Hard for me to believe, but this is my 100th blog post. That’s the equivalent of a 300 page book – with much less pain, and lots more fun interaction: thanks to all my loyal readers and responders! And hooray for the Internet for permitting easy and rich discourse about one’s ideas.
To celebrate I thought I would return to my favorite educational whipping boy* Algebra I. There are so many reasons for dumping on Algebra but a timely one (on this blog milestone) comes from the fact that one of my blog posts on math was quoted in the recently-released Publisher’s Guidelines on the Math Common Core Standards!
Before I get into my rant, let’s be very clear about a few things:
I love math. I taught math – Pre-calculus and Geometry – and I am good at it, my best grades in school. I think non-Euclidean geometry is one of the coolest ideas ever – no one should fail to learn about it in school, in my view. (The mathematician Morris Kline says that its development is one of the great ideas in modern history). I am thus not one of those Humanities types that thinks math is boring and pales in comparison with a good novel. Left to my own devices, I would rather read the Heath edition of Euclid’s Elements than any novel reviewed by Maureen Corrigan on Fresh Air.
Because of my education at St. John’s College, I have an unusually strong background in the history and philosophy of mathematics. We read Euclid, Ptolemy, Descartes, Newton, Lobachevski, Dedekind – and it was fascinating, because the meaning and purpose of the math was always in view.
So, I know, for example, one key reason why Descartes invented algebraic solutions to geometric problems and thereby gave us his coordinate system: to not only solve certain problems with conic sections more easily but to enable us to escape the arbitrary prison that you can’t have any number be greater than to the third power. Why? Because the universe only goes to 3 dimensions, so x to the 4th was nonsense in classic geometry.
I am not saying this to brag. I am saying this to make the point that I have some bona fides to justify what I am about to say:
Algebra is a dumb course.
It survives only by unthinking habit. It cannot be justified intellectually as a subject, really. It is just a set of tools, not an intellectual discipline with larger meaning and an ongoing scholarship.
Worse, it is an insidious dumb course because everyone must take it, and many people fail it – in part, because it is so dumb.
DO NOT MISUNDERSTAND ME! I said Algebra is a dumb course. I did not say the content called algebra is not worth learning. The distinction is critical.
Algebra, as we teach it, is a death march through endless disconnected technical tools and tips, out of context. It would be like signing up for carpentry and spending an entire year being taught all the tools that have ever existed in a toolbox, and being quizzed on their names – but without ever experiencing what you can craft with such tools or how to decide which tools to use when in the face of a design problem.
Algebra is thus like bad grammar teaching from yesteryear. Algebra I remains today much as it was when I took it in 1964 when I also had to slog through a year of Warriner’s Grammar, as I have found by being in many math classes in the last two years. Go over the home work, learn a new out of context tool like Systems of Inequalities, do some practice problems, quiz, repeat ad nauseum. The course has no big ideas, no direction, no purpose. And when was the last time you had to graph inequalities? (Pre-calc was worse, and I taught it: logarithms and other stuff made completely obsolete by graphing calculators.)
In fact, the course gets many things totally backwards intellectually – like graphing: why wouldn’t you graph to actually learn something that only graphing real-world messy data can reveal? Why wouldn’t you do lots of ‘best fit’ exercises with linear and non-linear data, for example, to begin to appreciate that graphing can help you find something of value? As it stands now in all algebra courses, it is the opposite: there is no reason to graph; you just have to learn set pieces with de-contextualized data already pre-made. Instead of finding cool patterns you merely learn connect-the-dots techniques for graphing.
Paul Lockhart brilliantly satirizes this routine-driven aspect of school math in A Mathematician’s Lament by imagining art taught as algebra is taught:
I was surprised to find myself in a regular school classroom— no easels, no tubes of paint. “Oh we don’t actually apply paint until high school,” I was told by the students. “In seventh grade we mostly study colors and applicators.” They showed me a worksheet. On one side were swatches of color with blank spaces next to them. They were told to write in the names. …
After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that.”…
“I see. And when do students get to paint freely, on a blank canvas?”
