What Basic Math Skills Should Be Expected of Students in a University-Level Linear Algebra Course?

Question

I am currently teaching a university-level linear algebra course and recently encountered an issue that made me question the assumed foundational math skills for students in this course. The issue arose during a quiz where students were required to find eigenvalues of a 3x3 matrix, necessitating the factorization of a cubic polynomial:

$$ −λ^3 + λ^2 + λ − 1 $$

One student challenged this, pointing out that extensive polynomial manipulation was not covered in our syllabus, which led me to reflect on what foundational math skills should reasonably be expected of students entering this course.

What basic math skills (e.g., polynomial manipulation, trigonometry, complex numbers) is reasonable for us to expect students to have when they enroll in a linear algebra course at the university level?

And when student challenges our assumptions by saying "it's not what the course is about" when they fail to carry out basic mathematics task, what should our response be?

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Update: The same student took another course I am teaching. This time I only give the students problems that requires factorization of polynomials of degree $2$. The student still does not know how to do it because there is a $9$ in $9 x^2$. 😃

Answer 1

What basic math skills (e.g., polynomial manipulation, trigonometry, complex numbers) is reasonable for us to expect students to have when they enroll in a linear algebra course at the university level?

Yes, all those things should be reasonable expectations for students coming into a linear algebra course. If it's a standard topic in Algebra 2 or lower, then by all means it is reasonably fair game for university-level Linear Algebra.

Here's the thing though: so many realities within math education are completely unreasonable. In particular, students often lack unreasonably much foundational knowledge coming into courses, when confronted with this issue, they often unreasonably argue that the course should drop to their level instead of them rising to the level of the course.

How to actually fix the problem

I've had to deal with something like your situation before. The remedy isn't easy. If you really want to fix the problem, then you have to hold your students accountable for learning the material in the course, including the prerequisite material that they are missing.

But here's the thing: it's not reasonable to expect your students to learn things on their own -- and it's entirely possible that they were somehow able to pass Algebra 2 without demonstrating sufficient mastery of the material. (The usual culprits are grade inflation and/or lowering of standards.)

So, if you want to actually fix the problem, then unfortunately you have to put forth a ton of effort supporting students through remedial assignments/assessments AND help sessions, while simultaneously "holding the line" in a hardcore way. You have to be a supportive hard-ass. It's a delicate balance.

What usually happens instead

What usually ends up happening instead is everyone turns a blind eye and pushes the problem further down the line: the instructor just gives the usual lectures/assignments, curves (or otherwise inflates) the grades, students go along with it, and the problem is left for their next instructor to deal with (or not deal with).

Response to a student

And when student challenges our assumptions by saying "it's not what the course is about" when they fail to carry out basic mathematics task, what should our response be?

"Go look at the standard linear algebra textbooks. Do they cover computing the eigenvalues of a 3x3 matrix? You bet. Why? Because that's a fundamental skill.

If I only give you 2x2 matrix problems and don't make you learn this fundamental skill, then not only am I neglecting my professional duty, but I'm also setting you and all your classmates up to fail in whatever math course you take after this. And I'm not going to do that.

What I'm going to do instead is support you in learning this stuff. I'm not going to lower the bar for you, but I am going to help you rise to meet it.

So why don't you come to my office hours on Tuesday and we'll go over how to factor a cubic by grouping. That was covered in Algebra 2, so you're probably just rusty. And even if you somehow got through Algebra 2 without having learned that, then don't worry I'm willing to work with you to get you back on track.

But remember, this is a two-way street: I'll coach you, but the only way coaching works is if you show up and put the work in. So I'm expecting to see you there Tuesday."

What about students who don't accept your support and still complain?

Not your problem. Don't worry about it. You can lead a horse to water, but you can't make it drink.

As you probably know by now, in the teaching profession there are going to be plenty of students who use you as a scapegoat for their own laziness. You just gotta develop a thick skin towards these things.

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Edit: In the advice above, I'm assuming that you covered the case of 3x3 matrices (that result in factoring by grouping) in class.

It's come to light in the comments that the exercises in your book were all 2x2 matrices, and that makes me question whether you covered the 3x3 case in class. If you didn't, then the student is justified in their complaint.

I agree with what JonathanZ says in the comments: "make sure that the material on the exam is a subset of the [types of] problems they've worked on during homework." This follows as a consequence of "it's not reasonable to expect your students to learn things on their own."

Answer 2

For what it's worth, when I cover eigenvalues of a $3\times 3$ matrix, I only assign matrices where either zero is an eigenvalue, or there is a helpful factor like $(1-\lambda)$ that stays on the outside when computing the characteristic polynomial. Either way, factoring is easily doable.

