The new weekly question is:
“If you could shake the public and make them understand one scientific idea, what would it be?”
I’m going to get all fundamentalist on this one. The one thing I wish everyone understood is…
Math.
I know, it’s cheating, and it’s a whole wide range of concepts rather than just one idea, but really—everything would be so much easier if they knew a little algebra, some basic probability theory, and a teeny-tiny bit of statistics. If only everyone understood probability, the opposition to evolution would decline rapidly (even the creationists who purport to be mathematicians seem to be innumerate), and of course, lottery systems would go bankrupt and Vegas and Reno would return to desert.
It really isn’t a lot to ask. I won’t even insist that everyone understand introductory calculus. I’m sure that all my fellow science academics will agree that at least it would make our disciplines easier to teach if we could put up equations as easy as Hardy-Weinberg or Nernst and not have half the freshman class stagger back, reeling as if we’d kicked their grandma by exposing them to grade school algebra.
Great answer.
Though I would say that you haven’t truly known pain until you’ve tried to teach methods and stats to a room full of psych and education majors.
Are bio majors really fairly math phobic too?
I’d go with making the public understand the definition of the word theory when it comes to science. Why is it they get it when it comes to art (music theory, etc) but can’t get their heads around that same definition for science.
I agree with probability being somewhere around #1, but I’d also put basic calculus near the extremum. Rates of change is a pretty steady requirement across sciences and real life.
I completely agree! Then us mathmeticians would rule the world!!!
PZ, don’t forget about no more “wave of the hands” tax cuts, no tax cuts when the government doesn’t make as much money as it spends, probably no budget deficits, no interest only loans, no … I could think of lots of things that would happen if people understood math!!!
I’d settle for arithmetic.
I think the world would be a better place if people used the careful LANGUAGE of science, which speaks in terms of data rather than truth, evidence rather than proof, and associations rather than causes (if there is insufficient evidence of causation). Think of the humility that might result if people viewed their positions as tentative and open to revision rather than beyond challenge and etched in stainless steel. Superstition and theocratic impulses would crumble into dust if such a shift in thinking were widely embraced.
Repeat after me:
Correlation does not imply causation
Shaking the public would require a massive earthquake.
(Yes, that was my first thought on reading the question.)
Math is a great answer, but also good (but more narrow) would be: nuclear energy.
Yes and no. As a genomicist trained in biology but whose work is essentially entirely computational, when I find myself having envy for other’s math skills, they almost always in discrete math. That’s because these days, tackling a scientific problem rarely is a matter of solving a differential equation (as important as they were historically), but rather is a matter of developing an algorithm. Discrete math is the math that helps there.
I’ll put my vote in for The Scientific Method. When people understand that, they can differentiate good and bogus science. After that, they’re on their own.
I tell all my sociology students that the most important idea I try to teach them is the logic of inference. If you don’t understand it, science appears to be a form of magic or merely opinion-trading. This is probably more of a problem in the social sciences than in the natural sciences, but I think it is the main stumbling block to learning and appreciating the amazing power of the scientific method.
Jonathan Badger:
I think that’s a funny comment. Differential equations are just as important as ever, just not (I’m assuming) in the field you happen to be working. I’m not saying this out of bias–I’m trained in CS and much better at graph theory and combinatorics than I am at calculus–but differential equations are the main way to model continuous dynamic systems and they show up all over the place, and will continue to do so until such time as the laws of physics change drastically. In fact, some areas in discrete algorithms such as adaptive mesh generation are primarily motivated by the need to approximate solutions to differential equations.
I will agree that many techniques used to attack differential equations are less important now, since you can use a fast computer to crank out solutions using finite element methods. But they are not primarily of “historical” interest (the way, say, quaternions are subsumed by more general matrix transformations).
PZ! My hero!
Now to grade that stack of calculus exams.
In keeping with the deeply religious tenor of this blog, a trinity:
1. “Theory” does not mean “entirely unsubstantiated idea promulgated by someone with no knowledge of the subject matter”;
2. Correlation does not imply causation;
3. An understanding that “twice as many people suffered side effects” means zip w/o knowing anything about the size of the test group or the number of people who suffered side effects.
