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This is something that I felt like was difficult for me in some classes, especially lower division differential equations and linear algebra classes. I know professors want to motivate certain topics but I feel like offering motivation honestly confused me more than helped, and I see this with some students I tutor.

Now, applications are good. Applications are great. Especially for non pure mathematicians, which is the majority of people taking these classes. However applications should be taught separately from the math, otherwise it can get really confusing. Most concepts in linear algebra and differential equations are honestly really basic straightforward definitions.

In particular, one that really stood out to me were eigenvectors/eigenvalues. I was confused until I stopped listening to all this nonsense about stretching, compressing, rotations, pictures of the Mona Lisa (https://upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Mona_Lisa_eigenvector_grid.png/320px-Mona_Lisa_eigenvector_grid.png), linear transformations, cute pictures of arrows getting stretched or turned around, differential equations, all of the answers in What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?, etc.

I remember SO MUCH hoopla about eigenvectors and eigenvalues. To be frank, it confused the hell out of me. Once I ignored the motivation and just accepted "an eigenvalue/eigenvector of a matrix $M$ are a pair $\lambda, \vec{v}$ such that $M\vec{v}=\lambda\vec{v}$", it got so much easier and clearer. Now I understand what everyone is saying but at the time what professors were saying seemed so much at odds with what I actually had to do. Pictures of the Mona Lisa, compression, rotations, etc. have nothing to do with what I had to do, and that is take the determinant of a matrix and solve for the roots.

I see this in students sometimes. They vaguely talk about these "motivators" but might not even be able to define what an eigenvalue is!, for example. It's not just linear algebra, but I feel like a lot of professors try to motivate and show connections too much, and it really is at odds with what students have to actually do.

I guess my question is, when is there too much motivation? Is it too much motivation, or just not enough basics?

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    $\begingroup$ Personally I'd say that if all you know about eigenvalues is to find the determinant and take the roots then you haven't fully learned about eigenvalues yet. $\endgroup$ – Jessica B Sep 3 '15 at 12:56
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    $\begingroup$ @JessicaB Why would you post that? Obviously I know more than that. I'm just talking about my experiences when learning. $\endgroup$ – user5108 Sep 3 '15 at 15:15
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    $\begingroup$ You give the impression that you think students should only be taught the calculation, and if they know that then the job is done. As I see it, `all this nonsense about stretching, compressing, rotations' is not motivation, application, random extra stuff... it is part of what you should be learning. You question sounds to me like you are advocating teaching to the test, which I would very much disagree with. $\endgroup$ – Jessica B Sep 3 '15 at 20:46
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    $\begingroup$ This is a good question. It is fairly easy to motivate eigenvectors geometrically but (at least for me) the subsequent applications didn't seem to follow the same geometric reasoning. I'd like to see eigenvectors used to solve simultaneous equations (decoupling) . This (I think) can be done by diagonalising the matrix an understanding of which depends on seeing how matrix multiplication acts on columns . I'm definitely not an expert but my point is the motivation needs to match the application which it often doesn't. $\endgroup$ – Karl Sep 4 '15 at 21:10
  • $\begingroup$ I agree. Even too much applications can be a distraction. I had the same exact experience in diffyqs. After all, I was at the Naval Academy and was going to see and use harmonic oscillator in physics, control systems, EE and quantum mechanics (chemistry). Four different areas. Not to mention that it is used in nuke school after grad. But in those courses, you are also learning what you do with the equation. Having a course where you learn something with just x and y was really helpful. $\endgroup$ – guest Jun 23 '17 at 15:34
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Let me propose a non-standard distinction between two terms (in the context of teaching):

  • An application is a problem or a task outside the main scope of the course with a solution presented using the present tools. When a teacher gives an application, the "applied problem" is actually solved in front of the students.
  • A motivation is a problem or a task outside the main scope of the course, which can be solved using the tools of the course but such a solution is not presented.

There are, of course, many things between and beyond these two concepts, but let me stick to these to make a point.

A motivation is vague and often plays with heuristic ideas, whereas an application actually shows how to use the course material for something else. Motivations and applications both present topics that are related to the course, but only applications make the connection explicit and clear.

A motivation is easy to give, since there is no need to go to the details and less time is required. The teacher doesn't even have to fully understand the motivation! A motivation is likely to be confusing, as it pours seemingly unrelated stuff into the students and helps them forget what is a definition and what is a heuristic interpretation.

