*Image: Integration by parts. Remember?*

Nearly every university science student takes 1^{st}-year calculus. The content is fairly standard: functions, limits, derivatives, max/min problems, working up to integration and often capped off by the powerful but counterintuitive trick of integration by parts*. I think 1^{st}-year calculus is widely seen (both by the Math departments that teach it, and the other science programs whose students take it, as the *sine qua non* of mathematical training for all scientists.

Is it?

Until about a month ago, I would have said so without hesitation. In fact, I did. I teach a population biology course** that’s cross-listed with our Forestry program, and when Forestry proposed cutting their 1^{st}-year calculus requirement, I screeched as loud as anyone (and a good deal louder than some). Surely you can’t do population biology without calculus! But I’ve been thinking about it, and I may have been wrong. Maybe biologists *don’t* need calculus.

I need to be careful about my argument here. *Biology* certainly needs calculus – in my own field, for example, we’ll always have theoretical ecologists using calculus to build and solve population-dynamic models. But that’s not the same thing as every biology *student* needing calculus. What do my population biology students need to know, mathematically? And what parts of calculus are on the list? A full answer would be too much, and too boring, for a blog post, but I think some attempt is important.

I’ll start with a claim that I would have considered, just as a month ago, heretical: I don’t think my students need to know how to take a derivative or to integrate a function. Whoa – there goes virtually all of calculus right there, you’re thinking! Not quite. What my students definitely *do* need to know is that *dN/dt* is a population growth rate, and more generally, that *d*(something)/*d*(something else) is a rate of change of the “something” with respect to the “something else”. We use such derivatives to specify the basic models of population growth (exponential growth, logistic growth), of metapopulation dynamics, of interspecific interactions (the Lotka-Volterra models of competition and predation), and much, much else.

So if we specify population models as derivatives, why am I exploring the notion that my students don’t need calculus? Here’s the thing: once we’ve written down the derivative, we do (at least in my course) the rest of what we need to do with it using only basic algebra. Take the Levins model for metapopulations, for example:

dp/dt = mp (1-p) – ep

where *p* is the proportion of patches occupied by a focal species, *m* is its migration rate among patches, and *e* is the per-patch risk of extinction. (If you aren’t familiar with this model, don’t worry; it’s just an example).

Having taken calculus myself, my first instinct is to integrate this to get an expression for *p(t)*. But I suppress this urge, because actually, I don’t care about *p(t)*. What I care about is whether there’s a stable equilibrium *p**, and to find this all I have to do is set *dp/dt* = 0 to find *p* = 1-e/m* (technically, this alone doesn’t assure stability, but I can verify that graphically). The two-species Lotka-Volterra competition questions are a bit more complicated, but the approach is the same. Here we specify *dN _{1}/dt* and

*dN*, but our real interest is finding conditions under which species 1 and 2 can coexist, and this question can be answered by setting

_{2}/dt*dN*= 0 and plotting the resulting “zero net growth isoclines”.

_{1}/dt = dN_{2}/dtThis is getting a bit technical, so I won’t offer any more examples. What’s important is the bottom line: I teach an entire semester of population biology without integrating or taking the derivative of anything. My students don’t need calculus – at least, not beyond the fact that *d*(something)/*d*(something else) is a rate of change, which might be zero.

This doesn’t mean that my students don’t need math. The “biologists don’t need math” trope drives me crazy***. I’ve heard it from well-meaning parents, high-school guidance counsellors, university admissions offices, physicists, you name it; and it’s just plain wrong. My students *do *need math; but I’m suspecting more and more that what they need is not what 1^{st}-year calculus delivers. My students need statistics (and they do take it, although the way we teach it is another issue). They need basic algebra, logarithms and exponentiation, simple matrix manipulations, and a few more things – but even as I type out this list, I realize it’s boring, that your list might differ from mine, and that it misses the point. What my students really need (mathematically) is just one thing: comfort.

What do I mean by “comfort”? I mean that my students need to see a page of quantitative material – an equation, a graph, a statistical table, an ANOVA table – without panicking. They need to be willing to write down an idea about how something works in the form of an equation, or to look at an equation and ask themselves what it suggests about nature. They need to understand that math isn’t a thing distinct from biology; it’s just a language we use to think precisely and to deduce the consequences of what we’re thinking. In short, they need to think of math as something quite ordinary, and something that’s their friend rather than their enemy.

