Some cognitive scientists think it's something schools could teach in addition to learning the rules and theorems:
The brain is a pattern-recognition machine, after all, and when focused properly, it can quickly deepen a person’s grasp of a principle, new studies suggest. Better yet, perceptual knowledge builds automatically: There’s no reason someone with a good eye for fashion or wordplay cannot develop an intuition for classifying rocks or mammals or algebraic equations, given a little interest or motivation.
“When facing problems in real-life situations, the first question is always, ‘What am I looking at? What kind of problem is this?’ ” said Philip J. Kellman, a psychologist at the University of California, Los Angeles. “Any theory of how we learn presupposes perceptual knowledge — that we know which facts are relevant, that we know what to look for.”
The challenge for education, Dr. Kellman added, “is what do we need to do to make this happen efficiently?”
The article discusses teaching children math, but I work at teaching adults law. In math, the problems have specific answers, in law, people disagree about the answers. When judges and lawyers disagree about how various texts apply to real cases, we tend to accuse each other of being biased in one way or another. But we see that bias — and our own supposedly
right answer — with the
eye that we have developed.
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Cognitive scientists are extremely earnest.
In math, the problems have specific answers..
1, 2, 4, 8, 16, _ ?
I hate when people try to reinvent education. This article is about rote memorization. That's not new or different at all, but an old method scorned by past reinventors of education.
Just remember these basics of problem solving: Who, what, when, where, why, how big, and it's difficult to go too wrong as long as these principles (Not mine but proven by many others over many long years.) are followed along with the understanding that facts are most important here.
These apply to most problem solving issues except for perhaps one major issue and that's why do some otherwise intelligent folks become Socialists?
Yeah, yeah, that last one is all about their preceived power desires.
If rote memorization can make a comeback in education circles by being reinvented with a progressive-friendly name, I'm all for it!
It is a tough road, though: once you train kids to see patterns ("the truth") all kinds of crazy things can happen. They begin to notice anomalies; they also have a basis from which to innovate.
I dunno--could be dangerous!!
He is stating the obvious. To live, humans need a Vision of categories so they can succeed at filtering out distractions. I notice that literalists become annoyed by fantasy which they perceive as enemy distractions. That may also be why Crack Emcee hates the occultists who create illusions that capture minds. Living one life at a time is hard enough work for my tastes.
They used these pieces to build a new block from the original one — for example, cutting a block that represented the fraction 4/3 into four equal slices, then making three more copies to produce a block that represented 7/3.
Hmmm. This seems to be a faulty example. 4/3 is one whole item plus a third of another item. If you slice an item into 4 pieces, you have 4 fourths. 7/3 would be accurately represented by 2 blocks plus a third of a block.
Or, is this new math where numbers can represent what ever we want?
How much of the practice of law is really problem solving? From what I see the problem solving aspect much of law is figuring out how to convict someone or sue them for lots of money, or preventing those things. In terms of trying to ameliorate a situation, law seems to have little interest.
Of the books and article I've read on problem solving, 4 steps are involved: 1)identifying the problem as specifically as possible; 2)coming up with a solution, i.e. brainstorming, idea creation, etc; 3)implementing the solution; 4)evaluation of the solution, i.e. is it working? Repeat if necessary.
Observing problem solving on the job and elsewhere, I see precious little of this going on. Most places fail at every step. 1)Can't/won't accurately identify the problem due to office politics, political correctness, higher ups biases; 2)True brainstorming rarely performed. Never witnessed it myself. 3)Inadequate implementation, not everyone cooperating; 4)No systematic evaluation of results.
Biggest problem I think most people have - and it is used to advantage frequently by lawyers and politicians (but I repeat myself) - is the inability to distinguish between correlation and causation.
Hmmm. This seems to be a faulty example. 4/3 is one whole item plus a third of another item. If you slice an item into 4 pieces, you have 4 fourths. 7/3 would be accurately represented by 2 blocks plus a third of a block.
Or, is this new math where numbers can represent what ever we want?
Ever bought a half-gallon of milk? Or a fifth of bourbon?
Ever bought a half-gallon of milk? Or a fifth of bourbon?
Your point?
Each of them is a whole thing of its own. Like each of the four thirds in the example from the article.
Yes, the brain recognizes patterns. But what makes pattern recognition possible is to have parts of the pattern already stored in memory. The more parts of the pattern, the quicker the recognition.
Part of my field in science is all about pattern recognition. So this story about teaching that sounds very familiar.
Is there such a thing as "cognitive science?"
If there is, and there are "cognitive scientists," is there any reason to believe these people are "cognitive scientists" other than that they say they are?
