What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?

Question

I have helped to TA and taught several courses with mixtures of advanced undergraduate and early graduate students in engineering/STEM. These courses are the classics: signal processing, control, optimization and (more recently) machine learning. The mathematics used in these courses run the gamut from linear algebra to multivariable calculus to probability and so on.

I am noticing that students coming into these courses (upper-year, shy of graduation, or beginning graduate/master students) basically have very poor ability in manipulating basic mathematics that they were taught and used over the years.

Some common problems:

1. No respect for dimensionality. Suppose $Av$ is a column vector, students will equate it to a row vector, a matrix, or even a scalar. This is by far the most common one.

2. Wild manipulations, such as freely dividing a matrix $1/A$ and basically treating a matrix as a scalar. Similarly, freely dividing vectors, raising vectors to powers, etc.

3. Incorrect beliefs, such as, "if a matrix $A$ is symmetric, then it has an inverse."

4. Some students are coming into advanced mathematics heavy courses without basically any background.

I am really at a loss as to why this is happening and what should be the correct remedy, as it seems that it is not easy to address this late into their study.

Also there seems to be no improvement despite feedback and I'm constantly being surprised by some "new mathematics" that the students would invent in the homework or tests.

Answer 1

I am really at a loss as to why this is happening and what should be the correct remedy, as it seems that it is not easy to address this late into their study.

Also there seems to be no improvement despite feedback...

Why is this happening?

It's likely the result of a lack of learning in prerequisite courses. It sounds like students were somehow able to pass the prerequisite courses without demonstrating sufficient mastery of the material. The usual culprits are grade inflation and/or lowering of standards.

What should be the correct remedy?

The "correct" remedy is to hold the students accountable for learning the material in your course, including the prerequisite material that they are missing.

This requires the instructor 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 (i.e. you have to be a bit of a hard-ass). And of course it requires a ton of effort from students because they have to put in the extra work to pay off their "learning debt," which is usually a rude awakening for them.

But 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).

Answer 2

I'd like to offer a bit of a frame challenge here. Whenever I made these sorts of mistakes during my mathematics education it was normally because I didn't understand what the symbols in the more advanced course were referring to. This often occurred when switching between courses which used different notations for the same material.

In all fields there is an implicit understanding of what type of objects are referred to by different classes of symbols. There is also a set of conventions about what symbols and operations do not need to be explicitly written down. Both of these seem obvious to experienced practitioners, but are really confusing to newcomers.

I'd go over all the symbols you use, reiterate what type of objects each one refers to and let your students know how you determined that.

Answer 3

I deal with a dash of this phenomenon in the context of teaching an "advanced programing techniques" course (i.e., 2nd semester programming) in a community-college CS program. There's frequently pretty poor preparation in the first semester, and I'm considered to have rigorous standards for our environment. Compare the current question to mine from four years ago: How to deal with radically different prerequisite preparation?

Here's a few things I do to try to help students get up to speed. They're not all usable in your situation, and none are a silver bullet, but together they give some incremental improvement.

• Give a brief overview of expected prerequisite knowledge in the first week (no more than that), referencing where to go for a deeper dive.
• Provide any summary material from prior courses (e.g., formula card or programming language quick reference card).
• Make digital lecture slides from prerequisite courses available for optional review.
• In early assignment specifications, explicitly call out book sections from prior courses they should review (e.g., "For proper input validation, you may wish to review Gaddis C++ Sec. 5.3").
• Find digital tools that can double-check work in some fashion (e.g., I wrote a basic C++ style checker and require students to use it, here).
• Uphold standards in grading, and give clear feedback to students about the parts at which they're not succeeding (again, referencing prior-course text sections as appropriate).

Full disclosure: One advantage I have is that the 1st-semester and my 2nd-semester course use the same textbook, and I've taught the prior course myself, so it's completely straightforward to expect students have access to the textbook and I can reference earlier sections at will. Maybe this convenience sample points the way to some things that help.

Answer 4

In his 2005 article "Reconceptualizing Procedural Knowledge," Jon Star argued that there is such a thing as "deep procedural knowledge," which he describes as follows.

