Category: math education

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A mom recently wrote in to ask this question about standard and expanded notation.

“How do you know when you are writing in standard form, expanded form? For example, is the expanded for of 30,048
30000 + 40 + 8 ?
Or for 29,486, the expanded form = 20000 + 9000 + 400 + 80 + 6 ?”

Professor Homunculus replies:

Precisely! Oddly enough, there is no “standard” for “standard.” What I mean is, for 30,048 the standard form could also be considered:

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The previous post was about the value of learning conceptually before you start practicing for skill.

There is an alternative argument that argues for the opposite. Many pedagogues try to plead the case that first you must teach the “basics” (meaning the basic skills, like the “multiplication facts”) before you can expect a child to acquire any meaning about it.

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There seems to be a big “fight” about “which should you teach first, math skills or math concepts.” A popular example is the “multiplication tables” versus the concept of multiplication (as repeated addition, for example).

It’s a pretty good bet to say that when memorizing things it’s easier if you can relate the objects. Like if you went shopping and had to get toothpaste, a toothbrush and dental floss, that would be easier to remember than if you had to get shoe polish, armadillo meat and an f-string for a lute (do lutes even have f-strings?)

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“In mathematics the art of proposing a question must be held of higher value than solving it.”

Georg Cantor

Here is an except from that comment:

“Frankly, I don’t care if an elementary school child can add long columns of numbers in their head - it is an almost worthless skill. I do care if they can think about mathematical concepts.

Better to teach them to come up with simple proofs (not memorized proofs) of basic facts in math.

Better that they should understand what a prime number is, and why we care about prime numbers.

Better that they should learn to enjoy slow, deep thought about puzzles and concepts.

That is where the gold standard in math education is.”

I wanted to revisit this thought, because Penny brought up some great points. I don’t disagree with any of them. But I must say that I, as well as a lot of the readers are coming from a different place. Penny is a brilliant research mathematician. A lot of us, on the other hand, basically have a history of thinking that we sucked at math (at least until we came upon Math Mojo, and learned that almost no one sucks at math, but some sometimes the way math is taught sucks.)

I wanted to address some of the points Penny made, because those points made me think a lot this month. Here’s

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With so many things in bloom here in rural upstate New York, I figured it’s time for some new Ideas with Math Mojo.

One of them is the addition of a new segment, called, “Strange Powers of the Mind.” Look for a lot more weird stuff like this at Math Mojo. Sign up for the Math Mojo Monthly (”Comes out Quarterly, Mostly!”) Newsletter to keep up with the additions.
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Surely most readers have had thoughts and observations about boys’ and girl’s differences concerning their typical learning strategies.

A reader (Susan G.) has been corresponding with Math Mojo for a week or so, and I’ve noticed that she’s made some great observations. She’s also written a e-book on word problems. I’m in the process of reading it, and it looks right-on-target. It will be a resource that teachers and home-schoolers will want to have.

I’ll be writing more about it in the future. In the meantime, if you want to find out more about it, please shoot me an e-mail (use the contact box on this page) asking about it, and I’ll put you in touch with her.

Susan had been kind enough to offer some of her expertise about education in general, and she’s been one of the readers who’s taken my request for proofreading to heart; she’s pointed out some typos in Math Mojo materials, which I’ll be correcting a.s.a.p. (Thanks for the heads-up on this stuff, Susan!)

She’s kindly given me her permission to publish some of the thoughts we’ve been sharing about math-ed.

Here’s and excerpt from a recent correspondence we’ve had:

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Today a concerned reader took issue with what he understands my methods to be. (See comment #4 at Augends, Addends and the Commutative Law of Addition.)
Fair enough, but I think he may have misunderstood my methods.

That could, of course, be due to the way I communicate them (or miscommunicate them). First let me say that none of the algorithms (ways of solving math problems) I teach are “mine.” “Math Mojo” is the name of my attitude, not the methods. The methods have been either gleaned from better sources than me (and most are hundreds, if not thousands, of years older), or I have “re-invented” them. That is typical for most people’s alternative methods.

Now to the issue; the reader stated:

After all these years (30) of struggling to teach children math, I finally realize why it is so difficult. A brief perusal of some of the mathematical girations you go through to multiply two numbers together explains a lot of why kids are poor at math. Commutative and associative properties are more easily understood when you have the basic tools to work with without adding zeros then subtracting the number from your cousins name on your mother’s side of the family. Teach the basics by rote then progress to the more abstract. Simple to complex seems to work.

