Stuff I'm Reading

"Preferred numbers" in engineering and product design; standardized near-logarithm increments to make both interchanging parts and mental arithmetic as easy as possible. (I've liked 1-2-5 for some time, myself.)

The "19-equal temperament" scale in music, a proposal to divide the octave (doubling of frequency) into 19 near-logarithmic parts. Argued in composer Joel Mandelbaum's PhD thesis that it's the only viable system with a number of divisions on this order of magnitude. Coincidentally matches the Hebrew calendar system and its pattern of leap years.

One-Day Calculus Lecture

Dan's brief one-day calculus lecture (1-page PDF): link.

Morse Code

Visited the USS Intrepid aircraft carrier museum yesterday (here in New York City). Among the numerous exhibits in the hangar deck is this one on Morse Code. The scrolling digital blue lines are supposed to be translations of each other. What's rather glaringly wrong with this?

Now I Use More Greek Letters

Here's one of the full-page ads currently running on the Virgin Mobile USA home page. Thanks, you jackasses.

Permutation Puzzle

Among the things I'm not good at are combinatorics and whatnot -- Let's say you have a fixed set of 18 numbers all in the range from 1 to 6 (so obviously a bunch of duplicates), and you need to put 3 of the numbers in each of 6 ordered bins. How many ways can the totals in all the bins come out?

On Corrupt Godless Programmers

There's a kerfuffle around Second Life at the moment and some shady antics of 3rd-party clients that are officially allowed to connect to the game. That I wouldn't care about, except that it motivated one of the fiercer critics to come up with this novel argument: computer programming (especially 3D) is inherently a godless and corrupt activity.

This completely short-circuits my usual "angry" filter. Is this a genuinely new idea in the world? Or is this the same as Dungeons & Dragons game religious criticism back in the 80's?

It's all about criminality.

And yes, I will say that coding as an activity does corrupt. I think it's because geeks as a class tend to be godless or agnostic. Sure, you will find the occasional self-professed believing Christian or Muslim or Jew, but by and large, coders do not recognize a Higher Power. They are not People of the Book, because they only recognize their own book, which is code. There are some that realize this manufactured, man-made thing is merely a creation, and not the Creator, and merely a bad imitation of the Creator's works in Nature. But most don't. Most think the coded artifacts are *better*.

This cult of the belief in code-as-law and coders as god particularly infects the virtual world industry, where people get to code not merely some word-processing application or processor of some function on the web, but get to control human beings very visibly, in the round, in 3-D. They love that.

I don't.

I think it's the beginning of their criminality, by which I mean their violations of the law and civilization norms to take, keep, and abuse power.

More Equals Signs

Previously I wrote about students not using equals signs properly. So apparently some guys at Texas A&M are getting papers published on this subject, and identifying it as a key way to distinguish between high-functioning and low-functioning math students and national education systems.

Reported is stuff like: "The two researchers suggest using mathematics manipulatives". I disagree. The problem is not lack of manipulatives. The problem is that nobody ever told students what the fucking equals sign means.

I'm semi-convinced that a greater emphasis needs to be paid on the physical syntax and grammar of writing (and as a result, reading) mathematics by students throughout the education system. But that's me.

Zimbabwe

Did you know -- Annualized inflation in Zimbabwe, late 2008, was estimated to be: 89.7 sextillion percent?

http://en.wikipedia.org/wiki/Hyperinflation_in_Zimbabwe

Godel's Naturalization

Back in college I heard this ridiculously awesome story of what happened when the mathematicians Morgenstern, Godel, and Einstein went for Godel's citizenship test. Just ran into a recollection of the event written by Morgenstern (type starts p. 2):

Read it here.

