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Quadratic Nightmares

Do you recognize that distinctive bit of mathematics? It’s something you probably last encountered in high school and, for some people, seeing it again now might even trigger a few nightmares.

Most American students learn the algorithmic process for deriving the quadratic formula, and how to use it to solve quadratic equations, in Algebra I. And likely forgot it not long after. It’s a process that hasn’t changed for at least several hundred years.

Maybe going back 4000 years to the Babylonians, according to a mathematician at Carnegie Mellon University who has discovered a simpler method, “one that appears to have gone unnoticed these 4,000 years”.

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Why Algebra?

New York state education officials are worried about the Algebra 1 Regents exam. They recently revamped the test and the passing rate fell, with less than 25% of students scoring at the “college-ready” level.

Those officials jumped into action.

This fall, they established a committee to study the results on the new exams to determine, among other things, whether the bar for passing, which students would have to meet starting in 2022, had been set too high. (They had originally said the class of 2017 would need the higher scores to pass, but last year decided to push that back.)

Their solutions, of course, focus entirely on test scores; how to get students to answer more of the questions correctly. Which means the same curriculum, taught using the same pedagogy, only more of it.

Among the ideas the city is considering: having fifth graders take math with a specialized instructor instead of one teacher for all subjects; teaming up with local universities to get more sixth- and seventh-grade math teachers certified in math instruction; creating summer programs for middle- and high-school students who are struggling in math; and training middle-school and algebra teachers in how to address students’ “math anxiety.”

In at least one high school, incoming students take Algebra for two periods a day, working with two different teachers. And, according to the very high percentage of passing scores on the Regents, it’s working. Even if, by devoting so much time to Algebra, “ninth graders are no longer taking art, music and health”.

With all the hand-wringing over test scores, few education administrators and politicians seem to be asking one basic question: Is completing Algebra really necessary for all high school students?

There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong – unsupported by research or evidence, or based on wishful logic. (I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame.)

We talk a lot about Algebra being a “gateway” course, one that students need to be a success in college. And how learning the discipline of mathematical thinking (from Geometry, as well as Algebra) improves students’ reasoning abilities in other areas.

But in most schools, students don’t really deal with the concepts of mathematics, much less any meaningful study of it’s application. Algebra I is taught in basically the same way it was long before computers were portable and inexpensive. We still emphasize the memorization of algorithms and test students on how well they can crank through the mechanics that their phone could do far faster.

Then there is STEM. We must have students study Algebra (and the S – T – E subjects) in order to fill all the STEM jobs that will go unfilled, causing great economic tragedy for the US. Which is a fallacy in many different ways.

Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.” Even in jobs that rely on so-called STEM credentials – science, technology, engineering, math – considerable training occurs after hiring, including the kinds of computations that will be required.

The bottom line is that, rather than worrying over largely meaningless test scores, we need to take a hard look at both whether we should expect all students to understand Algebra by the time they graduate,1 as well as how we teach the subject.

I think students would be far better served if they graduated with a solid, practical understanding of probabilty and statistics and how that math is applied – and misapplied – in the real world. Maybe we would have fewer adults wasting their money on lottery tickets and more of them questioning the sketchy numbers tossed around by business and political “leaders”.

Do We Really Need All This Math?

Also on the editorial pages of the Saturday Post, a professor emeritus of mathematics asks an excellent question: Do we really need all this math?.

How much math do you really need in everyday life? Ask yourself that — and also the next 10 people you meet, say, your plumber, your lawyer, your grocer, your mechanic, your physician or even a math teacher.

Unlike literature, history, politics and music, math has little relevance to everyday life. That courses such as “Quantitative Reasoning” improve critical thinking is an unsubstantiated myth. All the mathematics one needs in real life can be learned in early years without much fuss. Most adults have no contact with math at work, nor do they curl up with an algebra book for relaxation.

As someone who spent many years trying to market Algebra and Geometry to many students for whom the experience would be of little value, I must confess… I agree.

Kids certainly need to develop a basic number sense and some basic arithmetical skills during their time in school, especially how to efficiently using a calculator.

And their lives would be far better off if they graduated understanding the fundamental concepts behind probability and statistics to help them cope with the deceptive practices behind the way most polling is used, as well as the fraud run by many governments known as the lottery.

But for most students the four years of math courses they stumble through in high school, not to mention struggling with such incredibly useless skills as dividing fractions in earlier grades, is valuable time that could be far better used in their young lives.

The Answer is Imaginary

Michael Alison Chandler, a Post reporter who is spending the year taking Algebra II and blogging about her experiences, has arrived at the part of the curriculum that includes imaginary numbers.

Not surprisingly, she has questions.

First off, why is the square root of -1 “imaginary”? If nothing times itself can equal a negative number, than how can these numbers exist at all?

Second, Why do we need them? When, on earth, would you ever want to use them?

A few commenters offer answers to her first group of questions.  I’ll bet they’re also covered in the textbook.

Those in the second group are questions that all students should be asking – and getting good answers to – when studying math at any level.

Largely, however, they don’t.

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