I’ve given up on the math bit after dealing with a student cashier who could not make change, despite the fact that the register had “$13.21” on its screen…
Aaargh. Having received my bachelors in Psych, I have to say the methods and stats is the bread and butter of the discipline! On the other hand, if I think back to my stats class, I believe only myself and three other people in a class of 50 actually “got it”. Double Aaargh!
I wonder if that was why we were segregated? Stats for Psych Majors was a separate class
I hate math, and I especially hate calculus.
Einstein hated calculus: It drove him straight to the bottle.
All of his colleagues, back when he was still teaching in Munich, would mutter under their breathes, “Geht der betrunkene Professor und sein geliebtes Bier.”
He’d always have nasty accidents in the lab whenever he was drinking and deriving.
I think math is hugely important, but I don’t think that by itself it leads to scientific understanding. It actually places some obstacles in the way:
(1) Its standard of evaluating claims, the deductive proof, is too rigorous to say much one way or the other about most empirical claims–they’re all conjectures to a mathematician.
(2) It gets you from axioms to true propositions, but doesn’t test either of them against real data.
(3) A deep understanding of many parts of mathematics requires dedication and focus on specific formal systems, leaving little time for general critical thinking. E.g., suppose you’re Andrew Wiles in hot pursuit of a proof of FLT. If you have time to study recent scientific claims critically, that’s great, but the math isn’t helping you anymore than it would help you to become a concert pianist or chess grandmaster.
I think to do science you should learn at least a little math, with emphasis on elementary quantitative reasoning. In particular, you ought to be good at order of magnitude approximations. E.g., if someone told you there are enough used tires to form a planetoid the size of earth’s moon (which is not true) you should be able to give a back of the envelope volumetric proof of why it is not so, complete with your assumptions and their justifications. Quantitative reasoning is a huge part of critical thinking and one sadly missing in a lot of otherwise intelligent people.
Beyond that, though, it should be more than clear from the ID crowd that simply understanding advanced math is not going to turn you into a scientific thinker.
That would get my vote too, but I’ve got a better one: The imperative of reason as a decision-making process applies to all situations with empirical consequences. Even the ones you feel passionately about. Even your hot-button issues.
Far, far too many people seem to believe that there is some minimum threshold of emotional weight beyond which their position on the topic in question no longer requires rational justification. As just one example, I’ve lost count of the number of times where I’ve been in a conversation with someone who made a statement that implicitly assumed that exposing “minors” to fictional depictions of violence or sex would “harm” them, asked the person to describe in detail the supposed harmful effects and provide evidence that they in fact occur at all, let alone are caused by exposure to the content in question, and had them react as though I’d thrown a live tarantula straight at their foreheads.
Not a bad idea, actually, aside from being rather cruel to the tarantula… :(
“Repeat after me: Correlation does not imply causation”
Meh, but repeating that slogan alone implies paralysis. In the end, a whole bunch of stuff is just correlation, and yet causation is pretty accepted. Correlation doesn’t imply causation, but it does suggest causation or common factors as a possibility, can help confirm a theory of causation, works against a theory of independence, tilt the likelihood ratio from one hypothesis to the other etc etc… The phrasing ‘correlation does not imply causation’ is very often abused.
Really, it would be better to expand it into the subject of statistical inference, which is pretty close to what everyone really wants to write down – the scientific method.
lottery systems would go bankrupt and Vegas and Reno would return to desert.
Our biostatistician once told us he was going to a conference in Las Vegas. “But, they must hate it when statisticians come to Vegas, because the gaming tables are pretty empty.” We asked him if he was going to gamble himself. “Of course–I expect to be an outlier!”
Good choice!
When my son was five, I was having trouble getting him to focus on his arithmetic. That changed the day the register stopped working in our local convenience store. I handed the clerk a five-dollar bill and three pennies to pay my bill of $2.53. She stared at the money and then said, “I’m sorry, but I just don’t know what to do with that.”
