On many topics in education I hedge based on context. With this statement, I feel firm.
Only a surface understanding is provided. Being able to add and multiply matrices is only a minute portion of their meaning. It’s like looking at Calculus and deciding to teach only how to find the derivative of a parabola. Some classes may get to solving a system of equations as a matrix, which then solidifies the concept in the student’s mind in a row-context. However…
…the understanding provided is misleading. To work with matrices in a genuine context (i.e. a linear algebra class, not a side topic in Algebra II) requires an understanding of vectors and summation notation leading to concepts of linear independence and dependence and beyond. Students stuck in a rut of thinking of matrices as a clever way to arrange numbers will get lost and confused; I know this because I was one of those students. I had to “de-program” what I was taught in high school before I could grasp what was going on. Some students will continue to think of matrices as only systems of equations long into their education, leading to…
…a removal of geometric intuition. A true understanding of the determinant requires connecting it with the volume of a parallelpiped, but a student who continues on the row-thinking path and avoids column-thinking (vectors) will quite often get there. Determinants remain an meaningless value. Further developments (vector spaces, orthonormality, etc.) become even more so. Finally…
…all the material in high school gets retaught anyway. Not only is it retaught as a refresher, but the material needs to be recontextualized so it meshes with how linear algebra really works.
Filed under: Education, Mathematics
100% with you on this. It gets taught because it’s “neat.” I can’t think of any other reason, and there are a dozen topics I would go deeper into before I would even think of opening this one. Bad mistake.
Jonathan
I am not going to disagree with you about whether matrices should be taught in high school. However, when I do teach them, we explore transformations on the coordinate grid. These explorations help deepen an understanding of matrix operations. We also use them to solve equations – and again, I teach it in a way where the focus is on basic operations.
It is true that this preliminary work can be done in a college course; perhaps understanding of matrices can easily be put off until later. But it is, in my opinion, the height of intellectual arrogance when a college professor makes a statement that implies that a high school teacher can’t do justice to a specific topic at the high school level.
In fact, the most boring, tedious, and uninspiring math instructors I have have had have been college instructors. But I am not going to assume that all of them are similar, or that certain very exciting topics shouldn’t be taught a the college level because some college instructors lack the ability to teach to a range of learning styles.
Perhaps a better perspective would be to support high school teachers in a way that deepens their understanding of mathematics along with their ability to teach it more deeply.
So with transformations … you’re doing this post-trig? (That’s a little better sequence wise than putting them in Algebra I or Algebra II as is often done.)
I do think high school teachers are capable of “doing justice” to the topic. I just think in terms of raw topic sequencing it’s too early, and the background material to get things done in a way that won’t clash with later mathematics is to have at least vectors and summation notation, both things that don’t occur (if at all) until the end of Pre-Calculus.
I also understand some people have them on their standard school syllabus (we do; I teach high school, not college) so they have to reckon with them in some way. I suppose a guide to “everything you need to know about how matrices are used later” would be a good idea, then.
Sorry about assuming that you were a college professor. I should have done my research. I have just grown weary over the years of hearing “don’t teach this or that topic, you will only screw it up! Leave the conceptual stuff to us.”
Again, my apologies.
edavidt
Thank you for elaborating.
Agreed! We just axed em from our precalc curriculum. Hoo-freaking-ray.
Sorry, guys (he said, as he spit into the campfire) but I think that matrices can be a really useful tool to the pre-calc student in understanding 3d geometry… I suppose you could try to solve and explain all the possible intersections of three planes in space, but matrices are a way cool way to get students to work with finding the line where two planes intersect, and determinants are a nice way to find the equation of a plane that contains three points … so yeah, there are lots of things out there that would be pretty to cover, but I’m not thinking many that we usually leave out are much preferred to matrices.
Although I’m curious on the details (are you able to do the determinant in some way that makes it not just an arbitrary formula?) I don’t see how that evades my objection #2 above.
So we shouldn’t even introduce matrices at all? Do they show up on standardized tests?
I don’t believe it’s on the SAT, but don’t take my word as gospel.
It is on our state test and some other state tests. I don’t think it should be.
