We here at the MathBench team have prepared (or are preparing) about 15 modules relating mathematics to biology for your use, edification, and enjoyment. But just what is our agenda? Well, we're not secretive - we'll lay it all out for you. By making your way through these modules, we are hoping that you develop 10 key skills and concepts. These are:
10. Manipulate equations (i.e., plug in values, solve for a variable)
Equations are to math what sentences are to a conversation. They can provide information and answers (as when you plug numbers into an equation to get output) and if you follow the rules, you can get even more out of them. The rules of handling equations are called algebra, and the great thing about these rules is that they won't lead you wrong (getting them to lead you someplace useful, on the other hand, can be tricky, but that's what manipulating equations is all about).
9. Manipulate graphs (i.e., graph equations, interpret intercepts and asymptotes)
A graph is just an equation made pretty. Or not pretty, as the case may be. While an equation glares at you with slitted eyes, daring you to figure it out, a graph is flamboyently out there, just begging to be looked at. A graph can tell you at a glance what will happen when a particular input is considered, and what the trend is when the input gets very very very small. Or very big. A graph will tell you whether the output changes in a slow and steady kind of way, or whether it jumps around like a rabbit. Moreover, each elementary function has its characteristic graph, and with a little practice you too can force even scary-looking equations to show their graphical side.
8. Use elementary functions (linear and quadratic, exponents and logs, sin and cos)
Elementary functions such as polynomials, exponents, logs, sine, cosine, and tangent are the basic building blocks of mathematical conversation. Each function has its own "personality" -- including how it behaves, how it looks on a graph, and which values it works or doesn't work with. Get to know these basic functions, and mathematical equations will talk to you instead of just sitting on paper like so many wallflowers.
7. Scale up or down, using magnitude and significant digits
Scientists deal with a lot of numbers that are either very big or very small, and sometimes those numbers also seem very mysterious. How many disposable plastic bags are used every year in the U.S.? How many blood cells are running around your body? How much does a single molecule of sugar weigh? Amazingly, anyone can learn to come up with a rough guess for numbers like these, no PhD or memorizing-the-encyclopedia required. But, in order not really pull it off, you also need to know your own limits -- a/k/a report only significant digits.
6. Convert units and use unit analysis to check answers
Scientists also spend a good chunk of their time measuring things, and they have invented myriad ways of measuring the same thing -- that is, myriad "units" of measurement. Without conversions, it is often impossible to compare or analyze data. But that's just the tip of the iceberg. You can manipulate mathematical statements in much the same way that you manipulate numbers, and when you do, units can provide useful information, like "does my analysis make sense" and "I forgot my cheat sheet -- just how do these two measurements fit together in a single formula?"
5. Make simple probability calculations
Einstein may have said that God doesn't play dice with the universe, but now we're not so sure. At some fundamental level, the universe may be a probabilistic place. More to the point, there's a lot in biology that we don't know well, and we can't make more than an informed guess. What will the weather be in 5 days? How many cicadas survived their 17 year hibernation? How many flies can a fly-catcher catch? All these guesses pile up, and you need some way to extract the useful information and clear away the debris. This is where probability calculations come in. Learn to combine alternatives, string events together sequentially, and even leap probabalistic buildings in a single bound.
4. Understand and use statistical tests
Ahh, statistics, everyone's favorite punching bag. There are two possibilities when you use statistics. In a good world, your data might be so clear and convincing that the statistics is just icing on the cake, sort of like sticking out your tongue and saying "I told you so!". Or, in a not good (but much more likely) world, your data is not so clear and convincing. That's when statistics can really come to your rescue, by determining exactly where the dividing line between "yes" and "no" is, and how much confidence you have in that dividing line.
And, contrary to popular belief, statistical tests are not really hard to understand and don't need to be memorized. Here, we pull every test apart to see not just how it works, but more importantly, why it makes sense. Then when you put the test back together, you'll remember what you're doing and why.
3. Understand how mathematical models are structured and what flavors they come in -- discrete vs. continuous, stochastic vs. deterministic
Take one biological problem, add a liberal dose of mathematical equations, mix well, and what do you have? The answer is, a mathematical model. A model uses all of the ingredients above to create a way of answering a question and asking new questions. And while most of us are not chefs creating new models, we can at least follow the recipe in the cookbook. In addition, just as you know soup from salad from dessert, you should know the flavors that models come in. Discrete models consider real timesteps such as an hour or a century, while continuous models use the Magic of Calculus to create an infinitely small, "smooth" timestep. Deterministic models are control freaks -- they can tell you the exact response of every variable to every manipulation -- while stochastic models are more laid back, preferring to tell you things that probably will and probably won't happen.
2. Understand equilibrium and rates of change
Many equations, especially in biology, tell you what will happen as time passes. Concentrations go up or down, things get warmer or colder, mutations accumulate or get weeded outů When you look at one of these equations, three of the most useful questions you can are: how are things changing over time? is there a point at which change stops occurring? and, once you get to this state of changelessness, is it easy or hard to stay there?
And the number 1 megaskill or concept is...
1. Distill mathematical equations from a verbal description
Understanding biology is obviously why you're a biology major. Understanding math is something that is required as a pre-requisite. Math and biology should be like soulmates working in harmony for the greater understanding of nature. But sometimes it feels a lot more like math and biology are two separate countries, possibly with hostilities between them. You, the scientist, need to learn to be a mediator between the two. You can take a verbal, biological description and recognize where math could be a useful tool. If you're good at this, you can even look at a mathematical equation and see where it might have interesting applications to nature. Altogether now, in peace and harmony...
bonus: Use matrices to organize information
Let's face it, the world is full of information. Whether you're talking about a few mutations of a particular gene, or millions of species living in millions of places around the world, you need a way to keep track of it all. A matrix does just that -- like a portable database, you can use a matrix to organize information. How similar are A and B? how about B and C? C and D? B and D? wouldn't you rather just have someplace to put all those numbers? And a nice easy, efficient way to do math without having to look at each number inidividually? Matrices pop up in everything from models of population size to similarity between communities to genetic evolution to common statistics. Take a matrix home today and you'll wonder how you ever got along without them.