July 23, 2013, 8:00 am
Yesterday I was doing some literature review for an article I’m writing about my inverted transition-to-proof class, and I got around to reading a paper by Guershon Harel and Larry Sowder¹ about student conceptions of proof. Early in the paper, the authors wrote the following passage about mathematical proof to set up their main research questions. This totally stopped me in my tracks, for reasons I’ll explain below. All emphases are in the original.
An observation can be conceived of by the individual as either a conjecture or as a fact.
A conjecture is an observation made by a person who has doubts about its truth. A person’s observation ceases to be a conjecture and becomes a fact in her or his view once the person becomes certain of its truth.
This is the basis for our definition of the process of proving:
By “proving” we mean the process employed by an…
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 …
February 8, 2012, 11:00 am
Here’s a quote from Stephen Covey’s classic The Seven Habits of Highly Effective People, which I am reading right now. Covey here is talking about the difference between improving one’s life through the “Character Ethic” (think: Ben Franklin’s autobiography) versus simply changing how you interact with other people without a corresponding shift in your worldview, which he refers to as “technique” or the “Personality Ethic”. All emphases are my own:
To focus on technique is like cramming your way through school. You sometimes get by, perhaps even get good grades, but if you don’t pay the price day in and day out, you never achieve true mastery of the subjects you study or develop an educated mind.
Did you ever consider how ridiculous it would be to try to cram on a farm—to forget to plant in the spring, play all summer and then cram in the fall to bring in the harvest? The farm is…
February 6, 2012, 2:52 pm
This past Saturday I was paired with one of our faculty from the College of Nursing to interview several prospective students for academic scholarships. In between interviews, we had a great conversation about the inverted classroom. It turns out that the College of Nursing is implementing an inverted model in some of their classes, although they don’t know it by that name and are not trying to jump on an educational bandwagon. They are taking some of their courses, putting the “theory” (as it was called) online as audio files accompanied by sets of notes, and then using the class time for practica, labs, and discussion. When I described what I’ve written about here on the inverted classroom, my colleague readily agreed it was the same idea.
This makes a lot of sense in the health sciences because as a practitioner, theory and raw information are important, but it’s the practice…
January 18, 2012, 7:12 am
Here’s a previous article in an ongoing series of What I Learned in 2011.
While it was still on TV, the show LOST was a favorite of mine. No, that’s not strong enough — it was an obsession. I discovered the show about halfway through its fourth season when I downloaded the series pilot from iTunes on a whim. I was hooked. I proceeded to watch the episodes online at a rate of about one per day — sometimes two or even three — until I caught up. I read the blogs, edited the wiki, listened to the podcasts. I was completely and totally absorbed. And this is coming from a person who otherwise watches TV maybe about an hour a week (modulo football and kids’ shows).
What was it about that show that I found so engaging? For me, the main thing was the deep humanity of the characters. In the first few episodes, it was very easy to pigeonhole them all. Sawyer was the criminal you had to…
January 3, 2012, 1:59 pm
One of the main reasons I’m at the AMS/MAA Joint Meetings this week is to take an MAA short course on discrete and computational geometry. That course is wrapping up this afternoon, and it’s been a good experience. I came into the course with zero knowledge of computational geometry, a within-\(\epsilon\)-of-zero knowledge of algorithms, and an extremely rusty skill set in topology. But I’m coming out with an appreciation for this subject and, hopefully, a basis for pushing farther into the field and eventually contributing something new.
Teachers ought to take courses more often. Apart from being intellectually satisfying, it’s useful to be on the receiving end of academic teaching in one’s own discipline every now and then because it helps you remember what it’s like to be in the shoes of your own students. Here are some things I’ve re-learned about being a student in a math…
December 7, 2011, 8:41 am
Many indicators are pointing to a critical shortage of engineers among the current high school generation. What’s the cause of all that? A study (PDF) by the nonprofit organization Change the Equation (with backing from Intel), focusing on 1004 students between the ages of 13 and 18 with computer access, suggests two things: a perception of difficulty coupled with an overall lack of knowledge about what engineering really is in the first place.
The Intel survey showed 63 percent of the students ages 13 to 18 have never considered the career despite having “generally positive opinions of engineers and engineering.” The perception that engineering is difficult also played a part in the lack of job consideration.
But the teens were especially interested when they learned about the potential for engineering to help others, such as saving the Chilean miners who were trapped in 2010,…
November 22, 2011, 1:00 am
Tomorrow (11/23) is not only Fibonacci Day, it’s also the release day for the English translation of Math Girls, a mathematics-themed young adult novel (!) by Japanese author Hiroshi Yuki. Math Girls combines “the rigor of a mathematics text with the drama of teenage romance”, according to a press release by Bento Books (the US distributor of the novel).
I’d never heard of Math Girls before I saw that press release, but apparently it’s a cultural phenomenon in Japan, where it’s gone through eighteen printings, three sequels, comic book adaptations, and fan-created music videos. (The music video is here — I couldn’t figure out how to embed it. You should watch it, just because it’s the only Japanese pop video you will ever see that features a generating function.)
Bento Books describes the novel:
Math Girls introduces readers to a wide range of subjects, from the Fibonacci series …
November 8, 2011, 6:52 am
The title of this NY Times article making the rounds in the blogosphere is titled “Why Science Majors Change Their Minds (It’s Just So Darn Hard)”. But it seems like the real reason that 40% of university students today who plan on careers in the STEM disciplines end up changing into other fields or dropping out is only partly about the hardness of the subjects. What are the other parts? Read this:
But, it turns out, middle and high school students are having most of the fun, building their erector sets and dropping eggs into water to test the first law of motion. The excitement quickly fades as students brush up against the reality of what David E. Goldberg, an emeritus engineering professor, calls “the math-science death march.” Freshmen in college wade through a blizzard of calculus, physics and chemistry in lecture halls with hundreds of other students. And then many wash …
September 29, 2011, 4:20 pm
In case you didn’t hear, Amazon has announced a major upgrade to the entire line of Kindle devices, including a new 7″ tablet device called the Kindle Fire. The Fire won’t be released until November 15, but already the phrase “iPad killer” is being used to describe it. Wired Campus blogger Jeff Young put up a brief post yesterday with a roundup of quick takes on the Fire’s potential in higher education. One of those thoughts was mine. I’ve had some time to look around at what we know about the Fire at this point. I have to say I am still skeptical about the Fire in higher education.
It seems like the Fire is a very well-made device. I’m not so interested in getting one for myself — I’ve got a current-generation Kindle and an iPhone 4, and am very happy with both …