“You sound like one of my professors! They were always going on about expressing yourself and your feelings and things like that—really way-out-there stuff. I’ve got a degree in Painting myself, but I’ve never really worked much with blank canvasses. I just use the Paint-by-Numbers kits supplied by the school board.”
Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.
Algebra courses thus still make the same epistemological and pedagogical mistake that my grammar work once a week did 50 years ago: assuming that years of learning bits, out of performance context, is needed before you get to do the real stuff. Dan Meyer, along with others, has been debunking this idea for years on his great blog: here is an example of how problems, not tools, can drive a proper course.
We no longer make this spend-years-learning-inert-parts-out-of-context mistake in English. Even 1st graders learn to ‘write’ ideas right away. We appropriately do not now ask kids to first endlessly parse sentences, for example. So, why is algebra still doing the equivalent, with our official blessing? Learning bits of algebra out of context doesn’t make you a better mathematical thinker or problem solver – the supposed goals of math courses – any more than merely trudging through brush lessons in art makes you a better painter or going through Warriner’s, page after page, can make you, by itself, a better writer.
Again, I am not saying that the tools of algebra are useless. I am saying that all courses called ‘algebra’ are really badly designed. By divorcing content from use, by conflating exercises with problems, and by making it a tour of isolated tricks with no overall direction, we ensure that it is needlessly boring and abstract in the bad sense. Having year-long required courses called ‘algebra’ is as sterile and intellectually uninteresting as a required year-long course in C++ that never lets you actually program a computer to do anything.
To study “algebra” instead of real and interesting problems that require algebra is the nub of the issue. (Even in Greek you at least got to read great texts in the original in the end.) Read my interview with stellar mathematician (and former student!) Steve Strogatz on how unprepared HS kids are for genuine problem finding and solving in college math. And here was his NY Times column on algebra as part of his series on math.
Lets look at the matter as intellectuals. Here is a thought experiment: can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of intellectual priorities for the course but that have rich connections to other fields? Doubt it. Because algebra courses, as designed, have no big ideas, as taught, just a list of topics. Look at any textbook: each chapter is just a new tool. There is no throughline to the course nor are their priority ideas that recur and go deeper, by design. In fact, no problems ever require work from many chapters simultaneously, just learning and being quizzed on each topic – a telling sign.
Here’s a simple test, if you doubt this point, to give to algebra students in order to see if they have any understanding of what they have learned:
- What can algebra do that arithmetic cannot do or does very inefficiently?
- Is the order of operations a matter of core truth or convention? How does that compare with the Associative Property? What is and isn’t arbitrary in algebra?
- Why, mathematically speaking, are imaginary numbers and the inability to divide by zero wise premises?
- What is an equation that states the proper relationship between feet and yards? (60-80% of students will wrongly write: 3F = Y, showing that they have failed to understand the difference between English and algebra)
Think the first question is esoteric? A variant of it was asked on the NY State Regents Algebra test a “few” years back:
Even the widely-praised problem-based program in math at Exeter – including by me, here – errs on the side of asking narrowly-focused questions. Most problems are pretty small bore and there is little explicit attention to overall understanding.
Pernicious gate-keeper: algebra is the Greek of the 21st century. Worse, Algebra is a nasty gate-keeper course with no justification for it playing that role, just as ancient languages once were. Did you know that to get into good colleges not long ago you had to pass a Greek exam? In 1900 to be considered ‘educated’ and to be able to go to Harvard and such, you had to know Greek and take a test in it as part of the entrance exam. (Recall, each college had its own tests: the SAT hadn’t been invented yet). But who now thinks this is a sound idea – other than as a very crude sorting device?
Ah, that’s algebra’s nasty role now.
It’s pernicious and indefensible to fail students and deny a diploma over failure to master such a sterile course. I followed with interest the massive negative response that Andrew Hacker got to his NY Times essay entitled Is Algebra Necessary? Alas, he made the mistake of not sufficiently differentiating between the value of the content and the value of the course as traditionally designed and taught. But many of his points were spot on and overlooked by critics who thought he was calling for lowered standards and anti-intellectualism – especially the importance of access and respect of diversity of career directions:
Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry.