Your example polynomial is quite simple if one is comfortable with a guess-and-check approach, but the majority of students aren't. They need an algorithm.

Sure, one could remind them of Descartes' rule of signs and the rational root test (at least in the US, you can't count on students knowing these, because at most they are covered once in high school, briefly). But this doesn't feel like the best use of class time. And even if one knows these tricks, the case of a general cubic is still inaccessible.

More broadly, we have to keep in mind the actual purpose of by-hand calculations in the 21st century. It is not so much about getting the answer--we have convenient access to powerful computational tools that can give us the answer. Instead, solving problems by hand serves other purposes, including:

• Understanding how the method works, how it can go wrong, what kind of answer to expect, etc.

• General fluency and comfort with the relevant mathematical objects.

• A concreteness (for lack of a better term) to one's understanding, that can only be gotten by solving specific examples with actual numbers.

(Of course, some things are faster/better to solve by hand, if one knows how. But I'd argue that 3 by 3 eigenvalues aren't an example of that.)

From this point of view, students' lack of basic computational skill doesn't have to be a total disaster. You can compromise on computational stuff because they'll be using software to get the answers in the real world anyway. Compromise doesn't mean give up totally--for the reasons above, by-hand exercises play an important role, but don't need to be as comprehensive as they did 30 years ago.

Answer 3

One student challenged this, pointing out that extensive polynomial manipulation was not covered in our syllabus, which led me to reflect on what foundational math skills should reasonably be expected of students entering this course.

It sounds like you want to compile a list of what basic skills students will need to use in your course. Then instead of assuming they have those skills, you'll want to let them know what you expect. One option is to publish that list in your syllabus so (a) students can get ready, and (b) your conscience is clear in case of complaint. Another option is to do some just-in-time review when a "new" pre-requisite skill is needed.

Since your term is already in full swing, I would encourage you to use this semester to gather that list. Look at every assessment in the course and try to imagine a non-expert explaining their steps. They'd probably say things like "I used SOH-CAH-TOA" or "I had to factor a polynomial". That will take time, but having it will pay off in the long run.

The easy approach is to just put the list in your syllabus, and label it something like "Essential skills you need to have to be successful in this course". If you're feeling extra helpful, provide a resource for students to brush up.*

As for the particular list of skills, that depends on a lot of things (e.g. the course pre-requisites for your linear algebra, the textbook, your individual focus as an instructor, etc.). It's hard to imagine folks here being able to give you that list without knowing where your students are coming from. Yours is an important question in general, though -- what can we expect from students starting any class?

[Added later: After posting this, I realize it really isn't a proper answer since you are asking for the list of skills. [Downvote as needed.] However, I think it will be more helpful if you make your own list, based on what you are currently teaching and where your students are coming from.]

Answer 4

From the point of view of teaching, I would take a different tack. I'd say one of two things are likely:

1. The student was genuinely surprised by being expected to know something they did not know or had forgotten.

2. The student is grubbing for a higher grade.

A student might be naturally upset or frustrated in such situations; but the student might not be seeing the problem clearly, and the teacher does not have to address the issue on the student's terms. But each such occurrence should prompt the teacher to review whether they have met their responsibilites and whether they can help the student to better understand a student's responsibilities.

I had a student once complain that computing a difference quotient for $f(x) = x/(2-x)$ on the test was unfair because the homework had a problem like $f(x) = 1/(2-x)$ but not a problem with an $x$ in both the numerator and denominator. So I adjusted. Later I had one ask, "Where do I put the $+h$, with the first $x$ or the second one?" So I adjusted again.

I think both cases 1 and 2 can be handled in a similar way. However, in case 1, the teacher should understand their role. The student should not be surprised. They may be surprised if they did not pay attention (the student's fault) or if the course up to that point did not check their backgrounds (the teacher's fault). If it seems a problem — and the OP has identified it as such here — then the teacher ought to consider the examples covered in class and in the homework. In a comment, the OP says, "The exercises in the book are all about 2 by 2 matrices." A student might infer that's all they have to calculate.

The point is to head the problem off before it arises. This is best done if the student is shown (discovers) their deficiency through their own thinking and work in time for them to fix the problem. Good problem sets are important. As my example and the OP's quote show, textbooks cannot always be relied on to have good problem sets. I often supplement with my own problem sets (or a colleague's, whose happen to be excellent). But if you're lucky, you can find a good textbook. Or you can pay close attention to the problems in the book and keep within the confines inferred from them.

Answer 5

It's not reasonable to expect anyone to know how to factor cubic polynomials by hand. The fact is, you need to hand-pick the polynomial in the first place. In the general case, there is a cubic formula nobody memorized. You use a computer.

