Learning Density

Sent 3/1/2013 at 3:00am.

MITCET,

I think one of our goals should be to increase “learning density.”

I define learning density as learning per unit time. For me and many others, time is the key currency of MIT. In most cases, students are able to figure almost anything out, given enough time. However, I think we can reduce the amount of time it takes for students to learn the material by using data to reduce the small patches of frustration which students run into in classes.

Getting Stuck

In many classes, I understand ~80% of the material the first time I hear it. I understand a further 18% of material when I review it. However, for the other 2%, I get really confused and spend I can spend hours trying to attack the problem from different angles. This is a major time sink and source of stress for students.

Just yesterday, in Professor Orlin’s class I didn’t realize that we should still ignore negative coefficients when doing the min ratio rule when solving degenerate linear programs using Bland’s rule. I knew my answer for the question was wrong, because the objective value was not improving. I remembered that we should ignore negatives in the min ratio test. I was getting a min ratio of 0/-2, which I called a 0. 0 was allowed in one of the examples, so I didn’t think of anything it initially. The subtle answer I now realize is that we ignore the row when the coefficient is negative, not when min ratio is 0. This distinction is only important when you have degenerate linear equations, which was the focus of this week’s lecture. I spent 40 minutes figuring this out, and it was pretty frustrating. I searched the web looking for other explanations of Bland’s rule and I found several from various universities. All of them were slightly different, but none of them explicitly had this hint. All of them had a slightly different take on the material which I had to decipher. I tried reading the various definitions I had found closely and studying the different examples I had found. After trying 2-3 other things, I realized this was what I had done wrong. If only Prof. Orlin had included a small mention of this in his slides, or if Ebrahim (the TA) had mentioned it in recitation, or if one of the ~3 internet resources I looked at had the words “remember to skip negative coefficients in the min ratio test” it would have saved me 40 minutes. How many other people had the same misunderstanding?

It’s like a hard drive. Normally, it can read at a very respectable sustained read time. However, when it comes across a corrupt sector, the hard drive must try to attack the sector repeatedly from different angles. This series of seeks to the same spot is a several order of magnitude performance hit.

I believe this is one of the most pressing areas for improvement. One of the best ways to reduce this confusion is through TA office hours. Often a trained human can see the problem you are having and solve it in a small fraction of the time it would take you to solve it on your own. However these are either at inconvenient times that overlap with some class or activity, are a 20 minute walk or more round-trip, or are not until the next day when the problem is not fresh in your head. I think real-time (or close) help is critical. Piazza solves this problem very nicely, but response times can still be a significant barrier. This semester I am seeing ~30 minute response times. This is ok, but I think a ~10 minute response time is preferable. This is one area in which EdX can help, since it can provide a broad base of people to answer questions.

However, better yet, I believe, is fixing the source material to anticipate common problems students have and answer them before students stumble over them. I think this is where EdX can be particularly useful. Using data from EdX we can study pinpoint exactly where concepts didn’t make sense. We can see which quiz answers were wrong. We can see how long each person looked at each passage in the textbook. We can then make small tweaks for the next time the class is taught. (I think making these tweaks is critical. We can’t be lulled into compliancy by hard-to-update video.)

I think we should also have professors have students write on their homework which concepts they were stuck on to take an inordinate amount of time.

As I said at the Digital Leaning Symposium, the best classes at MIT are the ones which have been perfected over the years. These classes have refined their material over the years to the essential core that one needs to know. I am pretty sure that these classes use feedback to make subtle improvements over time. Somewhat relatedly, these classes are the most stable – with the same professor usually teaching every year. From a student perspective, having the same professor every year correlates strongly with a high quality class. I believe that the additional data that EdX/MITx provides can help us further refine and hone classes at MIT.

Downsides? The Distinction between Understanding a Concept and Problem Solving?

I could envision one arguing that the struggle helps MIT students learn to solve problems. However, although I am unable to articulate it perfectly, I think there is some distinction between struggling to understand a concept and problem solving. Yes, you need to use problem solving to understand where you have gone wrong. But this is a frustrating experience that is ultimately not very productive. Many of the P-Sets I have are “the right amount of hard.” These are not so easy that they are trivial, but there are no moments of getting stuck with the concepts. Instead it’s “smooth sailing;” they are still intellectually challenging in that they require an in-depth understanding of the material, and frequent references to class notes, and the occasional trying things a few different ways, but they don’t lead to extended periods of frustration over small issues. For example, this letter: I’ve had the key issues in my head for some time. A few days ago I thought about the specific subtopics I wanted to talk about. In writing the letter, I’ve had to articulate my specific thoughts onto paper. As part of doing that, I had to develop and flesh them out. I had to think about things I previously had not – for example this distinction. At no point was I super frustrated trying to understand something.

Here is one way of thinking about the distinction: would a TA explain it to you or would that give away the problem? Is it explaining a concept or giving a hint? The ideal is having just enough of a challenge to learn something inside and out and then layer on the next layer of complexity. If you start with too big of challenge at once, (like 18.01 for me) it all goes over your head and you don’t get any of it. The steps need to be the right size.

Nevertheless, I think we will still have bugs - meaning students would still need to use their problem solving skills to get unstuck. Plus, I figure if we made the easy things too easy to learn, MIT professors would just raise the bar by inventing harder problems in order to keep the average spread out.

I believe that these frustrating confusing moments add unnecessary stress – I sent Chancellor Grimson a preliminary version of these thoughts in response to his Stress Reduction initiative. Although this might be one of the most difficult of the ideas he is considering to achieve, I think it could be one of the most important. Although we might not want to, and can’t, eliminate it completely, I think we should reduce it.

-Michael