Andrew Hacker is back, now flacking the idea that algebra should not be required of high school graduates.1 I wrote about general algebra-requirement criticisms in 2006. But to Hacker’s argument: he proposes “citizen statistics,” to which I (and many others) respond, how you can teach such a course without algebra? “Effect size” and “meta-analysis” should both be key concepts in such a course, but you don’t get them without understanding what a standard deviation is.
When I read such arguments, I generally keep a few questions in the back of my mind:
- Is this an argument about what other people’s children should be exposed to? Often, arguments that propose limiting access to college or more challenging courses in high school are made by people with advanced degrees whose own children or nephews and nieces took high school math through calculus (or the equivalent for other subjects).
- Is this an argument about the subject as it should and can be taught or the subject as the author experienced it? If you had an awful algebra teacher, you might think that all algebra courses are horribly difficult. The same is true for any subject. The solution to bad teaching of an important subject is better teaching of the same subject, not the elimination of the subject.
- Is this an argument about the (ir)relevance of a topic that could likewise be applied to any academic subject? If this were an argument about geometry as commonly taught in high school (i.e., proof structures), I might be more sympathetic (though I would argue that again, the better path is to fix the subject). The argument that algebra is irrelevant to life is an argument that a significant portion of adults should not be able to analyze policy topics such as tax rates, fiscal and monetary policy, medical treatments, education policy, crime rates, or any public issue where empirical evidence, statistical relationships, and fundamental issues are much more easily grasped from an algebraic (i.e., general) understanding.
I lean strongly in favor of requiring algebra, but I am aware of the difficulty some students have with it. I may be the opposite of algebra-requirement critics in my personal life. I was extraordinarily lucky in having a string of good math teachers, with two exceptions, and that algebra was fairly easy for me with a combination of some earlier experiences in school and a good teacher. And while I am an historian, I continue to use my algebra skills in many ways.
Notes
- He is better known as the coauthor of a book critical of the value of college. On to algebra. [↩]
First comment –
I would be in favor of his idea under one condition: That the students and their parents both be required to sign a contract stating that they understand this decision will prevent them from getting a job as ….
The long list would start with the professions highest on a typical middle school wish list that depend on Real Math, like doctor, nurse, engineer, head of a big company, etc.
That might have the effect of generating motivation and support for better teaching of math.
Second comment:
They “commonly” teach proof structures in a typical (not honors) HS geometry class in your school system? Consider your children lucky, since my experience is that most HS grads have taken a pseudo-geometry class and know nothing about methods of proof. It drives the math faculty nuts because they can’t believe students in calculus don’t even know how to calculate the volume of cylinder, let alone the difference between equals and implies.
I read that article, and you absolutely know he is full of crap when he says Finland is so much better because of the student’s perseverance. They are better because they are older when they start school and have teachers who are compensated for having solid math skills. The problem in the US is that the kids don’t learn fractions in elementary school, and it is all downhill from there once kids who don’t know fractions become K-5 teachers who don’t know fractions and actively dislike math.
You are quite correct about statistics depending on basic algebra (that is, the use of symbols within otherwise simple arithmetic expressions) of the sort required for HS graduation. You can’t do statistics without equations that contain symbols, and the kids who enter college needing remediation are often ones who actually panic when they see the symbol x.
Explaining how a proof works is very different from asking students to construct proofs. As one of the few history Ph.D.s who took a real analysis course, I’ve had both strong and weak teachers in courses with expectations of proof construction. The weaker teachers I had were the math-course equivalents of illusionists who say, “You saw me perform this card trick, right? Make your own!”
Teaching geometry as a proof-oriented course is worshiping Euclid. There are alternatives.
I don’t see how you got that from my comment, which bemoaned the fact that they have never constructed a proof. (I have no idea if they have seen one, but there are indications from my math colleagues that they lack that experience as well.) It isn’t about worshiping Euclid, it is about learning how to construct a logical argument by actually doing what Euclid did. This lack also contributes to their inability to figure out what the area and volume of a cylinder might be when (not if) they can’t remember it or never learned it.
Yes, there are alternatives, but the don’t do those either.
If wanting geometry to be proof-oriented isn’t worshipping Euclid, shouldn’t we embed proof construction in all math courses or at least conduct research on where it is most effective to teach proofs? I’m all in favor of teaching logical reasoning, but it’s an empirical question whether it’s easier to teach proofs through geometry as you propose or through some basic number theory as ways to teach proof by induction or proof by contradiction.
