Down the Memory Lane: Math in K-12 Science Classes

Zinemin has a great post on understanding physics (and math) in high school.

I started writing a comment, then it got so long-winded that I decided (for once) to not hog other people’s comment threads with my verbosity, but to put it all in a post. Here’s what the comment would have been (Zinemin is a physicist, so some of the verbiage is more physicist-friendly than entirely general).

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I grew up and went to grade school and college in Europe, so my experience as a student is quite different from what I see that my kids and students experiencing.

I started having physics as a separate subject in 6th grade. I remember one of the most appealing aspects was the fact that I got to use my beloved algebra; we did the basic mechanics stuff — motion with constant velocity or constant acceleration; ballistic motion. We must have done the concepts of force and energy, because  I remember making my dad teach me some basic trigonometry during the summer after grade 6th because I wanted to do inclined-plane problems. The physics lab was beautiful, I still remember these posters with the basic SI units, derived SI units, common prefixes. The physics teacher was excellent.

In high school I had a great math teacher throughout, and a great physics teacher in grades 11 and 12. My physics teacher in grades 9 and 10 sucked, when we covered thermodynamics and much of electromagnetism, and I still feel like I don’t know them very well. This is of course ridiculous, since thereafter I won awards in all sorts of physics competitions, I went on  to major in theoretical physics and get a PhD in a related discipline. Still, there is a faint visceral insecurity about those particular classical physics topics stemming from this wobbly initial exposure, even though I use thermodynamics and electrodynamics all the time in research and teaching.

I started loving chemistry in high school, because I had several excellent teachers who showed us what the underlying laws were and why. I even went to chemistry high school competitions (I could titer with the best of them).  During high school, I developed a deep distaste for biology because all that my two high school biology teachers ever made me do is cram and regurgitate their lectures back to them; I still occasionally have nightmares about answering questions about the nervous systems of nematodes. I never particularly cared about the nature/outdoors (the kid of the concrete jungle and all that), so all the botany stuff was lost on me. I really enjoyed what falls under basic cell biology (e.g. what different organelle do, the role of RNA). At one point, in perhaps sophomore year, we were learning about neural synapses, and based on what she taught, it seemed to me like I could think of synapses as little capacitors that can get charged of discharged; I don’t remember the details other than that I came up with this simple circuit-level model of how information travels through a network of neurons based on how I understood what she had taught and based on what I knew of electrical circuits; the teacher was very rude and dismissive, she said something about not being interested in my silly ideas and to take the stuff to the physics teacher, and that she wanted me to learn the material exactly as she had lectured. So yeah. I don’t like biology because my fee-fees were hurt. Even though intellectually I recognize the importance and difficulty of problems in biomedical sciences, something deep inside me cringes and shrivels whenever someone proposes a collaborative project that veers anywhere towards bio.

These early exposures seem to have a pronounced effect on how much confidence we gain, and confidence appears critical for later achievement. But I digress…

My Eldest is like an education experiment for me and my husband, because the system is very different from what we are used to and we have no idea what comes next. Where I went to school, the system was challenging and very good for smart kids, while average and below-average kids were left to just get bad grades or flunk and generally never do well. The US does a much better job catering to the average future citizen, presumably because the above-average ones are expected to find a way to excel anyway; they sometimes do, but they rarely do if they are poor.  (nicoleandmaggie write a lot about challenges in getting access to education for gifted kids).

In connection with Zinemin’s post, I am witnessing my Eldest in the US pre-college education system and it is appalling how little connection is made between math and any of the sciences. Eldest is a freshman in high school, and they have integrated science (won’t have physics separately till junior or senior year, and even so only as an elective). This year, so far he’s had a unit of physics here and there, but they do not use math at all. You should have seen how they covered light that we observe from different stars, and inferences about star temperature or distance from color and brightness; it made my skin crawl. The math needed for the Stefan-Boltzmann law or Wien’s displacement law is really not that hard, a high school student could understand the power emitted per unit area of what’s essentially a generalization of the heater on the stove goes as temperature to the fourth, or why the intensity decreases as inverse distance squared from a source (such as a lightbulb; or a star). But it was all very qualitative, completely hand-wavy, with vague concepts such as perceived brightness and actual brightness (no definition of either and no textbook; based on the problems assigned, I managed to decipher the two to be, respectively, the intensity of light here on Earth (power per unit area) and total power emitted from the entire surface of the star; the fact that one is called perceived brightness and one actual brightness and they don’t even have the same units makes me want to break something. Once I deciphered what was meant, I was able to help my son with the work, but you should have seen his resistance. He is very good at math and can definitely do the manipulations needed for the calculations (it was a problem with three stars and their perceived/actual brightnesses and sizes and distances from Earth, so very simple algebra was all that was needed). Eldest just didn’t understand why I would want to inflict this math on him when the science teacher didn’t do it, it wasn’t necessary, and everything could just be handwaved. This is the only physics unit I saw him have this year; he might have had more, he just didn’t need help (he has excellent grades overall). But from what he mentioned  in passing, most of the integrated science focuses on biology, a little chemistry, some geology and some astronomy, but nothing with even with a little math.

