To be a fly on a great philosopher’s wa- … ceiling, actually

One night, the French mathematician and philosopher René Descartes was staring at a fly on his ceiling. (Ain’t no party like a mathematician’s party … we think it’s safe to say he leaned toward the introverted side.) He wanted to be able to describe this fly — or anything, really — given a fixed point in a two-dimensional plane. It was this seventeenth century eureka moment we have to thank for the Cartesian plane.

Now there are four flies on the ceiling! Must be August.

We also have René Descartes to thank for the variables we most commonly use in teaching algebra: x and y. By convention, x is the independent variable: the thing you “put into the function box,” and y denotes the dependent variable: the output once you substitute an x (“plugging it in”).

Why x and y? It was because they were close to the end of the alphabet. The letters a, b, and c often denote constants (known quantities). And just for fun, m for slope comes from the French: monter (pronounced roughly mohn-tay) means “to climb.”  

Think of a negative slope like this one as skiing “downhill” from left to right; positive slopes mean you montez the line upwards. 

Maybe you found Algebra I (oddly enough, the standard textbooks often use Roman numerals, rather than the more à propos Arabic numerals, in their course names) to be a breeze when you took it as a distinct course, probably in middle school, manipulating the variables until you found the slope of the line. Maybe you felt like Peppermint Patty, and just started writing 11. 

Regardless, this is a very North American, Anglosphere way of thinking about math instruction. In fact, when researching this topic, there seem to be four broad regions in terms of the when and wherefore of teaching algebra. Let’s take a stop in each, examining the pros and cons.

The Continental Way: L’art de la spécialité 

When I was teaching college admissions exams in Mexico, one line that always got completely baffled responses was that I put myself through a degree in French Literature by working in a DNA Sequencing lab. The US education system is one of the broadest in terms of not only not requiring specialization at a young age, but requiring a balance of courses in literature, mathematics, science, and social sciences through age 18, with almost all liberal arts colleges requiring the same type of broad coursework in your “generals” in your first year of university.

It is not thus among Western Europe (and Mexican universities, in this respect, most closely resemble the educational system of Spain). You get tracked much earlier, which would be like declaring a major or going into an apprenticeship in the Yankee context, between age 14 and 16. Take a look at this tree of educational tracks for French adolescents from a research paper by Megan McGurl at a liberal arts college in the U.S. state of Georgia). 

In the U.S., any student from any high school can apply to any university in the country. Once there, you can change your major several times. Your author recently appeared on Jeopardy! — the champ from my game is now a math teacher who, as an undergraduate at UCLA, cycled through such unrelated fields as biochemistry, philosophy, physiology, and finally ended up in statistics. This educational journey would never happen in France, especially at a highly selective university.

If you wanted to be a statistician, you’d have to be placed on the statistics track. Philosophy? One of the most competitive majors in the land of Sartre and de Beauvoir. And higher math would not be seen as necessary for a successful life outside the sciences.

So, at what age do Western European students learn algebra as a distinct field? The answer is … they wouldn’t recognize “algebra” as a distinct year that is divorced from other math courses. It’s all just … math class, taught in what we might term an integrated curriculum. In Italy, Germany, Switzerland, and yes, France, home of Descartes, algebraic skills are taught throughout the entirety of secondary school: and the stronger you are at STEM, the more likely you are to be encouraged on the STEM track.

How does Western Europe compare to the US in how its 15-year-olds perform mathematically? In the most recent PISA mathematics results, Switzerland was the top-scoring country outside East Asia, Italy was exactly the mean score for OECD countries, and the US was below average. It seems like integrated math is, at the very least, not correlated with lower performance, and very likely outperforms the standard U.S. curriculum. 

Our neighbors to the Frozen North 

What about Canada’s math results? ​​

I audibly said “wow” when looking at the following table, so you know I’m about to be invited to a wild fly-watching party à tout moment. Take a careful look at which province stands apart:

Prince Edward Island, well-known to bookish sensitive gingers for the setting of the Anne of Green Gables series (set in the 1870s), showed the greatest increase from the baseline. Anne herself became a teacher after going to a “normal” (education) school on a full-ride scholarship, and ended up marrying Gilbert Blythe, a doctor, with whom she frequently battled for top of the class. This excerpt shows that even in the era of puffed Gigot sleeves in the late 1800s, Canada’s English-speaking math curriculum was split into discrete courses that wouldn’t seem unfamiliar to a competitive secondary student today:

Oh, Diana, if only the geometry examination were over! But there, as Mrs. Lynde would say, the sun will go on rising and setting whether I fail in geometry or not. That is true but not especially comforting. I think I'd rather it didn't go on if I failed!

     Yours devotedly,
     Anne

Although rural PEI’s increase from 2010 to 2013 to 2016 in the pan-Canadian competition is impressive and would likely give Anne and Gilbert a run for their trapezoids, francophone Quebec is the clear standout in all three years.

In this article, Paul W. Bennet examines the reasons for the Québecois math advantage.

Fewer topics tend to be covered at each grade level in Quebec, but they are covered in more depth than in BC and other Canadian provinces. 

In other words, the original settlers — New France vs New England — brought certain attitudes towards math education to their new land that grew surprisingly deep. As long ago as 1827, Harvard required algebra to enter — Puritans didn’t like this as it felt too mercantile, but it stayed in the curriculum. For two centuries since, we have taught Algebra with a capital A as a separate “thing,” often forgotten afterwards, since our numeracy and abstract reasoning skills are below average.

