I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, there's a computation of sqrt(2) to 38 decimal places for no discernible purpose. He probably was merely exercising, just like a musician practices scales. Erdős was able to memorise 10 phone numbers at a time from a glance at a phone book, amidst other calculating feats.

I hypothesise that offloading this brain work to a machine atrophies your cerebral muscles. Exercises left to the reader are really exercises in an almost kinesthetic sense.

The computations don't have to be purely numerical: even a diagram chase in abstract nonsense is a valuable exercise. Computation frequently leads to insight. As children are adding two-digit numbers, they start to notice shortcuts about how to perform the computations. These are their own little private theorems, so to speak. So when Devlin here is talking about "mathematical thinking" and the less importance that performing algorithms has today, I hope we don't forget that just because it's less important, it doesn't mean it's not important at all.

I have a three year old son, and it is absolutely fascinating to watch his mathematical understanding develop. I've been a math teacher my entire adult life so I've had plenty of experience watching older students develop their understanding. It's entirely different watching your kid develop their understanding from scratch.

I recently looked for some kids' books that would focus on the more interesting problems in math, rather than just counting. I was happy to find a few books that have helped him see math as more than just counting. My favorite so far is The Boy Who Loved Math: The Improbable Life of Paul Erdos. [0] I knew of Erdos, but I didn't know much about him. I learned from reading this book, and my kid loves it as well. He is fascinated with aging, and he now sees it as normal that someone would spend their whole life focusing on numbers.

We are also starting to enjoy Bedtime Math: A Fun Excuse to Stay Up Late. [1] The idea is to give your kid some interesting math problems to think about at bedtime. We've found that it's a good way to help him think about things other than the dark, and strange noises while he's falling asleep.

It's fascinating to watch this development. A few nights ago: "Did you know that one of the oldest questions people have asked is, How many stars are there in the sky?"

"No, I didn't know that!"

"How many stars do you think there are in the sky?"

"Eight!"

[0] - http://www.amazon.com/The-Boy-Who-Loved-Math/dp/1596433078

[1] - http://www.amazon.com/gp/product/1250035856

I think that if you want to understand how ordinary people use math, a good place to start would be by looking at their Excel spreadsheets. Excel is arguably the most widely used tool for computation and even programming. I've seen some impressive spreadsheets for solving complex problems, written by folks with no more than a high school education.

With that said, I'd endorse a move away from solely algorithm based (standardized test based) learning.

And yet the way math is traditionally taught in schools and colleges, there is almost always is a unique right answer.

There are alternative approaches to teaching math such as problem-based learning, active learning, and inquiry-based learning, that help students not only understand the math better, but learn why the math is useful and how to apply it in the real-world (transfer of learning). Students also are much more likely to continue on and succeed in future courses and graduate.

Here are best practices for teaching calculus, for example: http://launchings.blogspot.com/2014/01/maa-calculus-study-se...

And research showing that in traditional lecture courses (vs. active learning courses), students are 1.5 times more likely to fail: http://news.sciencemag.org/education/2014/05/lectures-arent-...

Underrepresented populations (minorities) and females are much more likely to succeed in active and inquiry learning math courses, too: http://theconversation.com/who-learns-in-maths-classes-depen...

It is refreshing to find an article with a false title that contains it's own refutation within the first five paragraphs:

"The only career in which a high school graduate can expect to continue to work on [problems with a unique correct answer] is academic research in pure mathematics"

Unfortunately for this article, the premise that mathematics is the same thing as engineering is false.

If an engineering problem does not have a unique solution, its because of complications introduced by the real world. Any engineering problem which can be well-posed as a pure math problem, does of course have a unique solution; as the author concedes.

I agree with the author entirely. But I still feel it's important to be able to do computation in your head if just to develop that "number sense". As a physicist, I use computers to do all my real calculations. But when I get the answer that seems wrong, I can mentally do some basic calculations that convince myself the computer-answer I got is probably off by a factor of two, and look for the bug.

