitive math.   I tried my best to lay out problem solving so that you can see it is consistent with little children and consistent with the high school, college, graduate school and post doc experience.

Let me explain this bluntly.

You want your child to have problem solving skills.  This is much better than having to help with math or hire a tutor to spoon feed your child steps from question to answer.

But if you try to teach your child problem solving skills in the hopes that these spur cognitive growth, you will fail.  It's as bad as having your child memorize formulas and rote practice applying them.

Here's a brief history of skills.  In 1945, a researcher at Stanford named George Poyla took 3,000 years of research into how mathematicians solve problems from philosophers, ancient Greeks, and mathematicians themselves, and wrote a book called How To Solve it to help high school teachers mentor their students on solving geometry proofs.  The emphasis of How To Solve it is 'mentoring', not doing any work for the student or teaching problem solving algorithms or heuristics.

By the 1970's problem solving was turned into a pre-packaged, spoon feeding program to help students apply problem solving methods to pre-algebra and more advanced maths without the need to understand anything that they are doing, let alone math.

The #1 problem in problem solving is that the defective learning approach that emphasizes a speedy, correct answer that has been memorized and practiced has evolved into a defective learning approach that emphasizes a speedy, correct answer using a problem solving technique that has been memorized and practiced.

When I finished translating How To Solve it into a method suitable for parents of 4 year olds, I was stunned to find a solid approach that also works for graduate school.   I added a step that researchers at Berkeley identified as the #1 success factor for surviving their first year calculus courses.  The first experimentee of the program is now 9 year's old, and needs about 10 minutes to get a score of 50% on SAT reading comprehension tests.  Obviously, we have a way to go, but the method is so general that if pretty much works everywhere, including assembling Ikea furniture and fixing plumbing issues.  I would recommend it simply for the benefit of not having to call a plumber.

Here is the short version of the problem solving method:
  1. Be Baffled (thanks Berkeley math department)
  2. Spend a lot of time thinking about and exploring the problem
  3. Make mistakes and try again
  4. Check your work (I added this because it raises test scores)
In between #2 and #3 sit the process of problem solving.  In the last article, I demonstrated the most powerful problem solving techniques from the standpoint of a baffled parent trying to help their child learn some new material that is way beyond the child's skill level.  Think figure matrices, multiplication, fractions, exponents, algebra, trig or whatever.  I'm going to continue the numbering from the above list and explain why shortly.
  1. Start with a much, much easier version of the problem, like 1 x 2 = 2 and just keep adding to it and iterating until you are back to the original problem.  This can take weeks if you're trying to teach multiplication to a 5 year old.  In some cases, the child is missing something fundamental from material we skipped, so we just backtrack to an easier math book to practice the prior material and then come back to the problem.  Backtracking happens a lot in Math House.  Ironically, I can teach basic Trig in about 30 minutes, but it takes months to teach basic alegra.
  2. Translate the hard problem into 2 easier problems and solve the easier problems instead.  This approach usually involves decomposition or regrouping in the early years, and gets trickier in high school math.
There are other good approaches for more advanced topics outlined in Poyla, like solving the problem backwards, applying some theorem or proof that you just learned in the prior problem (which works for both Geometry and the COGAT), filling in the missing word or shape.  If you give the child enough space to explore the problem and make mistakes, the child will learn these methods on their own, or even better, make up their own methods however inefficient.

When I combine the two lists, which is why they are numbered contiguously, I end up with 90% of my teaching method for math until we get to Algebra and Geometry.

There is a great deal of contemporary discussion on the topic of why students are struggling in Physics.  The consensus of physics teachers is that students are more interested in getting to the solution (using the internet to find the method) and less interested in learning physics.   You can find many, many books written to demonstrate the step-by-step approach to solving every class, subclass, and subsubclass of algebra problem if you wish to be an algebra expert without knowing what you are doing.  If a parent would just take a step back from Teach To The Test, you'd find that it takes a fraction of the time to get a 99.9% based on thinking and learning than a 90% based on practice and memorization.  To emphasize this point, we tend to do 2 to 5 problems a day and make much more progress more quickly than children who do 30 or 40 easy problems a day.

Learning happens from the start of the first problem until the student realizes that there is a formula or method that can be used to solve problems of this type.  When the child struggles with 2/3+ 5/7, lots of learning is happening.  But once the child realizes that each fraction has to be transformed to share common denominators, we're done with learning.  Learning also stops when the solution is checked as well, right or wrong.

The biggest complaint I receive from parents who start down the path that I recommend is that it doesn't work.  By 'doesn't work', it means that their child is frustrated, lost, and getting nowhere.  To me, this is a description of the initial stages of the process a not a defect or shortcoming in the approach.  Some stubborn kids need about 6 weeks to undo the programming from school, programming that you must know what you are doing, do it quickly, and obtain the correct answer without effort or challenge. It takes a while for the child to realize that expectations have changed. 

Sometimes it takes 2 or 3 weeks on a half dozen problems to teach the child that we are going to go slow, think a lot, be confused, hit dead ends, have to backtrack, and get things wrong a lot. To accelerate this process (meaning show the student that the rules have changed), I'm usually confused, get the wrong answer, and don't check the solutions. Once the child gets past this hurdle, the pace begins to go very quickly, and if you stick with this approach, the child will in a few years teach themselves entire subjects very quickly, or if you insist on teaching your 9 year old algebra, not very quickly but adequately.

