One of the reasons I held off on commenting on the whole E. O. Wilson math op-ed thing, other than not having time to blog, was that his comments were based on his own experiences. And, you know, who am I to gainsay the personal experiences of a justly famous scientist?
At the same time, though, this is one of the big things that makes the original piece so frustrating. He’s speaking from his personal experience, but it feels like he’s chosen to draw exactly the wrong lessons from it. The relevant anecdotes are:
During my decades of teaching biology at Harvard, I watched sadly as bright undergraduates turned away from the possibility of a scientific career, fearing that, without strong math skills, they would fail. This mistaken assumption has deprived science of an immeasurable amount of sorely needed talent. It has created a hemorrhage of brain power we need to stanch.
I speak as an authority on this subject because I myself am an extreme case. Having spent my precollege years in relatively poor Southern schools, I didn’t take algebra until my freshman year at the University of Alabama. I finally got around to calculus as a 32-year-old tenured professor at Harvard, where I sat uncomfortably in classes with undergraduate students only a bit more than half my age. A couple of them were students in a course on evolutionary biology I was teaching. I swallowed my pride and learned calculus.
He obviously went on from here to have a stellar career, and from this he draws the lesson that math expertise isn’t actually all that important. Which, you know, is one way you could go with that, but I can’t help thinking that there are many more productive directions he could’ve taken from this starting point.
For example, having noted that his poor math background can be traced to his background in poor schools in the South, how about asking what can be done to improve the education at those schools? This seems like a possible starting point for a drive to improve the resources available to students in poor schools, so that the next generation of students won’t have to suffer the embarrassment of not having algebra before college.
Or, on a broader scale, the observation that otherwise bright students are put off by math seems like an excellent jumping-off point for some real inquiry. Why are these students put off by math? Is it something in their background, say, poor Southern schools? Are their skills actually weak, or do they just feel inadequate, due to some Impostor Syndrome kind of thing? Is there a way to either improve their skills or make math less scary for them, so that they don’t feel pushed out of science?
Those all seem to me like fruitful starting points for both scientific inquiry and policy activism. They’re questions whose answers could move you to try to change the world to make it better for everyone.
Instead, Wilson takes the path of least resistance. The most charitable interpretation is that he simply regards these as uninteresting issues; less charitably, he’s uncritically accepting the idea that math is something inherently unpleasant. In either case, mathematics is simply an obstacle to be worked around and it should be de-emphasized because his personal experience was that it never made much difference to him.
And that’s extremely frustrating, and kind of sad.
That article was everything I expected it to be considering it was written by an emeritus professor of biology.
Wilson: “I speak as an authority on this subject because I myself am an extreme case. Having spent my precollege years in relatively poor Southern schools, I didn’t take algebra until my freshman year at the University of Alabama. I finally got around to calculus as a 32-year-old tenured professor at Harvard…”
Novak: “In 1961, old man! Nineteen-sixty-fucking-one! My God, man, that was over fifty years ago! Some time between then and now, biology stopped being stamp collecting and started being science, and part of the reason is mathematical methods!”
Christ, when I’m in my 80s if I start spouting nonsense like that, someone have the courtesy to tell me to my face.
Having a relatively fresh degrees in molecular biology and electronic engineering (where I learned algebra and calculus pretty well) I can tell you that he is pretty much correct. While knowing a little extra math beyond basic arithmetic will help you here and there you will be perfectly fine without the vast majority of what I learned during EE studies as a biologist. The most useful part is probably statistics.
Physicists have a biased view of the usefulness of math in the real world. While I agree math is fundamentally important in many areas, there are also plenty of others where the most interesting problems are completely mathematically intractable. Calculus won’t help you find out what a particular protein does, or how a particular gene is controlled.
Also it’s good he didn’t talk about why some students are put off by math because it is pretty much obvious for the most part – people are not all created equal (shocking I know, but it’s true) and there will always be those who find a particular subject harder to learn and are put off by it. Math is no different, I find math relatively easy but there are plenty of subjects I find very hard to learn and not because of prejudices or impostor syndrome or some other nonsense but because I tried and the ratio of effort to progress is just much higher.
Another question that I should’ve added to the list of things that would be “Where are students getting the idea that they need to know math?” That is, somebody is clearly telling the students Wilson sees being “turned away” that they need strong math skills, and it’s clearly not him. I suppose it’s conceivable that Harvard biology majors actually take a bunch of physics and get scared away by the math-lovers there, but I suspect it’s more likely coming from Wilson’s colleagues. In which case, it would be worth asking why they don’t share his views regarding math.
