A Lot More to Find Out

How to understand math more deeply and have fun while doing it: tell yourself a story!

Sat, 17 Dec 2011 16:50:41 -0500

Do you think of math as “boring” or “too hard”? Or, if you’re a parent, does your child? Maybe you like math sometimes, or even a lot of the time, but now you’re studying something that hasn’t clicked. There are too many steps to remember, and you can feel your head aching just thinking about it.

Doing math by stimulus response is boring and unreliable.

If what I wrote above rings true for you, you’re probably trying to learn math by stimulus-response. You’re looking at a problem, thinking “where have I seen something kind of like this”, and then carrying out a bunch of steps that you’ve used before.

The most obvious example I see of students doing math by stimulus-response, without seeking true understanding, is when they look at a word problem, ignore the words in the question, add up the numbers, and put the sum as their answer.

Any time a child, or an adult for that matter, manipulates numbers without knowing why, he is doing math by stimulus-response.

There are two problems with doing math by stimulus-response:

It won’t work when the problem looks different.
It’s boring and unsatisfying.

Instead of offering more abstract explanation, let me dive into a specific example: division.

For every concept, make up a story out of an example problem.

I like to ask my students, “What’s your story for 15 ÷ 3 = 5?”

One story could be:

“When you divide fifteen people into groups of three people each, you get five groups.”

Another one is:

“When you divide fifteen people into three groups, there are five people in each group.”

Then, I’ll ask my students to draw a picture of the story.

Because stories aren’t just words; when you tell yourself a story, you can see it in your imagination. For some people, the pictures will be clear and distinct, full of visual detail. For other people, the spatial relationships the important part, and so their pictures are more like blobs floating around in space.

Either way, the reason you can do math better when you have pictures in your head is that pictures have moving parts. You can ask yourself, “If I changed this number, how would the picture look then?”

Here’s how this would work. Many students can do 15 ÷ 3, but don’t have any idea how to figure out 15 ÷ ½.

So, I tell them to go back to the story.

When I substitute ½ for three in the same story, I get:

“When you divide 15 people into groups of ½, you get __ groups.”

When you change the numbers, you may need to tweak the story and open your mind a little.

Of course, once you go back to the story, you may need to tweak it a little. If I get a student saying, how can you have ½ a person, I would tell her to make a more sophisticated story. Okay, it’s hard to think about half of a person—let’s pick something that it’s easy to take half of. We could change the story to be about cookies.

Now, the story is:

“When you divide fifteen cookies into groups of three cookies each, you get five groups.”

And it’s easier to ask:

“When you divide fifteen cookies into groups of ½ a cookie each, you get __ groups.”

Once they’ve gotten around to asking themselves that question, most students can clearly see what to do.

Stories within stories…

Let’s say we’d picked the other story about division:

“When you divide fifteen people into three groups, there are five people in each group.”

Now, we get:

“When you divide fifteen people into ½ a group, there are __ people in each group.”

Here, the confusion might, be, “What’s half a group?” Then, it’s time to ask, what’s half of anything? What’s the story for ½?

Maybe you think of it in terms of cakes:

“You get half of a cake when you cut it into two equal pieces. You can combine two halves of a cookie to make a whole.”

Now substitute:

“You get half of a group when you divide it into two equal sections. You can combine two halves of a group to get a whole.”

So, “If ½ of a group has 15 people, how many people are in the group?”

Once again, we’ve arrived at a question that’s a lot more straightforward and a lot easier to answer.

What now?

The next time you see a math problem and you start to think, “This is hard… It’s too complicated… ” The next time you feel yourself getting confused and frustrated, and just wanting to do something so that the problem will be over with, pull up this article and follow these step:

Think of an easy problem that uses the same concepts. (If you have to do 15 ÷ ½, an easier problem would be 15 ÷ 3.)
Make up a story about the easy problem. (If you divide 15 people into groups of 3 people each, how many groups are there?)
Once the story is solid, switch the problem back to the hard version. (If you divide 15 people into groups of ½ a person each, how many groups are there?)
Tweak the story if necessary. (If you divide 15 cookies into groups of ½ a cookie each, how many groups are there?)

Remember that the four step process for turning math into stories is a recursive procedure. As you go through the steps, there might be another math concept inside the story that you aren’t quite solid on. Make that concept into a story too.

Every wonder why some people seem to genuinely enjoy math? Have you ever looked at them and practically seen the gears turning in their head as they get excited about solving a new problem?

It’s because they’re telling themselves stories, filling in details, and then looking back at the story to see what happens next. If you learn to tell yourselves stories about math, your concepts will begin to click together. Math will stop being boring, repetitive, robotic, and frustrating (no wonder you don’t like it!) and turn into a creative journey of discovery.

Posted via email from More meaning than magic

Positive Vector Presents Nutrition 101: Cut through the Confusion and Optimize Your Diet

Mon, 05 Dec 2011 03:00:03 -0500

Click here to check it out!

Posted via email from More meaning than magic

what's on my mind

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If, in a dream in which you are fleeing from harm, you realize you are dreaming, you will still tend to continue escaping, even though you should now know that there is nothing to flee from. -Exploring the World of Lucid Dreaming

Posted via email from divia’s posterous | Comment »

"Speaking of livers,” the unicorn said. “Real magic can never be made by offering up..."

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“Speaking of livers,” the unicorn said. “Real magic can never be made by offering up someone else’s liver. You must tear out your own, and not expect to get it back. The true witches know that.”

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“Dear Noah, We could have sworn you said the ark wasn’t leaving till 5. Sincerely, Unicorns.”

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via scienceblogs.com

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via scienceblogs.com

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“

“How did it feel?” she repeated. It felt wonderful, but how do I describe that? “It felt—I’m not certain. I think I must have felt the way an opera singer feels, when everything comes together in a perfect performance. As if I was born to do this, as if nothing in the world was more natural or right for me. There was a joy, a feeling of completion, a feeling of coming home—” She shook her head. “All that, and more. I can’t describe it properly.”

He sat back, and there was no mistaking the satisfaction in his eyes. “You don’t have to. This was something du Mond never felt, and I should have known then that there was something wrong with him. The true Magician, the one who is born to it, comes to his work with pleasure, and not as if it is work. I suspect this must be the case with anyone who is doing what he is truly suited to, whether he be a Magician or a singer, a poet or a priest, or even a plumber. You had that joy about you; this is what you were born to do.”

”

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AI-Box experiment results

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He let me out! Yay! I’ll write more about this later, but for now I have to go to work.

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AI-Box experiment

Wed, 21 Apr 2010 15:23:00 -0400

I’ve been thinking about AI boxing for a while, and just for fun I’ve decided to try playing the AI against my friend Neil. We’re starting around now, and I’ll post the results when we’re done. (Just to note, I have pretty low confidence this will work, but I think it’s going to be fun to try!)

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- My Side of the Mountain by Jean Craighead George

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