Every science degree studies algebra in their first course, it should be regarded as something pretty basic. How comes people are still talking about this?
A 2020 Vision of Linear Algebra
181–190 of 205 posts
Re: A 2020 Vision of Linear Algebra
#182Earlier quoted context omitted.
A modest proposal: ban donating to them so they need to radically increase their student body (online or offline) to earn their keep with tuition, instead of relying on the donations of the extremely rich parents of legacy admissions students.
What problem do you think drastically increasing the student body will solve?
I'm mainly thinking of elite institutions. I'd first invert the question and ask what problem places like Harvard, Yale, MIT are currently solving. (This small set of elite universities is absorbing the vast majority of donations.)
As the number of student has exploded over the decades around them they've kept their student numbers the same as before even as they collect more and more money which they hoard with no concern for opportunity-cost-of-capital whatsoever. This has created an intensifying zero-sum battle for the admissions slots in these universities. Meanwhile the state systems are increasingly overwhelmed and sketchy private universities are increasingly scamming students on the edges.
By keeping their numbers so low, the Ivy League retains comfortably small classes, no major change in their overall mission and professorial lifestyle. At the student level they function to perpetuate privilege across the generations, especially via legacy admissions which seems like a gilded age concept I can hardly believe is still a thing in 2020.
I'm saying if the Ivy League were to change its mission to serve America and the world as much as they could instead, it would do a great deal to help the vast mass of striving students who are barely not making the cut, as well as the economy over the long run, and even reduce tensions in American democracy. And that change of mission probably can only happen if they start to ignore what their biggest donors think they should focus on.
Re: A 2020 Vision of Linear Algebra
#183Earlier quoted context omitted.
Most money which comes into MIT passes through overhead. That means if a foundation donates to MIT, a bit over 1/3 of that money might ends up with whatever they donated to. A bit under 2/3 might go into the general budget (overheads vary by funding source, but the numbers above are from one specific project). On paper, overhead is used for costs of running the place. In practice, it's used for things like upscale fa…
MIT’s current overhead rate is 50.5%, but that’s pretty standard. Here’s a scatterplot showing lots of research institutions’ rates. When this was published, MIT’s rate was slightly higher (54%). https://www.nature.com/news/indirect-costs-keeping-the-light... showing actual and calculated rates. This also applies to federal research grants and is meant to cover costs associated with actually hosting the research (ren…
The article quotes 56%, as the negotiated rate on NiH projects, not 54%, but the rate varies by funder and by project. The 2/3 number is from an actual project I had insight into as the base overhead rate.
What's a little bit hidden there is that MIT dips into these funds multiple times, in pretty complex ways. For example, a sponsor might pay overhead and graduate student tuition (which just flows into MIT's general coffers). Or graduate student tuition might be waived, and the sponsor pays just overhead. Or a donor might pay overhead when the money comes in, and again on specific purchases. Or capital expenditures might waive overhead. Etc. The level of complexity is high, while the level of transparency is low.
I'm not claiming any of this is unique to MIT by any means. It's just where I have the most visibility.
What's cute is that MIT claims to lose money on everything. In the article you cited: "'We lose money on every piece of research that we do,' says Maria Zuber, vice-president for research at the Massachusetts Institute of Technology (MIT)." You'll find similar statements about educating undergraduates and tuition. And just about everything else. I've worked through the numbers at some point, and MIT only loses money with clever accounts; it's good PR to say MIT subsidize everything it does, but it's often not a reality.
Re: A 2020 Vision of Linear Algebra
#184An interesting alternative to linear algebra is geometric algebra. I recommend googling around a bit for geometric algebra and trying out my implementation https://github.com/chakravala/Grassmann.jl
Re: A 2020 Vision of Linear Algebra
#185I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. Unfortunately I'm one of those people who tends to reject the…
Durable and flexible knowledge... After reading your comment and ansible's reply [0] I wanted to pause and comment on this. The United States Air Force Academy found that its cadets who took their first calculus class with a professor who focused on conceptual understanding helped those cadets create a durable and flexible understanding of the math [1]. The kicker is that the cadets got worse scores in Calculus I and…
When I taught Calculus, I taught understanding over memorizing steps to arrive at a solution for a particular type of problem.
The very best students loved it, but most of the people didn't like it at all.
With mathematics, like with gym, you gain when you put in effort. Most people don't enjoy either.
Re: A 2020 Vision of Linear Algebra
#186Earlier quoted context omitted.
> After watching this and having read the comment, I am quite puzzled by the approach American seems to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? The US is a very big place. I doubt there is an american approach to linear algebra. We really don't have a single approach to anything. Different schools and majors probably approach the topic differently. My college had a linear a…
> Different strokes for different folks. If it worked out for you that's all that matters. That sort of relativism is trite, specious, and insincere. If you don’t think it matters then why are you participating in the discussion?
> That sort of relativism is trite, specious, and insincere.
It isn't. I don't think you understood what I wrote. What's important is the ends, not the means. As long as the OP got what he needed out of linear algebra, then that's what matters.
