Purposeful Mistakes – Why REAL Learning is a Bit Messier than Memorizing

The new evidence from brain research tells us that everyone, with the right teaching and messages, can be successful in math, and everyone can achieve at the highest levels in school.There are a few children who have very particular special educational needs that make math learning difficult, but for the vast majority of children- about 95% – any levels of school math are within their reach…

No one is born knowing math, and no one is born lacking the ability to learn math”

~ Jo Boaler, from her book Mathematical Mindsets

Brains are not wired to learn in a time-efficient manner.  We learn from experience.  We learn from screwing up, reflecting on why that didn’t work, and trying something else.  We learn from making mistakes.

And we already knew that, right?

As a teacher, I’ve known that for years, but to be honest, I thought it meant to make the best of a bad situation – meaning the mistake wasn’t supposed to happen.  As a teacher, I still treated a mistake was an unintentional slip-up, and when we make them, it’s important to learn from them so we could learn how to avoid them.

There wasn’t an epiphany.  I had taken an online class offered by Jo Boaler for math teachers.  A year before that, I had read Mindset Theory by Carol Dweck, who was a mentor to Jo Boaler.  So, I didn’t have a brilliant breakthrough – I was actually a bit slow coming to the paradigm shift I’m about to describe.  But once it arrived – BAM!!! Things changed.

What if we asked kids to make mistakes ON PURPOSE?

Something happened when I was at the board one day showing the class a problem in geometry.  Kids weren’t getting it.  They didn’t know how to start the problem.

And then THAT kid – you know that kid – the comedian who always blurts out the most ridiculous answer – blurted.

That’s when it happened.

I looked at him.  I smiled that smile kids recognize when something really good just happened but they don’t know what.

And I ran with it.  The obviously wrong answer went up on the board and the energy in the class did a 180 (geometry pun intended).

As soon as I wrote the that answer on the board, I saw everyone relax and forget that they were just struggling to find the “right answer”.  That’s when they got interested.

They actively tried to figure out why the given answer was wrong.  Without much prompting from me, they found the mathematics needed from working backwards from the mistake we made on purpose.

“OMG you guys!!!!  You just invented a new way to solve math problems!!  I’m going to write about it and take full credit!  I’ll be famous!!!”

And now you’re reading about the technique I stole invented that day.  (Thank you 5 th period!!)

What we, the class and teacher, learned that day is that when you are frustrated, stuck, don’t know how to start, making a mistake on purpose gets you started and is a catalyst for analyzing the problem.

I have since used the technique of “Purposeful Mistakes” when tutoring students.

Let’s explore how making a mistake on purpose can be an effective way to start working a problem.  Here is how it could be used to tutor a student struggling with how to solve an algebra equation…


Me: “Ok, what do you do first?”

Sally: “I don’t know.I hate fractions.”

Me (noting that she can’t remember the how to do it, is anxious, and has decided to give up right away hoping I will tell her what to do step-by-step): “OK.  How about you just pick an x.  Make one up.  Let’s see what happens.”

Sally: “Huh?”

Me: “I’m serious.  Let’s make a mistake on purpose.  It’s ok.  It won’t be graded.  Nothing bad will happen.”

Sally: “Anything?”

Me: “Yup.  The crazier the mistake the better.  Go for it.”

Sally:Looks at it.  Thinks a bit because she’s not buying my ‘crazier is better’ direction, then says “x = 40”  (This is NOT a dumb mistake. She’s going for half of 80.  Excellent mistake!!)

Me: “Good! OK, if x is 40, what’s half of 40?” (Sally says “20”.) “Add 7 to 20, is that 80?”

Sally:  “No.  That’s 27.”

Me: “Ok, good.  Was your first x too big or too small?”

Sally: “Too small.”

Me: “Pick a new x.”

We will continue this process and my questions will become more refined as I begin to understand her thought process.  I won’t give the next step – she’s in charge and leading it.  Mistakes are encouraged.  Failed attempts become experiments and she will quickly adjust her strategy.

As messy and inefficient as this exchange may seem, it goes by quickly.  Many students, once told that we’re making the mistake on purpose, will relax, start to work things backwards, make a better guess for “x”, and will quickly arrive at the value that will work.

Often, when a student has previously learned the “correct” algorithm used to solve an equation, but has trouble accessing it, they will attempt another way to solve it after the anxiety of making a mistake has passed.  Most often, they will remember the “correct” way to do it.

That’s the teaching moment – that’s when the algorithm makes sense.  That’s when they see that they aren’t memorizing random steps – they are using a strategy to work backwards to solve for x.

(It’s important to note that many students can do multi step algebra equations in their heads – right up to the point when they can’t anymore because of the complexity of the equations.  We want students to learn the algebraic algorithms to navigate when their “I-Just-Know-It-Super-Power” vanishes.  Those algorithms are tools.  But the student who didn’t need the tool before has a hard time understanding why they need it now.  This student who has an intuitive understanding of the mathematics and their intuition is  turned against them when they are penalized for “not showing their work” without any further explanation.  Or, worse, they are penalized if they did show their thinking, but it wasn’t done exactly the way the teacher expected.)

Thank you for reading!  If you have any questions or comments,  please leave them below.  I am accepting students, so if you think I might be able to help your student, visit my full website, www.OnlineGeometryTutor.com to learn more about online tutoring.



One comment

  1. Oh yes Tammy. This is how I teach all my students too. Just let them try. However ridiculous the answer might be, they are learning from it. They learn much better than just us explaining the steps.
    You’ve explained the process beautifully.

    Liked by 1 person

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