What an excellent Hacker News Post. Please do write up a bit more about your method, what your students got out of the course specifically, and whether you've had success with it in other places. You have a clean, readable, and interesting writing style.

I think we can all recall a teacher or teachers in our life who truly made a difference in teaching us something - sometimes _despite_ our interest (or lack thereof) of learning it.

I was forced in University to take a course in Propositional Calculus (it was a breadth requirement for computing science) - it was mid-day so I attended the class (unlike early morning classes in which my attendance was abysmal) - I went in with zero desire - no coding, or even math. But, the timing was right so I just sat in the class.

The instructor spent thirteen weeks walking us through Reductio Ad absurdum proof, tautologies, conditional proofs, etc.. He gave us homework assignments that built on earlier knowledge, and just as we were getting into new material, he'd throw some earlier sequent proofs at us to solve and refresh our memories.

Here is the thing - I had _no_ interest in this course, the only thing going for it was it's timing, and repetition. I was _killing_ myself trying to learn integration by parts, linear algebra, discrete math, spending 5 hours outside of class for every hour in class trying to master those other course, and basically just sitting in this one distraction course. I was a C+ student in the math courses (after _massive_ effort) - I walked out of the Propositional Calculus course with an A+.

Sixteen years later I look back at University and can honestly say that the most useful course, that I _continue_ to have rock solid knowledge of PLUS use almost every single hour of the day was that stupid Philosphy 210 Propositional Calculus course. For the life of me I can't remember much of graph theory, and wouldn't be able to invert a matrix if you held a gun to my head, but, I'll sit in a meeting and see a ton of possible alternative and immediately start applying Disjunction elimination to break us out of a log jam.

There really is something to this spacing method.

If you made this story into a blog post, people would listen. (I'm actually serious, it's an interesting story and a lot of people might benefit from hearing more of the details and the thought process behind it.

When I was learning to play the violin, my teacher had a very strict method for going through the repertoire. You had your working piece, which was supposed to be hard - it was stretching the abilities of what you could do as a violinist. You had your polishing piece, which was your previous working piece where you were finessing all the fine points of technique and musicality. You had review, which was all the other pieces you'd learned so far. And you had your preps, which were little passages (4 measures or so) from your working piece that were so hard that you played them over and over again, way slowed down, until you got them right and could speed them up and incorporate them into your working piece.

I've tried to do something similar with my work so far - a working codebase that I'm just learning, a polishing codebase that I basically know my way around in, and various tweaks and bugfixes that I have to do for previous projects. It's a bit harder in the corporate world though - while Google engineers are given a lot of freedom to pick their projects, they're still subject to the needs of the business, and sometimes a project will come up that's such a great opportunity for professional visibility that I'd want to take it even if it involves working hard on something I already know well and dropping the polishing of stuff I just learned.

This remind me of a drawing book called "Drawing on the Right Side of the Brain". It teaches people how to draw realistically using techniques like contour drawing among other stuff.

However, middle school art teachers seem stuck in their little art project. I am not sure about high school, but it doesn't look like they teach the skills.

I'm just about to start lecturing the tutorials for a linear algebra class next week. The prof that is running the course taught it to me 4 years ago and is the reason I'm now in grad school. She did more or less the same thing that you mention, and it definitely works. I remember all of this material from back when I took the class; it's still familiar and I use it all the time.

But despite being very useful for a half to two-thirds of her students, there is always about 1/3 of the class who does terrible in this course. How did you get these people to keep up?

Anyway, I'm glad you posted this. It's a good pick-me-up motivator as I'm about to be trying to do the same thing!

I wish you were my teacher. That's a theory of education I've been pondering about for ages. Glad to see it formalized like this. Please consider running for office and fixing how we go about learning. :)

The question of why is there so much crappy education is interesting. I think part of the answer is that society is well-served by the failings of the educational system in that widespread educational crappiness helps support the class system. I think it would be fairly straightforward to turn 90% of the population into well educated upper middle class types, using techniques like btilly describes...

