Sunday, January 14, 2018

Struggling in Math

I have gotten a lot of questions in the last 2 months that I will summarize and then answer:

  • My child is struggling with their At Home Schooling math, which consists (usually) of me making them do a math work book that is 2 years beyond their grade level.
  • My child started school at 99% and is now at 85%.

I tend to stay focused on preparing for a strong high school math experience; neither of these two issues ever bothered me and your children are smarter and better than mine.  We did have a dip in test scores and I went into RED ALERT mode until it was corrected. Both of these topics have been covered over the years, but it's pretty hard to dig through my blog to find answers.  In addition, I already deleted the 300 articles that had a play-by-play of my struggles.

Both of these are linked, because in order to get to 99%, your child either has to go to expensive after school math programs that will gradually make them hate math, or your child will work ahead at home.

Struggling In Math
The answer to all of your struggling questions is called 'Backtracking'.  We do it all the time.  I can't imagine doing any math above grade level without a lot of it.  Here are some examples that I've written about while we were doing it:

  1. If we were doing EDM Grade 2 in Kindergarten 3 days a week, at least one day a week we did a first grade math workbook that was just adding and subtraction.  Some times this is a nice break, sometimes it's catch up, some times it's practice.
  2. Sometimes I take 2 or 3 weeks off to cover a concept that we never had or a concept that we just plane stink at.
  3. Sometimes an entire section in the workbook is almost all wrong.  Sometimes it's just a page or an important problem.  The kid just doesn't get it.  I circle the pages and we move on.  A month or 2 later, we'll come back to the circled pages and do them again.
  4. When the child is younger, there are some bad days because of hunger/sleep/sickness issues and we just do flash cards or arithmetic worksheets.  Bad days happen rarely at older ages (always the day after a sleep-over), but when they do, we do nothing at all that day.
  5. Sometimes we take time off from math and do projects like a puzzle or sewing something or a craft or a writing project or art, a comic book, whatever.  In each case, the child just starts doing it and I will not interfere.  I am convinced that these activities will produce a stronger mathematician than actual math.
  6. We like to do things backwards.  So if the book does it one way, we redo the whole thing backwards.
  7. We like to do things step-by-step.  Identifying the mini-steps helps you find backtracking material.  Here's a really simple example.  23 x 15.  This has 4 separate multiplication operations and 3 addition operations.   Maybe your child should just practice multiplying 3 x 4 and 30 x 40, 9 x 2 and 90 x 20 etc for a while before coming back, or 20 x 15 and 3 x 15.

There are two difference between you and me.  First, as previously mentioned, your children are smarter than mine.  Secondly, we back track a lot.  Why continue to struggle with the same material?  Do something else, practice something, come back to it later.  It will all get done in the end because we are both picky and uptight parents about math.

Test Scores

Lately I've been getting a lot of feedback from many parents that test scores are falling.  I get this from almost all parents (like 85% of the ones I talk to) at some point during grade school, usually right around the midpoint.  Here are the reasons:

  1. Your school program teaches and practices math at about the 85% level.  Over time 99% children will end up working at the 85% level.
  2. Your child is sick of doing math and needs a year off.
  3. You are not doing daily math at home at a suitable level and 15% of the country is.
None of this is a bad thing.  I think our program starts pushing math at the appropriate time and produces graduates who are really strong in math.   This will not make a parent happy in the following 2 circumstances:  #1  Your child needs a 99% right now on an annual standardized math test this year.  #2 You have some other objectives in mind that requires a 99%.


Here is my 3 part recipe:

  1. Get math at a suitable level.
  2. Do it.  Backtrack a lot.
  3. Focus on problem solving techniques and not math.  Math will take care of itself.
I can now see that I need another article because the leap between 3rd and 5th grade and it's called problem solving skills.   My particular approach can be summarized as focusing on nothing but problem solving skills during 2nd and 3rd grade and it works.  Not just any set of problem solving skills, but the core skills that are the foundation of all others.  That, in a nutshell, is 95% of the motivation behind Test Prep Math.  The other 5% is making math less boring than it normally is.  