Saturday, February 24, 2018

Skills in Action

A reader challenged me to explain how to do competitive math.   I'm excited about these problems because they demonstrate a fundamental skill set that is developed during learning to read at age 4.  It's very similar to the skills that cognitive skill sets like the COGAT teaches.  It's 100% learnable.

What I like about this problem set is that a parent can work through these and have the exact same experience that your child goes through.  I'm hoping that parents who take the time to work through this material will have better training when your child shows you a figure matrix that is baffling.  It's also a good opportunity for the parent who discovers this website 6 or 7 years too late so that I can show that it's never too late.

I've gotten numerous questions about what follows Test Prep Math 3.   I like competitive math as a warm up for SAT test prep.  We dabble with pre-algebra, but usually only in a Algebra 1 setting.  Sounds hard, but this is the skill set.

Here we go.  Picture a competitive math worksheet with 40 problems on it, that has a 45 minute time limit.  I suppose if we were serious about competition, we'd train for learned strategies to address the time limit, but we're not serious about competition, just doing a bit of daily math.  I think 5 problems is asking a lot of an 11 year old.

Question 1:  F - T - L - T - ? - ?   Find the last 2 letters in this series.
  1. I have no clue how to do this.  Anyone you has seen this question type probably doesn't have any clue because it has unlimited subjects.  But I have the most important skill of all, which is the proper way to be Baffled, which is to not care that I'm clueless.
  2. I think for a minute about adult IQ tests.  Friday, Tuesday, Something That Begins with L, Thursday.  Fail.
  3. F = 6, T = 20, a difference of 14.   L = 12, T = 20, - 8 + 8?  Fail.  Skill 2 - don't care how many incorrect answers I get.
  4. I stop and think about the question a bit.  Kids only know arithmetic, language, geometry, and a tiny bit of algebra.  Pre-Algebra is fair game.  In the real world, I should have used Skill #3 which is to spend more time thinking about the question and less time getting incorrect answers, but in competitive math with no time limit, lots of learning can happen in dead ends.
  5. Going the geometry route, all the letters have a single vertical line.  F has 2 vertical lines and T has 1.  That's 3.  L has 1 vertical line and T has 1.  That's 2.  I forgot to look at the answer set.  My skills are rusty because the answer set is part of the question.
  6. The answer set is:
    1. L - T
    2. L - B ( I think this B has no vertical lines.)
    3. L - M
    4. T - P
  7. If B is the answer, using counting horizontal lines in the series, we get 2 - 1 - 1 - 1 - 0, but if we take pairs of vertical lines, we get 3 - 2 -1.   B is the answer, I accidentally stumbled on it, and I have no clue why I am correct.  But it was the best of a bunch of confusing bad answers.
This problem took me about 15 minutes.  It's very similar to the type of work I do on a daily basis.  I wonder if competitive math tests are structured so that super duper problem solving kids prioritize the questions and their time before answering and just skip this one.  Probably.  

When you work with your child and do a problem that is really hard for their age and skill set, just like the one above, here's the benefit you both gain:
  1. You get used to working with baffling things and don't get put off.
  2. You make a lot of mistakes and don't get put off.  In fact, in my failed attempts (attempts not include above), I learned a lot of interesting things and picked up a few mini-skills on the way to dead ends.
  3. The solver is forced to think creatively and view the problem from different angels.  It will take a lot more problems to learn creativity, but since I am making a habit of baffled and mistakes as skills by force feeding my child these problems over and over again, we'll get their eventually.
  4. I never looked at the clock or the solutions.  This problem is kind of tricky and fun.  The solution will end the learning process and reinforce the Rule #1 that it's not about learning or getting better at something, it's about being right or wrong.  Rule #1 will destroy your child's ability to learn.  Rule #1 is an anti-skill.
When I work with kids, a team will really help, and I'm the only one available for the other team member, so in practice I ask a lot of questions (as needed) and make suggestions for the next attempt (as needed).  I'm always baffled.  In practice, I'm suggesting skills and approaches from my toolset of exactly 5 approaches to any math problems. 

Why is it that when your child comes to you and asks what 'dispersed' means, you're more than happy to tell him, in fact you're so happy your bright little child has an interested in vocabulary and is not skipping over unknown words when reading, but when your child gets a math problem wrong, you're disappointed?  What a horrible destructive way to teach children to hate math.  Adding a time limit makes it even worse, because then a teacher can mark of a series of unanswered questions.  This is why schools can completely eliminate tests through Junior year in high school and produce kids who blow away college entrance exams.

OK, let's see what we get out of more baffling problems.

