What to do about Algebra 2?

This week in Forbes magazine, Tom VanderArk renewed an ongoing and often intense debate  about the role of second-year algebra in math curricula in high schools and colleges. It's far from the first time that people have questioned whether Algebra 2 should be required of all students in a school district or college setting. There was pushback in 2013 from the state of Florida and more recently in California. In fact right around 2012-2013, when the Common Core State Standards were emerging (and largely because of those standards, which mandated some items from Algebra 2), resistance to "algebra 2 for all" also started emerging both in these specific cases and in more general op-eds, like this article in Slate and this one in the New York Times, and the issue doesn't seem to be losing any steam.

I think this is an important issue because it addresses a fundamental question about math education in this country, namely: What will the quantitative education of school kids and college students consist of? In that sense, the question of Algebra 2 morphs into a battle for the very definition and purpose of mathematics itself. No wonder it draws out strong responses from all sides. Because this is such a vital issue, I wanted to write this post in which I will probably make everyone who reads it at least a little upset: Because while I agree with much of what Tom wrote, at the same time there are significant issues in his article, and more broadly I think all arguments about this issue are starting to veer dangerously away from the people who can contribute the most: math educators. So in the spirit of an amicus brief, I want here to affirm some things and call other things out to try to guide this debate to a more productive place.

The Forbes article in a nutshell says the following:

• Algebra 2 is currently a gatekeeper course, used as a requirement for graduation in many school districts and colleges.
• Its role as gatekeeper is a holdover from a bygone time, starting in the 1990s with concerns brought up by "A Nation at Risk" which then morphed into Common Core standards. But whatever needs used to be in place for requiring Algebra 2, are today not as valid (if not completely irrelevant).
• In fact there is reason to believe that Algebra 2 perpetuates inequities among some student demographics, and anyway the "vast majority" (Tom's words) of people don't actually use the rote symbolic manipulation skills of Algebra 2.
• So instead, students should study coding and computational thinking. And there are some emerging examples in the real educational world where this is happening.

Before I get into the specifics, just a note: I met Tom VanderArk at the Steelcase Active Learning Symposium last fall, and I was really impressed by him. He's a smart guy with interesting ideas about the role of technology in our lives, including education, and I learned a lot from his keynote address. I was also grateful he took some time out to talk with me about my sabbatical at Steelcase, and he was  gracious and generous, despite dealing with the usual last-minute chaos of getting ready for a big talk.

So as I point out some issues in the article, this isn't personal. In fact I agree with many of his points. But if we are going to try to make a push for better mathematical education for young kids and college students today, we have to get the details right. Passion isn't enough. So I'll first point out the issues, then talk about what I agree with and where we should go from here.

In my view there are three major issues with the Forbes article.

Issue #1: Defining one's terms and using facts. The article leads off with an unfortunate quote from venture capitalist Ted Dintersmith:

The tragedy of high school math. Less than 20% of adults ever use algebra. No adult in America still does integrals and derivatives by hand - the calculus that blocks so many from career paths. It remains in the curriculum because it’s easy to test, not important to learn. https://t.co/DS52yevTbR

— ted dintersmith (@dintersmith) January 20, 2018

"Less than 20% of adults ever use algebra" is not a fact. It is a made-up figure without supporting evidence. (If there is research out there that supports this number, I am more than happy to be corrected.) Insofar as that "20%" number does have any basis, it's probably in Dintersmith's personal experience, so it's more likely to represent confirmation bias than anything else, a reflection of the people he's around rather than "adults" in general. If we're going to make a compelling argument, we can't lead off with opinions disguised as facts.

Of greater concern to me is the use of the term computational thinking. I have written a lot about computational thinking in the past (here and here for starters), but despite this and how much I've worked computational thinking into my teaching, the fact is that what we actually mean by this term is very fuzzy today. Jeanette Wing's paper is the usual go-to reference for computational thinking basics, and various folks (including Google) have built on this. But as Lorena Barba --- an honest-to-goodness computational scientist --- has pointed out, many of these recent formulations have drifted away from the original intention of "computational thinking" which is due to Seymour Papert. She writes:

The contemporary message takes an orthogonal direction to Papert, ignoring his Power Principle. It has happened to many powerful ideas when taught in schools (like probability) that they become disempowered: “reduced to shallow manipulation that seldom connects to anything the student experiences as important.” [...] Combing through dozens of articles on computational thinking, the emphasis is on problem-solving (low on the problem–project dimension), understanding (low on power), an operational perspective (low on the object–operation dimension), and content priming over media.

The "Power Principle", deriving from a 1996 paper by Papert, is in her words: "What comes first, ‘using’ or ‘understanding’? The natural mode of learning is to first use, leading slowly to understanding. New ideas are a source of power to do something."

