After the jump, some examples, both fictional and real world.

Kids today are still taught their times tables, but the curriculum makes room for other skills as well, skills that help students apply those math facts to real world problems. One skill taught in new math that is particularly upsetting to traditionalists is estimating. For example, besides memorizing that 7x9=63, students may also be taught to estimate the answer. 7x9 is pretty close to 7x10. The answer to that is easy, 70. So, the answer to 7x9 is going to be less than 70. If you think the answer is, say, 93, you'll know you are wrong. This drives some parents batty: what do you mean, my child is being taught that 70 is a close enough answer to 7x9?!?

Another skill taught is to perform a sanity check on your answer. In science class, a student may be asked to calculate the distance from the Earth to the Sun, given, say, the circumference of the Earth's orbit. A student applying math skills learned by rote memorization might busily calculate away and come up with an answer of 93 miles. A sanity check should suggest to him that the correct answer is a lot more than that: the Sun is farther away from Earth than, say, Houston is from Dallas.

Ironically, estimating and doing a sanity check used to be critical even in the pre-calculator days, when a slide rule hung off every engineer's belt. If you ever used a slide rule, you know they are very good at giving you the first two or three digits of a calculation, but they don't do squat in telling you where to place the decimal point. Good estimation skills can help you decide whether the answer is 93 or 93 million.

Estimating and sanity checks can help with another source of error, one that was popularized with the introduction of computers to do calculations, called "Garbage In, Garbage Out." Even if you can perform lightning quick, 100% accurate calculations, your answer might still be wrong if you start with inaccurate facts.

That brings us to a recent real life example. I blogged about Richardson's state representative Stefani Carter and her disregard for the sad state of Texas public school finance. That, in turn, led one commenter to lob an accusation that public schools are ridiculously way overstaffed:

Dallas schools employ 157,000 to process 20,000 students through the current education system. That's over a 7:1 ratio. Seven school employees for each student. How can the taxpayer not question the wisdom there?Source: The Wheel.

The reader claimed a DISD employee to student ratio of 7 to 1. A quick sanity check would have suggested the claim was preposterous. Go to a school. A ratio of 7 employees to 1 student would require the classrooms and halls and front office to be overflowing with employees. You'd have trouble picking out a student in the mass of adults.

The reader's error was not in dividing 157,000 by 20,000 (although that answer is much closer to 8:1 than 7:1, but no matter). It was in his starting assumptions. Is it reasonable to believe that there are only 20,000 students in the DISD when the population of the City of Dallas is over a million? In fact, the reader got his facts swapped. There are over 157,000 students in the DISD and 20,000 employees, not the other way around. He got the mechanical long division part right, but he failed spectacularly in applying those mechanical skills to the real world.

Why did this happen? Perhaps he is old enough not to have been taught math skills other than rote memorization and mechanical operations. More likely, the mistake (believing a 7:1 ratio of employees to students) aligned perfectly with his preconceived notion (schools are way overstaffed) and so it escaped normal sanity checks that would have easily revealed the answer to be preposterous. Ideology trumped reason.

Outrageously inaccurate errors like this influence public opinion all the time. Politicians exploit this to advance partisan ideology over wise public policy. State legislators like Stefani Carter act as if voter ID, sanctuary cities and carrying guns on college campuses are more important issues than solving the sad state of public school finance in Texas. Nationally, Congress acts as if the federal budget could be balanced if only we defunded NPR and foreign aid. Rather than letting the facts guide us to wise public policy, our ideology leads us to give uncritical acceptance to preposterous "facts."

I don't know how to solve this problem. Oh wait, I do too. It's called education. Someone please convince Stefani Carter.

## 8 comments:

I don't think you have the history quite right. New Math involved the introduction of theoretical mathematical concepts at an early age. These included things like the conversion of bases, set theory, and inequalities. In fact, when I read your piece, I went to the Wikipedia entry on New Math and it matches what I remember about the history of New Math from my university days.

I have a degree in physics and my father was an engineer. Obviously, I had mathematics ground into me. I do not recall converting bases until some time in high school and that was a "fun" exercise with friends when dealing with early computer ideas.

I do recall estimating and significant instruction on so called "word problems." Both are enormously helpful in the sciences. In fact, the quality of instruction that I was lucky enough to receive as a high school freshman laid the path to my degree. So by the late 70s there was emphasis on those topics. So does that mean New Math was abandoned by then? I'll let others judge.