Professor Homunculus’ reply:

I’m sorry you’ve come to that conclusion. If you’ve been teaching math for 30 years, you surely have some insights. But I can’t see see how you’d say, “simple to complex” seems to work. May I ask where it seems to work? And if it does, why is it a struggle for you, and why is it so difficult? Have you been teaching with the “girations” (sic) you say I use to make it so frustrating?

I’m not quite sure I understand the logic of your position.

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While surfing some of the other math blogs in the blogosphere, I ran across a post in Michael Paul Goldenberg’s Rational Mathematics Education blog. Rational Math Ed is a gem of a blog, and is a must if you are a public school teacher or administrator.

In a recent post of his he mentions an article written by Paul Lockhart entitled, “A Mathematician’s Lament.” It was written in 2002, but has only gotten mass coverage recently, since it was featured on a post at the website of Keith Devlin.

For those of you who don’t know Keith Devlin, you are in for a treat. His writings are among the most lucid you will ever read about math. He makes very complicated things easy to understand. Please do yourself a great favor and visit his website.

You may have heard his lilting voice on NPR as “The Math Guy.” Devlin has also linked to Lockhart’s article, which is available as a free PDF download.

If you have any interest in math at all, even if you are not an educator, you will truly enjoy the enlightening thoughts that Lockhart shares with you. His appreciation, and enthusiasm for math, and teaching it are joy to read.

So take some time and revel in the passion Lockhart will enchant you with. Download the article now from the bottom of the post at Devlin’s site.

Please read both the Goldenberg and the Devlin essays about the article. They say everything I’d want to say, only better.

Recently I got an request to review my booklet, “Numbers Juggling - Times without the Tables.” Request came from Sol Lederman, who runs the “Wildaboutmath” blog.

I’d heard that name before, but really couldn’t remember much about Sol, so I checked out his blog to see how serious it was.

Wow, it’s a great blog, full of lots of valuable information about math, how to learn and teach math, and the joy of math. You should definitely check it out.

Sol reviewed the booklet, and you can read his review here.

The review was generally positive, but Sol had a very valid and important criticism. Since the greatest value of the booklet is really in the seven follow-up e-mails in the e-mail course, it should be marketed as a course, rather than a booklet.

That got me thinking (as every good book-review should do). So now I am developing real, in-depth, home-study courses for each of the basic operations of arithmetic.

Each will be about thirty modules long. The modules will walk you through the basics to absolutely turbo-charged speed-math methods.

I’ll be telling you more about it as it develops. If you are interested drop me an e-mail. (Use the contact box near the upper right corner of this page).

Now on to the ADD part of this post. Many people who have problems with math have problems with attention, focus, concentration, etc. I am one of them. I have suffered with ADD for as long as I can remember. It was only “officially” diagnosed a few years ago.

As it happens, Sol suffers from it as well. Or suffered. He has a blog dedicated to journaling his recent “cure.” I have not met Sol, and cannot vouch for anything, but he seems very dedicated to describing his experiences honestly.

Let me say that I am a skeptic, down to my bones, and hope you take everything with a grain of salt. But I would investigate what he has to say. I have subscribed to the RSS feed to his site, and intend to look into the methods he as used. You might want to take a look as well.

Original Photo by Norsehorse Edited by Brian

Ah, I love it when readers beat me to the punch!

The comments to the original post pretty much sum up the paradox and it’s solution very well.

Khaled’s and Mark’s comments illustrate perfectly one of the things I wanted to point out about this puzzle. That point is:

Just because something is phrased a certain way is not reason to assume that that phrasing is the best way to represent the problem. And one way to critically examine the situation is to reframe it in a mathematical equation.

Khaled said, “Interesting how, once you assume that you can implicitly trust a given source, you can be led through any logic, or illogic, and have a lot of trouble pulling yourself back to a critical mindset.”

How true. Then Mark gave a good method to understand how to see where the paradox lies when he said, “I started to write an equation, because properly written equations can solve all counting problems, but then realized that this was pointless, because adding 2 dollars to the 27 dollars the guests paid did not reflect what happened.”

Exactly! The question was phrased to lead you to believe that because the facts were a certain way (which it accurately represented) you had to see it in a certain way (which was anything but accurate).

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