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Physicists avoid highly mathematical work despite being trained in advanced mathematics, new research suggests. The study, published in the New Journal of Physics, shows that physicists pay less attention to theories that are crammed with mathematical details. This suggests there are real and widespread barriers to communicating mathematical work, and that this is not because of poor training in mathematical skills, or because there is a social stigma about doing well in mathematics.
Dr Tim Fawcett and Dr Andrew Higginson, from the University of Exeter, found, using statistical analysis of the number of citations to 2000 articles in a leading physics journal, that articles are less likely to be referenced by other physicists if they have lots of mathematical equations on each page. [...] Dr Higginson said: "We have already showed that biologists are put off by equations but we were surprised by these findings, as physicists are generally skilled in mathematics.
"This is an important issue because it shows there could be a disconnection between mathematical theory and experimental work. This presents a potentially enormous barrier to all kinds of scientific progress."
http://phys.org/news/2016-11-physicists-mathematics.html
[Abstract]: Statistical Analysis of the Effect of Equations on Citations
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @05:26PM
No comment needed.
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @05:36PM
Maybe articles full of formulas are not as well written and thus not as highly cited?
Correlation =/= causation.
(Score: 2, Touché) by Anonymous Coward on Sunday November 13 2016, @06:34PM
Maybe articles full of formulas are not as well written and thus not as highly cited?
Correlation =/= causation
You see the irony, right?
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @06:37PM
No, because he merely listed an alternative possibility to demonstrate a point. He didn't write a serious scientific paper and then reach a potentially faulty conclusion.
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @06:52PM
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @08:15PM
He put forth a possible hypothesis, the basis of science. This isn't a scientifically rigorous blog and so everything is going to be OK. It's not going to be the end of the world.
Do I need to tuck you in too?
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @08:50PM
So what you are saying is that he put forth a completely unresearched, unsupported, and therefore not relevant point to try to discredit a properly researched and written paper. We could invalidate ANY scientific paper ever written if we want to accept that slipshod logic.
You're telling me that's not how it works? Man, wait until the anti-vaxxers, 9/11 truthers and "Moon landing is fake" crowd hear about this!
(Score: 1) by khallow on Sunday November 13 2016, @08:59PM
So what you are saying is that he put forth a completely unresearched, unsupported, and therefore not relevant point
What's up with all this anti-science crap? A point can be quite relevant even if it isn't a research paper. And let us note that "unsupported" can still be supported by reasoning and personal experience, even if you choose spuriously not to recognize those normal scientific processes.
Out of curiosity, do you believe reviewers of peer-reviewed papers conduct their own research in order to find typos, mistakes, and poor writing? I'll warn you right now that their criticism is usually unresearched and unsupported in your sense above, but it somehow manages to make a lot of scientific literature better.
(Score: 0) by Anonymous Coward on Monday November 14 2016, @01:10AM
He's afraid of a second opinion.
(Score: 0) by Anonymous Coward on Monday November 14 2016, @11:33PM
(Score: 0) by Anonymous Coward on Wednesday November 23 2016, @09:39PM
Well said meta-random-chatbot!
(Score: 0) by Anonymous Coward on Monday November 14 2016, @11:37PM
And let us note that "unsupported" can still be supported by reasoning and personal experience
You want to rephrase that before I rip you apart, or after?
(Score: 1) by khallow on Tuesday November 15 2016, @04:08PM
You want to rephrase that before I rip you apart, or after?
No need for me to rephrase. Go for the ripping apart, you just might learn something. It was quite clear that whoever wrote the earlier post wasn't counting reason and personal experience. And I did use "scare quotes" to indicate I wasn't taking the claim of "unsupported" seriously. Communication, you know.
(Score: 1) by khallow on Tuesday November 15 2016, @04:36PM
Those reviewers, unlike the GP, have actually read the paper they are critiquing and they are, at least usually, educated in the field they are reviewing papers for. The idiot above is just spouting off with the believe that his objection should hold the same weight as the published article.
And what makes you think that earlier poster isn't educated in a relevant field? And it's one opinion versus another opinion. Just because one opinion happens to come from someone who wrote a paper isn't very relevant. You have to go beyond just asserting without reason that one opinion is better than another opinion.
(Score: 3, Insightful) by khallow on Sunday November 13 2016, @05:40PM
So one of the things that "lots of mathematical equations on each page" may be selecting for is poorly written physics papers.
(Score: 2) by opinionated_science on Sunday November 13 2016, @06:01PM
agreed. And for those of us with fluid maths reading skill, finding mistakes first is the usual way of reading - Does the logical derivation fit "the rules". Then "do the assumptions apply".
But it depends on the mathematics - derivations of number theory proofs take some concentration. Differential equations are fun ;-) Especially numerically, with BLAS...
But if you're going to hide bad answers, this is where things get lost on the non-technically trained public.
The general lack of numeracy , allows politicians to be hoodwinked by unknown statistics...
(Score: 3, Informative) by mcgrew on Sunday November 13 2016, @08:26PM
The general lack of numeracy , allows politicians to be hoodwinked by unknown statistics...
On the contrary, they rely on the general public's innumeracy. Example: Illinois Governor Bruce Rauner's charge that "the average state worker makes $60,000 per year" whenhe was running for office. Accurate, but meaningless. The meaningful number is the median, not the mean. The average is WAY above the median. Don't forget, mathematicians have to eat, too, and it's as easy for a politician to hire one.
