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Friday, May 10, 2013


I have a confession to make.  When I was young, I hated math.  I didn't understand it and it frustrated me to no end.  

As a kid, it made me lose a lot of confidence in myself.  Why don't I understand this stuff?  What's wrong with me?

But now that I'm an adult, I've learned to love math.  

I'm not at all good at it.  But I still love it.  I feel so pleased when I am able to solve a problem.  I enjoy putting together a nice spreadsheet the way people enjoy putting together a nice garden.

I've been thinking about math recently - how we teach it, and how we struggle with it as a society.  This post is kind of just a smattering of thoughts and opinions I've on the subject.

So - if you hate math - please, read on.


Ok, this is a true story.

I was in Macy's the other day, buying some shorts (don't panic, its unlikely you'll ever see me in them).

As I walked up to the register, there was a woman ahead of me talking to the cashier.

The woman asked the cashier:  "I have two coupons, and I can only use one.  One is for $10 off my purchase, one is for 20% off.  Can you tell me which one would save me more money."

She rattled it off just like that, like a teacher reading a word problem to a school child.

The cashier pulled a little scanner gun out from behind the counter.  BEEP.  BEEP.  BEEP.

"The 20% off coupon would save you $18."

"Ok, I'll use that one, then" the shopper replied.

This exchange bugged me for the rest of the day.  In fact, a week has gone by, and I'm using it as a jump off point for a blog post.  It stuck with me.

And what is it that needled me so?

I couldn't stop thinking: Why the hell couldn't this poor woman reason out for herself which coupon was the better deal?

Faced with what amounted to very simple arithmetic problem (I'm going to explain how we can know that later), she either could not - or could not be bothered to - figure it out on her own.  She needed the cashier and her tools to work it out for her.

This shopper demonstrated a moment of what the neurologist Douglas Hofstadter dubbed "innumeracy,"  the inability to understand, work with, and extract meaningful information from numbers. 

Its like literacy, but with numbers and relationships in place of letters and words.  An illiterate individual sees a string of letters and has trouble decoding the information in them.  An innumerate individual sees a string of numbers and and has trouble decoding the information in them.
Now, I'm not suggesting its the lady's fault that she couldn't visualize clearly 20% of her purchase total, and compare that number to 10.  I think it has much more to do with how we educate people in America.  More on that in a bit.

Some of you are probably wondering "what does public numeracy have to do with economics?"  Well, just take a sec to think about this - how democratic can we claim economic and financial policy/products to be, when they revolve around a "language" many people can't read?

The problem the woman faced

First, lets take a moment to think about the particular problem the shopper was facing.

20% is the same as 1/5th.  That's easy to remember:

20 times 5 is 100.  100 divided by 5 is 20.  If a 20 cent coin existed, you would need 5 to make a dollar.  You need 5 pairs of dimes to make a dollar.  There's 4 quarters in a dollar, and if you took 5 cents from each quarter, each of the 4 quarters would be 20 cent pieces and the 4 extra 5 cent pieces would make a 5th 20 cent piece...

However you want to remember it - 20% is the same as 1/5th.  

20% of the purchase was $18, which is the same as saying $18 constituted 1/5th of the purchase.  So we know the woman's total purchase was worth $18 times 5.  She must have had $90 worth of stuff.

So the problem she faced boils down to "What's greater: '90 divided by 5,' or '10'?"  Not difficult.  Its a 5th grade math problem.

Now, I can understand asking at the register or getting out a calculator if it was a close call.  But the 20% off coupon provided almost twice the discount of the other.  Its not really close.  There's no reason she should have had to ask.

So what?

Yeah, I heard you asking that.  "Big deal.  The woman's got other stuff to worry about then doodley little math problems.  So what if she didn't bother to work out what coupon was worth more before she got up to the register?"

My point is, if this individual was completely numerate, she wouldn't have had to 'work out' the problem at all.  It should have been obvious at first blush.

90/5 > 10 should be a reflex.  Having to 'work out' a problem as simple as this is the nummeracy equivalent of having to 'sound out' a word.

Ask yourself this - if the shopper was at the counter asking the cashier to read some clothing labels and tags to her, would you have the same "so what?" reaction to the story?

Its the education, stupid

I'm a graduate of the University of Buffalo, a school that has a sizable international student population.

I did an M.A. in Economics there 3 years ago.  About half my cohort were foreign students.  Mostly from China, but a fair number coming from South East Asian and the Subcontinent as well.

