March 25, 2013, 8:00 am
I have a confession to make: At this point in the semester (week 11), there’s a question I get that nearly drives me to despair. That question is:
Can we see more examples in class?
Why does this question bug me so much? It’s not because examples are bad. On the contrary, the research shows (and this is surely backed up by experience) that studying worked examples can be a highly effective strategy for learning a concept. So I ought to be happy to hear it, right?
When people ask this question because they want to study an example, I’m happy. But studying an example and seeing an example are two radically different things. Studying an example means making conscious efforts to examine the example in depth: isolating the main idea or strategy, actively trying out modifications to the objects involved, making connections to previous examples and mathematical results, and – very …
March 8, 2013, 4:30 pm
I’ve been talking a lot with my colleagues about their teaching practices, as part of the NSF grant I’m working on. The inverted classroom (I used to call it the flipped classroom, but I’m going back to “inverted”) has come up a lot as a teaching technique that people have heard a lot about but haven’t tried yet — or are wary of trying. I’ve been wondering about the language being used, namely: Is the inverted classroom really a “teaching technique” at all?
My answer used to be “yes”. When I first started using the inverted classroom idea, I would describe the inverted classroom as “a teaching technique” that involves reversing where information transmission and internalization take place. Later I moved to saying that the inverted classroom refers to “any teaching method” that implements this reversal. Today as I was thinking about this, I think a…
December 31, 2012, 9:01 am
Here’s a piece of a conversation I just had with my 8-year old daughter, who is interested in becoming a teacher when she grows up.
Daughter: Dad, if you want to become a teacher, do you have to take classes?
Me: Yes. You have to take a lot of classes about how to teach and a lot of classes in the subjects you want to teach. You need to be really good at math to teach math, for example.
D: Then do you have to go out and teach in the schools, like Mr. D___ [the young man who student-taught in my daughter's elementary school this year]?
Me: That’s right. You have to take classes and you have to go into the schools and practice.
D: Do you have to practice with the little kids?
Me: That depends on who you want to teach. If you want to become an elementary school teacher you work with elementary school kids. If you want to teach in a middle school, then you work with middle …
November 9, 2012, 7:00 am
Speaking of faculty adopting research-based instructional strategies, Theron Hitchman (who blogs at Circles and Tangents) wonders aloud in the direction of math education researchers: Why didn’t you tell me? That is, referring to research-based instructional strategies that seem to work really well with students,
Why do I stumble on these things only to find that they have been understood for decades? Why didn’t someone knock on my door and tell me I was doing it wrong?
My basic point is this: If you do research on teaching and learning, you owe it to society to share what you know. Scholarly publication doesn’t count. The mathematics education community talking to itself is a necessary condition for sorting out the truth of things, but it is insufficient for educating the public and for changing practice on a large scale.
If you know that the standard lecture-homework-exam …
June 27, 2012, 11:52 am
Slate magazine has been running several articles on education this week, including two today that are of interest. This one by Konstantin Kakaes is worth looking at more closely, if only because it somehow manages to gather almost every wrong idea about technology in education in existence into a single, compact article.
The piece proposes that the effort to increase the use of technology in education “is beginning to do to our educational system what the transformation to industrial agriculture has done to our food system over the past half century: efficiently produce a deluge of cheap, empty calories.” I’m not sure which “effort” Kakaes is referring to, since there is no single push being coordinated from a secret underground bunker that I know of, and some efforts are better-conceived than others. But nevermind.
There are two overriding conceptual errors that drive this article…
May 4, 2012, 4:00 pm
Sorry for the boring title and lack of catchy image, but since my first post about the upcoming six-week Calculus 2 course, I’ve expended all my creativity getting the course put together and getting ready for Monday. In the earlier post I laid down some design ground rules for the course. Here, I’m going to say a little more in detail about what we’ll be doing.
It’s especially important on a highly compressed schedule like ours to use the class meetings themselves to jumpstart the assimilation process and then train students on how to carry that process forward as they go to work on the day’s material in the afternoon and evening. This is always an important goal of class meetings in any course — I’d go as far as to say that this is why we have class meetings at all. But when you cram a 14-week course into 6 weeks, it doesn’t take long for one incorrectly-assimilated concept to…
April 12, 2012, 9:45 am
This article in the Chicago Tribune talks about efforts to make math fun:
In the American drive to boost science and math education, it’s science that has all the kid-friendly sizzle: Robots and roller coasters, foaming chemical reactions, marshmallow air cannons.
Math has… well, numbers.
“America has a cultural problem with math. It’s the subject, more than any other, that we as a country love to hate,” said Glen Whitney, a passionate mathematician who worked for years developing algorithms for hedge funds. “We don’t see it as dynamic. It’s rote and boring and done by dead Greek guys a thousand years ago.”
The article goes on to talk about some efforts to spice up math, including MIT’s Labyrinth tournament, DimensionU‘s celebrity-driven “DU the Math”…
April 10, 2012, 8:00 am
In peer instruction, students are given multiple choice questions to consider individually, followed by an individual vote on the question using a clicker. That’s followed up by a small group discussion which is followed by a re-vote. Typically the percentage of students getting the correct answer to the question jumps, often in my experience with nearly the entire class converging on the right answer following discussion. But does that jump happen because peer discussion helps students understand the material better, or because students with a weaker understanding are socially influenced by students with a stronger understanding?
This research paper has some data that suggest the former. The authors administered 16 different sets of PI questions to a large-lecture (n = 350) physics class. The questions were given in pairs of “isomorphic” questions, having different contexts and…
February 23, 2012, 6:48 am
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…
February 13, 2012, 9:33 pm
A lot of my posts here are about alternatives to the traditional lecture-oriented classroom. Based on that, and on somewhat testy comments like these that I leave lying around the internet, you might get the idea that I am firmly anti-lecture. But that’s not entirely true. There are times and places where lecture works quite well, even better than the alternatives. Here are a few purposes for which I think lecture is well-suited:
- Modeling thought processes. The benefit of hearing an expert learner lecture on a subject is that, if the lecture is clearly given, the audience can gain some insights into what makes the expert an expert. A good lecture does more than convey facts or put problems on the board — it lays bare the cognitive processes that an expert uses to assimilate those facts or think his or her way through those problems.
- Sharing cognitive structures. Lectures provide…