Physics and math are fact. Law is fiction. That is to say somebody made the law up out of whole cloth and it can be whatever people agree it is. It can even be the opposite of what it was intended to be no matter how clear the language was that was used in its creation.
Physics reflects a reality that men did not construct. The math we use to describe it is correct in that it reflects a physical reality over which man has no control and which will not change regardless of his interpretation or biases.
This is why I went into engineering. There was no room for personal bias and the left could not pervert it so easily. The softer sciences and the faux sciences like psychology or climatology are another story. Those systems are so complex that you can't really make any reliable predictions about them so they are at best informed speculation rather than science.
There is something to be said for pattern recognition. When I studied it the teacher made several rounds of the lecture hall using all of the blackboards and when she was finished an hour later she said this is how we recognize the letter "r".
Humans do have an innate ability to look for patterns but this can just as easily lead us astray. We tend to see connections that aren't there just as we make faces out of clouds or constellations out of stars. This can lead to all kinds of trouble such as phrenology, or astrology, or Keynesianism and Marxism.
Each of them is a whole thing of its own. Like each of the four thirds in the example from the article.
A half gallon of milk is a half gallon. There is no way I can split a gallon of milk into 4 parts and get 4 thirds of a gallon, or split a gallon of milk in to 3 parts and get 3 half gallons. The fact that a half gallon of milk can be bought in a individual container makes no difference.
The same for fifths of whisky using 5ths instead of halves.
Milk and whisky aside, per the example I cited, you can't cut a whole into 4 pieces and get 4 thirds. Impossible.
Humans do have an innate ability to look for patterns but this can just as easily lead us astray.
Yes.
Because I found myself agreeing with this article in terms of the usefulness of pattern recognition, I was wondering how to distinguish it from reasoning by analogy, which is reliable only as a source of error.
It seems to me that the key difference is something like this: Pattern recognition is a way of finding that, say, 2X is another way of stating X+X. Reasoning by analogy is a way of claiming that shoes are like tits, in that ideally they come in pairs.
DAD, I'll try this once more, then walk away.
Let's suppose booze were commonly sold in thirds-of-a-gallon instead of fifths. Now suppose I went to the liquor store and bought four such bottles. Can you see that I would have 4/3 of a gallon of booze? Good. Then you can see that I can add to or subtract from my collection of liquor bottles in any combination of thirds that I want. That's the point of the math example in the article.
Numbers are not arbitrary in terms of their fundamental properties. But one of their fundamental properties is that 1 = 3*(1/3), which is to say that you can treat "1" as the whole thing you are talking about, or you can treat it as 3 units of thirds. It follows that you can also talk about 4 units of thirds, and each third can be combined with any number of other thirds.
DADvocate said...
...you can't cut a whole into 4 pieces and get 4 thirds.
True, but the example cited did not start with a whole, it started with a whole and a third. It might be easier if we attach units to it.
Imagine the original block is one and a third cm^3. Cut it into four equal pieces and each is 1/3 cm^3. Add 3 more and the total is 7/3 cm^3
True, but the example cited did not start with a whole, it started with a whole and a third.
The example says:
cutting a block that represented the fraction 4/3 into four equal slices,
"A block" is one block. They do say it "represents" a whole and a third, so I get your point. At best, they've found an even more confusing way to teach fractions. Closer to life examples would be better.
BTW - I'm involved in a project at work and some of my co-workers problem solving skills are terribly lacking. To the point where I'm not sure what they're trying to accomplish and don't think they know either. Two of us have given them a simple, clear way to manage this problem and they don't seem to grasp it at all.
Chip - I get your point. As long as you have 4 separate 1/3 gallon bottles, you're fine. To me, their example of "a block" representing 4/3 is like saying 1=1 1/3.
They seem to be trying to meld apples and oranges. I had a discrete math class where the prof mentioned how much patterns count in mathematical relationships and that Karl Friedrich Gauss is considered one of the greatest mathematicians of all times because of his ability to see those patterns.
But, as several point out, a problem in law doesn't have a finite answer and is subject to whim, perspective, and opinion.
I also agree with MarkG about "educators" trying to find a shortcut to make their work easier.
WV "moyer" Synonym for Lefty weasel.
Coleridge refutes pattern recognition (aka Hartley's Associationism), along with all of artificial intelligence and probably cognitive science, in Biographia Literaria, ch 5-9.
I wonder what Cordelia Fine thinks of using neuroscience theories to direct educational standards.
In my spacecraft design class, I call this "tuning your crap detector."
Math and law are both full of abstractions. An abstraction may be made visual by showing a particlar instance or example of the abtstraction, but, this involves moving from the general case to the specific.
What visualization of abstractions is particularly useful for is propaganda. Think, for example, of wartime posters.
From an educational veiwpoint, I'd say that students who cannot grasp the abstractions as abstractions are not learning the subject.