I asked for my money back, then handed her the three pennies. “Here’s a down payment. Now how much do I owe?” She stared at the pennies for a few moments, then her face lit up. “$2.50!” she said. I handed her the five dollar bill. “Take the $2.50 out of that, please.” She happily did.
My son commented, “So that’s why you keep telling me to learn my arithmetic.” From then on, he made a real effort. (I should probably confess that to keep him from getting bored again, we often practiced basic addition and subtraction using negative numbers.) In high school, he happily defended the local McDonald’s, where he worked part time, from a zoning board complaint by proving that the arches did not in fact exceed the total square footage permitted each business for a sign. While taking his first college math class, he called to tell me with excitement how useful calculus was turning out to be.
It all makes me regret that my own education contained so little math. Of course it’s never too late. I could start now.
I agree with PZ, primarily the statistics. If you understand how statistics work, than a lot of how we understand the world (and why we insist on repeatable experiments) makes a lot more sense. Also it will embolden newspapers to give more information on the statistics that they cite.
The previous post on the Vitamin industry I feel exemplifies the issue.
The one thing that I wish I could get into my poor head is:
math.
I do want to know introductory calculus! I don’t know what my problem is, since I actually retained a lot of algebra. I’m back to the “Trig for Dummies” book and I got tripped up right away. (It doesn’t help at all that there’s a typo–diameter instead of circumference–in one of the first problems, though! At least I caught that. I was so proud of myself.) I love math and always have; it just doesn’t love me. I’m trying! (Gabriela, are you reading this? Richard Cohen, don’t you dare feel sorry for me.)
I vote for classical logic training and critical thinking skills. You need those for math; you also need them to read the newspaper, decide which politician to vote for, figure out if the child or the dawg emptied the garbage all over the kitchen floor, etc.
Pity that people don’t need them to breed… *sighs* Others, let’s see…I’d also like people to realize that the bad habit of anthropomorphizing other species and non-living systems has “fallacious” coming out the ass… (ah, hell, now I’m doing it ;/)
What a surprise! Is this a math site?
I would never question the utility of the community of mathematics, nor the headiness of correct calculation or discovery, but as Wittgenstein communicated – seems so long ago now – math can’t touch God.
I like “scientific method” as an answer for the long run, but in this world, today, I would vote for evolution. If we, the human community, owned a clear understanding of the processes of evolution, we might be able to recognize ourselves as a community and initiate a reasonable process of interaction. After a few hundred years of negotiating, mediating and compromising, we might be in a position to turn our attention more fully to the beauties of math.
I guess I’m just frustrated in that I did end up taking quite a lot of calculus and it really isn’t the math you need in biology. Want to know where genes are? Gene finders use probability + graph theory. Want to know how genes are related? Phylogenetic inference algorithms are probability + graph theory, etc. etc. I’ve managed to pick up some of the stuff I need myself, plus I’ve had good relationships with CS folk, but the calculus based education I had seems like it was more geared towards the problems of 19th century physics than anything else.
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 meager 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, 19th-century British physicist
On teaching statistics – it has long seemed to me that statistics for non math majors should be more of a qualitative than a quantitative things. What non stataticians need to know, is not how to do statistics, but how to look at statistical information and evaluate whether or no it seems to have been done legitimately. While I do a fairly good job at evaluating statistics I know that I were doing anything of my own beyond a (very) simple set, I would simply ask for help from someone who really knows statistics. And a non math person does not gain that knowledge from 101. Rob
OK, I’ll pile on. Great answer.
When I saw the question, I thought “well, integration by parts, of course,” (half-seriously). But if we could get the public to get some kind of basic understanding of math, it would be marvelous. So many of the great stupidities of public policy are easily debunked by just a bit of understanding of math. (“Tax cuts increase revenue!” “OK, genius, cut the tax rate to 0% then!”) I disagree about the need to learn calculus itself. Basic understanding of statistics, proofs, and logical arguments would itself be tremendous.
Even a basic feeling for math would allow people to better understand the big numbers thrown around in discussions of public policy. Press accounts toss around “million”, “billion” and “trillion” without context, leaving the reader often with the impression that a government is wasting $x million on some arts program but spending $y billion on some weapon system is perfectly reasonable. And the increase in the national debt by _trillions_ of dollars is ignored.