But I'm hearing from parents of 3rd and 4th grade children that didn't go this route.  I've got some thinking to do.  It's solvable.  Anyone can catch up to any level you want to get to.



Friday, January 12, 2018

The Language of Math

There is a strong and important connection between math and language.

Think about a child learning language before the age of 2.  You point to a blue ball and say 'blue'.  The child sees round, blue, rubbery, your finger, you making some weird noise, you're looking at him or the ball or both, and you're probably smiling.   What is blue?  Then you point to a blue wall and say 'blue' and the kid is more confused than ever. 

In order to figure out blue, ball, yellow, green, box, toy, your child has a lot of confusion to sort through, is going to make 5,462,298 mistakes, and you're going to be smiling the whole time, and on top of that the child is going to have to identify patterns, sort through permutations and eliminate candidates until he comes down to blue is an attribute of color.   The child may not see round or plastic or squishy yet, maybe he can sense it, but when there is a word tossed out there for 'round', his ability to think logically will be substantially improved.

By 1915 or 1911, I'm still debating, cognitive psychologists determined that the process of reading uses 100% of all cognitive skills.  100%.  This will never happen again.

If you want to know why I'm so over the top obsessed with reading and vocabulary during age 4, so much so that I created Pre-K Phonics Conceptual Vocabulary and Thinking to jam as much 2nd grade material into the brain of a child who can't pronounce C-A-T, you now know why.

Don't Lose The Magic
Learning to talk and learning to read, not to mention learning to walk, are much harder by a factor of a gazillion than anything a child will learn thereafter, including Pre-Algebra.   But somewhere after learning to read, maybe around addition, the parent loses the Magic Learning Environment that allowed your child to overcome insurmountable learning objectives.  You used to sit there smiling dumbly mistake after mistake totally happy every time your child rose an inch off the ground and then fell.  Now you're yelling at your child for forgetting what 8 + 4 is or struggling with x-1.  At least I am.  We ALREADY discussed the exponent graph 3 times.  Would you just pay attention once?

The magic was that you were willing to try to teach your child what words mean, despite not having the slightest clue how this works, through mistakes and trying over and over and over again, usually smiling the whole time, and learning just exploded.

This is the first connection between language and math and it's pretty lame compared to what follows.

Reconnect the Two Dots
If math uses a certain sub set of cognitive skills, but learning to read (definitely) or learning word definitions (probably) used 100% of cognitive skills, wouldn't it be great if you could bring the missing cognitive skills back to the math learning process?

I think this is theoretically possible and in practice I just ask them to explain verbally to me how to what the question is asking, what do they know, is there anything they have learned before that can help, can you articulate your solution strategy?  I also throw in anything I can think of related to a problem, like 'Polyhedron' or some other word to get that verbal section of the brain working.

But mostly I like to talk through problems and concepts.

Recently, we came across this question:  What is 42% of 66?  This is an advanced post TPM problem.  I got it off a high school Algebra I final that has 190 questions and would be very hard for high school   We're doing about 5 problems per session and learning a lot.  This is an opportunity for a long discussion involving fractions, decimals, and %, as well as problem decomposition and lining up multiple steps, followed by cheating with algebra.  In other words, in addition to math, it's going to be about 25 minutes of talking.

Here's some fun verbal math discussions for a younger age.  In these cases, I did very little talking and just left key questions out there for 3 or 4 weeks while the math sank in.  Then we discussed, and I asked why? or prove it to me.