How is adding armor to the engine area not a unique right answer? I'll leave you to figure out why that is the best solution. If there is only a single best solution, isn't that then the unique right answer?

I'm a senior in college. I've learned about a host of mathematical concepts that are misunderstood by popular media (N dimensions, linear algebra, linear functions). Wanting to correct these topics, I wrote a blog post[1] about them.

But this blog post blossomed into much more than I was expecting. I was expecting to cover only those specific details, but then I got into nonlinear problems that have no closed form solution and got into what mathematicians do. While I'm still learning and they're an experienced mathematician, I like to think I can understand what everyone else thinks and what mathematicians think.

[1]:http://scottsievert.github.io/blog/2014/07/31/common-mathema...

If we randomly choose a matrix A of NxN coefficients, we are quite likely to end up with a linear independence, meaning that Ax' = 0 where x' is the vector [x^(n-1) x^(n-1) ... x 1] is a system of equations with a unique solution.

This is an example in which most problems derived from some real data in fact have a unique solution.

In case the above isn't clear what I simply mean is that, for instance, if we randomly choose the coefficients for a system of three equations in three unknowns, we are in fact statistically unlikely to end up with an under-determined system (multiple solutions). Two or more of the vectors in the matrix would have to point in the same direction, which is unlikely for randomly chosen 3-vectors.

Since uncountably many problems have unique right answers, I'd guess the cardinality of the set of problems with unique right answers is the same as the cardinality of the set of problems without. :)

He linked to DragonBox Elements which I had not heard of. It looks really amazing. This one is for geometry, similar to how the previous DragonBox was for algebra. Awesome find!

As for the armor plating question, did the plate the engine because of its size? The engines combined make up a smaller surface area than the other places but it was still shot up quite a bit. It would be less weight and not to mention it would be more 'mission critical'.

Most Math Problems Do Not Have a Unique Right Answer

The title is both right and wrong. You are actually comparing school or college math with math applied in real world. School or college math works with few variables, for instance, and we consider most others remaining constant. Rarely have I seen school or college level students working with, say, derivatives of more than three variables. School or college math is an exercise to establish the rule, the rigour and in most cases to create an appreciation of what math can achieve.

In real world, variables are plenty. If you take a handful to solve a problem considering other important ones to be constants, you will end up with one set of answers as against others if you had taken a different set of variables. Real world applied math is contextual. You remove context from the problem and real world math looks like school or college math. As demonstrated by the engine-armour-plate example, without the context of the airplanes returning after taking hits, the mathematicians would probably have gone with a statistical answer and would have been proven wrong!

However, I do agree that most math problems may not have unique right answer. Of course, we are not talking of,say, square-root-of-two having two different answers. However, take an instance where the problem is:"Find a number that is a sum of two infinitesomely large numbers one ocurring at an infinitely large interval of time from the other. Does it essentially fall on the numberline?" Well the first reaction to this question is: well, yes. Because if we are sure to find those two numbers then we are more likely to find their sum which has to fall on the numberline. Now, a more discerning reader might pause and ask: can you define infinitely large number and infinitely large interval. Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :)

Coming to a more basic argument: With math we are striving to arrive at a single agreeable solution. Whether it is statistics or calculus, we are interested in modelling the world to arrive at a set of recognizable pattern or a set of patterns. We apply the templates we learnt in school and college. For instance in arithmetic, numerals -- which are nothing but symbols -- help us reduce our problems into an expression which we can solve. The operations allow us to take these symbols through a set of processes that helps us model the problem.

But thanks to the author, what is clear is that applied math is contextual and answer may vary with the change in context. While school math is merely an exercise in familairising ourselves with a template.

The idea that mathematics as a whole is intuitive (as opposed to specific facets of its machinations) is as pervasive as it is misleading.

It's too bad most of the discussion here is centering around the title, rather than the post