Science and Mathematics

The other day, I had my students respond to a question about how mathematics relates to science.

In class, I had pointed out that mathematical definitions are very precise while scientific measurements can be rather messy. A mathematician has a very precise meaning when they say that two variables are proportional or have a linear relation. But when we get real data, even if they do not satisfy these precise meanings, we still gain significant information about the relation and might even say that the measured variables are proportional or linear. Unfortunately, many students seemed to think I was looking for a repeated discussion of this point.

Science can be thought of as the study of the physical world through the scientific method. Essentially, we make observations on what happens in the world (whether that be physical, chemical or biological interactions) and want to understand why that is happening as well as to predict what will happen in the future. In order to do this, scientists propose various hypotheses based on their observations (and past accumulated scientific experience) and then test those hypotheses. Experiments quite often include quantitative measurements, and part of the prediction is often to propose relationships between independent variables (the variables in treatments and control) and the dependent variables (outcomes). Experience may support a hypothesis or falsify the hypothesis, but it never can prove a hypothesis.

Knowledge based on patterns that we predict will continue, but which we can support but never prove, is called inductive knowledge. Science is an example of inductive knowledge.

Mathematics can be thought of as the study of structures that satisfy very specific rules. We have properties of arithmetic, algebra, and calculus. We establish specific axioms that describe our basic assumptions about the structures and then use logical argument to deduce the behavior of more complicated constructions. We might look at examples to see what ideas might be true or false, and in this sense mathematics can also take advantage of inductive knowledge. However, the objective in mathematics is not just to suppose that a pattern will continue; the objective is to determine conclusively if it must continue. We seek for proofs (that it is true) or counterexamples (break the pattern).

Knowledge based on basic assumptions (axioms) and logical argument that determines conclusively what must follow from these assumptions is called deductive knowledge. mathematics is an example of deductive knowledge.

Models form a connection between mathematics and science. Data often appear to follow a general trend, even in the presence of the noise of messy observation. A mathematical model takes that messiness and forms an abstract clean relationship that mathematics can work with. Based on the deductive approach of mathematics, we can often establish consequences of the assumed model form. We then apply those consequences as hypotheses in our scientific framework. The predictions from the deductive approach provide the predictions that can be used to falsify these hypotheses.

First Reading Assignment

Since I am not sure when Blackboard is going to be available for the class (I procrastinated asking for the two sections to be merged into a single section), here is the reading assignment and preparation for classwork for Wednesday.

Reading Assignments:

Online Calculus Textbook: Read Sections 1.1 and 1.2 (link below). These sections emphasize the idea that variables (which represent physical quantities) can be related, as independent and dependent variables. We want to think about related quantities throughout this semester. This text specifically asks for the web-browser Firefox version 3.0 or later (this is to render formulas correctly).

• Section 1.1
• Section 1.2

Come prepared to class having prepared answers for Problems 3, 4, 6, 7, 10 from Section 1.2 Problems. We will discuss these problems but will not turn them in.

How Students Learn: This book prepared by the National Academy of Sciences is actually written for teachers to focus on making courses better suited for students to learn. The first 12 pages introduce three concepts that students should be aware of in their own learning. The link below goes to the first page, and then follow the links to read through page 12.

• How Students Learn, page 1 through page 12.

Why is math fun? Why is math hard?

Today in class, I tried to help break the ice and reduce some of the anxiety related to taking a university mathematics course (Math 231). I asked students for examples of why they might find mathematics fun and why they might find mathematics hard. Here are some of the responses.

Why fun?

• It's fun when you struggle with a concept and then it finally clicks and you understand.
• It's fun to see mathematics actually being applied to a real problem.
• It's fun when you are able to spot trends and make predictions based on data.
• It's fun why you really understand why instead of just the "required" steps.
• When you understand, it becomes easy.
• It can be a lot like a game or solving a puzzle.
• It's fun to develop things logically.

Why hard?

• Later material builds on earlier material, so missing something early is permanent hardship.
• It can be really hard when the teacher goes too fast.
• It can be really hard when the teacher is unclear, especially if they can't give alternate ways of thinking about an idea.
• It can be hard if the teaching style is very different from your learning style.
• There are so many formulas, it can be overwhelming to try to memorize them.
• The theorems, rules and definitions are full of little details.
• It can be difficult to understand the many conceptual ideas that interact.
• Learning related technology can be challenging.
• Only one answer, so you can't fake it.
• Very hard to cram for exams.
• It can be really hard to find a little (stupid) mistake when proofing your work.
• Lots of homework, and problems can take a lot of time.

I'd welcome more comments, including examples of when you found mathematics especially exciting or examples of how your relationship with mathematics soured. Feel free to post a comment.

A New Semester --- Two Classes

This semester I am teaching a mathematical models in biology (Math/Bio 342) as well as the first semester of Calculus with Functions (Math 231). So I expect to have entries for both of these courses showing up.

Values, Equations, and Theorems

There is some confusion about theorems. For example, consider the Mean Value Theorem: If f is continuous on [a,b] and differentiable on (a,b), then there is some value c∈(a,b) so that f'(c)=(f(b)-f(a))/(b-a).

Some students think that the ratio (f(b)-f(a))/(b-a) is the Mean Value Theorem. But it is not; it is just a value that is called the average rate of change of f between a and b. You can compute this value as long as both f(a) and f(b) exist. It has nothing to do with derivatives or continuity.

Other students think that the equation f'(c)=(f(b)-f(a))/(b-a). This is closer to the truth, but still is incorrect. First of all, what is c? Second, this statement may not be true. For example, suppose that f(x)=-1 if x<0 and f(x)=1 if x>0. Suppose that a=-2 and b=+2. Then the ratio (f(b)-f(a))/(b-a) is equal to 1/2. But f'(x)=0 everywhere except at x=0, where f'(0) does not exist.

