# Should college mathematics always be taught in such a way that real world applications are always included?

I am teaching Linear Algebra this semester with the textbook Introduction to Linear Algebra by Serge Lang and most (perhaps all?) my students are not majoring in mathematics. As I was carefully introducing the vector-space axioms and proving the propositions and theorems, a student asked me the following question:

Shall we do less proving and learn some real-world applications?

I was shocked by the question. To me, I am teaching pure mathematics, not applied mathematics and by proving the results, I am aiming at the following:

1. To make the course logically self-contained. At least this is how I learnt mathematics: definitions, examples, theorems and corollaries, proofs, etc.
2. To train the students to think critically and logically. I believe that students should learn that they should not take anything for granted or by “blind faith”.
3. To build help the students to build a solid mathematical foundation “like a wise man who built his house on the rock”.
4. To follow the topics in the textbook. There is almost no real-world applications in the textbook that I have selected, which I believe is an excellent textbook for beginners in linear algebra.

About real-world applications? There are courses on topics such as mathematical modelling that are designed for those purposes. Now my question is: Could/Should courses such as linear algebra, calculus, differential equations be taught theoretically without introducing any real-world applications?

Edit: Thanks to everyone who commented or answered. What would the answer be if we consider other mathematical courses such as general topology, or functional analysis, or real analysis, etc.? Not sure if there are many real-world applications to those courses.

• Comments are not for extended discussion; this conversation has been moved to chat. – quid Feb 23 '18 at 22:13
• Zuriel, about your last edit: topology has applications in physics (e.g. in classical mechanics), functional analysis has applications in physics and engineering (e.g. quantum mechanics and electromagnetics), etc. Examples are endless: most if not all mathematical fields have applications, one should just have to look around. Both worlds, that of pure mathematics and that of applied sciences, would benefit from a more frequent exchange of ideas. – Massimo Ortolano Mar 4 '18 at 9:47
• Thanks @MassimoOrtolano for your comment! I never know the application of topology in physics at all. I once saw an application of point-set topology to prove the infinity of prime numbers but never thought it can be applied to classical mechanics. Do you have a link to more detailed information? – Zuriel Mar 4 '18 at 20:28
• I don't have a link, and they are connected to stuff I've studied around 25 years ago and never used again, so I'm a bit rusty on that. I have a few books, but need a bit of searching. But to give you a simple example, the celebrated hairy ball theorem allows you to prove that an integrable dynamical system – that is, a system which in a certain sense is well-behaved – is topologically equivalent to a torus (and it cannot be a sphere). – Massimo Ortolano Mar 5 '18 at 19:08
• I don't see the point of a proofs-based course in linear algebra for non-specialists. If the purpose is to teach logical thinking, why not just a course on logic? Not based on the standard FOL (a pedagogical nightmare IMHO), but a more streamlined version. – Dan Christensen Mar 6 '18 at 16:09

I have worked with a lot of students coming out of courses such as yours who:

• passed the course by blindly memorising proofs, theorems, and algorithms;
• learnt nothing (lasting) except solving some calculating exercises, a very vague idea of some terminology;
• had no idea what they just learnt, why they learnt it, and how it relates too their field of study; some had the idea that the entire point of the course was to teach them performing numerical calculations (which is nonsense, any computer is better at this than they are);
• had solidified or newly acquired math phobia;
• even if they learnt something, were unable to translate their knowledge in any way.

Courses such as these heavily rely on motivation, and most of your students have no motivation to do math for its own sake. This does not mean that you should only teach them applications or ditch your goals, but you should at least mention the applications (and motivate the general structure of mathematics).

For example, for vector-space axioms, you can motivate:

• The vast majority of what they will learn during your course is based on those axioms. This means that once they can show that some regular structure fulfils these axioms, they have the entire arsenal of linear algebra at their disposal.

• Many relevant structures can be described as vector spaces, e.g., geometrical positions, quantum-mechanical states, solutions to differential equations, Internet search engines, etc. Depending on the particular application and audience, you may briefly explain them or just leave it at mentioning them. But even the latter can be relevant, e.g., even students of physics who haven’t had any courses on quantum mechanics will likely be aware that it is a very central thing in many parts of modern physics and it may motivate them to view and learn linear algebra the abstract way and not just as a way of doing geometry.

