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A 2020 Vision of Linear Algebra

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Re: A 2020 Vision of Linear Algebra

#191

Post by pengaru »

dxbydt wrote: post #175
pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

> Maybe my understanding is all wrong, feel free to correct me, as my understanding on this subject is entirely self-taught. I wouldn't say it is all wrong. Just that the stuff you are talking about is a very tiny fraction of LA. I took a graduate class in LA, based on Strang's book. I have the book right here in front of me. So the stuff you allude to, i.e. rotation matrix, reflection matrix & projection matrix, is…

Thanks for writing this. I intend to pick up a physical copy of Strang's book as it's been repeatedly mentioned in a positive light on HN.

So does LA get substantially more involved than just lots of multiplications and additions or is it always at the end of the day still just bags of floats getting multiplied and summed? Is it just a fantastic rabbit hole describing what values you put where in those bags of numbers?

Re: A 2020 Vision of Linear Algebra

#192

Post by jkmcf »

romwell wrote: post #185
heymijo wrote: post #178

Earlier quoted context omitted.

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.

Did you happen to explain why you were teaching this way?

Re: A 2020 Vision of Linear Algebra

#193

Post by heymijo »

sriram_malhar wrote: post #190
heymijo wrote: post #178

Earlier quoted context omitted.

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?

Hi Sriram, a teaching system can incorporate both!

Apologies if my original reply made it seem like it can't.

Why don't teaching systems in America incorporate both the majority of the time?

Two major reasons:

1. Cultural inertia. Most teachers emulate the pedagogy that they experienced in their schooling. Some are aware that you can try to mix conceptual+procedural and try to. I call them the "notch generation"- trying to teach in a way that is different than they were taught. It's hard to do because...

2. The system is not designed to accommodate it. Incentives and higher order effects all conspire with cultural inertia to thwart it.

Re: A 2020 Vision of Linear Algebra

#194

Post by heymijo »

romwell wrote: post #185
heymijo wrote: post #178

Earlier quoted context omitted.

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.

You're between a rock and a hard place.

The rock are the incentives, how your performance is measured, and the short duration you will have teaching these students.

The hard place is students who have likely spent 13 years in K-12 learning without understanding and are now being asked to do engage in practices they have little to no experience with.* They also have incentives to get good grades and a good GPA, which can be at odds with actual learning.

*To get more concrete, the practices have a name--Standards for Mathematical Practice (SMPs). The National Council of Teachers of Mathematics developed them and considers them the "heart and soul" of the Common Core Mathematics Standards. Not only are these practices absent from most classrooms, all too many teachers are not even aware of them! (see my Notch Generation reply to Sriram to understand why)

https://www.teacherstep.com/breaking-down-the-common-cores-8...

Re: A 2020 Vision of Linear Algebra

#195

Post by ZeikJT »

heymijo wrote: post #193
sriram_malhar wrote: post #190

Earlier quoted context omitted.

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?

Hi Sriram, a teaching system can incorporate both! Apologies if my original reply made it seem like it can't. Why don't teaching systems in America incorporate both the majority of the time? Two major reasons: 1. Cultural inertia. Most teachers emulate the pedagogy that they experienced in their schooling. Some are aware that you can try to mix conceptual+procedural and try to. I call them the "notch generation"- try…

#2 bothered me so much in school. The system gauges success via tests that check short term learning. It really, really isn't good at measuring learning.

Re: A 2020 Vision of Linear Algebra

#196

Post by slavik81 »

BeetleB wrote: post #163
jacobolus wrote: post #45

Earlier quoted context omitted.

Gaussian Elimination is indeed from the 1820s. All the rest is more recent than that. The idea of matrix decomposition per se comes from the 1850s. The earliest work on something like the SVD is from the 1870s. You are onto something though. Strang is coming from a direction of numerical computations and algorithms for solving real-world problems. Pure mathematics departments for at least the past maybe 80 years ofte…

If you want to get into serious physics/engineering, the abstract aspect of linear algebra is much more important than the boring computational mechanics. And quite a bit of those computational mechanics lead you astray when you go to infinite dimensions.

How might they lead you astray? It almost sounds like you have an example in mind.

Re: A 2020 Vision of Linear Algebra

#197

Post by DagAgren »

madhadron wrote: post #122
DagAgren wrote: post #79

Earlier quoted context omitted.

I am convinced that monads induce a very specific kind of brain damage that makes a person incapable of ever explaining monads.

I've had good luck with explaining it as a characteristic of a programming language. In a language consisting of sequences of statements with bindings and function calls, we expect that f(x) is the same as a = x; f(a) and the same as g = f; g(x); That's the monad laws. Whatever craziness you want to put in the semantics, those are properties you probably would like to preserve in your language.

I think I understand monads less now.

Re: A 2020 Vision of Linear Algebra

#198

Post by DagAgren »

vajrabum wrote: post #149
DagAgren wrote: post #79

Earlier quoted context omitted.

I am convinced that monads induce a very specific kind of brain damage that makes a person incapable of ever explaining monads.

This explanation did it for me. https://www.reddit.com/r/math/comments/ap25mr/a_monad_is_a_m...

Pretty sure my point still stands.

Re: A 2020 Vision of Linear Algebra

#199

Post by heymijo »

ZeikJT wrote: post #195
heymijo wrote: post #193

Earlier quoted context omitted.

Hi Sriram, a teaching system can incorporate both! Apologies if my original reply made it seem like it can't. Why don't teaching systems in America incorporate both the majority of the time? Two major reasons: 1. Cultural inertia. Most teachers emulate the pedagogy that they experienced in their schooling. Some are aware that you can try to mix conceptual+procedural and try to. I call them the "notch generation"- try…

#2 bothered me so much in school. The system gauges success via tests that check short term learning. It really, really isn't good at measuring learning.

Hey man, that really sucks, and I'm sorry to hear it. I have a bunch of follow-up questions I'm curious about. I know HN isn't the best way to track replies. I've got heymijo.hn at gmail set up if you want to shoot me an e-mail.

I've worked in both K-12 and post-secondary education, studied the history of education reform in the United States, and visited schools/met teachers/students/etc that I've connected with across the U.S.

I'm always interested in hearing someone's story about school, how it did/didn't meet their needs, and how it has impacted them.

Re: A 2020 Vision of Linear Algebra

#200

Post by rapfaria »

katzgrau wrote: post #72

I 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…

At college I had a very poor algebra professor, and my first grade was 15/100. I didn't want to fail any classes, and since I only had them in the evening, I went on to take algebra with other two professors during the day. Those two were different, but I wouldn't say better than the first. But with determination, it finally clicked.

Second exam was 85/100, the highest between C.S. and Automation Engineer (both lectured by that first professor). While I do agree that a good teacher can pave the way for a good student, I think most of the work you have to do it yourself, as if your life depend on it (mine did).

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