A federal panel examining K-8 mathematics education in the USA has made some forthright recommendations, according to this article in the NYT today. Unlike many federal panels, this one has an uncommon amount of common sense in its conclusions. For example, this finding that is striking in the way it refrains from choosing sides in the math wars:

Parents and teachers in school districts across the country have fought passionately over the relative merits of traditional, or teacher-directed, instruction, in which students are told how to solve problems and then are drilled on them, as opposed to reform or child-centered instruction, which emphasizes student exploration and conceptual understanding. The panel said both methods have a role.

“There is no basis in research for favoring teacher-based or student-centered instruction,” said Dr. Larry R. Faulkner, the chairman of the panel, …

Following up on his three posts on classical education yesterday, Gene Veith weighs in on mathematics instruction:

I admit that classical education may be lagging in the math department. The new classical schools are doing little with the Quadrivium, the other four liberal arts (arithmetic, geometry, astronomy, and music). The Trivium, which is being implemented to great effect (grammar, logic, and rhetoric), has to do with mastering language and what you can do with it. The Quadrivium has to do with mathematics (yes, even in the way music was taught).

This, I think, is the new frontier for classical educators. Yes, there is Saxon math, but it seems traditional (which is better than the contemporary), rather than classical, as such.

Prof. Veith ends with a call for ideas about how mathematics instruction would look like in a classical education setting. I left this comment:

One of my linear algebra students is an education major doing student teaching. Today he showed me this method of simplifying radicals which he learned from his supervising teacher. Apparently it’s called the “Illini method”. Googling this term returns nothing math-related, so I think that term was probably invented by his supervisor, who went to college in Illinois.

The procedure goes as follows. Start with a radical to simplify, say \(\sqrt{50}\). Look under the radical and find a prime that divides it, say 5. Then form a two-column array with the original radical in the top-left, the divisor prime in the adjacent row in the right column, and the result you get from dividing the radicand by that prime number in the left column below the radical. In this case, it’s:

Here’s a problem I have with the way most calculus textbooks are written, and therefore by default the way most calculus courses end up being taught. Tell me if I am crazy or missing something.

We teach calculus from a depth-first viewpoint. That means that whenever we encounter a concept, we go as deeply as possible in that concept before moving on to the next one. There are some subjects where this makes sense, but in calculus we have a small number of main ideas that are made out of several concepts, and if we stop to attain maximal depth on every single thing, there’s a good chance that we never arrive at the main idea with any degree of understanding.

The big ideas of calculus — the rate of change (derivative) and accumulated change (integral) — are actually really simple if you consider them simply for what they are and what they were invented to do. Derivatives, for instance: …

Here’s a promotional video for a new math curriculum from Pearson called enVisionMATH. (It must be a sign of the times that grade school math curricula have promotional videos.) Watch carefully.

Four questions about this:

Should it be a requirement of parenthood that you must remember enough 5th grade math to teach it halfway decently to your kids?

Does the smartboard come included with the textbooks?

Did anybody else have the overwhelming urge to yell “Bingo!” after about 2 minutes in?

When will textbook companies stop drawing the conclusion that because kids today like to play video games, talk on cell phones, and listen to MP3 players, that they are therefore learning in a fundamentally different way than anybody else in history?

A. If math were a color, it would be –, because –.

B. If it were a food, it would be –, because –.

C. If it were weather, it would be –, because –.

I’m not sure exactly what the point of an exercise like this is — perhaps the curriculum is just trying very studiously not to get too deep into mathematics itself, thereby teaching math without the social stigma of being very enthusiastic about it. Or maybe the idea is to get kids to see math from a different point of view, as a sort of oblique path through math anxiety.

Either way, it’s the wrong approach. The only way to come to terms with math, conquer math anxiety, and appreciate (and learn) the subject is… to get good at …

Editorial: This is installment #6 in this week’s retrospective series where I’m reposting some classic articles with updated comments. For me, there’s a deep interplay between pedagogy and culture that we don’t make nearly enough of. I’ve posted many articles trying to make the point that many of the problems in education, particularly math education, that we try to treat with curricula or technology or pedagogy are doomed to fail because they are not problems with teaching — they are problems with the culture. In this article, I make that point and go one step further, bringing in a theme from my Calvinist/Lutheran religious beliefs.

Editorial: This is installment #5 in retrospective week. When I announced retrospective week, I said that some of the articles I would be highlighting may not have gotten many comments but started larger conversations — and this is certainly one of them, although the conversation went totally to places I didn’t want it to go.

Read the article for yourself, and you’ll see that it is a reflection on what makes a successful student in an upper-level math course, what education programs often cite as characteristics of successful teachers, how those two sets are often portrayed as mutually exclusive, and why math education majors have to work to possess both sets of characteristics as an integrated whole in order to become great teachers.

But that’s not how a lot of readers took it. In particular, some of the education majors at my college read this article and took it to be a public …

Midterms are coming up in a couple of weeks, and while most of the students in my precalculus class are doing reasonably well, some aren’t. Here are some questions I’ve struggled with every time I teach a freshman class, and maybe some of you out there have suggestions. If so, leave them in the comments.

How do you impress upon students (freshmen) the importance of coming to office hours? I don’t think I’ve had more than six distinct students visit office hours for help all semester long, and I’d consider this an active semester in terms of office hours. The rest go to the Math Study Center, study tables for football or fraternities, etc. but it does no evident good for a lot of them. I think it would do them good to come see me; but how to convince them of this?

How do you convince a student that their purpose for being here, their job, is to be a student? Some of the students don’t …

I am a mathematician and educator with interests in cryptology, computer science, and STEM education. I am affiliated with the Mathematics Department at Grand Valley State University in Allendale, Michigan. The views here are my own and are not necessarily shared by GVSU.

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