When I try to show my kid what I do for research, he zones out within 20 seconds because it is boring, and cannot understand why I would want to work on the stuff I work on because boooooring. This attitude appears common and continues into college. My undergraduate students still seem to think that they can be taught things in our physical science discipline without using math, as if math were some cruel curiosity that has no real use or connection to the concepts. It pains me when I hear this. Math is the language of nature and the fact that we can speak it is nothing short of miraculous.

1. physpostdoc says:

Great post! This explains why even bright US undergrads tend to switch off when tried to be introduced to concepts like calculus (based on my TAing experience in an Ivy League). I remember when I was TAing a lab course in my first year graduate school I tried showing how they can do error propagation analysis using logarithmic derivatives which makes it super easy and transparent, and yields exact results for independent random variables — but because of a perceived and irrational fear about partial derivatives most of them resisted and stuck to “remember” the cumbersome and tedious least square fitting “formula” written in the manual for every experiment/measurement they did. I also almost felt BAD for encumbering them with (un?)necessary details about sig-figs, “feeling” orders of magnitudes before even delving into exact calculations, and checking dimensional accuracy of their final answers — given that a typical US undergrad, especially at super-competitive Ivies, seem to be juggling at least 10^{4} extra-curricular activities with their coursework at any given time!

2. Pika says:

It’s not just US, I see the same perception of boringness and fear of math in UK&Irish students, but this was never there when I taught at universities in central Europe and Scandinavia. At a few occasions in UK/Ireland I had to give high-school-math-revision courses to MSc students and I modelled these courses based on high-school-math-for-studying-engineering-revision curriculum approved by UK universities. I was shocked to find out how lov level math this actually was (about what we learnt in 8th grade or year 1 of high school, where in my country I went through a similar system as xykademiqz).

So why such low level of math in certain (at the danger of overgeneralising, should I say English-speaking?) countries and not in others, I have no idea. I find it very curious.

3. jojo says:

Not entirely unexpected. That’s what happens when most of your high school science teachers teaching physical science are actually premed flunkees who switched to the teaching track after physics and O-Chem kicked their butts. In most states, you only have to have a bachelors in any science in order to teach all of them (usually you have to have taken at least 2 semesters of physics or chem to teach it, but you certainly don’t have to major in it).

My husband happens to be a high school physical science teacher and a physics major/math minor. He’s taught it all, and he’s even taught math some (currently teaching algebra). I really don’t think, though, that biology majors that last took calc in HS or as college freshmen should ever teach physics or physical science. Too many of them just plain hate math, my husband has (in horror) heard this from colleagues. Even worse hes’ seen some middle school SCIENCE teachers actually say this to their students (“Oh don’t worry, I hate math too!”) Worse, it’s typically female teachers saying this to female students. This is the kind of thing that should get you fired, but you can’t fire teachers in the US (except private schools).

I also want to put a little blame on the math teachers, though. Yes, math is the language of the universe and it’s supposed to be pure or whatever, but it would be nice if they would give some physical/practical examples once in awhile. I remember when I took calc it was immediately obvious to me that it had applications in physics and other sciences but the math teachers never made these connections explicit. Just “area under the curve” “slope of the line” etc. Worse, even senior physics didnt’ have any calc since some of the kids were taking it concurrently or not at all. I ended up asking my physics teacher to explain how to do the problems we were doing with calc which he was happy to do! But more integration would be great.

4. soggybomb says:

When I was in high school (in the US), my physics teacher was a biology major that had not taken a calc-based physics class. Let that sink in.

When teachers themselves are math-phobic, students pick up on that and can acquire a similar level of math-phobia as well.