What is the remainder when the 15-digit number 444444444444444 is divided by 9?
  1. Are you kidding - this is too big to fit in the calculator.  Curse you competitive math test author.  The answer pick list is irrelevant.  Again, I have no clue.
  2. Too hard of a problem.  So I fall back to how we tackled any math - starting at age 4, when it's too hard.  We start with the easiest version of the problem and work our way back to the harder problem:
    1. 4/9 ~ r 4
    2. 44/9 ~ r 8
    3. 444/9 ~ r 3
    4. 4444/9 ~ r 7  this is good practice for division but a fail in solving the problem.
  3. Then I remembered that when I teach division, I always make the student turn 36 ÷ 9 into 3*3*2*2/3*3.  Now were trying to turn this problem into a more solvable, easier version of this problem.  Here's goes:
    1. 4*111111111111111/9 = ?  Still hard.  Fortunately, I can look back on the first fail and continue.
    2. 1/9 ~ r 1
    3. 11/9 ~ r 2
    4. 111/9 ~ r 3.  Get it?  Light bulb. 
    5. Continuing, I get to r 0 at 1111111111 which puts 111111111111111 (15 digits) at r 5.
  4. Unfortunately, I'm stuck having to multiply the whole thing by the remainder.  This stinks, I stink, and your child stinks, so we're going to have to take baby steps.
    1. Since 1/9 = 0 + 1/9, 4*(0 + 1/9) = (0 + 4/9) ~ r 4, which is what I got in the first fail.  Notice I'm checking the answer, which is skill #4 at the base of the cognitive skills pyramid.  I suppose this requires some pre-algebra.
    2. 11/9 = (1 + 2/9), so 4*(1 + 2/9) ~ r 8, again, just like above.
    3. 111/9 = (12 + 3/9) but 4(12 + 3/9) is going to give us 48 + 12/9, slightly confusing, and I have to go read the question yet again.  Oh yea, we're dividing by 9, and trying to find the remainder, so I can write 48 + 1 + 3/9 ~ r 3 just like expected.
    4. At some point, the lightbulb goes off, and I can just jump to 15 ones's/9 = (something big + 5/9), and I multiply by 4 and get 4*something big + 20/9 ~ r 2, which is not even on the answer list.  The choices are 4, 5, 6, and 7.  
    5. So starting over, which I'm totally used to because we do it all the time, I note that the 9 digit number 111,111,111/9 = 12,345,679 r 0, duh, should have thought this though.  This makes 111,111,111,111,111/9 = something big 6/9 (since 15 digits is 6 more than 9 digits), and 4*(something big + 6/9) = 4*big + 24/9 = 4*big + 2 + 6/9, giving me the correct answer of 6.
We've got 3 big solutions approaches that we start using when the child is about 3 years old.  

At some point, your child is looking at * * * * * * of something and you ask her to count.  She answers 12 or 5 or gives up, so you start small, like *, then * *, then * * *.  I teach addition, fractions, and multiplication this way.  It works in graduate school and it was by experimenting that I found it works really well at the youngest ages.  It works on pre-algebra.  It works on all forms of high school math.  It's required for competitive math.  Math books do this from chapter 1 through chapter 15, but we do it in 5 minute increments and don't really need a math book.

Next, when a problem is too hard, turn it into an easier problem.  This is the foundation of algebra.  You might as well start now.

Finally, notice that there are 3 steps to this problem.  If you've seen TPM, you know why I think 3 is so important.  It builds working memory.  For the age group for the problem above, we're probably beyond working memory, and if not, doing these problems will bring it back.  But the working part in 3 steps is where the little brain turns itself into a big brain by defining relationships and patterns and working abstractions into algorithms from one part of the problem to the second to the third.  You see all three in the solution above.  A genius can do it in one step only under one condition:  the genius worked through enough of these problems to get really good at devising and applying algorithms. Don't be fooled into thinking it's genetic.  The rest of us are happy doing the 3 steps one step at a time.  One step at a time is good for 99%.

Moving on, how about this problem.  What is the value of 1 - 2 + 3 - 4 + 5 - 6 + ... + 81 - 82?

This problem not only demonstrates the value of spending way more time exploring the question than trying to answer the question, it also demonstrates the value of what I call "Seeing".  I learned it from the COGAT.  It involves looking at the problem from different perspectives.  

I checked to see that there were an even number of elements to this equation, all equaling negative one when paired, and came up with -41.  Eight minutes of thinking about the equation and 4 seconds deriving the answer.  With 40 questions and a 45 minute time limit, I would have come in last on the competitive math exam.  Can you picture me sitting with a bunch of 6th and 7th graders? 

This next question is my favorite and a really great exercise on it's own to teach exponents.  I love this question.  This differs in an important way from the math I would give a younger child but is identical in nature to the non-verbal section in TPM.  It involves doing a lot of work, organizing and thinking about it, and then answering.  

If a and b can take on the values in [0,9] (meaning that they can each be 0, 1, 2, ... 9), then the expression ab can take on how many different odd number values?
  1. To start, I just created a grid with 0-9 on the rows and 0-9 in the columns and started calculating the expression based on inputs.   In a competitive math situation, this is a waste of time and requires thinking, but with most kids (and 9 year olds), I make them use the brute force approach because they usually have never seen aoutside of 42.   I've got a whole exponent crash course (including negative and fraction exponents), but this seems to be a good starter exercise.  
  2. The rows are a and the columns are b.  I didn't calculate the *'s but I could have.  
  3. This seems to be a fail.  Too hard.  I did notice that only one zero in the top row and one from row 2 and column 2 are going to be included.  What is zero raised to zero?  It's either one, zero, or undefined, but if you read the question again (and you should because it's a skill), it doesn't matter to the answer.
  4. After rereading the question yet again, I noticed that I only have to deal with ODD numbers.  With the exception of '1', the rows with 5, 7, and 9 qualify, and since 3*3*3*3 = 9*9, the row of threes where the exponent is odd also qualifies but not when the exponent is even.  And we can add 1 only once and ignore zero.  And that gives the correct answer  of 27 (the whole row of 5,7,9) + 1 (from the one) + 5 (from 3 row where it doesn't repeat a value from the 9 row) =  33.
  5. It's possible to jump to step 4 as a competitive math coach, but not a regular bright kid doing competitive math coach.
I'm guessing the question needs about 5 readings before this work can begin.  I've watched little mathematicians create charts to answer questions and it's very gratifying.

