These two posts (here and here) over at Uncertain Principles are well worth reading if you like discussions of the divide between people who understand science and people who don't. Chad Orzel, being a physicist, instantly translates "doesn't understand science" to "doesn't understand math", which is fair enough, especially for physics. His analogy to the language of critical theory, as found in English literature classes and the like, has threatened to turn the comments threads for both posts into debates about that instead, but Chad's doing a good job of trying to keep things on topic.
What he's wondering about, from his academic perspective, is how to teach people about science if they're not scientists. Can it be really done without math? He's right that a fear of mathematics isn't seen as nearly as much of a handicap as it really is, and he's also right that physics (especially) can't truly be taught without it. But I have to say that I think that a lot of biology (and a good swath of chemistry) can.
Or can they? Perhaps I'm not thinking this through. It's true that subjects like organic chemistry and molecular biology are notably non-mathematical. You can go through entire advanced courses in either field without seeing a single equation on a blackboard. But note that I said "advanced". I can go for months in my work without overtly using mathematics, but my understanding of what I'm doing is built on an understanding of math and its uses. It's just become such a part of my thinking that I don't notice it any more.
Here are some examples from the past couple of weeks: a colleague of mine spoke about a reaction that goes through a reactive intermediate, an electrically charged species which is in equilibrium with a far less reactive one (which doesn't do much at all.) That equilibrium is hugely shifted toward the inert one, but pretty much all the product is found to have gone through the path that involves the minor species. That might seem odd, but it's not surprising at all to someone who knows organic chemistry well. A less reactive species is, other things being equal, usually more energetically stable than a more reactive one, and the more stable one is (fittingly) present in greater amount. But since the two can interconvert, when the more reactive one goes on to the product, it drains off the less reactive one like opening a tap. There's a good way to sketch this out on a napkin, where the energy of the system is the Y coordinate of a graph - anyone who's taken physical chemistry will have done just that, and plenty of times.
Here's another: a fellow down the hall was telling us about a reaction that gave a wide range of products. Every time he ran one of these, he'd get a mix, and bery minor changes in the structure of the starting material would give you very different ratios of the final compounds. That's not too uncommon, but it only happens in a particular situation, when the energetic pathways a reaction can take are all pretty close to each other. The picture that came to my mind instantly was of the energy surface of the reaction system. Now, that's not a real object, but in my mental picture it was a kind of lumpy, rubbery sheet with gentle hills and curving valleys running between them. Rolling a ball across this landscape could send it down any of several paths, many of them taking it to a completely different resting place. Small adjustments from underneath the sheet (changing the height and position of the starting point, or the curvature of the hills) would alter the landscape completely. Those are your changes in the starting material structure, altering the energy profile of all the chemical species. A handful of balls, dropped one after the other, would pile up in completely different patterns at the end after such changes - and there are your product ratios.
Well, as you can see, I can explain these things in words, but it takes a few paragraphs. But there's a level of mathematical facility that makes it much easier to work with. For example, without a grounding in basic mathematics, I don't think that that picture of an energy surface would even occur to a person. I believe that a good grasp of the graphical representation of data is essential even for seemingly nonmathematical sciences like mine. If you have that, you've also earned a familiarity with things like exponential growth and decay, asymptotes, superposition of curves, comparison of the areas under curves and other foundations of basic mathematical understanding. These are constant themes in the natural world, and unless they're your old friends, you're going to have a hard time doing science.
That said, I can also see the point of one of his commentators that for many people, it would be a step up to be told that mathematics really is the underpinning of the natural world, even if some of the details have to be glossed over. Even if some of them don't hit you completely without the math, a quick exposure to, say, atomic theory, Newtonian mechanics, the laws of thermodynamics, simple molecular biology and the evidence for evolution would do a lot of folks good, particularly those who would style themselves well-educated.