I strongly suspect that for most students, 1^{st}-year calculus doesn’t deliver mathematical comfort. If anything, it may take it away. How *can* we deliver that comfort? I don’t know. I’d hope to see students arriving from their K-12 education with it – but while some do, many obviously don’t. One thought I’m toying with: maybe trying to build on top of high-school math just takes students higher when they’re already afraid of heights. Maybe we should focus instead on knitting the math they already know into the questions they care about in their discipline – so they stop noticing the height in the first place. Later in their careers, of course they can learn more math – when they have some mathematical comfort, and when they’re motivated by a problem in their discipline. (That, after all, is how *I* learn.)

Speaking of comfort, the conclusion I’ve come to makes *me* uncomfortable. That’s partly because if I’m right, then I’ve been wrong for a long time. And it’s partly because no matter how clearly I think I’ve laid this out, someone is sure to assume I’m just rolling over on “biologists don’t need math”. So I’d be happy to be (re)convinced that biologists *do* need calculus. Bring it on.

*UPDATE: I like the look of this newish text for teaching calculus: Biocalculus, by Stewart and Day. I’m not sure it’s a first-year text; but the approach, thoroughly grounded in biological problems, would help my students a lot. Thanks to Amy Hurford for suggesting it in the Replies.*

*© Stephen Heard (*sheard@unb.ca*) February 9, 2016*

*Related post: Why do we make statistics so hard for our students?*

*^When I was an undergraduate, I was pretty good at calculus. My Engineering friends, though, thought I was *brilliant* at it. Here’s why. When they were stumped on an integration problem, they’d talk about in the residence breakfast line. I’d say something like “Stuck, eh? What do you need to integrate?”, and then pretend to listen while they told me. I’d gaze thoughtfully upward for a moment, then say “I think you need to integrate by parts”. They’d go off and try, **and** **it would always work**, and they’d assume I had solved the problem in my head. In fact, I knew just two things: engineers are really good at integrating, and engineers really hate integrating by parts. If they were stuck, they must have tried everything else. But the problem wouldn’t be on an assignment if it weren’t soluble, so it had to be one that requires integration by parts! There – I’ve pulled aside the curtain. There never was a wizard.

**^I would argue that population biology needs at least as much math as any other subdivision of biology, so it’s a good test case: if *any* biology student needs calculus, surely it’s a population biologist.

***^My mother, my aunt, two uncles, and my grandfather all taught math. In many ways, *I *teach math. I *like* math. It pains me to even consider reducing the world’s total teaching of it.

Andrew Jackson (@yodacomplex)I’m with you on this. I think that if biology students at least know that dN/dt *can* be integrated, and that you can get a computer to do it (either symbolically or computationally) then that in most cases is plenty! Indeed, many cases in biology quickly get to the point where symbolic integration is not possible, e.g. SIR models. I dont think biologists necessarily need to know how to solve these things analytically themselves, or via Newton-Raphson methods etc… We are currently re-designing our maths for biologists curriculum from 1st to 4th year undergrad and im glad of having this discussion here. Thanks for the post. – Andrew

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Markus EichhornThanks, this was an interesting read, especially from the British perspective where we teach almost no maths as part of our biology degrees. Most courses include a module in basic statistics, much to the chagrin of our students, who believe that in choosing biology they have made some kind of pact that ensures never seeing an equation again. I often see arguments from North American colleagues that their students need *more* maths, which makes me feel very insecure about the limited amount we deliver.

Do they actually need it? I’m sympathetic to your argument that the majority of students, even those who become research scientists, probably don’t require advanced calculus. The counter would be that by missing out on it they close the door to future avenues (and maths is particularly hard to catch up on later). Once they’ve been encouraged to believe that one can be a biologist without maths it’s hard to persuade them otherwise at a later date. One can also apply the same argument as for teaching Latin: for sure it’s fundamentally useless, but the diffuse benefits and ways of thinking are nevertheless valuable.

If you don’t mind then I’ll turn the question back on you: what would you teach instead? By clearing space in their schedule, what other subject would deliver greater direct or indirect benefits? I can think of plenty of things in our student program that I would scrap in favour of more maths!