Finally, the last question is this.  If x and y are integers and 360x = y3, what is the minimum possible value for x + y?  At this point, we left all kids under 4th grade behind and we're just looking at algebra.  Or are we?  Yes, I'm running out of steam and have already covered all the really great problem solving techniques. 

  1. After 30 minutes with the question, I decided that x is just a function of y, so forget about x.  Just find the smallest possible value of y.  Or do algebra.  It's late, I've exceeded the maximum good thinking time of a grade school child of 25 minutes, and the Olympics are on.
  2. But I don't like 360, so I wrote 2*2*3*3*2*5x = y3.   Then I rewrote it to be 2*2*2*3*3*5x = y3.  You can see that if y is an integer, x has to be 3*5*5, making y = 2*3*5.  So x = 75, y = 30, and the answer is 105.  
I got the entire solution correct by following these steps:

  1. I had no clue what to do.
  2. I went off in the wrong direction by trying to use algebra, which I can, but doesn't solve the problem for a kid who doesn't know algebra.  Fail.
  3. I tried again.
  4. I spent more time looking at the question and eventually started to rearrange it in the hopes of finding an easier problem.  (I.e., I used one of the big five 5 math problem solving techniques.)
  5. I looked at it, specifically looking at the root primes against the exponent on the other side of the equation.  I used my power of seeing things differently.
  6. The answer emerged with no effort.
This is why studying for the COGAT is so critically important.  It's the easiest way to get the skills.  If you missed this opportunity, there are other opportunities including competitive math.  It seems harder and more complicated because your child is older and the math more obscure, but it's about the same.   If you did this when your child was younger, you would have blocked out all of the tears and frustration by now and just remember how it all worked out.  Same with bed wetting in the middle of the night.  Remember that?  Of course not.

Is there anything different between a child who does this problem successfully and one who gives up?  Not mathematically.  It's all in these base skills which are 100% learnable and needed for high school math.   If you want a strong competitor in a math contest, you'll need interest and a lot more practice, but if you just want a five on the BC Calculus without having to nag your child or hire a tutor, do a few problems and focus on the skills.

Saturday, February 17, 2018

Totally Doable If Done Right

In the last few weeks, I've stumbled across a whole new group of people who are suddenly concerned about their child's education either because they decided it would be nice to have an actual child in the next few years, or they have an actual child and just found out about the COGAT, or they are getting COGAT scores back and deciding that it's time to get serious.

My Power Mom's Group, or PMG, from last year is officially demoted to Last Year's Power Mom's Group because your kids all met their ridiculously high cutoff goals (and are solidly on their way to additional goals).   There's one more item on the todo list for the next few months and then I'll declare a 100% success rate based on selection criteria that includes a) great parents and b) capable kids.  The new members of LYPMG are going to get heavy doses of my super secret program to crush the MAP test in the coming years.  How similar are the COGAT and the MAP?  COGAT skills are a prerequisite of the MAP, but the COGAT type math isn't what people generally consider to be math and the MAP has way more math than anyone realizes.  If you are not in LYPMG, then you'll read about my super secret MAP program but you won't realize that you're reading about it until I can get everyone in the house past the 7th grade MAP.

For newbies, I've been working on a less insane sounding description of my math approach, with a nice sounding title like Easy Fun Math*.  (*Also known as Ridiculously Hard Insane Math until you get it, and then it's just Ridiculously Hard Math.)

Here goes.

First, read read read read.  If your child only has 60 minutes per day of at home schooling, devote 40 minutes to reading.  If your child has 6 hours a day because it's Saturday, devote 5 hours and 40 minutes a day to reading.

Secondly, do not, under any conditions, every teach math.  The skills your child needs to excel in math are organizing, seeing patterns, trying again, iterating, comparing, trying out different options, defining, extending, explaining, rethinking, simplifying (ie organizing), decomposing (ie organizing), and not being put off by mistakes, lack of information and clarity, and total confusion because if your child isn't working in on a math problem that starts with mistakes, lack of information and clarity, and total confusion then they are not working on a math problem that will develop the skillset.   The super advanced skill set for math includes good executive skills and a lot of Grit.  If your child develops these skills under your guidance, your child will excel in math.  If you teach math, your child won't need any of these skills, won't develop them, and then someday will fail at math.

Look at 'First' and 'Second' again.  Higher order math skills are developed by reading.  This really matters when your child is 2 and 3.  By 4th grade, it will be assumed but not a major factor in the program.

Third, at the 99.8% level, which is totally doable if done right (Totally Doable If Done Right, my new motto, and this just replaced the original title for this article which was Advice for Newby Math Parents), there are a lot of parent skills involved.  While the child is learning each new skill, you will be learning a new skill.  Your child will see math in a different way, and you will see coaching math in a different way.