I am self-studying linear algebra as an adult using Strang's textbook, having taken linear algebra in high school after getting a 5 on the AP calc exam, and taking Calc 3. I am pretty sure even decades ago, factoring cubics was not taught in my algebra classes (for the smart kids who would go on to finish AP Calc BC in junior year). If they came up, the answer was, graph it, and find the intercepts. That's real algebra, by the way, graphing it and seeing what the answer might look like.

Even Strang shies away from 3x3 matrices in the eigenvalues chapter. As he says in one of his lectures, life is only so long.

Are tricks for factoring some special polynomials part of linear algebra? No. Not really. Will it help them write computer algorithms for computing eigenvalues? No. Will it help in writing proofs in the second linear algebra class, or analysis, or abstract algebra? Maybe?

Answer 6

This answer assumes that the student was acting in good faith and has an honest interest in mastering the material. I leave it up to you to decide whether that is the case here, but I would urge you to always assume best interest on behalf of your students unless you have strong evidence to the contrary.

The issue arose during a quiz[...]

This is an essential part here. Apparently the student was unaware that the skill that was required to solve the exercise (factorizing a polynomial) was required for this particular quiz (about matrix eigenvalues). This can be a particularly frustrating situation for a student, as even if they did spend considerable effort into mastering the material, you managed to hit a blind spot here by assuming knowledge of a skill that the student was not aware of would be required for solving the quiz.

There is several questions that arise from this:

Is solving polynomials actually part of the set of skills that you wanted to cover with the quiz?

If not, then simply give the solution to the solved polynomial as part of the text of the question. This is similar to how physics exams sometimes give solutions to particularly tricky integrals in the question text - the skill that you are trying to get across is how to model a particular physical problem, not how to solve a nasty integral.

If it is actually a relevant portion of the topic you are teaching, then it was obviously not clear to this student that that was a required skill, which is a problem.

Did the student pay attention in class?

I'm going to assume here that they did. If not, that's not your problem, but this doesn't sound like that. This sounds like a student that did their due diligence by reviewing the material and now felt sucker-punched by a topic that they thought to be unrelated to the contents of the course and failed to review.

The easiest way to avoid these kinds of problems is by making sure students have seen the technique that is required for the solution in the course - if you cover a similar or different exercise in the lecture that requires the same skill, students will have seen the technique and know about its importance in the context of the course. It's okay if you don't have time to explain the whole technique from scratch, but if you do decide to gloss over, your students may appreciate you giving a reference to a text book where they can look up the details behind the technique again in case they forgot (or never learned it in the first place).

Where to draw the line? I can't repeat/give references for every single thing I show!

This is the truly difficult question. There's certainly some knowledge you have to take for granted, you can't start every course at Algebra 101. But there are certain topics that may be worth additional effort:

• Is this something that students certainly have seen before, but maybe only once? Even if you had a good grasp of a technique when you wrote an exam on it a semester or two ago, our memory is not perfect. Without sufficient repetition, we forget. Cut your students some slack for things that they may have only seen once in their studies and didn't have a lot of opportunities to apply since.
• Is this something that students typically struggle with in the course where the skill is introduced? We all know that there is certain parts of a curriculum that come easier to students than others. Some parts of the exam almost all the students get right, while in others, even the good ones make mistakes. Keep in mind that you don't have to master the entirety of the material to pass the exam. Be wary of those places where students have previously been prone to taking shortcuts by not reviewing difficult sections of the curriculum. Give them a chance to fix the gaps in their knowledge by mentioning it again in the course.
• Are your students just not as good as you thought? Not all classes are the same. Sometimes the whole class is just not very good on average, sometimes you have a few students that are struggling. You can't save everyone, and your job is not to make sure that everyone passes. But ignoring the specific situation in a class is also not right. Be transparent about your expectations and be helpful if students are struggling and asking for help, even if their tone when asking strikes you as inappropriate. Acknowledge that no student is perfect and make it a priority to help them overcome their shortcomings.

Answer 7

Some calculus courses, often two or three, are generally required to take linear algebra even though it's sort of a different subject. I didn't know one could pass Calc 1 and not have enough facility in algebra to notice the factorization of the given cubic. Calc 1 is a filter for those students who lack basic skills in high school algebra.

If it's a question of dealing with a troublesome student's objections (that's not fair! rather than being embarrassed and asking for help with this problem they'll see was simple in retrospect) I think one can still use phrases like "mathematical maturity" expected to be gained by completers of such courses, and point out that any class won't teach every single exact question you might need to answer with the material. I generally try to be somewhat unhelpful to such students, because I can't imagine facilitating them going on and bringing their helpless attitudes to even further courses. My intention is that the answers I give will be useful if they have a decent attitude, useless otherwise.