Before I retired, my boss held a PhD in statistics, and I worked for Illinois. As tRUMP said, "I love uneducated voters." He would not have been elected without them, and neither would Rauner have.
No one born who could always afford anything he wanted can have a clue what "affordability" means.
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @06:36PM
(Score: 1, Insightful) by Anonymous Coward on Sunday November 13 2016, @06:40PM
Well, programmers who write code similar to how math is usually written are called shitty programmers for producing nigh unreadable, difficult-to-maintain code. It's possible to understand it, but it's an unnecessary waste of time. It's laziness at its finest, but the tradition would be too hard to break.
(Score: 2) by EvilSS on Sunday November 13 2016, @06:54PM
thanks for clearing that up.
(Score: 1) by khallow on Sunday November 13 2016, @09:42PM
So using math in a math-heavy field is lazy. Man, there are a ton of lazy fucks in the field of pure mathematics then.
It can be actually. If I copy my math work directly from my notes into the computer with little to no explanation, that's going to be pretty lazy right there. The laziness here is not the using of math, it's not doing the work to make the math more understandable to the reader.
(Score: 2) by EvilSS on Monday November 14 2016, @06:53PM
(Score: 1) by khallow on Monday November 14 2016, @11:51PM
Ah, and you can state, as fact, that this is the case with every paper that the study in question evaluated then?
No. But it would be a fact that your demand would be irrelevant to my earlier observation since I wasn't characterizing papers but rather a common and lazy technique for slamming out papers which just happens to result in high math content.
(Score: 2) by EvilSS on Tuesday November 15 2016, @02:27PM
(Score: 1) by khallow on Tuesday November 15 2016, @03:48PM
OK, then you have experience in publishing scientific papers?
I even have experience in publishing scientific papers where I copied the math from my notes into the paper in said lazy way that I just described. In my defense, I was a lot younger then and didn't realize what a mess I was making or the enormous patience of my advisors. It takes a lot of learning and experience to write decent math IMHO.
Have anything to back up that argument?
Here's the abstract [projecteuclid.org] and paper [projecteuclid.org] in all their shining glory. Tell me I'm wrong. Even worse, a year earlier I had made up overheads of that work and presented those in public. That was even worse since I learned a bit about writing and presentation of math afterward.
And LaTex is a life saver here. I can't imagine how this paper would look written in Word, but it would be even uglier.
(Score: 1) by Ethanol-fueled on Sunday November 13 2016, @10:29PM
If you worked for ID and you asked Carmack to explain his fast inverse square root [wikipedia.org] in his code comments, the whole shop would have laughed your ass right out the door.
(Score: 2) by LoRdTAW on Monday November 14 2016, @05:33PM
Not too hard to grok this. The evil floating point bit level hack accesses the float memory as a long int and stores it to i. Then it's shifted right by one and subtracted from the magic number 0x5f3759df. It's then converted back to a float and stored to y. Then its off to a bit of math to take the original number (divided by two), multiplied by the square of y, subtracted from 1.5 and multiplied once again by y all as float. An optional second iteration perhaps further refines the answer. Then the result is returned. The magic number is probably a constant and easily conveyed in hex form that allows the result to properly align to the IEEE float format to avoid NaN errors.
(Score: 1) by khallow on Sunday November 13 2016, @08:35PM
So what you are saying is that the majority of physicists lack the math skills to understand their own field. Got it.
Well, you could try reading my post to see what I actual said. Got it?
(Score: 2) by JoeMerchant on Sunday November 13 2016, @08:46PM
So, in school, I spent about 10-20% of my time in math and math heavy coursework, learned some cool stuff, then graduated and went to work in bio-informatics analysis software development, which is more or less where I've been since 1990. In this relatively math-heavy line of work, I break out the heavy math tools about once every 5 years - I'd say, honestly, less than 0.2% of my work involves any kind of math more complex than a column sum on a spreadsheet. My younger colleagues shy away from the "heavy stuff," preferring to go shop for a library package that has already worked it out for them (which may sometimes be an efficient alternative to "roll and validate your own".)
Occasionally I run up against intractable math problems (care to simulate the heating at the tip of an arbitrarily bent coil of wire placed in an MRI receiving 64 or 128MHz RF excitation pulses?... nope, and neither does anyone else as far as we can tell.) Sometimes it's just hard, maybe a dozen lines of algebra - once I had to fill a sheet of paper with the equation solution (taking the better part of a full day, plus another day to find the mistake and verify the final solution was correct.) In between, literal years go by with no reason to do anything more than a summation, or possibly run a signal through somebody else's FFT library.
For all the study of math, it doesn't appear to be "where the work is at" in real life. Maybe 0.2% of the working population does heavy math >20 hours a week, the rest of us just have meetings where we talk about how to rate something 1-5 for a decision making table, even if half the people in the room don't understand the math behind the table, that doesn't stop them from voicing their opinion, at length, about how the topic du jour deserves to be a 3 instead of a 4.
🌻🌻🌻 [google.com]
(Score: 1, Touché) by Anonymous Coward on Sunday November 13 2016, @09:36PM
I reject your claim; it is usually the mathematics that clarifies the blathering prose.