Deep procedural knowledge would be knowledge of procedures that is associated with comprehension, flexibility, and critical judgment and that is distinct from (but possibly related to) knowledge of concepts.

I would argue that mistakenly thinking $Av$ is something other than a column vector demonstrates a lack of deep procedural knowledge of matrix multiplication. Similarly, mistakenly "dividing by $A$" demonstrates a lack of deep procedural knowledge of matrix inversion and solving matrix equations. Students may have seen the ideas of "linear algebra," but they haven't deeply learned the procedures of "matrix algebra."

What's the remedy? In addition to Justin Skycak's suggestion to hold students accountable for learning prerequisite material and to support them with remedial practice, I would recommend that, in your instruction, you work through examples that feature the kinds of procedures that students struggle with. As you do these examples, you should describe your thought processes out loud. This is a kind of cognitive apprenticeship.

In addition to any extra practice resources you decide to provide for students, I suggest incorporating procedural practice on homework—a few short exercises at the start of the assignment, for example. Exercises help students develop and maintain fluency, and you can also ask questions that push them towards deeper knowledge of procedures. You might ask students, "Given the matrix equation $AB=C$, if $A$ is a $3 \times 4$ matrix and $C$ is a $3 \times 1$ matrix, what do the dimensions of $B$ have to be?" Or "Does the matrix equation $Ax=b$ always have a solution?"

Answer 5

In my experience mathematical tools are pretty much like learning a language. You won't get fluent if you don't practice a LOT. In high school this is enforced with constant homework, tests, solving exercises in class etc. At university level there is no such baby-sitting, but the students still have more or less the same mindset and maturity level so they do the bare minimum to pass the classes. If the only solution the university has is to inflate grades and make the exams easy, you have your problem.

Give them a homework sheet at every class + a short test similar to the previous sheet (making sure they don't rely on sharing answers for the home stuff), both rigorously graded. I see no way around. A few months of this treatment should help them tremendously.

Answer 6

Here's a breakdown of the problems:

Some common problems:

1. No respect to dimensionality. Suppose $Av$ is a column vector, students will equate it to a row vector, a matrix, or even a scalar. This is by far the most common one.

One thing I would suggest is if this is a common problem, then why don't you show them why $Av$ isn't a row vector, matrix, or scalar.

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For example: Consider the column vector $Av$ with $A=\begin{pmatrix}6\\-5\\4\end{pmatrix},v=2$. Then, part of your problem is done (they shouldn't possibly be able to equate it to a matrix of some sort, since this is a $1\times3$ column vector.) Then, if they have taken some sort of Linear Algebra class, then it shouldn't be hard to get them back on track from there with a quick review on how column vectors work.

2. Wild manipulations, such as freely dividing a matrix $1/A$ and basically treating a matrix as a scalar. Similarly, freely dividing vectors, raising vectors to powers, etc.

Again, it shouldn't be that hard to review with them why they can't freely divide any matrix $1/A$. An easy way to review is by showing that if $\operatorname{det}(A)=0,A^{-1}=1/A$ is indeterminate. For example, if we take the $2\times2$ matrix$$A=\begin{bmatrix}2&3\\4&6\end{bmatrix}$$since we know $\operatorname{det}(B)=ad-bc$ for any $2\times2$ matrix $B$, and then inverse of any matrix $B$ is defined as$$\dfrac1{\operatorname{det}(B)}B$$we have that $A^{-1}=1/A$ does not exist since we have$$\operatorname{det}(A)=ad-bc=2\cdot6-3\cdot4=12-12=0$$

3. Incorrect beliefs, such as "if a matrix $A$ is symmetric, then it has an inverse.

Again, you can review with them that if the determinant of a matrix is equal to $0$, the matrix does not have an inverse.

4. Some students are coming into advanced mathematics courses with basically no background.

I honestly feel like this is way above my jurisdiction to provide an opinion on. You should probably discuss this with the school board if you feel like there should be more restrictions on taking these classes that lean heavily on advanced mathematics, however that is only my opinion.

Also there seems to be no improvement despite feedback and I'm constantly surprised by some "new mathematics" that the students would invent in the homework or tests.

