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There is a smart way of learning, and it consists in first finding out what are the most valuable pieces of knowledge to acquire. The ones that will give you the highest value for your investment in attention and time. For instance, when you are learning a new language, it is typical to find that 2% of the words make 95% of conversations. You'd do far better if you knew what those words are when acquiring vocabulary.

My question is whether there are mathematical concepts, results, techniques, tricks, etc., that give such high payoffs. It could be because they are used everywhere, or because they are really easy to learn considering how much they worth, etc. As was pointed out in the comments, the meaning of "payoff" depends on the goals of your mathematical tools. Techniques like frequency analysis might bring the bread to the table of an engineer, but be irrelevant to a number theorist. I, in particular, plan to become a sort of applied mathematician in neuroscience and perhaps other "soft" sciences. I don't know beforehand exactly what I will be doing. That's why I want to gather a set of skills that are useful in a broad, applied, sense.

You might express the efficiency of a bit of knowledge in the following way:

$$\frac{\text{how much is it worth mathematically}}{\text{how much do I need to invest to learn it}}$$

The "how much is it worth" would be the weighted sum of

  1. The extension of mathematics in which is required to know it
  2. The number of applied problems whose solutions require it
  3. The work time you could save by using it, instead of acting in a naive way
  4. The average amount of errors you could stop making if you use it

The "how much do I need to invest" is the time it would take me to master this skill, trick, knowledge, etc. provided I already have all the prerequisites assumed in a standard introduction to the topic. Avoid considerations about how much money books or papers would cost, that is not the issue here.

Current List (to be filled with the answers)

  1. Visual evaluation of 2D and 3D graphs (see Jasper's answer)
  2. Dimensional analyis: you will have an edge in problems related to physical magnitudes, with little extra effort. You could also find the functional form of some solutions without calculations
  3. Logic: most common and dangerous fallacies
  4. Algebra: Elementary Algebra (how to manipulate expressions, and doing it without errors) Linear Algebra (Matrices, Vector Spaces, etc.)
  5. Calculus: The very vital concepts of limit, derivative and integral. Numerical procedures
  6. Statistics: How to lie, Resampling techniques(from Richard's answer)
  7. Optimization:
  8. Approximation methods: like in Mahajan. Street fighting mathematics, they are easy to use, but can give you a decent answer to a hard problem
  9. Fourier Transforms: are used in a lot of areas and not hard to grasp
  10. Elementary Programming: It's not properly maths, but needed everywhere. You could learn high level open source languages like Python and R in around a semester, and use them for a lot of things.
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    $\begingroup$ The answer depends greatly on what you want to do. There is a difference between proving a famous old conjecture and excelling at applying mathematical tools to engineering problems. Can you tell what you are trying to achieve? (This is true for languages as well. Do you want to write the greatest poem of all times or be able to order food in a restaurant?) $\endgroup$
    – Joonas Ilmavirta
    Mar 24, 2016 at 20:43
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    $\begingroup$ At first glance, this looks broad to the point of being intractable. But here is at least one initial question: What is the goal? The goal of wanting to score high on the Putnam vs become an actuary vs become a Physicist vs (etc) affects the (perceived) worth of mathematical resources. For the other part of your ratio: The (time, energy) investment required is, in a nontrivial part, a function of your current status (even "status" interpreted broadly: knowledge base, free time, access to materials, etc). This question would need to be refined a fair bit to fit... $\endgroup$
    – Benjamin Dickman
    Mar 24, 2016 at 20:45
  • $\begingroup$ @BenjaminDickman very much appreciated your comms, you are quite right, I will try to refine and edit the question $\endgroup$
    – Way Too Simple
    Mar 24, 2016 at 21:01
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    $\begingroup$ What math are you assuming that the learner already knows? What convenient computing ability are you assuming that the learner already has access to, or can easily purchase? $\endgroup$
    – Jasper
    Mar 24, 2016 at 21:17
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    $\begingroup$ @Armando -- Schools used to spend a lot of time teaching mental arithmetic techniques that many people now use pocket calculators or smart phones to do. Some people use graphing calculators, Maple, or Matlab to do algebra and calculus homework. Your assumptions about what computing tools a learner has affect what mathematical tools the learner can avoid intensively studying. $\endgroup$
    – Jasper
    Mar 24, 2016 at 22:46

4 Answers 4

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I guess that (by the question's metric) the most valuable things to learn are visual and conceptual, instead of the ability to perform detailed computations. Some concepts are very easy to learn, and have substantial value, so the ratio is very high.

For example:

  • Some of the most common logical fallacies.
  • Some of the most dangerous logical fallacies.
  • How to Lie with Statistics.

  • Given a 2-D graph:

    ** Is it a function?
    ** Where is the slope zero? infinite?
    ** Where are the inflection points?
    ** Does it have asymptotes?
    ** Is it symmetrical? even? odd?
    ** Is it periodic? What is the period? frequency? amplitude? phase shift?

  • Given a 3-D shape:

    ** What portion contributes the most to its volume? its stiffness? its strength?
    ** Removing which material would weaken the shape the most?
    ** How does doubling any dimension change the load the shape can carry?
    ** How does doubling any dimension change how much the shape will sag under a load?

  • Always keep track of units when doing word problems.

