[katzgrau]

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 process until I understand why it works.

If it wasn't for Strang's thoughtful and sometimes even entertaining lectures via OCW, I probably would have failed the course. Instead, as the material became considerably more abstract and actually required understanding, I had my strongest exam scores. I didn't even pay attention in class. I finished with an A. Although my first exam was a 70/100, below the class average, the fact that I got an A overall suggests how poorly the rest of the class must have done on the latter material, where I felt my strongest thanks to the videos.

So anyway, thank you Gilbert Strang.

[heymijo]

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 gave professors who taught in this way worse ratings.

Ansible's anecdotal reply is what a lot of students experience. A feeling of initial success with the material, but they later find that their knowledge of it was fleeting and inflexible. What the Air Force Academy study found was that professors who taught in the manner ansible described, that resulted in fleeting and inflexible knowledge, were rated higher by their students. Those students got better initial scores in Calculus I, but went on to do worse in later calculus courses and related courses.

I encourage you to read the study. It is as good of a study design and execution you can get in the social sciences.

David Epstein also discusses the study in Chapter 4 of his book, Range [2].

[0] https://news.ycombinator.com/item?id=23154241 [1] http://faculty.econ.ucdavis.edu/faculty/scarrell/profqual2.p... [2] https://www.goodreads.com/book/show/41795733-range

[romwell]

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.

[thr0w__4w4y]

Yes indeed. Outside of work, I'm an endurance sports person, so basically performance is correlated strongly with training hard and suffering. There is a saying, "Pain is weakness leaving the body", I first heard it in high school (team went on to win a state championship in a highly competitive state). When I was suffering on workouts I just pictured myself getting stronger.

OK hopefully I didn't get too far afield. To me, the analogous concept in learning, particularly in technical fields, is that "learning is ignorance leaving the mind".

In college, particularly math and physics, I /always/ focused on understanding the underlying principles. Initially it was out of fear that if I forgot the formulas, I could re-derive them. But a strange thing happened... through that process, I developed an intuition and an ability to "see" what formulas and concepts to apply when. Once I got to that point in a problem, "seeing it for what it was", finishing to the solution became busywork.

[heymijo]

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...

[jkmcf]

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

[sriram_malhar]

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?

[heymijo]

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.

[ZeikJT]

#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.

[apricot]

I always did very well on tests at school, but I wasn't really learning anything, or more precisely, I wasn't learning how to learn. I was learning how to pass tests, but that's a rather useless skill to have. I had to learn learning as an adult, and it was more difficult than if I had to learn it as a child.

[heymijo]

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.

[ansible]

> 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.

I had a similar, though sort of opposite experience.

In high school, I breezed through the material, and started teaching myself calculus during the summer to prepare for university. Other than being a lazy student, I had no problems taking the 2nd semester advanced calc 2 and 3 courses my freshman year. I totally get what's being taught. There weren't a ton of practical examples, but I can easily see (for example) what the purpose of integration is, and how and why you'd do it in two or more dimensions. I could work the equations, no problem. Everything is great.

Along comes sophomore year, and still thinking I am hot stuff, I take advanced linear algebra and differential equations. More of the same, I thought.

Well... we seemed to spend the entire semester just solving different kinds of equations. No explanations given as to what they are for, where they are used, or what the point of any of it was. I struggled, for the very first time.

I either got a D or F for the mid-term exam, which was shocking to me.

We had one chapter where we were doing something practical. This is where you have a water tank, and a hole in to bottom. Because the pressure lessens as the tank empties, the flow rate is not constant. However, you can solve this via diff equations, and I really grokked it. I finally saw the point for some of what we had been doing. But it was just that one chapter, we skipped any other practical aspects for what we were studying.

I did end up pulling out a 'C' with that class, to my relief. Sure, most of the blame for my lousy performance must rest with me, because of my poor study habits. And a little blame can go to the TA, who wasn't a good communicator, so that hour every week was kind of useless. But I also blame the material and how it was presented.

