When Will They Learn Algebra?

Themistocles, Thucydides, Peloponnesian War

I have intended to follow up on my post on primary education with one on secondary education. The key issue that must be addressed before anything else can be said is tracking, a word toxic to the political discussion.

Certainly tracking exists; nobody expects doctors, lawyers, engineers, and Samaritans to each receive the same education. Yet nobody dares point to any specific place where it happens. The system for distinguishing educational success¹ is a gateless gate². Surely it happens somewhere between the ages of 8 and 18, but the details get blurry quickly. Any visible methods of distinction are ostracized, harangued, and subverted.

The most visible conflict regarding tracking relates to the teaching of algebra, and that is where we will focus our readings today.

What is algebra education?

Unfortunately, Wikipedia has no article on algebra education. If you think that’s an oversight, you can write one yourself, or comment here.

For the purposes of this article, we will generally focus on the American “Algebra I” - a course that covers mathematical topics including graphing functions, factoring polynomials, and explaining the quadratic formula. The State of California's curriculum is a representative outline.

When does Algebra Education occur?

How long should a man’s legs be? Long enough to reach the ground.

In short, either 8th grade, 9th grade, or 8th or 9th grade. Note the formatting; some (most) school systems offer a choice, while some do not.

The US Department of Education website has a whitepaper about access to Algebra I in 8th grade. “Only 59 percent of schools offer Algebra I in 8th grade”, and 24 percent of students take it in 8th grade.

In a recent controversial move, the San Francisco school district decided to eliminate 8th grade algebra. Their website quickly explains that they didn’t really do that, but rather they restructured 8th and 9th grade math into a different arrangement; many “Algebra I” topics are still taught in 8th grade.

How important is Algebra?

Four years of math in high school, with a strong foundation in algebra that builds from middle school, is key to higher education access. Therefore, ensuring that middle and high school students succeed in math — and in algebra in particular — is an important issue for policy and practice. (Snipes and Finkelstein)

Algebra acts as a gatekeeper for high school graduation and post-secondary success. Students who pass Algebra 1 by the end of ninth grade are more likely to take advanced mathematics courses, graduate from high school, and succeed in college. Yet persistent inequities in access to rigorous algebra due to issues of placement, preparation, and quality of instruction have kept the gate closed for a large proportion of students, particularly minority and low-income students. (Stoelinga and Lynn)

To a certain extent, these takes are making a category error. At the extreme, they point to Algebra I as the gatekeeper of the gateless gate, and feel that if they can “solve” that one issue, every student will be able to succeed at college.

Yet algebra is not a magic potion. Even if every student mastered algebra by the end of the 9th grade (and I believe it would be reasonable to expect 85% of students to do so), many would still be ill-suited to various educational tracks.

Lowering the Bar

Another take is that if some students can’t do algebra, we simply shouldn’t require it. This makes the opposite error from the previous section. By lowering the gate, we can let everybody through! But when the idea shows up in the Opinion pages of the New York Times, it must at least be acknowledged.

To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason. … Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. … it’s not easy to see why potential poets and philosophers face a lofty mathematics bar (Andrew Hacker)

There is at least the kernel of one good idea in the article: it is probably more important for the 10th percentile³ student to understand statistics than to understand algebra. But for the “college-bound” student, the suggestion that algebra is unnecessary is simply wrong.

If you can’t pass Algebra I by the end of tenth grade, you are bad at math. You should not be sent through the educational system to become an engineer. You probably shouldn’t become a lawyer or a business executive. You definitely shouldn’t become a philosopher. You should be able to go to art school, assuming you are good at art. Ergo, we need ability tracking and need an algebra requirement for many of those tracks.

The hard problem of how and when to explain to students (and their parents and advocates) that they are bad at algebra and they cannot become a lawyer … is left to a future essay.

My Take

The power of instruction is seldom of much efficacy, except in those happy dispositions where it is almost superfluous. - Edward Gibbon

In the post on primary education, I propose no “math class” at all through the 4th grade. However, mathematics is still taught. Some mathematics is simple language fluency; can you understand the sentence “I had 7 action figures and bought another one and now I have 8 action figures”. Other parts are “financial literacy” (understanding that if you have $10 you can buy 5 ice cream sandwiches for $2 each).

I will be proposing a standard “fifth grade gap year”. Well designed systems allow for slack in the system. If, say, a pandemic occurs and months of school are lost, it can be made up. Students can pursue the arts, music, sports, or foreign languages. Certain topics like sex education that don’t fit well into the standard structure of school can be handled sui generis. We probably need “internet education” in this year as well.

And then secondary school. In sixth grade there will be “Sixth Grade Math”. Students who progress through the material faster … will progress through the material faster. (I am skeptical that tests of arithmetic ability will be a reliable predictor here.)

I estimate⁴ that the 90th percentile student can finish Algebra I material by the end of 7th grade, the 50th percentile student can finish Algebra I material by the end of 8th grade, and the 15th percentile student can finish Algebra I material by the end of 9th grade. As my dad would say, the exact values are an empirical question, one that can be determined by scientific research.

In the end, any classroom system that gives substantially worse results than self-directed study must be re-examined. And that bottom 15 percent … should target a high school diploma not endorsed for math. More on that later.

1. There is a deeper philosophical debate whether “achieving an educational track for a higher-status job” should be considered success at all.

2. You may recognize “The Gateless Gate” as a translation of Mumonkan.

3. The 99th percentile is the top 1% of students according to a metric; the 10th percentile students are worse than 90% of students on that metric. We use “abstract math ability” as the metric here. This is an abstract and approximate metric, I make no claim that this can be measured by a screening test, but neither does it need to be measured.

4. I have absolutely no research data guiding these estimates, only my instincts on how smart teenagers are and how hard mathematics is.