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ScientistSeesSquirrelPost authorMarkus – thanks for commenting! I did not know that British programs were comparatively math-light, and that’s interesting given that I have a number of friends who are British-trained evolutionary biologists and very strong in math.

To answer your question: I have recently come to think that what our students need most is formal training in how to be a student! We have courses like that (usually called some flavour of “University 101”, but they’re widely considered by students and faculty alike to be peripheral, not foundational. And yet students need things like study skills, library skills, time-management skills, etc. far more than (I think) they need another biology course or (maybe) calculus. (Although a large part of me still hopes to be outargued on that one). A few thoughts about this in these two old posts: “What if we flip the students instead of the classroom” http://wp.me/p5x2kS-cG and “Where the heck are my students” http://wp.me/p5x2kS-iB. But I’ll have more to say/rant about in a future post.

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B. DoyleI don’t quite understand the dispute. Firstly, I was required to take 4 math and science courses (or the equivalent 2 years) to graduate from high school. I took Biology, Earth Science and Physics, in addition to 3 math courses, viz. Geometry, Trigonometry and Pre-Calculus. Then I got to college and found that math majors are required to take a science sequence, i.e. two back-to-back courses of Biology, Chemistry or Physics.

If I am required as a math major to take _two_ science courses, why can’t science-related majors make it through Calculus I, which isn’t that hard anyway?

And I disagree with this notion that “a computer will do it all anyway” because the ability to problem solve, which is largely introduced in mathematics courses, is extremely important in practically everything we do in everyday life.

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devinbloom“what would you teach instead?”

A: Organismal biology! It doesn’t matter which group or topic, great research ideas come from knowing something about organismal biology and natural history, and there is such a tragic shortage of such courses today.

As someone who is not very good at math, I would still encourage students to take calculus. I agree it does not need to be mandatory and you clearly can succeed without it. But, it is a really, really great skill to have. This post actually reaffirms my convictions that calculus (and math in general) are important in biology because without strong skills in calculus, Stephen would not have been able to clearly articulate why students don’t need calculus for population growth models (oh the irony)!

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Lior PachterAt UC Berkeley we have a new courses (instituted about 4 years ago) called Math 10 that replaces calculus for freshmen biology majors with a two-semester sequence that, while retaining some calculus, also teaches statistics and probability and combinatorics. See this post about it https://liorpachter.wordpress.com/2013/08/22/catching-a-bus/ as well as https://math.berkeley.edu/courses/choosing/lowerdivcourses/math10A and https://math.berkeley.edu/courses/choosing/lowerdivcourses/math10B

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ScientistSeesSquirrelPost authorThanks, Lior! I think this kind of broader course that stresses math more generally as a tool might be an improvement. Thanks for leaving the links.

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Jessie MathisenI’d really like to see more subject-specific math classes. In other words, I think biology for math majors should be a common class. That probably wouldn’t fly in a lot of places because it sounds watered-down, but I’m not thinking of a watered-down class. Rather, I’m thinking of a class that focuses on specific math skills most often used by biologists. In my imagination, this class would be taught largely with case studies – pick a research problem, talk about how it was addressed mathematically, and do the math. The main advantage I see with this approach is that it would help ground math in physical reality. All too often, students have trouble linking what they learn in math class to real-life mathematical applications. That’s not to say there isn’t value in abstract math, it’s just that a majority of students don’t particularly need to go there.

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Tony DiamondAs an ancient British-trained population biologist most assuredly NOT “very strong in math”I am delighted to see your conversion Steve! You may not remember but I used to argue in Science Faculty against calculus being regarded as a sine qua non for ALL scientists, but got tired of being shouted down and eventually learned to keep quiet. Your post is spot on and your dN/dT example is classic because even such long-gone luminaries in pop biology as David Lack would use and understand that without going near calculus. I was wondering how long I could continue being paid to do ecology without my lack of calculus being shown up. Looks like I might make it to retirement with your post to back me up..