> If you don’t think it matters then why are you participating in the discussion?
What is this "it" you are talking about? Be more specific and precise otherwise you don't add anything to the discussion.
Re: A 2020 Vision of Linear Algebra
#187Earlier quoted context omitted.
> Different strokes for different folks. If it worked out for you that's all that matters. That sort of relativism is trite, specious, and insincere. If you don’t think it matters then why are you participating in the discussion?
> That sort of relativism is trite, specious, and insincere. It isn't. I don't think you understood what I wrote. What's important is the ends, not the means. As long as the OP got what he needed out of linear algebra, then that's what matters. > If you don’t think it matters then why are you participating in the discussion? What is this "it" you are talking about? Be more specific and precise otherwise you don't add…
I'm sorry, but I really dislike the "whatever works for you" sentiment. It's antithetical to the spirit of engineering / technical discussion, which are predicated on the idea that there are better and worse ways to do the thing in question.
> As long as the OP got what he needed out of linear algebra, then that's what matters.
I don't think I even agree with this. I might agree with
> As long as the method of teaching linear algebra improves the mean educational outcome in a statistically significant fashion then that's what matters.
But that's what Strang's getting at -- better and worse ways of teaching Linear Algebra. It's not the case that any method is OK just because it worked for some European dude on the internet with poorly concealed anti-American feelings (I'm from Europe, I'm allowed to say this :) ).
Re: A 2020 Vision of Linear Algebra
#188Earlier quoted context omitted.
> That sort of relativism is trite, specious, and insincere. It isn't. I don't think you understood what I wrote. What's important is the ends, not the means. As long as the OP got what he needed out of linear algebra, then that's what matters. > If you don’t think it matters then why are you participating in the discussion? What is this "it" you are talking about? Be more specific and precise otherwise you don't add…
I'm sorry, but I really dislike the "whatever works for you" sentiment. It's antithetical to the spirit of engineering / technical discussion, which are predicated on the idea that there are better and worse ways to do the thing in question. > As long as the OP got what he needed out of linear algebra, then that's what matters. I don't think I even agree with this. I might agree with > As long as the method of teachi…
> I'm sorry, but I really dislike the "whatever works for you" sentiment.
It's your right. No need to apologize.
> It's antithetical to the spirit of engineering / technical discussion, which are predicated on the idea that there are better and worse ways to do the thing in question.
Sure. But that's the point. There are better ways ( as in plural ). Meaning there is more than one right way.
> It's not the case that any method is OK
Nowhere did I say any method is OK. You are building a strawman here. I didn't just write "Different strokes for different folks." I wrote "Different strokes for different folks. If it worked out for you that's all that matters." The key being "If it worked out for you that's all that matters."
> because it worked for some European dude on the internet with poorly concealed anti-American feelings
Nowhere did I sense any "anti-American" sentiment from OP.
> (I'm from Europe, I'm allowed to say this :)
So maybe it's a language issue. English as a second language is great but leaves open the possibility of misunderstanding. Honestly, I still don't know why you got so upset about such an innocuous comment. I think what we had here is a failure to communicate.
Re: A 2020 Vision of Linear Algebra
#189Earlier quoted context omitted.
> Different strokes for different folks. If it worked out for you that's all that matters. That sort of relativism is trite, specious, and insincere. If you don’t think it matters then why are you participating in the discussion?
> That sort of relativism is trite, specious, and insincere. It isn't. I don't think you understood what I wrote. What's important is the ends, not the means. As long as the OP got what he needed out of linear algebra, then that's what matters. > If you don’t think it matters then why are you participating in the discussion? What is this "it" you are talking about? Be more specific and precise otherwise you don't add…
You're right that I was a bit rude, so I am sorry.
> My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.
The OPs referring to an intellectually totally different approach to Linear Algebra. One coming firmly out of pure mathematics. Whereas Strang's notion that one should teach matrices first is aimed at students who are probably never going to conceptualize linear algebra in terms of abstract vector spaces and linear operations on them. So, I do still take issue with what you wrote. "Different strokes for different folks" really doesn't cut it here, IMO. Being tendentious, I'm tempted to say it's more like "If you're happy getting practical calculations right but never really understanding what mathematicians mean by linear algebra then Strang's approach might be OK for you." But the point is that the approach to linear algebra that the OP referred to, versus Strang's, are not just equally valid alternatives and whichever one works for you is the best by definition. They are routes to two different places.
Re: A 2020 Vision of Linear Algebra
#190I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. Unfortunately I'm one of those people who tends to reject the…
Durable and flexible knowledge... After reading your comment and ansible's reply [0] I wanted to pause and comment on this. The United States Air Force Academy found that its cadets who took their first calculus class with a professor who focused on conceptual understanding helped those cadets create a durable and flexible understanding of the math [1]. The kicker is that the cadets got worse scores in Calculus I and…
Very interesting. I don't understand why a teaching system cannot incorporate both a conceptual understanding as well as hands-on applied knowledge. Is it a matter of the time available?