(Score: 1) by khallow on Sunday November 13 2016, @09:46PM
(Score: 0, Flamebait) by Francis on Sunday November 13 2016, @10:01PM
That's really not true. Mathematics is much more precise when it comes to describing things than English is. What's more, since many people in physics are not native English speakers, using math is preferable as that's something that they've all got in common. If you want to describe the motion of particles, or forces, the natural way of doing that is via some form of mathematical notation. Doing so with English leads to a huge loss in both precision and readability.
Having lots of mathematical equations on each page for a field like physics isn't really a problem. It would be a problem for a field like sociology where there's a less precise level of understanding available.
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @11:28PM
It would be a problem for a field like sociology where there's a less precise level of understanding available.
What exactly are you saying? Are you implying that sociology is not a precise science? Do you not realize that social sciences deal with actual data, facts from the real world, and statistics, not the imaginary mumbo-jumbo world where theorists posit things like "particles" and "dark matter" and then use "numbers" to say stringy theoretical things about them!
Doing so with English leads to a huge loss in both precision and readability.
This is completely wrong! The only reason that English does this is that the majority of people who speak English are idiots, idiots and Americans. But I repeat myself. Perhaps you should speak auf Deutsch, for great science! Who do you think invented sociology?
(Score: 1, Interesting) by Francis on Monday November 14 2016, @04:26AM
It's not completely wrong, you'd run into the exact same problem in German, French, Mandarin or Swahili in terms of trying to describe things in a language that's designed for communicating other things. We chose English as the standard because that was the dominant language at the time when it was standardized. It had been French, Latin, Greek and various other languages depending upon the era, but English was spread widely enough that it was standardized to. The same way that if you're a pilot, then you'd better know how to speak English.
As for sociology, no it's not a precise science and the whole idea that it is is rather ridiculous. Apart from age and a few extremely rudimentary indicators, none of it is quantitative data, it's qualitative data which gets interpreted as best as possible, but lacks a uniform measure. It's not like a mile or a KG which are defined units that can be used as such, most of the data you find cropping up in the social sciences is heavily dependent upon the conditions for replication and as such, much of the time fails because of the lack of reliability in the measures.
(Score: 1) by khallow on Sunday November 13 2016, @11:48PM
That's really not true. Mathematics is much more precise when it comes to describing things than English is. What's more, since many people in physics are not native English speakers, using math is preferable as that's something that they've all got in common. If you want to describe the motion of particles, or forces, the natural way of doing that is via some form of mathematical notation. Doing so with English leads to a huge loss in both precision and readability.
Sure, you have to keep in mind that your paper may be read by someone who can barely read English. But it's still better than solid math. Sorry.
I might add that I speak from experience here. My first math paper was utter crap. It was an interesting though relatively simple result on shoehorning the fractional part (the remainder after you subtract off the integer part) of geometric like sequences (the sequence increases in each term by a factor that is bounded from below) into a narrow slot (say between 0 and some epsilon greater than 0). I wrote it as part of a master's thesis on such things. The problem though was that it was a poorly written wall of math and had a lot of extraneous detail as well.
Sure, the person who doesn't understand English could eventually puzzle out what was in the paper, but it would have taken a while. I could do a lot better, such as taking one or two pages to describe what earlier took me half a dozen. So seriously, I believe even for people who have trouble with written English, an approach which is more sparing of math formulas would be better.
(Score: 0, Troll) by Francis on Monday November 14 2016, @04:42AM
I don't agree with that at all and decades of trying to read as various authors try to convey in English what should be conveyed in math just reinforce the notion that it's sheer madness to try and use language designed for people to communicate such ideas.
The point is, that if the math is written in a way that requires people to puzzle their way through it, then either the math was improperly written or was targeted at the wrong audience. It's not the fault of the math as a language anymore than it would be using Shakespearean prose to communicate the rules on the customs and immigration forms. One shouldn't blame English for that, they should blame it on the idiot that thought it was a good idea to use archaic words and grammar to express something that requires a lower level of English.
Same thing here. If you're going to comment on the equations, that's fine, but you're going to lose a ton of precision in doing so and if you don't, that suggest that you probably fucked up the math in some way. Either you reduced it too far, or you didn't use sensible notation during the process. Either way, that's not the fault of the math, it's an indication that the people using it didn't know what they were talking about.
Day after day, I run into books written by learned morons where they removed information by simplifying things that shouldn't be simplified. They're still mathematically valid equations, but they've removed so much of the information, that you can no longer understand it, you just have to memorize it. And that's a shame.
In this case, why on earth would anybody in their right mind throw out all the information by trying to shoehorn things into English that make much more sense being expressed using math? Right now I'm reading an engineering textbook and the author has done a heroic job of making the English comprehensible and relevant, but it's still not as good as just using the appropriate math to communicate most of the ideas he's looking to communicate.
Anybody wishing to get into a field like physics are engineering ought to be prepared for the relevant abstract thought, both in terms of reading and writing. Blaming the math for operator error is rather immature and isn't to be encouraged.
(Score: 1) by khallow on Monday November 14 2016, @09:38AM
I don't agree with that at all and decades of trying to read as various authors try to convey in English what should be conveyed in math just reinforce the notion that it's sheer madness to try and use language designed for people to communicate such ideas.