By the way, mentioning applications can even be beneficial to mathematicians. I once sat in a course on differential geometry full of mathematicians who didn’t know that the determinant can be used to quantify volume. Moreover, a good mathematician should be able to notice whether their finding is relevant on the next layer of application, whether it is inner-mathematical or in a different field – because the people in those fields will probably not notice.

Another sidenote: If you don’t know good (educational) applications of a certain piece of mathematics yourself, you can ask on this very site or Mathematics SE.

• Comments are not for extended discussion; this conversation has been moved to chat. – quid Feb 23 '18 at 22:16

At my University, there are four different first-semester Linear Algebra courses taken by Undergraduates:

1. Math 214, Applied Linear Algebra, is "an introduction to matrices and linear algebra... The emphasis is on concepts and problem solving. The sequence 214-215 is not for math majors. It is designed as an alternate to the sequence 215-216 for engineering students who need more linear algebra and less differential equations background."
2. Math 217, Linear Algebra, is for "Mathematics majors and other students who have some interest in the theory of mathematics", and is "explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved... Throughout there will be an emphasis on the concepts, logic, and methods of theoretical mathematics."
3. Math 417, Matrix Algebra I, is "an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics majors, who should elect Math 217."
4. Math 419, Linear Spaces and Matrix Theory, "covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary."

Students may only receive credit for one of these courses, and they are designed to appeal to students with different needs and interests: most Engineering students take 417, but those majoring in IOE or EECS take 214; most math majors take 217, but Honors math majors may take 419 instead.

Notice, however, that all of those course descriptions include some reference to applications. Indeed, the textbook for Math 217 -- the course intended for most math majors -- is Linear Algebra with Applications by Otto Bretscher.

To claim that all mathematics courses must have real-world applications is obviously absurd, and is a straw man. But perhaps this mathematics course should. In general, I think it's really important to ask: Who is taking the course, and why?

Edited to add: A few further thoughts.

1. On re-reading the OP it strikes me that the 4th reason given for avoiding applications ("To follow the topics in the textbook") is really quite circular, as the OP chose the textbook, despite the fact that it avoids applications, because it is (in the OPs opinion) "an excellent textbook for beginners". If the OP thought that beginners should encounter applications, presumably a different textbook would have been selected, and then the same reasoning could be used to argue that one must include applications in order "to follow the topics in the textbook".
2. What counts as an "application" is highly subjective. I know people who regard the theory of linear ODEs as a very practical application of linear algebra, and others who regard that as a purely theoretical subject in its own right. Similarly, projective geometry can be taught as an "application" of linear algebra, but many of your students might regard it as purely abstract and theoretical.
3. The idea that pure logic is the foundation on which mathematical applications are built is ahistorical. The reason why linear algebra flourished in the 20th century is precisely because people (mainly Physicists, but also Computer Scientists and others) found it extremely useful. Without that "foundation" of useful applications, it is possible that it would have remained a marginal topic of interest only to a few specialists; instead it has become a cornerstone of mathematical thinking.
• Thank you for pointing out my flawed argument! Anther teacher in my college taught the same course with the textbook titled "Linear Algebra and its Applications". It is I who decided to change to a book titled "Introduction to Linear Algebra". He apparently favours applied maths while I love pure maths. – Zuriel Feb 20 '18 at 17:58
• +1 Beautiful, insightful, and evidence-based answer. – Daniel R. Collins Feb 20 '18 at 19:03

I believe you need to listen beyond what your student is saying. Your student is not saying "I want to do some applications in class." What your student is really saying is "I'm bored and lost and this is rapidly becoming a waste of time for me, so I'm making this suggestion because I care enough about my own learning and trying to connect with you is how I am taking charge of my own education."

Your student saying this is a good thing, because your students want to connect with you and sense that you will listen if they make suggestions for how they'd like to learn or say that they aren't learning. That is better than the alternative. I bet you the student who asked you this is not the only student who feels this way. That's how classroom dynamics often works. Teaching is hard and when done well is a two way street. This is an opportunity for you to learn from your students and discover new ways to connect the material to them. I can tell your goals are born out of a desire to see them as independent, critical thinkers. That's very wise and caring of you.

