Most Math Problems Do Not Have a Unique Right Answer
devlinsangle.blogspot.com
devlinsangle.blogspot.com
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.
So I disagree with the OP. The fact that some mathematicians brains work that way is not evidence that they all do, nor is it evidence that it is helpful or preferable.
Of course, that does not mean kids should not be taught arithmetic. It just mean that we should distinguish between mistakes caused by not understanding the topic and those that are just result of doing a lot of operations with similar numbers. the former should be dealt with as bigger deal then latter.
A while back I read about a mathematician taking an informal survey of other mathematicians. One of the questions was if they knew how to manually calculate square roots. (Apparently, the mathematician was part of an older generation that was taught this procedure in class.) Most mathematicians did not know how and a few responded with commentary such as "I don't need to know it in my work and if I had to, it's a procedure I could look up."
Unfortunately, I don't remember if I read about that in a book or a webpage (and google search doesn't find any hits) so I can't give a cite.
The mathematician doing the survey was including it as part of a larger essay. He was explaining a similar theme to the article's author: tedious manual computation exercises are overemphasized in the typical math education.
(1) Goal: compute square root of x.
(2) Make a guess, g.
(3) Compute an updated guess, g' = (g + x/g) / 2.
(4) Repeat step (3) until you achieve the desired accuracy.
It's not incredibly quick, but it works and it's easy (though tedious) to carry out.
Lots of other fun approaches here: http://en.wikipedia.org/wiki/Methods_of_computing_square_roo...
The mathematician was probably talking about this procedure that does not require convergence:
http://en.wikipedia.org/wiki/Methods_of_computing_square_roo...
This is the method I also learned (at one time and then forgot). Apparently, this procedure was part of a typical math curriculum many decades ago. Like Latin and cursive handwriting, it's been deleted as a mandatory skill.
In the square-root algorithm we remove the largest square possible at each stage, and work on the remainder. There's a hitch, because there's more detail to carry over at each stage. We do that with a linear factor of what we've already removed (that's the bit where we multiply the quotient so far by two) and then add the "epsilon" for the next lump to remove.
This can all be derived from first principles using the fact that (x+a)^2 = x^2 + 2ax + a^2. I did that when I was about 13.
#define DECK 11
double z() {
const double s=2.0, c=0.1;
double arr[DECK], low=1.0, up=s, inc=c;
int j, k;
for(j=0; j<15; ++j) {
for(k=0; k<DECK; ++k) {
*(arr+k)=low+(double)k*inc;
}
for(k=1; k<DECK; ++k) {
if( *(arr+k-1)*(*(arr+k-1)) <s && *(arr+k)*(*(arr+k)) >s ) {
low=*(arr+k-1);
up=*(arr+k);
break;
}
}
inc*=c;
}
if( s-low*low < up*up-s )
return low;
else
return up;
}
Personally I think this is a myth and the brain is not like some muscle that needs to be kept in shape with a specific set of exercises. Certainly stimulating the brain has been shown to have positive effects in older people, but it hasn't been proven that there's a difference between learning a new language or playing bridge. And nothing proven was found to prevent Alzheimer.
Generally speaking, offloading computations to a machine allows one to concentrate his gray matter on other things and before we come to a conclusion that this leads to brain atrophy, evidence needs to be presented.
But I believe the parent's point was more that what is taught is becoming fundamentally different. If you learn how to add using a calculator are you learning to add or learning how to use a calculator? It's a subtle distinction but in practice the ramifications become important.
> One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”
( taken form http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf )
Btw, 57 is now jocularly referred to as the "Grothendieck prime".
One of the key differences (being a mathematician myself) is that the big leaps of progress often come at the high level, often talking with others, and ignoring computations. Then when you have three hours to sit down and calculate, you go back and make sure your high-level ideas pan out. And you worry that you're making mistakes the whole time :)
The point is that you don't just sit and compute for its own sake, nor is there a time crunch, nor is there even a single "right" way to get your computations to go through. Often you can choose one of many routes to bound some quantity, or many different proof techniques that involve very different computations.
People really don't compute things "on the fly" in mathematics. When you hear stories about Gauss these are folk legends about people with extraordinary practiced abilities, and the majority of the mathematical world doesn't work like that. People do the grunt calculations offline so they can spend their time in talks/discussions doing actual work.
I totally agree. To me, there are 2 kinds of algorithms that one should get used to. 1. is computing fast enough (that doesn't mean super fast, but sometimes it means hours instead of months), and 2. is spotting out errors, both evident and subtle.
This said, I'm certainly not in the set of great mathematicians.