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ScientistSeesSquirrelPost authorUm, I might have been one of the ones shouting you down… 🙂

To be clear, I don’t think Lack (e.g.) used and understood population biology without “going near” calculus. There’s no question he and many others had the mathematical comfort I’m talking about, and could make, use, and interpret models – without necessarily doing a lot of integration himself. I would also draw a line (perhaps not clear enough in the post) between what we need to teach every undergrad, and what grad students and researchers should learn!

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YolandaProvocative post… and I wholeheartedly disagree! Natural selection is an optimization process, so we must understand optimization to fully understand adaptation (I’m speaking as a behavioural ecologist). I doubt vague concepts help with understanding. Instead, very concrete derivations or proofs can really help and can build confidence. Case in point… we are discussing Alerstam & Lindstrom’s Optimal Bird Migration chapter from 1992 in my lab meeting, and we’re making sure we understand each step in the process, so we can critically evaluate the logic. For those who are interested, getting from eq5 to eq6 requires the quotient rule for differentiation. Of course the paper doesn’t state this (how many times have you read “It can be shown that…”). Once one remembers the quotient rule and how to apply it, algebra gets you the rest of the way to the marginal value theorem, and the logic of finding the tangent to the curve of the gain function makes sense. Yay! By learning first year calculus, students are much better prepared to read and critically evaluate the primary literature and acquire deep understanding of key principles in evolution & ecology. It is fun to problem solve, and we should be encouraging this in our students.

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ScientistSeesSquirrelPost authorI’m so glad that you’re willing to wholeheartedly disagree! Although perhaps I’ll disappoint you by _agreeing_ wholeheartedly with your last sentence. But I wonder: the example you chose is from your lab meeting, so you’re talking about your grad students. What fraction of the 1st-year Biology cohort do they represent? Does the argument extrapolate to the (1-f) who are on other paths? Can you give an undergraduate example where understanding of biology is fundamentally improved by knowing the quotient rule for differentiation? That’s a real question, by the way, and I’d actually be quite happy to have to back down on this one!

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YolandaI think the fun aspect is often overlooked…I also find that my undergrad students smile and have a sense of achievement when they overcome an R programming bug.

I don’ t think I agree with the question. If the product rule for differentiation applied to biological questions but never the quotient rule, would we never teach the quotient rule? Do we only teach what is directly applicable – how do we know when we will need algebra or calculus? I will turn the question around instead, and ask when would it be valuable to teach a biological concept by showing the underlying calculus or algebra?

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Erol AkcayI tweeted a few immediate reactions but needed to write in complete sentences to express my thoughts as someone who just started teaching ecology at the undergrad level. My first reaction to the question in the title was “of course they do,” so I expected to disagree strongly, but in fact the title turned out to be a bit of a clickbait, since Stephen actually doesn’t argue for doing away with calculus entirely, but for teaching it differently. I still disagree with how he wants it to be taught, but I don’t think we need to debate whether students need calculus or not: they do. Question is 1. how much, and 2. how should it be taught.

Stephen says they just need to know what a derivative means, and be comfortable with expressions involving it. I full agree with the second part, but I don’t see how the approach of “just know this means rate of change” without actually knowing where it comes from (as a limit of secants) or how it can be computed (using a few fairly simple rules) helps the comfort level. Maybe I am weird, but I certainly don’t feel comfortable dealing with objects I only know at a vague interpretative level. Same goes for the integral. In the words of the bard: http://pbskids.org/video/?guid=be225c89-3fc7-436d-b175-5ec16e087d20

Now, do I think existing calculus courses do a good job at this? Perhaps not. It’s a slightly outdated cliche that calculus courses are designed for physical scientists, but only slightly outdated. I certainly hated my calculus class, and I was a Physics major! And it is true that calculus is actually less central to college biology than it is to college physics, and teaching only calculus at the expense of other topics like linear algebra and probability theory is very bad. But there are now quite a few efforts to remedy these problems, e.g., Berkeley’s sequence mentioned above by Lior, or this book by Bodine, Lenhart and Gross: http://press.princeton.edu/titles/10298.html. There are sometimes complex departmental politics, but I don’t think we can say anymore that we don’t know how to teach students math that is useful and essential for biology. At this point, retreating to “you don’t need to learn how to take derivatives” is a big step backwards, I’d say. By all means let’s do fewer volume integrals (though they were fun!), but let’s keep the chain rule (and integration by parts — I LOVE integration by parts!) and probability theory, and eigenvectors, etc..

