A few days ago, Andrew Hacker, an author and former professor of political science at Queens College in New York City, created quite a stir with a *New York Times* op-ed entitled Is Algebra Necessary?, in which he argues that it is no longer necessary to expect the vast majority of K-12 students to study algebra, geometry or calculus.

Hacker argues that the teaching of mathematics takes a toll beginning with students in junior high or middle school. "Algebra is an onerous stumbling block for all kinds of students," ranging from the disadvantaged to the affluent. Hacker acknowledges that basic "quantitative literacy" is important for everyone, but he asserts that "there's no evidence that being able to prove (x^{2} + y^{2})^{2} = (x^{2} - y^{2})^{2} + (2xy)^{2} leads to more credible political opinions or social analysis." He further argues that even in jobs that require science-technology-engineering-math (STEM) credentials, considerable training occurs after hiring, and (by implication) mathematics training in school is not that important. He concludes:

Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves. So why require it, without alternatives or exceptions? Thus far I haven't found a compelling answer.

Hacker's article has already attracted several persuasive responses, including, for example, an op-ed on the *Scientific American* website. Some other responses include blogs by Rob Knop, Daniel Willingham and RiShawn Biddle.

### Deficiencies of K-12 education

The present authors (DHB from USA, and JMB from Australia) fully acknowledge that there are many problems with present-day K-12 mathematics education worldwide, not the least of which is the relatively poor training of teachers. As William Schmidt and Curtis McKnight note in their book *Inequality for All*, only 35% to 40% of the 7th and 8th grade teachers they surveyed in Michigan and Ohio had either a major or minor in mathematics, and only half of teachers in the 9th and 10th grade.

Other nations outside the U.S. are not standing still. Finland for example, has ranked at (or near) the top of the OECD nations in educational performance for more than ten straight years, due in large part to its strict requirements for teacher training (all teachers must have at least a masters degree). See also our recent articles School maths is failing children and Yes, there's a numeracy crisis -- so what's the solution? in the *Conversation*. Along this line, in Australia the New South Wales provincial government has just ruled that all aspiring teachers must specifically study mathematics and science, and further must meet minimum entry scores to even qualify for educational programs at state-operated universities.

### Why pick on mathematics?

One fallacy in Hacker's reasoning is clear: Why single out mathematics? Yes, a knowledge of calculus may or may not help one negotiate through traffic or connect one's computer to the Internet, but the same could be said for many other disciplines. How does knowing whom *Hamlet killed accidentally* help one be a better consumer? Does knowing the history of the Spanish-American War help one complete one's tax return? Many other examples could be listed.

### Why algebra is necessary

With regards to Hacker's article, while we agree with many of his specific points, we respectfully disagree that one can be a well-functioning adult in the 21st century world without at least the rudiments of algebra. Consider a very common, simple question: You bought an item at a store, and although you didn't keep the receipt, a charge of $173.15 was posted to your credit card account. The sales tax (or VAT) rate is 8.25%. What was the pre-sales-tax price of the item? With algebra, it is easy: x + 0.0825x = x(1 + 0.0825) = 173.15, so x = 173.15/1.0825 = 159.95. How can you possibly solve such a problem without at least a rudimentary mastery of algebra? As another example, if you get a 20% pay cut, followed by a 20% pay increase, will you be better off, worse off, or the same? (Answer: worse off).

We also agree with Hacker that specific topics taught in a typical K-12 mathematics curriculum need to be rethought in a 21st century economy. Detailed calculus-based graphing techniques (which can now be done better by computer) could give way to a greater emphasis on "discrete mathematics," including aspects of probability and statistics, which might better be termed "reasoning and reckoning."

But a basic facility with algebra is essential to do justice to such material, even at the high school level. And then there is the indisputable fact that a good facility for algebra, geometry and yes even calculus is absolutely essential for that subset of students who aspire to careers in STEM fields. Yes, it is true that even those who major in STEM fields in college and/or graduate school still have significant skills to learn once on the job. But without a strong background in the topics that Hacker derides, they will be hopelessly lost.

### Mathematics for minorities

As mentioned above, one of Hacker's points is, in effect, that the requirements for algebra and the like serve as an "onerous stumbling block" for disadvantaged students. But we argue the opposite: a system that "tracks" students into mathematically less challenging courses early in junior or middle school, or even in high school, runs a very serious risk of derailing talented young minds from disadvantaged environments. One of us (DHB) has a daughter with several years experience teaching at a very disadvantaged high school in Southern California. She agrees that a "tracking" scheme of the sort implicitly proposed by Hacker would lose a large number of poor black and Hispanic students -- these students need "more opportunities in later grades than other students."

Yes, many disadvantaged students score less well than others, but in many cases this may be due to the fact that they lived in a foster home, or in no home at all, for a year or two. And "dumbing down" the program does not do these students a favor either -- it only places them further behind and deeper in discouragement at the prospect of ever catching up with their more privileged peers. But with dedicated teachers, numerous such students from this daughter's high school and others in the area have gone on to major and excel in mathematics, science and engineering. Some have even received scholarships to institutions such as Stanford University and U.C. Berkeley. None of this would have happened if Hacker's proposed changes had been implemented.

### It's a personal matter

Finally, we must add a personal note. DHB still recalls that his fifth grade teacher reported to his parents that he was a "good B student," and even after he exhibited significant precocity in mathematics, was told in the ninth grade by a well-meaning school counselor that "he didn't really have the aptitude" to be a professional mathematician. If DHB had taken this advice, this article would never have been written!

Similarly, JMB recalls the terror of the British eleven-plus test that he took at the tender age of 10, which would irrevocably decide whether he would be assigned to a precollege track or a vocational track. Introduced as part of the well-intended Butler educational reforms after WW2, the eleven-plus had the effect of consigning 75% of students to a non-academic stream, thereby determining much of their future life. JMB did pass the exam, but what if he had a bad day (say due to a toothache or recent family trauma)? Again, this article would never have been written! Incidentally, his wife, who has a hard time with numbers, thrived mathematically after algebra was introduced in high school.

### Summary

While we generally agree with Hacker that significant improvements must be made, dumbing down the curriculum, or tracking students at an early age into college-prep or non-college-prep mathematics is not the answer. Hacker may well be right that only 5% of the workforce *needs* rigorous STEM training, but there is no way to know which 5%. We also know that making all mathematics seem applicable has been a pedagogical disaster. In the 1990s the Dutch did just that (as part of a larger battles over Realistic Mathematics Education): mathematics was to balance checkbooks and understand mortgages. No wonder the university mathematics cohort was cut by two thirds when such miseducated students reached college. To misquote Peggy Lee, "if that is all there is then keep on dancing."

Mathematics really does matter in the 21st century, and better mathematics teaching is desperately needed. Our decision-makers need to acknowledge that some things that need to be fixed are costly. There are no quick smarter-and-cheaper alternatives to investing in training and in paying specialist teachers better.

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