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September 18, 2007


The issue of answers first is based on an opinion that the vast majority of students are incapable of dealing with process; plus the fact that answers are sufficient to train technicians, including most of the practicing professionals important to our our lives and deaths. It is an elitist opinion, but largely unchallenged by the very students it degrades. Maybe some day.....

Dewey offers a different explanation for it in one of the passages I replaced with [...] :

There is a strong temptation to assume that presenting subject matter in its perfected form provides a royal road to learning. What more natural than to suppose that the immature can be saved time and energy, and be protected from needless error by commencing where competent inquirers have left off?

From the point of view of the specialist (in my case, the practicing scientist), what you are excited about and want to tell people about often is the ANSWER -- "hey, look what I learned!" -- and not so much how you got there. (In fact, it's common practice in seminars and in papers to fudge the real history of the experimental road one took, to make the story "flow" better... something one would never do with the results themselves.)

There's also a large (largely right-wing) contingent that derides the process-based approach as fuzzy and not fact-based, their criticism summed up by the not-at-all-right-wing Tom Lehrer's song "New Math": "in the new approach, as you know, the important thing is to understand what you're doing rather than to get the right answer."

You're likely right that an underestimation of student abilities may be part of it, but there's probably more than just that going on... as one colleague used to say, "Our textbooks all go 'we know this, we know this, we know this.' We ought to write one that goes, 'we don't know, we don't know, we don't know' -- but nobody would buy it!"

Just to note that The Scientist has a list of favourite blogs of top science bloggers here: http://www.the-scientist.com/news/home/53596/
(H/T Pharyngula)

I'd like to reapproach process in light of this blog.
so much of what we cite is content- answers.
but so much of what we respect is process- emptywheel figuring shit out.

I think the point I am trying to make is that in the end, answers are what matters to those who can't or won't participate in the process.
Compare dKos to TNH. What makes a reclist diary is largely content; but here, it seems the draw is as much the visualization of process as the obviously good content.

Now stepping into your world- your colleagues respect of your analytical skills is well and good, but if you don't have the answers, you are hosed wrt to tenure. And if someone forgets to cite your results, your answers, you are personally more concerned than if they fail to recognize your process. No?

In the end, many scientists themselves defer to the value of answers....

Interesting stuff and interesting posts from empty pockets. One pretty decent general medical resource (although the site may be changing a bit) is
Patient Care which is the site a "throwaway" (controlled circulation) journal most doctors get with articles from top people at tertiary centers, good graphs and illustrations.

It has search of its archives, and arranges articles by topics to a degree.

A great list of philosophy videos.


i'm starting to like this gig you are doing a lot.

enjoyed the cites from you and commenters last week -

booked 'em.


in science education

the first question to students should be

why bother?

why do we care about this topic?

what can we learn from this?

whether it mechanics in physics


invertebrates in biology


polysacharides in chemistry.

this introduction to a particular aspect of science used to be routinely missing.

maybe it's there now.

in short, put the knowledge to be learned in a historical and social context,

before going on to the facts, the theories, and the formulas.

i would argue this is equally true for math education.

thanks to all for the additional links. keep 'em coming. those philosophy videos look particularly interesting to me.

drational, it's a mistake I think a lot of scientists (myself very much included) often make. We see the big picture of where the results lead, and see the technique as just a way of getting there, not that interesting in itself. We tend to explain things the same way. It's going to take some work for me to appreciate that, for the novice, understanding the means is as important as understanding the ends. There's an added complication that often the means are really messy -- sometimes we do experiments for no good reason, or (frequently) because we think we're working on question A, and end up chasing down an unexpected result that leads to Z. It is easier to set up the result by saying "We set out to understand Z..." than to explain the tortuous and mistake-filled path we really took. Although my favorite seminars are always the ones where the speaker does just that.

Orion, I omitted some of Dewey that's very relevant to your comment, about putting material in a context that is meaningful to the everyday life of the student and not simply as acquiring knowledge for its own sake:

[...]because the material is stated with reference to the furtherance of knowledge as an end in itself, its connections with the material of everyday life are hidden. [...]
The pupil learns symbols without the key to their meaning. He acquires a technical body of information without ability to trace its connections with the objects and operations with which he is familiar[...]