Monkeynomics

Laurie Santos giving a TEDTalk on the results of an economic experiment with a "monkey market":

http://scientopia.org/blogs/thisscientificlife/2010/08/10/laurie-santos-how-monkeys-mirror-human-irrationality

The "Take home message of the talk" (as she says at 16:47) is that the choice to take risk differs on whether the situation is perceived as a gain or a loss -- regardless of the risk/reward being exactly the same in each case. When presented with the option of either (a) 2 grapes, or (b) 50/50 chances for either 1 or 3 grapes:

- Monkeys take the safe choice (a) in a gain situation, i.e., start with 1 grape and possibly add some more later,
- Monkeys take the risky choice (b) in a loss situation, i.e., start with 3 grapes and possibly take some away.

That being the same as humans tend to do on analogous tests. My personal interpretation is that this points out how negative numbers are actually a very sophisticated, hard thing to deal with for most people (and other organisms). Most of the time in a natural community you'd be taking actions to gain things -- the "loss" scenario is somewhat artificial and abusive, and we're not set up naturally to deal with that well (i.e., we don't have a natural built-in processor for negatives, and for most brains things just kind of go "kablooey" when forced to deal with them).

Wow, That's Freaking Cheap

Seen on the ride into school.

Similarly, don't forget about ye olde "Verizon Math Fail" recording from a couple years back. Glad that wasn't me.

This Is The Dumbest Goddamn Thing You Can Say About Statistics

"A large population size must require a larger sample size."

This -- or any iteration thereof -- is the dumbest goddamn thing you can say about statistics. While it's a clear demonstration that someone's missed the whole point of inferential statistics, it's also one of the most common things you'll hear about them. (Often in the form of "That sample is only a small proportion of the population.") Here's some of the varieties of this statement that I've encountered over time:

How do they project statistics like that? I'm trying to imagine what kind of sample size you'd need to represent, well, everything in the universe. [In regard to matter/anti-matter ratio in the universe as researched at Fermilab; comment posted at Slashdot]

Adobe claims that its Flash platform reaches '99% of internet viewers,' but a closer look at those statistics suggests it's not exactly all-encompassing... the number of Flash users is based on a questionable internet survey of just 4,600 people — around 0.0005% of the suggested 956,000,000 total. [News summary at Slashdot]

That poll doesn't convince me of 4e's success or lack thereof. Also, there's only 904 total votes while ENWorld has over 74,000 members, so that's only a small fraction of forum members (addmittedly many of those 74,000 are probably inactive). [In regard to the popularity of the D&D game's 4th Edition; comment posted at ENWorld]

You get the idea. To save some writing time here, I'll use n to indicate the sample size and N to indicate the population size. For any statistical inference, if n=50 is an acceptable sample for N=1,000, then it's also acceptable for N=10,000, N=1 billion, or N=infinity. In particular, one thing that never really matters is the ratio of sample to population.

Brief illustration: Let's say that you're using a sample mean to estimate a population mean (much like in a scientific opinion survey, etc.). As long as you have a sample size of at least 30 or so, you automatically know what the shape of all possible sample mean results is: a normal curve, as per the (mathematically proven) Central Limit Theorem. And then you can use that curve (via some integral calculus, or a resulting table or spreadsheet formula) to calculate the probability that your observed sample mean is any given distance from the population mean. Does the size of the population have any bearing on this sampling distribution shape? No. Does the CLT make any reference to the size of the population? No, not whatsoever. You have a moderate-sized sample (30+), you know the shape of all possible sample means, you calculate your probability from that (or some equivalent process), done.

Exception: In calculating sampling distribution probabilities, you'll use something like the fact that its standard deviation is σ/√n. (Here the σ indicates the standard deviation of the whole population.) Now, if the population size happens to be exceptionally small (like, N≤20n), and you're sampling without replacement, then you can improve the estimate a bit by instead using the correction formula √((N-n)/(N-1)) * σ/√n. But why bother? (a) You're almost never in that situation, (b) it rarely makes that much difference, and (c) you're just making extra number-crunching work for yourself. So you're actually better off assuming that the population is really huge or even infinite (as is actually done), thereby saving yourself calculation effort by way of the simpler formula. For any N>20n, the difference is negligible anyway (which is to say: lim _N→∞ √((N-n)/(N-1)) * σ/√n = σ/√n). Run some numerical examples (pick any σ you like) and you'll see how little difference it makes.