Some basic math intuition would also release the plague of the permanent state of fear that seems to have engulfed Americans in the past five years.
Proof by induction includes a step where you have to go back and test a starting case. It’s no good building a sequence which theoretically works but still starts from a false premise, ie a base value which doesn’t work.
Thank you, Jonathan. That is exactly right, and I had to find it out for myself as well.
Kristine –
Are you keeping a diary or other record of your math struggles? I ask because as a TA I saw many students struggle in ways that just boggled me. Occasionally (signficiantly less often than I would have liked) I’d see a student return to a concept later that had completely escaped them and be able to handle it, but I never understood what process lead from point A (complete vexation) to point B. (understanding)
There have been concepts I haven’t understood, and I can remember struggling with various things (algebraic geometry kicked my ass hard), but I don’t think I’ve ever experienced the kind of panic-inducing lostness I occasionally saw in some of my students. I’d really like to know how one gets past struggles with math at the undergraduate level or earlier.
My suspicion (based on, so far as I can tell, no actual evidence) is that when this happens there’s some very basic hole in the student’s background that no one can identify because they don’t know where to look. Occasionally, enough other concepts crowd around that hole that the student experiences an “aha” moment and fills in the background hole themselves. If they ever try to communicate their great insight, they receive blank stares from people who never had that particular hole, so they end up feeling silly about ever having had that hole and shut up about it. The people doing math education never find out that this spot is a spot people have trouble with, and so the overall condition doesn’t improve.
How about: the idea that things which are alive have no special “life force” as compared with things that are dead. On other words, that vitalism is wrong.
I’ve got to admit that mathematics is a great choice, but, if I could shake the public and make them understand one scientific idea, it would definitely be evolution – in sufficient depth to address the issues raised by evolution deniers. Of course, no one can sway them, but understanding the flaws in their claims well enough to refute them before a rational audience means that one must be quite scientifically literate. Remember, this is my fantasy, and the idea of smacking the general public into having a reasonable understanding of the world around them is actually exciting.
A fundamental contemporary conception of evolution requires one to have a sound grasp, but not expert knowledge, of many scientific disciplines and the associated math pre-requisites. The age of the earth is more than 4 billion years, fossils are aged in the millions of years, and modern humans are a few tens of thousands of years, so one needs enough chemistry/physics to understand radioisometric dating techniques. Genetics give us insight into common ancestry, mechanisms of inheritance, source of variation, etc. Geology tells us environments are not static, so organisms must adapt or risk extinction. Biochemistry tells us about commonalities among essentially all lifeforms and the specifics of physiological processes. These and other disciplines would give John Q Public the intellectual resources necessary to make informed decisions for a modern technologically-oriented society.
But, more than anything else, understanding of evolution is also understanding that we humans are not separate from the natural world but are simply one small part of it. We are powerfully influential here on earth, but we are indeed, only one small speck in the overall scheme of things.
Oh, the world would be so much easier and better off if everyone had a decent understanding of basic math. For a while after college, I tried my hand at teaching, substituting in both middle and high school, as well as private tutoring in math. Also, with a math teacher for a mother, I saw first hand the complete lack of understanding and failure to teach math in our schools.
At least half of the students still did addition by counting on their fingers, and less than 10% could do any arithmetic without the aid of a calculator. Almost nobody knew their most basic multiplication tables, and concepts like fractions and decimals were beyond even the high school students. And they certainly weren’t getting any help at home.
One great (terrifying) story from a student’s mother: she was attempting to help her daughter with her homework on decimals, and told her, “Oh, I didn’t know what those little dot things were, so I just had her erase them.” Dot things!!!!!!!!
There have been concepts I haven’t understood, and I can remember struggling with various things (algebraic geometry kicked my ass hard), but I don’t think I’ve ever experienced the kind of panic-inducing lostness I occasionally saw in some of my students. I’d really like to know how one gets past struggles with math at the undergraduate level or earlier.