Great example of advanced students who have compartmentalized their textbook learning from their real-world experience, via Richard Feynman:

We first took two strips of polaroid and rotated them until they let the most light through. From doing that we could tell that the two strips were now admitting light polarized in the same direction -- what passed through one piece of polaroid could also pass through the other. But then I asked them how one could tell the absolute direction of polarization, for a single piece of polaroid.

They hadn't any idea.

I knew this took a certain amount of ingenuity, so I gave them a hint:
"Look at the light reflected from the bay outside."

Nobody said anything.

Then I said, "Have you ever heard of Brewster's Angle?"

"Yes, sir! Brewster's Angle is the angle at which light reflected from
a medium with an index of refraction is completely polarized."

"And which way is the light polarized when it's reflected?"

"The light is polarized perpendicular to the plane of reflection, sir."
Even now, I have to think about it; they knew it cold! They even knew the
tangent of the angle equals the index!

I said, "Well?"

Still nothing. They had just told me that light reflected from a medium
with an index, such as the bay outside, was polarized; they had even told me
which way it was polarized.

I said, "Look at the bay outside, through the polaroid. Now turn the

"Ooh, it's polarized!" they said.

After a lot of investigation, I finally figured out that the students
had memorized everything, but they didn't know what anything meant. When
they heard "light that is reflected from a medium with an index," they
didn't know that it meant a material such as water. They didn't know that
the "direction of the light" is the direction in which you see something
when you're looking at it, and so on. Everything was entirely memorized, yet
nothing had been translated into meaningful words.


Not all the best was written then. The year 1901 is still a bit early.

Marcy,OFFTOPIC, but I wanted to be sure you saw this post - I do hope you are sending your detailed info about links in justice/Gillespie/Keisler and wonderful US of AT & T article to Conyers, Waxman, Feinstein, Leahy and any others that might need reminding of how important it is to NOT DROP the investigations that have been ongoing AND to be fully aware of the undertext of what has been layed out before them...Of course, no one understands that better than you! Vickie


One of thing best parts of 'new math' was learning how to do it right in order to get the right answers. (I used my seventh-grade math methods to get the right answer on a college midterm, because I knew how it worked and the professor's method was a whole lot more complicated and harder to understand.)

Process is important, and the people who keep telling is isn't need to go take a long walk off a short dock.

PJ, yes but as Lehrer's tune satirizes, it's also a lot harder to learn "why" and "how" than just to memorize "what." Better, but more challenging. (for the teachers as well -- or especially, as his song goes...)

An additional thought on why it has been difficult to get what Dewey calls "psychological" -- what we're calling process-based -- teaching to replace "logical" (answers-based) teaching: it's MUCH more difficult to test kids on whether they know the process, than whether they know the results. Standardized testing isn't compatible with reasoning-based learning which, as Dewey notes, ends up with the student knowing less but knowing it better.

When I was in grade 3, we learned some of the mechanics needed for set theory (really elementary stuff like union, intersection, elements of a set, etc. (but most of all we learned, through rote practice, how to draw kick-ass curly braces!)). It was not until many years later when I studied the foundations of arithmetic that I learned what the hell that nonsense was supposed to be used for. The very idea that grade 3 students were going to become little Bourbakis is so utterly bug-fuck-crazy that it boggles the mind! The idea that grade 3 teachers, damn few of whom had studied rigorous math, would understand how set theory relates to arithmetic is just as crazy. In a field rife with bad ideas, enacted on a grand scale without any empirical support from pilot programs, the "new math" stands out as one of the dumbest ideas in the history of education. If this is an example of process based education, then I say it has no place in classrooms. Process based learning comes from an apprenticeship; learning by doing under the guidance of a mentor. Large classes under a single instructor are far more amenable to fact based teaching, even though some sort of practical, mentor-based, learning will have to follow on if the students are going to be able to use their factual knowledge in a useful way.

Ken, I don't understand what you're saying. You're saying that you were taught by a reasoning-based method that was beyond your skill level in third grade, and so you think all reasoning-based learning is beyond the skill levels of all students in all grades?

I think you need to consider "advanced vs basic skill level" and "reasoning vs fact based approach" as independent axes.