They want to see how your and their goals align, and that's not unreasonable. In fact, they are already critically thinking. The difference is that your teaching is the topic. Part of critical thinking is reflection and they are doing that. By asking these questions, they are testing you because they do not want to have to take the material you are presenting with the "blind faith" that it'll be useful to them someday. Those are already some excellent students you have to be so thoughtful!

However, know that these are your unique students and random people on the Internet can't tell you what they need to hear (and neither can your textbooks.) You can't connect with them if you don't know who they are or what they know. Talk with them, not us.

P.S. I'd reframe your question. Instead say this: Will the students in my math course this semester succeed more if real world applications are included?

• As a "bad at math" student, I find it much easier to figure out how what I'm learning is going to be applied. Never went beyond basic college algebra and stats, algebra was tough until I realized that f(x) is really public int f(int x){return x*x;} I managed. In stats, I excelled - it was all applied, with real world examples. I'm sure that the students taking calc2 & diff eq will have applications for it - modeling physics, cryptography, etc. other than it acting as a weed out course for the engineering school, etc – ivanivan Feb 22 '18 at 3:04
• There exists another possibility. The student is making the comment to be disruptive. I can't know without being in the room and knowing the student, but if the course is clearly not geared towards applications then it is just plainly disrespectful to confront the teacher in the midst of class with a demand to complete change the direction of the lecture. If the student sends an email to privately ask for that then sure, I can completely agree with your POV. In the same direction, if I was in a physics class and I asked if we could please do more careful math it would be equally absurd. – James S. Cook Mar 16 '18 at 3:16

I challenge the assertion that students need to see applications in everything.

When I first started teaching I labored under the delusion that I should explain connections to physics whenever I could (in calculus, DEQns, linear algebra etc.). Now, I think my efforts do have an audience, but not the main audience that I find in my classes. Personally, I've always had a fairly big majority of engineering students with a sprinkle of math, math-education, comp-sci, various science majors. It turns out, the fear of math is only outdone by the fear of physics. I've learned to go easy on those comments in recent years. If I was to teach an audience of math-physics undergraduates then you would hardly recognize the course in comparison to what I teach now.

My approach is simple. In math class we talk about math.

I will make time for applications in maybe 10% of the time, but I refuse to let it take too much time. If you pick a particular application then that audience will love it, but the rest of the kids are then bored, frightened or annoyed. The common thing all students have in taking a general education math course is the need to study math.

To my thinking, homework is the place for applications. How can they use math from class to solve those beloved real world problems? If this homework is recommended and not required then the audiences which express this supposed love of applications can pursue their passion individually.

There is plenty of theory in linear algebra to fill the semester. I think by in large homework is the place to play with applications. For example:

• in class I teach the method of row reduction
• in homework I ask them to find polynomials to fit data

Now, I will talk applications in passing, but not as motivation. Motivation is by in large given from mathematical questions. It might be born of an example, but not an application. Motivating a general mathematical topic from a particular application can give a very skewed origin story.

All of this said, if I was teaching "Applied Linear Algebra" then my focus is quite different. So, the question is, what is the course description?

Anyway, the assertion that math should be taught always without applications is just as absurd as the incessant drumbeat to include applications at the cost of technique and analysis. Balance, moderation, academic freedom for a better future.

EDIT: after reading some answers I noticed I missed a major point in responding to your question. By your account, you were in the middle of Lecture stating the central core definition of the course when you were interrupted with this purportedly genuine question from the student. It occurs to me it doesn't actually matter if the question is sincere or not. The fact is that this student has violated a basic part of the student/professor relationship. Asking you to change the direction of a lecture mid-stream because he is not particularly excited about the core of a class? What if I did the same in an applications class? How about in the middle of my physics professor working an example I interupt and ask why we are not focusing more on theorems of physics which apply to all problems as opposed to this particular problem which has no bearing on the larger picture? Is it ok for me to demand more theory in the midst of a carefully prepared difficult example the professor has labored to prepare? Of course not. That would show a fundamental disrespect for the professor. I'm not much for holding up professors in terms of academic rank, but for the student/professor relationship to work it is also necessary for students to consider the humanity of professors. In short, I think the best response to such a student when the question happens is: "you are welcome to visit office hours, I'd be happy to discuss applications with you there. Our focus here is theory and in time I'll show you how this theory allows endless application to applications..." something like that.