An application actually works (ideally) with the very tools presented in the course, so it both provides a connection to something else and gives an example of the use of the main tools and definitions. Applications take more effort to give, but they give more to the students. A student feels much more empowered if they can, instead of just listing related problems, actually solve a related problem with their newly found tools.

In my opinion one should be very sparing with motivations (in the sense defined above). Convert, if possible, all motivations to applications or at least give the students a chance to do it themselves. If you wish to list several related topics to make the course seem more relevant, try to keep it separate from actual mathematics. When preparing lecture material, make a distinction between vague motivations and well explicated applications, and think which ones are more appropriate. Ask yourself: "Can my students really see how to connect this other thing to this concept from my course? How much do I need to explain to make them see it?" Don't answer too idealistically.

To answer your question (now in the standard meaning of "motivation"): I would say there is too much motivation if its connection to the course material is not made clear. Throwing a bunch of ideas in the air is not very helpful unless you take them (or make the students take them in exercises) and show how to actually work with those ideas and the course material. Rotations and arrows and distorted paintings can be useful in teaching eigenvalues if they are properly connected to the core content of the course.

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Yes, I agree that there is too much motivation... in the sense of sloppy presentations where it is very unclear to students exactly what is what. I think a lot of the time the refrain "is this on the test?" is a mangled sense of confusion over whether what's currently happening is a skill they're expected to master, or motivation for a skill, or justification for a skill.

Personally as a halfway young lecturer I've evolved to a fairly formal and ultra-old-school presentation format of definition, theorem, proof, and applications (specifically in that sequence). While there's nothing new and sexy in that with which education schools can win grants and publications, I constantly get students coming up to me and asking "how did you develop this revolutionary teaching style". (!) I think that that pattern keeps people aware during any lecture cycle of exactly where we are and where we're about to go next.

But it still gets moderately little traction from other instructors that I share it with. The current dogma tends to be "problem-based", i.e., the opposite, and the conference presenters and so forth are in favor of pitching poorly-defined problems at students working in groups and hoping for the best. Like you, I think that the essential strength of the math discipline grows out of careful and unambiguous definitions, concise theorems, clear proofs, and then well-defined applications as a test of the subject matter at the end. I'm reminded of this quote in the preface to Stein and Barcellos, Calculus and Analytic Geometry 5E (1992):

At the Tulane conference on "Lean and Lively Calculus" in 1986 we heard the engineers say, "Teach the concepts. We'll take care of the applications." Steve Whitaker, in the engineering department at Davis, advised us, "Emphasize proofs, because the ideas that go into the proofs are often the ideas that go into the applications." Oddly, mathematicians suggest that we emphasize applications, and the applied people suggest that we emphasize concepts. We have tried to strike a reasonable balance that gives the instructor flexibility to move in either direction.

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    $\begingroup$ I recently did an "example" in my calculus III course where I discussed the relation of momentum, angular moment, force, torque and gravity ala Newton's central force law. All to showcase how the rules of differentiation work for space curves to say interesting things about physics with minimal effort. After this, I get the question: "what problem did we solve ?" I think this over-emphasis of problem solving is absolutely stifling as an instructor who wants to share more than just canned solutions to standard problems. I whole-heartedly share your feeling about teaching style. $\endgroup$ – James S. Cook Sep 8 '15 at 2:00
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One problem with "motivating the subject matter with concrete applications", specially in first courses, is that the students have no clue (yet) on the "concrete applications" to their main course of study. Explain e.g.somebody in the first year studying chemical engineering that differential equations model the kinetics of chemical reactions, or to a mechanical engineering student that the system of equations models the forces in a bridge, and they won't know what you are talking about. Double confusion: the subject matter of the course and the motivation. Also, when the time later comes to apply said half-understood tools, they will be forgotten.

One way out is some sort of "just in time" teaching: Teach the tools with their use, hand in hand. That is hard to organize given standard curricula. But it did work very well in some courses I took, which required mathematical tools well outside the standard fare. The rigor and completeness suffered a lot, but as I got interested I filled in much of that later on by self learning.

