## Commentary on “Wrong Answer: The Case Against Algebra II”

There’s a new article by the author Nicholson Baker that is not raising as much a fuss as “Is Algebra Necessary?” from The New York Times last year, probably because it’s at Harper’s behind a paywall. Also, as I write this children are fleeing from algebra all over magazine stands:

The title strikes me as odd, given that the general argument is aimed at all algebra, and the text constantly references algebra as a whole rather than the quite US-specific version of Algebra II. (As far as I’m aware no other country has such a thing; most of them integrate all forms of mathematics together.) What is Algebra II anyway, and where is the cutoff point where Nicholson Baker considers algebra to be too hard to handle? He hints at what he’d like as a replacement, which strikes me as a math appreciation analogous to music appreciation courses:

We should, I think, create a new, one-year teaser course for ninth graders, which would briefly cover a few techniques of algebraic manipulation, some mindstretching geometric proofs, some nifty things about parabolas and conic sections, and even perhaps a soft-core hint of the innitesimal, change-explaining powers of calculus. Throw in some scatter plots and data analysis, a touch of mathematical logic, and several representative topics in math history and math appreciation.

This seems like a truly random collection of topics, but it sounds like Mr. Baker is basing his list off the notion of using The Joy of x by Steven Strogatz as a possible text.

I have had some personal experience with using popular texts; when I team-taught statistics at the University of Arizona we used The Drunkard’s Walk, Fooled by Randomness, and Struck by Lightning: The Curious World of Probabilities. While it led to interesting discussions, there just wasn’t enough meat to have students do problem solving based solely on the text. The math from the books was purely a passive experience. (We still did just fine by augmenting with our own material.)

I still hold forward the absurd idea that students still solve math problems in a math class. If you’re designing a freshman mathematics-teaser course, I might humbly suggest Problem Solving Strategies: Crossing the River with Dogs, which has the virtue of steering away from algebra as the sole touchstone for problem solving.

Back to Mr. Baker’s attempt to define Algebra I:

Six weeks of factoring and solving simple equations is enough to give any student a rough idea of what the algebraic ars magna is really like, and whether he or she has any head for it.

Mr. Baker himself seems to have a confused idea of what algebra is like, but he’s not alone. (See: my prior post on what is algebra and why you might have the wrong idea.) Also of note: some countries don’t bother to factor quadratics. (I haven’t made a map comparing countries, but it seems to be continental Europe that ignores it and just says “use the quadratic formula”.)

I could see solving equations managed in six weeks, but in a turgid, just-the-rules style that would be the opposite of what we’d want in this kind of appreciation class in the first place. Just the concept of a variable can take some students a month to wrap their head around, making me disturbed by the notion that six weeks would be enough for a student to find “whether he or she has any head for it”. (This isn’t even touching the issue of just how much is internal to the student. I’ve heard dyscalculia estimates of up to 7% of all students, but excluding that group a great deal of the attitude seems culturally specific. Allegedly in Hungary [I don’t have a hard research paper or anything, this is just from personal anecdotes] it is quite common for folks to say mathematics was their favorite subject in school and the level of disdain isn’t remotely comparable to the US.)

They are forced, repeatedly, to stare at hairy, squarerooted, polynomialed horseradish clumps of mute symbology that irritate them

The article’s invective along these lines makes me wonder again how much the visual aspect to mathematics is the source of hatred, as opposed to the mathematics itself.

This picture is from the Adventure Time episode “Slumber Party Panic” and is supposed to represent the ultimate in math difficulty. However, the math symbolism is strewn truly at random and there is no real problem here. I am guessing a good chunk of the audience couldn’t even tell the difference and for them, any difficult math problem looks like random symbolic gibberish.

This is related to another issue, that of bad writing. Here’s Mr. Baker quoting a textbook:

A rational function is a function that you can write in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those for which Q(x) = 0.

I claim the above definition is simply bad writing, and a cursory check of the Internet reveals several definitions I’d peg as clearer in a students-who-don’t-like-math-are-trying-to-read sense:

any function which can be written as the ratio of two polynomial functions.

a function that is the quotient of two polynomials

“Rational function” is the name given to a function which can be represented as the quotient of polynomials, just as a rational number is a number which can be expressed as a quotient of whole numbers.