I'm enough older than Ken that we didn't get set theory until jr high - and then not every class, only the 'new math' classes (which, to be honest, were two-thirds of them). It isn't easy stuff to understand.
I still don't know why they want younger kids to learn this stuff (and some of the other stuff that's being put into the 'national standard' category for testing).
It remindes me of the moves to get every kid a calculator, before they learn arithmetic (aka, what the calculator is useful for). (BTW, one of my college math teachers said that calculus is where you learn arithmetic!)

I love this series emptypockets, looking forward to the next episode! :)

~Pockets, no, I'm definitely not saying that. First, I'm saying that the implementation of the "new math" program was dumb beyond measure. Second, it is foolish to try to teach rigorous math based on set theory in K-12, outside of some advanced placement high school classes. A far, far better example of math education is the Euclid course that used to be taught in junior high where you take axioms and definitions and build up theorems to prove triangles have 180°, etc. Geometrical constructions are far more tangible than sets. Around here, though, they have removed all the theorem-proof content from the Euclid section, so even this good example must have had its detractors. But even with this more accessible instruction in the method of proving theorems, one only learns how to follow the proofs and analyze them for correctness. I'm not a mathematician, but it is very rare to find someone with an undergrad degree who can actually construct a proof from scratch. This is something you learn through an apprenticeship because nobody has any idea of how to teach it formally. How many 1st year grad students do you get in your lab who already know how to do independent research?

Can you give an example of a reasoning based approach (at any skill level, but it has to be formal classroom education, not a mentor-based apprenticeship) that has been successful? Perhaps the Feynman in Brazil anecdote is one such. Here (and in the US), students learn word-problems, where they have to map verbal descriptions of problems onto a physical or mathematical model, solve the equations, then map the solution back onto a verbal solution to the problem. The Brazilian kids didn't learn that, so they weren't able to follow Feynman's hints. But speaking personally, all three of my daughters learned how to solve word problems by going through their homework problems with me and getting hints and suggestions until they finally caught on (i.e. this was mentor-based learning). The formal process of teaching a reasoning-based approach to word problems in school was of very little use (and I don't think this is because my daughters are particularly dull; many of their classmates simply didn't learn to solve word problems (at best they learned to recognize enough typical problems to get by - basically they learned some "facts)). In the cases of all three of my daughters, the teachers were dedicated and earnest, but this is really a tricky thing to teach by way of formal classroom instruction.

Another point that I only touched on in my rant (really, that's all it was...an opportunity to rant against the new math) is the question of why educators don't implement radical new programs as controlled trials? If you can do it in medicine, where people's lives are on the line, you can surely do it in public education! I would insist that no changes be introduced to the curriculum without being evidence based. You can easily argue that reasoning-based learning is better than fact-based learning, but that is no basis to implement wide ranging reforms. Prove that a *particular* reasoning based course actually teaches students to use reason, and then we can move forward with the changes.

PJ, the set theory we got in grade 3 wasn't really much of anything, just unions and intersections and that sort of thing. Utterly trivial for someone in grade 3, but also very obviously useless. I also remember slogging though arithmetic in different number bases (grade 3 stands out because I moved between grade 2 and 3; one year I'm happily doing normal arithmetic and then all of a sudden I'm surrounded by eggheads doing arithmetic in base 6 -- a year from the twilight zone). I like your college math prof's bit about calculus and arithmetic.

Ken, "psychological" or reasoning-based teaching in math is difficult because it's the most abstract of the sciences, but I'll try to think of a couple of examples suitable for the lower grades.

First, I'd say that I don't really see how set theory is reasoning-based, especially the way it sounds like you learned it. Dewey again: "It is, nevertheless, a frequent practice to start in instruction with the rudiments of science somewhat simplified. The necessary consequence is an isolation of science from significant experience. The pupil learns symbols without the key to their meaning." It sounds like they were teaching set theory as if it were an end in itself ("behold the beauty of set theory!") rather than presenting a problem you want to solve, letting you struggle a bit, then revealing set theory as the only way to approach it -- or better, guiding you to re-discover set theory on your own. Similarly I remember when I took calculus it was just a meaningless set of (somewhat interesting) exercises, but it never occurred to me why anyone had bothered to invent it -- it wasn't until I took physics years later that I understood why Newton (Liebniz?) had invented it, or even really appreciated that an individual HAD invented it. What if that historical understanding came first? What if I didn't learn subjects until I had a reason to use them?