  • The definition of 'square root' is this.  2 is the 'square root' of 4 because 2 x 2 = 4.  What is the square root of 9?  Does 10 have a square root?  (Not yet, but it will later).
  • What is the square root of negative one?  It's call 'i'.  What is i * i?  Why is this important (because the Fundamental Theorem of Algebra does not hold true without i in case you're wondering).  What is the square root of - 4?  
  • What is 2 - 5?  I love this discussion.  It goes like this: "Three".  If 2 - 5 is 3, what is 5 - 2 and why are they both 3?  This can't be right.  If you have 2 and you give away 5, what do you have left?  "You can't do it".  Oh yes you can my friend, yes you can.  
If I can't find something to discuss in math work, I'll start looking for more math.  y = mx  + b and f(x) = mx + b are great topics for discussion and not writing.   That's why we end up covering advanced math at a young age, simply to have something to talk about.  How's this going?  About as well as learning how to talk in the first place.


Is Any of This Going To Help?
I'm not 100% sure yet, but it might help with math learning.  It's definitely helping with writing.  Trying to compose an explanation for a complicated mathy topic just learned is really hard.  It is a foundation leadership skill.  It's similar to a reading comp skill, but only vaguely.  It's easier than any classroom speaking task.  I'm certainly not going to end up with a wall flower, what with me demanding a thorough explanation to a complicated explanation.

Product Recommendation
I highly recommend IQ Twist or IQ Puzzler Pro.  We've had these sitting around for the last few years and my kid and his class are now obsessed with them.   His 4th grade teacher is buying them for the classroom.

It wasn't until I solved a problem myself that required turning and flipping multiple shapes when I realized that it's NNAT and somewhat COGAT training.  We started talking through the solution to one tough problem and how one shape could only go in one certain place before I realized that this is all logic, visualization and math.  If you run out and buy these for a 1st grader like I did, feel free to reach out for help because it took me a few years to figure out how to use these with a younger child.


Monday, January 8, 2018

Real Math

My son complained about his daily math.  It was some problems from two Pre-Algebra topics.

If we do pre-algebra every day it's going to get boring.  I refuse to do either decimals or long division or math facts or anything between kindergarten math (totally engrossing) and pre-algebra (marginally useful) because it's all boring and useless. 

I need a fall back plan. He's been playing IQ Twist lately (highly recommend this game even though I don't get paid for any of my recommendations) and that got me thinking.  There is this great math book called Mathematics 1001 that has 1000 math topics in addition to 2 pages on trig that allowed us to cheat our way through it.  One of the topics in this book called 'Net's looked like the shapes in twist, and a little reading later uncovered this idea.

Here are two Nets for a triangular pyramid.  If you cut out either Net, you can fold it into the triangular pyramid. 


There are 11 nets for a cube.  Draw them.

I watched two sets of skills in action.  First, there was geometric visualization, including rotating, flipping, and 3 dimensional manipulation of shapes which trumps the two dimensional manipulation on a cognitive skills test.  If we were facing a test this year, I would have only shown the diagram on the left above and asked for 2 more nets for the triangular prism (even through there is only one because cognitive skills tests test your ability to come to terms with incorrect questions).

Secondly, there were budding permutation skills at work, which is an extremely important math skill.  Since no kid is going to get to 11, this gives me the opportunity to suggest permutations.  "What's a permutation?"  Well, take the letters a, b and c.  I can write them as abc, acb, bac, bab, cab, and cac.  There are 6 permutations of the letters a, b and c.  Please give me the permutations of 1,2 and 3.  This should be pretty simple.  Then look at the basic T shaped Net for a cube, and start permuting the squares, one square at a time.

We got to 7, which is pretty good for my 25 minute time limit.  I need to stop at 25 minutes to save room for follow up questions, like telling me the rules for building a Net while staring at the 11.

Real Math
I expect this child to go far in math.  He's not going to go anywhere without some intervention.  Here is my intervention.

I showed him a web diagram of the 11 nets for a cube.  I stated that some guy (Albrecht Durer) asked how many ways you can create a folding diagram for a cube, and he came up with 11.

Then I showed my kid the pre-algebra worksheet of about 20 equations. 

