Tuesday, June 20, 2017

Dealing with Mistakes

A reader posted a comment on the last article asking for clarification on my comments about mistakes.  I'm probably not going to get this right on the first try, but here is my Policy Statement on Mistakes.  I owe everyone this because I have been advocating doing math at grade level + 2 starting in Kindergarten or 1st grade (if you're late and catching up) and of course no kid can actually do that.  Until they do.

My Policy Statement on Mistakes

When I sit with younger children, I witness numerous skills emerging that I see fully matured with older kids on more advance material.  I've got 2 kids who are 3 years apart, and I work with each of them one after the other on Saturday and Sunday so I see it every week. Before I did any of this, I read George Poyla's manual on how to do high school Geometry proofs and rewrote it for a 5 year old.

I see some really cool skills develop.  I need to get these skills to a very high level so we can do very cool things by high school or later.   Also, these skills are needed to survive in our GAT program and to get really high scores on tests.

Go www.IXL.com/Math.  You can click on each grade and see a list of things that you probably think are skills because school refers to these as math skills. (You can do this for other subjects as well.) They are not skills. Comparing a rational number is not a skill (if you picked 6th grade).  Almost nothing your child is going to do in a math workbook is even remotely a skill. I'd like hack into their website and leave a virus so that after each item, for example "Division facts for 2,3, and 4", the virus would append "is not a skill".

Let's say I sit down with your 5 year old and give her this problem:  XII - IV = ?. I actually did this with a table full of Kindergartners.   They all got it wrong. One kid wandered away from the table.  What I did next depends on the kid and where they are with their own personal skill set level.  I brought out little bags of Doritos.

Some kids think that math is all about a problem that already has a solved solution.  I'm going to assume they get this from their overly stressed out helicopter parents.   These kids are a mess.  This is level 0.  These kids hurry through problems and will do anything to get the right answer so they can move on to the next problem.  If they get it wrong, they are upset.

For these kids, I change the question.   I want to know how they solved it.  I want to know how XII and IV work.  What are the rules?  It turns out each of these is a mini math problem in disguise.  What are the mini math problems?

I'm usually working with material that is more complicated that a one step 3 + 7 = ? problem so I want them to restate the problem into 2 mini problems.   With something that is way over their heads, I want to start with something easier like I + II = ? and just work our way back to XII + IV even if it takes many days, which it usually does until they really get it.

After a few weeks of this, it should be clear to the child that we're working with material that they are not the slightest bit competent on so I'm not really interested in XII + IV = ?  I'm interested in how to solve this problem even though you don't know what XII + IV is.  We're lucky to get through 2 or 3 problems in one sitting.  Maybe even 1 problem.

If the child has a working memory deficit (thanks to school curriculum), the child has a really hard time keeping XII (the mini problem) and IV (the other mini problem) and XII + IV (the third mini problem) in their brains the whole time and is always lost.   This is going to take at least 6 weeks for working memory growth to kick in.  This is a big problem on GAT test prep.  If you jump ahead with a 5 year old to EDM Grade 2, every problem is 3 problems in one for a 5 year old.  This is why we always work ahead.  Once they get XII + IV, it's only one problem to them and they are no longer using skills.

Never, ever, ever in my whole coaching career have I cared if the child can do arithmetic properly.  3 x 7 = 21 will come on it's own because we are going to encounter this a lot in the next few years.  By the time they master 3 x 7 = 21 we'll be on to 1/3 + 1/7 = ? and then 1/3(x + 1/7) = ? so they'll never actually be at the point where they are zipping through problems they mastered.  They are struggling the whole time trying using survival skills, and this is where skills develop.

So if the kid struggles with 21 + 13 and actually solves it, but gets 45, and it's because they added 20 + 10 and got 40, and 1 + 3 = 5, we'll write 20 + 10 and 1 + 3 together and do it again.  Or not.  Depends on my mood.  I'm in no hurry.  I know with 100% confidence that the skills will kick in during the next few months and we'll probably stop doing 2nd grade math by about 8 months later because it will no longer be challenging.

After a few weeks of this (maybe 6 weeks), some rudimentary skills have developed, and I can move on to the next stage.   I call this the "Wrong" stage. This is the most awesome stage where the majority of skills develop.   Here is my dialog:
  • Kid: 45
  • Me:  Wrong.  Try again.
  • Kid:  33
  • Me:  Wrong.  Try again
  • Kid:  43
  • Me:  Wrong. Try again.
  • Kid:  35
  • Me:  Show me how you are doing this step by step.
  • Is it 44?
  • Me:  Your just guessing.  Prove it to me.
This dialog is not possible with a child who is at level 0, but it is possible at level 1.  You'd think it would be more productive if I just asked them to explain it to me after the first wrong attempt and you would be right.  But I need to practice saying "Wrong" with no emotion because I don't care.  This is hard to do.  I still need practice after all of these years.

If the child doesn't figure this out after a month, I'll just explain it to them, it being this: It's much easier to do math if you do it right the first time and check your work (aka do each problem at least twice before asking if you got the right answer).  The stubborn ones don't believe me, but eventually I catch them cheating (aka doing it twice) in order to save time.

Once the kid can explain what they are doing, as in "I add 20 and 10 and then I add 1 + 3, and then I add 30 and 4", they are at level 2.  At level 2, they spend a lot more time thinking about the problem and less time worrying about the solution.