When the Forbes article speaks about "computational thinking", what does it have in mind? Is it Papert's idea of "progressively deepening understanding" --- and if it is, what exactly do we want students to understand? Or is it just teaching kids how to make animated cats in Scratch? Or something in between? VanderArk writes:

That starts with problem finding--spotting big tough problems worth working on. Next comes understanding the problems and valuables [sic] associated--that’s algebraic reasoning. But rather than focusing on computation (including factoring those nasty polynomials), students should be building data sets and using computers to do what they’re good at--calculations. [...] A little coding can be useful to set up big tough problems. A basic coding class or two can be helpful in this regard. The new approach, exhibited at Olin College and signaled by the launch of the Schwartzman school at MIT, is just-in-time coding, a computational resource available across the curriculum--learn the right coding to apply the right tools at the right time to solve the right problem.

To me, this feels poorly-defined and impoverished. Solving important, relevant problems is of course very important and criminally underused in many math classes today. But there is a lot more to learning mathematics than just having the ability to pick the "right tool at the right time to solve the right problem". Ironically, being able to choose the right tool at the right time for the right problem is exactly the same goal that some teachers have for learning the very rote symbolic manipulation that this article denounces. (Think "techniques of integration" in Calculus 2.) Do we really want a "computational thinking" that merely swaps one form of rote symbolic manipulation for another? Where's the four-part foundation of decomposition, pattern recognition, abstraction, and algorithm design in other formulations of this idea? And also, where's the beauty that is threaded throughout mathematics? Does everything have to be useful in order to justify keeping it around? What a dull world that would be.

The article mentions Olin College and the MIT Schwarzman College of Computing, This website at Olin gives a little more insight on what "computational thinking" might look like. But the Schwarzman College example seems hardly informative. Long story short, MIT is spending $1 billion to create a new college which will "lead the world in preparing for the rapid evolution of computing and AI." Fifty (!) new faculty positions will be created and it promises to "educate students in every discipline to responsibly use and develop AI and computing technologies to help make a better world". But how this will be done and whether it's an externally valid approach to teaching "computational thinking" (whatever this means) is unclear. For example, those faculty --- will they be actually teaching, or just doing more research? And students --- what will they be actually learning, when will they learn it, and how will they learn it? A $1 billion dollar price tag doesn't give the answer and throwing that much money into an already-rich institution for unclear learning outcomes doesn't strike me as a decisive blow toward making math education better in this country.

To sum up: If we are going to make a convincing case for a better approach to math education, it has to start with facts, clear definitions, and transparent strategies and learning outcomes. That gets me to the next issue:

Issue #2: Conflating "Algebra 2" with symbol manipulation. What exactly is Algebra 2? Usually it is a big course with a lot going on inside it. The most memorable (for better or worse, mostly worse) thing that happens in Algebra 2 is a lot of rote symbolic manipulation, like factoring cubic polynomials or other actions which can, I will readily agree, be charitably categorized as party tricks (for very special kinds of parties) and nothing more. In the article, and in many fellow-travelling articles, Algebra 2 is equated with symbol manipulation.

The problem is that this is too simple. Take a look around at people's Algebra 2 syllabi, and you will find Algebra 2 tends to be a mix of topics, some of which are neither clearly interesting nor especially useful (e.g. factoring cubics), along with others that many Algebra 2 detractors would say are vitally important for people to know today, such as:

• Probability and statistics
• Basic function types (linear, power, exponential, etc.) and transformations of, and modeling with those basic types
• Linear programming and linear systems
• Matrices

If you get rid of Algebra 2 to remove the burden of symbolic manipulation skill, you also get rid of the above ideas, unless you create something else in its place. But I don't see these ideas coming up in what's called "computational thinking" here, which remember is characterized as "understanding problems and valuables [sic] associated" along with "just in time coding". How is a student supposed to "understand" the structure of a problem without some formal study, at least self-study, of these Algebra 2 topics?

The quote from Ted Dintersmith at the beginning of the article said that very few people "use algebra", but what does "using algebra" mean? If I think critically through a complex problem in life, am I "using algebra"? If I am looking at a data set and trying to decide what kind of regression model to use --- linear vs. power vs. exponential etc. --- am I "using algebra"? Because I learned to think critically through complex problems and learned about function types in algebra.

To sum up: Algebra 2 is not just one thing, or as the article puts it, "a course on regurgitated symbol manipulation (Algebra 2)". It can be just mindless symbol manipulation, but it often isn't, because there are teachers out there who have the vision to take it to another level.

