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"Lately, my students keep telling me why what we are learning is not important. They ask me when will we use this in the real world?" There's a quick reply to this that I think people won't like, and you would have to be very careful using, but actually makes a more serious than it first appears. One possible answer is: "You're quite right. ...

40

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; ...

37

Math is just as useless as almost any other subject As a math tutor, I've thought about this a lot over the last 15 years or so. Aside from tutoring, I don't use my math education in "the real world". Here's a list of educational requirements that I have also never used in "the real world": American literature (Hemingway, Flannery O'...

29

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 ...

21

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 ...

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I've never had success with giving a list of applications to such students - because, realistically, we don't use most of the math we teach. For example, I teach early equations in one of my classes, things like "solve the equation $5x = 15$". If asked why this question is "useful", I could give this answer: Well, suppose you were ...

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 ...

16

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 ...

15

The American Mathematical Society provides posters promoting awareness of mathematics, its beauty, and applications. Here's a quote from the AMS Posters website: "Students frequently ask when they will use the math we learn in real life, and your posters provide great visuals to support the answers to this question." The AMS also have "...

13

Consider something besides an "all or nothing" approach. Here's what I did a couple of times when the topic was optional and I didn't have much time, but I still wanted to give students an introduction to the method. Simply restrict yourself to introducing the method by defining the Laplace transform, computing it for some simple examples, explaining what ...

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 ...

13

There's a reason why you can't find a good non-geometric example: when the dimensions of the axes on a graph are distinct, perpendicularity is units-dependent. There is thus no natural aspect ratio with which to draw the graph. If you change the scale for just one axis, you destroy any perpendicularity that was present, thus demonstrating that it was an ...

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 ...

11

Symmetries give profound insight into structure. This is demonstrated by Noether's theorem, proved just under 100 years ago by Emmy Noether (probably the greatest female mathematician in history). Noether's theorem states that every symmetry of a physical system is there because something is being conserved. Examples: Translational symmetry -> ...

10

Another classic is the following: A rectangular floor measures $300 \text{ cm} \times 195 \text{ cm}$. What is the largest square tiles that can be used to cover the floor exactly?

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Trigger warning: math enthusiasts do not like this answer. They ask me when will we use this in the real world? When students ask when they will use something in "real life", they are rarely expecting an actual answer to how something is used. They are telling you: "I don't understand this." Rather than coming up with try-hard ways to ...

9

People tend to use the presence of symmetry in a phenomenon to simplify a model of it. For example, in physics, systems with certain symmetries are the easiest to model (e.g., an infinite plane, an infinitely long cylinder, a sphere, etc.). But we have to be careful here. Ian Stewart has a very nice article on symmetry-breaking. (The article is from his ...

9

There is Methods of Modern Mathematical Physics by Reed and Simon, which is a 4-volume book which teaches functional analysis, with a focus on operators in Hilbert spaces. Its main aim is to provide a sound mathematical background for the methods used in quantum mechanics, but it serves well as a textbook on functional analyis (for example, it covers some ...

9

At the introductory level, Erwin Kreysiz's Introductory Functional Analysis with Applications is excellent. See the reviews at amazon.com. It's probably a bit elementary for you at this point (still, it could be very suitable for others reading this thread for recommendations), but I recommend at least looking through a copy at the library. I've actually had ...

9

Going back to Euclid, I have found questions such as `Given a large supply of rods of length $15$ and $21$, what lengths can be measured?' can appeal to students. This also motivates the result $\gcd(a,b) = ra+sb$ of the (extended) Euclidean Algorithm.

8

Galois Theory is the place where insights from one field (structure of groups) impacts another field (study of solutions of polynomial equations). I think it's the only time undergraduate students study such a phenomenon- certainly it's a classical and profound example of the interconnectedness of ideas. I would argue on that basis that Galois Theory is in ...

8

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 ...

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

Some nice geometrical applications arise in the analysis of periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Concrete experience with implementations in toys like Spirograph also provides excellent motivation for more abstract concepts such as cyclic groups.

8

Given a set of sites determine which points are closer one of the sites than any of the others? If closer is measured by Euclidean distance then given the distinct points A and B the points equidistant from A and B lie on the perpendicular bisector of the segment A and B. The regions one obtains are known as the Voronoi diagram associated with the sites, and ...

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

Some examples: It's much easier to develop a symmetric plane or boat than to calculate all the forces regarding an asymmetric hull. Car keys being symmetric it's great usability feature: saves time and frustration. Pots are symmetric, because they are easier to produce and nicer to use. Wheels wouldn't work if not for symmetry. Games are often symmetric to ...

7

Also having a short period of time to introduce my DE students to Laplace Transforms I began with two 'Axioms': (1) $\mathcal{L}\{c_1y_1(t)+c_2y_2(t)\}=c_1Y_1(s)+c_2Y_2(s)$ (2) $\mathcal{L}\{y^\prime\}=sY(s)-y(0)$. From these two we derived the Laplace transforms for polynomial, exponential and sinusoidal functions and proceeded to use those results and ...

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