At level 2, we can dive into the solution more and I start to demand that I want more than an answer.  I want them to figure out how to do each problem in a cheaty way on each and every problem.  I'm not longer happy with 20 + 10 = 30, I want them to explain how to do 20 + 10 without thinking.  My own kids actually counted on their fingers 10 times from 20, which is great, but now I want them to look at it and think about it.  Let's try 20 + 20, 20 + 30, 20 + 40, 30 + 30, etc. until they see patterns.  This is listed on the IXL page as a skill, but the skill is actually the kid figuring out how to do this without being taught.  I have to provide a) the opportunity, b) the request to do it and c) no help whatsoever.   I usually have to do this again next week and again the week after this until they get it on their own.

As a parent, I'm so tempted to explain it to them.   As an older parent, I know that if I break down and explain something, I undercut learning and will end up having to explain math the rest of their lives.  So I don't explain, and 2 days of work is suddenly 2 weeks of work.  It pays off in the end but for those 2 weeks you're thinking that it's a waste of time and your child should just learn a trade and forgo college.

At level 2, the kid has already been through the "Wrong" dialog enough times to know where it's heading and spends more time redoing the work and breaking it down and explaining it to himself to reduce the "Wrong" iterations.

In the course of doing this from 1 + 1 through calculus, I haven't found a problem yet that can't be solved by Poyla's two most powerful tools: break the problem down into 2 smaller problems or find an easier version of the problem and just work your way up from there back to the other problem.  Sometimes we come across a math concept that the kid needs but doesn't know at all, and we'll go to IXL or Khan and just look for easy spoon feeding problems for the next week or so.   Eventually the kid gets to the point where they can devise clever ways to solve a problem on their own on the spot with a brand new complicated problem.  But this is level 3 and they won't get there until after TPM.

It's rare that we get past 3 or 4 problems in one setting.  The EDM Grade 2 workbooks have 6 problems per page, but this pace is not sustainable for more advanced math, and we didn't actually get up to 6 problems per day in EDM for a few months.  Test Prep Math is heavy on logic, working memory, and problem solving skills and ends up being 1 or 2 problems a day.

In EDM, there were days at level 2 where I witnessed exemplary problem solving that resulted in 50% incorrect answers.  I grade the work, announced good job and moved on.  Some days we had 100% wrong answers and we went out for ice cream, which is the house rule.  In subsequent years, I could raise the bar and we don't call it quits until all mistakes are corrected.  After 4th grade, I started to apply this approach to reading, science, and writing and it worked.

If your child is doing math worthy of thinking, your child will be making a lot of mistakes.  If your child is not making a lot of mistakes, then they are picking up the bad habits that I described in the beginning of this article, so they are actually getting dumber so move on to harder material.

If your child is making a lot of mistakes, and you and your child are having a hard time dealing with this, then you can concentrate on overcoming your bad attitude toward mistakes and not worry about any other types of learning.  This is the big step, a key difference between GAT and average, and the roadblock to the other skills which I've explained in past articles.

There are great ways to deal with the pain of mistakes, including the pain of making a mistake because your child in fact doesn't know how to do the problem and won't ever know without some help.

  • Ignore the mistakes and just move on.  This is a signal that it's not about right or wrong anyway, which it isn't.
  • Do the work over.  You should do the problems and get them wrong too.
  • Split the problem up into mini-problems and do those.   My kids needed a lot of help learning to split problems up.  We had to take time off to do all of our math facts the splitting way.  (Eg, 6 + 6 is really 2 hands and the 2 invisible fingers, as in 5 + 5 + 1 + 1.)
  • Take time off and do something else.  Like an IXL page that practices the thing they should have learned but didn't because you skipped it, or a grade level workbook.
  • Try an easier problem and make it harder and harder until it looks like the one in the book that's causing trouble.  
Because of those last 3 bullet points, my kids are really good at problem solving even though we took so long to get through math workbooks.  The silver lining is that eventually your kid gets really fast.  I'll talk about that later.


  1. Interesting! Will try it out with my 5 year old. How does a child figure the solution to a problem XII + IV, if he has never been taught the roman numerals? Or similar such concepts that are admittedly difficult to figure out , if they've never seen it before. Curious about how achild's brain works to make sense of this?

    1. I don't cover the how in prior articles, but I cover the why. I can see the how taking place but it always happens so I don't worry about it. 'Mistakes' is only the #2 or #3 skill in the GAT arsenal. The #1 skill is getting used to working on new and confusing things. In this case, like others, the internet is standing by, and we start with I, II, and III, make no progress on IV, then V, VII, etc V and X take a few days to sink in. IV and IX took me a week or 2. Then L, C, and M seem to fall relatively quickly. Recently we did dx/dy (derivatives) and I pointed out that you can derive the equation from the derivative. 30 minutes of staring and my son said "you can't figure this out because the constant could be anything". I told him it was called 'C' and I was too choked up to say any more. 6 years of this approach really pays off down the road. 3 weeks into EDM at age 5, he announced 'I can't do this and never will be able to do this.' I remind him annually.

  2. Interesting! Will try it out with my 5 year old. How does a child figure the solution to a problem XII + IV, if he has never been taught the roman numerals?


    1. This is the most important thing you'll ever learn, from roman numerals to calculus. Start with I, II, and III and go from there. We often come across unsolvable problems, and we try to solve them with really easy numbers (like 2 and 3) and work our back to the harder version of the problem, or just try out part of the problem instead of solving the whole thing. It gives your child something to do instead of cry, and it's an advanced graduate level solution strategy that works in a variety of situations at much younger ages.