Even closer is to say that f'(c)=(f(b)-f(a))/(b-a) for some c between a and b. This is actually the conclusion of the Mean Value Theorem. It requires the entire statement, particularly the statement that the equation is true for some c and that the value c must be between a and b.

But my example above provides an example where the conclusion of the Mean Value Theorem is false. That does not mean that the Mean Value Theorem itself is false. After all, it is a theorem, and that means that it has been proved to be true always. The part that is missing is the hypothesis for the theorem. The conclusion can only be guaranteed to be true using the theorem if the hypotheses are all satisfied. In this case, you must also check (or give a reason why) the function f is continuous and differentiable on the interval from a to b, including the endpoints for continuity.

Similarly, ∫_a^b f(x) dx/(b-a) computes the average value of a function f on an interval [a,b]. The value can be computed anytime the function is integrable over the interval [a,b]. The Mean Value Theorem for Integrals has nothing to do (in principle) with this calculation.

However, if f is continuous on [a,b], then
f(c)=∫_a^b f(x) dx/(b-a)
for some c∈(a,b). This entire statement comprises the Mean Value Theorem for Integrals. The hypothesis that must be verified to use the theorem is that f is continuous on [a,b]. The conclusion is that you are guaranteed that
f(c)=∫_a^b f(x) dx / (b-a)
for at least one value c between a and b.

Limit or Function Evaluation?

I've noticed that some students are perplexed about when they use a limit or function evaluation. I presume that the cause of this confusion is that students have learned that you evaluate a limit by plugging in a value. But this is only because nearly all functions that they work with are continuous.

You use a limit evaluation when you need to know what the value of the function should be by using information from the side of the point of interest. When using a limit, you must use limit notation: lim_x→c f(x). Then you use the appropriate rules of limits to evaluate (and hopefully, the function is continuous).

You use a function evaluation when you need to know the value of the function at an actual point. There is no limit involved, just function evaluation. You just use function notation, say f(c), and compute the value defined by the function.

For example, suppose you are calculating an instantaneous rate of change as the limit of an average rate of change. The average rate of change only makes sense when the interval of interest includes two points (endpoints of an interval). The instantaneous rate of change is found by seeing what the value of the average rate of change does when the two points move closer to each other, or more particularly, as the second point approaches the first point.

On the other hand, suppose that you know the derivative, which is itself a function. Then the instantaneous rate of change is calculated by function evaluation using the derivative function. (The limit was already used to create this new function.)

In particular, I have noticed this problem when dealing with finding extreme values of a function. When the interval of interest is an open interval, we are acting as though the domain does not include the endpoints. So, with this restricted domain, evaluation of the function is not possible (since the points are not in the domain). So we must use the information about the function immediately adjacent to the endpoints, and this is with a limit. In this context, the value found in the limit is not achieved at the endpoint, although it might be achieved somewhere else in the domain.

On the other hand, if the interval is a closed interval, then the endpoints are included in the restricted domain. If the function is continuous at these points, then evaluating the function directly is appropriate, since the point is in the domain.

Final remark, if the function is discontinuous at some point in the interval, you must also check the limits at that point for consideration when looking for extreme values.

Differential Equations Project Tips

As some questions are asked more regularly, I thought I'd provide some general discussion here.

(1) Start with the proposed form of X(t).  Compute X'(t) and X''(t) based on that form.  Then use those calculations to discover when X'' + k/m X = 0.

(2) The question "mean physically about the mass on a spring" is not asking you to think about the mass (as in measurement) but is asking you to think about what the statement X(0)=1 means about the state of the mass at time t=0 and what the statement X'(0)=0 means about the state of the mass at time t=0.

(3) X(0) is a constant and has derivative of d/dt[X(0)] = 0.  Recall that dX/dt > 0 implies that X is increasing, dX/dt < 0 implies that X is stationary (instantaneous rate = 0)

(4) An arbitrary quantity A is proportional to some other quantity B if it is always that case that A = k B for some constant value k (the constant of proportionality).  Now interpret the statements to identify what pieces of the equation are proportional to what.

(5) Although you have studied e^x in precalculus, you will not use the logarithm at all in this work.  Instead, I just want you to consider some function that has the special property that exp' = exp.  (This sentence is analogous to sin'=cos and cos'=-sin.)  However, you do need to think about the chain rule: X(t) = A exp(rt)  (Since the argument is not simply t, you must use the chain rule.)  This problem is exactly analogous to Step 1.

(6) You will get something like

dX/dt = "formula involving X and a, b, and m"

X is increasing when "formula" > 0, decreasing when "formula" < 0, and stationary when "formula" = 0.  So use your skills with algebra (think sign analysis) to find conditions when these are the case.

(7) You need to understand the relationship between a rate of change and an actual change.  To understand this as well as possible, see the section we skipped in Chapter 3 (last section).  But what you essentially need is that we will follow the tangent line for the time increment Δt.  How much change is there when the rate of change and the duration of time are both known?

(8) The new version of Excel has some unanticipated differences from what I had when I wrote the project.  The labels are not assigned from a menu anymore.  Instead of the 3-step process that is described, you just click in the label field in the header section of Excel and type in the new label and then hit enter.

The calculations you see in the first few lines should exactly match your hand calculations in part (7).  

Do not print the spreadsheet (it takes WAY too many pages).  That is why I ask you to submit your spreadsheet on Blackboard as part of the project.

(9) I must receive a print out of the graph -- hand drawn figures are not acceptable.   Ideally, this entire project report would be typed (perhaps using Equation Editor for the equations), with the figures naturally fitting in.

(10) Make hypotheses and test your hypotheses.