  • You can always multiply by one. (This lets you do most unit conversions.)
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  • $\begingroup$ Claiming that "slippery slopes" are usually examples of the so-called "slippery slope fallacy" is itself a dangerous fallacy. Modern history is filled with examples of people pointing out slippery slopes, being laughed off as hyping "slippery slope fallacies", and being proven right 20 years later. $\endgroup$
    – Jasper
    Mar 24, 2016 at 21:48
  • $\begingroup$ Thanks a lot Jasper, I think you really understood the question. I was starting to think nobody will consider it valuable. I posted it in MathSE and was kicked out. By the way, could you provide references or links for "most common fallacies" or "most dangerous fallacies"? $\endgroup$
    – Way Too Simple
    Mar 24, 2016 at 22:00
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Estimation: +1 for previous mentions of estimation / Street Fighting Maths / Dimensional Analysis. To the list I would add numeric techniques for Calculus and resampling for Statistics. When you combine the basic estimation techniques with these numeric techniques the range of what you can intelligently estimate greatly expands.

Basic numeric techniques can be used with only HS pre-calc maths combined with a few weeks of programming knowledge (how a basic loop operates and how variables work). This is a very low time investment.

As for what you get - numeric calculus moves static and linear estimation techniques into the dynamic world. High school science usually tops out at idealised parabolic and simple harmonic motion - with numeric techniques you can overlay as many forces as you want and get a picture of real world motion. You have problems with 3 bodies?... how about examining the motion of 300. Struggling to find the right U substitution to integrate a shape - set $\Delta x$ to a micro-metre and let the computer do the hard work.

Something else I really value is the sensitivity analysis provided by numeric techniques. Using basic estimation techniques is fine, but by augmenting them with numeric techniques you can find out how much more information you actually need before you get a good enough estimate.

The complementary techniques of numeric statistics allows you to work with real rather than idealised distributions. You can calculate probabilities by directly (re)sampling the problem space rather than working with p-values of some idealised problem.

When students learn more advanced maths, the numeric techniques continue to magnify the power of their knowledge. The cost is very small, the pay off is huge.

Added reference and a few other concepts I've found useful:

Resampling a basic introduction: http://www.dummies.com/how-to/content/the-bootstrap-method-for-standard-errors-and-confi.html This page also links to a computer program and text about resampling techniques. I am not a specialist in statistics, so if anyone can recommend a text, I'll add it in.

Wider than just resampling, there are other amazing Monte Carlo techniques built on the same premise - that by doing something simple a huge number of times we can get a good estimate of complex behaviour. They can also apply to calculus for estimating areas etc. and their application in statistical mechanics really opened my eyes.

I have especially found an understanding of Markov Chains and random walks to been very useful. It is fairly simple to understand, but as with other techniques, all you need to do is estimate the model, and an algorithm does all the hard lifting to estimate your distribution.

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  • $\begingroup$ Thanks a lot for your contribution, do you know of some good book on resampling? $\endgroup$
    – Way Too Simple
    Mar 28, 2016 at 14:42
  • $\begingroup$ And, by numerical techniques in Calculus, do you mean it in a broad sense, or just differential equations integration techniques? $\endgroup$
    – Way Too Simple
    Mar 28, 2016 at 14:49
  • $\begingroup$ @Armando Sorry, I don't know of any books on resampling - for teaching I've used High School workbooks/websites for very basic bootstrapping as it is part of the curriculum. For everything else I picked up online as it was not taught in the few stats classes I did at university. I'll put a link to a good description in the text. $\endgroup$
    – Richard
    Mar 29, 2016 at 1:56
  • $\begingroup$ @Armando I had been primarily thinking of techniques for integration and DEs (Euler's method and extensions), as for me personally they seem to pop up regularly. Thinking about it I remember that I've also used Monte Carlo techniques for Calculus, but there are likely other numeric techniques I've used that currently slip my mind. $\endgroup$
    – Richard
    Mar 29, 2016 at 2:38
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So, you want a skill-set which allows you to solve a wide variety of problems. You want to develop mental toughness. The few, the proud, the physics major I think. That's my recommendation, the study of physics exposes you to mathematical techniques and concepts of modeling which are on target with your goals. My evidence? I would point you to the many jobs outside physics which physics PhD's tend to thrive in. I have a friend who pursued a PhD in physics and now is successful in a job which is essentially actuarial in nature. You could say similar things about a PhD in math, but, it seems to me Physics as a discipline is more on target with your goals. In short, I would add:

  • develop physical intuition

As far as I know, besides intrinsic capability, the way to do this is to take courses and listen to physicists. Beyond books and homework problems there is a culture of physics. This is part of what forms the intuition. For me, being part of the physics major was a big part of how I developed what physical intuition I now possess. Short of pursuing a degree, at least take the major courses in physics.

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  • $\begingroup$ @James, I´m a physicist, although don´t have a PhD yet, and I think you have a point. However, could you point to some specific resource to develop "quick" physical intuition? I consider I don´t need it myself, but perhaps other users might $\endgroup$
    – Way Too Simple
    Mar 28, 2016 at 14:45
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Some points not already in the other excellent answers:

Basic algebra! Most students in elementary university courses are still struggling with the elementary algebra, if you really can do this without problems you will have an edge/learn faster in all other courses. Then, even if not really a mathematics topic, learn programming. Whatever you will do within applied mathematics, programming will be an important part. Language is not so important, but learn it in connection with some mathematics course, such as for example numerical analysis.

Then, linear algebra. This is used in aboslutely all parts of applied ( and pure) mathematics.

Basic statistics. especially visualization.

Probability.

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