[crispyambulance]

I think that whether or not students do well, there's a common theme in university math curricula for non-math majors. Basically, math gets taught as a kind of "toolbox" of techniques. Unless there's a strong follow-up in subject matter courses (for example in engineering coursework), those math skills effectively evaporate.

Some places use a rigorous "proof-theoretic" approach in math curricula. It's much harder and takes more time, but it's better than merely grinding on hundreds of easy calc-101/diff-eq problems, because students gain an understanding that doesn't erode as easily once they forget "the tricks".

More CS, engineering and science students, IMHO, should dabble in math department courses beyond the the usual "required" sequence for their majors. It can be eye-opening and provide long lasting benefit to take a hardcore real-analysis course, abstract algebra or a number of other courses in math.

[dunefox]

> More CS, engineering and science students, IMHO, should dabble in math department courses beyond the the usual "required" sequence for their majors

That was absolutely not allowed at my faculty (admittely computational linguistics, but I would have massively benefited from math courses). No courses other than the predefined ones, no matter how relevant. Now I have to learn so much afterwards, it's not even funny.

[crispyambulance]

> ...have to learn so much afterwards, it's not even funny.

It's true.

The sad thing is these problems start well before university when high schools pressure students into "advanced" math coursework without demonstrating mastery of previous topics. It builds a shaky foundation and sets the student up for a lot of needless difficulty later on.

Much better to slow down, focus on fundamentals early on and then build breadth in university coursework.

[sandyarmstrong]

Oh man, Differential Equations. After doing well in Calc 1-3 I thought it would be no big deal. I paid attention in class and barely did the homework because it all seemed so straightforward but it was boring and I was not engaged.

I came in for the first exam, sat there for maybe 15 minutes reading the questions, and realized I had no idea how to solve any of them.

Luckily it was before the drop date! That was a turning point where I decided to only take classes that seemed fun. For me that was discrete math, number theory, abstract algebra, etc.

[hhmc]

It's probably an oversimplification, but differential equations -- as a field of study -- tends to be much more a grab bag of tricks than many branches of mathematics.

[sh-run]

I took linear algebra through a community college and had one of those rare, really awesome CC instructors. He had spent most of his career at Cray and later Raytheon and then semi-retired as a community college instructor. He took time to make really great interactive Jupyter notebooks. That combined with 3 brown 1 blue videos really made linear algebra click for me.

My only regret is that I took the class as a six week short course. I think my recall would be better if I had taken the full semester. We covered all the material, but missed out on the longer spaced repetition. Linear Algebra was by far my favorite pure math course, I hope to revisit it soon. Maybe Strang's lectures are the way to do that.

[exabyte]

There is a linear algebra series on Udemy called "Complete Linear Algebra: : theory and implementation" by Mike Cohen that I really enjoyed doing because he walks you through Matlab demonstrations (code included for Matlab and python. I adapt to Julia using the PyPlot wrapper for Julia).

I particularly like his videos because he breaks them down into small bites that are easy to work into your day and he's a great teacher.

https://www.udemy.com/share/101XOWAkYTd19WTQ==/

[jlelonm]

Has he published these notebooks online anywhere?

[rapfaria]

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).

[rcthompson]

I had a very similar situation in my linear algebra course: in hindsight, I would literally have been better off teaching myself the material than listening to the professor. To this day it's still the main weak spot in my math/stats knowledge base. I'm really interested to check out these lectures.

[metreo]

Intermediate Stats writing out Chi-Squares by hand on exams literally ended my academic inclinations. I had been using software to do this for a while and something about the process of being forced spend hours memorizing how to by hand just to "earn" a letter rubbed me the wrong way. I absolutely know much more about Chi-squares then I'd ever need to, possibly an imprinting of the bad experience.

[cat199]

haven't watched but based on summary these seem more about pedagogy than the subject - that said there is a full course worth of videos taught by same professor that are pretty good

[qorrect]

Oh yeah Gilbert Strang's original course is amazing , https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...