My suspicion (based on, so far as I can tell, no actual evidence) is that when this happens there’s some very basic hole in the student’s background that no one can identify because they don’t know where to look.
————————————————-
I think Daniel Martin is correct. I have both seen and felt this feeling of utter panic. (I have gotten over my own feelings enough to get a masters degree in Statistics and a PhD in Mathematics.) However, I have seen people who are completely unable to grasp concepts I think are trivial.
I frequently saw this in elementary calculus. I have a friend who used to say that no one ever failed calculus; people failed algebra while taking calculus. I think this is correct. The extra knowledge that elementary calculus adds to algebra is quite small. However, the type of calculation you do becomes much more difficult. Any problems a student had with algebra stand out in high relief. I frequently think we should not fail students in mathematics; we should send them back to take the previous course until they have really understood it. I know this has real practical consequences for the educational system but, if you want people to learn, I think it is a reasonable idea.
I also have a bone to pick with physical and biological scientists who think they understand mathematics well-enough to teach it to their students. As an undergraduate Mathematics major who stared blankly at many scientists fouling up Mathematical concepts I can assure you that you do not. I got news for you. If the Mathematics majors in your classes do not understand your Mathematical explanations, it’s your fault, not their’s.
–PatF (Former Mathematics professor spoiling for a fight.)
PaulC puts my attitude best
I think to do science you should learn at least a little math, with emphasis on elementary quantitative reasoning. In particular, you ought to be good at order of magnitude approximations.E.g., if someone told you there are enough used tires to form a planetoid the size of earth’s moon (which is not true) you should be able to give a back of the envelope volumetric proof of why it is not so, complete with your assumptions and their justifications. Quantitative reasoning is a huge part of critical thinking and one sadly missing in a lot of otherwise intelligent people.
although my impression of how people deal with stuff is that everybody, not only science practitioners, would benefit from training a size gauge reflex into their daily lives.
And this is, in the end, a trained habit (as PaulC explains); of stopping and “figuring” for yourself. And it’s something that is worked against in this society, where all of us, as consumers, are massaged with advertising and spin in order to provoke a quick decision reflex whether in the supermarket or in the voting booth.
Probability’s a good basis for dealing with stuff, but it requires a deliberate suspension of the impulse to quick decision-making and even in some more involved medical studies it can be fairly grossly misinterpreted by practitioners who’ve passed their stats modules by learning how to plug formulae in.
If you want a better grounding in stats you just have to come to grips with the calculus and for my druthers I’d like to have every person on the planet inducted properly into the fabulous ways of e; this properly grasped, and banks (and the first world economies that they interact with) would have to work a little harder for their monstrous profits without the miracle of compounding interest to prop them up, and there might just be a better grasp of the carrying capacity (wrt to human population growth) of this globe.
And everybody should be given the maximum chance to appreciate the beauty of my favourite equation
e^(i * pi) = -1
It makes complex exponential functions expressible in geometry and thus “seeable” much more quickly.
Math is good for modeling, and so for basing decisions. But if I had one idea to put over, it would be the basic method of science and its epistemology.
By formalising parts of science one can use deduction and inference to look for true or false statements.
But the fact is that the methods of science, that has proven themselves by being used successfully, use observations to find facts about phenomena, and consequently theories validated by observations.
Facts are not exactly truths. Observations, theories and facts have uncertainties, randomly choosen limits for decisions, and may possibly be falsified later.
It would be great if people understood that deduction, induction or inference is mere means to create hypotheses. And that science is about facts and not religious or philosophical Truth. It would help with many problems that people have with science. (It is of course permissible to take theories basic phenomena to be Reality. There is no valid problem with that.)
“Correlation does not imply causation” would be my second choice.
Daniel, that’s a great idea for learning math. My SO and I do this with chat transcripts, because he doesn’t know what I don’t understand, and I don’t know what I don’t understand, either. Dyscalculia is a real bitch. Numbers (and spatial relations and sequences and all that stuff) just don’t work the same way for me as they do for everyone else. I’ve been struggling with algebra for almost two decades now. One thing that really helped me was to find out what you could use it for, because the way it’s presented in most math classes I’m familiar with, it just sort of hangs there in space and isn’t relevant to anything except solving the practice questions in the textbook.