I feel like this is definitely something you should expect if you have students with no background in mathematics coming into class that lean heavily on, of course, courses that rely heavily on advanced mathematics. I'll even admit, I've even done that before on M.SE when, of course, I was trying to do something that I didn't know what to do. (An example of this would be on this question of mine) I would recommend checking in with each student individually (if you could do that feasibly) and ask about what they might have meant by the "new mathematics".

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So, what's the correct remedy? Most likely just reviewing with your students the basic mathematics for each section that they will need to know throughout the course to make sure you really drill it in their head. Otherwise, you're probably going to be having this problem. If any of the students does some "new mathematics" that you don't understand, then try to chat with them privately at some point to ask about what they meant by their "new mathematics" that they put on their homework/test.

Answer 7

The problem is with the prerequisite courses, so if you want to fix it, start by talking to the people who teach the prerequisite courses. If you're not comfortable with that, raise the issue with a trusted colleague who has more seniority and/or is on an appropriate committee (e.g. undergraduate education committee).

Answer 8

A lot of students can graduate STEM having taken only a single Linear Algebra course. Yes there’s some LA in differential equations or abstract algebra (less sure about the engineering side), but suffice to say this is not a lot of practice. And it’s practice from a single main source. It could be that almost every matrix in their differential equation course was invertible, their LA text focused on symmetric matrices for a whole chapter, they worked heavily in Matlab or similar, etc.

I was never taught what a matrix was until college, not a single lesson. I’m about to finish STEM undergrad and that single LA class is doing a lot of heavy lifting. Since all of your points revolve around LA, I’m willing to believe they just need a refresher/reinforcement, especially since the points you make aren’t too advanced. It’s a lack of exposure. Another answer gave a quick fix for most of your points. That’s probably all they need, a small refresher and transition to new your class’ notation.

Answer 9

Having taken courses that require linear algebra ~10 times, I can tell you that even performant students need a refresher on linear algebra when they are coming into a class that requires linear algebra, no matter how many times they have had the refresher, or how recent their linear algebra class was.

There is so much jargon, gotchas, caveats, and theorems in linear algebra that it isn't reasonable to expect students to recall these the moment they need them months or years after they last heard the words.

Every such course should start with one or two weeks of refresher on basic linear algebra, covering every topic needed for the course. This refresher should be based on a concise, clear, well-formatted minimalistic source that they can reference throughout the course. Such linear algebra refreshers are common place. My personal favorite is the first chapter of Griffith's Quantum textbook, but I'm a physicist.

Answer 10

Your first three examples are linear algebra. Example 4 might or might not be, hard to say...since it is not an example but a vast generalization.

Probably one issue is that most ee students don't really need or do much LA. At least not to the extent that they do normal algebra. And not to the extent that mathy types think they do. Many EEs don't get a full semester, just a few weeks within an engine math survey class.

CS probably do get LA course but are less bright and less mathematical in general than EEs. And then you're talking MS students, not Ph.D. add onto that the hype around machine learning. It's zero surprise that if you have hard-core LA multiple regression stuff that they will choke.

In terms of the solution? I would say either weaken the curriculums. Don't scoff, look at stats or even physics which can be taught with lower math demands versions and still significant useful learning. Or...keep the MS as is and just let the weeding sort itself out. I doubt remediation, at that level is practical on a population basis, i.e. in general. Nor is fix the pipeline a practical option for this dealing with the output.

Answer 11

It seems to me that your students have not performed too much matrix multiplication by hand (using pen and paper). Perhaps it is time for some computer assisted algebraic operations.

Also, the fact that students mix so easily different types such as scalar, vector, matrix, made me think of an analogy to data types in programming languages. In a programming language, one cannot mix different data types, unless these types are somehow compatible.

This leads to the suggestion of having some practical sessions on matrix multiplication in a programming language. (Certainly, your students must have come across at least one programming language, given their technical background)

The advantage of a programming environment is that you get an error when performing some illegal operation, such as multiplying matrices of incompatible dimensions, or the division 1/A, where A is a matrix. Basically, it is the software that tells you when you are doing something wrong.

A second advantage is that, after declaring a variable of some type, you can display it (print it to the console), thus allowing students to notice that a matrix is a totally different thing from a scalar.