Well, I didn't say avoid math altogether. But a lot can be and should be communicated by normal written language because it's a better tool for the task than math is.
(Score: 2) by driven on Monday November 14 2016, @03:54AM
In software development, programmers use comments to clarify what a section of code does.
Then there's the school of thought that if you break your code into well-named functions and use well-named variables, comments are often not needed.
With mathematicians using single letter 'variable' names (often not even stating their purpose) and strange symbols, mathematical formulas look more like poorly written Perl code than something that is meant to be picked up and used by someone other than the author or someone trying to learn about math.
Formulas aren't even that long - why not expend a few extra letters to clarify meaning? Hard for me to believe that today's mathematical representations can't be improved upon and made more tractable to more people (including myself).
(Score: 2, Insightful) by khallow on Monday November 14 2016, @08:50AM
Formulas aren't even that long - why not expend a few extra letters to clarify meaning?
Because it means a lot more work and overhead for the researcher. In mathematical work, these symbols are used far more often than in computer programs. The researcher is trying to construct a variety of certificates demonstrating certain things and there is plenty of backtracking, dead ends, etc that aren't present in computer programs. I think it would be more like requiring computer programs to make their computations human-readable. That would introduce a fair amount of overhead.
Similarly, it generates work and overhead for any readers confirming the results of the paper. And if the researcher uses one set of symbols for research efficiency and other for discussion of the work, that's more work for everyone.
Finally, what actually is worth communicating in this way? Math is notorious for being a place where meaning creates problems rather than solves them.
A common source of new math is abstraction of a math description of some physical system. The variables and concepts to the physical system lose a lot of their meaning when so abstracted. But often phrases and variables are retained, such as with Lagrangian [wikipedia.org] and Hamiltonian [wikipedia.org] mechanics, where the labels that originally corresponded to physical parameters such as "action" get reused even in contexts where they don't make sense. Compatibility with related material turned out to be more important than a paper that was somewhat more self-contained. (I guess that's another problem with math, its concepts and variables tends to have an overly broad scope compared to programming.)
Alternately, one might wish to strip away meaning. A classic example of this is the Lorenz system [wikipedia.org], something I'm studying at the moment. Edward N. Lorenz, a mathematician and meteorologist was studying chaotic behavior of a model of weather systems he had constructed, and attempted to abstract out what was causing the chaotic behavior. The result is expressed in terms of three completely abstract functions x, y, and z of a single independent variable t (which managed to retain its meaning of "time"). Presumably, the three variables originally meant things like wind speed, temperature, or moisture content, but that has been completely stripped away.
When one assigns a meaning to variables, the reader will usually do so as well. That usually aids in understanding, but it can hinder it as well. Math is patterns promiscuously applied. If a pattern appears or is represented in a system, then the consequences of that pattern appear as well, even if no one is aware of the existence of the pattern in the system!
(Score: 1, Insightful) by Anonymous Coward on Sunday November 13 2016, @06:31PM
Idiot: "Teacher! Will we ever use any of this algebra?"
Teacher: "You won't, but one of the smart kids might."
http://www.smbc-comics.com/comic/why-i-couldn39t-be-a-math-teacher [smbc-comics.com]
(Score: 2) by ledow on Sunday November 13 2016, @07:13PM
Because God forbid we expose children to things that are difficult for them to learn.
(Score: 3, Interesting) by RamiK on Sunday November 13 2016, @08:19PM
Studies show 5-12 y/o have very limited capacity to comprehend math. What they do well at those ages is rote memorization. As such, if one were to optimize the curriculum, those early years could be better spent learning an extra language or two, and maybe a musical instrument if they're so inclined, while following the fashion of religious schools and having the children memorize mathematical tables and identities without any attempt at actually explaining or exercising what they're learning.
Overall, there's plenty of time to teach everything. Much like a good gym routine for novices, it's simply a question of efficiency rather than priority.
compiling...
(Score: 1) by Ethanol-fueled on Sunday November 13 2016, @10:12PM
I am now a math retard, but once upon a time I was 2 grades ahead in math and learning imaginary numbers -- and how those were taught remains to this day one of my largest pet-peeves. First of all, they're not "imaginary" because they're used every day (particularly where I work).
Perhaps I'm biased, but I believe that they should be taught alongside electronics where inductive and capacitive reactance represent complex impedances (sounds fancy to the layman but it's a pretty simple concept). And they should stop calling them fucking "imaginary." I believe every student should have to take an electronics class, because it is a practical science and at its most basic levels can be taught with arithmetic and hands-on experience.
My former professor, who had a degree in pure maths, told us that the running joke was that it was 99% pure -- and I believe it. Even in babby's first discrete math class one gets different answers about what zero divides (just itself, or all integers?) depending on which interpretation one subscribes. Some people think zero is one of the natural numbers, others don't. In calculus II here students are taught notation abuse, separating the dx from dy in dx/dy to solve basic differential equations.
(Score: 4, Informative) by bzipitidoo on Monday November 14 2016, @01:17AM
I'm pretty good at math, but I could be better. Math is poorly taught. Made much harder than it actually is. To add to your complaint about imaginary numbers, the meaning of such words as geometry (earth measure) and trigonometry (triangle measure) and even "tri-angle" is mentioned once or twice, then forgotten. Politicians understand that voters need to hear simple slogans over and over for them to sink in. Math teachers seem to feel that students will just magically understand the big picture if they're buried under technique after technique.
