Post written by Chris Buddle and Carly Ziter (MSc student at McGill – you can follow her on twitter)
Population and Community Ecology is an introductory undergraduate course at McGill University and each lecture typically starts with an x-axis and a y-axis drawn on the chalkboard – something like this:
The course is taught from a quantitative perspective, and it uses equations, models and graphs to cover concepts ranging from logistic population growth, to metapopulation ecology and estimating species diversity. The class uses Gotelli’s “A Primer of Ecology” as the text – a book that walks through many ecological concepts from first principles. It includes calculus, probability theory, statistical distributions, and null models.
It was therefore fitting that the ‘E.O. Wilson versus Math” debate was discussed during lecture last week. Students were asked to read Wilson’s piece in the Wall Street Journal, and read some of the blog posts that reacted to this, including Jeremy Fox and Brian McGill‘s posts on dynamic ecology. Students were also asked to look at some of Terry McGlynn’s writing over at small pond science, and to come to lecture prepared – to have opinions and be willing to discuss these opinions.
For those not fully aware of this debate, here it is in a nutshell: Wilson argued that a ‘deep’ understanding of math may not always be prerequisite for doing great science, or at least may not be required for generating big ideas and concepts. Wilson was in part trying to encourage people who are ‘math phobic’ that this phobia needs not be a reason to stay out of science. Not surprisingly, this stirred up a lot of debate (and some of it was rather harsh!), and the debate was particularly interesting from the perspective of Ecology since this discipline has always struggled with this topic (see Terry’s excellent post about tribalism in ecology for some perspectives on this).
Here is a summary of the key points that were discussed during lecture – and let’s just say that a 50 minute lecture slot was NOT enough time for this topic! (by the way, there were between 50 and 60 students who attended this lecture, and the class is comprised primarily of students studying environmental biology).
Many of the students were surprised at the tone and overall discussion points that emerged from Jeremy Fox’s post – they argued that when they read Wilson’s piece, they didn’t feel the intended audience was ‘established’ ecologists – but rather the post was meant for students at the start of their careers. Some of them found the blog posts way over the top, and the academic discussions took away from the main message. Some felt that Wilson was arguing in part about the need for freedom to think without any boundaries (mathematics, or anything else). Creative thought need not be constrained, and students coming up through the system, whether they are math literate or not, should never fear heading into science (indeed, some confessed that an increase in math courses may have driven them away from biology altogether). Related to this, mathematical models all require assumptions (we talked a LOT about this when working through Gotelli’s book!), and any assumptions are limiting and could distract from thinking out of the box about any topics, including ones that are ecological. These students worried that the constraints imposed by math could force ecologists to view the world through a particular lens.
That being said, many of the students also agreed that a deeper understanding of mathematics was absolutely required for ecology – especially since the world is complex, with complex problems – problems that require multiple disciplines to solve. However, while these disciplines include mathematics and biology, they also include literature, history, environmental policy, and more. What a solid argument! And it was great to see that argument expressed by 20 year-olds. Yes, math is important, but it is one tool that we need in this world, and it’s not necessarily more important than other tools. While some ecologists are strong in math, others may prefer to hone their policy skills, for example. Ecology’s strength, in part, is in its ability to bridge different disciplines and students expressed how ecology is actually a ‘great uniter’ of biology and math (and other fields, certainly some areas of ecology draw upon a range of ideas from sustainability science, medicine, economics, history, etc).
The students also expressed concern about how mathematics is taught, from elementary school all the way to University – they expressed how learning mathematics in isolation of other topics is ‘ok’ for individuals with an intuition and natural ability with math – but many students felt that a better way to learn about math was applying it to the ‘real world’. The application of mathematics is the best route to learn mathematics. Ecology was again touted as a perfect example of a discipline in which application of mathematics is clear – from predicting distribution of invasive species to modelling species richness in fragmented forests. For some students, math was not a subject they initially enjoyed, or strove to learn – it was ultimately through their study of ecology that they began to value math as a tool they could use to support their discoveries, and lend credibility to their work.
By in large, students agreed that mathematics was required for ecology, but there was certainly debate about how much was enough – whether it was enough to use mathematics as a tool, or that perhaps mathematics was more like a language. A language in which fluency is required so all the nuances can be understood and that the full meaning is in place. From those advocating mathematical “fluency”, there was a strong opinion that like languages, mathematics can be learned with hard work and focus (yes, they agree with Wilson on this point!) – this opinion comes with a wealth of experience in the classroom at McGill since many of the students are mother-tongue French and have learned English after coming to McGill. In other words, if you can learn a language you can also learn math.
The final argument put forward by students was that this entire discussion about Ecology was from a very narrow perspective – what about the role of traditional ecological knowledge? Ecology is a much older discipline than Clements, von Humbolt, Haeckel, or even Aristotle. Throughout history, humans have been interacting with their environment, and have been observing nature. By this act, humans have been counting, developing models, and making predictions… for thousands of years. Linking mathematics to nature is very, very old. Ecologists ought to pay more attention to other ways of looking at the natural world, other ways to visualize, predict, observe and count. Although this is certainly not the same kind of math as presented by Gotelli, perhaps it could be as insightful.
In sum, the discussion with undergraduate students on this topic was insightful, fascinating and important. There was clearly a strong appreciation for the role of mathematics in ecology, but also different ideas about the degree to which a deep understanding of math is required – which often related back to the students own struggles with, or aptitude for, math earlier in their studies. It was validating to hear that they appreciated using Gotelli’s book to learn the foundations of ecology, and recognized that ecological models can be both limiting and liberating.
12 thoughts on “Ecology and Mathematics: perspectives from undergraduate students”
I majored in math but now study spiders and write patents. I’ve noticed one big problem with most of the discussion about the role of math in biology: most people equate math to calculus and its preconditions. Math is so much more than this. This assumption is reasonable considering my own life, where despite taking AP calculus in high school, I didn’t learn what math was until my second year in college as a math major.
Math is the formal study of abstract relationships. It seeks to classify every kind of relationship possible and to study their properties. Calculus can be thought of as just the study of relationships among continuous ranges of numbers, or more broadly, properties of measurement. But learn set theory and you will see new ways to organize sets or learn graph theory and you will see new ways to structure nodes or learn group theory or model theory and you will see new ways to produce equivalent models. There is not end to the new kinds of relationships you can learn about. And the more kinds of relationships you’re aware of presumably, the more you’ll be able to see in nature or think to test nature against.
I suspect the only reason the biology community is having this argument at all is partly because high schools introduce students to little more than calculus and partly because our culture equates math with numbers.
Joe – great comments, thanks for taking the time to write it. It reinforces many things that the students said – the way math is taught in high school is often unidimensional and doesn’t do the discipline justice – irrespective of how math is integrated into other disciplines!
It’s nice to see that these students are discovering subtleties more quickly than many professionals in the field. Good on ya. If I did have to identify one book for students to spell out how math describes ecology, in both amplifying and limiting ways, it’d be Gotelli’s. Its clarity is gorgeous.
Thanks Terry – what is amazing to me about Gotelli’s book is that MANY students hang on to it and tell me (long after) that they found it incredibly useful, and many that go on to grad school still refer back to that book – in many cases, it’s because of HOW the material is presented in addition to the content itself. I think if Gotelli’s book (or an analogous one – is there one?) were used more often in UG ecology courses, the discipline would be better off in the long run…
Great to hear some students’ perspectives on this, the responses I read were all angry ecologists/mathematicians. Wilson’s WSJ piece is essentially a summary from one of his book chapters, which I’m reading through right now, and I’ve gotta say–I agree that some of the responses to his piece were just vicious. His piece was focused on encouraging young scientists to enter the field/stick with it, so of course he’s going to argue that you don’t need to be a wizard of math to be a scientist: and that is EXACTLY what budding scientists need to hear.
I heard similar views from my professors when I was realizing I wanted to study biology, and I was worried about the math, and it helped me gain the confidence I needed. Math is certainly important, but so are other skills. It’s truly a shame that teaching math through primary and secondary school is so fragmented, and I think that’s why people shy away from it. Each year is supposed to build upon the previous one, but that connection is never explicitly stated or reviewed (in my experience at least), so students are just expected to remember many different concepts that might not be closely connected in their minds. In high school, I went from Geometry to Algebra II, then to Pre-calculus, where I needed to remember some stuff I learned in Geometry but hadn’t thought about for a year! It was very confusing.
It sounds like you have some astute students–the angry responses were distracting from the take home message of Wilson’s article: if math isn’t your best subject, don’t give up. It’s not an insurmountable obstacle and can be learned, just like any other subject. That is some encouraging advice for struggling undergrads.
Derek – thanks for the perspectives – they are personal and insightful. Connections between content in different courses is key, and on the whole, I don’t find this is done well, at high school or in University. Remembering piece x, y and x without seeing clearly how x, y and z fit together is a real problem.
I do have some very astute students in the class, and they really spent the time to read the material, digest it and think carefully about it. It helps that the entire course is quantitative, so they have had an entire term to appreciate (or dislike!) the way that math was fully integrated into ecology.
Math and sciences go hand-in-hand, they always have and always will. IMHO – math is a skill that can be taught, fostered, and practiced. Passion, curiosity, and a love of research or science are traits that are harder to teach. Thus I fully stand behind EO Wilson when he encourages students that if they love science to go for it and not stay away if they fear their math skills are not up to snuff. I have had many students with inquisitive minds and passion for natural sciences confide in me that they dream of research or grad school but ‘suck at math’. What a shame it would be to lose these future bright scientists due to their apprehension of math.
Personally I was always pushed into advanced math in high school/university due to my grades but the way that it was taught it never really stuck with me. The statistical tools I learn now that I can see applied to my own work – now that sticks! I wish more undergrad classes would follow your example Chris.
Thanks Barbara! Great comment. I think you have captured effectively a lot of the sentiment from the class – you don’t want to turn people off science because of (potential) issues they may have with math. It’s just such a shame that a bad experience with math, perhaps during high school (or earlier) can really turn people off from pursuing disciplines that use math. This is the heart of the problem, I think. We need really great teachers – and ones that show the beauty of math…in all its applications but as well as in its fundamental sense.
i didn’t make an aeronautical engineer because I had real problems learning calculus. I made a D in college algebra the second time through, so that was not surprising. Anyway, I ended up a biologist, and was able to do enough math to do it myself, or find someone who could. I had productive and enjoyable career as a biology professor, and build and fly model airplanes as a hobby. So I am a living example of what Wilson was talking about, I suppose.
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The concern about how mathematics is taught really resonated with me.
Although at some level, mathematics need to be taught as a separate field on its own (when you focus on proof), the things taught in high school and introductory University courses should not be independent of application. Most of the math a typical scientist needs to learn can be taught in physics, ecology, and other applied classes and not separate from them. In most intro math courses, the profs already try to “motivate” the math they are teaching with physical intuition, why not take this to its logical conclusion?
I definitely learnt more math trying to solve physics problem in high school than I ever did in my math classes.
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