Tag Archives: Critical thinking

December 16, 2010, 2:30 pm

A problem with "problems"

I have a bone to pick with problems like the following, which is taken from a major university-level calculus textbook. Read it, and see if you can figure out what I mean.

This is located in the latter one-fourth of a review set for the chapter on integration. Its position in the set suggests it is less routine, less rote than one of the early problems. But what’s wrong with this problem is that it’s not a problem at all. It’s an exercise. The difference between the two is enormous. To risk oversimplifying, in an exercise, the person doing the exercise knows exactly what to do at the very beginning to obtain the information being requested. In a problem, the person doesn’t. What makes an exercise an exercise is its familiarity and congruity with prior exercises. What makes a problem a problem is the lack of these things.

The above is not a problem, it is an exercise. Use the

Read More

April 25, 2010, 1:47 pm

The case of the curious boxplots

I just graded my second hour-long assessment for the Calculus class (yes, I do teach other courses besides MATLAB). I break these assessments up into three sections: Concept Knowledge, where students have to reason from verbal, graphical, or numerical information (24/100 points); Computations, where students do basic context-free symbol-crunching (26/100 points); and Problem Solving, consisting of problems that combine conceptual knowledge and computation (50/100 points). Here’s the Assessment itself. (There was a problem with the very last item — the function doesn’t have an inflection point — but we fixed it and students got extra time because of it.)

Unfortunately the students as a whole did quite poorly. The class average was around a 51%. As has been my practice this semester, I turn to data analysis whenever things go really badly to try and find out what might have happened. I …

Read More

April 4, 2010, 8:30 pm

Turning questions into learning

Attacking Difficult Questions
Image by CarbonNYC via Flickr

The hardest thing about teaching the MATLAB course — or any course — is responding to student questions. Notice I do not say “answering” student questions. Answers are not the issue; I’m no MATLAB genius, but I can answer 95% of student questions on the spot. The real issue is whether I should. If my primary task is to teach students habits of mind that translate into lifelong learning — and I earnestly believe that it is — then answers are not always the best thing for students.

I’ve noticed four types of questions that students tend to ask in the MATLAB course, and these carry over pretty seamlessly to my other courses:

  1. Informational questions that have nothing to do with the problem they’re working on or the material. Example: When are your office hours? When is this lab due? When is the final exam?
  2. Clarifying questions that seek to make sense…

Read More

March 1, 2010, 9:21 pm

MATLAB and critical thinking

My apologies for being a little behind the curve on the MATLAB-course-blogging. It’s been a very interesting last couple of weeks in the class, and there’s a lot to catch up on. The issues being brought up in this course that have to do with general thinking and learning are fascinating, deep, and complicated. It’s almost as if the course is becoming only secondarily a course on MATLAB and primarily a course on critical thinking and lifelong learning in a technological context.

This past week’s lab really brought that to the forefront. The lab was all about working with external data sets, and it involved students going to this web site and looking at this data set (XLS, 33 Kb) about electoral vote counts of the various states in the US (and the District of Columbia). One of the tasks asked students to make a scatterplot of the land area of the states versus their electoral vote count…

Read More

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…

Read More

February 19, 2008, 10:29 am

What is a classical education approach to mathematics?

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:

I…

Read More

February 18, 2008, 2:19 pm

A trifecta on classical education

Gene Veith, one of my favorite religious writers and the proprietor of the terrific Cranach blog (and provost at Patrick Henry College), has three quick posts today on classical education. He touches briefly on teaching content rather than process, and how classical education teaches bothl; on critical thinking; and on learning styles and the teaching of “meaning”. Some clips: 

The key factor in learning is grasping meaning, a concept that evades any of these sensory approaches. (While cultivation of meaning is what classical education is all about.)

and:

More substantive scholars say that being able to think critically requires (again, see below) CONTENT. You have to think ABOUT SOMETHING. Whereas much of the critical thinking curriculum is all process, trying to provoke content-free thinking. (The classical solution: DIALECTIC, featuring questions AND answers, as in that great…

Read More

November 2, 2007, 2:00 pm

Retrospective: Critical thinking, visualization, and physical intuition (10.11.2006)

Editorial: This is the penultimate article in the retrospective series we’ve been doing all week here at CO9s. This one takes us back to 2006 one more time.

One of the things that fascinates me most about teaching math is seeing how people acquire and use problem-solving skills. And one of the things I like to think and write about the most is how people can approach problems in different ways — especially when those ways are not the standard ways of doing so — and why students make various conceptual mistakes when they try.

This article was written after a calculus homework set involving a pretty standard intro problem about the velocity of an arrow shot straight upward on the moon. (Where the **** do we math people get these problem ideas?) I was reading James Gleick’s biography of Richard Feynman at the time and was very keen on how important visualization is in problem solving. I…

Read More