• I love your answer. Though I came across one of my students just now and he told me "I would LOVE to see applications in this course". I am struggling between doing what I love and doing what the students love. – Zuriel Feb 20 '18 at 20:18
• What kind of applications? Real applications involve a lot of computer work. Do you want to do a computer project? Is there something which meaningfully involves what we cover here and the application you hold dear? Make them do the work for thinking about it. For example, in a college algebra class after introducing functions I was once asked for an "application". I said something, I forget what, but in retrospect what I should have done is put the ball in the student's court. Tell me a real world process or problem then maybe I can tell you how functions have everything to do with that.. – James S. Cook Feb 21 '18 at 2:48
• @Zuriel make them define what constitutes "real world". See, a real "real world" example is not an example in a course. It's a job. Real applications are not so easily communicated. Sure, textbooks are packed full of baby applications and physics which is close enough to foundational to have easy math, but real processes are ugly. For example, contrast accelerator physics with representation theory. Reality, thousands of particles, statistical analysis a total mess. Representation theory: we can teach in a semester. The rep theory is used in the real world ap, but it's just a part of the real – James S. Cook Feb 21 '18 at 2:53
• @TheChef What application of representation theory to accelerator physics are you referring to? I'm no expert on accelerator physics, but from what I do know, it relies primarily on classical analysis, PDEs, and the like. – Logan M Feb 21 '18 at 20:27
• @LoganM as I understand it, all of modern particle physics in the standard model rests on various representations of tensor products of special unitary groups. There's the SU(2) electroweak stuff, the SU(3) color or flavor behind quarks and gluons and the like etc. These govern the possible interactions which are possible, sort of the high-energy analog of the allowed transitions of electrons in chemistry (which incidentally also happen to rest on representation theory of angular momentum...). I don't know precise details these days, but, that's roughly it. – James S. Cook Feb 22 '18 at 3:10

I'd like to expand a bit on The Chef's answer. Specifically, there's no need to require any kind of emphasis on "real world" applications. That is: Generally "real world" applications refers to some kind of industry mechanism in which some piece of mathematics is directly used (circuits, GPS, etc.).

A student uninterested in a particular aspect of mathematics won't suddenly gain interest if one relates said aspect to a microchip an ATM—unless the student were already interested in microchips or ATMs.

I think applications of mathematics are always worth while, but an application doesn't necessarily mean some industry is using it. An application can be something as simple as why this is a cool piece of mathematics. Does it have any other mathematical or even philosophical implications? Is there some crafty riddle that can be written or solved using this? Are there any non-obvious relationships this elucidates?

(The following is a bit of a philosophical note, but it's the reason I put this answer down: I fear that it's all too common to want to justify the study of mathematics by appealing to some kind of industrial production of something. I'd like more of an emphasis on the beauty of mathematics—its art and its cleverness. When a student asks "Why am I learning this?" he/she isn't asking "What can I manufacture with this proof?" or "What market can I enter with this equation?" but "Why should I be interested in this?" )

Should they be taught without any applications? (Note the lack of the over-limiting adjective "real world".) I would say no; at the very least, applications give us the answer to the question "Why these axioms and not some others?"

I understand why you think a "pure" mathematics course should avoid "applications". But applications don't have to be in the "applied mathematics" sense of the term; they can be examples of how studying one area of mathematics helps you understand another, or even just be interested in another. For example:

• To fully understand things like Cauchy-Schwarz, you need an intuition for quadratic functions;
• Understanding projection in vector spaces helps you understand least-squares fitting, provided you realise a set of functions can be a vector space;
• Linear regression, as long as it has multiple predictors, lets you work with any model of the form $$y=\sum_ia_if_i(x)$$, or even one that's transformable to that, such as a power law;
• Least-squares regression (again dependent on quadratics) is equivalent to MLE with Gaussian errors, and the study of Normal distributions also hinges on $$\pi$$, and it's worth understanding why;
• If you also think of differential operators as vectors, it makes sense not only of the solution spaces (and dimensions thereof) of homogeneous linear differential equations, but also how we can construct rival orthonormal bases for them;
• In many problems, the possible terms are limited to a surprisingly small set of options just by the need for indices to contract, and in particular to achieve not just a law in terms of vectors, matrices etc., but to have such laws have the right kind of invariance;
• Once you think of polynomials as vectors, you'd love to have an inner product with respect to which monomials are orthonormal, and before you know it you're defining $$\langle f,\,g\rangle:=\frac{1}{2\pi i}\oint_{|z|=1}f^\ast g\frac{dz}{z}$$, which leads you into complex analysis;
• Sooner or later, anyone who thinks about vector spaces and complex analysis will think about topology, and how much of the geometry of vectors survives there, and why certain parts don't.

I could go on.

Now, obviously you don't have time to flesh out all that in lessons when someone asks why we study vectors. But if you remind yourself of points like this before you enter the classroom, you might find one or two worth mentioning.

One of my favourite things about mathematics is that not only can two seemingly disparate areas have interesting connections, but to explain a connection you might need a third or even a fourth area. For example, why do the squares in $$\color{brown}{\text{number theory}}$$ have reciprocals summing to something connected to $$\color{limegreen}{\text{geometry}}$$, $$\color{red}{\frac16}\color{\limegreen}{\pi}^\color{blue}{2}$$? One can explain the $$\color{blue}{2}$$ in terms of calculus, but one can explain the $$\color{red}{\frac16}$$ in terms of combinatorics. (In particular, that's the right way to think about it if you want to carry it over to $$\zeta(4)=\frac{\pi^4}{90}$$, $$\zeta(6)=\frac{\pi^6}{945}$$ etc.)

• Thank you for your insightful elequoent answer, but I'm still lacking any connection to the real world with your answer. – user924272 Oct 5 '20 at 23:17
• All the things in this answer are more real than most of the things which occupy the real world. Poltics, commerce, law all subjective and based on the whims of a particular society which will soon pass away. The relationships given and discussed, the application of one math to another, these are more real than real world applications. These are lasting applications that every thinking generation of humans can enjoy. +1 – James S. Cook Oct 9 '20 at 15:28

I studied pure and applied math at A level nearly 30 years ago, and I can honestly say that although I could do the exercises, I never had it explained to me the usefulness of what was being taught. Applied was easy - there were applications for that branch.

Pure math? Well... the domain of the abstract is purely abstract.

A syllabus tells the teacher what students are expected to know / exhibit knowledge / "understand" / "utilise" / "solve" a problem, in order to pass a test.

Thats a box tick, right? Well, kind of...

Actually anchoring the syllabus to concrete use-cases seems to be a rare teaching skill in itself. The "what" is easy for any teacher. The "why" may have even eluded them as students.

Differential equations? Factorisation? Integration? Turning points in graphs? Polynomials? If only the explanations given to me actually resonated... I mean, why would anyone look at a graph and say ah, this is an X^2 behaviour, or say this is a polynomial. WHY would I need to know where the turning points are? What's that got to do with anything?

In my experience, the context of the explanations were so narrow, I simply couldn't connect.

My advice to those teaching Mathematics: Real world examples please, and have a number of them at the ready - you never know which one will resonate with the student, and when the penny drops, it will make it all worth it!

• Just to add a bit more weight to this..... Clearly I've missed a fundamental step in appreciating Maths for what it is. How can I "fix" myself to appreciate the beauty of maths? Or is it a Marmite thing - you get it or you don't? I have a good imagination, and a fair level of intelligence.... but I still feel stupid because I don't get how this knowledge will ever prevent me from getting eaten by lions. Hungry lions I get.... teach a lion math? You'll get eaten. Cheersthanksbye – user924272 Oct 5 '20 at 23:36

I think the emphasis should be on calculational manipulation, not on proofs. This is true for linear algebra, but especially true for diffyQs. The vast majority of students are not math majors and needs these tools to follow derivations and do homework in physics and engineering classes.

Proofs should be included more along the line of derivations than proofs. To motivate and justify the calculational tools learned. Not for their own lofty purpose.