I will say that interpolation of trigonometric functions, exponentials and other tedious but rigorous "by hand" methods are extremely important to those kids who have the aptitude to "get it." The emphasis on calculators is a mistake and probably used just to sell calculators. High school level students

canlearn how the calculator gets its answers and those lessons carry huge importance. Your discussion of using estimation as a sanity check is spot on. I saved myself in many an exam with that, and ball parked professional measures in my later career so I could narrow down to the real answer.One of our real problems in mathematical and science education is that policy makers think more is better. Not so. Not by a long shot. Requiring huge amounts of math and science for every student is foolish. Not every student has a natural aptitude to succeed in mathematics and nor should they. We have de-emphasized more classical concepts like geometry by Euclidean proofs, and even verbal ones like philosophy. The world needs all kinds of people and we shouldn't be concerned that every student can deal with trigonometric identities.

Andrew, thanks for the feedback. You are probably right that some of the skills I mentioned were not introduced with "New Math." As I suggested in the original blog item, the abilities to estimate and do sanity checking have always been valuable. In my own early education, I received plenty of both rote memorization drills and lessons on concepts like set theory and number bases. ("There are 10 kinds of people in the world. Those who understand binary and those who don't.")

I think the idea behind "New Math" was to give students an understanding of *why* you "put down the 4 and carry the 1." From that Wikipedia article, Richard Feynman is quoted, "subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting." For readers who don't know him, Richard Feynman was one interesting physicist.

Re: Feynman. I am curious about the source of that reference. None is really given by a link and I cannot find the quoted source except to refer to the Wikipedia article. Feynman did tell the story of being on California's textbook committee in "Surely You're Joking, My Feynman" and on one of the Safe-cracker Suite CDs. It is a hilarious story. Some things haven't changed.

I'll have to say, I'm one of the people who find the idea of teaching "right" and "wrong" estimating as kind of absurd. Frankly, rather than teaching estimating I think there should be more focus on generic "problem solving" and that estimating would fall out as a subset of that.

I'm not certain I can agree with your final comment regarding more education - I think there may be a number of very well educated people that are guilty of exploiting via ideology. So I'm inclined to believe there needs to be some other mechanism to rectify the polarizing impacts of ideology. Let me give an example . . .

Dallas county has had more documented cases of guilty verdicts overturned due to DNA testing than has (to my knowledge) anywhere else in the nation. In some of the overturned cases, there has also been documented suppression of evidence. (In my humble opinion, many of these verdicts were racially driven, supporting an ideology of race bias) And yet, even now when these verdicts have been overturned and proven beyond question to be wrong - and were in some cases known by prosecutors to have been "wrong" at the time, there have been no professional repercussions of this egregious situation. Apparently, no prior District Attorney or Assistant District Attorney is "responsible" and held accountable, arrested, convicted, and serving time for what amounts to criminal fraud. (not a legal term, my term.)

I believe this same lack of accountability makes it simple and easy for politicians to use ideological demagoguery for their own ends. After all, the penalty for it appears to be non-existent.

Gary, thanks for the feedback. I didn't mean to imply that politicians like Stefani Carter need more education. She's Harvard-educated already, so she knows exactly what she's doing. It's the voters who don't recognize the absurdity of claiming that the DISD has an employee to student ratio of 7:1 who need more math education, or at least better math education. I'd support your call for more focus on generic "problem solving" skills.

Mark, The underlying factors were stated incorrectly, not the math. Even after data error was corrected, and it was, you still insist the math was wrong.

True, the misinterpretation of the underlying data was incorrect. But, that's not an error in the math. It was unreliable base information, not bad math. Understanding the is a difference is another lesson to learn here.

Nathan, a ratio is a mathematical concept. The commenter (curiously, you don't say who it was) claimed an absurdly incorrect ratio as fact. The ratio presented was off by about 50x. That's an error. That's math. If it makes you feel better that the error was caused by "Garbage In, Garbage Out" and not by a calculation error, go ahead and feel better. Either way, the claim was garbage. Garbage that the commenter didn't catch, even though it would have been obvious to anyone with a basic math education and an objective attitude towards the subject of public school finance. What caused him to make the error is an interesting subject. I suggest it's because his ideology trumped his reason.

Mark, The ratio 127,000:20,000 approximately equal to 7:1 is not incorrect. That's math. The numbers are what they are. As you say, approximately equal to 8:1, that is also correct.

If you want to say the numbers represent something, incorrect, that's a different matter. But the math remains correct. Numbers don't change with what you call them.

I didn't make up these basic rules. They've been around longer than public school finance problems. Maybe this is a lesson in why simple concepts get misconstrued in perceptions of modern day political ideology.

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