# Tag Info

56

Some of your students will become engineers, and engineers use complex numbers all the time, e.g., to represent impedance. This kind of thing is by far the most common application. Complex numbers are also used in quantum mechanics. after algebra II, they never use complex numbers until pretty much complex analysis. I assume you mean "they never use ...

39

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; ...

28

We owe students a presentation of the Fundamental Theorem of Algebra -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of nth degree has precisely n roots (including multiplicities). When made, it serves as a capstone and culmination of all the work that the student has done in elementary algebra. Of ...

20

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

17

In the first place, it is "only" common course-naming conventions (and the AMS and NSF subject classifications) are to blame for the impression that there is some meaningful schism between something called "pure" and something called "applied" mathematics. Second, as noted, as much as anything people rationalize their own limitations or failings by blaming ...

16

I'm just finishing up a graduate course in computational topology which could be adapted very effectively for this purpose. We're focusing on topological data analysis and computational homology. All the topology in the course has been self-contained, meaning that essentially no previous experience in topology was required. The book we're using is ...

16

I am surprised that nobody has mentioned differential equations. If you know about complex numbers and Euler's formula then there is a beautiful unified theory of linear differential equations with constant coefficients in terms of the characteristic equation. Without complex numbers the theory becomes somewhat ad-hoc, with different solutions depending on ...

15

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 ...

14

When in doubt, I often decide simply to quote others! A nice choice, in this case, would be someone who started as a pure mathematician, then worked in applied mathematics, and ultimately moved into mathematics education. Luckily, precisely such a person exists in Henry Pollak: Ph.D under Lars Ahlfors, then Director of the Mathematics and Statistics Research ...

13

I think the fundamental tenet of this question is simply false. Here are some of the many encounters that undergraduate college students have in my classes: In Calculus II, as an application of power series, we discuss Euler's identity: $$e^{i\theta} = \cos(\theta)+i\sin(\theta).$$ Still in Calculus II as an application of the preceding example, we derive ...

12

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 ...

10

I do not have a true answer, but only a personal anecdote that backs up the point of teaching applied maths to mathematicians-to-be. I work on fundamental mathematics, mostly differential geometry. My best work to date (in collaboration with Greg Kuperberg) has a differential geometric statement, but uses heavily linear programming which, as a fundamental ...

10

If you looked at those other topics, you saw the links to the books page at my blog, Math Mama Writes. There are a number of books there that would work for a teen. One of my favorites is Carry On, Mr. Bowditch, by Jean Lee Latham, the (slightly fictionalized) biography of Nathaniel Bowditch, who modernized navigation in the 1700s. But maybe that has too ...

9

Perhaps the experience of just such a student may help you? As a student, when I chose my second-year maths courses, I compared the list of topics that were in second-year courses to the topics I studied in first-year courses and I decided what I wanted to do based on the things I enjoyed most in first year. It seemed to me that the courses labelled as "...

9

Question: What's different about applied mathematics that you don't enjoy? Notice the difference between that and asking why they don't like applied mathematics. One demands an accurate description of applied mathematics, the other can be answered in terms of one's aspirations or perception of value, and that distinction can be explained if it's unclear or ...

8

This quote from Jerry Marsden's 1980 AMS Bulletin review of Dieudonné's Treatise on Analysis is relevant: History screams at pure mathematicians not to ignore applications; the origins of such important topics as calculus, Fourier series, operator theory and dynamical systems were all closely related to applications. Some of the greatest mathematicians,...

8

Wow, thanks for the recent shout-out. I hope this is the right place for me to add a few references that might be useful and haven't already appeared in the answers. Rob Ghrist has just written a fantastic new book called Elementary Applied Topology which provides a soaring and current overview of the field. There are no exercises (yet!) but the figures are ...

8

Just another question from a math guy showing his ignorance of math as a service course. 2nd order diffyQ with constant coeffiecients (most important diffyQ for applications) has complex roots in the characteristic equation. [IOW very standard part of standard ODE course; OFTEN part of even a second semester calculus course during the diffyQ survey...was ...

8

If I use your simplification that $f_0 = 0$, then I suggest just choosing a real eigenvalue $\lambda$ and writing out the relation for the other parameters: $$-\lambda^3+f_1s_0\lambda + f_2s_0s_1 = 0$$ Now isolate a parameter, say $s_1$: $$s_1 = \frac{\lambda^3-\lambda f_1 s_0}{f_2 s_0}$$ Then just find values that satisfy your requirements. It should be ...

8

Air speed/direction on a weather map) is a very intuitive one. There's also other fluid velocity (and flux) vector fields in various chemE, mechE, and nukeE applications. I personally think the air speed is most intuitive as something where you really need speed and direction (i.e. a vector, not a scalar) and it's something people encounter in daily life. ...

7

In my experience teaching undergraduate engineering students, key topics include at least some calculus, linear algebra and differential equations. Exactly how much depends on the field/subfield of engineering. A sampling of other topics includes: Boolean algebra (digital electronics) Graph theory and algorithms (networks) Probability and statistics (...

7

First give me an example of real life then I can tell you about the math in that activity. Now, is math central to the activity? From my perspective, it is likely. From the perspective of the person you might seek to teach math, probably not. Much of math has been black-boxed in our daily life. Cash registers are automated to the point that at least 1/4 ...

7

First: there seems to be a traditional belief that "pure" math fusses over tiny uninteresting details that "applied" math takes for granted, etc. Sure, we can operate this way, and make "pure math" as irrelevant as we want, or caricaturize it as such. Oppositely, if we try, we can caricaturize "applied math" as slip-shod fuzzy-thinking. :) As I've ranted ...

7

I would start with a discussion with the Engineering department about what it is they want the students to learn. Teaching different skills will need different methods, so you might as well start with learning what will help you most in the short term. Personally, my experience suggests proofs are likely to be very low priority, with a focus on calculation ...

7

"(Books in German or with German translation are a plus)" and "she is particularly interested in geometry." The OP's notes suggest: Ziegler, Günter M. Do I Count?: Stories from Mathematics. CRC Press, 2013. CRC link.                   Ziegler, Günter M.: Darf ich Zahlen? Geschichten aus der ...

7

I wouldn’t feel bad about leaving it out, but I think it’s a valuable conceptual example for understanding matrix algebra. Computing the QR decomposition is equivalent to applying Gram-Schmidt orthogonalization to the columns, and I think it’s really instructive to see how this corresponds exactly to the fact that Q is orthogonal and R is upper triangular (...

6

This may not be a direct hit, but since you mentioned "emphasis on visualization," may I suggest you investigate Tristan Needham's Visual Complex Analysis. E.g., see this MSE answer:

6

Most people are not going to use most of what they use in high school. High school (in the US, at least) is about building a broad foundation for students to be able to jump into any specialization when they go to college. Some examples of things I haven't had to do since high school: Name the 5 (I think 5, maybe it changed?) biological kingdoms Diagram a ...

6

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 ...

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