That said, if I could impart any scientific wisdom to the general population at large, I’d give them the basics of linguistics, semiotics, and discourse analytics. Until people start communicating precisely, and realising that words (and other symbol forms) have discrete, contextually-defined meanings, they’re never going to get much of anywhere with anything else. If everyone understood that stuff, it would eliminate the “fair and balanced” problem right there, among other things.
I would go with Ockham’s Razor. It is the strongest argument against the existence of God, and the one most often misused by Christians who think “simple” means “easy for an idiot to understand.”
I guess you could argue that Ockham’s Razor falls under the category of math, since it’s sort of a qualitative way of saying you can judge a hypothesis by multipling the probabilities of your assumptions. Either way, I can’t see how anyone who properly understands it could possibly believe in God. Atheism would get a big boost.
Plus, I like the irony of an argument against religion named after a devout friar. I guess it was hard to know any better in the 1300s.
The imperative of reason as a decision-making process applies to all situations with empirical consequences.
EvenEspecially the ones you feel passionately about.EvenEspecially your hot-button issues.Fixed it for you, Azkyroth.
The strike tag didn’t work. For anyone’s future reference.
I noticed the strike thing. And yeah, you’re right.
Speaking of which, I’m very impressed (how sad is this) by the fact that so far no one has cited the morass of circular reasoning and “everyone knows”es or the array of badly and fundamentally flawed research that purports to show a simplistic and predictable cause-effect relationship such as I describe, tried to turn the question around on me, or accused me of being a sociopath/pedophile (yes, all of these have happened in previous conversations on this topic).
Zeno, you’ll never finish them. Wait until you’re half way through, then you’ll see…
Bob
simple logic?
About the gaping hole in education idea: I was in a graduate seminar with another doctoral student in biological anthropology who didn’t understand a graph we were discussing. The professor asked her a couple of questions and discovered she didn’t know y=mx+b, the simple equation of a line! Turns out she had moved several times during her childhood and somehow missed this little item, but still, how did she manage to pass algebra without it? How did she do well enough in high school to go to college? How did this woman get into grad school? I mean, when I took the entrance tests there the ACT and the GRE had math sections.
On education and psychology majors’ stats classes, I was in a grad level course on stats with a bunch of struggling education grad student. Guess what concepts gave them problems…..mean, median and mode! Mean was the basis for all the inferential material we learned the rest of the class and they just didn’t get it….I think basic arithmetic is a gaping hole for many people, and it’s getting worse. My friend’s second grader uses a calculator for math at school. The teacher says it helps them check their work.
UGH! I could go on for hours about the complete lack of basic math, more less the problems with calculus, statistics, etc.
There are so many good suggestions, I must revoke my second choice. I missed Azkyroth’s (idea of) bad habit of anthropomorphizing. It isn’t really primary science, but it is farreaching.
Ockham, logic, fallacies, causality, universality, unaccuracy, the dealer has the advantage, … There are so many goodies.
I’m really rather behind the pack when it comes to math. Not really because I want to be, I’d actually like to learn more about math, and I hope that soon I’ll have the time to do so, but for a long time I was really put off from math. When I was in high school, my math classes consisted mainly of “when this, do that.” I could never retain it, because my teachers never told me WHY you do that when this. Since I’ve moved to Japan however, one of my good friends is a math teacher, and one of the ways we practice his english is by teaching me math. It’s been great. He knows that I’m not very good at math yet, and he wants to practice his grammar and expand his vocabulary in english, so he always tells me WHY we’re doing what we’re doing in every problem. I’ve gotten more out of the last 4 months of math with him than I did in 3 years of high school and 2 semesters in college…
Precisely. That is the ultimate demonstration of God’s nonexistence.
Are bio majors really fairly math phobic too?
