Pythagorus (& the cognitive unconscious)

I've just come across this post by Gary Rubenstein: You-reeka-math.

Here's a question.

For 8th grade students, Rubenstein prefers a visual proof of the Pythagorean Theorem to the algebra-based proof in Eureka Math/engage^ny math.

I follow the engage^ny math proof, but I can't make head or tails of the visual proof Rubenstein argues is more appropriate.

Am I missing something?

Is the visual proof easier for the teacher to explain, but harder for me to explain to myself?

Related: I've been meaning to read Rubinstein's Reluctant Disciplinarian forever.

I remember a fabulous passage (which I don't have time to fact-check at the moment, so if I'm remembering some other book, I'll have to come back and revise): something to do with novice teachers (Rubinstein) asking the old hands how they dealt with things like spitballs and talking out of turn and the like.

The old hands' advice: "I don't put up with that."

"I don't put up with that" is a classic example of experts having no idea how they do what they do.

Filed under: cognitive unconscious.

The cognitive unconscious is a sadly neglected concept in K-12.

K-12 education: way too much thinking, way too little knowing.

Google speaks...

...through Tom Friedman, no less:

The least important attribute they look for is “expertise.” Said Bock: “If you take somebody who has high cognitive ability, is innately curious, willing to learn and has emergent leadership skills, and you hire them as an H.R. person or finance person, and they have no content knowledge, and you compare them with someone who’s been doing just one thing and is a world expert, the expert will go: ‘I’ve seen this 100 times before; here’s what you do.’ ” Most of the time the nonexpert will come up with the same answer, added Bock, “because most of the time it’s not that hard.” Sure, once in a while they will mess it up, he said, but once in a while they’ll also come up with an answer that is totally new. And there is huge value in that.

How to Get a Job at Google
FEB. 22, 2014 | NY Times

I am praying our superintendent doesn't see this.

UPDATE: Daniel Willingham on the actual research. Thank heavens we have him.

Should math students explain their answers in words?

Complex information, such as that required for motor skills, can be learned implicitly, without awareness.

[snip]

Imagine you are riding a bicycle, and you start falling to the right. How would you avoid the impending crash? Many cyclists say they would compensate by leaning towards the left, but that action would precipitate the fall. When responding to the same situation while actually riding a bicycle, these same cyclists would turn their handlebars in the direction of the fall. The example (from Ref. 1) highlights the distinction between implicit and explicit knowledge*. Implicit learning refers to the ability to learn complex information (e.g. skills such as bicycle riding) in the absence of explicit awareness. Anecdotes such as the bicycle example offer subjectively compelling demonstrations for the existence of implicit forms of knowledge that are distinct from (and possibly in conflict with) explicit knowledge, but the existence of such learning without awareness has been difficult to prove scientifically.

Implicit learning revealed by the method of opposition
Tim Curran
TRENDS in Cognitive Sciences Vol.5 No.12 December 2001

It's an article of faith, inside schools of education, that procedural learning is dumb.

Having spent a great deal of time immersed in the literature on the basal ganglia, which handle  procedural learning, I'm pretty sure that assumption is wrong. Possibly very wrong.

The emerging research on the intelligence of nonconscious learning and the cognitive unconscious hasn't surprised me at all, mainly because, years ago, I read Arthur Reber's Implicit Learning and Tacit Knowledge: An Essay on the Cognitive Unconscious. Reber's book was so world-altering that I have kept it on my desk ever since.

As I recall (it's time to re-read), Reber opens his book with the question of expertise: how to transmit expert knowledge from one generation to the next.

In other words, he opens with the question of education.

e.g.: We have, today, people who know how to perform open-heart surgery. We will need, tomorrow, people who also know how to perform open-heart surgery. Since the people who know how to perform open-heart surgery today will grow old and die, we need to transfer their knowledge to the next generation.

How?

