kitchen table math, the sequel: the learning spike?

Sunday, April 17, 2011

the learning spike?

As I mentioned the other day, C. and I have suddenly jumped a good 50 points in our math scores on Blue Book tests.

On April 6, C. missed 7 and skipped 1 on a 20-item math section.

Four days later, on April 10, he finished all 8 questions in an 8-question multiple choice section and missed just 1. He got 7 of the 10 grid-ins right, missed 2, and skipped 1.

And: all three of the questions he missed were dumb mistakes. He knew how to do the problems, and did them quickly enough to finish the test.

I've had the same experience. Last summer I couldn't hope to finish a math section; yesterday I finished early enough to go back and check my bubbles.

At first I thought the higher scores were a fluke. But C. has now turned in the same performance on 5 math sections in a row, and for me that number is probably 6 or even 7.

the power spike of learning?

I'm surprised. I don't remember ever experiencing a sudden jump in learning like this, and my understanding of the "learning curve" is that it's a power curve (if that's the right term), not a right angle. You make more gains early on than you do later.

C. and I made practically no gains early on. While C and I weren't doing a lot of SAT practice fall semester, we have been working with some regularity since January, and in that time we've gotten nowhere. He's been stuck in the high 500s, and I've been stuck in the low 600s. (Very low.)

In fact, I've been stuck in the low 600s for a good two years now. Not that I was practicing SAT math per se -- I wasn't -- but I have been studying high school math off and on during that period, and I've seen no transfer to SAT math at all.

Yesterday, my score on all 3 sections of Test 2 in the College Board online course was 690. C.'s score was 640.

C. said, "It's like I jumped over a wall."

If you graphed our scores on an xy plane, it would be more like we leaped a tall building in a single bound. 

Now we have to leap another one.

arguing in French

The other night at dinner we were talking with our friends about whether they'd had this experience. One friend, an attorney, said tax law was her version of SAT math. She didn't get tax law at all until one day she did. 

Then Ed remembered learning French in France. He was doing what C. and I have been doing: grinding away, putting in the time, having nothing much to show for it. 

Then one day he was sitting around with some friends, and one of them made a provocative statement about something or other. Ed disagreed, an argument ensued, and at some point Ed realized he was arguing in French.

Arguing in a foreign language is the equivalent of an 800, I think.

10 tests

What does this mean, if anything?

Well, first of all, I have to see whether C. and I really are stable at this new level. I suspect we are, but we'll see.

Second: start early. I have no idea why it's taken us so long to experience this leap, but no one becomes an expert - or even a proficient novice (which is probably what we are now) - in a day.

Third: The Blue Book has 10 real SAT tests. C. and I have taken all of the math sections in all 10 tests, and I have taken all 3 math sections in Debbie Stier's January 2011 test.* Our scores jumped somewhere in the neighborhood of 30 math sections taken over several months' time.

My current thinking on SAT prep is that students should do all 10 sample tests in the Blue Book at a minimum and should spread that work out over at least 4 months.

I'm also thinking it would be a good idea to do the 9 tests College Board offers online for $70.

reading and writing

I'm going to start paying attention to the reading and writing sections. We've done far fewer of those because C. is a very good reader and has been since he was little. He was one of those kids who taught himself to read. That's a funny story, which I know I've told before. C's Kindergarten teacher called us in for a parent-teacher meeting and told us C's handwriting indicated that he was at risk for a reading disability, which was true. Very bad handwriting is a flag.

Naturally, I figured: we've got two autistic kids so now we're going to have a dyslexic kid, too. Just our luck. 

Two weeks later, C. could read. All of a sudden. He went from not reading to reading.

(Another case of a power right angle?)

Back on topic: because C's SAT reading scores are routinely in the low 700s, there's not a lot to learn about SAT prep from observing him, I don't think. He misses or skips questions when he absolutely does not know a vocabulary word and can't figure it out from context. I told him yesterday he has to get back to memorizing his SAT vocabulary words, so the challenge will be remembering to nag him to do it.

("Have you studied your SAT words?" "No." "Do you know where your SAT flash cards are?" "I have an iPod app I use." "Fine, but do you know where the cards are?" "They're in the family room." etc.)

The writing multiple choice questions are more interesting; he misses more of them and presumably will benefit from more practice. (We've done very few writing sections.)

I'm going to start paying attention and will report back.

* We had also re-taken all 3 sections of the first test in the Blue Book as well as 1 section of the first test in John Chung's book.
The Official SAT Study Guide, 2nd edition

Dr. John Chung's SAT Math


Catherine Johnson said...

Yesterday's dumb mistake: I added 29 and 5 and got 30.