According to the report, Quebec schools feel no particular compunction to graduate every student  — they have much lower graduation rates than the rest of Canada. In other words, more positive attitudes to math and more comfort with “tracking” could help explain the difference in outcomes.

Ontario, British Columbia and the other English-speaking provinces have all been greatly influenced by American educational theorists, most notably John Dewey and the progressives.

My very American small-l liberal arts education in French literature (with a long stop at molecular biology) means that I in fact knew Dewey in Jeopardy!: the true mark of a well-rounded mind in our English-speaking society is, perhaps, knowing a little about a lot of subjects. In Quebec, their version of Jeopardy! lasted only two years. 

2200 years of history: teaching algebraic thinking early

九章算術- The Nine Chapters

Above Switzerland are the usual suspects: East Asian countries, including Japan, South Korea, and (arguably) China. The history of algebra is not one of a singular moment of revelation, but rather an observation in the world of amounts that gets increasingly abstracted. If I say that the amount of butter in the recipe for cupcakes will need to be doubled because the guest list for a birthday party has twice the number it did before, that’s algebraic thinking. The translation of the “word problem” to figures is where many English-speaking students get slowed down significantly.

Mainland China has a mixed record on the apples-to-apples comparison in international surveys: they have a large amount of internal migration, and have even more extreme tracking than Europe. Even at the college level, your major is completely determined by entrance exam scores — according to a recent New Yorker piece by an expat writing teacher, students were almost universally disappointed with their results.

When a student applies to university, scores are all that matter—no teacher recommendations, no list of extracurriculars.

Students in China are taught math facts such as 4*8=32 at a young age — 7 — and the curriculum emphasizes memorization. Take this analysis from a UK perspective (hence “maths” and “different to”):

The goal of maths education in China is to develop conceptual and procedural knowledge through rigid practice. In comparison, the UK maths curriculum is less focused and consistent. China uses whole-class instruction, engaging all students in the material and prompting feedback. This is different to the UK model teaching of maths, which is more focused on small groups and individual attention.

Chinese students are taught to understand numerical relationships and to develop and prove their solutions to problems in front of the whole class. This means students understand whole concepts of maths, allowing them to apply previous knowledge to help them learn new topics.

Proofs in the English-speaking world are confined to Geometry class, if they’re taught at all. By contrast, the nine chapters form the basic math curriculum that has remained unchanged for 2200 years. So, algebraic thinking is introduced from the very earliest ages in China, and you’re put through a zero-sum competition that makes Anne Shirley’s geometry tests in her late teens look like a cakewalk.

Is algebra necessary for a well-lived life?

The U.S. from the very beginning has tended to prioritize skills divorced from their intellectual traditions — in a society where there is an ideal of limitless self-invention, and a sideways glance towards authority, something as seemingly rigid and uncreative as y=mx+b can induce a yawn, at best, and Don’t Mont-ay On Me at the far end. Peppermint Patty wouldn’t just randomly put 11 on her exam were she in Shanghai, and she would probably be a high school dropout in Quebec. Even de Tocquevile would recognize her anti-authoritarian rebel streak as uniquely American. 

So, if we want to improve our results in algebra from San Diego to Charlottetown, one idea would be to bring back its mercantile usefulness. Algebraic thinking can help you in economics, balancing a checkbook, and the favorite advice to confused Gen Zers — learning to code.

Additionally, Americans have built some of the most selective and prestigious universities in the world — if we get the word out that Algebra is not just a textbook for 13-year-olds, but an entire way of thinking that can be introduced in childhood and reinforced throughout one’s education, then we will be more competitive on the world stage and lift the floor for all students.

At Socratica, we agree with this take in Cal Matters: 

“There’s a huge problem with math instruction right now. The way things are set up, it’s not giving everybody a chance to learn math at the highest levels.”

REBECCA PARISO, MATH TEACHER, HUENEME ELEMENTARY SCHOOL DISTRICT

Anne Shirley was a 19th century orphan who nevertheless was deep in the college admissions arms race — as evidenced by her single-minded competitiveness with her classmate, Gilbert.

"I'd rather not pass at all than not come out pretty well up on the list," flashed Anne, by which she meant—and Diana knew she meant—that success would be incomplete and bitter if she did not come out ahead of Gilbert Blythe.

(Turns out they tied for first, helping her to secure a place in college she couldn’t have afforded otherwise.) Although fictional, their story is a compelling argument to introduce algebraic thinking early in publicly funded K-12 schools. If she didn’t have the chance to learn algebra and geometry when she was clearly ready to do so, she would have been constrained by the traumatic circumstances of her childhood abandonment. Isn’t it better for the Annes of today to have access to a curriculum that challenges them? Students like Gilbert, from comfortable middle class homes, have always had more access to higher education, regardless. 

Descartes’ demise

A sad postscript about why Descartes perished in his 50s, a cautionary tale for night owls:

His habit of sleeping until 11am had been brutally disrupted by Queen Christina of Sweden, who persuaded him to go to Stockholm in 1649 and wanted to do maths with him at 5 o'clock every morning. Descartes endured the early mornings and the Scandinavian cold for a few months, but eventually contracted pneumonia and died. Source 

Sleep as late as you need, bundle up, and learn algebra with our videos here.

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Charly Kuecks

Charly Forrester Kuecks is a writer, test prep tutor, and marketer. She works as an educational consultant for Socratica.