Excuse me if someone already addressed this (I can only read so many comments), but, yes, bio majors are very math phobic. A boatload of them study biology because they like science, but don’t want to do math. Or, even worse, they want to become doctors. Little to they realize that you can’t do good biology without a solid understanding of basic statistics and calculus. I’m not saying you need to be a math wiz, you just can’t be math phobic or you are are destined to fail.
I’ve been thinking about this thread’s question for quite some time now and find I cannot answer it because anything I answer presupposes other things. For example, answering math presupposes some sort of sensitivity to plausible hypotheses about the world (assuming we don’t want to turn everyone into mathematicians, that is) and so on. The only answer that I can give is a vague one: humans can and must understand the world rationally.
Yeah, the math thing bugs me too. We would think people were ignorant if they said “I hate literature” yet otherwise intelligent people say “I hate math” every day. Parents tell their kids that they hated math in school. We perpetuate the math hating from generation to generation.
When my niece and nephew were young I made math fun for them. We’d do little tests and when I came over they’d say “Can we do math?” They are now many grades ahead of their class in math, having done things like exponents and negative numbers while the rest of their class is learning how to add.
Math is fun and it does teach logic and inference and a lot of other skills necessary for critical thinking.
Sorry, I tried. I can’t get further than the quadratic equation. I took statistics in college, did my homework, got a decent grade, but I still don’t get it. I’ve always had problems translating English into mathematical formulas.
It has also been my experience that bio majors tend to be quite math phobic. However, math phobes are not necessarily doomed to fail as professional biologists. It all depends on the kind of biology they end up in. My SO is a cell biologist (figuring out how certain organelles work) in a lab with a world class reputation, but she uses no statistics at all in her work. When I saw her producing a graph for a paper she was writing, I commented on the lack of error bars in the graph (each plotted data point was a mean value of several measurements). She didn’t even know what I was talking about. So I showed her how to calculate standard errors and how to plot them. When she presented her results in her lab, she was hailed as a math genius by her colleagues.
I don’t think there’s any one thing that could make the ignorant and uneducated capable thinkers.
The one thing that I would cause people to understand, then, is that they have no right to hold opinions on subjects or make policy that affects or is affected by those subjects until they are knowledgeable and educated.
Hmm. It wasn’t my experience at all that bio majors are math phobic. We were, however, foreign-language-learning phobic (one of the perks of being a science major at my college was I could skip the foreign language requirement to take 4 more science classes for a BS instead of a BA in biology)
Also, an easy way to force people to learn math better: take away their graphing calculators. I honestly think the reason I did well in math is because I lost my graphing calculator during high school and was too cheap to get another one. I consequently focused on solving things the equation-and-graph way much more, even discovered that was sometimes easier and led to more accurate answers, and causes you to think about the way math works mechanistically more.
I’m generally with Azkyroth, but even his carefully-worded concept presupposes that people know how to reason properly!
For example, the scientific method isn’t just some arbitrary selection of rules. It represents what may well be the simplest “practical” protocol for collecting knowledge about the natural world. (Practical in this context includes allowing for realistic limits on our abilities, resources, and trust in each other.) To understand the confidence that scientists place in the scientific method, you need to be able to understand *why* they do their work that way, and not otherwise. To understand the limits of science, you also need to know what prices are paid, for doing things that way.
In short, Keith Douglas has a point about implications…. ;-)
I teach Chemistry and Physics in High school. I would agree that I would like people to understand math better.
The problem with math, in my experience as a student and teacher, is the way math is taught in math class. Students who understand the equation of a line in a math class, don’t understand how it is connected to Temperature and Volume when pressure and moles are constant. I was mad at my math teachers when I got to college chemistry and started to learn stoichiometry and realized numbers are connected to units.
drag64 brings up another thing in science teaching that has been bugging me for ages (since high school classes, actually) – the sloppiness in elementary science discussions about the difference between units and properties. For example, in my grade 10 physical science class, we were taught about (what is correctly) the elementary relationship between potential difference, current and resistance of a DC circuit – Ohm’s law. This is often written V=IR. Now, we were told that we could call I “amps”. This is wrong – amperes (or “amps”, in the slang) is simply the SI unit for current. Failure to teach this correctly makes dimensional analysis and other useful skills very confusing. (In this case this is made worse by calling the potential difference “voltage” even in careful contexts.)