The other option is to use examples at hand, using concepts they already have from high school or everyday life. Hard to come up with reasonable applications, that don't look contrived (or aiming cannons at flies). You have to be careful to get problems that are easy to understand and to model. A colleague around here had a resounding success with a course for second term students in modelling all sorts of situations mathematically, using a CAS or other tools to solve the equations and focusing on the building of models.

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In the preface to Intro to Applied Maths, Gil Strang says 'the equations of applied mathematics are better understood when the algorithms that solve them are presented at the same time'. In the preface to my lecture notes, I've extended this to 'in engineering, both equations and algorithms are better understood when they are driven by the need to model engineering systems'. One of the commonest complaints I hear from student engineers - of whom I've taught over 10,000 - is 'why are we doing this?' For them, the application is the motivation and comes first. Of course, pedagogically (if not in real life), the application must be appropriate to the level of knowledge of the student; that's why we have structured curricula.

In short, student engineers need an application to motivate the maths. In early years, this must be something quite simple; in later years, a more complex application might motivate deeper study of the same mathematical topic (eigenvalues are a good example). This is not, I think, how mathematicians approach a topic.

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  • $\begingroup$ This is true. Teaching math majors is quite different from teaching engineers. If lots of applications are presented to the engineering students, care should be taken to actually connect the applications to the mathematical content. If these connections are left too vague, the students may fail to find motivation. $\endgroup$ – Joonas Ilmavirta Sep 18 '15 at 9:57
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I am a retired educator. I have found that we do not really teach subjects; we teach STUDENTS. We actually present some of the content of our subjects and we study those subjects ourselves, and how best to present their content. I find it true that we REALLY learn what we prepare to teach.

Attempting to motivate students, is not the same as successfully (two letters "c" IS correct! ...contrary to the spell-checker) motivating them.

As teachers, we may have subjectively idealistic goals and measurable objectives. Students may have very different goals-and-objectives from one another as well as ourselves. What stimulates and motivates one student may confuse and demotivate another. That is just the way it is. Yet, there is more, both hope and complexity…

There is also the issue of "personality conflict" between students as well as between student and instructor. Teachers usually address their own styles of learning when presenting subject material. Some of us are visually oriented others are subjectively abstract or objective literalists. The real problem is providing for students who perceive and think differently than we do. This is now being better provided for in public schools and many undergraduate programs. It still requires getting to know our students ( > sigh < ) which is time consuming.

Personally, I did better in Geometry and Trigonometry than in beginning algebra. Algebra made much more sense to me AFTER studying Geometry and Trigonometry and doing real navigation with an E6B. For others it was quite different; algebra prepared them for Geometry and Trigonometry. We all enjoyed studying together anyway.

My solution was “over-preparing” or being ready to meet a multiplicity of needs when they surfaced without presenting everything “I-had-up-my-sleeve” each and every time I presented. Presenting too much tends to confuse many students while others need more as stimulus and clarification, which I offered individually.

Some times, if your faculty is large enough, you can swap students with a professor, instructor or T.A. who is very different from yourself and all of you may get really wonderful results!

I hope some of this is helpful....

Doc

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    $\begingroup$ Docendo discimus? How true! $\endgroup$ – rdt2 Sep 19 '15 at 11:38
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In addition to the other good answers... and as they say... "motivation" and "application" are not necessarily the same, at all, so to accidentally/implicitly impose on students/listeners the idea that they are is confusing and dis-crediting.

Given that, one might (as I would) further attempt to debunk/disclaim the apparent distinction between a mathematical idea and its "application". Although the attempt at such a distinction has historical and philosophical interest, I do seriously think it is misguided (although sanctified in many standard mythologies of mathematics and its history).

(And then there is the non-trivial issue of bad teaching, bad communication, ... The trouble this causes is endless. It might be useful for any specific individual to try to ascertain the extent to which their attitudes and tastes about (external? Objective?) mathematics was in fact strongly influenced, or entirely determined, by good/bad, hostile/friendly teachers in related courses.)

After some of that noise is sorted through, I'd like to make the claim that there is no meaningful distinction between "theory" and "applications", although there are many reasons that people would like to claim so. That is, although at a naive level it is plausible that there could be significant distinctions between "applications-oriented" and "theoretical" mathematics, upon closer inspection the claimed distinctions are ... content-less... unless one grants the reality of one of several possible wildly simplistic world-views. :)

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