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When I took precalculus, we learned about polynomials and how to factor them, we learned about trigonometry and lots of great and useful identities there, and we learned about matrices. They didn't call it linear algebra, they just called it matrices. We learned how to multiply them, we learned a whole bunch of ridiculous complicated methods to calculate determinants. We learned to solve systems of linear equations using Cramer's Rule, which is the mathematical equivalent of scrubbing a floor with a toothbrush. Never once did we actually learn what matrices were, how they would be useful, or why we would want to know any of this. I found the whole exercise to be tedious and pointless and quickly braindumped all of the matrix stuff as soon as the class was over.

Now, 20 years, a degree in math, a degree in physics, a career as a software engineer and data scientist, using linear algebra in building neural networks and many other places later, I am now TAing a precalculus class, and I still have no idea what the point of teaching about matrices in it is. Like maybe some basic matrix/vector multiplication would be useful, applying them to systems of linear equations, but why do we force our students to painfully calculate determinants by hand when literally no one does this and the only reason to study these algorithms is if you are implementing a computer program to calculate them.

Don't get me wrong, I am a huge proponent of learning fundamentals and practicing working through problems by hand before you give it to your calculator. It is important to understand the concepts behind what you are doing and how they applies to the world around you so you can use them to solve problems in applications and they aren't just pure abstractions. Except that doesn't really work for matrices because we never bother to explain what they are or how to apply them, even less so for determinants.

Linear algebra is a very useful field used quite a lot in physics, engineering, computer science, and any number of other areas. And it has nothing to do with calculus. When I took a linear algebra class, I had to relearn all of these fundamental operations anyway, except there weren't any lengthy exercises in calculating determinants by hand because no one ever actually does that outside of a precalculus class. I have students asking me why we are learning this and I honestly don't have a good answer to give them. Why is it there?

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    $\begingroup$ Perhaps better asked in matheducators.stackexchange.com $\endgroup$
    – GEdgar
    May 11, 2023 at 0:52
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    $\begingroup$ "doesn't really work for matrices because we never bother to explain what they are or how to apply them" That sounds like a failure on the teacher's and your part, not a reason to wait to introduce matrices to students of that age. $\endgroup$
    – JMoravitz
    May 11, 2023 at 1:14
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    $\begingroup$ I think there is a belief among mathematics educators that more advanced topics in calculus are best understood through the lens of linear algebra (multivariable calculus especially). Apostol's two volumes on calculus may be of interest to you, because he decided to teach calculus and linear algebra simultaneously, and gives his reasons for doing so throughout the books. $\endgroup$ May 11, 2023 at 1:26
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    $\begingroup$ Part of your story does not hold together. Are you saying that when you got around to the point where linear algebra began to be relevant to your work on neural networks etc., the matrix knowledge you got in your precalculus class was useless to you? And so presumably you had to learn everything in linear algebra from scratch, including all the matrix stuff that occurs in linear algebra? $\endgroup$
    – Lee Mosher
    May 11, 2023 at 2:47
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    $\begingroup$ @LeeMosher I do think it's fair to say that students have a tendency to braindump anything that won't be useful to them in the near future. I've taught matrix theory at the college level a few times, and while most students can remember doing "something with matrices" in high school, that's all they really got out of it. I still have to go through and teach them the basic operations all over again for a second time. $\endgroup$
    – Michael
    May 11, 2023 at 3:03

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"It is important to understand the concepts behind what you are doing and how they applies to the world around you so you can use them to solve problems in applications and they aren't just pure abstractions."

I guess this is what separates the people who go into some area of applied Mathematics from many pure mathematicians. Just to play devil's advocate, one could very well argue that studying the abstractions is an end goal in and of itself that develops mental flexibility. When thinking about, for example, matrix multiplication, the question "when will this be useful?" is certainly fair, but from another point of view, one could say it gives students an example of an algebraic object that can be multiplied that is not a real number. It's also a first example of a type of multiplication that is non-commutative. Some students may care about this, others may not - but I do think it's important to make the point that the study of Math, like so many other fields of study, has value separate from its real-world utility. I think it's a mistake to think of it merely as a tool to be exploited by other disciplines whenever the need arises. After all, many mathematicians are perfectly content to define some purely abstract object and ask "what are its properties?"

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