What does surprise me, however, is that you say that your undergraduates who are pursuing physics/engineering degrees have this attitude as well. I did my undergrad at a big state school in physics, and all of my peers were like-minded in that we were tired of having things hand-waved. We were motivated to get down and dirty and do the hard business of learning physics. Since it seems from your posts that you work at a similar type of institution, I wonder why your students aren’t similar.

5. I loved maths at school, but I am one of the last to have gone through the UK system when exams at 16 were actually split into two tracks, one more academic and one more practical-oriented. I found the mechanics stuff in physics really boring because it was more simple than the things we were doing in maths, and clearly too abstract to be useful (and I also have a complete visceral twitch around biochemistry and magnetics, because of bad school teaching – SUCH an important job). I still enjoy maths, I just know better how little I know about it and how poor I am at it which takes off the gloss a bit. But I love rearranging equations and doing algebra still! It’s like distilled essence of puzzle, pure fun!

I face a LOT of maths phobia among my undergrads, who are in an environmental discipline (I teach STEM, but our department is ‘integrated’ and ‘cross-disciplinary’) – and what is worse among grad students. A large part of my recent research involves very simple mathematical models (there is an integral in the maths, but we explain it and usually code it in the clunky computer-power-using ‘work out each slice and add them up’ way because it works for what we want), and I’ve only ever had one undergrad willing to do a project with me on that (German-educated), and one post-grad who really got into the maths without fear (Scandinavian educated).

6. I was blessed with excellent math teachers. Science, not so much (integrated science = making paper DNA chains for bio for an entire week, listening to sexist fly stories for physics, and doing nothing at all for chemistry).

But hey, heaping on the math was good preparation for social science in college and grad school, right? If I’d had better science teachers, I might still be a post-doc somewhere with blatantly sexist senior professors instead of a tenured professor with only implicitly biased colleagues.

7. pyrope says:

@Jojo – your comment about female teacher’s influence on the perceptions of young girls reminded me of this study: http://www.pnas.org/content/107/5/1860.short

It is amazing how much our early teachers influence our later trajectories. For me, I loved math through 11th grade calculus, but had an abysmal physics teacher who just sent everyone to study hall each day…then for 12th grade, I moved to a different school system where they didn’t offer calc II except as an independent study where the teacher routinely told us how much smarter she was than us. Second semester I took calc II at community college where the prof was completely incomprehensible. At the same time, I had the most wonderful, inquisitive and creative teacher for AP Biology and after school field ecology…guess what I do now.

8. I hated bio in high school for the same reasons that you mentioned. Now work in a related discipline and I can tell you with certainty that language of bio is Math as well. Not many bio folks would want to admit that for now though

9. @Funny Researcher: Actually, “math” in the sense physicists use (mostly differential equations) is still not very important in biology. What is important is statistics, and not just the lightweight high-school version that most biologists take, but heavy-duty statistical inference techniques for dealing with high-dimensional noisy data. One needs some calculus to understand the statistics, but don’t really need ODEs and PDEs.

10. @xykademiqz: I’m sorry that your son does not share your love of math. It can be disappointing to have a child not share your interests (and often hurt their future prospects by doing so). I was fortunate that my son shares many of my interests, and that he is really good at some of the things I’m most interested in. He’s not a copy of me by any means (he has a lot of talents that I don’t share), but we can talk about a lot of things where we have common interests.

11. The people the next seats over on the plane just now were lamenting students’ inability to interpolate. Or to even use the interpolate functions on their calculators.

12. Among educated people, if you say you “hate art” or “hate architecture” you will be perceived as weird. But if you say with a passion that you “hate maths and physics” people will generally react indifferently or even with sympathy. I find this very weird and I am sure it would not have to be this way. It is just a cultural thing, people pass their hate of these subjects on to their children and entire families seem to be proud to say “we are just all bad in maths”. It is very sad if even schools are propagating this kind of mindset.

I am astonished how often I get the response “oh I hated nothing ever as much as physics” when I tell people I teach physics. First of all, it is actually kind of impolite to blurt that out to me, second, it annoys me greatly that people seem to be proud of missing out completely on one of the greatest human achievements, that our civilization is built on, actually.

I think it is a bad sign for our future. Everyone just wants to work in the service sector and let China and India do the engineering and production, and long term also the innovation for us. Last week I talked to a philosophy major who teaches philosophy and literature at high school. He told me he “hated physics” and that “science was just another religion, and it was laughable how science major never were able to see this”.