People's first thought (again, I haven't read Reber's book in years, so take my summary with a grain of salt)…. People's first thought was that experts should simply tell novices how they do what they do.

Simple!

But that didn't work out.

The reason it didn't work out: experts don't know how they do what they do.

That fundamental insight into the nature of expertise has never left me.

Experts don't know how they do what they do.

The fact that experts don't know how they do what they do has made me highly skeptical of "explain your answer" questions as the sine qua non of math achievement and comprehension. As far as I can tell, adults who are really good at what they do have an enormous amount of nonconscious knowledge and comprehension, so shouldn't that also be the case with children who are good at math?

I don't think the second proposition necessarily follows from the first. Perhaps math students should be able to explain, in words, why they did what they did in order to arrive at a correct answer. However, anecdotally I do see "math kids" who "just get it" -- and, anecdotally, those kids always look like the good-at-math kids to me.

In any event, I was tickled to discover that people who know how to ride a bicycle not only don't know how they do what they do but in many cases consciously believe they do exactly the opposite of what they actually do.

why there are no child prodigies in literature, history, or philosophy

Abstract: Writing skills typically develop over a course of more than two decades as a child matures and learns the craft of composition through late adolescence and into early adulthood. The novice writer progresses from a stage of knowledge-telling to a stage of knowledge-transforming characteristic of adult writers. Professional writers advance further to an expert stage of knowledge-crafting in which representations of the author's planned content, the text itself, and the prospective reader's interpretation of the text are routinely manipulated in working memory. Knowledge-transforming, and especially knowledge-crafting, arguably occur only when sufficient executive attention is available to provide a high degree of cognitive control over the maintenance of multiple representations of the text as well as planning conceptual content, generating text, and reviewing content and text. Because executive attention is limited in capacity, such control depends on reducing the working memory demands of these writing processes through maturation and learning. It is suggested that students might best learn writing skills through cognitive apprenticeship training programs that emphasize deliberate practice.
Kellogg, R.T. (2008). Training writing skills: A cognitive developmental perspective. Journal of writing research, 1(1), 1-26

uh-oh

"when A.P. testing began in 1956, memorization was not yet a dirty word"

Rethinking Advanced Placement
By CHRISTOPHER DREW
Published: January 7, 2011

In theory, the new A.P. courses are going to replace "memorization" with "critical thinking."

In reality, critical thinking depends on memorization: you can't think critically without something to think about, and that something has to be stored in long-term memory. If you're going to think critically, you have to know (i.e. remember) what you're thinking about.

What happens when you try to think critically about a subject without memorizing its terms and concepts first?

What happens is that you can think about 4 items at most. That is the number of new, discrete elements you can hold in conscious, "working memory"^* at one time. Four. And four may be pushing it.

Of course, when it comes to critical thinking, 4 is a tiny number. Experts think critically about far more elements at one time; being able to think about a vast amount of complex material is pretty much the definition of an expert, as a matter of fact:

The sine qua non of skilled cognitive performance is the ability to access large amounts of domain specific information [i.e. knowledge]. For example, it is estimated that chess masters have access to as many as 100,000 familiar configurations of chess pieces (Chase & Simon, 1973). As another example, in order to make sense of what he or she is reading, a reader must have access to information gained from previously read text. This is particularly true when reading complex technical material filled with jargon.
summary of Ericsson, K. A., & Kintsch, W. (1995). Long-term working memory. Psychological Review, 102, 211-245.
David Zach Hambrick, 1998, gt8781a@prism.gatech.edu

basal ganglia lollapalooza

Here's an example from my own life.

As a nonfiction writer, I'm essentially a permanent student: I am constantly trying to write interesting articles and books (mostly books) about material that may be brand-new to me. My current project involves the basal ganglia, which I knew nothing about going in. The vocabulary alone is overwhelming: nucleus accumbens, orbitofrontal circuit, putamen, striatum --- and that's just for starters.