Forth, your child's math score is going to be constrained by working memory.  I can't stress this enough.  School math needs one or zero working memory buckets in the brain.  Think 'Ann has 2 apples and Bob has 5 apples.  How many apples do they have together.'  Test Prep Math starts with 2 and ends up with 3 working memory buckets - or more - on every problem.  I've settled on 3 since it appears after 3 a pencil is needed.  

Test Prep Math emphasizes messy, sometimes unanswerable problems (in clumps of 3, all mixed up and interspersed with vague words and ridiculous plots).  Now you know why.

There is an ongoing debate on whether or not children should memorize their math facts.   Teachers who need to get all 30 kids in the class past arithmetic errors in 2nd or 3rd grade are generally stuck with memorization exercises  - even in GAT classes.  Researchers who are figuring out how to get kids to the upper levels of math excellence can explain why memorization is counter productive. 
If you search 'Boaler Memorize Math Facts' you should find a few really good articles explanating why memorization is a bad idea by the leading researcher in this field.  You may also come across an counter argument from Greg Ashman that totally misses the point, but get's so close with this diagram that he's one sentence away from solving his own problem.  Look at this diagram:

Note to Ashman, the goal here is not to use long term memory to help with the math facts but to triple working memory.  Also note that this diagram makes me want to sneeze.

Boaler attributes number sense to strong math skills.  Number sense and math fact memorization are two exclusive roads to math, and memorization falls short.   In my ground breaking research I found that use of working memory isn't just a tool for math, it's a math skills generation factory.   The child learns the next level of math skills while working arithmetic in working memory.  When people see the term 'Working Memory', they see 'working MEMORY'.  It's more accurate to view this as 'WORKING memory AND MATH SKILLS GENERATION FACTORY'.  Please note that Boaler's research concerns making math accessible to everyone, but my research concerns a child who just blew away the COGAT and is looking for the next big leap in skills.  

Maybe groundbreaking doesn't cover it.  Here's what we got out of the workings of working memory in action:  an 8 year old who is solving problems off of middle school competitive math tests.

When I wrote, and rewrote, and refactored and added to Test Prep Math, I met my goals to tackle working memory, base skills, and no math if it can be helped.  I failed on the no math part because I couldn't help sneaking in math.  A little geometry, a little algebra, and if you look closely, you'll see the makings of other maths, but I generally avoided division, and avoided decimals and anything else that is on a Common Core list.  This approach doesn't work for everyone.  Some people are short sighted and think of math as topics from a math book.  Others already taught their kids math and the horse already left the barn.

One of my favorite exercises is to do Every Day Math Grade 2 in K.  For those that missed the opportunity in K, it's simply known as Current+2.   I think of this as an exercise in Grit and not math, kind of a warm up to the challenge that will follow.   Last week, a reader shared her child's current math situation which sounds so dire, what with mistakes, frustration, and not getting it.  Once again, my children are even worse in comparison, but we manage to score consistently in the high 90's (like 99, which is what I expect) and do almost no work at all.  One year ahead in math for us and maybe 40 to 60 minutes during the week.  That leaves plenty of time for reading, crafts, and projects.  My secret isn't smarter kids but kids who don't quit.  And we do things totally different, like work smarter and not harder.

After successfully avoiding the memorization of math facts, I've extended the counter cultural approach with not really ever learning math or being remotely competent in any one math topic.  Focusing on underlying skills for years at the expense of math has really paid off in a big way.

You'd think the next step after Test Prep Math would be learning actual math, maybe tackling Pre Algebra.   Instead, we took a detour into competitive math, not really like school math at all, and then I've opened up 7th through 12th grade math topics for any given weekend.  I think we have about 3 20 to 30 minute sessions each week, and the topic could be a first look at derivatives, exponents, polynomial zeros, 'What is sine and why am I making you go through this pain?' or anything else.  One week it was exponents, and the next week my older kid saw logs for the first time and had to invent and derive formulas for logs that corresponded to the exponential formulas that we worked in the prior week.  When this child sees logs again in a month, he will have remember exactly zero of it, but he's got the tools to make short work of it.

After 4th grade, the little one will spend the next year or two working through SAT books.  Other parents will try this and find that it's a disaster.  Our experience will be even worse, but we'll plod on come out with 2 completed books, about 18 practice tests in all, and then move on to the reading comp sections.  I've recently summarize the parent coaching skills needed to get through this approach successfully.  When the 9 year old gets through the first page, 3 problems attempted, 3 wrong answers, and a lot of complaining and tears, I'll wonder why the heck I'm doing this.  Then I'll remember that I've done this type of thing many times before, and it will magically work out in the end.

Saturday, February 10, 2018

Visual Math Et Cetera

For years, I have been asked for a recommendation for 4th grade math.  I now have one, and one for 5th grade as well.  It's called Visual Math.  These are not expensive books.  The authors are from a ground breaking group of researchers that I've been following since the beginning of  Back in January, I wrote an article where I said that our current math curriculum needs to be flushed as an artifact of the Industrial Revolution.  There is equally challenging, more engaging, more pertinent math to the information age.  Visual Math.

