You are here: Home / Archives for Unpacking the Three Rs

How your brain’s desire for efficiency can hurt you on standardized tests

March 1, 2012 By Leave a Comment

Think of a fruit. What’s the first thing that pops into your mind?

Think of a bird. What’s the first thing that pops into your mind?

If you’re like almost all of my students, when I asked for a fruit you thought of an apple. Maybe you thought of a banana or an orange. I very much doubt you thought of an avocado, and I’m even more confident you didn’t think of a miracle fruit.

I don’t know which bird you thought of, but I bet it could fly, that it probably didn’t live in the water, and wasn’t huge. Maybe you said eagle or sparrow. I bet you didn’t say duck, flamingo, or penguin. You very likely thought of a regular bird.

Whether we like it or not, when we prompt our subconscious mind with query, it will reliably spit back the most common, regular, typical, central example of whatever it was you said.

As well it should! Common things are, well, more common than rare ones. Language is for efficient communication too, and if everyone’s brains are collectively spitting back “apple” when we think “fruit”, it helps us coordinate.

We should all say “thank you” to our brains for for being so good at giving us central examples when we query them.

And then we should ask them to do something different on standardized tests.

On the SAT, if a problem says, “x is a number”, our ever-so-helpful brains will spit back numbers like 3, 10, and 50. Not -3, 1/10, or -1/50.

Similarly, if a problem says, “A is a set of numbers whose median is 5”, our brains like to suggest sets like {2, 3, 4, 5, 6, 7, 8} and {1, 5, 7}. Less so {-100, -100, -100, 0, 10, 12, 12, 12}.

But the way the problems are written, you won’t necessarily get them right if you just think of all the central, regular examples that are the first thing your brain pops out. You need to be able to see the whole space of possibilities.

Next time you see a standardized test question, remind yourself, “My brain is giving me the robin. I need to think of the penguin too.”

Further reading: Typicality and Asymmetrical Similarity

Filed Under: Posts Tagged With: , ,

How to Understand Math More Deeply and Have Fun While Doing it: Tell Yourself a Story

December 19, 2011 By Leave a Comment

This post is part of a series, Unpacking the Three Rs where I teach you how to watch your thoughts in the context of reading, writing, and arithmetic.

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.

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.

Doing math by stimulus response is boring and unreliable.

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

  1. It won’t work when the problem looks different.
  2. 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.” 5groupsof3

 

Another one is:

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

3groupsof5

 

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 a person each, you get __ groups.”

30halfpeople

 

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.”

Thirtyhalfcookies

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?”Halfgroupofthirty

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:

  1. Think of an easy problem that uses the same concepts. (If you have to do 15 ÷ ½, an easier problem would be 15 ÷ 3.)
  2. Make up a story about the easy problem. (If you divide 15 people into groups of 3 people each, how many groups are there?)
  3. 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?)
  4. 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.

Ever 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.

Filed Under: Posts Tagged With: , ,

Watch Yourself Think, See Yourself Grow

December 19, 2011 By 1 Comment

“You can do something destructive to yourself (feelings, beliefs, values, behaviors, etc.) over and over as long as you do it unconsciously (without continuous conscious awareness). But once you begin to do the non-resourceful feeling, behavior, belief, value, etc. consciously, it will begin to fall away. You just cannot do something that is not good for you and also do it consciously.”

–Bill Harris

“There are many words that can skip over mysteries, and some of them would be legitimate in other contexts – “complexity”, for example. But the essential mistake is that skip-over, regardless of what causal node goes behind it. The skip-over is not a thought, but a microthought. You have to pay close attention to catch yourself at it. And when you train yourself to avoid skipping, it will become a matter of instinct, not verbal reasoning. You have to feel which parts of your map are still blank, and more importantly, pay attention to that feeling.”

–Eliezer Yudkowsky

We can’t see most of what we do!

How many of your thoughts are you skipping over? When are you engaging in self-destructive behavior because you’re running on automatic?

There are many, many processes running in our brain and body, and most of them are happening unconsciously. Which is as it should be. Our conscious mind only has so much bandwidth. We can’t be constantly thinking about breathing, or the muscles we’re using to walk, or planning every word we say before we start talking.

In the case of walking, we may consciously think, I’m walk across the street. Our mind already has a whole concept of “walk” that involves making lots of muscle movements, using our eyes to make sure we don’t bump into obstacles, and balancing using information from our inner ears. When you think about everything involved, it’s hard not to be impressed with walking—it’s such a complicated and sophisticated concept!

However, one of the most useful skills for personal growth is being able to pick something that you’re doing unconsciously and unpack it. You want to be able to zoom in and watch the sequence of thoughts, feelings, and actions going on in your mind to create an outcome, whether it’s walking across the room, or not feeling understood by your partner.

Unpacking is hard when it’s something you don’t like about yourself.

And it’s particularly hard to chunk down and really look at the details of what’s going on because we’ve afraid to see what’s there. If we’re doing something that’s causing ourselves pain, physically or emotionally, it’s going to be at least a little painful to look at. Doing an original seeing on the way we relate to other people may threaten our identity.

My mother sometimes makes this one face—not sure how to describe it—but she sort of scrunches her face up—that she hates. Whenever she catches herself doing it in a mirror, and sometimes when she sees other people making the face, she thinks, “I would never do that if I realized what it looked like!”

And she’s right—she wouldn’t! But the fact that she doesn’t like that she does it makes it harder to notice when she is doing it.

Here’s the good news: you can train the skill of closely watching what’s going on from your head in whatever area is easiest, and then transfer the skill over to look at the hard stuff.

Practice unpacking neutral content to train the skill.

Unpacking is hard. I’ve never known anyone to be good at it who hasn’t practiced a lot. Then again, I’ve also never known anyone who practiced unpacking their concepts a lot who was bad at it. Everything I’ve seen points to the conclusion that unpacking mental representations is a trainable skill. And once you’ve trained it you’ll feel the confidence, pride, and growing sense of personal power that comes from being excellent at anything.

Once you’re skilled at unpacking, you’ll be in the habit of asking yourself a whole new set of questions!

  • What exactly am I think when I say ____?
  • How do I know whether fits into category?
  • What was I picturing just before I said those words to myself?

It will seem perfectly natural to ask yourself the same set of questions when you’re procrastinating at work or arguing with your spouse.

And when you have that habit—of responding to confusion and frustration by looking more closely at your own thoughts, you won’t be able to stop yourself from growing in response.

Introducing “Unpacking the Three Rs”

I want you to have the experience I just described. Because I want you to have it, I’m going to write a series of posts about observing you’re thinking about some of the stuff you learned in school, focusing on the so-called Three Rs: Reading, Writing, and Arithmetic.

I’ve been a professional private tutor for more than ten years (since I was in high school myself), and even before that I spent hours on the phone with my friends explaining math concepts. My mother’s a learning specialist and professional teacher and tutor too. This stuff is in my bones. But in all my experience, it’s been quite rare that I’ve heard educators talk directly about how to watch our thoughts when we’re learning, and how to represent material in our head so that we can understand it deeply, which is why I’m going to start doing it!

Filed Under: Posts Tagged With: ,