As an undergraduate alumnus of the University of Alabama at Tuscaloosa I could observe, from my time, that those who majored in biology (not education biology) were largely those who lacked the maths skills to do chemistry or physics and the practicality to be successful in the business college. Additionally, the number of education biology majors outnumbered the (real?) biology majors by about an order of magnitude.
Another lesson he could have taken from his experiences is that at some point he felt the need to sit down and learn calculus. Why was that?
“Novak: “In 1961, old man! Nineteen-sixty-fucking-one! My God, man, that was over fifty years ago! Some time between then and now, biology stopped being stamp collecting and started being science, and part of the reason is mathematical methods!””
So Darwin wasn’t a scientist, but merely a stamp collector?
Maybe the problem revolves around the difference between being good at math and being good enough at math. Our culture puts a barrier around math, arguing that the ability to learn it is innate, so if you lack that talent, don’t even bother. There are other cultures where the attitude is that math might take a bit of work, but everyone can learn it well enough. (e.g. This is the case in much of Eastern Europe, China and India.) I’ve seen people who are actually fairly good at math simply choke when faced with a math problem.
This leads to math avoidance followed by avoidance of fields where math is actually quite important. WIlson, as you noted, eventually had to buckled down and take a few courses. One possible approach is to restructure the science curricula to sucker people into learning the math they’ll need. Start teaching science, but give them enough “just in time” math to get through the next round of material. Obviously, this would be easier for biology than for physics, but it could be done. The math would be much less terrifying if it was taught as part of the student’s field of interest.
The way a number of eventual biology, chemistry, kinesiology, neurobiology, etc. majors enter college is as pre-meds. Intro Chem requires basic algebra skills and ideally some calculus (integrated rate laws really do make more sense if you know how to integrate & what it means when you do). Too often I find my colleagues teaching it as plug and chug. Here is an equation, look at the variables you know and which one you are solving for. It is boring and not really teaching the chemical concepts. You have a student who is weak in math and add the boring delivery , you have a recipe for students fleeing. A number of students have similar complaints about the physics they take. It is part of viewing these courses as “weed-out” courses to shrink the number of pre-health students.
The issue is then how these courses are taught not the math. You can teach differently and still be rigorously quantitative.
My alma mater, no longer requires physics as part of the biology major. I see that at a number of places. I think it is because those courses are viewed as math-heavy weed-out courses. It is a shame. If you are carrying out gel electrophoresis, it is useful to know what an amp, watt, and volt are.
The *important* lesson is that, if one is to be a biologist, one can learn the math when one needs the math.
Something I was told was that physicists and mathematicians, for reasons cultural and for all I know biologically developmental, peak early. And you can’t even make a good start in physics without a chunk of mathematical tools. Given those things, it might really make sense to have to learn the math before you learn the science. Although, personally I found the calculus I did learn enormously more interesting/easier to think about after I had some physics. I think teaching math and science in a more integrated fashion has a lot of promise, myself.
However, being adept at math is simply not a prerequisite for being a biologist. Telling students they have to be good at math before they can start on science- even if well-meant and surrounded by good support for learning mathematics and somehow delivered in a social context that doesn’t perpetuate existing differential representations seen in science today- is simply factually wrong (it conflates ‘science” with “physics”… tempting for physicists, I know).
It is, indeed, pernicious to set math up that way if some people really do develop the cognitive ability to understand certain types of math later (either chronologically or in relation to other cognitive skills). While it’s likely nobody has to wait until they are 32 till they will be able to learn calculus, to take a smart kid and tell them in 8th grade that if they don’t learn algebra NOW they will never be a scientist, is both cruel and incorrect. And believe me, they DO tell people that. There is also some data I’ve heard about on learning algebra that suggests that not everyone is going to be ready for it at the same time. It may be actively harmful to introduce it too early (what happened to test scores in California when they started trying to get algebra to students in earlier grades is the relevant dataset).
Futhermore, Kaleberg is right- a growth mindset for math is not the default in our culture, to our detriment. Maybe the discouraging feedback I got about pursuing science because of my ‘lack of math skills’ wouldn’t have been as problematic if math hadn’t seemed like an innate talent?
there are also plenty of others where the most interesting problems are completely mathematically intractable
Even here, knowing some mathematics helps. It is useful to be able to calculate an approximate solution, because often the terms that make the problem intractable are small and the full solution will not look much different. There are other times when this procedure fails spectacularly (fluid turbulence is a notorious example from physics; in this case the small term comes with an additional boundary condition). You need some mathematical intuition to tell you whether your situation is in the former or latter category.