So here's the question. How exactly am I to (a) understand and (b) think critically about a passage that contains these four terms if I haven't memorized what these terms mean and how they are related to each other first?

The answer is: I can't.

If I don't memorize vocabulary, I have to look up the definitions and then try to hold the definitions in mind while also reading and trying to think about what I'm reading.

It can't be done, and the reason I know it can't be done is that I've spent a lot of time trying to do it. I always make the same mistake with each new project I tackle. Somehow I think I can just look things up (Google!) and remember them while I read a complex study or article.

But I can't. No one can. Looking up four new words and remembering four new meanings maxes out working memory. There's no capacity left to read and understand a text using those four new words and four new meanings, let alone think.

I don't know why this is. Logically speaking, shouldn't it take just as much working memory to hold 4 memorized terms in mind as it does to hold 4 non-memorized terms in mind?

The answer is no: knowledge - content stored in long-term memory - extends working memory.

When you know a lot about a subject - when you have a great deal of knowledge stored in long-term memory - you can think about more than just 4 things at once.


blackboards vs PowerPoints

^* Working memory is essentially consciousness: it's what you're thinking about and/or remembering right now. When you hold a phone number in memory while dialing it, you're using working memory.

spelling alone

Deliberate Practice Spells Success: Why Grittier Competitors Triumph at the National Spelling Bee

what is skill?

Skill is a level of performance in any given task that can be acquired only through practise. Indeed, one can consider any skilled professional as a person who has had the motivation to practise one thing far more (for approximately 10,000 hours extended over more than 10 years²⁰) than most people could endure (BOXES 2,3). Across a wide range of tasks, the relationship between one measure of skill, the speed of task completion and the number of practise trials is well approximated by a power law²¹ (FIG. 1). This implies that performance continues to improve with task-relevant practise indefinitely, although the rate of improvement declines over time. Of course, most of the relevant data comes from tasks learnt for short periods of time in the laboratory. However, it is worth highlighting one classic study that reported performance of an industrial cigar rolling task²². The study included workers who had produced in excess of ten million cigars over seven years of work and they were still getting faster!

Inside the brain of an elite athlete: the neural processes that support high achievement in sports
Kielan Yarrow, Peter Brown and John W. Krakauer §
Nature Reviews Neuroscience Volume 10 | AugusT 2009
p. 587

Cool!

Independent George on learning the martial arts

I started taking martial arts classes over the summer. Before doing anything fun, we spent the first couple weeks entirely on footwork. Two weeks teaching educated adults how to walk. It was not only boring as heck, it was sometimes painful and exhausting.

Nevertheless, everyone stuck it out because it was well understood that we had to master these basics - how to advance, retreat, and sidestep without falling on your face - before we could get to the fun stuff. Moreover, we kept practicing footwork drills even as we moved to the more advanced techniques - because it wasn't enough to simply know how to move, it needs to be automatic.

Even after it's automatic, you continue to practice the fundamentals constantly because it's really easy to get lazy. When you get good at throwing punches, you tend to think only about the punch, and not about returning to your guard position afterwards. You learn to guard against a takedown, and you forget to protect your head.

The importance of fundamentals became obvious when my work schedule eventually prevented me from attending class regularly - and my performance dropped precipitously. The things which had once been automatic now required my conscious attention. Instead of learning new skills - or applying them in new combinations - I had to spend twice as much time re-learning the old ones that I "understood", but clearly hadn't mastered.

But, of course, all of this is totally different from academic subjects.

I began tennis lessons last summer, and did not have my first "fun" lesson until just a few weeks ago. That's around 7 months of distinctly not-fun weekly lessons and practice before I experienced the slightest feeling of fluidity and simple enjoyment. Up to that point it was a challenge keeping myself going to lessons, in fact. I felt trepidatious before each one.

Knowing what I do now, I'm happy we had the wit to force C. to carry on with his tennis lessons. He's now quite a good player, and has tremendous fun playing with friends and classmates. That has happened only because we forced him to keep going to lessons during periods when he would have preferred to stay home and pursue the sport he's (apparently) great at: video games.