Btw, I greatly sympathize with feeling a bit insecure forever about topics that you heard about first time with a bad teacher. I have the same thing, unfortunately with mechanics. I had a mean physics teacher in all of high school and a very mean professor in general mechanics, so these topics always stress me out a little bit.

13. Interesting – I had no idea that “integrated” science was taught at the high school level. I had biology, chemistry, physics, and then AP versions of all these.

I was fortunate to have excellent math teachers across the board, I do wish there was more integration of math into science courses. There are so many places where it’s possible but most of my courses either ignored the connection or would say “there’s no need to worry about these equations”. Sadly I am guilty of both in my courses.

14. gasstationwihoutpumps, of course, there is no sharp distinction between which fields need PDEs and ODEs vs which need stochastic methods.

For example, there are many physical science fields where stochastic processes and extraction of information from noisy data are critical (signal processing, imaging science). In physics, any problem that involves many-particle systems (e.g. plasmas, charge carriers in semiconductors, ionic transport in solids, molecular dynamics in liquids and gasses). On the flip side, even for systems where relevant properties are described by partial or ordinary differential equations, powerful stochastic numerical techniques are sometimes used to solve these equations.

While my knowledge of math in biological sciences is very limited, certainly differential equation are not entirely absent. An example that readily comes to mind are the beautiful coupled nonlinear differential equaions known as the predator-prey equations.

15. anon says:

Integrated science in high school!? I am American, and my school was probably above average in quality, but not super awesome. Integrated science was for the students at the lower end of the bell curve. If you were going to college at a level above community college you were expected to take Bio, Chem, and Phys, in that order. If you were at the upper end of the bell curve you took those three classes plus one AP version of those classes. Or if you were a freak like me, you decided that order was stupid and took Phys, Chem, AP Chem, and then AP Phys and Bio at the same time. (Can’t tell I thought highly of myself.) All of those were only algebra-based though, which definitely meant college calculus-based physics classes kicked my butt (didn’t stop me from majoring in it though). I guess I’m glad my school was flexible enough to accommodate me.

In any case – I am not sure if I ever had a “bad” teacher. I certainly didn’t like geometry because I didn’t like proofs. I blamed it on the teacher at the time, but thinking back, I can’t see anything she did wrong (and I did plenty well in the course). I guess the most frustrating class I took was the advance-track integrated science in middle school – it was hand-wavey physics, and while I can see what he was trying to do now (developing intuition without hiding behind technical stuff), at the time I was definitely on the “get on with it and show us the math!” team. I have always thought of math as a useful tool. I feel like most of the examples were practical until I got to high school, and even then some of the example were practical. It was a little dry at times and many of the examples were lame… but still. Maybe I could see the big pixel and liked puzzles enough that the boring parts could be overlooked? I feel like among my peers there was still a good number of “not good/don’t get math” people despite having the same education as me.

I wonder if I lucked out? Or my parents were just really good at making sure I took classes with the right teachers? Maybe I was just one of those kids who was going to succeed no matter what the system gave me?

Sorry for the navel gazing, but it is an interesting topic and I do wonder (as a future educator – finger crossed) why students turn out so differently in terms of math phobia.

16. anon, my son is a highschool freshman, he is taking integrated science honors (regular and honors were the only two options available to freshmen; there is no science acceleration). He will start taking individual science courses next year, I think only one in sophomore year and will be able to take more than one junior and senior year, in part because he’ll be able to get a PE waiver on account of swimming.

He liked 7th grade science (did extra stuff at the university as well) but hated 8th grade science, said it was stupid and boring and didn’t learn much. This year he’s happy even if I am not. 🙂

17. Another thing I didn’t mention: when I was growing up, there were extracurricular math and science meetings, especially if the kid wanted to partake in competitions. They were meetings with your regular teachers for extra stuff before or after school. I went to these a lot.

18. I agree that there is some use of differential equations in biology (not just predator-prey relations but things like modeling fluid flow in the heart), but these are not central to biology the way PDEs are central to physics. There is also a fair amount of statistics in physics, though it is not central the way that PDEs are. All kinds of math finds uses in all kinds of fields, but what is important in a field does vary, often quite a bit. Combinatorics and mathematical induction are central to computer science, statistics are central to biology, calculus is central to physics, …

19. This is why it never occurred to me to major in physics as an undergrad (I was EE): math classes were great but the science classes were not and contained little quantitative work. Physics in particular was incredibly dull w/out calculus. Now I’m an astrophysicist but I came into it the roundabout way.

20. The above should have read that high school math was great and science poor.