Some simple, common applications should be included. The majority of the class though should just be calculation manipulation. (As word problems are harder.) The kids will get a lot of practice with equations for circuits, harmonic oscillator, feedback control, etc. later. But at this point, it is nice that they get some familiarity with just the x/y manipulations. That way it is less a challenge when they learn new technical topics and have some math chops, versus only learning it in math class. Also some techniques (e.g. second order diffyQ with constant coefficients) have key applications in multiple areas.

But a few applications should be included--emphasize those that are more easy to conceptualize. So NOT quantum mechanics. But dilution or simple circuits. (I am giving diffyQ examples as I know the topic better than linear algebra.)

P.s. The tone of your question makes me wonder if you even KNOW the applications of linear algebra or diffyQs. Maybe there is a gap in your own technical training. Also, I find a lack of empathy in that you don't think of the motivations of your students and how they may differ from yours. (Not even if you or they are right, but different strokes for different folks.)

• I know some applications of linear algebra and differential equations as I have taught both courses using more application-oriented textbooks. Now the dilemma is, I only have limited class periods. The more application I include in the course, the less time I have for introducing the theoretical part of the subject. – Zuriel Feb 20 '18 at 18:32
• I see three parts (theory, calculation, applications). That is how I framed my answer. You just keep responding as if there were only two. – guest Feb 20 '18 at 22:52
• Regarding the postscript, you can stop wondering: matheducators.stackexchange.com/questions/13628/… – Ben Voigt Feb 21 '18 at 1:45

Much of the mathematics in undergraduate use have direct application to physical applications, and I would highly recommend using them as homework examples. When they fit in naturally, it's quite beneficial to the learning of many students to link the mathematics they see to physical problems and concrete examples. This, however, is most appropriate in math courses that are given to non-math majors as preparation for physics/engineering.

The goal of most of the lower-division math courses is to get students to understand the theory, but at a bare minimum be able to use it. And on the "solve something" problems I think it's natural to set up or describe the equations or mathematics use, and then tell the student to solve it. I'd also take some pains that the problem is solvable without any of the physical descriptions (here's an equation, solve it). These lends itself probably most naturally in calculus and differential equations, but in almost every course there are generally natural applications.

In matrix algebra I do remember that many of the examples and connections to real world use are forced, and the course is fundamentally mostly one in theory. But for the nuts/bolts problems where students are introduced a topic and asked to compute something to demonstrate knowledge of the basics I think homework problems or examples can be given that are practical. In the very early parts of the course when students are first asked to actually perform a matrix multiplication it could be good to note that these can be used for scaling, shearing, inverting or rotating objects. When students are first taught to do determinants, you could give a homework problem that asks them to compute a 2x2 and 3x3 determinant that happen to be the Jacobian matrix for polar and spherical coordinates (you could also if desired mention that the matrices are Jacobian matrices for transformations for interested students to look up, but these are not the focus of the course). To mathematicians I suppose this wouldn't be the point, but for the future engineering student this is literally what they need to know.

But at some point in a course in linear algebra it descends into honestly, a course in theory, and there will be fewer and fewer direct computation, and concrete examples will start requiring domain specific knowledge and distract from the point of the exercises.

Keep in mind a lot of courses at the first and second year (and sometimes third) are taken by scientists/engineers, and the goal is mostly to teach enough mathematics that they can perform calculations. While these are math courses, they're used to serve their needs as well.

I will dodge the "should" part of the question, and try to address the "objective" issues.

If you want to teach a class with fewer proofs and more applications,"Serge Lang's book is not the way to go. The math books that are most concerned with "applications" are the ones written by and for engineers.

Serge Lang's Linear Algebra book, like all his other books, is more about "why" than about "what." That is, they have a lot of proofs and relatively few real world applications.

Students in my gen ed college math class often ask me "what is math good for?" What they mean is "what information will this class provide that I can use in other classes, or at work, or at the casino, or managing my 401-K" I have come to the conclusion that there is nothing that I can teach them that wouldn't be better served by giving them an appropriate app for their smart phone. I have put together a power point presentation that illustrates this that helps them to learn critical thinking skills they can use in the class as well as after they leave the class. Of course most of my colleagues want to burn me at a stake for disrespecting our discipline. I hope it doesn't upset you too much: http://faculty.ccc.edu/jnadas/FDW-2019.pdf