I wish to heck I'd figured this out where music lessons are concerned.

Somehow I thought it was right to be your child's frontal lobes for academics but not for hobbies and leisure activities. I didn't think it through. Not sure why I managed to put my foot down on sticking with tennis. It just came to me one summer day, picking C. up from the local tennis camp, that this was it: he liked tennis, he was capable of playing tennis, so tennis it was going to be.

inflexible knowledge

sigh

I've told C. he has to start doing a page of problems in Greenhall's Acing the New SAT I Math, which I ordered after comparing Amazon reader reviews for the various prep books. 

Today both he and I were stumped by this problem:

8. S is the sum of 4 consecutive integers, the smallest of which is n. In terms of S, what is the sum of four consecutive integers of which the greatest is n?

I solved it via "mathematical reasoning," a phrase I'm putting in quotation marks because while I do possess sufficient mathematical reasoning abilities to solve this problem via number lines and "n + 1s," apparently I do not possess sufficient mathematical reasoning power to solve it via substitution. (Or, rather, to recognize the fact that I am looking at a simple substitution problem.)

This is maddening.

However, the book is obviously going to be good for me to work through. 

Fun, too.

Does it take 10 years for inflexible knowledge to start easing up a bit?

Do we know?

how to grow an athlete

I still haven't read this story, but this paragraph caught my eye:

If I gave in to the uncontrollable Ericssonian urge to put Spartak's success into a formula, it would read something like:

Intense Parents + Young Kids + Rigorous Technique + Toughness = Talent

How to Grow a Super-Athlete
by Daniel Coyle
NY Times Magazine
March 4, 2007

I should probably go post that in the Comments on Concerned Parent's blog.


the "naturals"

Speaking of how to grow an athlete, I learned yesterday, while spending FOUR HOURS out on in a CVS out on Long Island trying to get Andrew's prescription medications labeled in compliance with state law for autistic kids attending 7 days of summer camp for the first time EVER, that there's a fellow in Switzerland who may be able to distinguish between the "naturals" and the "works hards." (You can get a lot of magazines read waiting 4 hours in a CVS.)

I wish to heck Psych Today had the print story posted online. The neuroscientist was an aspiring chess grandmaster who worked his tucchus off, was sent to special schools, etc. He finally gave up when he was beaten by an amateur player who "just liked to play."

Here's the blurb they do have posted.

The difference he's seeing has to do with memory, but I haven't figured it out yet.

out with the old

The old in this case being teachers in their 50s:

My school is a good school. Attracting quality teachers should not be a problem. But, in a system that does not value senior teachers this does not happen. Seniority transfers are no longer available. Administrators want the newbies who take a much lower salary than the experienced teacher and jump to follow all their directives. Experienced teachers in the schools already are being harassed and told to transfer (as if some other school would take them.) Only the young are valued.

When I started teaching there were many experienced teachers around. I needed to learn and they were around to help. Sadly, this is no longer true. Sadly, even the older teachers still around are so overwhelmed with work that there is no time to help the new ones. Soon the only teachers left will be the ones who don't know the material and can't teach. These teachers will embrace the horse shoe seating arrangement and they will embrace group work so my guess is they will be embraced by the administration.

Limitless Inability

This is what we see here, although the high school did recently hire a science teacher with 10 years' experience.

Primarily, though, we hire novice teachers. In math. I know of one instance in which the district refused even to interview a superb, experienced teacher qualified to teach h.s. math and physics. In another case my district told an experienced teacher that if he were to take a job here he would have to take a 5th-year teacher's pay.