Even more absurd exception: One requirement that the Central Limit Theorem does have is that the population standard deviation must be nonzero, i.e., σ>0, which does rule out having a population size of just one. But, c'mon, if that were the case then what you're doing isn't really sampling or inferential statistics in the first place, now is it?

In summary: If anything, a larger population size makes the statistics easier, and the math is simplest when you assume an infinite population size in the first place. Other than that, population size has no bearing on the math behind your estimation or surveying procedure.

One final, really simple observation: If an opinion poll is performed at the standard 95% confidence level, then its margin of error can be basically calculated by: E = 1/√n. (Compare to the formula for standard deviation above; the σ disappears due to a particular very convenient substitution and cancellation.) Does the population size N appear anywhere in this formula? Nope -- it's fundamentally irrelevant to the process.

(I've written about this before, but I wanted a version that was a bit more -- ahem -- direct, for posterity's sake.)

Stuff that Shouldn't Work

Making practice or test exercises is harder than you might first think before becoming a teacher. If you ever make a problem up on the fly while lecturing, it's highly probable that you'll create something with hideous fractions, irrational or imaginary numbers, extraneous solutions, etc., that you didn't want, which winds up sidetracking you from the point you were trying to make.

Another pitfall is creating problems that are singularities, i.e., the correct answer can be produced by some completely incorrect process, one that won't work for any other problem of the same nature. For remedial math students, this is almost a nightmare scenario, since their capacity to correctly generalize from the specific to the abstract is already shaky and confused as it is.

As just one example, here's one of my favorites from the algebra workbook we use at my school (custom edition produced by other teachers in same department):

If 1.05x = 22.05, then x = ?

Now, the correct process is to divide both sides by 1.05, and see that x = 22.05/1.05 = 21. But horribly, if a student mistakenly subtracts 1.05, then they also get the same answer! Say x = 22.05 - 1.05 = 21. Thus, this exercise allows a student to "submarine" a totally broken process (answers are multiple-choice in the book), giving them apparent confirmation that they're doing the right thing when they're absolutely not. (Note that this particular exercise was changed in a newer edition after I pointed it out.)

Enough prelude. The thing I'm trying to get around to is that last night I saw the "crown jewel" for this kind of problem, as part of a set of practice problems for the ACT Compass Test in Algebra. (You can actually see it here: "Sample Math Test Questions: Numerical Skills/Pre-Algebra and Algebra", Algebra item #14). Something like this:

For x ≠ 3, reduce (x² - 9)/(x-3).

Now, the point of this exercise is to practice factoring (in this case, the top is a "difference of squares") and then cancel like factors on the top & bottom. Write: (x² - 9)/(x-3) = (x+3)(x-3)/(x-3) = x+3.

But last night my students got all weirded out when I was writing that much (there's an additional wrinkle in the Compass problem, but it's not germane to my point) and said they got the right answer with a lot less work. They explained: "Divide x² on top by x on bottom and get x. Divide -9 on top by -3 on bottom and get +3. There's the answer, x+3."

Now obviously this is a horribly mutilated process (and not uncommon!), thinking that you can divide individual terms in a rational expression piecemeal. (My best explanation, not that it gets fantastic traction, is always "Division distributes across addition, so if you divide by x, you have to divide every term by x." ) But the really crazy unique thing about this problem is that the broken process actually works for every possible problem of this format!