Except that I'm stuck on fractions, polynomials, mononomials, exponents, algebra, trig and calculous because darn it, they show up everywhere in math and all fields whether you're doing machine learning, number theory, or Hollywood CGI.  I guess I'm always one rebellion ahead of the next trend.

I don't face the same broad classroom education challenges that the authors of Visual Math face.  I face the challenge of a single kid.  My idea of visual math starts with COGAT test prep, Building Thinking Skills, and the rest starting ASAP, like age 4.  See my curriculum page.   In a house enriched with crafts followed by Minecraft, visual skills are overdeveloped.

But the genius of Visual Math isn't just a much better more appropriate visual (and thus more timely) curriculum, it's the approach outlined by Jo Boaler years ago that is question heavy and solution light.  In other words, spending time understanding and defining the problem, whatever that may be, in the process really learning math, and as an after thought deriving a solution.  You've heard it before from me, and this is where I got it.  There is much more to the approach beyond this.

I'm a big fan of a single problem that is hard, multi-step (working memory intense) and requires a lot of time to solve, preferably something goofy or non-sensical, if that's what it takes to turn a predictable answer into an argument.  I don't want a child to come out of this having mastered 3 x 5, which is useless, but having mastered getting there from the unknown, or better yet, an unknown mess.

And that brings us to 1/2 and 2/3.  A few months ago, a reader asked what to do about struggling with fractions.   I offered to get on Skype, but since I'm insane, and can turn any 30 second problem into a 30 minute challenge, the reader declined.  Too bad.

There are 2 parts to a good fraction problem. 

The first part is 1/2 takes about 3 brain clicks to understand.  I think 98% of the problem with fractions is that kids expect 1 click, they don't get it on one click, and they are frustrated or worse.   I watch this with the brightest children trying to tackle fractions at a totally inappropriate age.  The second part is the fraction in a more complicated setting of a pre-algebra problem.  Too hard for younger kids, but doable at a pace 10 times slower than a 5th or 6th grader.  Solving a fraction problem is multi-step.  When I work with fractions and children, or algebra, or exponents, I expect a few weeks to get them to admit that they have to work the problem step by step.  They are determined to do one single step, because it's one problem after all, and if they have to do 3 steps, then it becomes three times the work.

Kids who are trained in math hit a wall with fractions.  Kids where are 99.9% wizzes hit a wall for the opposite reason.  Both groups underestimate the problem.

Lately I've been working on the next challenge.  How quickly can I get kids to be adept with pre-algebra, exponents/logs, functions, geometry proofs, algebra, trig and calculus?  By quickly, I mean a small number of problems and weeks per topic.  My group is 4th to 7th.

In each case, a few problems can be used to explore the basics.  During this time, there is wonder involved with the new syntax and the concepts that it articulates.  Like the first time a child stumbles on negative numbers or square roots.   A few problems get the job done.  To take the next step requires a special problem solving approach for each field.  We avoid the complicated applications that fill 90% of a decent text book and just stick with the basics. 

I've come up with a one session introduction to trig that addresses many of the questions (about 25%) on a good trig final.  One session for a 9 year old.  I remember struggling with this exact same material for about a month in high school, trying to remember formulas.   I'm really disappointed about how bad the course was and how unprepared I was (not having studied math between 1st grade and trig). But I'm mainly disappointed with the approach to math from the 1920's which I used in high school. 

The last thing I'm going to do is explore the other 75% or so of each of these topics.   I think this will be an 8th grade exercise.  Is it possible to send a child to high school prepared to be bored with A/B calculus or chemistry?  Can this be done with almost no work whatsoever?  I'm starting to think so. 

I enjoy getting articles from readers that include an age and a topic and a description of how much they are struggling.   I think, wow, we struggled much worse.  I can tell them that and actually solve a problem.  I can also state, if needed, 2 or 3 ways to get past it and how long it will take (longer than you think.)  In some ways, this is just like potty training.  Some parents wring their hands over every trip to the potty, and others let their kids poop all over the place until the problem takes care of itself.  The only thing I did differently was discuss plumbing while cleaning the poop off so that I'd have someone I could count on someday to clear clogs.

Someday is almost here in math.  In plumbing, my 13 year old routed the pipes right before his birthday.

Saturday, February 3, 2018

Innovations in Math Education

I promise as soon as I complete this article I'm going to start populating the reading list in the prior article on reading.  But first...

My 8 year math program is about to come to fruition.  In April, math Experimentee #1 (a newly minted teenager who started K with a bunch of my newby missteps) is going to take the MAP test, and after a long interval of not caring, this score counts and it has to be a 99%.  In this article, I'm going to summarize where we are, demonstrate the leap in math skills that happens in 4th grade, demonstrate how my math program is dramatically different than regular programs, and present it in such a way that I lose most readers before I get to the end because that 99% is competing against about 10,000 other kids in Chicago who's parents are all googling 'How To Get 99% on the MAP So My Child Gets Into A Decent High School'.  Also, I'm going to discuss my approach in purely in 4th grade terms to help parents of younger children plan ahead, and explain why Test Prep Math is the way it is. 