In addition, blindly applying numerical methods is an excellent way to get into trouble. If you don’t have any idea what sort of solutions to expect, you can’t tell when your numerical method is behaving badly–for instance, if your fluid simulation allows waves to propagate faster than one grid point per time step, you are likely to get nasty results.
I’d agree with Becca #9, that the growth mindset is the most important lesson.
It wasn’t that E.O. Wilson didn’t need math. He needed it enough to spend the considerable time to go back and get it. It’s the mindset that if he missed it the first time, he had the ability to put in the work to go back and get the math, and the time spent wouldn’t torpedo his career.
It’s two different things if the message is “you aren’t good at it, so find something else,” or if it’s “you need to spend the time to get good at it,” and the students don’t want to or don’t understand that they can.
I wrote a blogpost on this topic a while ago, and since it’s a click away, let me just quote for you the summary
“No, you don’t need to know maths to plant a flower, to admire a night sky, or to like a crystal. But as in the arts, getting to know the artist and his techniques add to the appreciation and understanding of her work – may that be the Fermat’s principle, data compression, self-organization, Noether’s theorems or chaos. Mathematics is the language of Nature and learning it is your connection to the universe. No more and no less.”
http://backreaction.blogspot.com/2011/03/what-is-mathematics-good-for.html
Chad,
I don’t know that you correct to suggest him or his colleagues are telling students they better be good at math or won’t make it as a scientist (just my feelings & I wouldn’t be entirely shocked if I was wrong). For me, my earliest memories as a kid regarding scientists: crazy hair, glasses, and math equations. How many times have you seen a scientist on a cartoon, in a movie, TV show, etc. with a blackboard dominated by math equations? I think most of us just grow up believing that a lot of scientific fields requires a strong understanding of math. I think its just something we assume based off how scientists are portrayed to the general public. At least for me that’s how I came to think in that way and I never heard anyone say otherwise, but how often does the avg. kid find themselves in a discussion regarding scientific fields and the importance of mathematics for each. Nice blog post.
Sorry, just to tie things together… I don’t think many kids enter college even considering science as a career b/c they have this false assumption (formed at an early age) that science requires extremely hard math. I feel this leads some of them to not even consider careers in science early on, which means they won’t ask what is really required mathematically for a specific scientific field because they rules that out as a career shortly after their 1st year of algebra.
Wilson is making an important point that everyone is missing, that it is more important to be able to formulate the problem and from the formulation conceptualize the broad outline of the answer.
At that point you can go out and learn the math, or learn who the mathematician is who can help you. If you think about this in physics story terms, this is pretty much Einstein. His real strength was the ability (and the guts) to par the problem down to its basics not his skill as a mathematician.
Yes, but Einstein’s math skills and understanding helped bring him to his conclusions. I agree that it is not, in every case, going to be absolutely essential, but it sure as hell helps.
If you want an example of a physicist who made major contributions without much math, Einstein’s not your guy. You want either Faraday or Bohr.
Of course, Faraday’s lack of mathematical training severely limited the impact of his ideas– his picture of magnetic fields was fantastic, qualitatively, but came in for some ridicule because it was unfamiliar and kind of fuzzy. It wasn’t until Maxwell put the whole thing on a firm mathematical footing that E&M as we know it came together.
Einstein didn’t know Riemannian geometry before he started working on General Relativity, but he reinvented/ learned it fairly quickly. His mathematical abilities were not really an issue.
Interesting comment about Niels Bohr not being good at math as his older brother Harald Bohr was an eminent mathematician. Einstein received mathematical assistance from Marcel Grossmann but I don’t know how important to his work this assistance was.
I’m not claiming that Bohr was lacking math ability in an absolute sense– honestly, I don’t know much about his work in that area. His major contributions to physics, however, were surprisingly non-mathematical, and this was remarked on even at the time. For a theorist, his papers include very few equations, and rely more on insight and intuition than formal mathematics. Heisenberg famously described him as “primarily a philosopher, not a physicist.”
Faraday arguably was short on mathematical ability, but that was more a matter of background, I think. He came from humble origins, and didn’t get much training in mathematics. His entry into science was as a lab assistant for Humphrey Davy, doing experimental work.
My understanding (mostly from Pais’s biography of Einstein) is that Grossmann’s help was essential in getting Einstein started on the right path. I don’t think it was ever the case that Grossmann did the math for Einstein, though– they did early work on general relativity together when they were in the same city (Zurich, I think, but might be misremembering), but the final theory was completed when Einstein had moved on. But again, I’m going off memories of Subtle Is the Lord…, so I could be mistaken.