Consider all possible ways of constructing a "difference of squares" on top, and one of its canceling factors on the bottom. Case 1: Say you're reducing (a²-b²)/(a+b). Correct process: (a²-b²)/(a+b) = (a+b)(a-b)/(a+b) = a-b. Incorrect process: (a²-b²)/(a+b) = a²/a - b²/b = a-b (same answer). Case 2: Say you're reducing (a²-b²)/(a-b). Correct process: (a²-b²)/(a-b) = (a+b)(a-b)/(a-b) = a+b. Incorrect process: (a²-b²)/(a-b) = a²/a - b²/(-b) = a+b (also the same answer).

So not only does the "broken" process work for all permutations of this kind of problem, it even manages to get all the signs correct regardless of how those have been set up. Arrrghh!!!

(Silver lining: At least two of my students had the courage to tell me that's what they'd done, and I had the presence of mind to listen to it last night. I've used this practice test for about 5 years without anyone pointing out how they were doing it like that.)

Number 91

What's wrong with the number 91?

Consider this: With just three exceptions, all of the composite numbers up to 100 have a factor of either 2, 3, or 5. So, for all of those numbers, you can pretty much immediately see that they're composite, via your simple, standard divisibility-identifying tricks.

The three exceptions, things that have a "7" built into them as their smallest factor, are: 49 (7*7), 77 (7*11), and 91 (7*13). Of course, 49 and 77 are pretty obviously composite (knowing one's squares and what divisibility-by-11 looks like).

So 91 is the only composite number up to 100 that I can't immediately identify. I tend to incorrectly conclude that it's prime if I'm just working at it mentally.

Counterexample to the CLT!?

The introductory statistics text that I teach out of presents the Central Limit Theorem this way:

The Central Limit Theorem (CLT): For a relatively large sample size, the variable x' is approximately normally distributed, regardless of the distribution of the variable under consideration. The approximation becomes better with increasing sample size. [Weiss, "Introductory Statistics" 7E, p. 341]

In other words, any distribution turns into a normal curve when you're sampling (with large sample sizes). I also know off the top of my head that the formal CLT is talking about a distribution that's been standardized (converted z = (x-μ)/(σ/√(n))), and how its limit as a function is the standard-normal curve (centered at 0, standard deviation 1).

So one day I'm walking around sort of half-dozy and I'm thinking, "Wait a minute! What about a constant function? If you had a distribution that was fixed with one element (say, 100% chance that x=5), any conceivable sample mean would just be the constant x'=5, and there's no way a graph of that looks like a normal curve, right?"

Meditating...

Well, the thing that I didn't immediately have in my head, and is also entirely left out of the Weiss text -- There is one single fine-print requirement to the CLT, and it's that the standard deviation of your variable must be nonzero (and also non-infinite), i.e., 0<σ<∞. Which is sort of obvious from the fact that you need that to standardize with z = (x-μ)/(σ/√(n)), it being used to divide with in the formula and all. And of course a constant function has zero deviation, so it's indeed outside the scope of the theorem.

Guess I can't get too mad at the Weiss text for this... chances of it being useful for an introductory student to spend time parsing that is about nil. (Obviously, it hasn't come up in 5+ years of teaching the class.) Still, it might be nice to put it in a little footnote at the bottom of the page so I don't go daydreaming about possible counterexamples on my commute.

Sets as Plastic Baggies

Many times I've seen the use of set-braces (roster notation) compared to an "envelope". Here's one example (speaking of empty sets and the null-set symbol):

To help clarify this concept, think of a set as an envelope. If the set is empty, then the envelope is empty. On the other hand, if the set is not empty -- that is, it contains at least one element -- then there are items in the envelope. One such item can be another envelope. Using this analogy, the symbol {Ø} specifies an empty envelope contained within another envelope. [Setek and Gallo, "Fundamentals of Mathematics" 10th Ed., p. 74]

Now, what I think is the jarring discordance in this analogy: You can immediately see the contents inside a { } symbol, but not so with envelopes (being opaque and all). That's probably why the whole metaphor always feels foul in my mouth, and might be part of the reason I get a poor reaction from students when I try to use it in class.

