# Category Archives: Critical thinking

June 25, 2013, 12:04 pm

# What math topic do engineering faculty rate as the most important?

I’m at the American Society for Engineering Education Annual Conference right now through Thursday, not presenting this time but keeping the plates spinning as Mathematics Division program chair. This morning’s technical session featured a very interesting talk from Kathy Harper of the Ohio State University. Kathy’s talk, “First Steps in Strengthening the Connections Between Mathematics and Engineering”, was representative of all the talks in this session, but hers focused on a particular set of interesting data: What engineering faculty perceive as the most important mathematics topics for their areas, and the level of competence at which they perceive students to be functioning in those topics.

In Kathy’s study, 77 engineering faculty at OSU responded to a survey that asked them to rate the importance of various mathematical topics on a 5-point scale, with 5 being the…

July 3, 2012, 9:08 am

# The trouble with Khan Academy

At some point around the beginning of February 2012, David Coffey — a co-worker of mine in the math department at Grand Valley State University and my faculty mentor during my first year — mentioned something to me in our weekly mentoring meetings. We were talking about screencasting and the flipped classroom concept, and the conversation got around to Khan Academy. Being a screencaster and flipped classroom person myself, we’d talked about making screencasts more pedagogically sound many times in the past.

That particular day, Dave mentioned this idea about projecting a Khan Academy video onto the screen in a classroom and having three of us sit in front of it, offering snarky critiques — but with a serious mathematical and pedagogical focus — in the style of Mystery Science Theater 3000. I told him to sign me up to help, but I got too busy to stay in the loop with it.

It…

February 23, 2012, 6:48 am

# What Happens if We Just Ask Questions?

Someone asked me recently what was the one thing that’s changed the most about my teaching over the last 10 years. My response was that I’m a lot more likely now than I was in 2002 to organize my classes around asking and answering questions rather than covering material. Here’s one reason why.

The weekly Mathematica labs that we have in my Calculus 3 class are set up so that some background material (usually a combination of math concepts and new Mathematica commands) is presented in the lab handout followed by some situations centered around questions, the answers to which are likely to involve Calculus 3 and Mathematica. I said likely, not inevitably. There is no rule that says students must use Calculus 3 to answer the question. The only rules are: (1) the entire solution has to be done in a Mathematica notebook, and (2) the solutions have to be clear, convincing, and…

May 18, 2011, 9:53 am

# The "golden moment"

We’re in final exams week right now, and last night students in the MATLAB course took their exam. It included some essay questions asking for their favorite elements of the course and things that might be improved in the course. I loved what one of my students had to say about the assignment in the course he found to be the most interesting, so I’ve gotten permission from him to share it. The lab problem he’s referring to was to write a MATLAB program to implement the bisection method for polynomials.

It is really hard to decide which project I found most interesting; there are quite a few of them. If I had to choose just one though, I would probably have to say the lab set for April 6. I was having a really hard time getting the program to work, I spent a while tweaking it this way and that way. But when you’re making a program that does not work yet, there is this sort of golden…

January 11, 2011, 10:14 pm

# The inverted classroom and student self-image

Image via Wikipedia

This week I’ve been immersed in the inverted classroom idea. First, I gave this talk about an inverted linear algebra classroom at the Joint Meetings in New Orleans and had a number of really good conversations afterwards about it. Then, this really nice writeup of an interview I gave for MIT News came out, highlighting the relationship between my MATLAB course and the MIT OpenCourseware Project. And this week, I’ve been planning out the second iteration of that MATLAB course that’s starting in a few weeks, hopefully with the benefit of a year’s worth of experience and reflection on using the inverted classroom to teach technical computing to novices.

One thing that I didn’t talk much about at the Joint Meetings or in the MIT interview was perhaps the most prominent thing about using the inverted …

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

November 29, 2010, 9:00 am

# What correlates with problem solving skill?

About a year ago, I started partitioning up my Calculus tests into three sections: Concepts, Mechanics, and Problem Solving. The point values for each are 25, 25, and 50 respectively. The Concepts items are intended to be ones where no calculations are to be performed; instead students answer questions, interpret meanings of results, and draw conclusions based only on graphs, tables, or verbal descriptions. The Mechanics items are just straight-up calculations with no context, like “take the derivative of $$y = \sqrt{x^2 + 1}$$”. The Problem-Solving items are a mix of conceptual and mechanical tasks and can be either instances of things the students have seen before (e.g. optimzation or related rates problems) or some novel situation that is related to, but not identical to, the things they’ve done on homework and so on.

I did this to stress to students that the main goal of taking …

August 8, 2010, 12:47 pm

# Calculus and conceptual frameworks

Image via http://www.flickr.com/photos/loopzilla/

I was having a conversation recently with a colleague who might be teaching a section of our intro programming course this fall. In sharing my experiences about teaching programming from the MATLAB course, I mentioned that the thing that is really hard about teaching programming is that students often lack a conceptual framework for what they’re learning. That is, they lack a mental structure into which they can place the topics and concepts they’re learning and then see those ideas in their proper place and relationship to each other. Expert learners — like some students who are taking an intro programming course but have been coding since they were 6 years old — have this framework, and the course is a breeze. Others, possibly a large majority of…

May 12, 2010, 12:52 pm

# Boxplots: Curiouser and curiouser

The calculus class took their third (and last) hour-long assessment yesterday. In the spirit of data analytics ala the previous post here, I made boxplots for the different sections of the test (Conceptual Knowledge (CK), Computation (C), and Problem Solving (PS)) as well as the overall scores. Here are the boxplots for this assessment — put side-by-side with the boxplots for the same sections on the previous assessments. “A2″ and “A3″ mean Assessments 2 and 3.

Obviously there is still a great deal of improvement to be had here — the fact that the class average is still below passing remains unacceptable to me — but there have been some definite gains, particularly in the conceptual knowledge department.

What changed between Assessment 2 and Assessment 3? At least three things:

• The content changed. Assessment 2 was over derivative rules and applications; Assessment 3 covered…

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 …

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