Most importantly, you don't even need to go deep into the language syntax. In fact, you barely need the basics of a language in order to perform these algebraic operations.

A demo with Julia:

Answer 12

There are many good suggestions made concerning this topic .. What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations? ..

Some common problems:

• No respect for dimensionality. Suppose Av is a column vector, students will equate it to a row vector, a matrix, or even a scalar. This is by far the most common one.

• Wild manipulations, such as freely dividing a matrix 1/A and basically treating a matrix as a scalar. Similarly, freely dividing vectors, raising vectors to powers, etc.

• Incorrect beliefs, such as, "if a matrix A is symmetric, then it has an inverse."

• Some students are coming into advanced mathematics heavy courses without basically any background.

I am really at a loss as to why this is happening and what should be the correct remedy, as it seems that it is not easy to address this late into their study.

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As I have ADHD, I have found that I have had to build my own math books. There are symbols that I don't understand in math, and converting them takes time, thus by building my own math book using a clearer / cleaner language of symbols makes my life easier when it comes to math.

It takes longer to learn math concepts. This is due to how my brain processes math information. The brain processes language in one area of the brain, and math in another. Language is processed as a series of pictures, like asking a child to spell a word and sound out the word. They are deconstructing the word to understand how to say it and what it means.

Math requires the pictures of equations to be deconstructed. The more letters in the picture / equation, the more frightening the equation becomes. Most programmers misunderstand math equations as assignment statements, and should be reminded that they are relationships. This will confuse them until you show how the equation can be solved for different questions based on which variables are defined.

Another difference between the math center and language center is that the math center is a true false engine. If you give a student a rule and ask them to apply it, they won't know if they are applying it correctly until after several hundred problems have been completed. You can see this with 1st and 2nd graders working out simple math problems, ask them .. is "1 = 2 - 1" the same as "2 = 1 + 1" or "1 + 1 = 2"? ..

The next part is that physics problems require multiple equations manipulated in the right order to get the right answer. This requires a strategic thinking, which many students are not exposed to at all. To help with this, ask the student to write down the problem, the equations they think are needed to solve the problem, all the constants that they know, and their strategy to use this information to solve the problem.

As for linear algebra, I found this to be fun. I liked the repetition of doing the problems using the same basic strategies, over and over again. One of the things I really wanted to explore with linear algebra was kirkoffs rules (sp). I had a horrible time trying to complete the equations by substitution, but you needed calculus before you could take linear algebra.

To summarize ..

• ask the student to write down a strategy to solve the problem

• realize that math center in our brain is a true false engine that requires reinforcement learning

• provide a lot of similar problems for the student to work on

• help them to learn and differentiate the aspects you are seeing lacking.

• build a separate book for them to work through, if you're going to be teaching the same class over and over for a few years.

Hopefully, this will help you understand why students are behind and how to bring them up to speed.

Answer 13

Have you actually looked at linear algebra textbooks? As far as I can tell, they are uniformly bad. It is a really easy subject made incomprehensible by authors. I never took the course, but I got a perfect score on my prequalification test by teaching myself—mostly by solving lots of problems.

Answer 14

I am an engineer who drove math professors and tutors insane. I understood some difficult concepts but struggled with easy details. I did not know it was possible to simultaneously be gifted and disabled.

You will have an easier time teaching engineers if you can identify their learning style(s).

Learning Styles—Visual

Here's one technique based in neuroscience:

Ask the student to visualize a tree and describe their thought process out loud. What did they see or hear?

If they are like me, they saw nothing. One of my disabilities is aphantasia. I have never seen an image in my mind. It is impossible for me to visualize any math concept like you or 97% of other people.

Before I knew I had aphantasia, I would've described a symbolic tree. I "visualize" by feeling coordinates in the space around me. The result is similar to an outline minus visual detail.

To visualize a math concept, I need an external tool. I could draw it on graph paper, use software to model the result, or watch a video which shows visual models of the concepts. Interactive three-dimensional models online seem especially helpful.

If the student describes a super detailed individual tree, they are probably a visual thinker. An engineer who thinks this way can potentially imagine an entire machine in their mind and disassemble it to test each part. If this is your student, try to describe every concept in significant visual detail.