I’m not really sure of the argument about linear algebra, but matrices per se should really not be taught at high school level. I mean, the high school teacher who taught matrices ruined my life. Really.
Because of my poor performance multiplying matrices and calculating determinants by hand, I was led to believe I wasn’t good at mathematics, and what’s more, that I dreaded it. My parents are musicians, so I went for film school. Half-way through that I was helping roommates with elementary calculus and computer programming and it hadn’t yet hit me that I had made a terrible, terrible mistake.
Still, I was too cautious: I thought I couldn’t handle math, physics or a real engineering, so I took economics, hoping the maths wouldn’t be too much. I’d soon be taking classes from the mathematics department, and when I wandered into graduate school (mostly from not knowing really what I wanted to do with my life) I went shoulder-deep into econometrics, avoiding all economics as much as possible. When it was time to write my master thesis, I couldn’t be bothered to research anything about actual, concrete economics. I eventually walked out of it without ever completing a degree (actually by never visiting the institution’s facilities ever again).
Somehow I landed a job in a large thin-thank that involves doing some econometrics, some statistical analysis and recently some simulation work. But that’s mostly because someone on the inside knew me as a teenager and didn’t know much about my meandering ways.
I hope this sounded appropriately harrowing, as opposed to sappy. Failure to draw the damn number columns straight — it’s not that I didn’t understand the algorithms, it’s that I garbled those number bunches –ruined my life. I coulda, woulda, shoulda done something useful with my life — I could be _doing_ something instead of just _analyzing_ reality I can’t control.
Amen!!! Everything you said rings dead-on true to me—especially the part about “de-programming,” which is exactly the word I would use. From all the lessons on matrices I got in my high school math classes, college physics classes, and lower-level college math classes, I learned two things:
1. Matrices are clever arrangements of numbers that help you do certain calculations by memorizing pithy rules.
2. Once you’ve memorized all the rules, you’ve learned all of linear algebra.
The second thing turned out to be even more dangerous than the first, because it meant there was no reason for me ever to take a linear algebra course. If one of my friends hadn’t convinced me to take an intro-level class for math majors in my junior year, I might still have no idea what’s going on!
This experience has convinced me that abstract linear algebra—maybe just with real vector spaces, instead of over arbitrary fields—should be a freshman math requirement for pretty much everyone. I majored in physics, and the intro math requirements for that were vector calculus and differential equations, both of which made FAR more sense to me once I had taken abstract linear algebra! Differential equations is one of the reasons I think abstraction is crucial. If you’ve been trained to think of vectors as lists of numbers, you’re stuck with the idea that linear equations are analogous to vectors is some sort of ill-defined way, and you’re always uncertain about how far the analogy goes. If you’ve had an abstract treatment, you know that linear equations *are* vectors, and you always know exactly what you can do!
I think my first encounter with tensors is another great illustration of why abstract linear algebra is so important, although maybe this one is more relevant to physicists than anyone else. My freshman physics teacher mentioned tensors in passing one day, so I decided to look them up on the internet. I quickly gave up, because I didn’t have enough geometric intuition: I couldn’t figure out how to write a four-dimensional grid of numbers on a piece of paper! Unfortunately, it seems like a lot of grown-up physicists actually teach tensors this way…
Anyway, that’s my rant. Sorry it’s so long! As you can tell, linear algebra is one of my soft spots, and I’m glad to have met someone who feels the same about it as I do! ^_^
Couldn’t similar arguments be made for most of math education, though? Multiplication tables shouldn’t be memorized in elementary schools because it takes away from the true nature of symbolic, abstract algebraic notion of binary operations? Or less abstractly, we have to reteach all the basic operations once we hit fractions anyways.
Geometry as a whole doesn’t mean much without measure theory.
etc.
I think exposure to some of these subjects prior to a “true, deep understanding” of the material is ok. Most people don’t learn things the first time they see them anyways.
I would not call what gets learned in geometry and arithmetic “only surface meaning”. There’s quite a bit a meat there, and none of it is misleading later, because later systems (measure theory and so forth) simply extend what is known with, say, Euclidean geometry as a special case.
Teaching just matrices is more analagous to, say, teaching just lines in geometry. Euclidean Geometry is a complete mathematical system; matrices aren’t.