Numbers Instead of Variables

Ever tried to teach someone how to remember this math rule:

Or tried to remember it yourself? Do the letters trip you up? Or your students?

How about this one:

I did a video on this some time ago. It has more Xs than an adult movie theatre. Here’s a still from the video:

In general, it’s a good way to remember the rule.

But what if variables freak out your students?

I’m the only person I know that teaches a kid to count, “1, 2, 3, x, 5, 6, …” (And I’m saving both for Daughter’s college fund and her psychiatrist.)

Most people aren’t even told about letters in math until they’re well established in doing stuff with numbers.

So why do we use letters to explain stuff?

A tiny detour…

I just geeked out and bought a course on google analytics through App Sumo. Andrew Warner with Mixergy, in the first video, was interviewing Justin Cutroni.

In less than two minutes I had to pause, rewind and listen hard to what Justin said:

Actual data makes a huge difference when you’re teaching.


Rewind. Re-listen.

What I heard was, “Hey! Knock it off with the letters already! Use some actual numbers when you’re teaching stuff!”

Hey! Use some numbers!

Theory is great. If you’re into that kind of thing.

But when we’re learning, we need something to hold on to. That’s both “we” as grownups in Justin’s Web Analytics class, and “we” as kids in Miss Kelly’s Algebra class.

We need data. Numbers.

Something that feels good, makes sense and is easy to wrap our brains around.

So what’s up with the letters?

Math books and math courses are written by mathematicians. Folks who are as comfortable with letters as they are with numbers. People who can take theory to new heights of abstractness. And never need a beer doing it.

Everyone else, well, they’re just unfortunate casualties.

But you can change that!

You don’t have to be a mathematician to teach math. In fact the less of a mathematician you are, the more likely you are to succeed in teaching math.

Pull yourself out of “teach like a mathematician” mode and think about what a variable is.

It’s a number. It’s data. Just an average ordinary thing that you can represent on your fingers.

So now, what does this mean:

Maybe this:

Or this:

Or even this:

And after you and your child play around with these and other examples using regular old numbers, your kid will say out loud:

You just gotta add up those number that are flying in the air.

And he’ll be right.

Actual data.


Whatcha think? Share in the comments or via twitter: agree or disagree“].

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10 Responses to Numbers Instead of Variables

  1. Hi Bon,
    Math was not my best subject in school. I would like to have learned more about Algebra and other types of Math equations back then but I just couldn’t wrap my mind around it.

    As an adult though I am finding that learning from better teachers makes things a lot easier.

    • Justin,

      It is indeed important to have a good teacher – especially in a subject that you’re less than excited about.

      Perhaps someday, with parents like you starting to say positive things like this, kids will go into math class with an “of course I can do this, everybody does” attitude.

      Thanks so much for stopping by!

  2. I absolutely agree! I have seen so many homeschool programs that start with early algebraic notations or … letters …. in both the lesson examples and practice problems.

    While many will not cough or even hiccup, what ever happened to teaching with the numbers?

    We end up with a child that never masters basic arithmetic. We also have a child still struggling with the “letters” in Algebra based on their previous foundations and experiences.

    Letters have their place, and it is not in basic computational math. 🙂

    • It’s crazy, huh? I didn’t even notice it until recently.

      I wonder how many other things I’ve been blind to. And how many other professional math teachers are just as blind. (One of the reasons the best math books are written by homeschool parents!)

      Thanks for you comment, Chrissy!

  3. Well…it’s about time! I guess it takes longer to get basic concepts inside a math teacher’s head than it does a regular person, lol. I have always found it bizarre, confusing and cumbersome to mix letters and numbers, but of course, I have always been dismissed as “making excuses, being hard-headed, afraid to learn something new….blah,blah,blah. “That’s the way things are here in River City (now the Bayou City, lol) and you just gotta accept it and learn it the way we tell you.”

    • You have hit the nail on the head, Vikki! Math teachers rarely stop and look at the world (or the classwork) from the student’s perspective. It took me a while to start doing this, but it is still hard.

      This video shows the most amazing student perspective that I ever noticed:

      Totally nuts!

      Since then I stopped trying to “give info” and started trying to share bits and watch how people take them. I’ve been doing this almost 20 years and I’m still learning new things!

      Thanks so much!

  4. I don’t think she is off her rocker at all. I was surprised to see a large nmber of my classmates in Math 0306 to be kids fresh out of high school. That indicates a problem to me. If the old math methods work so well, why haven’t some 12 grade students “gotten” it yet?

    • Indeed, Vikki – it’s time to see that we’ve been working on a basis of insanity…

      Doing the same thing over and over and expecting different results.

      Weird that we haven’t seen this.

  5. Algebra is all about abstraction. However,abstraction is sometimes too abstract to have true meaning. Moving from computational math with numbers straight into abstraction based algebra with letters is like a bait and switch with students minds. It’s like we are saying now that you understand numbers let me take them out of the picture entirely. Even with the first instances of algebraic reasoning, letters were not used solely. It started with the use of sentential logic which utilized full sentences to derive the properties that we now describe with abstract variables. For example the communicative property of addition started as a sentence a little more abstract but similar to “if 1 plus 2 equals 3 then 2 plus 1 must also be equal to 3” Where now we have the more abstract “a+b=b+a” which is less obvious and has less meaning to those not versed in this form of thinking. (sorry for the long comments I have to stop myself sometimes… math consumes the 90% of my brain that I dont have control of)

    • Thanks for the comment, John – and no more apologizing, I love them!

      I never thought about it as a bait and switch. Interesting.

      Since you can reverse it, too (switch and bait…?) would that matter?

      Like if I can’t remember the “rule” I can convert letters to numbers and see what works. Then I switch back.

      I’m gonna have to ponder this one more.

      Thanks so much!

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