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.
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The Number Warrior 
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.
I just fininshed teaching a mini-unit on matrices with an honors Algebra II course. We covered addition, scalar multiplication, multiplication of matrices, solving matrix equations using the inverse, and solving matrix equations using row reduction.
The textbook presented solving matrix equations using the inverse by giving a formula for the determinant, then giving a formula for the inverse (of a 2 by 2 matrix), then showing how to solve a matrix equation, etc. If I had merely introduced the material in that way, almost all of the kids would have forgotten it (or worse, thought that they knew all there was to know) and I agree that it would be a waste of time, or worse. As a class, though, we derived the formula for the inverse of a 2 by 2 matrix using matrix multiplication and algebra. Thus, the determinant has meaning.
The textbook introduced solving matrices using row reduction by establishing the “row operations.” I insisted on making the connection to algebra so that they could see that all row operations are legal only because the corresponding algebra moves are legal.
I know that we only scratched the surface of the vast subject of linear algebra, I would contend that these kids will benefit greatly from the challenge of looking at real mathematics. I do agree that the surface level (i.e., looking at the surface, not scratching it) look that most textbooks give is useless.
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.
If you are being serious, what your story conveys to me is an amazing lack of agency and responsibility in your own education and professional life. You let an inability — or, much more likely, an unwillingness — to write matrices neatly enough to be able to multiply them — stop you from pursuing a career path you say you wanted. It’s hard for me to believe that you would have been successful if only matrices had not been part of your high school curriculum.
Indeed, you tell about how you walked right out of a master’s program in economics and/or econometrics. I wasn’t able to gather exactly why this happened, but I presume that matrix multiplication was not the culprit this time?
You’ll probably not read this, let alone understand it, but let me try and set the recordf straight.
(1) I *couldn’t* write matrices neatly. Roll your eyes all you can, butI had an undiagnosed attention-deficit disorder, and besides, _my handwriting was clumsy_. I was clumsy. I took martial arts, acting classes and drawing lessons in order to overcome clumsiness, which did work at least partially. Life moves on fast, though.
(2) I walked out of graduate school because I loved econometrics, but was in a heterodox economics program. Academia wasn’t really for me anyway — writing the same paper over and over with incremental additions.
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.
I was lucky enough to be able to take a linear algebra class as a sophomore in high school. (Hanover High, Hanover, NH; this was quite a while back, though.) We used a college textbook, titled something like “Linear Algebra”, or maybe “Linear Algebra and Affine Geometry”. I loved it. (I can visualize the teacher, but can’t remember his name.) Linear independence was certainly taught. Geometric interpretations of the “dot product” as well. Solving linear equations, row reductions. Maybe I just have a mind that loves linear algebra, but maybe this had something to do with my going on to specialize in theoretical quantum physics. (It’s just linear algebra…especially the way I do it.)
Now of course this was taught pretty much like “real mathematics”, and the geometric connections were made very clear. The idea that the solutions of a set of equations Ax=b could constitute a line, a plane, in fact an affine subspace of any dimension, was made quite clear. (I *think* even the word “affine” was used and defined.)
So in this sense, I’d say, every high school should at least give people the chance to see matrices!
On similar lines, I went to elementary school in the era of “new math”…in first grade, I thought it was really dumb being asked to make a big distinction between three apples, saws, etc… and “the set of three apples, three saws, etc…”. But on the other hand, even learning it from relatively uninspired teachers, I got quite a bit from learning the definitions of ring, field, etc… in 5th or 6th grade. (In fact some of the vague mental images that I seem to have attached to modular arithmetic, are of that elementary classroom and worksheets).
More linear algebra, not less! But do it right. Surely even in a unit in a high school course, one should be able to do some justice to this.
Thank you: ever since I saw the appendix of Common Core standards suggesting a Linear Algebra class, I wondered if an actual one had been taught at the HS level.
So in this sense, I’d say, every high school should at least give people the chance to see matrices!
Specifically in the sense you cite, I’d say sure. With a full semester it’d be possible to teach things correctly and not create misconceptions. At present this is a rare case.
I’m still curious what would be removed from the curriculum to make room for a full linear algebra class.
This is Brazil, but I was taught complex numbers in high school, including their polar form. I mean, it helps that I recognized that kind of reasoning instantaneously when I first saw ODEs, but as noted in a comment above, having a proper linear algebra course being replaced by a course in matrix algorithmics *ruined my life*
I disagree. I have found that students find matrix topics more interesting and useful than much of the rest of their high school math experience. If your teacher taught you cramer’s Rule and made you invert and find the determinants of large matrices, then you were indeed taught an outdated and relatively useless curriculum, Things have changed,
Markov Chains, matrix message coding, and Leslie matrices can be used to show them real-world applications without the tedious task of doing the computations by hand. Solving systems of linear equations using the inverse of a matrix is a heck of a lot easier than most other tedious methods now that we have the software to do it. Matrices make transformations come alive – you can now apply a matrix to an object as well as to a point and see the effect.graphically. Plus – all of this interesting stuff lays the groundwork for the small but significant percentage of students who will need to use matrix mathematics at a higher level.
