July 23, 2013, 8:00 am

By Robert Talbert

Yesterday I was doing some literature review for an article I’m writing about my inverted transition-to-proof class, and I got around to reading a paper by Guershon Harel and Larry Sowder¹ about student conceptions of proof. Early in the paper, the authors wrote the following passage about mathematical proof to set up their main research questions. This totally stopped me in my tracks, for reasons I’ll explain below. All emphases are in the original.

An observation can be conceived of by the individual as either a *conjecture* or as a *fact*.

**A conjecture is an observation made by a person who has doubts about its truth. A person’s observation ceases to be a conjecture and becomes a fact in her or his view once the person becomes certain of its truth.**

This is the basis for our definition of the process of proving:

**By “proving” we mean the process employed by an…**

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October 26, 2010, 12:00 pm

By Robert Talbert

Here’s another question about the same enVisionMATH worksheet we first met yesterday. Take a look at this section, and think about the mental processes you’d use to answer each of these problems:

Got it? Now, let me zoom out a little and show you a part of the worksheet you didn’t see before:

If you’re late to the party and don’t know what’s meant by “near doubles” and the arithmetic rules that enVisionMATH attaches to near doubles, read this post first. Questions:

- Now that you know that these are supposed to be exercises about near doubles, does that change the mental processes you selected earlier for working the problems?
- Should it?

June 22, 2010, 7:42 pm

By Robert Talbert

Yesterday at the ASEE conference, I attended mostly sessions run by the Liberal Education Division. Today I gravitated toward the Mathematics Division, which is sort of an MAA-within-the-ASEE. In fact, I recognized several faces from past MAA meetings. I would like to say that the outcome of attending these talks has been all positive. Unfortunately it’s not. I should probably explain.

The general impression from the talks I attended is that the discussions, arguments, and crises that the engineering math community is dealing with are exactly the ones that the college mathematics community in general, and the MAA in particular, were having — *in 1995*. Back then, mathematics instructors were asking questions such as:

- Now that there’s relatively inexpensive technology that will do things like plot graphs and take derivatives, what are we supposed to teach now?
- Won’t all that technology…

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February 7, 2008, 9:12 pm

By Robert Talbert

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:

\(\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & \end{array}\)

Now look for a prime that divides the lower-left term…

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February 2, 2008, 4:33 pm

By Robert Talbert

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: …

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October 30, 2007, 2:19 pm

By Robert Talbert

**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 …*

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