Final thought: I also agree with Andrew: it is very important to teach students how to do computer math, especially numerics. But I’d argue you still need a good foundation of “pen and paper” math for that to build on. After all, a computer just does what you tell it to do, you still need to know what you are doing.

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ScientistSeesSquirrelPost authorThanks, Erol! I’ll plead guilty to your charge of clickbait… And I love integration by parts, too; it’s just that I’ve never yet, in 30 years, come across a situation where it helped me understand any biology!

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William Kenneth Wayne GodsoeSteve i loved the post and will have a few more things to say later, but i thought i’d mention funnily enough that “i” had to do integration by parts just last month. I had a student who was trying to combine parameter estimates form experiments to create a model of logistic growth. Because we were using a non-standard version of the model (dn/dt=rn-an^2) I could not, for the life of me figure out the units, and spent several hours with terms like snails/hour^2/meter piling up on my white board. Ultimately I had to sit down with my wife, who is a high school calculus teacher, and integrate the problem to see where the units came from. I believe that integration by parts may only be useful for exponential/logistic growth, but for those two problems…

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Elinor“Maybe I am weird, but I certainly don’t feel comfortable dealing with objects I only know at a vague interpretative level.”

I agree with Erol’s sentiment here. In high school I learned physics before calculus, but even with an excellent physics teacher some of the basic concepts didn’t click until I was taking calculus.

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lindsaywaldropI agree that biology students need to be comfortable with math as a language in order to do well, but I don’t think the answer to comfort is to take less math. If math is the foreign language in which science is written, let’s compare it to another foreign language like Russian. How would you rather deal with learning it: two years of written and spoken training then immersion in a Russian-speaking country or being handed a dictionary in an alphabet you barely can read and dumped in the middle of Moscow? I mean, you may find helpful strangers here and there who speak English, but probably not many, and they may not be interested in helping you understand basic road signs and asking for a bathroom. Yet, if students don’t have at least some training in calculus (and other types of math, I agree with Lior that all math =/= calculus), they are being set up for the latter situation. Comfort comes with practice and time. Sure, you can teach them a derivative means a rate of change and they can understand the one example, but they will understand it too superficially to be able to do anything other than answer questions on a test with it. And by the time they are ‘ready’ to learn more math, they will be surrounded by people already fluent in the language who may or may not care/have time to catch them up. After all, how are you suppose to make an intellectual contribution to a reading group discussing the character intricacies in War and Peace when you are still stuck sounding out the words to See Spot Run?

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TobiIn my first year as an undergrad I had a conversation with a student about the usefulness of calculus and for a brief moment I agreed with him that it was quite useless. But later on, I pondered on the usefulness of all other subjects as well; from biology to physics. I think there’s a faulty assumption here, which is that because students major in Biology or whatever, they would necessarily end up in related professions. This is not peculiar to Steve’s position. Universities are meant to deliver universal education, where along the line, students can find their core interests. That’s why we grade students from 0-100%.

My undergrad university would take every student admitted into the university through all the departments that exists there; from physics to crop ecology (math and computer science students hated this), sociology to political science regardless of the program they are in. Strange enough, I’m talking about a university that only delivers science programs. As I advance, I have come to appreciate this curricular structure the more, because I’m quite adaptable and I can readily summon my basic understanding of other fields when there’s a need for it. Although I can understand Steve’s position, it nonetheless oversimplifies the value of those “irrelevant courses”. The point is, when universities teach courses like calculus, physics etc that have no immediate usefulness to students, they are only creating adaptable individuals. Therefore, an argument for or against calculus in Biology is meaningless if it all boils down to students’ interests.

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rcannon992I think that the average biology student only needs to understand the concepts – of things like rates of change and integration – of calculus. But certain career paths, like epidemiology or population biology, will surely be closed to them without a grounding in calculus. Although there’s probably an App for everything these days!