Incidentally, this seems to be a problem in less developed scientific fields, too. What units (not properties) are used in, say, psychology? For example (and I am genuinely curious) what is a typical unit used in the Fechner-type psychophysical laws?
“I would never question the utility of the community of mathematics, nor the headiness of correct calculation or discovery, but as Wittgenstein communicated – seems so long ago now – math can’t touch God.”
Precisely. That is the ultimate demonstration of God’s nonexistence.
And answers like this one are the inevitable demonstration of a pervasive inability to grapple meaningfully with the human problms of our world.
RavenT quotes Jonathan Badger:
Again I just have to insist that this is a very narrow view of what problems are currently of interest in science. Calculus is a powerful tool for abstracting continuous rates of change and finding the result of accumulated change. Differential equations are the way to express relationships between these changes, and if you cannot set up a differential equation, you are greatly hobbled in your ability to analyze continuous dynamic systems (even when you simulate a system in an “intuitive” way, e.g. adding acceleration*delta t at each step to velocity, velocity*delta t to position, you are setting up and solving a differential equation. You can do this without realizing it, I admit, but roughly in the sense that M. Jourdain was unaware he had been speaking in prose).
Now you can dismiss this as the “problems of 19th century [what about quantum?!] physics” but these are also problems of 21st century engineering. Finite element methods show up in all kinds of CAD tools, and these are merely algorithms for solving a differential equation numerically to some level of approximation. It’s not just “old economy” stuff like jet planes. I don’t know how you think someone designs the audio circuity of an iPod without diff eqs. Even the digital circuitry has to be understood and analyzed in terms of voltages and currents at some point in the design.
Engineering is not science, of course, but there are areas of basic science (e.g. atmospheric science) that are not physics but require some physics to make progress. Diff eqs are still the tool of choice in many such areas.
Two implied points I agree with:
You can do a lot of science without calculus, and discrete math–graph theory, combinatorics, and inductive proof–is underemphasized in the high school curriculum.
Calculus is taught in an archaic way that puts emphasis on techniques that were more useful in the 19th century. A closed form solution will always be of great aesthetic value, but there is no longer the urgency to simplify solutions this way, and we know that many systems can only be solved numerically. So many of the complicated techniques in advanced calculus (e.g. integration by trig substiution+parts, solving differential equations by Laplace transform) should not be given the emphasis they have now in my opinion, while the ability to set up equations and solve them numerically should have more emphasis.
There is math and there is math.
Using math models, including unit and unit analysis, is applied math in other areas – but so are differential equations. If we go further on that route we could add some group theory, topology, differential geometry and what not, to prepare for modern theoretical physics. It would help the math subject too.
Speaking of math itself it is an excellent way of learning how to set up formal theories and actually prove stuff formally. I can remember what a relevation that was – and how hard it was to unlearn it later to make use of ordinary unrigorous or heuristic unformal theories and models again. :-)
Looking at modern solution possibilities, numerical theory and practice are essential. But perhaps also looking at modern variants to calculus such as nonstandard analysis – we can use Newtons intuitive differentials rigorously, for example. A little effort in order to be lazy later.
True, but I was speaking solipsistically of what problems are currently of interest in science to me. :)
That would be a fairer way to state what I meant, yes. I spoke to a group of high school students at Math Day about how I use math in my research, and graph theory was definitely alien territory to them. Even if they don’t want to study calculus, at least that’s familiar to them because it’s highly quantitative, so it “looks like math”.
You’re right; I’d like to see discrete math emphasized more in high school, and by intro biology advisers in college as well.
Did I mention that the class I TA’ed for most (and therefore what I am basing my observations on) was “Calculus II for the Biological and Social Sciences”? (as opposed to “Physical Sciences and Engineering”) And that I TA’ed at Johns Hopkins, which probably has a higher concentration of I-wanna-be-a-doctor undergrads than most places?
The number of my students who were planning on medical school in the future is something I always tried desperately hard not to think about.