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November 23, 2010

Problem Solving A Primer

Nick Diaz

Recently I met the parent of a student of mine, who introduced himself and proceeded to engage me in pleasant conversation about the scope and sequence of my mathematics classes. His child’s class is now learning to write and solve some complicated equations; so he wondered what topic was next in the program.


Is it problem-solving, he asked?


No, I don’t teach a unit on problem solving; instead, problem solving is a recurring theme, not one to be taken up for a couple of weeks of school time, then laid by the wayside and forgotten. Problem solving is what any mathematics teacher, especially at the middle school level, should have students do every day, every week, every month, and every class.


Problem solving is what math is all about. Math is not about learning algorithms, or memorizing multiplication tables (incorrectly labeled “times” tables). That is arithmetic. I don’t teach arithmetic; I teach mathematical problem solving, every day, every week, every month, and every chance I get.


The goal is, and should be, to create and develop problem solvers, not blind arithmeticians – in other words, to make real students out of children, and to turn them into effective communicators of mathematics.


A concurrent goal of any mathematics teacher is to foster and encourage students to acquire and develop quantitative reasoning.


Problem solver:


1.      Someone who questions, finds, investigates and explores solutions to problems.

2.      Demonstrates the ability to stick with a problem to find a solution.

3.      Understands that there may be different ways to arrive at an answer, and applies math successfully to everyday situations.


Any parent can encourage your child to be a good problem solver by including him in routine activities that involve math – for example, measuring, weighing, figuring costs and comparing prices of things he wants to buy.


To communicate mathematically:


1.      To use mathematical language, numbers, charts, or symbols to explain the reasoning for solving a problem in a certain way, rather than just giving the answer.

2.      It also means careful listening to understand others' ways of thinking and reasoning. You can help your child learn to communicate mathematically by asking him to explain what he must do to solve a math problem or how he arrived at his answer. You could ask your child to draw a picture or diagram to show how he arrived at the answer.


I become annoyed when I ask a student to explain how he arrived at his answer, and then I must insist, in a stern tone of voice, that the entire class listen to the student in question. For some reason, it seems as if students don’t value each other’s explanations as much as they should. It isn’t only good manners, but a necessary academic habit for students to develop – listening to each other communicate about mathematics.


Mathematical reasoning ability


1.      Thinking logically, being able to see similarities and differences in objects or problems.

2.      Making choices based on those differences and thinking about relationships among things. You can encourage your child's mathematical reasoning ability by talking frequently with him about these thought processes.


Some Important Things Your Child Needs to Know About Mathematics


You can help your child learn math by offering him insights into how to approach math. He will develop more confidence in his math ability if he understands the following points:


Problems Can Be Solved in Different Ways.


Although most math problems have only one answer, there may be many ways to get to that answer. Learning math is more than finding the correct answer; it's also a process of solving problems and applying what has been learned to new problems.


Wrong Answers Sometimes Can Be Useful.


Accuracy is always important in math; sometimes, however, you can use a wrong answer to help your child figure out why he made a mistake. Analyzing wrong answers can help your child to understand the concepts underlying the problem, and to learn to apply reasoning skills to arrive at the correct answer. Ask your child to explain how he solved a math problem. His explanation might help you discover if he needs help with number skills, such as addition, subtraction, multiplication and division, or with the concepts involved in solving the problem.


Take Risks!


Help your child to be a risk taker. Help him see the value of trying to solve a problem, especially if it's difficult. Give your child time to explore different approaches to solving a difficult problem. As he works, encourage him to talk about what he is thinking. This will help him to strengthen math skills and to become an independent thinker and problem solver.


Being Able to Do Mathematics in Your Head Is Important.


Mathematics isn't restricted to pencil and paper activities. Doing math "in your head" (mental math) is a valuable skill that comes in handy as we make quick calculations of costs in stores, restaurants or gas stations. Let your child know that by using mental math, his math skills will become stronger.


It's Sometimes Okay to Use a Calculator to Solve Mathematics Problems.


It's Okay to use calculators to solve math problems – sometimes. They are widely used today, and knowing how to use them correctly is important. The idea is for your child not to fall back on the excuse of "I don't need to know math – I've got a calculator." Let your child know that to use calculators correctly and most efficiently, he will need a strong grounding in math operations – otherwise, there’s no way to tell whether the answer displayed on the calculator is reasonable.


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