Let's try a better metaphor: A clear plastic baggie (with a zip, perhaps?). These you can, like the set { } symbol, instantly see into. If you put one plastic baggie inside another, then you can immediately see it sitting inside there... just like the frequently-misused {Ø} notation.

So let's use a metaphor that shares in the transparency and clarity of our precise math notation.

... In New Jersey

So I got a few inquiries about the prior post on how a Psychology Today article could make me yell out loud. Yes, I was sitting in my apartment alone and hollered out, "Oh NOO!!!" at this part:

In an article published in 2005, Patricia Clark Kenschaft, a professor of mathematics at Montclair State University, described her experiences of going into elementary schools and talking with teachers about math. In one visit to a K-6 elementary school in New Jersey she discovered that not a single teacher, out of the fifty that she met with, knew how to find the area of a rectangle.[2] They taught multiplication, but none of them knew that multiplication is used to find the area of a rectangle. Their most common guess was that you add the length and the width to get the area. Their excuse for not knowing was that they did not need to teach about areas of rectangles; that came later in the curriculum. But the fact that they couldn't figure out that multiplication is used to find the area was evidence to Kenschaft that they didn't really know what multiplication is or what it is for. She also found that although the teachers knew and taught the algorithm for multiplying one two-digit number by another, none of them could explain why that algorithm works.

Holy god, that is insane. I have a hard time imagining anyone being unable to find the area of a rectangle, never mind school staff actually teaching arithmetic, to say nothing of going 0-for-50 in a survey on the subject. I mean... unbelievable! Maybe I should take a poll of students in one of my own classes. Is it possible that people had just forgotten what the word "area" meant?

Just to complete the progression in the article:

The school that Kenschaft visited happened to be in a very poor district, with mostly African American kids, so at first she figured that the worst teachers must have been assigned to that school, and she theorized that this was why African Americans do even more poorly than white Americans on math tests. But then she went into some schools in wealthy districts, with mostly white kids, and found that the mathematics knowledge of teachers there was equally pathetic. She concluded that nobody could be learning much math in school and, "It appears that the higher scores of the affluent districts are not due to superior teaching but to the supplementary informal ‘home schooling' of children."

On the larger thesis of the article, that current math instruction in K-6 is doing more damage than good, and could and has been dropped successfully at least once... you know what? I can potentially believe that. It's possible. If the quality of math instruction is truly that atrocious, I wouldn't want children subjected to it -- of course the only consistent result would be crippling lifelong math anxiety (per Dijkstra's, "as potential programmers they are mentally mutilated beyond hope of regeneration," and all that).

But far more important than my own best-guesses would be: This would take a lot more research before going forward with a plan of that nature. And no way in hell would we either get (1) the research approved, or (2) the program implemented here in the USA (what with the READING AND MATH UBER ALLES!! approach to education that comes down from the political wing these days).

Psych Today Article

Any article that has me yelling out loud while I read it must be a good one. Highly recommended for several reasons:

http://www.psychologytoday.com/blog/freedom-learn/201003/when-less-is-more-the-case-teaching-less-math-in-schools

Still Amazed

Oh, and I got to do this again today. Still amazed.

Equals Or Not Equals

In my remedial algebra class, I assign these stupendously short homeworks, but require precisely written work -- professional format, one full operation per line, justification in words at each step. (I don't have much evidence that it helps the students, but at one point last year I had an emotional meltdown trying to read students' normal math writing, so there you go -- it's quite literally defending my own sanity.)

So on a lot of the homework, inevitably, some fraction of the class leaves out all of the equals signs. And since I go through and just check off each line as being correct or not, this results in every single line being wrong, resulting in zero points for the entire assignment. Even if everything else was correct, even if all the final results are the same as the answer sheet.