I will stake out a good Popperian stance and assert my strong supposition that if done well, research will show that geometry is the least effective place in the K-12 curriculum to teach proof construction. The world is now welcome to demonstrate that I am dead wrong.
You hit this one out the park, too. Especially your comment about other people’s children. That should sting.
Sherman,
After I drafted my own reply to Hacker’s piece, Robert P directed me to yours since our views are quite congruent.
Scanning the Higher Ed? website, I saw Hacker’s partner and co-author Claudia Dreyfus states that she failed geometry four times in high school (“a state record”), in her ‘official’ bio.
Personal attacks have no value in a reply, especially when I consider that I barely squeaked out of college calculus with a ‘c’. But I tend not to tout my calc grade as an ‘accomplishment’ and have tried since then to remedy my knowledge gaps. Dreyfus plays her failure differently.
That said, I suspect that it’s hard to avoid one’s worldview being influenced by the experiences of our loved ones. Aesop’s fable of the fox and the grapes came to mind – things that are too hard to reach probably weren’t worth grasping for. And Dreyfus certainly seems to have lived a fulfilled life, despite her geometry struggles.
ML
You all are obviously talking about abstract reasoning here, and do not appreciate the cognitive scaling that Ruby Payne finds along the SES continuum. Her critics refer to this as the “deficit” model.
I have recently become a big fan of Ruby Payne, and you can click on some of her materials here.
Search on “Payne” — links at the bottom 2/3rd of the page.
http://wvde.state.wv.us/institutional/PD/filecabinet.html
Glen,
Do not trust Payne’s deficit orientation. Period. For a range of accessible criticisms of her for-profit consulting work and generally research-free claims, see Larry Ferlazzo’s list of readings on Payne.
Starting with Adam Smith (earlier?), and Marx especially, attention has been paid to effects of division of labor and CLASS on consciousness. Deny Payne, and you deny these as well.
So, where is the research on the social cognition of poverty? Is Ruby Payne left to conquer this field all by herself? As ill equipped as she is?
This is where I come into it — from socially-constructed cognition. And the differences are “real” to me — I work with these clients, and it helps me to navigate their reality.
(This is why your take on algebra is so funny. It is so “rich” people.)
Not “blaming the victim” here (any more than it is blaming the “rich” for being the way that they are) — critics are just blaming the messenger!
And here (because I am also a big fan of Saul Alinsky) from Larry’s website — “Saul Alinsky, the father of modern-day community organizing and the founder of the organization that I worked for during my organizing career, once said, “The price of criticism is a constructive alternative.”
There needs to be an epistemology of poverty, and of wealth.
There is none!
“I believe that those of us who are critics of Ruby Payne need to do a far better job of offering constructive alternatives that teachers can use today and tomorrow — right in their classroom — if we want more to see the fallacies of Payne’s approach.” Agreed.
Ruby Payne isn’t the first, second, or thousandth person to look at class. That’s part of the problem with the worshipping of her consulting gigs; districts too often buy into her shtick as a package to “solve” problems of inequality as if no one else has looked at class, cultural capital, and whatnot.
It’s facile to say that education experts only want other people’s children to forego Algebra. I would have like my youngest daughter to forego it, in favor of a better grounding in K-8 math — which might have led her eventually to Algebra, at least Algebra 1. Instead,she was given Algebra 1, Geometry, and Algebra 2, none of which she mastered even remotely. What a crock. Give kids, all kids, Algebra when they demonstrate readiness for it, whenever that might be. And push harder on the low-income kids with extra tutoring, to make sure they’re not slacking off just because they don’t get much push from home.
Dear E.B.,
You’re absolutely right that it would be facile to assume that everyone who is skeptical of an algebra requirement is doing so for other people’s children only. That’s not what I said, but it is a question I ask when I read this and similar arguments by those with doctorates, and I think it’s a fair question.
A little late on this thread… But on the “is geometry the least effective area of math to teach proof construction” question, I would argue that algebra is less effective because its more abstract. When you define a point or a line in geometry, you’re defining something that you can approximately visualize — less so with the definitions needed to construct algebra. I think you need to understand the idea of mathematical proof in order proofs in algebra not to seem like strange semantic games.
Plus, the the fact that modifications of Euclid’s 5th postulate that lead to coherent geometries that describe non-planar surfaces is one of the clearest demonstrations of what mathematics is and how it relates the physical world.