He's teaching in a private school now.

how to be a genius

"It's complicated explaining how genius or expertise is created and why it's so rare," says Anders Ericsson, the professor of psychology at Florida State University in Tallahassee who edited the handbook. "But it isn't magic, and it isn't born. It happens because some critical things line up so that a person of good intelligence can put in the sustained, focused effort it takes to achieve extraordinary mastery. These people don't necessarily have an especially high IQ, but they almost always have very supportive environments, and they almost always have important mentors. And the one thing they always have is this incredible investment of effort."

[snip]

"It's funny, really. On one hand it's encouraging: it makes me think that even the most ordinary among us should be careful about saying we can't do great things, because people have proven again and again that most people can do something extraordinary if they're willing to put in the exercise. On the other hand, it's a bit overwhelming to look at what these people have to do. They generally invest about five times as much time and effort to become great as an accomplished amateur does to become competent..."

[snip]

Take intelligence. No accepted measure of innate or basic intelligence, whether IQ or other metrics, reliably predicts that a person will develop extraordinary ability. In other words, the IQs of the great would not predict their level of accomplishments, nor would their accomplishments predict their IQs. Studies of chess masters and highly successful artists, scientists and musicians usually find their IQs to be above average, typically in the 115 to 130 range, where some 14 per cent of the population reside - impressive enough, but hardly as rarefied as their achievements and abilities."

[snip]

A sober look at any field shows that the top performers are rarely more gifted than the also-rans, but they almost invariably outwork them.


the ten year rule

This has led scholars of elite performance to speak of a 10-year rule: it seems you have to put in at least a decade of focused work to master something and bring greatness within reach. This shows starkly in a 1985 study of 120 elite athletes, performers, artists, biochemists and mathematicians led by University of Chicago psychologist Benjamin Bloom, a giant of the field who died in 1999. Every single person in the study took at least a decade of hard study or practice to achieve international recognition. Olympic swimmers trained for an average of 15 years before making the team; the best concert pianists took 15 years to earn international recognition. Top researchers, sculptors and mathematicians put in similar amounts of time.

[snip]

The same is true for Tiger Woods. He seems magical on the golf course, but he was swinging a golf club before he could walk, got great instruction and practised constantly from boyhood, and even today outworks all his rivals. His genius has been laboriously constructed.

[snip]

...the subjects of Bloom's study, like most elite performers, almost invariably enjoyed plentiful support in their formative years. Bloom, in fact, came to see great talent as less an individual trait than a creation of environment and encouragement. "We were looking for exceptional kids," he said, "and what we found were exceptional conditions." He was intrigued to find that few of the study's subjects had shown special promise when they first took up the fields they later excelled in, and most harboured no early ambition for stellar achievement.

[snip]

Michael Jordan, widely considered to be one of the world's greatest athletes, struggled horribly when he moved from basketball to baseball, where he was routinely flummoxed by minor league pitchers.

[snip]

Eric Kandel of Columbia University in New York, who won a Nobel prize in 2000 for discovering much of the neural basis of memory and learning, has shown that both the number and strength of the nerve connections associated with a memory or skill increase in proportion to how often and how emphatically the lesson is repeated.

[snip]

Genius must be built.

[snip]

These disciplines all but insist that the traditional distinction between nature and nurture is obsolete. What we call talent or genius illustrates vividly what the past 25 years have taught us about gene expression - that our genetic potentials are activated and realised only through environment and experience.

[snip]

We should probably shelve the notion of genius as an innate, almost irrepressible gift and speak instead of expertise, talent or even greatness - terms that hint at the work underlying supreme accomplishment.

source:
How to Be a Genius

I wonder if spending hours and hours and hours of your life trying to figure out the connection between stereotypy and novelty so you can FINALLY finish a draft of CHAPTER ONE counts towards the 10-year rule.

Or do you have to stop the clock every time you get stuck?

...................

Don't answer that.

..................

I like the idea of replacing genius with greatness.


do what you love (may have to hit refresh a couple of times)
expertise and deliberate practice
freakonomics post on Ericsson
Cambridge Handbook of Expert Performance (terrific)