How to Teach Elementary Math

342 Pages·2009·4.68 MB·English

How to Teach Elementary Mathematics

© 2009 Christopher Thomas

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Table of Contents

Introduction 4
Chapter 0 Mathematics 5
Chapter 1 The Natural Numbers 6
Chapter 2 The Number Line 12
Chapter 3 Cuisenaire Rods 16
Chapter 4 Equality 19
Chapter 5 Inequalities 22
Chapter 6 Basic Addition 26
Chapter 7 Basic Subtraction 35
Chapter 8 Units 40
Chapter 9 The Base Ten Number System 45
Chapter 10 The Addition of Large Numbers 49
Chapter 11 Basic Geometry 55
Chapter 12 The Subtraction of Large Numbers 64
Chapter 13 Models of Multiplication 74
Chapter 14 Multiplying by Powers of Ten 86
Chapter 15 Basic Areas 92
Chapter 16 The Multiplication of Large Numbers 99
Chapter 17 Mental Math 110
Chapter 18 Division 116
Chapter 19 Dividing Large Numbers 121
Chapter 20 Factoring 131
Chapter 21 Multiplying and Dividing Factored Numbers 137
Chapter 22 Multiplying and Dividing with Units 142
Chapter 23 The Symbols Game 148
Chapter 24 Fractions 156
Chapter 25 Reducing Fractions 168
Chapter 26 Comparing Fractions 175
Chapter 27 Adding and Subtracting Fractions 181
Chapter 28 Proper and Improper Fractions 191
Chapter 29 Multiplying Fractions 197
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Chapter 30 Dividing Fractions 204
Chapter 31 More Areas 212
Chapter 32 Decimals 222
Chapter 33 Adding and Subtracting Decimals 228
Chapter 34 Multiplying and Dividing by Powers of Ten 237
Chapter 35 Multiplying Decimals 244
Chapter 36 Long Division with Decimals 250
Chapter 37 Large Numbers and Scientific Notation 262
Chapter 38 Circles 271
Chapter 39 Word Problems 280
Chapter 40 Conversions 288
Chapter 41 Ratios, Proportions, and Percents 294
Chapter 42 Probabilities 301
Chapter 43 Angles 305
Chapter 44 Similar Triangles 317
Chapter 45 Volumes 325
Chapter 46 Planar Nets 333

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Introduction

This book has been written as a resource for elementary school teachers. The concepts
and calculations of basic arithmetic and geometry are laid out in a gradual, step-by-step fashion.
Emphasis is placed on how to explain the usual procedures of arithmetic with toys, games,
pictures, exercises, and illustrative examples. The general goal is to give students very simple
and basic exercises to conduct and discuss. In the process of the class discussion (prompted by
the teacher, as necessary), the hope is that the class will notice the patterns that form the basic
rules of mathematics. When children make discoveries like these, they develop interest,
excitement, and a very solid understanding of the underlying principles. The key is for the
teacher to use exercises and examples that most clearly illustrate and explain the usual
procedures and algorithms of elementary mathematics.
The ideas in this book are not intended to replace a teacher's curriculum. Every teacher
has certain favorite examples, exercises, and games that give character to his or her teaching
style and should not be abandoned. Also, there are a nearly endless variety of different types of
mathematical problems. No book would be capable of covering all the possible problems and
situations that an elementary school teacher might face. Instead, this book focuses upon the core
basics of elementary mathematics: whole numbers, addition, subtraction, shapes, multiplication,
division, fractions, decimals, area, and volume. When students have a solid understanding of
these basics, their teachers will not have much difficulty teaching them more advanced topics
and problems that might appear in textbooks, exams, or real-world situations.
There is no reference to grade levels or ages in this book. The only requirement for each
section is that the students be comfortable with the preceding material. In an ideal situation,
those children who find the material easy should be allowed to move on to more complicated
mathematics. Those children who struggle should be taught from the last place where they fully
understood the material. Of course, few educators find themselves in ideal situations. It is
hoped that this book will at least provide some ideas and insights for those front-line warriors
who daily struggle with the challenges of elementary education.

I would like to extend my gratitude to all of those without whom this work would not be
possible. My mother taught me enthusiasm and daring. My father taught me patience and
reasoning. Richard Lavers and Ernst Fandreyer showed me the fun of teaching math. Loring Tu
taught, by example, how to write a book. Mauricio Gutierrez and Zbigniew Nitecki taught me
the necessity of hard work. Most of all, this book would not be possible without Mahesh
Sharma, who taught me the area model of multiplication, the components of the concepts of
whole numbers and fractions, and many other ideas which were essential for this book.
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Chapter 0: Mathematics

