New textbook teaches students about matrix methods and their real world apps | Hacker News Reader

New textbook teaches students about matrix methods and their real world apps

ece.engin.umich.edu

77 pointsteleforce33 comments

People have different learning styles, and for whatever reason the general population is better at 'concrete' thinking than at 'abstract' thinking (research sez).

When I read about that research, I remembered when I first taught a 9th-grade general math class. I tried teaching fractions the way I was taught ... but after several days, a test showed it just wasn't working at all. Frustrating for most too.

One day about a week later I drew a box on the blackboard, and called it one dollar (still worth something at the time.)

I then drew a single vertical line through it, pointed at the left half and asked 'what's this worth?' Slight delay ... '50 cents?' Then I added a horizontal line, pointed at the upper right corner: 'what's this worth?' Soon it was becoming easy! Sixths, eighths... 'take away 3 sevenths, what's left? ... After a couple more hours of this, most of the class easily solved 'simplify 3 sixths' or 'how many quarters to make 2 and a half'. Every few weeks later, we revisited to see what stuck.

Works the same in the sciences, for many anyway.

> first taught a 9th-grade general math class

This blows my mind. The kids in the US don't learn/know simple fractions until they are ±15yo? In Europe I believe it's around 9-10 years old which is still late considering I've had it when I was 7 or 8.

Yeah, there's something off in the story. I went to public schools in one of the most poorly educated cities in the US and of course we were doing fractions when we were 7. 9th grade (admittedly "honors", so about a year ahead of non-honors students) was geometry / trigonometry. For non-honors I believe that was algebra.

My kids started learning fractions at age 7 in a very under resourced, rural school.

Something about op's story smells funny.

I went to a rural public school in the US, we did fractions in 4th grade (~9-10 years old). 9th graders learning fractions sounds highly unusual.

Introducing fractions in 9th grade is not typical in the US either. I would charitably guess the commenter is mis-remembering or there were special circumstances, like it being a remedial course for kids who were far behind in their academic achievement.

Usually it’s around 2nd or 3rd grade, though I don’t recall exactly as I’m not an educator and my kids aren’t that old yet.

As a former high school math teacher, I interpreted "general math" as a class intended for students who have not been successful with the typical grade-level math curriculum. That is a situation where teachers need to get creative with instructional approaches.

Fractions were certainly in the curriculum earlier, but were they learned? I'm sure my high school calculus cohort understood them, but I'm not sure the average adult can add fractions confidently. Those who don't are going to weigh down any math class after that point.

(I'm not blaming people traumatized by early math schooling. I just think there are many.)

I have a friend who is a high school math teacher and I was talking to them about how linear algebra and trigonometry are crucial to 3D graphics and rendering. I told them how going from a set of vertices of a cube to showing several cubes in different orientations and positions on a screen is mostly just a series of matrix multiplications, and why the geometric properties of a triangle make it the ideal primitive to use for rendering.

The details about computer graphics and such was new to them, but very quickly it made sense to them (mathematically).

Later on when they had started a section in their class about triangles they used al this to illustrate and relate the subject to something they were all familiar with.

I could only find a (legal) pdf[1] for $80. Pretty steep price when I'd probably only want to read 2-3 chapters.

[1]: Swedish site: https://www.bokus.com/bok/9781009418133/linear-algebra-for-d...

You could also check the world catalog to see if a library near you offers the ebook for lending. Universities typically allow the general public to walk in and look at books without registration.

https://search.worldcat.org/title/1409698868

I’ve been surprised recently to discover that matrices are not now taught in English schools unless “extra” qualifications (GCSE further maths at 16, A level further maths at 18) are taken, whereas when I did maths in the mid-80s they were included in the main syllabus.

How are matrices taught at school (if at all) in other countries?

University level only in portugal. None in highschool. There's also no advanced programs.

In US schools at least (a couple of decades ago for me), I think we did Kramer's rule or something like that. Not a lot of matrix math though until university.

Edit: Cramer's Rule lol

In France, matrices are taught in an optional mathematics class for students that want to go further.

In the Netherlands, it's not included in the 'standard' curriculum but it's optional if you take 'Wiskunde D', i.e. basically what elsewhere is called 'advanced placement math' or some such. ('Wiskunde D' / 'math D' is basically a 'capita selecta' of various math topics for those who want to prepare for math-heavy/STEM university studies, whereas math A, B and C have fixed curricula and each lay a different foundation for various follow on studies. Math A has the stuff useful for (amongst others) medical studies, such as statistics but less calculus, math B is the most 'rigorous' and math C is for those who can't tell a fraction from an asymptote but since math is a required subject still need a way to pass their high school final exams).

When I was in highschool in Belgium 30 years ago, linear algebra (well, 'matrix calculations' is was called then) was part of math (depending on your electives) but I can understand why it's not taught today any more by default; there's just too much to teach, so you have to make choices. And I imagine that today they say 'we better do fewer things well than a lot with insufficient depth', which is what my experience is from how I took it 30 years ago. What's the point in doing linear algebra when it's only the purely numerical operations without teaching why or when you'd want to take a cross or dot product? And it's only a one semester course when you go to university anyway, so it's not like it's necessary to include early on or that any other high school subjects need it.

I attended a decent high school on the U.S. in the late 1980s.

IIRC, I took 5 math courses high school: algebra, geometry, precalc, calc 1, and calc 2.