Then they don't even pick the best techniques. My pet peeve is the "slope of the line" formula y=mx+b taught in geometry. It is a terrible representation. Can't handle vertical lines, and is a general pain to work with. The Linear Algebra method of keeping lines decomposed into x and y portions and dropping that miserable 'm' for slope is much superior. Two of Linear Algebra's basic operations, the dot product and the cross product, are awfully handy for geometry problems. Another blunder is going straight to the analytic methods, skipping the more intuitive and visual ways used by such ancient mathematicians as Archimedes. Yes, analytic is more powerful, but visual is easier to understand. Geometry class is "analytic geometry", but too often abbreviated to just "geometry", and the students never get any explanation why that "analytic" part is there and what it means.
I find it hard to believe that high schools can burn up a whole semester on trigonometry, when the essentials can be taught in a few days and fit on a business card. Half of trig is "sine is opposite over hypotenuse, cosine is adjacent over hypotenuse", and it is sometimes useful to remember the law of cosines and the trig version of the Pythagorean Theorem, sin^2 + cos^2 = 1.
Calculus is vewy scawy. Most students never get anywhere near the stuff. Those that do don't come away with a good feel for how to apply it. Students are drilled on meaningless formula, learn how to take the derivative of polynomials, trig functions, logarithms, etc. but not what to do with any of it, and that's a tragedy.
(Score: 0) by Anonymous Coward on Wednesday November 23 2016, @10:47PM
With regard to trig, I really struggled to learn it*. (Math "genius" here, btw.) I later understood more fully why. A lot of the complexity of trig is artificial. It has more to do with history and naming of things than anything else. I understand it extremely deeply today, however, and think it's awful that we force it on kids the way we do. There are alternatives. I particularly like the approach of Norman Wildberger--Rational Trigonometry. I feel this is a "sweet spot" of understandability and insight into the actual mathematics, that is actually something accessible to students, without forcing concepts that are terribly artificial. But I strongly agree that analysis is conflated with trig unnecessarily.
Imaginary numbers are a holdover of misguided assumptions in mathematics, and could also be purged if formalized appropriately. It is somewhat arbitrary what mathematicians have chosen to accept as primitive, and imaginary numbers happen to fit that mold. That's not to say they'd disappear. They'd just have a different representation.
I'm less inclined to agree with you on slope/intercept. Unlike the others I've mentioned, that's less of a reach (much of the complexity is necessary), and more of just a practical considerations problem. Yes, linear algebra is powerful and unifying. But to ask middle schoolers to understand it is too much.
*Fun story: After two days of panic trying to learn trig, I did finally get it. It took three more days.
(Score: 2) by RamiK on Monday November 14 2016, @01:32AM
they should be taught alongside electronics where inductive and capacitive reactance represent complex impedances
This will (and does) help address comprehension which reduces antagonism. But when people are complaining about the whole trig \ imaginary(complex) \ (partial) differential equation \ Laplace transforms calculus subject, they're really complaining about the endless memorization of identities at an age when they really shouldn't be learning via memorization.
but once upon a time I was 2 grades ahead in math and learning imaginary numbers
You might have noticed this already, but this is the same with most EE students: They've all been either on an accelerated private school program, come from high achieving families with academics parents (ideally a stay-at-home mother) or from a culture that hasn't stigmatized memorization and repetition for early tuition (Jewish religious schools, private boarding schools and Asian cram schools immediately pop to mind).
There are rare top-of-the-crop exceptions that can still memorize as adults. But those are so few and far between that they're trivial to single out earlier in life and don't need to be considered as part of a standardized program.
So, I stick with my original suggestion: Get those 10 year old sheets of identities and let them hammer at them. They don't need to solve anything or nothing like that. Just cram it all in. When they'll hit ~16ish, you can expose them to circuit solving and physics side-by-side with the actual explanation of the math and solve some problems. Once it's just about applying what they already have in their heads, the process should be fairly painless.
compiling...
(Score: 1) by khallow on Monday November 14 2016, @09:59AM
In calculus II here students are taught notation abuse, separating the dx from dy in dx/dy to solve basic differential equations.
Differential forms [wikipedia.org] (what is represented by such things as "dx" and "dy") rocked my world. It isn't just notation abuse, but turns out when applied correctly to be a better way to do calculus over multidimensional spaces (especially when the space is squirrelly, like it's curved, crossed over itself due to some weird embedding in another space, or something). What's interesting is that the basic symbols of modern differential forms were developed by Leibniz back in the 17th century, while the full math itself came around at the beginning of the 20th century.
(Score: 0) by Anonymous Coward on Monday November 14 2016, @10:29AM
Wait, you weren't learning complex numbers, of which the imaginary numbers are a mere subset?
No. That's what they are named, and there's no reason to change the name. Just as we don't stop calling irrational numbers "irrational" despite nothing about them being irrational in the common sense of the word. And we don't stop calling transcendental numbers "transcendental" despite them in no way related to religion or spiritualism.
One of the things you have to learn when you learn mathematics is that the mathematical terms generally do not mean the same as the common terms, even though they may originally have been derived from that term.
Also note that from the common meaning of the term, all numbers are imaginary, even the natural numbers. In nature, you may find five apples or five oranges, but never the number five. The number five is an abstraction that lives purely in our mind.
