# 32 Mathematical Ideas: Problem-Solving Techniques

Jenna Lehmann

**Solving Problems by Inductive Reasoning**

Before we can talk about how to use inductive reasoning, we need to define it and distinguish it from deductive reasoning.

**Inductive reasoning** is when one makes generalizations based on repeated observations of specific examples. For instance, if I have only ever had mean math teachers, I might draw the conclusion that all math teachers are mean. Because I witnessed multiple instances of mean math teachers and only mean math teachers, I’ve drawn this conclusion. That being said, one of the downfalls of inductive reasoning is that it only takes meeting one nice math teacher for my original conclusion to be proven false. This is called a **counterexample**. Since inductive reasoning can so easily be proven false with one counterexample, we don’t say that a conclusion drawn from inductive reasoning is the absolute truth unless we can also prove it using deductive reasoning. With inductive reasoning, we can never be sure that what is true in a specific case will be true in general, but it is a way of making an educated guess.

**Deductive reasoning** depends on a hypothesis that is considered to be true. In other words, if X = Y and Y = Z, then we can deduce that X = Z. An example of this might be that if we know for a fact that all dogs are good, and Lucky is a dog, then we can deduce that Lucky is good.

**Strategies for Problem Solving**

No matter what tool you use to solve a problem, there is a method for going about solving the problem.

*Understand the Problem:*You may need to read a problem several times before you can conceptualize it. Don’t become frustrated, and take a walk if you need to. It might take some time to click.*Devise a Plan:*There may be more than one way to solve the problem. Find the way which is most comfortable for you or the most practical.*Carry Out the Plan:*Try it out. You may need to adjust your plan if you run into roadblocks or dead ends.*Look Back and Check:*Make sure your answer gives sense given the context.

There are several different ways one might go about solving a problem. Here are a few:

**Tables and Charts:**Sometimes you’ll be working with a lot of data or computing a problem with a lot of different steps. It may be best to keep it organized in a table or chart so you can refer back to previous work later.**Working Backward:**Sometimes you’ll be given a word problem where they describe a series of algebraic functions that took place and then what the end result is. Sometimes you’ll have to work backward chronologically.**Using Trial and Error:**Sometimes you’ll know what mathematical function you need to use but not what number to start with. You may need to use trial and error to get the exact right number.**Guessing and Checking:**Sometimes it will appear that a math problem will have more than one correct answer. Be sure to go back and check your work to determine if some of the answers don’t actually work out.**Considering a Similar, Simpler Problem:**Sometimes you can use the strategy you think you would like to use on a simpler, hypothetical problem first to see if you can find a pattern and apply it to the harder problem.**Drawing a Sketch:**Sometimes—especially with geometrical problems—it’s more helpful to draw a sketch of what is being asked of you.**Using Common Sense:**Be sure to read questions very carefully. Sometimes it will seem like the answer to a question is either too obvious or impossible. There is usually a phrasing of the problem which would lead you to believe that the rules are one way when really it’s describing something else. Pay attention to literal language.

*This chapter was originally posted to the Math Support Center blog at the University of Baltimore on November 6, 2019.*