Actually, if you are going to use software to solve systems of linear equations, finding the inverse of the matrix is clearly the wrong way to do it. You should do an LR decomposition or an SVD. Luckily, most software systems don’t expect you to know what you are doing, and provide a black-box “solve” that does something more reasonable than inverting matrices.
[…] NOTE: I still think matrices shouldn't be taught before vectors. I'm making my best go at the curriculum I need to do, though. Bert Speelpenning's series on […]
As matrices were taught in my high school algebra II class, and as most modern algebra II syllabi suggest, the subject is not worth covering. The course doesn’t facilitate understanding of matrices, and what is learned is of no help in later work with matrices.
I remember well learning to use 2 x 2 and 3 x 3 matrices of coefficients and augmented matrices to solve systems of simultaneous linear equations. This involved learning a new natation to do what had already been learned. Then we “learned” to used determinants. We learned arbitrary rules for calculating 2 x 2 and 3 x 3 determinants (rules which did not generalized to 4 x 4 and higher.) We were told (without derivation, or without being encouraged to derive for ourselves) the forumlae for solving inhomogeneous systems of 2 or 3 equations. Excercises consisted of tedious plug-and-chug solutions of numerous systems. The ONLY thing we learned was another (underived) way to solve for variables. We learned nothing about determinants: Nothing about determinants as volumes, nothing about LaPlace expansions or calculating any but the special cases of 2 x 2 and 3 x 3 determinants. We did “learn” how to add and subtract matrices, but didn’t even learn to multiply them. But since we also didn’t learn about matrices as linear transformations, it hardly mattered that we didn’t learn how to compose (multiply) two or more of them.
I short, matrices and determinants were presented soley as a trick to be used as another cumbersome and tedious way of solving systems of linear equations–but only of order 2 or 3. We did do some tedious (non-matrix-based) elimination solutions of 4 x 4 equations, but it was never made clear how matrix methods might be generalized to solve them.
If all that is done with matrices is tedious arithmetic, then what good are they? The whole power of matrices and determinants is their generality of applicability to n-dimensional space. If you can’t get at least some of that into an algebra II class, then what is included will be a complete waste of time.
I agree. It would be better to teach rotation matrices for animation or setting up and using a computer to solve large systems of linear equations for electric circuits. The algorithms for inverting matrices and computing determinants are pretty useless in algebra class, but learning how to set up large systems and have computers solve them would be useful, as would matrices as transformations (not just rotations but perspective transforms, if you have time).
There are two faulty assumptions in most of these arguments: (1) all high school math students will go on to a math related field in college (assuming they go); (2) we should learn all applications of a topic while or even before learning how the concept works. If we follow the above reasoning, we shouldn’t really teach elementary school students multiplication since field axioms are not taught nor the concept of a group. Commutative properties are left out as are concepts of binary vs. unary operations. Nor should they learn about line numbers or triangles because they don’t understand the basic axioms of geometry.
I agree that depth is important and seeing matrices as a novel way to arrange numbers may be useless – if we are only considering the first assumption, for example. But math is not only a tool for engineers, it is also a tool of thought and even more important a form of art. It should be taught so that every student can find a purpose and get excited about the mathematics.
Sure, relearning things in a new way is always part of the learning process. Every year I find geometry students who are led to believe that geometry is about shapes and perimeter and area. This doesn’t mean that these topics should be stopped at the elementary level; it means that in high school geometry, students need to be open to changing their perspective in order to gain a deeper insight.
“Teaching just matrices is more analogous to, say, teaching just lines in geometry. Euclidean Geometry is a complete mathematical system; matrices aren’t.” That’s right Jason. And geometry is taught after many years of seeing just lines, or basic math operations in elementary school that may have seemed irrelevant. Matrices in themselves are not a complete system, neither are vectors nor functions. They are, however, important tools, around which further theories have made them important concepts. Sets are another such examples. I teach sets and functions to all of my students, from Algebra to Geometry and on up to Calculus. However, my expectations and applications are different depending on the level. But they are used throughout the course.
Matrices as with many other mathematical constructs are not written in stone or sent from heaven, they are the creation of human minds throughout history. As such, they are not static and can be used for further applications in the future. The point of precalculus mathematics (algebra 2, trig, and analysis) is to skim the surface of higher mathematics – hopefully with some depth and connections. It is the student who goes on to college level math that needs to be open minded and ready for greater abstraction whether they study linear algebra, abstract algebra, or topology. Even if a teacher is so bad as to just say that a matrix has rows and columns for arranging numbers or information like a spreadsheet (and that’s it!), it’s still better than not giving the students a chance to see the construct at all.
It’s one thing to say matrices shouldn’t be taught at the high school level, but another to say that they shouldn’t be taught poorly. But that goes for all of mathematics. (You would be surprised : I have heard of teachers teaching geometry without proofs – and I don’t mean students don’t do it, I mean that even important theorems are not proved by the teacher!)
“The understanding provided is misleading” is still my main argument here. I was misled. I had issues later due to this. I do not know any way to fix the problem other than incorporate vectors.
Vectors can be taught at an earlier level, but that would be a significant change to American curriculum (other countries do them earlier, so it’d be possible to pull off).
