By way of Majikthise, I found this excellent post by Abbas Raza about the problem of mathematical illiteracy. But to step back a bit, this trail of links began with the release of new teaching guidelines by the National Council of Mathematics Teachers:

The report urges teachers to focus on three broad concepts in each grade and on a few key subjects -- including the base-10 number system, fractions, decimals, geometry and algebra -- that form the core of math education in higher-achieving nations.

I think this is exactly the right approach. It's more important for students to develop specific competencies, such as fractions, decimals, geometry and algebra, than to develop the fuzzy skills often described in state educational standards--'critical thinking' being the worst of these. A story by Abbas describes exactly what I mean:

I sometimes tutor students for graduate admissions tests like the GRE or GMAT, and the first time I meet with them they often show me algebraic word problems they got wrong in a practice test. I ask how their junior high math is, and no one ever admits that they can't do 7th or 8th grade math. Then I ask them to subtract one number from another for me, using a pen and a piece of paper I hand them: say -2and7/8ths minus 1and3/17ths. You'd be surprised how many of them are tripped up and make a mistake in a simple subtraction that any 8th grader should be able to do. The problem is they really cannot do ANY algebra until they are consistently and confidently competent in such simple tasks as adding, subtracting, multiplying and dividing numbers, and yes, this includes fractions, decimals, and negative numbers, but even these college graduates generally are not.

I noted a similar phenomenon when columnist Richard Cohen argued that algebra might not be necessary for a student who failed the class six times. That she could not calculate the correct change without the aid of the cash register, indicated that the problem wasn't algebra, but *subtraction*.

Before I get jumped all over for advocating 'drill and kill', I'm not. *How* one teaches these skills can vary, and, given, that people are different in many ways, there probably is no one right way to teach. I myself learned arithmetic without reciting "1+1 is 2, 2 + 2 is 4." I had all sorts of groovy tactile things in my class including an abacus. You can also teach the notion of different base numerical schemes--creative kids like that.

(**An aside:** the whole different bases thing also allows you to work in some very cool Mayan archeology. To digress even further, I'm always astonished that archeologists were able to figure out that the Mayans used a base five system without understanding their language.)

What I am advocating is a content-based *outcome*, which for a long time has been overshadowed by jargon-laden buzzwords (you need to know something before you can critically think about it). Given the importance of basic math for, well, everything, certain content-related skills and knowledge must be mastered. These guidelines sound like a step in the right direction.

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Mike, your post is right on the mark. It seems that the academics who come up with teaching standards, often assume that the best way to learn something is for the student to first learn the fundamentals of why something works, before learning how to use it. For example, it's like requiring calculus students to master epsilon-delta proofs before they are allowed to do integration, or requiring primary grade students to understand number theory before learning arithmetic.

Looks like Gabriella needed help in English, as well as basic arithmetic:

"I don't want to be there no more," she said.

I've often found that the most significant "light bulb moments" in my life are the ones that surround understanding the principles behind something that I learned how to do mechanically. I can appreciate trying to learn theory before practice at the college level and beyond, but I agree that kids can learn their mathematical operations without first appreciating the formal axioms behind them or "thinking critically" about what they're doing. In my experience, it's easier to say, "Remember how you do X whenever you want to accomplish Y? Well here's what X does and why it allows you to accomplish Y," *after* they've learned how to do X.

I remember being taught about the ones' place and the tens' place and so on in elementary school, and I really didn't figure out why they bothered teaching it to us until I was introduced to the concept of bases other than base 10. I could certainly have done without those lessons early on and spent more time mastering the practice of arithmetic. There's plenty of time later for learning why we carry the one after learning when and how to carry the one.

Also, a related issue is the use of calculators in math class. While a calculator is a needed tool in a science class, I think they should be banned from most high-school math courses. Kids need to understand what calculators do, and I think having calculators in math class gets in the way of learning the basics. For example, my son is required to have a graphing calculator for his geometry class. I see almost no justification for this somewhat costly requirement, and I think kids will learn far more geometry with the old straight-edge and compass. Geometry is usually the course where kids are introduced to mathematical proofs, and no technological gizmo is going to teach that.

I teach math (to college kids), and my experience is that learning math is exactly like learning a sport -- students need coaching in the ideas, and drilling in the actual skills.

If we focus too heavily on just the skills, we get students who can perform mechanical procedures without being able to generalize (e.g., when you teach polynomial long division you have to start all over again, since they don't really understand the general principle of long division). If we focus on the principles, nothing sticks for very long.

My approach? Less lecturing, more practicing in class as material gets discussed (seriously, who learns basketball by listening to the coach talk?). Daily homework problems, discussion follows. Teaching standards and trashy textbooks can be huge obstacles, but good classroom practice can still prevail.

Mike, my admittedly vague memory of the Mayan numeral system tells me it is often described as base 20, on the grounds that each position represents a power of 20. The individual numerals are made up of bars and dots (except for the shell-like glyph representing 0), with the bars representing 5 and the dots representing 1, but that aspect of Mayan numbers is not base 5 in the same sense Arabic numerals are base 10, as the value of a symbol depends on whether it is a bar or a dot, and not its position. An additional complexity is that for some calendar related calculations, Mayans used base 18 for the second position, possibly because 18 * 20 = 360, close to the number of days in a year.

I learned base 8 counting from watching The Simpsons. For me, calculus classes made a whole lot more sense after I had spent time studying ground-water movement and potential fields. But my all-time favorite pedagogic trick was following the lesson on dividing fractions with pineapple upside-down cake.