And that gets me to the third and perhaps most important issue:

Issue #3: The lack of math educators' voices. I haven't read every single article about Algebra 2 in the news, but in the ones I have reviewed for this post, there is a distinct and alarming absence of the opinions and voices of actual math educators in these public debates --- i.e. people who are now and have been in the trenches of education, working with students and dealing with the on-the-ground issues of teaching mathematics. The closest we get to talking to an actual mathematics teacher in the Forbes article is a quote from Dan Meyer (which was misattributed before it was fixed); venture capitalist Ted Dintersmith, who has to the best of my knowledge never set foot in a math classroom as a teacher, is quoted twice as much as Dan in the article. The main character in the Slate article is a professor emeritus of political science with zero math teaching experience. And so on.

This doesn't mean that Tom VanderArk or Ted Dintersmith have nothing useful to say about math education. (Far from it; keep reading.) But it does signal a frustrating and dangerous trend where the opinions of those who know best about the nuances of mathematics education in this country and who work daily on the human level to work those nuances out, are being ignored in favor of the opinions of venture capitalists, Silicon Valley types, and sometimes even celebrities.

To sum this up: If you want to make a compelling case for improving math education in this country, talk to math teachers and see what they are doing, what's working, and what's true. Linking to news articles about schools doesn't count. Get in there, up close with teachers and let them inform you.

And now that I've called out numerous aspects of the Forbes article, let me express how much I agree with its basic message. (I told you, this post will make everyone at least a little mad.)

Algebra 2 enjoys a privilege in our school and sometimes college curricula that it didn't earn and largely doesn't deserve. While it can contain some ideas that are vitally important (see above for a list) and while you can make an argument that the symbolic manipulation portions of the course are good for building critical thinking skills, or willpower or character or grit or whatever you want to call it, the fact remains that in many people's experiences, Algebra 2 is all about rote symbolic manipulations, and whatever else of value may be in the course are drowned out by the sheer pointlessness of having to factor cubics, simplify $(x^4)^{100} (x^6)^{1/2}$, and so on.

Algebra 2 enjoys this privilege because Calculus enjoys a similar state of unearned privilege, and in many cases Algebra 2 is a requirement because Calculus is seen as the apotheosis of the high school math curriculum; it's also usually the core of the college math major, although courses like linear algebra are probably more important today. David Bressoud among others has written powerfully about the drive to calculus that has reached a fever pitch at the high school level and consequently put college calculus at a crisis point. Calculus has a central position that is a holdover from the Space Race days; we math teachers tend to approach calculus primarily as symbol manipulation; neither of these two defaults are often questioned, and since Algebra 2 is considered the place where kids are supposed to learn the basic symbol-pushing skills "needed" for calculus, guess what Algebra 2 typically looks like.

But, there's hope: Bressoud also has been involved in changing calculus at his home institution of Macalester College so it's now a modeling-centered course with an increased focus on conceptual understanding and applications and a reduced emphasis on symbolic manipulation. It's possible, in other words, to dial back the rote mechanics of Calculus, reframe the subject as a way of understanding the world through modeling in context, and still learn all the things we say that symbolic skills teach --- critical thinking, attention to detail, grit, whatever.

If it can be done with Calculus, it can also be done with Algebra 2.

It's also very important to note Tom's reference to Conrad Wolfram and the injunction to "teach as if computers existed". I actually wrote about this over eight years ago, and reprised the idea in this post which eventually turned into a TEDx talk last year. No biology teacher would dream of teaching that subject without a microscope out of the belief that students need to learn critical thinking by doing observations without technology. Similarly, no astronomer would teach her courses without a telescope. And yet when it comes to the computer (and particular tools like SageMath, Python, and Jupyter notebooks) which is a basic tool for exploration and analysis in the study of mathematics --- the microscope/telescope of the math world --- we fear its use and worry that if we teach with it, if we teach students how to use the tool like a professional, that they will somehow be worse off. This is crazy and unprofessional and possibly unethical, and we need to turn it around and be braver and more creative than that.

The way forward from here on this debate can proceed like this:

• Let's be clear about what Algebra 2 is and is not, and also be clear about what we would like to replace it with, if anything --- or what parts of Algebra 2 should be phased out and what parts should remain.
• Let's focus our decisions based on sound research, enriched by the lived professional experiences of math educators who are getting it done with real students every day. Bring others into the mix in this debate but resist the temptation to be over-awed by rich people and thought leaders.
• Let's avoid buzzwords like the plague and insist on concepts that mean something.
• Let's explore how other people have solved this problem. At my university, for example, students have to complete a quantitative reasoning unit to graduate; College Algebra (basically Algebra 2) is one way to satisfy it, but there are eleven other courses that do, including courses in logic or computer cartography. So perhaps there's a middle ground between forcing all students to take Algebra 2 and forcing them to learn to code.
• Finally, let's try to get students a little more involved. Missing from much of this debate is a diverse collection of student voices, some of whom might want math completely gone while others find something meaningful in that which we want to discard. Because if you center the debate on "What serves students the most?" then you will never go too wrong.