## Is Algebra Necessary? Yes and No.

Political scientist Andrew Hacker recently asked “Is Algebra Necessary?” and the response has, unfortunately, been predictable.

Those in society’s minority who did well in math courses are “shocked” at the suggestion that we change the typical math curriculum. The teaching may be “dismal” but algebra is a “foundation stone” in developing critical thinking skills. “It teaches one how to think.” It’s a little amusing but mostly disheartening to see folks who claim to support more challenging math standards fall back on strawman arguments, condescension, sarcasm and, my favorite, math errors in their arguments.

Those in society’s majority who did poorly in math tended to respond with relief at the suggestion of dropping algebra, although there are a few PMSD (post-mathematics stress disorder) victims whose career paths were altered by failing math and who still carry the associated baggage and resentment.

Let’s set aside the hysterics (“We are breeding a nation of morons“) and give both sides of this debate a fair shake, shall we?

**Arguments in Favor of Eliminating Algebra as Courses Required for All**

We definitely teach too much algebra and do so mindlessly, without considering whether it’s useful. As a teacher of math courses from arithmetic through calculus at a community college, I fought the losing fight to remove useless topics from the curriculum. For example, Cramer’s rule is a relic from the days before computers and is as practical as a slide rule, but trying to remove it from the required topic list elicited resistance and deep resentment from many of my fellow faculty. Hacker’s suggestion that we reconsider the requirement for so much symbolic manipulation is sensible.

While teaching algebra I tried to limit my syllabi to (1) the topics that would be used in later courses, (2) topics that might be useful outside the classroom, and (3) some examples of true beauty. I emphasized (1) but snuck in some of (2) and (3). Still, I did teach some material that would *only* be useful in later math courses and never in *any* kind of applied setting. We could easily cut more topics by curtailing the length of the required math sequence, at least for the math subjects taught.

Algebra is not the only way to teach disciplined thinking. One can teach precisely the same thinking skills while removing the abstraction that makes math seem useless and difficult to many students. One idea (not mine): we could integrate math beyond middle school into the science curriculum and use applications as motivation. Then there’s no need to learn how to apply math to story problems; the stories *are* the original problems. This also prevents folks from running around with “hammers” looking for “nails”. Unfortunately I am not sure how to get there from here; teachers of other subjects would have to cover the needed material and that would require revamping the way teachers are taught.

We don’t need to learn algebra to develop our intuition about rates of change, interest rates, probability, statistics, and other topics that typically follow algebra. There are ways (videos, interactive widgets, simulation using simple programming) to develop the intuition of calculus — often the only calculus needed in a job like medicine — without approaching it the rigorous, analytic, symbol-pushing way we typically do. Even for those who eventually will need algebra, we can teach more advanced symbol manipulation skills later as needed.

Nobody (well, almost nobody) is saying that learning algebra has no value whatsoever. However, as long as we have limited resources the pertinent question is, does algebra give us more benefit than spending that time elsewhere? I suggest that **programming, statistics, and finance** are better uses of most students’ time. Programming is how the nearly countless computers in our lives work; a basic understanding of how they do their magic would be invaluable. Statistics are essential for making sense out of the sea of information around us. Finance is challenging and vital for artists and engineers alike as long as they want to buy a home or save for retirement. If we remove the symbol-pushing exercises of algebra and replace that class time with simple programming, statistics, and finance, we’ll gain more than we’ll lose.

**Arguments in Favor of Keeping Algebra a Courses Required for All (with occasional rejoinders)
**

Let’s be more specific about “algebra”. A first course (“Algebra I”) often includes basic linear algebra (lines, graphing them, solving systems of linear equations, and matrices) plus evaluating polynomial functions, graphing quadratic functions, and solving single quadratic equations. A second course (“Algebra II”) builds on this with lots of factoring polynomials, exponential and logarithmic functions, quadratic inequalities and other algebraic prep for calculus.

An Algebra I course like this is incredibly useful. The concepts generalize to every science, from physics to political science. With this foundation a student can learn the intuition behind calculus, statistics, and other tools that are *useful to have seen* but usually *not useful to retain.* **I strongly recommend keeping Algebra I part of the core curriculum.** With moderate resources, this material can be covered in middle school for most if not all students.

Algebra II is where common responses to “Why am I learning this?” jump the tracks.

- “It builds your brain like exercising build muscle.” I used this one regularly; it’s true, but only a half truth. Programming, statistics and finance can do the same with the added bonus of being unquestionably practical.
- “It helps students understand where more advanced math comes from.” Playing with simulations is even better for most students in understanding why more advanced math techniques work the way they do.
- “It teaches structured thinking.” Programming is even better for that
*and*it is easier for students to see what why structure matters. Mess up the structure and programs do odd things.

Algebra II is full of topics that you don’t need in order to *understand the intuition* of common useful advanced ideas; you need Algebra II when you will try to *master* more advanced ideas. **I strongly recommend making Algebra II something that fewer students take.**

**Known Unknown**

My recommendation to remove Algebra II from the universal curriculum is contingent on at least one assumption:* Students who need more algebra will have sufficient time to learn it later.* To see if this is true we would need to take average to bright students interested in technical fields and wait until college to teach them algebra. This is not common. Interestingly, I have some experience that is close to this: teaching math to political science graduate students. While there are a few notable exceptions, most political science undergraduates take very little math. Graduate students work very hard to learn the math necessary for advanced statistics and game theory, and generally they succeed. This evidence is circumstantial but shifts the burden of proof onto those who might argue college is too late to learn algebra.