C. said, 'That's why I always use the calculator."

Catherine Johnson said...

C. got that question right, I am happy to say.

Anonymous said...

I am like that with names. At the start of the school year, I need to learn 150 names. For the first week it is slow going and mostly by memory tricks. Then all of a sudden, I know almost every name automatically.

I wonder if your speed has increased because you have become better at instantly categorizing the math questions.

Paul B said...

I'm relearning the piano after 50 years away from it and I experience a similar effect. First I learn the melody. I find this the easiest, probably because it's already familiar to me. Then I learn the left hand, also pretty easy, but dull since harmony sounds pretty dreadful by itself.

Then I hit a wall and for a week I have to really push myself to practice at all. I'm playing the song but it's not music. Finally I reach a point where points in the song are being played with what I'll call 'emotion'. I get to a point where I start to play with it a bit, modifying parts to suit my mood.

When I get to that point I know I'm on top of wall. To jump over I just work on the parts that are still mechanical and dull. It's all about practice, practice, practice.

In a strange way I always knew that practice was the key to proficiency in math but I could never really say why. With the piano I can 'feel' it happening. What's new to me at this stage of the game is that I'm no longer reading the music, or even thinking about what keys to strike. It's like I've freed up the part of my brain that works my eyes and my fingers to just hear the music. My ears have my entire attention.

I've even learned a trick to speed the process (at least it works for me) where I play with my eyes closed. I'm convinced that it has to do with minimizing what's taking place in your short term memory in order to focus on the important stuff.

I perceive a plateau in this process when I'm in that painful part of the practicing. It doesn't seem like anything is happening. It's very frustrating and I just have to push through it. I suspect there really is steady progress being made but it just can't show itself until every little brick has been laid. Every day there's a different bar to screw up until finally you've paid the price of admission.

Practice is what transforms the difficult into the thoughtless.

Allison said...

The "learning curve" shape people refer to is usually considered some form of a sigmoid:

It's shaped like an S, sorta.

It's flat-ish for a while, then steep, then flat-sh again.

Depending on the parameters, those flat parts can be almost entirely flat or can have more slope to them, and the upward curve part can be sloped or nearly straight up depending on parameters.

The major feature is that you go through a period where you're not getting that much bang for your buck, then you are, and then you aren't again.

For the flat-steep-flat versions, you expend a lot of effort for a while and seemingly make no progress. But then suddenly, finally, you take off, and your improvement is phenomenal for just a bit more effort. And then, you level off again.

The whole thing repeats then--the top flat part is a new bottom flat part, and you'll be there a long time before you get on the steep part again.

To learn something, you need to do all of that hard work to get off the flat part. And once you're on the top flat part again, there's really a question of whether expending more effort is worth it.

Catherine Johnson said...

That's not the curve I have in my books.

The curve is exponential (I think that's the term).

It's not an S-shape.

It has a steep curve in the beginning and a much flatter curve later.

Catherine Johnson said...

Here's an example:

Catherine Johnson said...

My tennis teacher loves this hierarchy:

* Unconscious incompetence
* Conscious incompetence
* Conscious competence
* Unconscious competence

Catherine Johnson said...

"the more you practice, the less you gain from each trial"

Allison said...

So, I think cog sci is pretty clear that the "flat" parts are times during which you're accumulating information/skill before you've gotten to where you're building an efficient schema. The steep part is where the schema gets built or is built, and then you're just placing stuff into that schema as fast you can. Eventually, that schema fills, or that schema's correctness/value reaches a limit. Then you're on the flat part again.

What happens next is what's interesting: the schema itself gets chunked, and so you can build a new schema with the old-chunked schemas hanging off, filling it up, repeating the process.

Without practice you just can't get enough of your mind to know how to chunk, how to convert that information into compact subroutines of knowledge. And after a certain point, the kinds of stuff you're practicing are changing. So in music, you stop having to consciously translate the notes on the page into your fingering. Later, you stop even the subconscious translating to finger because you're hearing it in your head when you read it. In arithmetic, you stop having to check your arithmetic and are able to start tackling "what do I know about triangles". After enough practice, you've chunked all you know about triangles and are now practicing translating a problem about trapezoids into triangle rules unconsciously.

Allison said...

No, that learning curve is only showing the top half of the S.

Learning curves have inflection points. The curve shown in that graph sets the inflection point at the origin and only shows beyond that.

That's wrong. The bottom half is where we feel the most pain, and where the stuff matters most. If you're on this part of the learning curve, your bang-for-buck is low and falling. But what's most interesting is what happens BEFORE that inflection point. That's the part Paul's talking about being the frustrating part, and it's in the neighborhood of the inflection point that you recognize how dramatically you've improved.