Archive for: September 2008

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Division by zero is one of those basic concepts that confuses the poop out of people.

You’re taught, “You can’t divide by zero.” But are you taught why? Adequately? Nah. That’s one of the fundamental goobers of elementary school. They give you rules to memorize, but even the teachers are unclear of why those rules are rules.

It’s not too tough to understand why division by zero (in arithmetic) is “verboten.” You just have to get out of the mindset of “well, it doesn’t make sense.” It does make sense. It just doesn’t make sense if you only think about it with your brain-stem. You have to break out of the “intuitive” mindset. (Intuition is also not all it’s cracked up to be. Untrained intuition, that is.)

If you are a person who has a hard time letting go of the notion that division by zero (in arithmetic) cannot be done, consider this:

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Tags: divide by 0 , divide by zero , dividing by 0 , dividing by zero , division by 0 , division by zero

Recently I got a question from read that I think a lot of other readers would also like to ask. You may have a student, a child, or it might even be you that could use some Ideas to help with this issue. Here’s what the reader asked:

I was looking online for suggestions about how to teach kids to check work, when I came accross your website.
 
I liked the tone of what you wrote, and am writing you because of the experience I think your site conveys.  I am a volunteer tutor working with a 14 year old student.  He is a very fast thinker, but has very little experience being careful in doing almost anything. He either gets things quickly or drops the ball.  I am working with him on being persistent and checking things (which he probably needs to learn more than math), but it is tricky.  

He gets going so fast, and then sort of gives me a look of “I’m done” after finishing problems.  I find myself correcting him on each issue, but the bigger issue is not the mistake that he makes on any given problem, but instead his math style.  I recognize this, and try to address the issue, but I’m still having problems.  He reverts very quickly to his old ways and I feel overly critical always saying “you see you missed that.”  I want to develop a suggestion that he can use to slow himself down to be systematic, but other than “going slow and looking” I can’t.  And that isn’t very concrete.
 
I am working on getting him to write much neater (he used to barely look when he writes) and to focus on the page and being systematic, but it is really hard.  Any suggestions? 

 -J

Professor Homunculus’ answer:

Hi, J, 

That is a great question. I actually have a response from personal experience. I have some of the same study tendencies as your student. I didn’t really even learn any math (not even long division) until deep into my thirties. 

As I learned, I followed my progress closely, and being a magician, I tried to keep in touch with whatever my mind was doing that actually helped. 

One way to change the thinking patterns when doing math is to do your writing with your non-dominant hand. There are a lot of theories why this sometimes works, and non of them are as important as actually trying it. 

There are other ways to “trick” your brain into performing the operations less sloppily - writing backwards, speaking your answers in another language than your mother-tongue (if you know one), or writing in calligraphy come to mind right away. 

You could also read the problems in a mirror, or only do them as they are read to you, and be allowed to write the answer - you have to speak it back. 

All of these are worth a try, but none of them will be any good if your student tries them as just a “parlor trick” and doesn’t stick with them and notice any differences. As he notices them, he should write them in a notebook. 

This way he will be learning more than math - he will be learning about the personal style of his own mind - and he will be able to positively affect it. 

Let me know how it goes.

Tags: checking your answers , learning techniques

Not much of a post today, but there are two things I’d like to bring your attention to:

If you’ve read the recent posts about bases, you might find this comic in today’s Sunday supplement of many newspapers interesting. It’s from “Foxtrot” by Bill Amend. He often references science and math in his comics.

This week there was an article in the NY Times Science section about what has been theorized to be the two different “number senses” humans may posses. Here is an except:

“a host of new studies suggests that the two number systems, the bestial and celestial, may be profoundly related, an insight with potentially broad implications for math education.”

As far as I can tell, the theory references subitizing (a two-dollar word for a nickel concept), which the article does not mention. Subitizing is the Idea that the mind can count or estimate small numbers of items without engaging higher parts of the brain. It is the “sudden knowledge” of amounts. There is more to it than this, of course. The article gives you a pretty good Idea of what it is.

Tags:

Thought you might enjoy this bit of math “logic.”

Tags: math humor

I read this in a recent blog post from The Economistblog.com:

“From the National Mathematics Advisory Panel’s Final Report , released yesterday. It is entitled: U.S. Math Education is “Broken and Must Be Fixed”:

 

“International and domestic comparisons show that American students have not been succeeding in the mathematical part of their education at anything like a level expected of an international leader. Particularly disturbing is the consistency of findings that American students achieve in mathematics at a mediocre level by comparison to peers worldwide.”

 

As always, there’s something to this story, but it’s deeper than a short blog post can explore adequately. You just have to be mildly conscious to notice that American schools are generally doing a pathetic job of math education. But it’s not that easy to assess “blame,” or “what needs to be done.” There’s no simple answer to such a complex problem.

That it’s “broken” seems clear. But what is “it?” Each state, district, and even type of school is different. One thing that is clear is that creating simplistic “standards” (which are supposed to cover all federally funded schools) is just jaw-flapping by people who understand neither education, math, nor children.

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Tags: math ed , math education , math pedagogy , math wars , public school math