Vectors should definitely be taught before matrices. I was agreeing that matrices should be taught with things they are good for, not as abstract ways to manipulate numbers. The applications of matrices are more important than the hand-algorithms for manipulating matrices. (Matrix multiplication is worth teaching, but inverting matrices and doing Gaussian elimination by hand probably is not.)
I think now I see your misleading issue – which is seeing matrices as simply a tool for systems of equations. This would definitely deprive you of seeing it as a complete system. I agree, this would be the worst way to teach matrices. Perhaps, now, when I hear a teacher decides to skip the matrix chapter, I will feel comfort that at least the students won’t see the brief & incomplete treatment given by the textbook (because that is the most they would have received).
As for vectors, it can easily be taught in geometry as a directed line segment, used for transformations, and applied to area and distance. Detaching matrices from systems of linear equations alone is important – as is the treatment of vectors with or prior to matrices. Although I have taught vectors after matrices and brought in matrices as a review with vectors – I see merit in teaching vectors and later seeing matrices as a more powerful system where vectors are just one application. Also, once matrices are seen as a separate mathematical system, then systems of equations can appear as an application as well. If students already grasp the algebraic and graphic meaning of independent vs. dependent systems of equations, then they will be more excited to see that (and hopefully why) determinants can be a powerful tool.
I would also argue against matrix inversion except for showing that matrix as a system in itself has an inverse that produces an identity matrix just as functions and their inverses leading to the identity function when composed or multiplicative inverse of numbers leading to the multiplicative identity. To get into gaussian elimination and further, would mean at least one semester on matrices or as was mentioned before a complete linear algebra course.
I make neither assumption, but I do presume that we DESIRE whatever we teach in high-school mathematics to be beneficial to at least one of (a) those students who will go on to further mathematical study in college, and (b) those students who will not.
The standard high-school treatment of matrices fails group (a) because it tends to mislead students. It isn’t just incomplete treatment. It leaves mistaken impressions about the mathematical importance of the topic. The treatment does offer a computational trick (albeit usually, as in the case of my course, without derivation or justification), but students already have other computational methods for solving problems that they are taught to solve using determinants. Even without considering the misleading features of determinant instruction, its lack of utility for those students who will go on to further mathematical study makes it suspect.
As for students who will not go on to further study of mathematics, I cannot think of many topics we could include that would be more useless to them than determinants. Those students are scarcely likely ever to apply any computational strategy they may have learned to solve three simultaneous linear equations in three unknowns, let alone a second technique. Unlike the group (a) students, group (b) students are unlikely to be misled as to the significance of determinants; but that is only because they will never apply, and will probably forget, whatever they have “learned.”
In short, studying determinants in algebra II is useless for most students, and misleading for the rest. That study should be removed from the high-school mathematics curriculum in favor of some other topics helpful to at least one of groups (a) and (b).
If only determinants are taught, then I am in agreement with you. See my reply just above as well. My original comment was about matrices as a whole, not just the determinant operation. As for a 3 by 3 system, I would avoid determinants except to show them it is possible to solve this way; I also point out to them that there are much better ways but they are beyond high school and not necessary at this point. I would rather spend more time on matrices (since that is truly advanced algebra) and less on functional analysis and trig in algebra 2, but not visit it again – except for vectors – in precalculus where functional analysis and trig should be developed in greater depth.
I, too, was given the standard treatment of high school matrices: systems of equations (no vectors), determinants, cramer’s rule, but this was more than 20 years ago. In college, I discovered very few had even seen matrices until then while I felt at ease with them.
Anyway, if we leave out matrices, I would push for a good treatment of statistics instead.
There are a lot of elementary school age kids that have no trouble working with vectors; even four and five dimensional vectors. I would think that most can grasp the intuition of computing the volume of a parallelepiped. I have six kids, and the three oldest (ages 7, 8, and 10) have no trouble giving a cursory explanation of linear independence. When a child has some geometry foundations (this includes basic trigonometry) and can work efficiently with functions and vectors, then they should certainly not be barred from learning linear algebra. Not only is there no need to have a foundation in calculus to benefit from a course in linear algebra, I believe that learning the fundamentals of calculus is a richer experience for those who approach it with an understanding of matrices.
In short, linear algebra ABSOLUTELY should be taught to high school age kids (or younger), because vectors and functions and basic trigonometry should be taught to elementary school aged children.
It’s funny I was recently thinking about this now five-year-old post. Vectors can be introduced much sooner (and is in most countries) so that you can put linear algebra sooner.
Still, whenever we push something in curriculum down, we have to ask the question (which doesn’t get asked enough): what do we cut out?
So, what do you want to cut?
So I’m only a few years late to the party on this. I teach high school math. I disagree that matrices must be taught after vectors. I learned them in the same order you said you did, and I found that knowing the mechanics of row operations and matrix operations made learning vectors much easier.
Also, with app design being so easily accessible (now as oppose to 7 years ago), students can actually use matrix multiplication when creating an app.
I think the main problem is having a math teacher teach matrices without having the bigger picture in mind.
I honestly don’t mind. I’m still in high school but I have no trouble understanding inverse matrices.