**Bottom Line: Question Mathematical Authority**

Hacker thoughtfully asked a good question: are we teaching what we should be teaching? One cannot decry the educational establishment as ossified but resist any attempt to change for the better. Hacker may be throwing out the baby. My ideas might not be the best. What are your ideas? Let’s discuss it.

* This blogger is an award-winning teacher of math, statistics, and programming at the high-school through graduate levels, holds graduate degrees in math and political science, and works in the defense industry as a data scientist.

Dear Steve: Could you kindly send us an email address for you? Andrew Hacker and I would like to touch base. We are doing a book on the politics of math education and we’d like to talk with teachers. Best, Claudia Dreifus

Concerning this section

“Algebra II is where common responses to “Why am I learning this?” jump the tracks.

“It builds your brain like exercising build muscle.” I used this one regularly; it’s true, but only a half truth. Programming, statistics and finance can do the same with the added bonus of being unquestionably practical.

“It helps students understand where more advanced math comes from.” Playing with simulations is even better for most students in understanding why more advanced math techniques work the way they do.

“It teaches structured thinking.” Programming is even better for that and it is easier for students to see what why structure matters. Mess up the structure and programs do odd things.”

Perhaps programming/simulations, statistics,and finance might be better for those skills.

My Statistics class, which admittedly is probably the most useful math class I took (challenged only by my game theory class) took 1 full year. However, n the first quarter, we learned only the z curve. But yes, a full year of stats has more practical significance than a full year of algebra.

I’ve seen simulations and I know a little programming as well, and they definitely do teach structured thinking. And with it’s more concrete (less abstract) structure, it probably is easier to learn than algebra. So yes, a full year of programming and simulations (these are so closely related that they could be combined, don’t you think) has more value in teaching structured thinking that a full year of algebra.

However, the question we need to ask is “Does algebra have a greater TOTAL value, compared to these other courses for the same amount of time (in academic years).”

In a more mathematical sense…

Let

A= total value of algebra

B= total value of statistics

C= total value of programming/simulations

d= value of “brain building” of algebra

e= value of structured thinking of algebra

f= value of groundwork for future math of algebra

g= added value of statistics “brain building”

h= added value of programming/statistics structured thinking

j= added value of programming/statistics groundwork for future math

Assuming we are only talking about one year, The system becomes

If A= d+e+f

AND

B= 1/2((d+n)+e+f)

AND

C= 1/2(d+(e+n)+(f+n))

(Note that the 1/2 is due to the fact that they both need to fit inside the same year, and so they are being taught 1 semester each).

THEN

Does A > B+C

Or does A < B+C

Or does A = B+C

Side note: Finance is incredibly easy to teach. Depending on how deep you are going with it, it can be taught anywhere from 2 weeks to 2 years. I am of the opinion that it should be part of a "practical knowledge class."

Opps, I realized I changed my variables without changing the “n”

The first n is g, the second n is h, and the third n is j

Factoring is one simple concept that Algebra I taught me that has been useful throughout my life. It has made my computer programs shorter and has allowed me to simplify arithmetic calculations in my head. Many intelligent, non-mathematicians are delighted to learn that a price discount of 15 percent is the same as finding 85 percent of the original price. Or that a 15-percent tip added to the price of a meal gives the same total as multiplying the price of the dinner by 1.15. I might have come up with the same shortcuts without algebra, but they remained with me because I did.

A major problem with the Algebra II course is its split personality. It tries at once to be the terminal mathematics course for most students, and the foundation course for others. It fails both kinds of student.

The material of the course may be divided into three parts, that which was already covered in Algebra I, work unique to the Algebra II course (at least insofar as high school is concerned) and material covered again later in pre-calculus or calculus. Since most Algebra II students have taken geometry between Algebra I and II, they will need considerable review of much, if not most, material covered in Algebra I. Students having little interest in mathematics will generally need more review. Many capable students will find the review pointless and boring. Some will be turned off to mathematics by the tedious review.

The second caegory, matrices, determinants, systems of three or more simultaneous linear equations, is unlikely to be of much use at all for mathematically-challenged students. For those who will study mathematics further, those sections are almost worthless. I recall from my Algebra II days learning Cramer’s Rule for solving systems of equations (only in 2 or three variables). But the rule was never derived (even though it es easily done for systems of 2 equations, and is a challenging, if tedious, algebra problem in 3 variables.)

Finally there is material taught better and in more depth later in a pre-calculus or analytic geometry class. Why spend lots of class time barely covering the conic sections in the coordinate plane when it will be covered more in depth the next year with a more elite student cohort? The material is necessary for those who will pursue mathematics at a higher level, but pretty pointless for those who will not. Why give a poor emaciated presentation that won’t help those interested in mathematics, while completely bewildering with its uselessness those who are not interested?

Remove Algebra for non math/science majors!