Let's start at the beginning.   My first goal back in K was to conquer Every Day Math.   We didn't have to pick everything up at once, just a lot of hard work to show 'You Can Do This'.  My goal was simple.  For Experimentee #1, the goals focused on entering a GAT program in 1st grade, given that we were totally behind because we did nothing to prepare for it, not even phonics or learning to read, but at least the math would be familiar (it would be EDM Grade 2 - a complete repeat) and he would have some confidence.

After crying, forgetting, getting them all wrong, spending a week or two on a single 6 question worksheet page, having to find 1st grade books to practice concepts and skills we never saw before, I transformed the following survival skills into Academic Coaching Skills that we would use for the rest of our lives and pass down many generations (hopefully) of bright descendants until one actually wants to study math in graduate school.  Here they are:

  1. Set Your Expectations To Zero.  Don't expect your child to get anything correct, understand it, remember it, work on their own, or anything.  Even if you do the same problem every day for a week and it's 7 + 6 = ?  This is the parent skill.  The child-parent team skill is to enjoy 'Being Baffled' on totally hard work that has never been encountered before that will take a lot of time to sort out.
  2. Make Mistakes.  Mistakes are the key.  After a while I stopped looking at solutions because I expected mistakes.  
  3. Take A Long Time.  When we slowed down to 30 minutes per problem, we started making progress.  This is also known as 'Read The Question' where we spent more time thinking about 7 + 6 and what it could mean and how to work it before solving it.  
  4. Other tips I put in the blog over the years, but the top 3 were the key.
So here's what we got.  At one point, we sat down and looked at Student Journal #1, with every single problem answered.  Every single problem.  No child anywhere does every single problem in a math book, or every page or even every chapter.   This is a rare and invaluable life lesson.   Experimentee #1 has an extremely high tolerance for work, chores, painful work, hard chores, ridiculously hard chores.  Even better, Experimentee #1 is not put off in the slightest by being totally confused on material that is way beyond his abilities.

Somewhere during this process, the speed of learning and work accelerated to match the challenge, and by about 1/2 through Student Math Journal #2, we quit because the challenge was gone.

Experimentee #2 experienced hard core phonics (Pre K Phonics Conceptual Vocabulary and Thinking age 4.0) and hard core math (Shape Size Color Count age 3.9) because I wanted to address any gaps I found in GAT preparation and more importantly COGAT prep, and did it with a sledge hammer the size of an SUV.  Experimentee #2 has math skills that Experimentee #1 will never have, like a child who learns to play the violin from birth will always outplay a child who picks it up at age 6, but Experimentee #2 has a completely different work ethic.  Experimentee #2 will sit down with something quietly for hours and master it, but not without a lot of complaining about the fact that he can't pick it up immediately.  Experimentee #1 never complains.

There is a completely different path for K and 1st grade that will produce almost identical short term results.   Many parents enroll their children in an after school math program.   In a good program, the child learns problem solving skills and solution strategies as well as practices math daily.  This is not a bad approach, but it is not consistent with the goals I mentioned above and a few I am going to add shortly.

After 1st grade, we stopped learning math and went more hard core into Test Prep Math.   This series is not about becoming adept at advanced math topics, but becoming adept at navigating convoluted questions, staying in the 'math game' because the questions are somewhat on the goofy side and don't include boring, manufactured math book type questions, and building working memory.  This book is not designed for children already at the 99% level for math, it's designed to get them there shortly thereafter.  I've had a few parents who's kids finish 2 years of after school math (and are at 99% already) complain that the beginning of the book is too easy.  Kind of a 'duh' moment for me, but one I need to mention for those kids, Test Prep Math Level 3 in 2nd grade is preferred.  The purpose of this book is to lay the groundwork for 99% thereafter, not to put a 99% kid at 99.9%, except by accident (which is what we experienced, by the way). 

Instead of more math, we went directly from Test Prep Math into reading comp questions.   This should be obvious from the problems in Section 1.   Section 2 takes us directly into competitive math questions (because I need something to fill the gaps before ramping up real math in 4th grade).  But the MAP score is only half math; the rest is reading comp.

For those of you who get SAT books when searching for Test Prep Math, here they are:
From 1st through 4th grade, we only stayed a year ahead in math while I put together the basic skill set that we need.  This basic skill set is very similar to the skill set that kids would use to survive an advanced engineering or abstract math course in college but it's missing formal solution strategies.  College is the other goal, and I'm thinking ahead as usual.

At inappropriately young ages, while we were biding our time putzing around with current + 1, I started introducing advanced topics, just for fun, just to exercise thinking and start to explore the wonder of math. It was enormously enjoyable to surprise a kid with these types of questions:
  1. What is 5 minus 3?
  2. What is the square root of 4?  9?
  3. What is 1 divided by 2?
  4. If 1/3 of my donuts are chocolate, what percent of these are not chocolate?
If your child sees any of these questions for the first time in school, I guarantee the wonder, fun, learning and enjoyment of math will be totally crushed out of the experience because your child will be presented with definitions, comprehensive examples, and a long list of routine problems that have nothing to do with anything.   It then just becomes a pattern matching and lookup referencing exercise.  The child will 'learn' math, but not know how to learn.

Sometimes we would resort to backtracking, which is finding a workbook or online resource to practice the material during the learning process.  If we got '1/4' kind of but not really, a worksheet might fill in the gaps.  If a concept (fractions in this case) requires an understanding of division that is not there, we would certainly backtrack to a division worksheet and then come back to fractions.