(Score: 0) by Anonymous Coward on Monday November 14 2016, @02:04PM
I actually don't even agree with how you describe the names.
Imaginary numbers are called such, because they came to be that way: a "value" that was invented because it made the formulas easy to solve. That makes them fairly imaginary, even if we just accept them like we do "normal" numbers (admittedly I am here ignoring your comment about all of them being imaginary which surely is a good point :-) ).
There is a lot of things "irrational" to the human mind about irrational numbers, though I guess it comes rather from "ratio" (fraction), so that is likely just a misunderstanding of a name, and can be easily explained.
Similarly for transcendental, though I am not sure if that is not even a mistake translation, as the texts that started using it were in Latin and German I believe.
Most of these issues with naming really would IMHO be easier to fix by adding a few sentences on where this came from before teaching the details. At reasonably tiny bit of history in their mathematics won't hurt anyone, and actually quite a few people will enjoy it. And even in mathematics one can learn from history (like all the cases where major flaws in proofs that were considered solid went by unnoticed for ages).
(Score: 0) by Anonymous Coward on Wednesday November 23 2016, @10:15PM
I am a mathematician, but hardly a mainstream one. To my knowledge the abuse of notation you refer to could probably be justified rigorously under a framework like non-standard analysis. In other words, it may not be strictly valid using the axioms the student has been provided, but it is probably still justifiable under a similar, albeit more technically precise framework.
(Score: 3, Interesting) by ledow on Monday November 14 2016, @08:10AM
Math nerd here.
At 12 I was writing computer programs to calculate pi, from a formula based on a Taylor-series expansion.
My primary school teachers had said I was all-but useless at maths because I didn't sit down and learn the timetable and - at their demand - recite each perfectly without error in front of the entire class. Literally, by the end of the year, everyone else had their 12 stars for doing so and I think I'd done the 1's and 2's.
In secondary school, on the first day there, a teacher grabbed me (kindly, but immediately upon seeing me). That teacher INSISTED he was my only maths teacher until age 18. Why? Because he taught the top-set.
I was soon top of that school, of about 1000 kids. I now have a maths degree.
The reason - teaching by rote in primary school is a lot of shit, especially for maths. That's not maths. That's arithmetic at best. It's an awful way to learn. Just because you/your parents learned that way means nothing. It's an awful way to learn, still, and an awful way to teach maths.
If you are memorising, you are not doing maths.
If you do not derive, you're not doing maths.
If you do not work out from first principles, you're not doing maths.
If you do not tie it into areas that you learned yesterday, a year ago, 2 years ago, etc. all at the same time, you're not doing maths.
To this day, my arithmetic is average. I can do any sum or multiplication or division, but I do it by going back to first principles, not rote. I can do it from first principles in my head often faster than people can do it any other way. Fortunately, I'm surrounded by machines which can do arithmetic much better than I can - and that would have been true 200+ years ago too.
But my maths? I work in a private school - as an IT guy, my degree is maths and computer science - and I have embarrassed the maths teachers more than once. I like to walk up and correct their formulae on the board while they're teaching. I've had to sit and explain public-key encryption properly when teachers could only skip over it and say it was "something to do with primes". And I did it at a level that 10-year-olds got it. When you're an IT guy, people don't expect you to know maths inside-out.
And, actually, my strength is midway between maths and computer-science. I love graph theory, coding theory and others that are basically mathematics applied to computing principles. I can't program a game without finding the optimal mathematical algorithm for things. I just enjoy doing it.
The reason my teacher chose me from the crowd? I was doing maths FOR FUN at that age, 10, having come from a school where that kind of independent thought was all but openly discouraged and they had "generic" teachers who did not teach subjects but the whole class for the whole year. And because maths is what I did when I was bored.
To this day, I wonder what could have happened if that teacher had worked in my primary school too.
Every time I see "maths is too difficult", I mentally append "for me." You're not the only one in the school. And not teaching maths by rote helps immensely for understanding. I still encounter maths graduates who have NO IDEA where sine, cosine, etc. actually comes from - it's just a magic "by-rote" function for them that does something useful.
(Score: 3, Interesting) by RamiK on Monday November 14 2016, @11:46AM
How is your anecdotal exceptionalism relevant to the general population? How's having a teacher singling you out as a prodigy and mentoring you applies to everyone's curriculum?
This is the same mistake every novice athlete makes: They look at routines of exceptionally talented, advanced and often juiced athletes and mimic their routines. Ask any music prodigy that's successfully teaches music for a living and they'll confirm the same: While they didn't need nearly as much drilling and memorization, the vast majority of their students do.
This is exactly what's wrong here. People look at high achieving youth that are being schooled privately by excellent teachers and think this should somehow apply to their children. Worse, those youth later reach positions in academia that deprive them of the experience of teaching normal people and influence the curricula based on their anecdotal evidence.
And let be clear, we're talking about engineers, not theoretical physicist and mathematicians. People who actually do end up using calculus to design high-power infrastructure and machines on a daily basis NEED to know those identities. Programmers making algorithmic choices need to have all that "arithmetic at best" linear algebra and combinatorics as second nature so when they work on the code they be able to shift between representations on the fly like a musician switches keys reading sheet music.