So, those examples. At the third grade level, I would have appreciated learning multiplication if I had a real need to quickly total sets of equal number (say, in a game) rather than rote memorization of tables. Closer to your set theory example, there was a computer program in the early 1980s called Rocky's Boots that was a series of puzzles you guided a little Raccoon through using AND, OR and NOT gates to control circuits (to turn a switch at the end of a long circuit on or off and get to the next level). Surely that teaches the basic concepts, but in a puzzle-based, reason-it-out-for-yourself way. As you astutely identify, this kind of teaching is mentor-based (here the mentor has been built into a machine) and not suited to a traditional "teacher in front of a room of kids" environment. Since a lot of reasoning-based thinking is the act of reasoning -- something that happens at the personal level -- I think you're right that it lends itself best to personal musing or one-on-one teaching. That's a big challenge. I'm not sure it means it's not worthwhile to try to use these principles though...

I think the approach can be extended to history and geography too, and already is being used that way to a large extent. I think there's a move away from memorizing map lines and state positions (although as any Miss Teen America can almost tell you, some memorization wouldn't hurt) and towards teaching WHY political boundaries are where they are, what the differences between people who live there are, and teaching history more like the way emptywheel writes it -- as a series of actions by mostly self-interested individuals moving in a particular political environment -- than as a series of dates, Acts, and "great men" who swoop onto the scene and change the course of the nation.

When I was teaching a few lectures to advanced high school students on evolutionary genetics, I started out asking them "How do we know all life on Earth is related?" and "What if 99% of life here is related, but there's another 1% that's totally unrelated -- how would we find it?" and then let them reason out the importance of DNA for understanding evolutionary relationships. It worked well, I think better than just telling them "DNA is the code of life blah blah...".

Re: your last point, I'm completely on board with asking for more evidence-based theories of teaching. But of course, the problem there is how do you test how "well" the students learned it...? The test has to be written in a sophisticated way to distinguish real understanding from memorization (similar problem to distinguishing a human mind from a computer!)

~Pockets, I think that we are in agreement (not at all surprising). My rant was triggered by the suggestion that new math had been a reasonable attempt to introduce reason-based learning into K-12 classrooms. If you want to see how set theory is used to understand arithmetic from a reason-based perspective, have a look at Whitehead and Russell's Principia Mathematica next time you're in the library. You won't have to read very far to see why this approach was ridiculous for school age kids.

An example from molbio might be that you want to teach some kids how DNA strands stick together, so before talking about hydrogen bonding you first introduce the 6-12 law for molecular interactions. Now it's obvious that these kids will not know enough E&M to derive a 6-12 energy law, so instead you get them to practice drawing little "6" and "12" exponents. By the time they know enough to actually derive the law, apply it to the case of a hydrogen being attracted to an oxygen with a free lone pair of electrons, there will be no benefit from having practiced drawing those little exponents when they were young. To suggest there would be a benefit is ludicrous, and obvious to even the most casual observer. Yet this is exactly what was done with the new math and set theory. How the Bourbakists pulled it off is a mystery of the highest order.

I agree that it is difficult to measure the effectiveness of teaching in a classroom based system (compared to a one-on-one mentoring relationship), but it still needs to be done. In medicine there are many diseases and complaints that have no identifiable pathophysiology associated with them. So we question patients about pain, daily activities, physical capabilities, etc. to try to measure a quality of life that can be used to compare treatments. We follow patients for many years, even decades when possible. This is difficult and the results are often fuzzy and hard to interpret, but it does no good to choose a simple assay if you know the measurement tells you nothing about the intervention. Standardized, multiple-choice tests are the simple assay, but they hardly exhaust the options that one might pursue.

So we're in agreement that students need to learn to think (as well as learn a bunch of facts to help guide their thinking). If I have a point, it's that rhetorical means can not be used as the arbiter of whether to change the curriculum en masse. An experiment must be done, using proper controls, blinding where possible, and appropriate assessments, before any change can be contemplated. If an education researcher has a great idea for some new program that will get kids to use reason to solve problems, but no proposal for an experimental test with appropriate outcome measures, then we respectfully tell them to go back to their workshop and complete the proposal before their next resubmission.

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