Category Archives: Critical thinking

April 17, 2010, 6:15 am

Active learning is essential, not optional, for STEM students

This article (1.2 MB, PDF)  by three computer science professors at Miami University (Ohio) is an excellent overview of the concept of the inverted classroom and why it could be the future of all classrooms given the techno-centric nature of Millenials. (I will not say “digital natives”.) The article focuses on using inverted classroom models in software engineering courses. This quote seemed particularly important:

Software engineering is, at its essence, an applied discipline that involves interaction with customers, collaboration with globally distributed developers, and hands-on production of software artifacts. The education of future software engineers is, by necessity, an endeavor that requires students to be active learners. That is, students must gain experience, not in isolation, but in the presence of other learners and under the mentorship of instructors and practitioners. …

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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…

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March 23, 2010, 1:25 pm

Programming, lectures, and the inverted classroom

Punch card from a typical Fortran program.
Image via Wikipedia

We started programming in the MATLAB course a couple of weeks ago. It’s been… interesting. Keep in mind that 75% of the students in the class have never written a program of any sort before; half the class rates themselves below a 6 out of 10 in “comfort level” in using computers at all. As with everything else in this course, the audience is everything.

I started this three-week unit last week with a minilecture on FOR loops. But wait, you say: I thought you were using an inverted classroom model for the MATLAB course, where students are assigned reading and viewing tasks outside of class, accompanied by homework assignments designed to help them extract the relevant information and then do simple applications of what they’ve learned. Well, yes, that’s been the plan, and the practice up until now.

But I decided to go with a minilecture/activity model for the…

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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…

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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…

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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…

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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…

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October 16, 2007, 9:32 pm

Fear, courage, and place in problem solving

Sorry for the slowdown in posting. It’s been tremendously busy here lately with hosting our annual high school math competition this past weekend and then digging out from midterms.

Today in Modern Algebra, we continued working on proving a theorem that says that if \(a\) is a group element and the order of \(a\) is \(n\), then \(a^i = a^j\) if and only if \(i \equiv j \ \mathrm{mod} \ n\). In fact, this was the third day we’d spent on this theorem. So far, we had written down the hypothesis and several equivalent forms of the conclusion and I had asked the students what they should do next. Silence. More silence. Finally, I told them to pair off, and please exit the room. Find a quiet spot somewhere else in the building and tell me where you’ll be. Work on the proof for ten minutes and then come back.

As I wandered around from pair to pair I was very surprised to…

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October 9, 2007, 9:33 am

This isn't a political blog, but…

…comments like this one from Anderson Cooper’s blog just make me wonder what people are thinking. This is in answer to the question, “Have you decided for whom you are going to vote in the 2008 presidential election?”

Hillary has won my vote! She is the smartest woman I know and I think it is about time we get a woman running this country! Men have been in charge far too long and look at the mess our country is in. If the country seriously wants change I think Hillary is the only one that can make it happen.

It’s not so much the choice of candidate here as the reasoning. I understand that being smart is a good trait to have if you are going to be President, but honestly, what does the level of intelligence have to do with it? Should we simply administer an IQ test to each candidate and pick them that way? And what on earth does gender have to do with it? Should we just pick a candidate…

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July 18, 2007, 10:07 am

More about funding and achievement in public schools

D-Ed Reckoning has some of the stats I referred to in this post which discredit the supposed relationship between public school funding and student achievement. In particular, check out this scatterplot for the 501 school districts in Pennsylvania:
 Blogger 7617 77 1600 Ses2

An R2 value of 0.01? Ouch. And, as Ken notes, even at the highest levels of funding you don’t have a consistent trend of high student achievement but rather two trends, one going up and one plummeting down. So, one last time: Higher funding for public schools does not correlate with higher student achievement, despite what our intuition (and various advocacy groups) would suggest.