We had one professor from Portugal.  He had a habit of writing out any mathematical argument, function or model out on the board twice - once as a chart or graph, and once entirely in mathematical symbols.

The reason why?  As he bluntly put it "The Asians understand calculus symbols but not charts.  Americans understand charts, but not symbols."  He wished we'd all understand both.

Obviously, there's nothing that genetically predisposes Asians to better understand mathematical languages, or Americans to understand graphics.  But there are differences in our educational systems, and I really think that's where the root of the problem lies.

The culture and custom in US when it comes to math education is to teach kids to memorize rules.  When I was in 4th grade, we just memorized "six times six is thirty-six."  We didn't worry about why.

My impression has always been that in non-US educational systems (not just Asian, but Europe and Australia as well), there's more of a focus on understanding concepts then just learning calculation tricks.

Kids are made to understand what's going on behind the symbols, ultimately making their education more salient and usable outside the classroom.
For more on the differences in educational culture in the USA and abroad, check out:
Remember, numbers and notations are just symbols that represent real concepts.  Math is about how those real concepts relate to each other, not just about squiggles on a page.

Popular protests

There's some popular protests I've heard to the assertion that math education and public numeracy is so vital.

"You don't need to know math if you have a calculator"
A calculator is a tool.  If can help out, save some time, and avoid errors, but it can't tell you how to best use the numbers it process to inform decisions.

You can't lay a calculator on top of a pile of bills and pay stubs, turn it on, and let it sort out your finances.  You can't place your calculator on top of a budget proposal and tighten up its gaps.

A pile of hammers can't build a house, and buying a stove doesn't make you a master chef.  If owning a calculator is enough to make you nuerate, then having a dictionary around is all it takes for an new born baby to reach literacy.

"You only need to learn basic math - enough to get by"
The NYT ran an op-ed last year called "Is Algebra Necessary?"  In it, the author suggests that public schools need not even bother teaching kids algebra or trigonometry.  Just teach them what they need to survive - how to balance a checkbook, some basic statistics, etc.

So where is the op-ed entitled "Is 'To Kill a Mockingbird' Necessary?"  How about "Are Essays Necessary?"  "Are Book Reports Necessary?"

To suggest that students shouldn't be exposed to anything more than basic math feels a lot suggesting they shouldn't be exposed to anything more than basic reading.  If they can read a Facebook status and a road sign, they're literate.  Forget books.  They'll never read one after graduation so why waste their time with reading.

My dad often says "Math isn't about notation and numbers.  Its about logic and reasoning.  The notation and numbers are just the material you learn to reason with."

He's dead on.  We expose children to algebra and geometry and calculus not just to teach them math but to train them how to think rotely.  To hone their ability to work with concrete concepts.  To practice reason, logic and analytical thinking.

That is the important skill.  Don't miss the forest for the trees!

"You'll never use math outside the classroom"
We don't see ourselves using math in day to day existence because we think math is just a series of tricks and rules to be memorized.  We think of it as a self-contained game of symbols and values.  We don't realize how it relates to the world off the page.

Our teachers got so hung up on teaching us how to multiply by 3 or divide by two, that we missed the point of those exercises.  What does it look like to triple something?  What does it mean to cut it in half?  That's what's important.

Rearranging furniture?  Making an Excel file?  Planning your finances?  Shopping for a car?  Looking to invest money?  Building a shed?  Thinking about economics?  Buying groceries for a dinner party?  You need math.  You need numeracy.

You don't even need to really be working with actual integers per se to be flexing your numeracy.  Understanding spatial relationships, estimating sizes and distances, comparing trade offs, visualizing what you're going to do before you do it, working with objective information, it all requires numeracy.


There's a huge stigma in this country about literacy.  We don't like it, and we've worked hard to eradicate it.

We've been pretty successful.  We are, by and large, a very literate society.  However, as a developed society, with a well educated citizenry, we aren't the most nummerate one.

The crazy thing is we already know how to address the problem.  We need to teach kids why half of 5 is 2.5, not just teach them to memorize little computational tricks.

We need to get the math curriculum right.  And we don't have to invent or discover one!  There's so many models out there we can emulate!  We just have to do it.

If it was obvious that our schools were failing to teach kids to read, there'd be a riot.  But failing to produce an adequately numerate society gets no reaction. 

Were's that riot?  Cuz I'm ready to join it.

Dan Meyer: Math class needs a makeover

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