I asked this question.  If there is a mathematics professor and researcher at some university asking questions and writing papers and going to conferences and helping his colleges in the Physics and Information departments apply abstract math to their work, which math is this professor doing right now?  (And by way of association, which math are the physicists and computer sciences clamoring for?)  Does it look like this (pointing to pre-algebra) or does it look like this (point to the 11 nets for a cube.) 

The Answer
The answer is the net stuff.  And why is it that your school curriculum looks like pre-algebra, the type of math that mathematicians don't do?

Here is my (mostly inaccurate but totally) true history of math curriculum in the United States.  In 1930, a vice president at Ford Motor company created a list of skills needed by factory workers and accountants and dealers to create and sell cars.  This skill set was widely applicable to industrial work of all types.  A curriculum was created to teach it and used throughout the United States.  Lots of cars were produced and everyone was happy.  This curriculum is still used in 2017 in the midst of the Information Age.

Of the 96 maths out there, school is going to consist of the 5 that would help you build cars by hand or build a bridge, which you are never going to do.  The maths that you actually need to get through your life - starting now - are not taught at all.

What I find most interesting is that the 5 maths taught in US curriculum are almost devoid of skills compared to the maths that could be taught. 



Tuesday, January 2, 2018

The Train Wreck Revisited

The 'Train Wreck' is one of the 5 things that should keep a parent awake at night worrying.  Three of these are reserved for children over the age of 12, and that only leaves 'Lacking Motivation to Read' as the two things you need to worry about right now.

The term 'Train Wreck' is used in situations where a child who previously got all A's in math is now getting a C.  This most commonly occurs between 4th and 8th grade.  It can also occur in Algebra or Geometry.   This term also applies to a child's test scores falling from 99% to anything below 90% and is somewhat related to regression to the mean.

None of this is very shocking.  You want shock?  Let me define this term formally.

Train Wreck:  At one point, your child held a formidable skill set and did well in math.  A few years later, when your child faces a new math, the child doesn't do well because the child does not possess the skill set required to do well.  You are left wondering what happened and either correctly blame yourself or incorrectly blame the teacher.

The most common cause of the 4th grade Train Wreck is a child who is overly endowed with skills entering 1st grade and spends the next 3 years at school not thinking.  By 4th grade (depending on the school district and curriculum), there is a jump in complexity, and the child has no tools in the tool shed.  The train wreck in middle school or freshman year is usually caused by a catastrophic failure of curriculum, but can also be the result of a bright child languishing in an average curriculum.

Regression to the mean is an empirical consequence of the level of instruction in school.  Kids who score below the mean usually catch up test-score-wise while experiencing instruction at the mean, and kids who do much better than the mean usually slow down while experience instruction at the mean.  I'm waiting for the field of cognitive psychology to have a 'duh' moment and figure this out, but tat the time of this writing, they are still baffled.  Anyway, Regression to the Mean is a less dramatic version of the Train Wreck but is caused by the same factors.

There are at least 2 leaps in cognitive requirements that take place in grade school math, and at least two in language arts.  In high school, a really great curriculum will have at least one leap every year (most don't).   Are you happy with your child scoring well this year, or are you really concerned about their score in 2 or 3 years?  Thanks to No Child Left Behind, teachers are  mandated by law to be concerned only about this year at the expense of next year.  Thanks to having 30 kids with a variety of skill sets, the teacher can only do so much.  You're going to have to pick up the slack. 

Happily, I've found that being only concerned with 2 years from now tends to take care of this year and next year for no additional effort.  By when we work ahead 2 years, I stay focused on the skills, not the math.

There's a really great book by a psychologist to deal with the Train Wreck.  There's a lot of great 'Yoda' in this book, but it's downfall is that the author doesn't address the skills issue.   He has a valid excuse because he has a PhD in Psychology, a field who thinks IQ magically happens. I've done a little work in this area, but not enough to write a book.  My market is shooting for 99% (if you're reading this, this is now your official goal if it wasn't before) and we're going to need the whole bag of skills to get from X% to 99%.