Hardy’s A Course of Pure Mathematics, page 38:

the quotient of one polynomial by another

Godfrey Harold Hardy admittedly goes on from there, but his text is written for mathematicians. High school texts have no such excuse.

In a similar vein, the article later quotes a 7th grade Common Core standard:

solve word problems leading to equations of the form px + q = r and p(x+q)=r, where p, q, and r are specific rational numbers

which I suppose is meant to be horrifying, but in this case the standards are written for the teachers and aim to remove any ambiguity. Here’s problems that matches the standard:

1. You bought 3 sodas for 99 cents each and paid 10 cents in tax. How much money did you pay?
2. You bought 4 candies for 1.50 each and paid \$6.20. How much was tax?

The standards can’t simply produce many examples and gesture vaguely. They must be exact. Just because 4th grade specifies that students “Explain why a fraction a/b is equivalent to a fraction (n×a)/(n×b)” does not mean students are using variables to do so. (In case you’re curious what it does look like to the 4th graders, Illustrative Mathematics has tasks here and here matching the standard.)

There’s lots more to comment on, but let me leave off for the moment on this quote, because I’m curious…

Math-intensive education hasn’t done much for Russia, as it turns out.

…is this statement (in the last paragraph of the article) accurate?

### 18 Responses

1. Thank you. I wondered about that last statement too. I guess he thinks he can say that because things have gotten kind of messed up in Russia. I believe that has more to do with political changes than with the education people are receiving.

The math-intensive education may not have solved every problem (that’s not what education does, contrary to Obama’s and Duncan’s beliefs), but the Russians I know seem to have fun with math, and have some good, deep understandings.

I didn’t give the article’s rant enough thought to see what you saw. I agree with you – whatever becomes of math classes, they need to be about problem-solving.

2. I wrote a response as well, but didn’t focus on any of the same points. I really like that definition comparing a rational function to a rational number. Not only does it give justification for the terminology, but it would help students remember the definition.

• It’s nice writing trick. They were thinking about how it would be understood and not just if the defintion was correct.

3. […] another point of view, you should definitely read Jason Dyer’s thoughtful commentary on Baker’s […]

4. One of my former colleagues, a math teacher, used to say this about the Soviet Union: “They all know calculus yet they can’t make a toothbrush that lasts more than a week.” He didn’t have much love for the Soviet Union: his last name was Tatischew, and his ancestors used to be one of the most prominent families in Russia, chased away after 1917. I am still not sure what his point was. (I am not sure what mine is, either… 🙂

• Is that an indictment of the education system, or of the system of government?

(It does touch on one of the issues I’d like to get into sometime, which is how much of student motivation is based on economics. In a particular country, if high math scores are the only way to get into a university/get a good job, then scores will go up. In a a system with more routes to wealth / success, scores will go down. In the case of Soviet-era Russia, experimental funding was low so pure mathematics was able to blossom. That doesn’t improve on toothbrush-construction any.)

• To be honest, it was not an indictment at all – if anything, it was an indictment of an argument. To say: “Even though A is good, B is bad – therefore, A is not important” is an argument not worthy of a math-educated person. In Soviet Russia, math and physics were strong because it was a government policy to make them strong – and, unlike the US government, the Russian one had (and still has) nearly total control of the entire educational system, from pre-school to grad. school.

• A very good point. In France, academic achievement in Mathematics is needed for advancement in government positions. It is considered the hallmark of logical thinking, which since Descartes has been highly valued. No member of the French educated class would produce an article like Mr. Bakers or Dr. Hacker.

• I’m not sure what the point is either, I’m still trying to calculate if 21 of 30 free throws is 70%, how to prove 6 more makes 75%?

• I’m not sure what the point is either, I’m still trying to calculate if 21 of 30 free throws is 70%, how to prove 6 more makes 75%?

5. I think it is unfortunate that, almost by definition, the vast majority of journalists, writers and the educational policymakers (that is, politicians and EdD’s) – that is, the *opinion-makers* – do not have math/science backgrounds and have an almost visceral hate/distrust of numbers. This article is just one of the examples. Another one is Harry Potter: the math education/teacher is almost non-existent and the chem. teacher is one of the most hated characters in the books, his late redemption notwithstanding.

