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I'm working in the field of applied mathematics (optimization and numerics) and I meet a lot of students saying that they are allergic to applied mathematics or that they hate it or some quotes like "I'm here to learn how to think and become a thinker and that's why I only want to do pure mathematics", "Can't the engineers do all that?". I think, the students split in two groups: One group contains poor students who somehow worked out to pass basic courses, but are not able to apply the basics and want to blame something for that. The other group consists normally of very good students who like the concept of Definition -> Theorem -> Proof and miss that concept when it comes to modelling, looking at specific algorithm, comparing ideas etc.

Of course, due to the curriculum, they have to attend lectures about applied math, so I could say that and be fine. But somehow I feel that one can argue at least with the second group of students.

How should one react to such quotes? Should one even react or could it make the situation worse? If yes, what are good arguments?

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You could tell them about John von Neumann who I understand started as a pure mathematician then became an applied mathematician later. –  Joel Reyes Noche Apr 5 at 15:21
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Perhaps use the mixture of ignorance and arrogance which frames their question against them. Ask them, do they really want to do pure math without even having the basics any applied mathematician knows? –  James S. Cook Apr 5 at 20:18
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Perhaps there should be a similiar question: "How to react to students saying that they are allergic to pure mathematics?" (For the record, I usually teach mathematics to engineering students.) –  J W Apr 6 at 11:00
    
Show them (undergraduate) Linear Algebra with absolutely no applications. Eventually they'll be so bored with it, they'll ask "Why is this so important? What's the point?" –  Aeryk Sep 24 at 16:24

8 Answers 8

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 something external. And identifying an "other" and vilifying it has a long tradition in human history.

You could explain to the students that the definition-lemma-proof-theorem-proof picture is usually created with much hindsight, after many bouts of experiment, heuristics, revision. So our modern definition often came last in real life, while the desired/needed proto-theorem was presented to us by exterior circumstances. That is, "axiomatic mathematics" is most often done after a long period of experiment and revision.

Sure, some parts of school-math present only the end-product, but it would promote a methodological error to allow people to take a "Whiggish" view of these developments. That is, it's not that pre-axiomatic ideas were foolish or became irrelevant or obsolete. Rather, they constitute the phenomena-of-interest that were later organized by axiomatization.

Terry Tao https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ and others, such as http://m-phi.blogspot.com/2013/03/terry-tao-on-rigor-in-mathematics.html have reasonably labelled developmental stages of mathematical thinking as pre-rigorous, rigorous, and post -rigorous, in application to individuals. This has observable substance in application to individuals, but application to the larger enterprise is less clear.

And we can point out to our students that few of the historically well-known mathematicians made a distinction between "pure" and "applied", nor did they artificially confine themselves either in action or self-labelling. Most of the really interesting developments in mathematics arguably help us understand both very literal objects as well as idealized objects, and the whole range between.

And there is also the caution that some of the mid-to-late 20th century sense of "applied math" really meant numerical solution of PDEs, whether truly directly "applied" to create marketable products and local jobs or not. It may be that the WWII and study of non-linear waves to make big bombs leant special gravitas to numerical solution of PDE. But by this year we find that nearly all parts of mathematics have bluntly crass utility in business settings. Certainly it is not the case that these applications necessarily follow anyone's whimsical set-ups of axiom systems, but mathematics itself never really did random things, either.

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Very good answer! Applied mathematics is just pure mathematics which happens to have some applications. So if the students are really strict about this, they should stop learning any field as soon as it has an application. –  Mon Kee Poo Apr 16 at 17:32
    
@MonKeePoo: I respectfully disagree with your first statement. Applied mathematics is the label for certain branches of mathematics that are traditionally or usually associated to applications. That is to say, applied mathematics can have no applications at all. PDE and graph theory are usually classified as applied mathematics but there are plenty of results in those fields without any real world applications. –  timur Oct 9 at 2:09
    
I have to admit you're right. I started following a master in applied mathematics this year, and the classes I'm taking (measure theory and functional analysis) are definitely the most abstract courses I've ever followed (and I can't really imagine a situation where I would actually apply measure theory). –  Mon Kee Poo Oct 18 at 19:08

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 Center at Bell Laboratories, and, along the way, wrote a fair bit about mathematics education (and has been a long-time "visiting" professor at Teachers College Columbia University).

There is a wonderful interview with Pollak that touches upon some issues related to applied mathematics. Before pasting a few relevant excerpts, one citation of the interview is:

Pollak, H., Albers, D. J., & Thibodeaux, M. J. (1984). A conversation with Henry Pollak. The College Mathematics Journal, 15(3), 194-217. Link.

Three excerpts that I hope can help shape your responses to such students:

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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 mathematician, I have never been taught.