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mooerslabSteve, I also think I agree. For instance, I can’t do calculus to save my life, but like you, I do try to give my students that feeling of comfort with relevant concepts in my biodiversity and evolution classes (you know, exponential growth, logarithmic distributions, equilibria and graphical first and second derivatives, RESIDUALS). We need students to be comfortable with relevant math and stats, and maybe that does not include integration in first year. Of course, if we went that route, we would need to offer “advanced” maths ourselves to honours students. Why don’t biology departments don’t take the bull by the horns? Everyone claims is it just because the math department “won’t let go,” but I wonder. Heads and deans could consider this question.

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Brian McGillI tend to agree for undergraduates. I think the core issue is the notion of service courses. We farm our math teaching out the math department. Our chemistry out to the chemistry department etc. And the end result is our students get taught a ton of stuff that is NOT relevant to undergraduate biology but is taught either because it IS relevant to some other discipline sharing the service course (e.g. physics or engineering) or simply because that is how the home department (math or chemistry) likes to approach the world.

Assuming students come in with appropriate pre calculus (algebra, geometry and trig), I suspect a good one year sequence (somewhat like Lior mentioned) could indeed cover all the math and stats an undergraduate needs. And math could be better integrated into biology courses. But then we would have to teach math instead of biology, and the math department would get upset at losing students. But the students would be oh so better served. University silos strike again!

I make a very similar argument for graduate ecology students here: https://dynamicecology.wordpress.com/2014/10/20/what-math-should-ecologists-teach/

And I’m sure the same thing applies to chemistry (although my chemistry is so far in the dim past I am less qualified to pronounce why). But does every undergrad biology student need to know the intense details of specific organic reactions? I doubt it. I’m sure one year of biology-centered chemistry could do it. Maybe even one year of biology centered chemistry and physics.

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seeddispersalThank you for this post. I have a unique perspective in that I was trained as a biologist as an undergrad., completed a Ph.D. in ecology (empirical), and I am now a postdoc in a is strictly theoretical lab. I loved math as child and almost majored in it, but I chose biology because I was fascinated with snakes (they have no arms or legs–how do they survive!?). I took three statistics and two required math courses as part of my undergrad. curriculum (the courses were calculus “for the life sciences” and algebra). Both of these math courses and the statistics courses were useless to me in grad. school when when the expectation was be literature-literate. As I progressed in my graduate studies, however, I wished for a deeper insight into my field–I wanted to understand the theoretical foundation. I have spent the last 18 mos. learning all of the math that I wished I knew in grad. school–those years would have been an entirely different experience. I feel like a bit of a `convert’ and I am happy to have made the choice to go theoretical/mathematical. I do feel much more comfortable even though my skill set is still relatively rudimentary. To echo Erol Akcay, I have found my computer invaluable as math tool. It’s helped me immensely in both checking my pen-and-paper analyses and as a heuristic tool.

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Gavin SimpsonSomeone in the audience said exactly this during a lively Ignite session at ESA last August (2015), to much applause and general, but not unanimous, approval of those gathered in the session.

I’d spend the time freed to cover probability, applied statistics & statistical thinking, data munging, and general computing/programming skills. These things are just generally more useful for undergraduate biologists. I’d much rather students understood probability better and could think about data and problems in a proper manner than have them be able to integrate or differentiate up the wazoo. I’m also amazed by how many students, who are surrounded 24 hours a day by powerful computing devices, are almost completely incapable of getting a desktop to do anything more than deliver up cat photos or send snap chat messages.

I doubt many biology undergraduates will ever have to integrate or differentiate anything post-graduation. I’m reasonably sure most will need to use a computer or look at data or even just be confronted by the media/cranks over lotteries/drugs/diseases/super-foods.

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William Kenneth Wayne GodsoeShould we remove some calculus from the curriculum, I think it should be replaced with a review of lines. In my experience lots of undergrads need careful revision of how slopes and intercepts connect a graph of a line and a linear equation. I am always shocked at how often this concept becomes useful. Want to explain a null hypothesis? Slope=0, Want to describe biological significance? Picture a graph including the relationship between the variables of interest, measure the slope. Have you skipped calculus in favor of ZNGI plots? Great, but if students feel intimidated by slopes and intercepts, it will be hard picture what the plot is telling us.

Most importantly students tend to forget the concept of a linear equation, but many have seen it before, and the concept is easy to teach and visualize.