For some reason I developed the habit of sketching an Escher Cube* (with a cross in the middle) when I was in highschool, I still do it when my mind is blank.
*Perhaps it isn't Escher after all, I cannot find an image online. It is an impossible 3d cube where each corner crosses through the centre.
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
With that said, I'd endorse a move away from solely algorithm based (standardized test based) learning.
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...
"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.
But I would include all of the following as math problems: -Prove that some equation actually has solutions -Find some bounds on solutions to that equation -How can I compute an approximate solution to that equation? -How good is that approximation going to be? -What's a way to try to separate large data sets into clusters?
With regards to the first four: many equations people are interested in simply have no hope of getting a solution you can write down. Simple example: sqrt(2). A more complicated example: solutions to the Navier-Stokes equations.
He says that they checked to see where the most damage was, but really they were checking to see where the least damage was. If the planes were able to fly back with terrible damage to everything except for one part, then adding armor to that part is a pretty obvious solution.
On the other hand, figuring out exactly how much armor to add is much more complicated, because you have the constraints of weight, production time, etc, plus you have to have some kind of estimate for the effectiveness of the armor in actual combat. I would probably go with "Most Real-World Math Problems Do Not Have a Unique Right Answer".
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...
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.
There are infinitely many addition problems involving two numbers, and that's a subset of the set of problems that the article claims to be smaller.
> a question that a mathematician might ask
I agree that this is not a very useful definition, by virtue of its extreme and probably excessive inclusiveness, but I think that it's hard to do any better without using words like 'interesting' that themselves need careful definition. (It's fair to argue that my definition in turn requires clarification of the term 'mathematician', but I can weasel my way around that by replacing it with 'person', or else just declaring that anyone interested in trying to ascertain the normality of a number is mathematically minded enough to be called a mathematician.)
By this definition, I think that it is hard to argue with my claim to have produced an uncountable family of problems—simply because, at least classically, to do so you'd have to produce a specific number x about whose normality no mathematician could ever ask. I could then demolish that counterexample by asking you if that particular number x was normal. :-)
For example, why is the set of all questions utterable by humans not a countable set? By that reasoning there are more numbers than questions!
(I am only saying that because I was told that the intuitive answer of "armor plate the most struck parts" is wrong.)
In fact, the mathematical work was done by Abraham Wald¹, a Jewish refugee from Romania, at Columbia University in 1943. A reprint of his memos is available², as is a more accessible article about his work³; the specific one is Part V Subdivision of the Plane⁴ Into Several Equi-Vulnerability Areas:
“Thus, for the observed data of this hypothetical example [italics added], the engine area is the most vulnerable in the sense that a hit there is most likely to down the plane. The fuselage has a relatively low vulnerability.”
I haven't seen evidence that his work was applied to actual aircraft design using real data.
¹ http://en.wikipedia.org/wiki/Abraham_Wald
² http://cna.org/sites/default/files/research/0204320000.pdf
³ http://people.ucsc.edu/~msmangel/Wald.pdf
⁴ the flying kind
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.
Well, yes it does - but so does the tire pressure. If you add weight to a car, the tire pressure increases. So does the pressure it exerts on the road. Noticing that vehicle weight is a common scaling factor in two places tells you there's probably a simple relationship between the tire pressure and the road pressure. And it leads you to the counterintuitive conclusion that yes, if you increase the pressure in a tire, and keep vehicle weight constant, it increases the pressure the tire exerts on the road. Pressure's tricky and counterintuitive like that.
For sure, the road pressure and tire pressure aren't necessarily equal - not all of the weight of a vehicle is borne by the column of air between the contact patch and the wheel hub, some is transferred through the sidewalls, some through the tire rim to the air above the hub, and so on, and if you are a tire manufacturer or a formula one race engineer you will want to take those things into account. But for arguing about what the pressure on the surface of the solar panels in a road surface would be, tire pressures are -a- right answer.
I'm not sure if this is what you're describing, but many nonlinear[1] math problems have no closed form solution[2]. That means you can't use any regular function, all the operators and the infinitely real numbers to describe every solution: you can only use the infinitely real numbers to describe one solution.
I've written a blog post on this topic[3]; that blog post works through all the underlying stuff before getting to these closed form solutions.
[1]:https://en.wikipedia.org/wiki/Nonlinear
[2]:https://en.wikipedia.org/wiki/List_of_nonlinear_partial_diff..., https://en.wikipedia.org/wiki/Closed_form_solution
[3]:http://scottsievert.github.io/blog/2014/07/31/common-mathema...