That is, visualizations might be used as a larning aid (esp. for slower students) but they are not a substitute for actually learning a subject that deals in abstractions.
DAD, imagine a container whose volume is 4/3 of a gallon. And you want to manage the milk distribution in units of 1/3 of a gallon.
Pattern recognition in this context: How do you know how to spell the word cat? Well, you memorize the fact that it starts with a 'c' and not a 'k'.
A woman at work wrote a memorandum including the word pneumonic.
We need more of those in education.
I have always thought that there was a "duality of views" that is a useful secret for children to have in learning math. Rote is extremely important on one level, and seeing the patterns is important for anchoring that view.
In 1968-69, I taught a 4th grade class in Newark, New Jersey. New math was all the rage, but I personally thought it was crap. Though I had not been trained as a teacher, I did know math. The difficulty some of my kids had been having when I took over the class in late October, was with their times tables, and I knew that was going to become the basis for learning long division.
I made them memorize their times tables, and we even had a "times-tables bee." But one day I also took an overhead projector and turned thumb tacks upside down on it to form black dots on the light pattern shining on the wall. I then began arranging the dots in simple numerical patterns, showing them that three dots in a row constituted "a three," or "one three." And then I showed that I could add another row, i.e., another three, which would make "two threes," totaling six. Then, I went on to three threes, and so on. When I got to "five threes," which I explained could also be seen as "three fives," little Dwayne, who was sitting next to the projector, suddenly, got a look of discovery on his face. It was a look I had never seen coming from Dwayne before; a look, by the way, that I can still see in my mind's eye today . . . with great joy!
"Hey," Dwayne exclaimed, jumping to his feet and pointing at the light projected on the wall. "This is just like arithmetic!"
"That's right, Dwayne," I said. "It's just like arithmetic! In fact, it IS arithmetic . . . just another way of looking at it!"
Little Dwayne had suddenly seen the pattern. Others had too, of course, and we spent the next hour or more working our way through the times tables, each of them developing a second way to see math.
The problem with problem solving is that it doesn't give the student practice in choosing which problem to solve. In math, in particular, I think a better approach is to study and think about what one is most interested in, and then if one senses one has an insight that further results lie somewhere, go look at that, taking just a little step over there to see if that insight holds up. Picking a question to prove or disprove at random or because it would be glorious to solve is not a very effective approach if for no other reason than that what one could be led to try to solve might be too difficult to solve or might be unsolvable. (Godel showed that any consistent axiomatized theory of math strong enough to give the results about the natural numbers will have sentences neither provable or disprovable.)
As for theory vs. application, a good theory is mostly always better than a good application. It's just that, oh, I don't know, maybe 90% of theory is not good, but stuff that people have dreamed up so they can publish something that looks impressive. And it's hard to tell the difference between good theory and bad theory. Using the right definitions in math is more important probably than actually proving things. But even a subject being applied doesn't necessarily mean that it is going to be down-to-earth. Electron physics would be a very useful branch of physics, but the quantum mechanics that is used to describe it is one of the sketchier more incoherent areas of physics, at least to me.
Maybe part of the reason students don't like theory is that they don't have faith that the theory is worthwhile. An entirely reasonable faithlessness in a world replete with Emperors who have no clothes. There are fairly simple areas of math I know are typically done wrong, notwithstanding the right way of doing things has been out there for years. So when a complex subject makes me feel like it has definitions or an approach that are just wrong, as I do about algebraic geometry and quantum mechanics, I mostly avoid thinking about them. Even though I like commutative algebra (which is supposedly closely related to algebraic geometry), and quantum mechanics would be greatly useful to my understanding of things if I could ever make sense of it (because it be not incoherent), I have essentially given up on them. But the subjects are so hard, you never know for sure whether you are just being stupid for not appreciating them.
This isn't about "rote" memorization at all -- not that there's anything wrong with that. Six year olds, for example, absolutely adore memorizing things. Maria Montessori recognized that over a century ago and used it to good benefit in her program.
No, this is about pattern recognition -- making analogies if you will. If this catches on, do you suppose the SAT will, once again, introduce the analogies section?
If I recall, it was abolished because black students didn't do well and it was deemed "racist". Too bad, it was, as reinforced by this article, an indicator of well-developed cognitive skills.
Is one issue that modern education tends to treat everyone as being alike in all ways? People learn in different ways and some at different times in other ways.
So, what is the objective to asking this question? Is the objective to point out something that's being missed, and by whom?
There have been great insights mentioned above and some that were at best, tedious and banal or worse. Still, so what in so many ways.
Yet I feel that public schools today suffer from pressures to be socially correct and not denigrate others, so we wind up lowering all to a base level.
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