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drake1987I enjoyed your read, but had several issues. I will work my way from the bottom of your post back towards the top. The idea of “comfort” for students appalls me. I am a student myself right now. I read your blog because it presents new ideas and perspectives on old ones, because I want my status quo to be challenged, not comforted. I have gone to school to push the bounds of my knowledge, not to have easy classes. Forgive me if I have mistaken the meaning of your use of the word comfort, but that isn’t a good reason to remove a class. Like you pointed out, many students don’t show up with a good working knowledge of stats or other mathematics. The answer isn’t less maths, but more (including calculus). I don’t like most math myself but I continue to take math classes: statistics, experimental design, and yes, calculus. It is uncomfortable for me because I struggle, but I end up with a better and ultimately more comfortable understanding of math when I finish a class. Also, for me, calculus was the place where math finally made sense and it unlocked the rest of the math world for me. I understand this won’t be true for everyone, but taking away calculus is like taking away a mechanics spanner. It is a basic tool that although might not get used all that often on fancy new models, is still necessary from time to time. Without it, the job might not get done at all.

Secondly, I will quote Fred Guthery’s “A Primer on Natural Resource Science” because he has written it more succinctly and clearly than I ever could: “Training in mathematics [pertaining to stats, algebra, and yes specifically calculus too] has at least 2 positive influences on the analytical mind: it fosters the ability to think at extremes and to think the arbitrary…..A complete scientist of natural resources needs training in at least these courses: algebra to manipulate equations and handle calculus; calculus to understand functions and to handle derivatives, integrals, and limits at extremes; linear algebra to handle matrix mathematics; and probability with ample exposure to mathematical statistics to understand the probability models that undergird a great deal of natural resource research.”

Maybe I am just standing in the way of progress but I think it is important to teach, exactly because it hasn’t been exposed to many students. I would argue it helps in the long run even if it doesn’t get used by every student. Maybe I am biased, but I am still a student and wish I had more time to take more math classes, not because I enjoy them, but because the are hard and push me to think.

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drake1987Also, thanks for writing and bringing things up that make us challenge our assumptions!

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ScientistSeesSquirrelPost authorThanks for the detailed comment! I think I’m actually using “comfort” in a way you should like – I mean that students should think of math the way you think of math, as a tool they can and should use, and don’t need to be scared of. Having to struggle is not a virtue; being willing to struggle because you come from a base of comfort and understand the utility is a virtue! So that’s what I meant. That leaves as our major difference (and I am not saying I’m right!) whether a traditional course in taking derivatives and integrals is what biology students need most – that is, as I say in the post, I don’t want to teach less math; but I want to achieve better results with my students, who are currently getting little out of the math they’re taught. I realize it’s quite heretical to think the path to that outcome lies in taking any particular bit of traditional math out of the curriculum! Particular bits, not math itself…

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drake1987Thanks for the reply! It definitely cleared up some of your meaning to me. I appreciate it a lot and thanks for writing great stuff.

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Staffan LindgrenVery interesting topic. I will add my 2 bits as a naturalist with (at least partial) ‘mathophobia’. My feeling is that it is not THAT calculus is taught that may be a problem, but HOW it is taught. For the most part it is taught without a biological context, so students don’t understand WHY they are learning it. Consequently they may not apply themselves as much as they should. The same can be said about physics and chemistry. When I was an undergrad in Sweden (at about the time T. rex went extinct), a 3-year biology degree had a full year of chemistry in the curriculum. Do biology students need chemistry? Of course they do. But do they need 1/3 of their education to be chemistry? For most of them probably not although I stumbled into chemical ecology – go figure. (Incidentally, I ‘escaped’ with a new program that compressed the full year chemistry to one term – a brutal term because the instructors essentially crammed the full year into half the time). At UNBC, where I taught for 21 years, math, stats, chemistry and physics were generally taught by faculty unable to provide the context (i.e., using biologically relevant examples and problems) that would engage biology/natural resources management students. We had highly qualified faculty in my department, but we were not allowed to deliver “statistics”. This had nothing to do with what would be best for the students, but everything to do with how different departments were funded. The problem as I see it is that 1st-2nd year ugrads don’t know what the future holds. I wish I had been stronger in math and stats, but I remember nothing in my undergraduate education that explained to me how I might be able to apply math, chemistry, and physics skills as a biology professional later in life, so I never pursued that knowledge. Do biology students need calculus? I think they do, but I am not sure that the format of delivery is adequately meeting the need.