So in one sense, kind of brutal. In another sense, this requirement was made abundantly clear in class, including a fully written example from me on the submission page that they're turning it in on. So if you can't follow that simple requirement, one might argue it's an eminently reasonable end result.

But here's another way of looking at it: Each line of math is, effectively, a sentence. (A highly condensed sentence in specialized notation, but the same nonetheless. It can be re-hydrated back into normal English at any time.) And the equals sign is the verb "to be". It's the most important verb in any language! What if someone were in a writing class and submitted a paper without any verbs? What if they were entirely unable to say "you are", "I am", "he is" anything at all? Would an English teacher totally flip out? You bet they would.

And that's exactly my reaction when I see a paper like that.

Complete sentence: 5 + 3 = 8 ("Five plus three is eight.")
Sentence fragment: 5 + 3 8 ("Five plus three eight.")

Repeat that sentence fragment for 10 or 20 problems per page and see what happens to your eyeballs. Slings and arrows and all that.

Lottery Puzzle

Let's say there's a lottery run once every month that costs $100 to enter. Chance of winning is 1 in 20,000. The prize is 5 million dollars. Should you play this lottery?

"Community" TV Show

The NBC comedy "Community" gets way more things right than they have to. For example: The sexpot statistics professor.

But wait, there's more. I've long held a theory about performance that, "No actor can play more intelligent than they are in real life". Largely this is a matter of diction: I hear actors stumbling over, or putting incorrect emphasis on, pieces of vocabulary that they don't really know or use themselves in daily life. "Community" places really great actors throughout the ensemble; they're delivering lines like "as long as we keep the work/fun ratio the same I want to keep seeing you", and "the deception is making the sex 36% hotter" so fluidly that they slide by almost without me realizing that they were meant to be jokey.

Elementary Teachers

Did you know?

Beilock, who studies how anxieties and stress can affect people's performance, noted that other research has indicated that elementary education majors at the college level have the highest levels of math anxiety of any college major.

This in a larger article finding:

But by the end of the year, the more anxious teachers were about their own math skills, the more likely their female students — but not the boys — were to agree that "boys are good at math and girls are good at reading." In addition, the girls who answered that way scored lower on math tests than either the classes' boys or the girls who had not developed a belief in the stereotype, the researchers found.

http://news.yahoo.com/s/ap/us_sci_fear_of_figures


Edit: The article linked above is gone from Yahoo News. But you can still see that via the Internet Archive's Wayback Machine. And here's a citation and link to the original academic article:

Beilock, Sian L., et al. "Female teachers’ math anxiety affects girls’ math achievement." Proceedings of the National Academy of Sciences 107.5 (2010): 1860-1863. (Link)

Phonics and Bases

Speaking of the "Math Wars", here's an observation I made about teaching the most basic elementary-school subjects.

When I was very young, we were taught reading by "phonics", i.e., sounding out the letters of unknown words, and then thinking about how those sounds related to words we already knew. Sometime after that, phonics was dropped for a "whole word" approach, but it seems like the pendulum might be swinging back these days.

Why do "phonics" make sense as an instructional strategy? Because our system of writing is a technology based on exactly that principle. The whole point to our alphabet is that it is phonogrammatic, i.e., written symbols represent spoken sounds. This system of writing is meant for there to be an obvious connection between what we write, and what we say. (Of course, this is totally different from logographies such as Chinese whole-word characters and Egyptian heiroglyphics, but that's an entirely different story.) In teaching a child to read, why would you not use the language tool the way it was designed to be used?

Similarly, the traditional way of teaching arithmetic is to memorize a small number of basic facts (addition and multiplication tables), and then learn fundamental written procedures for adding, subtracting, multiplying, etc., large (many-digit) numbers. More recently, we've had to deal with the "Math Wars" is which training in time-tested prodecures has been frowned upon as too authoritarian (or something). Rather, there appears to be extensive time spent in base-10-system conceptual understanding, and then inventive or creative exhortations to make up your own multifarious addition, subtraction, etc., algorithms.