I don't think we ever covered matrices, except as a condensed format for solving systems of linear equations.

I graduated in 2021, and my experience corroborates yours. I took that same sequence of math courses, up to calculus 2. If I had been on the more advanced path for math, there was exactly one more class; I don't remember what it was, but it wasn't linear algebra. I only took linear algebra in college.

Mine was slightly different with geometry (middle school) > algebra (high school) > trig > calc 1 > calc 2, then in college they had calc 3 (which in hindsight was linear algebra with a heavy physics lean), then differential equations. That was all of my required math, I took two stats courses and a quantitative analysis and a game theory course as electives that all where under math but not required and didn't seem to be in any real order after prereqs.

In Germany simple basic matrix algebra is introduced in the last two or three years at school (grades 11/12 and 13 where that exists)

Thanks - sounds like Western Europe and the US arent that different from England.

Anyone with experience from an Asian country or Eastern Europe (eg Poland Hungary)

In India, matrix basics are in the standard high school curriculum (multiplication, associativity, adjoint, transpose, determinants , inverse etc).

But the curriculum doesn't pay any heed to vectors or the notion of linearity, so we come away with no use or intuition for these curious objects.

To the best of my knowledge:

* In Israeli high-school the system requires from students to take a number of credits in various disciplines. Some number of credits for certain disciplines is mandatory, while most are elective. Taking the maximum number of credits in math will include some matrix algebra. Taking maximum math credits isn't uncommon as the school graduation exams will have a major influence on college admission.

* In Ukraine the equivalent of high-school is optional: a large fraction of students goes to an equivalent of a trade school instead. For those who stay, the program may vary greatly based on the teacher's own preferences, what they believe would be beneficial for the college entrance exams. So, it could include sometimes even somewhat advanced college-level math, but not necessarily uniformly so. I remember that our algebra teacher put a huge emphasis on solving problems from Skanavi's book (some scary-looking multiple layers of arithmetic expressions that need a lot of effort and creativity to simplify them), but there was very little attention paid to matrices (perhaps only mentioned in the context of linear equations systems).

The content of the book looks really nice, but I'm not sure why one needs a book that explicitly aims to "engage students" and is also targeted at upper-level undergraduates or graduate students. Surely students at this level have a sufficient level of self motivation to get by... would have preferred to see the same content targeted at first year undergraduates.

As a person who just dives into things, these sorts of resources are helpful to bridge many gaps. It's a nice way to learn more than one thing at the same time. Theory and rote learning has its limits, and the expectation shouldn't necessarily be towards perfect engineers. Breadth can live alongside depth in a wonderful swirl, yo.

As a person teaching information technology to students for more than 20 years now, I can assure you that only fraction of students are that interested. Perhaps many seem interested due to peer or economical pressure.

Meanwhile economy needs proper new recruits which is impossible statistically at this point.

We also, back in the day, did not immediately grasp the beauty, and also the need, for programming logic (prolog) or certain discrete structures - things people learn in the universities.

Yet we learned ourselves to debug and code - by trial and error, by working in friends groups, and loved it. Loved to write code and see things move on the screen. We were just kids in high schools, right, makes us what - 14 at the time? Nobody has ever pushed me or chased me do that.

Since I started teaching Perl in 2003 (and did so for 10 years on university grounds) I got absolutely convinced that people learn better when they can see the result, when they entertain and generally when something gets to convince them the matter is not so hard at all.

So, kudos for the book.

Nope. Many engineering students take maths courses purely because they are compulsory, and many have no interest in theory and struggle with abstraction (which is often how they view maths in general, even though engineering maths is very concrete relative to what many mathematicians do). The thing is, you need proficiency in the basic tools and skills to be able to do the interesting applications later, and tools and skills in isolation seem abstract. Working to keep students engaged in a maths course when they're not doing a maths degree is an important part of that teaching.

There is another University of Michigan professor who created an Introduction to Computational Linear Algebra course: https://grizzle.robotics.umich.edu/education/rob101.html

with free textbook: https://grizzle.robotics.umich.edu/files/ROB_101_Textbook_W_...

All math books should follow this example. All people who took calculus because they had to, how have you used it? Yes, I know, most of you haven't.

I once used it to get reimbursed properly for local salary taxes I had to pay while at a job site.

Converging series. Shocked the scumbag HR guy into submission

I mean all STEM need to take introductory physics because they will need to understand the basics of how the world works. You can't understand Classical Mechanics and Electromagnetism without calculus.

Now there are many universities and majors which will give you physics before calculus (forgetting that the original argument were in reverse) or will give some majors what is called Algebra based physics. Which is kind of trust me these are the equations to solve this problem, you just need to solve this algebraic manipulation of numbers. So if I give you any physics problem that requires any further manipulation or derivation then you will realize you did not understand physics well.

There is also benefits of studying calculus in itself. For example, a lot of optimization techniques that is being used by programmers and ML are rooted from calculus methods.

>All people who took calculus because they had to, how have you used it?

Although I create software, my degree is in finance. Calculus is everywhere, and I mean everywhere, in finance after you get past the basic intro courses, for school at least. However, in real life I have only used it not that often and mostly related to machine learning and optimization. Even then, I didn't do it by hand, I just knew what to do and let computers calculate the rest.

With coding problems in Julia, nice!

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