Keeping it closer to home, pick up Zed Shaw's books and look up his essays. He's discussing these very same issues as relevant to teaching programming. It's elementary stuff. But so is high-school.
compiling...
(Score: 1) by Ethanol-fueled on Sunday November 13 2016, @10:21PM
The problem with younger people is that they don't understand why learning that stuff is so important, and it is a tedious grind in the context of passing a math class. Sometimes people just have to wait until they're ready to learn on their own terms. It doesn't help that a lot of math teachers are monotone Ben Stein-tier autists.
I spent my entire algebra II class in high school ditching when I wasn't pinching my girlfriend's braless nipples and playing Led Zeppelin on my guitar in class, and the teacher let me get away with it because he was one of those guys who didn't bother to motivate such blatantly unmotivated students.
Well, when I actually decided I wanted to learn math around college age, I had to start from -- no joke -- introductory plane geometry because I sucked at math so bad. If there was anything I could say to any math newbie it's to learn the fucking symbolic manipulation of intermediate algebra, because that's all math fucking is all the way to quantum physics and beyond. "Calculus" is just a fancy way to introduce new algebra techniques. It's all fucking algebra, and if you blow your algebra class you're gonna be set at least 5 years back and be in a world of shit should you decide you want a degree that will make you some money.
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @07:38PM
I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.
- Lord Kelvin
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @07:49PM
And those who depend only on numbers for their reality miss a significant portion of the human existence. They don't even realize when their numbers are hindering themselves...
Although hey, for hard sciences that quote holds up pretty well, but when you get far enough along it breaks down again...
(Score: 1) by khallow on Sunday November 13 2016, @08:41PM
And those who depend only on numbers for their reality miss a significant portion of the human existence. They don't even realize when their numbers are hindering themselves...
I saw that movie.
(Score: 0) by Anonymous Coward on Monday November 14 2016, @03:46PM
Please share, never heard of that plotline.
(Score: 1) by khallow on Friday November 18 2016, @06:57PM
(Score: 1, Informative) by Anonymous Coward on Sunday November 13 2016, @07:59PM
For me, the problem is that very abstract maths is contrary to the reductionist approach that has been very successful in modern physics. Even the hardest theories, like general relativity and quantum field theory, only have a small dose of tensor algebra as the hardest thing.
If a theory is highly reliant on very abstract mathematics, it often means the science is flawed. If you can't present something in a reasonably simple way, you are probably wrong.
ps: I am a professional physicist
(Score: 2) by dingus on Sunday November 13 2016, @09:58PM
On the flipside, if you present things too simply, you're also probably wrong.
(Score: 0) by Anonymous Coward on Monday November 14 2016, @05:35AM
(Score: 0) by Anonymous Coward on Monday November 14 2016, @01:20PM
Quantum field theory actually makes very heavy use of group theory,
And then we had String Theory ....
(Score: 3, Insightful) by mcgrew on Sunday November 13 2016, @08:08PM
Not all of them. Isaac Asimov was a biochemist and wrote a book called "Asimov on Numbers". However, one chapter in that book showed WHY scientists fear math--even the best get it wrong. One chapter in Asimov's book, "A Piece of Pi" got the math right, but he treated a three dimentional object--a big vat the ancient Hebrews made, where the instructions gave both diameter and radius. Asimov held that such a vat would have six sides (I'll have to drag it out and re-read it), forgetting that vats' walls have thickness; a pot's rim is not zero cm thick. So one measurement indicated the outside walls, the other the inside wall.
I argue online with folks who are far better at math than me about the Nyquist equation [wikipedia.org]. Some have come out and said that it says that resulting waveform will be perfect, but it doesn't say that at all. In fact, there is alias distortion. Whether or not it's audible I'm not sure , but at a 44k sampling rate, a 17kHz tone has only three samples. With only three samples it's impossible to discern the difference between a sine wave and a saw tooth wave. I always propose getting a bunch of teenagers and use analog equipment to generate a 17k sawtooth and a 17k sine and see if the kids can tell the difference.
So far, no one has done that as far as I know.
No one born who could always afford anything he wanted can have a clue what "affordability" means.
(Score: 1) by khallow on Sunday November 13 2016, @08:50PM
Whether or not it's audible I'm not sure , but at a 44k sampling rate, a 17kHz tone has only three samples. With only three samples it's impossible to discern the difference between a sine wave and a saw tooth wave. I always propose getting a bunch of teenagers and use analog equipment to generate a 17k sawtooth and a 17k sine and see if the kids can tell the difference.
The next harmonic for sawtooth waves after the 17kHz base tone is at 51kHz. Someone might be able to hear that (there apparently are some people who can hear sounds well above the usual range for human hearing), but they're not going to be your average teenager.
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @08:57PM
What about a square wave?
(Score: 2) by rleigh on Sunday November 13 2016, @09:40PM
It doesn't matter. Nyqvist is all about sampling sine waves, so sawtooth/square waves are irrelevant in the context of digital sampling. The reconstruction filter is sinc (or approximating sinc), so you'll get a sine wave back out irrespective of what goes in. If you want to reconstruct or at least approximate a sawtooth or square wave, you need a much higher sampling frequency. And if you take it into 2D with images (Raleigh and Airy discs), it's still all sine waves.