Over time, however, I discovered the power of bucketing, which I subsequently labeled 'Power Bucketing'.  This is very similar to what I witnessed with Experimentee #1 going into 1st grade and being handed the same EDM Grade 2 workbooks that were completed the previous year.   Math is much easier to understand the 2nd or 3rd time than the first time, and quick mastery is the likely result.

With '3 - 5', I would just leave the question out there and not answer it.  Or maybe I would answer it, but then a month later I would ask it again and watch the same process starting over again from the start, but going a bit faster and progressing a bit farther.  When this come up again out of nowhere the third month, we might end up with mastery with almost no work and exactly zero practice.  Even better if the child sees negative numbers on his own in a book, he dives right in and the result is not only self-mastery, but he owns it.

SQRT(4) and also 5x - 13 = 2 will demonstrate the leap in skills that takes place around 4th grade.  Kids coming off work like Section 2 of TPM can calculate both of these without understanding how they do it.  Good little mathematicians iterate through possible solution values until they arrive at the answer, and great little mathematicians add weighing with high-low bands that narrow to the solution strategy to arrive at the answer more efficiently.

After 4th grade, when the brain is capable of higher order thinking, these two exercises gain new meaning.   The definition of SQRT(4) is the number when squared that equals 4.  In other words x^(1/2) is solved backwards.  Square roots present the opposite syntax of squares, and the solution is to back into the answer.  This is critical for topics that are going to come later. 5x - 13 = 3 is a simple introduction of y = mx + b, which is an important framework for characterizing more complicated problems, and the elements of y = mx + b have additional meaning besides finding a number.

There are also new skills that come with these math concepts.   A 3rd grader will jump in and solve either problem to get a number.  It's all one step.  A 4th or 5th grader will decompose the problem, spend more time analyzing the question, and learn more during the problem.  I've introduced younger children to the next level of math skills, like problem decomposition and making a hard problem easier; this exercise can take 20 minutes and is really good for thinking.   It requires a lot of working memory which is why in 2nd and 3rd grade working memory is most of the focus.  But older children do this intentionally, quickly, and know why they are doing it.

Let's look at some pre-algebra concepts that have been a real struggle for me to teach.   

First, x^2x^3 versus (x^2)^3.  Per formula, the first is x^(2+3) and the second is x^(2*3).  But we're not interested in formula's, because formula's produce math dummies.  

The way to do these problems is to work the question and not solve the problem.  x^2x^3 is simply (xx)(xxx) = x^5, and (x^2)^3 is (xx)^3 is (xx)(xx)(xx) = x^6.   Eventually, the child will memorize the formulas in the same way they used to count on their fingers for 5 + 3 and eventually knew that 5 + 3 is just 8.  Before 4th grade, the best I can do is lay the ground work for decomposition, restating the problem, multi-step solution operations, but they still jump into more advanced problems trying to get to a number in one (hard) thinking step.   I've noticed that after school program kids are drilled in multi-step solution strategies, but I don't want a child trained in math solving.  I want a thinker.

This is the biggest difference in my goals and methods.  I don't want a child who is trained in math, a a child good in math, a child who knows (advanced) math topics or a child who is 99% because of this training.  I want a child who does really well in math he has never seen before or mastered because he is a thinker and a learner and can apply thinking and learning to math.  I've always said if you need a 99% because it is required for GAT entry, do what ever it takes this year and forget your principals.  In 7th grade, I can't say this; it is not possible to short cut your way into a 99% without a solid learner-thinker.  Also, we've never actually deviated from principals or practiced rote math and we have always either been at 99% or been within striking range (in a bad year).  I will say that it's never too late to start.  There are advantages to starting early, but starting late does not preclude achieving the ceiling on a test.

The most challenging topic using my approach on pre-10 and post-10 children is parenthesis.   I will illustrate with this problem:  (6^2 + 18 + 2 + 4^2)) - 2^2.  This is not a complicated problem, but it is not possible to do a page of these problems with a child still learning exponents and parenthesis without writing down at least 3 or 4 steps in order to check steps for errors.  In other words, it's faster and easier to let the pencil do the work than the brain.  Before 4th grade, I'm happy to endure 4 or 5 wrong answers from mental calculations because the impact to working memory (not to mention arithmetic practice) is useful.  But with the problem above, working memory gets a work out and the child still has to write down each step to survive the problem.

In our 1 year ahead math program, it is common for kids to fall to 85% by about 6th grade.  The program administrators - geniuses way ahead of their time - are focused on the final result and this interim dip is a researched based way to achieve the final product.   The extra 14 points are achievable with a bit of extra work.  If you review this article from the beginning, you'll see 3 or 4 math education concepts that all work together to produce 99% without a lot of extra effort.  I don't think this approach would work very well in a classroom situation without modification, but it certainly can at home.  Once any topic above is presented above, the next step depends on the child's response in the context of the child's individual skill set.  A parent who gets to know their child and experiments a little with backtracking, repetition, exploring the question will stumble toward success.

Now back to the 7th grade challenge that introduced the article.  We have a very ambitious goal but not a lot of time to achieve it given homework and nonschool activities.  The topics, approach, learning environment, and general mess of our preparation is in my opinion an exact mirror of the test.