Why do "procedures" make sense as an instructional strategy? Again, because our system of written numbers is a technology based on exactly that principle. The whole point to our place-value system (base-10) is that it is intended to make the specific procedures for adding, subtracting, and multiplying simple, straightforward, and consistent for everyone! Consider the history of written numbers: With ancient systems like, again, Egyptian numerals or Roman numerals, there was no way to get addition or multiplying accomplished by simple writing or mental effort. Without a fixed base system, numbers don't "line up" the way they do for us. To get any arithmetic done (including total sales or tax calculations), you had to go to a licensed counter (think: public notary) with their counting board or abacus tool for use as a mechanical calculator, and trust that the computations they did there were correct. You were entirely at the mercy of this elite, cryptic profession, just to do simple addition.

At some later point, our Hindu-Arabic numeral system was invented and made its way to the West. This system of writing numbers is meant for there to be an obvious connection between the numbers we write, and how to add and multiply them, via a specific written procedure. Kings and princes were astounded at the prospect of people being able to do arithmetic on paper, or in their head, without using an abacus as a calculator. It's like magic! It's not some kind of accident that we use a positional-number system, it was engineered that way only so that we could use a specific adding and multiplying algorithm. In teaching a child to do arithmetic, why would you not use the written number tool the way it was designed to be used?

In summary, it's fascinating that both reading & writing instruction have taken almost exactly parallel paths in the last few decades in America. In each case, they have abandoned the rationale for which the writing tool was designed in the first place. It's like showing someone a power saw for this first time and them asking them, "Can you think of something you might use this for?" To leave out the intention of the tool is to miss the whole point of it. We should play to the strengths of our writing technology, and not frustrate ourselves fighting against it.

Math Wars

Recently I've had some discussions about the wacky stuff being taught in elementary math classes these days. Not something I deal with directly in the college classes I teach, but turns out there's a whole history to the current situation actually referred to as the "Math Wars"!

http://en.wikipedia.org/wiki/Math_wars

Basically, there's been a dispute over whether to emphasize "procedural" (algorithmic, memorized step-by-step processes) or "conceptual" (creative, inventive, big ideas) skills in the earliest grades. In the last 2 decades or so the "conceptual" camp has basically won the debate in teacher education schools, claiming to have research backing up the approach. Recently there have been calls for a more middle-ground approach.

Interesting articles in this month's American Educator magazine. One by cognitive psychologist Daniel T. Willingham: http://www.aft.org/pubs-reports/american_educator/issues/winter09_10/willingham.pdf

In cultivating greater conceptual knowledge, don't sacrifice procedural or factual knowledge. Procedural or factual knowledge without conceptual knowledge is shallow and is unlikely to transfer to new contexts, but conceptual knowledge without procedural or factual knowledge is ineffectual. Tie conceptual knowledge to procedures that students are learning so that the "how" has a meaningful "why" associated with it; one will reinforce the other. Increased conceptual knowledge may help the average American student move from bare competence with facts and procedures to the automaticity needed to be a good problem solver. But if we reduce work on facts and procedures, the result is likely to be disastrous.

And another article by professor E.D. Hirsh: http://www.aft.org/pubs-reports/american_educator/issues/winter09_10/hirsch.pdf

The victory of the progressive, anti-curriculum movement has chiefly occurred in the crucial early grades, and the further down one goes in the grades, the more intense the resistance to academic subject matter with the greatest wrath reserved for introducing academic knowledge in preschool. It does not seem to occur to the anti-curriculum advocates that the four-year-old children of rich, highly educated parents are gaining academic knowledge at home, while such knowledge is being unfairly withheld at school (albeit with noble intentions) from the children of the poor. For those who truly want equality, a common, content-rich core curriculum is the only option. It is the only way for our disadvantaged children to catch up to their more advantaged peers.