Ben Tilly

One of my favorite stories of basically successfully implementing this in practice was this anecdote of teaching linear algebra at Dartmouth in the 90s: https://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html

That is a worthwhile option, but parallelization hasn't generally offered me nearly that much of a win when I'm limited by memory access performance. In my experience, you're best off optimizing relatively limited pieces of a system rather than big applications. And only consider the rewrite when you've looked at optimizing it in place. This means that the thing that needs to be optimized with a rewrite often has a chance of not having any giant stupid mistakes. For example see http://bentilly.blogspot.com/2011/02/finding-related-items.html for a case where I found a thousandfold speed increase by rewriting from SQL to C++. As much as was reasonably possible, I did not change the basic algorithm.

These cultural differences even determine how you eat your corn: http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eati… (I know this is weak evidence that has been debunked by stronger studies.)

But how do you eat your corn? http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eati…

You might find this interesting -http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eati…

The corn eating division, https://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eat…

OK, I've done some reading on this, including a several year old blog entry on this topic by the person who brought it up here [1], and it looks like I misunderstood what is required to consider the problem solved. I took it as requiring that volume, shape, and orientation of the water all be correct in the tilted cup, and assumed that people who fail are probably getting shape and orientation right but are not getting the volume right. Apparently, the failure mode is typically to get the shape and/or orientation wrong, and you pass if you get them right. Volume doesn't have to be right. [1] http://bentilly.blogspot.com/2010/08/piaget-water-level-test.html

On a funny note, for mathematicians, "analysis vs algebra" bend predicts if they'll eat corn depth first or breadth first. http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eati…

> Seeing proofs as programs really helped me come to terms (heh) with mathematical logic. I suppose this underscores how some people think algebraically while others think analytically. I am an algebraic thiner. I like to break apart algorithms into a mathematical calculus to understand them. Humorous: http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eati…

This link was in the comments on that article with some speculation on how it might work http://bentilly.blogspot.de/2012/11/speculating-about-hyperloop.html

You might like the dy/dan blog http://blog.mrmeyer.com His focus is on elementry/high school but I think his approach would also make college-level classes much more interesting. There's also this delightful post from btilly http://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html that I found wonderfully informative. Make sure you read the q/a in the comments.

This is one of the reasons why we need human teachers: http://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html more at : http://news.ycombinator.com/item?id=850485

One thing I think would be interesting to research is how different people are good in different areas of math. There's lots of anecdotal evidence[0] of people saying they tend to be good at either more algebraic subjects (say group theory or algebraic number theory) or more analytical ones (PDEs or measure theory) likewise I would extend this to also include geometric/topological subjects. In general especially as once gets into more advanced mathematics the abstraction level gets much higher quite quickly so it would interesting to know if there is any connection between brain structures and ability to grasp abstract mathematics. As an example I know a physicist who can tell you everything about general relativity and has a solid grasp of differential geometry but one time I was trying to teach him some mathematical logic and he just couldn't grok it no matter how many hours we spent on it. Likewise subjects that use category theory heavily (homological algebra/algebraic geometry for example) are known to be difficult to understand even for professional mathematicians in other areas. [0]: https://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eat…

This "naive reductionism" looks to me like the difference between the way that analysts and algebraists work on problems [1]: "If I have to wave my hands and explain it, I would explain it like this. In algebra there are sequences of operations which have proven to be important and effective in one circumstance. Algebraists try to reuse these operations in different contexts in the hopes that what proved effective in one situation will be effective again. By contrast an analyst is likely to form an idiosyncratic mental model of specific problems. Based on that mental model you have intuitions that let you carry out long chains of calculations that are, in principle, obviously going to lead to the right thing. Typically your intuition is correct to within a constant factor, and you're only interested in some sort of limiting behavior so that is fine." How else is a person supposed to intelligently explore the solution space of a problem other than through some kind of "naive reductionism"? [1] http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eati…

Interesting. It might make be fun to try and look for "process oriented" vs. "existence/properties" oriented people around, and will sure make for some entertaining lunchtime discussions... Like the thing with algebra-oriented vs calculus-oriented people eating corn: http://bentilly.blogspot.ro/2010/08/analysis-vs-algebra-predicts-eatin… :) I tend to be process oriented by default I guess, even your sine example... The sine function only "clicked" for me after I saw it as the y projection of the radius of the trigonometric circle and imagined that if you attached a pen to this point moving on the y axis while a paper scrolled at constant speed behind it you'd get that ondulating line. Now, I understand that this way of thinking quite biased (I tend to get very good intuitions about "how to build things" but not very good at explaining why they work afterwards), and it can make me miss obvious shortcuts, like looking for what properties and relationships conserve etc. and this is why I make an effort do more "what is" type of thinking about stuff... but "how to" thinking just seems more natural to me. Maybe it's "scientists" vs. "engineers" thing and programming just happens to be a middle-ground where both kinds of people are frequent :)

Wow you quoted me :) (I kind of feel embarrassed, its an essay I through together and then basically retracted because it was so thrown together, and Iv'e been to lazy to fix it...) Some thoughts on what you said: "Smeder misses the main point, which isn’t identifying the problem; a gazillion people in the Chronicle of Higher Education have said virtually the same thing at various times. The real problem is solving the problem, which requires changing the incentives that drive professors. " Your argument is that we need to put the carrot in front of the teachers. Give them incentives to be better teachers. I think this is true. But it only solves part of the problem. Your solution is analogous to a magazine company giving its sales team commissions in a world where sales teams don't earn commission. Which I agree is smart. In fact I hate it that teachers don't have incentives to be good teachers. ... continuing the magazine company analogy: Yes, I agree sales reps should be paid on commission. However, if the magazine sucks and the company is poorly run fixing the issue with the sales team isn't going to fix the company. - The point of my essay was to say that the problems with college are out of the hands of teachers. It is an organizational problem. In a magazine company a organization problems would be: The magazine has a bad art department or the main editor sucks at his job. These are issues that correcting the mistake made with the sales team wont…