Tuesday, November 17, 2009

Concerns about Derivatives

Okay, I have just finished grading HW #4. This assignment had some differentiation rules. But many of you are not comprehending the purpose of a derivative rule.

For example, we had the special rule for squares of functions:

d/dx[f2(x)] = 2 f(x) f '(x)

This means that any time there is a formula squared and you need to take its derivative, you can apply this rule.

(2x+3)2 corresponds to the function f(x)=2x+3 being squared. So since f '(x) = 2:

d/dx[(2x+3)2] = 2 (2x+3) (2) = 4(2x+3)

Similarly, (x2-4x+5)2 corresponds to the function f(x) = x2-4x+5 being squared, and f '(x)=2x-4

d/dx[(x2-4x+5)2] = 2 (x2-4x+5) (2x-4)

You need to be an expert at identifying the form of an expression in order to apply appropriate rules of differentiation, and next semester, rules of integration (anti-differentiation).

Thursday, September 3, 2009

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.

Monday, August 24, 2009

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).
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.

Tuesday, January 13, 2009

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.