Of course, I managed to learn what I needed for the project (and I had already met with LP in previous research), but I now consider a bit sad not to have been more educated in what is often called applied mathematics, because it is really mathematics (and often very beautiful mathematics at that).

I should stress that the use of LP in our work is related to the so-called "Delsarte method". Delsarte was French, one of the founders of Bourbaki, but it is Greg that learned me about him. He was really amazed to see how little known he and is work is in France, among pure mathematicians.

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A couple of decades ago, I taught a linear programming course (cross-listed between mathematics, engineering, and the business school, so it really was applied math). I already knew a little about the subject (e.g., duality) but learned more when preparing to teach the class. I was pleasantly surprised to learn a proof (that Bland's rule prevents cycling) that I found really beautiful, by the standards of pure mathematics. (Unfortunately, it was beyond the scope of the course, so I didn't get to teach it.) –  Andreas Blass Sep 22 at 0:31

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, Gauss, Hilbert, Poincaré, and von Neumann, were all rooted in applications.

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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 not acknowledged.

Motivation:

Deep theorems often come from intuitions. Intuitions have to come from somewhere. Sometimes you get them from working elementary examples' features into your brain, but this can be near-sighted. Instead, they often come from an appeal to a natural connection between two subjects, and the overdetermining of properties induced by the multiple sets of semantics - every group is a permutation group, for example, is a formalization of the intuition about groups being transformers as well as algebras. The Sylow theorems and several properties of interest are capturing periodicity and self-similarity decompositions of group structure arising from viewing them as or expecting them to be the collections of symmetries of an object. Applied mathematics functions always in a context, and cross-pollination of semantics is an inherent part of such naturalistic environments. As such, it's an important source of intuition.

Projective geometry, for instance, arose from applied mathematics and the intuition it brought, and its importance in the structure of abstract objects (algebraic geometry, projective representations, manifolds, homology, hyperbolic geometry, continued fractions, the modular group) is worth the thinking it takes to justify without any trace of solipsism.

Applied mathematics can lead to these new fields because it poses problems that wouldn't be considered problems or be thought up by someone working only off their existing knowledge. It can also pose problems that are unmanageably difficult, which in mathematics suggests there are underlying paradigms left to be uncovered.

For a modern example, crack analysis can now be done using fractal measures and other abstract ways of reasoning, except when it comes to intersections of propagating cracks, these theories tend to break down abruptly and the finite-element method must be used to proceed with the description of the cracks approximately.

It's also one of the best ways to motivate a special case, informing the specific calculations that will allow one to become adept at a field. Most (classes of) differential equations are posed as tasks for their representation of a physically observed behavior, and in that way arbitrary calculations obtain a life of their own through application. So, for a pure mathematician - and by that I mean one which isn't strictly concerned with real-world tasks, applied contexts are rich interfaces to mathematics. Knowing any field in such detail helps gauge your depth of knowledge in other fields, and an application will evade giving a sense of security until that point is reached and the motivating problems have been solved.

All mathematics is descriptive of something, so how is non-applied mathematics to be intrinsically distinct from applied mathematics? It might be they disagree with having a non-mathematical goal, or have the impression that someone primarily does applied mathematics when mathematical goals aren't inherently valuable to them. It might be because educators often treat students as if mathematics is opaque to them whenever it isn't shrouded in a real-world scenario. These are cultural reactions. Perhaps their problem is actually with the experience they expect of applied mathematics, or the type of output they expect to be giving. The diagnostic should come before the prescription. Much of what people are trying to avoid by professing these alignments can be a problem in pure mathematics, in a generalized form, and if you have the wrong reasons to avoid something you narrow your tools for gaining experience and proficiency in a field in which you're mostly left to your own devices.

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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 "applied maths" contained mostly things I did not enjoy in first year maths. That is, heavily computer-based things, numerical things and differential equations. So I chose not to study any of those courses and did statistics and pure maths instead.

It wasn't until much much later that I thought about why I didn't enjoy those topics in first year.

Firstly, the maths-on-computers thing: In first year, this was mainly just frustrating. I would regularly get error messages for missing what seemed like small things, and I got told off by the teachers for not using the "most efficient" methods in my programs even though there didn't seem to be any instructions in how to make them more efficient. It seemed like a wasted effort when most of the problems we did were just as easy for me to do by hand. Moreover, my way of understanding things is very much visual and geometrical and the computer programming was very much text-based and just didn't feel the same.

Much later, I discovered that I could use the computer to simulate things and create lots of data that allowed me to investigate things I could never have achieved by hand. For example, I could simulate a board game I was designing to see what it did, or decide if a hypothesis test in stats really did seem to have 80% power. This power of the computer was not really apparent to me in first year. I also discovered the pleasure in having to make a program do something and checking and rechecking and editing it until it did what I needed it to well. This pleasure of a job well done was also lost on me in first year.