(Score: 2) by mcgrew on Sunday November 13 2016, @11:19PM
Nyqvist is all about sampling sine waves
You think that electric guitar played through a fuzzbox is a sine wave? It sure looked like a sawtooth on the oscilloscope in that physics class (regrettably decades before digital music). If nyquist was only sine waves, you'ld seldom hear Jimmy Page's guitar from a CD or ogg.
Nyquist applies to any shape of waveform.
No one born who could always afford anything he wanted can have a clue what "affordability" means.
(Score: 2) by jcross on Monday November 14 2016, @02:13AM
A sawtooth wave can be decomposed into a series of sine wave harmonics, with each having a relative amplitude of 1/n, where n is the number of that harmonic. The really high harmonics mostly just make the points of the sawtooth sharper, and beyond the Nyquist limit it's not possible to make it any sharper. If the Nyquist frequency is above your threshold of hearing, you wouldn't be able to tell the difference anyway.
So applying the limit to sine waves doesn't prevent it from being applied to other waveforms as well, it just affects the apparent shape of the wave via its harmonics. If Jimmy Page were to somehow manage to play a 20 kHz sawtooth on his guitar, and we sampled it at 44.1 kHz, the corners of it would be so rounded that it wouldn't be distinguishable from a sine wave anyway. Wave shape just doesn't matter at the edge of the limit because you lose all the higher harmonics that would distinguish one shape from another.
(Score: 2) by rleigh on Monday November 14 2016, @07:32PM
Yes, it applies to any shape of waveform, but that sawtooth is made up of multiple sine waves of varying frequency and amplitude, and that's what's being sampled. Same with a square wave.
http://mathworld.wolfram.com/FourierSeriesSawtoothWave.html [wolfram.com]
http://mathworld.wolfram.com/FourierSeriesSquareWave.html [wolfram.com]
(Score: 1) by khallow on Sunday November 13 2016, @09:53PM
(Score: 2) by tfried on Sunday November 13 2016, @09:29PM
Not all of them.
And wouldn't that be a truly strange world, where an interest in subject A totally precludes an interest in subject B. But one interesting bit along your line of thought may be, that there is a niche for active contributions by a mathematician in - probably - any field of science.
I always propose getting a bunch of teenagers and use analog equipment to generate a 17k sawtooth and a 17k sine and see if the kids can tell the difference.
So far, no one has done that as far as I know.
So, did you? Neither did I, but seriously, the kids would not be able to tell the difference. The ear separates frequencies spatially. Among other things that's important because any single neuron cannot fire more than around 100 per second, so the incoming signal absolutely has to be decomposed. So, to get a pretty good analogy, feed your 17k sawtooth through an FFT. You'll get a 17k sine and a lot of overtones at much higher frequencies. You'll hear the 17k sine component alright (if your ears are young enough), but those higher frequencies will not be representable - neither in a 44khz sample, nor in a human ear.
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @08:25PM
Math is hard
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @09:16PM
So is Ken.
(Score: 2) by rleigh on Sunday November 13 2016, @09:56PM
There's nothing wrong with mathematical equations. But the point of scientific research is to communicate ideas, and often a graph or other visualisation of an equation can greatly help understand what's going on in addition to equations. A paper which is nothing but dense equations is vastly less readable than one which goes the extra mile to present the information in a more easily digestible form. I'm a scientist, but not a mathematician. It can take a lot of effort to plough through all the working and fully appreciate what is being written. Sometimes a few paragraphs of explanatory text or a figure can do the same job. I'm not saying equations are bad, just that if you wish to get your ideas across to a wider audience, they might not be sufficient on their own.
I've been to some meetings about modelling complex biological systems at the cellular scale. Some people had a few slides of equations and then showed renderings of their models so we could see in realtime the dynamics of the systems they were modelling. Others had 40 slides of dense equations, with several pages of terms at the start. I was lost by slide 6 when I'd already forgotten what the terms were on slide 3, as had the entire audience. 30 pages of difficult equations aren't something an audience can grasp in an hour. A paper is different, since you can go through it at a much slower pace. But I still think some considerations for a wider (non-mathematical-genius) audience can be very helpful. Even amongst scientists who are good at mathematics.
(Score: 1) by shrewdsheep on Monday November 14 2016, @10:49AM
If an article on soylentnews contains an equation, hm... reference to a scientific article, it gets fewer comments than otherwise, you see?
(Score: 0) by Anonymous Coward on Monday November 14 2016, @02:41PM
I'm a Ph.D. physicist. There's no social stigma against using math. Physics is a really big field. A lot of us are experimentalists. We know our math well enough, especially those things related to data analysis, but we don't do a lot of the mental masturbation some of the theorists do. If there is a paper that looks like it should be in a mathematics journal, that's not the paper I'm interested in reading unless it is directly relevant to my work. I'm not going to cite what I haven't read. And if I need to wander afar into the theoretical side, that's what collaboration is all about. There are plenty of practicing physicists that I wouldn't trust to know which end of a wrench to hold, much less trust them to walk around unsupervised in a lab with optics or high voltage.
It is no surprise that biologists are put off by math because very advanced math is not used as much in that field unless you are working in the bio-quantum arena.
When I was in grad school, there were three levels of physics offered to the undergrads: physics with calculus (science and engineering majors), physics without calculus (pre-med, nursing, etc.), and physics without physics ("Science 101" to satisfy a "science" graduation requirement).