California’s two university systems, for instance, consider applications only from students who have taken three years of mathematics and in that way exclude many applicants who might excel in fields like art or history. Community college students face an equally prohibitive mathematics wall. A study of two-year schools found that fewer than a quarter of their entrants passed the algebra classes they were required to take.
“There are students taking these courses three, four, five times,” says Barbara Bonham of Appalachian State University. While some ultimately pass, she adds, “many drop out.”
Another dropout statistic should cause equal chagrin. Of all who embark on higher education, only 58 percent end up with bachelor’s degrees. The main impediment to graduation: freshman math. The City University of New York, where I have taught since 1971, found that 57 percent of its students didn’t pass its mandated algebra course. The depressing conclusion of a faculty report: “failing math at all levels affects retention more than any other academic factor.” A national sample of transcripts found mathematics had twice as many F’s and D’s compared as other subjects.
Hacker and I worked (separately) on a project entitled Quantitative Literacy, an MAA intended antidote to many of these problems. The phrase “quantitative literacy” tells you much of what you need to know about a better direction for math programs.
Please do not write in saying how important algebra is for learning high-level sciences or how wonderful it was for you personally. I know it works for those who like it (me included) and we know it is needed for high-level science work. But that addresses the needs of only about 20% of the school population. Our student surveys consistently rank HS math as their least favorite course, by a wide margin – though 22% list it as their favorite. 46% don’t merely like it least: they hate it, and their words are telling: “makes me feel stupid” and “seems utterly pointless.” You can download here sample student responses as to why math is their least favorite subject here: Math HS least favorite responses.
So, the fact that some people need it and like it doesn’t justify making every American child learn it in such a dessicated and demeaning way (and fail to receive a diploma if they don’t learn it to boot). I can make a better case that learning logic and Boolean algebra is a far better pre-requisite for both professional and adult life. Even so, I would never require it of everyone. Why can’t your math requirement in high school be met by statistics and probability? Better yet, why can’t students choose math electives in high school, just as they do in college?
And please do not write in saying that this criticism of algebra will lower standards. I am happy to make the stats. course very rigorous. I am also happy to require a course in philosophy and logic as a substitute for the critical thinking that you may (falsely) claim that algebra develops. I taught philosophy at the high school level for over a decade and truly believe it deserves to be a graduation requirement more than algebra, just as it is in France and in all IB high schools today.
No, algebra hangs on like any other bad habit in schools that we then rationalize, like corporal punishment. It’s time we faced up to the fact that the course called algebra is intellectually and pedagogically bankrupt. In my next column I will propose a completely different way to embed algebraic (and geometric) tools into far better – more engaging, meaningful, and intellectual defensible – courses.
PS: Since the post, the Publisher’s Criteria for HS Math and the Common Core came out. This quote could not make the matter clearer:
“Fragmenting the Standards into individual standards, or individual bits of standards … produces a sum of parts that is decidedly less than the whole” (Appendix from the K-8 Publishers’ Criteria). Breaking down standards poses a threat to the focus and coherence of the Standards. It is sometimes helpful or necessary to isolate a part of a compound standard for instruction or assessment, but not always, and not at the expense of the Standards as a whole. A drive to break the Standards down into ‘microstandards’ risks making the checklist mentality even worse than it is today. Microstandards would also make it easier for microtasks and microlessons to drive out extended tasks and deep learning. Finally, microstandards could allow for micromanagement: Picture teachers and students being held accountable for ever more discrete performances. If it is bad today when principals force teachers to write the standard of the day on the board, think of how it would be if every single standard turns into three, six, or a dozen or more microstandards. If the Standards are like a tree, then microstandards are like twigs. You can’t build a tree out of twigs, but you can use twigs as kindling to burn down a tree.”
* from Wikipedia: A whipping boy was a young boy who was assigned to a young prince and was punished when the prince misbehaved or fell behind in his schooling. Whipping boys were established in the English court during the monarchies of the 15th century and 16th centuries. They were created because of the idea of the divine right of kings, which stated that kings were appointed by God, and implied that no one but the king was worthy of punishing the king’s son. Since the king was rarely around to punish his son when necessary, tutors to the young prince found it extremely difficult to enforce rules or learning.
PS: Article published after this post on just how unneeded algebra is: