At the NJIT conference two weeks ago, I was asked to sit on a panel on “Future roles for mathematics and statistics within the biological sciences”. To start the discussion, the moderator Farzan Nadim asked two questions. The first was on whether biology needed mathematics (the implication being that mathematics was not the basis of biology as it is for physics) and the second was how to educate biology students in mathematics and statistics. Mathematical ecologist Stuart Pimm of Duke, immediately jumped in to protest that in evolutionary biology and ecology, mathematics has always played a dominant role. He also made the tongue-in-cheek quip that we didn’t need to train Americans in mathematics or science because we could always import them. Robert Miura followed with some personal observations that there was a cultural divide between mathematicians and biologists, one example being that mathematicians look for dimensionless quantities to determine what’s big and small (e.g. important and unimportant) whereas biologists think in terms of dimensional quantities that make sense experimentally (e.g. it’s important to know if the current you are measuring is in picoamps or microamps).

I started by agreeing with Stuart that evolutionary biology and population genetics had always been mathematical but that there was a paradigm shift after Watson and Crick. With the advent of molecular biology, mathematics began to play a much less important role because there were so many mechanisms to be discovered. We don’t know what most of the twenty thousand or so genes in our bodies do and how they are regulated. A biologist can easily have a productive career doing single gene knockout experiments and see what happens to some system. A more ambitious one can study double knockouts or triple knockouts and trying all combinations can keep all biologists occupied for much longer than the age of the universe. Then there are the effects of small RNA molecules and epigentic effects. Basically there is no end in trying to catalog all of the possible mechanisms. Thus, I think that there is a challenge to convince biologists, at least molecular biologists, that mathematics is useful.

However, I do believe that mathematics can be helpful for biology. For one, it may never be possible to understand how a biological system works by simply doing gene knockouts. I used the analogy of trying to figure out how a car works by doing knockout experiments. First you find that if you knockout the door lock, the car won’t work. Then you knockout the door and it works again. Knockout the ignition switch, doesn’t work, and so forth. Some car biologists may focus on the radio, others the gas tank. Some may try to figure why there is a large sound when the car turns on. In essence, it takes more than knowing what the parts are and even what they do individually to figure out how a car works. Clearly this is already accepted in bioinformatics and so-called “systems biology”. Biologists realize that computational and statistical methods are necessary when confronted with millions of bits of data. Mathematics has also taken hold somewhat in systems neuroscience, where concepts from probability theory, information theory and dynamical systems are used by experimentalists, often in collaboration with theorists, but sometimes not. Thus, I think that as biologists start to grapple more with systems with feedback and multiple dimensions, they will begin to realize that their internal models and intuition are insufficient and they will be forced to turn to mathematics.

Now the question is how do we educate biology students in mathematics. My answer was simple. We make it a requirement that all biology students take mathematics courses. I suggested that they should at least know 1) calculus up to differential equations including some qualitatitve theory of differential equations and phase plane analysis, 2) statistics, and 3) linear algebra. I think with a background in these areas they should have some sense of when something they are puzzled by could have a mathematical answer and also be able to converse with mathematicians. I actually went further and suggested that an educated citizen of the modern world should know some mathematics because many decisions that directly affect our lives are based on mathematics and especially statistics.

In fact, we should re-evaluate what it means to have a liberal arts education. What is the canon of knowledge that an educated person should have? I think everyone should have a basic understanding of science and mathematics. Maybe we should design a required one semester or year course called “Science and mathematics survival tools for the modern world.” It could include basic statistics and mathematics that would be useful in everyday life, like knowing what statistical significance means, or what the probability of running into your hairdresser’s boyfriend’s son at the bottom of the grand canyon is? (The chance of running into someone with some connection to you unexpectedly is quite high because you know a lot of people. It’s the multiple testing problem in statistics). They should know how to solve n linear equations in n unknowns. They should know about the central dogma of biology and the standard model of physics. They should learn some basics about molecules and thermodynamics (e.g. there are no perpetual motion machines and water cannot have a memory of a solvent that has been diluted away). They should learn about ethics. I think this would be an interesting course that students could really enjoy. These topics don’t need to be covered in depth; the students just need to know that they exist and how to get more information if they want it, including a follow up course.

I believe that was the sound of millions of English majors howling in protest. I have long supported the idea that a stronger background in math and science is important to any liberal arts education. If for no other reason, it should hopefully reduce the absurd number of science conspiracy theories (perpetual motion, vaccines, etc. ad nauseam…).

Given how much biology students wail about taking calculus, how much coursework do you think you could actually get implemented? In particular, at a research university? or a liberal arts school?

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Hi Daniel,

I think it would be possible to make the science/math course for the English majors fun. They wouldn’t be expected to do that much actual math. It would be more like watching Nova episodes but with more information and homework. As for biology students, I think they already have to take Calculus, don’t they? In any case, I don’t think any science major should be allowed to hide from taking a year of math courses. You could water down the courses they take perhaps.

Carson

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The flip side of the question is, What non-science things should every scientist be required to know?

The reason I ask is that we tend to want the general public to be educated about science, but unfortunately are not always quite so educated or sophisticated in reaching the general public…

I definitely support the science ethics education, not just for non-scientists but for scientists as well. Coming from a liberal arts education… I was actually horrified that a surprising number of people with science/engineering backgrounds (particularly from the British-style system of a very focused undergraduate study) actually had a grand total of zero classes in ethics, philosophy, history, etc, particularly in regards to science. I would definitely hope that science majors from non-liberal-arts schools would be required to take courses in those areas.

The difficulty about trying to require advanced mathematics from everyone is that math is a skill learned sequentially, and if someone missed out on essential building blocks way back whenever, it may not be possible to bring them up to speed in a single semester. However if the course was like an interactive Nova show then it might not be as much of an issue. If it was interesting it might even inspire some study more mathematics after seeing that it is indeed useful, relevant, and necessary to their lives – something that may not have happened if they had poor instruction in the past (or lack of motivation).

On a side note: required mathematics courses for biology majors does not necessarily result in biology majors understanding math…unfortunately depending on the skill of the teacher and how motivated the students are, instead of ending up with a big picture framework of how math is used and useful in biology, and learning how to go about finding those tools or people who know how to use them, they may end up with a jumbled mass of equations and a permanent math phobia, especially if they don’t have a good foundation in the area and just barely pass the course.

Or, there might just end up being a whole lot more pre-med psychology majors….LOL.

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I would make everyone take ethics, especially the science majors. I would also force all social science majors to take the science/math course as well. There can be no escape. I think the biologists should take the math for biologists course. It’s okay if they don’t all understand it. They should at least see it. You can bring a horse to water but …

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So much to discuss. I agree with the general sentiment here … as do many of my colleagues with whom I’ve had similar conversations. Though I am left with two questions. First, how to we (as educators and/or scientists) overcome the stigma against mathematics? For instance, there have been all kinds of studies showing gender and ethnic sentiments towards the hard sciences generally negative (e.g. see Danah Boyd’s blog). My feeling is (say what you will) that people are generally intellectually lazy and we need to learn to overcome that first, then we can teach more generally and equip people with these basic tools – this seems a more fundamental problem. My second question is when do we begin? My feeling is that starting earlier is better. Having such courses in college is way way too late … by then, biologically brain development is largely done, our prejudices towards what we finding intellectually pleasing, etc are all already in place. My feeling is that there is no reason why we can’t have students learning calculus by middle school, with a carefully planned and tested curriculum.

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My comments at NJIT were in fact driven by the fact that there is a stigma towards mathematics. Part of that is certainly that it takes effort to learn and the payoff is not clear. I believe the current wisdom is that we make math more appealing by putting in real world examples and so forth. From my experience teaching calculus, I found that this “new age” approach does not work. The students do not make the connection to the real world. They just get more confused. What they really want to know is what do they need to know for the exam. So my strategy is that you force everyone to take math and you don’t try to dress it up. What you do is make them practice, practice and practice until they can do elementary things. Everyone in middle school should know how to add fractions. Everyone in high school should be able to do simple geometry and algebra. I don’t think giving calculus to middle schoolers is the solution although I do believe that students that are ready should be allowed to take it. I think we just make it clear to them that this is what an educated person should know and don’t try to gloss it up. I don’t think I would get far with the education board though.

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I have to agree with Tirstan that a lot of what needs to be fixed as far as teaching mathematics needs to be fixed long before college. From my (admittedly limited) experience teaching elementary algebra and calculus, many students have not gotten over the fundamental hurdle of associating mathematical symbols and objects with their own experience. Hence, real world examples make them more confused because they don’t understand how to ‘map’ math into anything other than math.

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That’s why I think that we give up on trying to teach the mapping. We should focus on the process. To learn how to add fractions, you need to do many examples. h Then if and when they do make a connection to the real world at least they can do the calculation.

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I think if/when/how to teach that mapping is the fundamental point of interest here. Without such a mapping, mathematics becomes a (sophisticated) logical exercise. There are certainly many reasons why such an exercise is worthwhile in and of itself. I would certainly argue that merely the practice of thinking and communicating in a logically rigorous manner in mathematics improves one’s ability to do so outside of math. However, a purely logical enterprise is likely to continue to generate scorn in many students leading to the original problem.

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I think teaching logical thinking is the main goal. Mapping to the real world can only come after that. That is also why we need to force everyone to do it. Then the scorn doesn’t matter. I know people who hated piano as a kid and resented having to learn it but, they now can read music and play piano.

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Simple – but not easy – answer: catch kids when they are young and still want to learn (i.e. 6th grade or earlier), then drill them until they are good at math. It is the rare person that dislikes something that they are good at. So the solution is to make sure kids are good at it, and then many will like it. If you wait too long they will have failure and the concept that they are stupid ingrained in them, and it will be extremely difficult to reverse that because then they will probably be very far behind and not particularly motivated to improve.

The reason it’s so difficult to catch people up later because all the skills are sequential, so after a certain critical point missing foundational building blocks will pretty much doom how far someone can go.

Side comment: As one of those people with a not-so-great musical education experience (I didn’t figure out the goal was to make music till I was almost in college), I’ve noticed two camps of people: those who seemed to be making music from the start and those who never quite caught on to that fact.

One big difference, at least from my observation, is that those people who loved it from the start generally had families that made music together and whose parents loved it. Those whose parents couldn’t play or weren’t particularly musical… well, it’s a toss-up whether we ended up liking it or not. My siblings quit completely – but admittedly, they do have some basic level of musical skill.

The take-home lesson for mathematics being, of course… family (or the type of people you are around) matters. Being around people who can appreciate something doesn’t guarantee appreciation or excellence, but it makes it easier. If kids come from families where the parents are essentially mathematically illiterate, barring outside intervention or incredible innate predisposition, it’s very difficult to achieve comprehension, much less excellence, if there’s nobody around to help you.

The moral of the story: if we want people to appreciate and understand mathematics, it’s got to start at the elementary or grade school level…. understanding and mastery breeds liking, whereas lack of understanding creates a mental block that is reinforced in future encounters.

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I just want to add two comments:

– first of all, I am not sure whether it would be good that everyone knows at least some amount of maths. Learning maths can be very demanding and may prevent from learning lots of other interesting things (the converse is true as well). I guess there should be non “canon” as Carson says. Having distributions of levels of competence in a given domain ensures a society as a whole works well. (To make a naive comparison, bees are starting to go and collect food at different levels of food shortage. It ensures that the right number of bees is collecting food. Whereas, if they all had the same behavior, they would all find themselves in or out at a given level of food shortage.). Having people highly specified in one domain, thus deprived from any knowledge in maths (and the inverse as well) may be a good thing after all.

The essential thing may be to know what you really know. And on this point (even if every mathematician has once in his life to come across a guy who pretends to have a revolutionary theory even if he admittedly knows no maths) there is much more work to do on politics, where most citizens believe they have the solution to the financial crisis even if they have never studied politics at all.

-Second point, which is more sociological and close to what Karen wrote. One thing that can make someone loves maths is to like someone who already loves maths. How many people eventually got a MSc in maths because they played funny maths games with their parents, brothers, sisters or friends when they were kids? Then, growing up, trying to solve harder and harder (less and less funny alongside…depending on your point geekness), the knowledge in maths increases. So, basically, if you hate your teacher, this may be a reason not to study maths. The solution may be to train maths teachers to be cool guys. ;)

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Hi Romain,

I think you misunderstand me. I’m not suggesting that everyone know how to prove the Bolzano-Weierstrass theorem. I’m just saying everyone should know basic math such as how to add fractions or know what statistically significant means. I’m willing to bet that a third or more of the population can’t add a third to a half. Our society is increasingly becoming more mathematical so the population needs to drift to accommodate. There will always be a variance around this mean. All I’m suggesting is that make the cutoff on the lower end higher. I don’t think making people know some basic amount of science and math will be so demanding that they can’t learn something else. I also don’t think making people love math is the issue. We expect everyone in our society to be able to read and we don’t think that we have to make people love to read to be literate.

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So what should people with dyscalculia do? Kill themselves?

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Obviously, there are always exceptions to any rule. If a person has a disability then they should certainly be excused.

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Dyscalculia…[…]Math and biology (and what an educated citizen should know) « Scientific Clearing House[…]…

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[…] the concepts of arithmetic, statistics and quantitative reasoning. I have even posted before (see here) that I thought mathematics should be part of the accepted canon of what an educated citizen […]

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