The Mathematics That Every Secondary School Math Teacher Needs to Know

Alan Sultan, Alice F. Artzt

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781315391885
• Edition: 2
• Topic: Education
• Subtopic: Teaching Mathematics

Designed to help pre-service and in-service teachers gain the knowledge they need to facilitate students' understanding, competency, and interest in mathematics, the revised and updatedSecond Edition of this popular text and resource bridges the gap between the mathematics learned in college and the mathematics taught in secondary schools. Highlighting multiple types of mathematical understanding to deepen insight into the secondary school mathematics curriculum, it addresses typical areas of difficulty and common student misconceptions so teachers can involve their students in learning mathematics in a way that is interesting, interconnected, understandable, and often surprising and entertaining. Six content strands are discussed—Numbers and Operations; Algebra; Geometry; Measurement; Data Analysis and Probability; and Proof, Functions, and Mathematical Modeling. The informal, clear style supports an interactive learner-centered approach through engaging pedagogical features:

• Launch Questions at the beginning of each section capture interest and involve readers in learning the mathematical concepts.

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• Practice Problems provide opportunities to apply what has been learned and complete proofs.

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• Questions from the Classroom bring the content to life by addressing the deep "why" conceptual questions that middle or secondary school students are curious about, and questions that require analysis and correction of typical student errors and misconceptions; focus on counter intuitive results; and contain activities and/or tasks suitable for use with students.

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Changes in the Second Edition

• New sections on Robotics, Calculators, Matrix Operations, Cryptography, and the Coefficient of Determination

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• New problems, simpler proofs, and more illustrative examples

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• Answers and hints for selected problems provided

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Chapter 1
Intuition and Proof

1.1 Introduction

The history of mathematics is replete with examples where observation and intuition led mathematicians to correct conclusions. However, there are just as many cases where it led to incorrect conclusions. For example, for many years mathematicians believed that there was only one kind of geometry—Euclidean. That proved to be false. They also believed that negative numbers had no meaning. Yet you know from your studies that negative numbers are essential in real-life applications.

As a secondary school student, you were probably only given the correct final results, like a negative number multiplied by a negative number is a positive number, and not made aware of the bumpy path it took for results like this to be discovered. This most likely left you with the false impression that mathematics evolved in a systematic way in which mathematicians created only correct results. To get a true understanding of the work of mathematicians, and the need for proof, it is important for you to experiment with your own intuitions, to see where they lead, and then to experience the same failures and sense of accomplishment that mathematicians experienced when they obtained the correct results. Through this, it should become clear that, when doing any level of mathematics, the roads to correct solutions are rarely straight, can be quite different, and take patience and persistence to explore. We begin this process by exposing you to some of the instances in history where intuition led mathematicians astray and give you a chance to test your own intuition on these problems. Hopefully, by the end of this chapter, you will understand why proof is so important. These types of situations are what account for the present-day rigor that is part of today’s mathematics curriculum. In the second section of this chapter you will experience the variety of methods that you can use to either prove that your own mathematical observations or intuitions are correct, or possibly even incorrect!

1.2 Can Intuition Really Lead Us Astray?

Launch

Evaluate the expression N² + N + 41 for integer values of N from 1 through 5. Do you believe that this expression represents a prime number for all positive integers, N? Justify your answer.

Most people who see this problem for the first time easily verify that the expression is prime for each of N = 1, 2, 3, 4, 5. After checking N = 6, 7, 8, …, 20 and seeing that we still get prime answers, our intuition starts to kick in and tells us “Maybe this really is prime for all values of N.” How can we be sure? How many cases must we take before we know with certainty? We will re...