If the student begins to visualize a tree by hearing their own voice describing the tree in words—or they begin talking to themselves and are clearly thinking in words first—they may be a verbal thinker. People who think in words tend to be linear, sequential, and detail-oriented—Step A goes to Step B. They may need you to specify every detail in a step-by-step order, including details which might be obvious to most other people.

Learning Styles—Kinesthetic

My primary learning style is "kinesthetic"—I learn with my hands. Here is an example of a kinesthetic method for learning calculus—three-dimensional printable calculus models

Kinesthetic learning could involve using the human body to understand math concepts. Here are some examples: I might draw math with my hands in the air, form an odd shape with my hands or legs, or dance the patterns. I could play an instrument like a piano or keyboard in a specific way related to math concepts. I could make visual art which is based in mathematics like origami or data visualizations.

Video games where my character walks around in multiple dimensions (three-dimensional, etc.) to explore concepts are very helpful. I enjoyed studying math concepts which related to the physics in video game environments.

Learning Styles—Visual Spatial

My thought process is apparently called "visual spatial" or "pattern thinking"—common in engineers.

If the engineer happens to be female like I am, she may also have an inverse learning style. Women are more likely to be "holistic thinkers". I need to understand the big picture of any topic before the details.

I expected my first programming class to start with an overview.

• What are all the languages we could use?
• When would we choose X language over Y?
• Which approach is the most efficient to solve the problem?
• Which is easiest for another person to understand?
• How difficult are these languages compared to each other?

I did not begin to understand math until I read highly theoretical math proofs and lectures by PhDs about topics which are far beyond my level of formal education or knowledge. Math classes did teach me some details, but I hadn't understood how those individual symbols and concepts connected to the big picture.

If engineers do not understand the patterns which unite all concepts you are trying to teach them, some may have an unusually difficult time remembering easy details.

How did I end up learning any advanced concepts without a foundation in the basics? Funny story. I'd never taken math beyond algebra. On my first college placement test, the result said calculus.

I can almost always pick the correct multiple choice answer on even an advanced math test. If you ask me the answer to a difficult math problem, I can sometimes tell you the correct result.

Unfortunately, I frequently cannot explain how I know the answer. I don't understand the details of math languages very well yet.

I also have dyscalculia. If a student often accidentally transposes numbers, they may have this challenge too. It's frequently impossible for me to do basic easy math in my head. Dyscalculia can result in some challenges with understanding left / right or east / west concepts and aspects of directions in the real world. Since I used software for basic math, like calculating tips at a restaurant, I didn't understand that this was one of my challenges.

You can thank hyperlexia for the length of this post—"too many words". If your student has hyperlexia, they usually talk a lot. They may struggle to summarize math concepts in a linear way like Step A leads to Step B. My thought process is associative, meta, and theoretical. It is a bit similar to the branches of a tree or a fractal pattern. To summarize in any language, I often need to see every detail visually first. Afterwards, I try to arrange those details in order like solving a puzzle.

Neuroscience research indicates that disabilities are highly related to stress. The more your student feels like they are failing, the more they might struggle with any unknown learning disability. Unfortunately, I was so frustrating to educators that even incredibly calm and patient TAs and professors sometimes began shouting profanities at the top of their lungs. I used to joke to friends that I must have a superpower which made otherwise rational mathematicians lose their minds. Please try not to give up on the idiot engineers because apparently a few of us might be secretly good at math—we just think differently than most. ;)

I hope any of this was helpful. If anyone reading this might like feedback on specific math learning materials, feel free to contact me. Nobody knew I had any learning disability, but I have at least six, if not more.

I think more students might love math if they were asked to write their own equation. I fell in love with math last year when I tried to write a symbolic equation which physicists apparently call the "theory of everything".

Since I'm challenging for math educators, I'm hoping to eventually design a custom degree and teach myself with expert guidance. However, if anyone reading this might be interested in trading your math teaching skills for a website / social media / marketing or other technology-related tasks, I love the idea of a trade with a real mathematician. I am a usability engineer, but I studied many topics, so I could likely find something useful to offer you. =]