... But if we did that, who would drink Coors and drive forklift for Walmart and not complain about it?

So I think that a huge function of the educational system is to educate a large part of the population badly. I think that the teacher training system serves this is as well, by selecting for mediocre teachers and then making them more mediocre via training. Additionally, in the schools themselves a lot of effort is made to classify young people into dumb and smart categories, usually unfairly (http://en.wikipedia.org/wiki/Pygmalion_effect), and this classification stamps them for the rest of their lives and creates a population of hopeless lower class workers.

If there are SOME good teachers in the system like btilly, then SOME kids go on to get out of their class, which is perfect in that (1) we need to claim that it is possible to pull yourself up by your bootstraps, and (2) we need to recruit SOME kids to grow up to fill management positions, but not too many that they all can't find careers. Historically, underemployed educated people are the people who become union leaders.

The beauty of it is that the people who fail think it is their fault!

(Sorry to rant about the educational / class system in general, rather than the topic of how to teach advanced math, but I couldn't stop myself.)

Call me paranoid, but there you go.

Could you go into more details into SRS for math? "Fact Memorizing" type systems seem to fit well with things like supermemo (languages, law, history, biology, a bunch of sciences), but not so much with things like math (other than memorizing formulas). And did you really add homework questions like find the kernel of this subspace in the first few weeks of the course?

I recently heard that newly educated teachers come to schools with a strong desire to teach their specialties. And to transmit some of their true passion over to their students. But since being a teacher (here in Norway) means they have to administrate and organize the kids so much, they have little energy left for teaching. Meaning the teacher appears as a boring, pale fuck (not very inspiring). This is of course a organizational problem, and not a pedagogical one.

Seems related to this Wired article that recently was reposted on Reddit:

http://www.wired.com/medtech/health/magazine/16-05/ff_woznia...

Bottom line is that to maximize retention, there's an optimal time for being re-exposed to information: right before you're about to forget it. Seems like re-covering the previous material in the class may have accomplished this.

I think I learned more about retention (for me personally) by reading your comment than by reading the article.

Thanks for sharing.

Hey OP, was your technique influenced by Mihaly Csikszentmihalyi's Flow? I'm referring to the relationship between ability level and challenge. Do I see similarities with your clearly effective teaching style and some of the elements Video Game designers have almost unanimously implemented to great success?

It's sad that professors are promoted for research, rather than either research or teaching. It amazes me that the private U.S. universities that are most highly ranked for undergraduate education are the ones with the best research.

is this smart.fm?

http://www.techcrunch.com/2009/09/10/cerego-raises-3-4-milli...

Please do a more detailed version, if it makes sense to do so. You actually managed to say a lot in this little space. I had a teacher too who did this on a small scale - the first 5 mins was a summary of the previous calls. The first day of each week he summarized the entire chapter or section in the first 10 mins. This bird's eye view of the material went a long way in keeping the concepts 'live',

That's really awesome. I'm a huge proponent of finding the real, actual insights and lessons behind a subject (plug: I blog at betterexplained.com).

The dirty little secret of education is that it's "ok" to cram and forget. Most classes reward that; education is about "being able to do something once" vs. actually remembering and applying it.

I hope you write more about your findings!

I'd suggest the book Why Don't Students Like School? (http://www.amazon.com/Why-Dont-Students-Like-School/dp/04702...) to those interested in applying the findings of cognitive science to their own learning. I'm finding it very useful in that regard.

I think a good self-teacher would be better a teacher than who just hold some fancy degrees. Since good self-teachers always know what students need, and where they failed to grasp the essential, and most importantly how to get them out of obstacles.

And I bet you must be a good self-teacher. I wish I have ever had you as my teacher!

I'm wondering what type of university you were at that let you write a final. The most flexibility given to grad students in introductory courses I've seen is, at most, to write a quiz or two. Most don't even let you write homeworks.

I've toyed with the idea of keeping a practice journal, but I've always been too lazy to stick to it. Your comment has inspired me though! This way, I can go back X number of days and review whatever I was working on then.