Subtest II: Mathematics Study Guide for the CSET Multiple Subjects Test

Math Skills

Problem Solving

While you have no doubt done a lot of math problems in your life, to a math person the word problem implies reading a description of a situation, understanding the given information, knowing what you are looking for, figuring out the steps to find it, and doing the calculation(s). In short, it is a “word problem”—a fearsome term to every math student. This kind of problem is very common in most math subjects, as they are, in a sense, the intersection of math with the real world, and the very reason we learn math.

Securing Information

Problem solving essentially boils down to three main ideas:

What do you have?
What do you want?
How do you get there?

Be organized about this. List the relevant given facts with their units. In a math class, everything given is generally relevant, but you never know. Watch for unnecessary information. Write down what you are looking for. Include units. Know that there may be relevant information not given and it may take an extra step or two to come up with that.

Solving

It may take a lot of thinking to make sense of a problem, but persevere and very often you will be rewarded with a sweet “aha moment” when it all becomes clear. Complex problems can often be broken down into two or three simpler problems that you will recognize. If you can’t figure out the final answer, see if there is something you can figure out. That may well be the missing link to finding the answer.

A trick that can sometimes help if your problem has weird numbers is to change them all to simple numbers. That way, your brain doesn’t get distracted by the numbers and can better cope with the rest of the problem.

Sometimes trying to work backward works. You might think “If only I knew fact 2, I could see how to get to fact 3,(the answer). OK, how can I get fact 2?” Probably from fact 1.

Modeling

Models in math can mean different things. An equation using words or symbols can be a model. Labeled diagrams are models that can be a huge help to make sense out of a problem. Is something falling off a cliff? Draw it, using words and/or symbols to label distances, times, and velocities. Brains work much better with pictures of some sort than they do with just numbers and words. Sometimes, an actual hold-in-your-hand model will be useful, especially with three dimensional problems. Software is available to model 2-D and 3-D shapes.

Using Other Information and Strategies

A conjecture is a statement made without complete information. For example, draw a lot of examples of different looking right triangles. Look for patterns. Try to come up with an idea about them that you think may be true, such as A right triangle may be an isosceles triangle. That’s an example of a conjecture. It doesn’t have to be true to be a conjecture, but the hope generally is that it will turn out that way.

Reasoning

Reasoning is a combination of recognizing patterns, using problem-solving strategies, being creative, and above all using logic to draw correct conclusions.

Reasoning Resources

These are the resources in your head that make use of essentially all the skills you have learned in mathematics. Abstract and quantitative reasoning in all areas of mathematics should be in your skill set for solving math problems. Being able to recognize types of problems and knowing the tools needed to solve them is important, and always try to see if there is some alternate way to go.

Accuracy Checks

Checking your results can be done a few different ways.

Simply repeat the solving steps you used to see if you get the same answer. This will often catch simple mistakes.
Try rounding off all values in the problem to get a quick estimate and see if it is close to your answer.
Substituting your answer into the original equation to see if it works can often tell you if you are right or wrong.
Sometimes just having good number sense can help you decide if your answer is reasonable. If you’re trying to come up with the radius of the earth and your answer is 520 m, that tells you that you may have messed up a unit conversion somewhere. It’s common in geometry to ask if a statement is always, sometimes, or never true. For example: A right triangle is an equilateral triangle. (Answer: never)

Explanation

Explanation is generally carried out using the well-organized modeling techniques listed above to make your reasoning clear to others. You want to show a step-by-step listing of your thinking. Adding comments can usually make it even clearer. An ultimate example of this is a formal mathematical proof.

Academic Language

When making comments, you’ll need to use appropriate language for the situation, showing an understanding of mathematical terms and procedures. You’ll also need to apply this language to your evaluation of a given argument…did the creator do it well? If not, be able to point out where the argument went wrong. It’s also related to the mathematical notation section that comes next which cautions you to be careful to use accepted, well-defined terms in your arguments.

Mathematical Notation

In short, be very careful and precise in what you write and say. One incorrect or misplaced symbol changes the numerical representation.

Relationships

Explain how a result is related to other ideas. Mathematics is logically built, layer by layer. It’s important to understand how each idea is supported by the ideas below it, and how, in turn, it supports ideas above it.