# Category Archives: Problem Solving

January 4, 2010, 7:00 am

# Wolfram|Alpha as a self-verification tool

Last week, I wrote about structuring class time to get students to self-verify their work. This means using tools, experiences, other people, and their own intelligence to gauge the validity of a solution or answer without uncritical reference an external authority — and being deliberate about it while teaching, resisting the urge to answer the many “Is this right?” questions that students will ask.

Among the many tools available to students for this purpose is Wolfram|Alpha, which has been blogged about extensively. (See also my YouTube video, “Wolfram|Alpha for Calculus Students”.) W|A’s ability to accept natural-language queries for calculations and other information and produce multiple representations of all information it has that is related to the query — and the fact that it’s free and readily accessible on the web — makes it perhaps the most powerful self-verification tool…

December 31, 2009, 5:30 am

# How to convert a "backwards" proof into a "forwards" proof

Dave Richeson at Division By Zero wrote recently about a “proof technique” for proving equalities or inequalities that is far too common: Starting with the equality to be proven and working backwards to end at a true statement. This is a technique that is almost a valid way to prove things, but it contains — and engenders — serious flaws in logic and the concept of proof that can really get students into trouble later on.

I left a comment there that spells out my  feelings about why this technique is bad. What I wanted to focus on here is something I also mentioned in the comments, which was that it’s so easy to take a “backwards” proof and turn it into a “forwards” one that there’s no reason not to do it.

Take the following problem: Prove that, for all natural numbers $$n$$,

$$1 + 2 + 2^2 + \cdots + 2^n = 2^{n+1} – 1$$

This is a standard exercise using mathematical…

December 30, 2009, 10:19 am

# Resisting the urge to verify

When I am having students work on something, whether it’s homework or something done in class, I’ll get a stream of questions that are variations on:

• Is this right?
• Am I on the right track?
• Can you tell me if I am doing this correctly?

And so on. They want verification. This is perfectly natural and, to some extent, conducive to learning. But I think that we math teachers acquiesce to these kinds of requests far too often, and we continue to verify when we ought to be teaching students how to self-verify.

In the early stages of learning a concept, students need what the machine learning people call training data. They need inputs paired with correct outputs. When asked to calculate the derivative of $$5x^4$$, students need to know, having done what they feel is correct work, that the answer is $$20x^3$$. This heads off any major misconception in the formation of the concept…

October 18, 2008, 7:48 pm

# It's Tricki

Tim Gowers gives a lengthy report here on the development of the Mathematical Tricks Wiki, which he is now calling the Tricki. The Tricki will be a wiki/database of mathematical problem solving techniques that will, if development proceeds, eventually be something like an expert system that mimics how a human mathematician’s brain works when solving problems. In Gowers’ words:

The main content of the Tricki will be a (large, if all goes according to plan) body of articles about methods for solving mathematical problems. Associated with these articles will be many qualities that will vary substantially from article to article. For example, some will be about very general problem-solving tips such as, “If you can’t solve the problem, then try to invent an easier problem that sheds light on it,” whereas others will be much more specific tips such as, “If you want to solve a linear…

October 15, 2008, 8:54 pm

# Technology in proofs?

We interrupt this blogging hiatus to throw out a question that came up while I was grading today. The item being graded was a homework set in the intro-to-proof course that I teach. One paper brought up two instances of the same issue.

• The student was writing a proof that hinged on arguing that both sin(t) and cos(t) are positive on the interval 0 < t < π/2. The “normal” way to argue this is just to appeal to the unit circle and note that in this interval, you’re remaining in the first quadrant and so both sin(t) and cos(t) are positive. But what the student did was to draw graphs of sin(t) and cos(t) in Maple, using the plot options to restrict the domain; the student then just said something to the effect of “The graph shows that both sin(t) and cos(t) are positive.”
• Another proof was of a proposition claiming that there cannot exist three consecutive natural numbers such that the …

September 1, 2008, 8:28 pm

# What part of (f(x)-f(a))/(x-a) don't you understand?

Why is the concept of the difference quotient so hard for beginning calculus students to handle? The idea is not as hard as some other concepts at this level that students have fewer problems with. You start with a function f and a point a. You are asked to write, and then simplify completely, the fraction

$$\frac{f(x) – f(a)}{x-a}$$ or $$\frac{f(a+h)-f(a)}{h}$$

This involves four clearly-defined steps. (1) Compute all the function values in the numerator. (2) Perform the subtraction between the two objects in the numerator and simplify. (3) Factor the result out completely, and (4) see if you can find a common factor to cancel. And there’s a step (5): Since you know that every time you’ve done or seen a problem like this, there’s a factor/cancel step at the end, you know you screwed up if there isn’t one.

But somehow, the fact that this is a totally algorithmic, almost…

August 13, 2008, 5:15 pm

# What are some fatal errors in proofs?

The video post from the other day about handling ungraded homework assignments went so well that I thought I’d let you all have another crack and designing my courses for me! This time, I have a question about really bad mistakes that can be made in a proof.

One correction to the video — the rubric I am developing for proof grading gives scores of 0, 2, 4, 6, 8, or 10. A “0″ is a proof that simply isn’t handed in at all. And any proof that shows serious effort and a modicum of correctness will get at least a 4. I am reserving the grade of “2″ for proofs that commit any of the “fatal errors” I describe (and solicit) in the video.

March 12, 2008, 8:39 pm

# What is creativity?

Phil Wilson has an interesting article at his place today, reporting on a lecture by cognitive scientist Margaret Boden on the nature of creativity. Here’s a sample of Phil’s notes:

She [Boden] began by assuring us that creativity is not magic or divine, neither is it a special faculty possessed by an elite, but rather an aspect of general intelligence. Most importantly, an understanding of creativity is not beyond the reach of the scientific process. What, then, is creativity? Her answer: coming up with ideas which are new, surprising, and valuable.

Phil then goes on to list the meanings of those latter three terms in the context of Boden’s ideas on creativity. It’s very interesting stuff and highly useful for teachers, and you should go read the whole thing.

One part of Phil’s post/Boden’s talk that struck me in particular was this bit about what Boden calls “exploratory creativity”…

March 10, 2008, 8:16 pm

# Suggestions for an adult math convert?

I got an email this afternoon from a reader who is interested in learning mathematics — as an adult, post-college. The reader has an advanced degree in a humanities discipline and never studied mathematics, but recently he’s become interested in learning and is looking for a place to start.

I recommended The Mathematical Experience by Davis and Hirsch, How to Solve It by Polya, and any good college-level textbook in geometry (like Greenberg, or for a humanities person perhaps Henderson). I felt like these three books give an ample and accessible start at — respectively — the big picture and history of the discipline, the methodology of mathematicians, and a first step into actual mathematical content.

But what I thought this was an interesting question, and I wonder if the other readers out there would have similar suggestions for books, articles, movies or documentaries… anything…

December 13, 2007, 1:51 pm

# A resolution about group work

One of the things I have learned this semester (which is now officially over, having turned in my last batch of grades this morning) is the following lesson which I am convinced I must implement immediately: Group work has been playing far too great of a role in my student’s grades. From this point forward, assignments which could conceivably be done in groups — not just those that are designated for group work — will count for no more than 10-15% of the grade in my courses.

I like collaborative learning. I think, in fact, that working with other people on math can be not only a highly effective way of doing so but also carries with it a powerful pro-math socialization effect. The best personal friendships that I had during my college + grad school years were those that I formed with my classmates in my various math classes, as we struggled through material that, to us at the time,…

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