Secondly, the differential equations: In first year, differential equations was very much a procedural sort of topic. It was all very much of the form: this particular situation goes with this particular solution because it just does, so get good at recognising it. This went against everything I felt about understanding maths. I felt that there ought to be unifying principles and big picture ideas. I also liked the problem-solving aspect of other bits of maths where you had a small amount of tools and you worked out yourself how they fit together to show/prove/find something and there were often different methods of doing it, all of which were kind of clever. I disliked it when people said that maths was a list of problems and how to solve each one, yet this was precisely my experience of differential equations in first year.

Much much later, when helping students to study the higher-level courses, I saw more methods and how they fit together. I saw how quite deep bits of theory were brought to bear on these problems, such as treating functions as a vector space, or representing functions as infinite series. I also saw how very clever some of these methods were, and was able to stand in awe of the cool problem-solving aspect of the discipline. I wished then that more had been said about this side of it when I was in first year.

Finally, the numerical methods stuff: In first year the extent of this was calculating integrals by the trapezoidal and Simpson's rules. We brushed over this stuff very quickly and only did it on functions where we knew how to calculate it another way anyway. I certainly never had a feeling of a bigger picture. Also, they taught us these crazy error formulas which were just made me think "where the hell did this 2880 come from?" and "surely if I know the fourth derivative of a function I can do better than using an approximation?". Finally, it was just a lot of hard work.

Much later, I realised that there were unifying principles in these methods (such as the area being related to the average value, so it's reasonable to average some values). I also realised how supremely clever it was that you could figure out an error formula, and I wished I knew more about where those sorts of formulas came from.

There is one final aspect that is worth mentioning: the applied-maths-ish topics were the ones I just didn't understand so well, probably because they didn't fit with my natural way of thinking. So another reason I didn't enjoy them was simply because I didn't enjoy the sensation of feeling confused and they were a lot more like hard work and a lot less like play!

So to sum up: it wasn't so much that I didn't like the applications, but that the style of teaching, learning and problem-solving in the applied maths topics in first year was something that I didn't enjoy. Later, I realised that there were aspects of it that I could enjoy. (Though to be completely honest, when I watch people doing it, it often still doesn't look like fun to me.)

And what can you take away from this as a teacher?

Well perhaps you should ask them about the topics they have seen as applied maths in the past, and whether they enjoyed it or not. And indeed, ask them about what particular aspects of other maths topics they did enjoy. Then you might have a different sort of picture of why they feel grumpy.

You could also try to include elements in what you're doing that make it feel more like the other maths topics. Talk about the big picture structure. Talk about the clever maths problem-solving aspects of it (as opposed to the solving real-world issues). Talk about the pleasure of a job well done with the computing stuff.

Finally, you can make it really really really clear. There is nothing like simply not understanding that gets students looking around for reasons not to do something!

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First, rather than "pure" and applied mathematics, I prefer "theoretical" and applied mathematics. What is "impure" mathematics?

Second, many students are put off by the "word problems" they see in K-12 since if these problems are typical of the "power" of applying mathematics, then they are skeptical. (If Mary is twice as old as her brother was when ..... How much does Mary's dog weigh?)

In my experience the best way to expose students to applications of mathematics in lower grades is to show them how powerful graph theory is as a tool for getting insight into the world (garbage collection and mail delivery routes, routing packets on the internet, and software that provides driving routes from A to B, etc.) and examples that show the role of mathematics in the technologies that are so much part of their worlds. Thus, the cell phone in your pocket (or pocketbook) is made possible by error correction and data compression ideas, frequency assignment algorithms, as well as GPS technology.

Sadly, the recently adopted CCSS-M do little to change the perceptions of mathematics that many students get, and I think are likely to make these perceptions worse.

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Impure mathematics :-) –  Willie Wong Sep 25 at 14:07
    
A google search with the string: impure mathematics yields some curious items. –  Joseph Malkevitch Sep 27 at 19:04

If you want to motivate your students, you have to address that issue.

Maybe, you can motivate them via modelling. Show them, that what they thought is pure math, is already modelling.

  • Set theory is modelling the human psyche.
  • Peano is modelling linear, stepwise algorithms.
  • The axioms and basic theorems about real numbers are modelling human reception of the surrounding.

Maybe you can catch them with Gabriel's horn or Banach-Tarski-paradox and the gap between math and physics and the need to close that.

Then you can continue with "truly applied math" by motivating with the need to model something.

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Would you please explain more about what you mean by "Set theory is modelling the human psyche"? It is interesting for me. –  Saint Georg Apr 6 at 14:35
    
@SaintGeorg Human psyche is capable of differentiating between a set and it's individual elements. Some animals cannot, that's why fish swarms are a defensive measure against attackers. Other example: Discrimination, alliances and other social activities work because we treat other humans in sets with union, intersection, complements, … –  Toscho Apr 6 at 16:24

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