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Amy HurfordGreat post. Regarding “maybe we should knit the math into questions they care about in their discipline”, this new book is fantastic for doing that with Calculus and Biology: http://www.amazon.ca/Biocalculus-Calculus-Sciences-James-Stewart/dp/1133109632

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ScientistSeesSquirrelPost authorThanks, Amy! I’ve updated the post to include a link to that book, in a more prominent place than the Replies. Glad you liked the post.

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ash khanyeah very interesting ……….i have done BS in Maths n teaching mathematics to biotechnology students they usually ask the question where is maths uses in biology?????????they are in the first semester they dont have read the models n laws as you describe there? so how could i satisfied them????????????plzzzz ans

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ScientistSeesSquirrelPost authorWell, not to be dismissive – but I’d think answering that question would be pretty important for someone hired to teach math to biologists! Math (not just calculus) is everywhere, and opening a biology textbook or two would be a good start to finding it.

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Gregory Walsh… I’ve been thinking about this all day which has led me to stumble upon this post. I’m going to spit this out fast due to time constraint so excuse spelling and grammar errors.

This may sound “out there” or “extreme” but it’s only a thought exercise. So don’t let your emotions get the best of you. This is just what I believe but I would love feedback from other biologist.

Imagine we are conducting a coin toss. When we flip a coin, we say the probability of it landing on heads is 50/50. An actual person may flip a coin 100 times and roughly half of the time it will land on heads. Now remove the person flipping the coin and replace them with a calibrated machine which flips the coin with the exact same force from the exact same height each time. Now when the coin is flipped 100 times we may find that only 5% of the time the coin lands on heads. This may lead us to think there is no longer a 50/50 random occurrence. Why is this? With the machine, there is now a constant casual element effecting the outcome of the coin flip. The more we know about a coin toss the greater probability we can assign to the outcome of the toss. If we know every physical detail of the toss and we are able to identify them as variables of an equation, then we can accurately predict the outcome.

We’ve been using math this way for thousands of years. Think of all we’ve accomplished and all we’ve learned about our universe. If we agree to the quote “The book of nature is written in the language of mathematics” why would the physical makings of biology be excluded from mathematical interpretation? Are we really that special? Its hard for me not to get fired up about this because it seems very arrogant. I have always felt that indisputable mathematics act as our sensory system for knowing what is and what is not. Mathematics simply remove subjective thinking.

I feel that this is a philosophical issue and not a technical issue. Instead of thinking: “How much math do I need to know for biology?” It’s more logical to think: “How can I apply all the math I know to support my findings in biological research?” Instead we have a academic culture of students dreading the idea of taking calculus because somebody has led them to believe that “it’s to hard” or that they just don’t need it. I wonder what Newton and Einstein would think of this?

We know more about the laws of nature that make up all the stars and galaxies in our universe than we know about our own selves. We are quantitative of events we may never actually observe but in biology we take a qualitative analysis before a quantitative analysis. Without studying mathematics, the latter of the two will be out of reach.

In science, we should be meta-cognitive of ourselves. Science isn’t about being right, its about discovery. I know this was blunt and don’t want to be offensive. Just throwing an idea out there with hopes of stirring up a response. Feel free to rip on the thought. Its all good. I really just want to know what others think?

My closing statement just for my own personal amusement. If you want to know the answer to biological issues such as cancer…. judging from every other scientific discovery we’ve claimed… it’s probably a math problem lol.

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Israel ValenciaThank you for your opinion, I felt understood with your post. I started a calculus course last semester thought by a physicist. I have learned to derivate many kinds of functions in more ways than I remember, however, I feel disappointed of me because even I have invested six hours every Saturday, I can’t apply derivates to biological systems freely. I mean I can derivate functions, but I can’t abstract the reality to apply derivates to a biological system. I have learned more about the applicability of derivates by my own searching about its logic and utility and no to much about its solutions. I think some teachers teach too much about how does maths works, but too few about its logic and applicability. Finally, I feel lost and disappointed by knowing too much and at the same time being able to apply too few.

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