6. All the commentary regarding these articles – Mr. Baker’s and Dr. Hacker’s – while addressing specific issues and contentions presented by the articles, miss on the most important aspect of the necessity for educating our children in mathematics. Simply stated, the body of knowledge that we have today called mathematics (I am specifically NOT calling it ‘Modern’ mathematics) represents the greatest intellectual achievement of the human race! All cultures have contributed to it, all people can understand it, and it transcends any competing scientific or artistic body of knowledge. In fact, mathematics may be considered the most perfect art. To quote G H. Hardy, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” For any person to be considered educated without extensive knowledge of mathematics is laughable.

7. What about the people who want to become auto mechanics, hairstylists, or electricians? How would Algebra II help them? It may be true that for high end positions mathematics is seen as desireable. But how is mathematics used in the French government? Is it just a badge of honor or does it have practical use? And in many of the other countries students are “tracked” with many not in college prep courses, yet everyone is tested for Algebra in the US.

8. Baker’s article presents one side of the paradox with Common Core – the standards are both too demanding for a large percentage of students but are completely inadequate for the math/engineering track. Here in California, the state has moved from one of the most ambitious programs “1997 Standard” to CCSS. Math levels in elementary and middle schools dropped off a cliff — perhaps as much as one and a half academic years. Our local district made the tactical error of changing curriculum before the state realigned the tests. The result: one our best elementary schools dropped two ranks in math achievement. What happened was simple – many 4th graders for years had been introduced to simple variable algebra (if x is a number, what’s the next number in a counting sequence). Common core repeated concepts like simple arithmetic from third grade. When faced with letter x and simple probability, they scored low.

There of course is a much deeper question here in education whether kids have abilities for math as they might for music or baseball. We can’t all be world-class hitters or pianists and nobody is suggesting we try. If we treated math the same way, we would spare both sides of the ability curve the pain and uselessness of the Common Core.

• “There of course is a much deeper question here in education whether kids have abilities for math as they might for music or baseball. We can’t all be world-class hitters or pianists and nobody is suggesting we try.”

I would argue that in order to sustain economic and technological development, society must have enough good engineers, scientists and technicians; also, in order to avoid economic chaos, society must be made of the individuals who have at least some money-management abilities. Therefore, while it’s OK to have very few baseball or violin players, it’s not OK to have only a small fraction of people competent in math. Therefore, it makes sense to teach everybody math with some rigor. Sure, those who are innately more able will achieve more – but even those “at the bottom of the curve” will undoubtedly achieve more if compared with a system where they are taught math without rigor or not at all.

“…the paradox with Common Core – the standards are both too demanding for a large percentage of students but are completely inadequate for the math/engineering track.”

As for the Common Core standards (or any other national- and state-level standards), they are *meant to* establish reasonable competency levels for the “bottom 95%”, NOT to challenge the top 5% which will always find the ways to challenge themselves, as they always have. So… I wouldn’t worry about the top students and I would hope that the not-top (as opposed to the “bottom”) kids will, if taught with more rigor, achieve the Common Core standards, thus increasing the pool of *reasonably math-competent* young people in this country.

• Perhaps since my exposure is only as a student and now parent in a top performing district, I see the distribution differently in that CCSS would not satisfy maybe the top third of math students here. Assuming engineering schools are not reducing admission requirements, high school math should include Calculus I to at least the AB level. Working backwards, that means algebra in middle school. The “acceleration” is very inefficient and causes unnecessary pressure on students and districts with limited budgets. In the end, the parents with resources will work around the teaching in public schools with tutors, outside programs and summer school or they will seek private school alternatives. Families that trust and follow the curriculum will find their access to top colleges much more limited than before.

9. I think that some of the social sciences suffer from physics envy and have a pretension to mathematics which causes them to unnecesarily mathize their curriculum. A case in point is finance. i wrote a paper to show that most of the math and scary equations in undergraduate finance courses can be replaced with simple simulation models in Excel. Please let me know what you think. Here is the link:
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2512091

10. Thought-provoking piece – I was fascinated by the points ! Does someone know if my assistant can obtain a sample UK Skandia Disinvestment Request example to fill out ?