Mathematics is the study of relationships. Arithmetic studies the relationships among the
numbers, the way numbers can be combined and compared. Geometry studies the relationships
between lines, shapes, lengths, areas, and volumes. Algebra is nearly the same as arithmetic, but
with some of the numbers unknown and replaced by variables. Trigonometry studies the
relationships between the angles and lengths of triangles. Calculus studies the relationships
between the slopes of curved lines and the areas beneath them. In more advanced mathematics,
the objects which are compared and related can be extremely abstract, but the overall process is
much the same.
First we describe the objects and properties which we are about to study. Next, we talk
about how they can be compared. We try to find natural ways to combine the objects. At every
step, we see if it is possible to undo what we just did, if we can reverse each process of
combining things. Finally, we make entirely new objects and start all over. This, in a nutshell, is
mathematics.
If mathematics is the study of relationships, then it would be most interesting to jump
straight to the most important and useful relationships of all: human relationships.
Unfortunately, human beings change a lot and are quite unpredictable. It is hard to say anything
certain or definite about all people, much less make predictions about what an individual will do.
Thus, we will leave the study of human individuals to psychology and the study of groups of
people to sociology. Instead, we will begin mathematics, the study of basic relationships, with
objects that are basic, universal, and do not change. We begin with numbers.

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Chapter 1: The Natural Numbers

Our first objects of study will be the natural numbers, also known as the counting
numbers, which begin with 1, 2, 3, 4, and so on. When you teach children about numbers, there
is something important to remember: numbers are at the same time very weird and very natural
and innate.
Why are numbers weird? Well, what are they made of? Are numbers made of wood or
metal? Do numbers have weight or height? Can we pick up a number? Can we sit on a
number? The answer to most of these questions is "no." Numbers, we see, are not physical
objects, and so they are quite different from cats and buttons and most of the other usual things
in our lives. If we want to explain what an awl is, all we need to do is find one and then show
how it can be used for making holes. If we want to explain what the number 5 is, however, we
cannot bring one out because it is not a physical object.

This leads to a place where mathematics is often taught incorrectly. Many
people confuse the number five with the symbol that represents it. Each of the symbols
illustrated on the left represents the number five (with a Hindu-Arabic numeral, a
Roman numeral, and tally marks respectively), but none of them is actually what the
number five represents. A number is a concept, and should not be confused with the
manner with which it is represented.

We shall get into the meaning of a number shortly, but first let us look at an
analogous situation. Love, like the number 5, is a concept and not a physical
object. You cannot weigh love in pounds, cannot measure its height, or any other
such thing as that. Suppose a person told us that he knew all about love and that it
was the object illustrated on the right. We would immediately realize that this man does not
really know what love is. Sure, he might be able to read a sentence written like "I ♥ New York,"
but he does not fully understand the concept.
The concept of love can take a very long time to properly understand. Rare are the
teenagers who understand the extent of their parents' love. Couples who have been married a
long time often say that there was much they did not know about love after only five years of
marriage. This is natural because love is a complex and abstract concept.
Fortunately for teachers, recent studies have hinted that numbers are innately known and
understood, even by infants. Newborn babies were shown cards with various numbers of objects
on them. At regular intervals, the cards would be replaced. The researchers kept track of how
long the babies paid attention to the cards before they looked away. It turned out that the infants
paid the most attention when the number of objects on the cards was changed. A baby shown a
card with three dots, for example, would pay more attention to a subsequent card with two dots
than a card with three rectangles or a card with three dots in a different arrangement. Similarly, a
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baby shown a card with two dots would react more significantly to the subsequent sound of three
drumbeats than to the sound of two drumbeats.1

The most natural conclusion from these experiments is that people are born with an
innate ability to recognize different numbers. If babies did not have the ability to discern the
difference between small numbers, they would not be able to react to these differences. A small
percent of people, however, seem to be born without these basic and innate mathematical
abilities. Just as people with dyslexia read more slowly than most people, people with
dyscalculia have impaired number sense and take much longer than most people to count the
number of dots on a page.
It is for these reasons that we can say that numbers are weird (they are abstract concepts
made out of ideas and not tangible objects) and yet natural and innate (people are generally born
with the ability to recognize the differences in small numbers). Now, let us examine what
numbers actually are. For the sake of simplicity, we will use the word "number" to mean
"natural number" up until the point where we begin to examine fractions.

A natural number is a concept that has three parts:
(1) A number is a property of a group of objects.
(2) Two groups have the same number if their objects can be paired up.
(3) Numbers can be put in order.

The first aspect of numbers is that every group of objects has a number. Thus, given any
group of objects, it is reasonable to ask "what is the number of these objects?" Note that the
number is a property of all the objects put together and not something individually owned by any
one of the objects. The group of circles below has a number, as does the group of triangles and
the group of squares.

1 "Easy as 1, 2, 3." The Economist 30 Dec. 2008
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Analogously, the property of being lined up is another group property. The triangles are
lined up and the squares are lined up, but the circles are not. This property of being lined up is a
collective one; if even one square were moved up or down, then the group of squares would no
longer have the property of being lined up. Similarly, there is no circle in the group that has the
number 5; the number 5 is a property shared by the whole group.

The second aspect of number is very intriguing. It states that even though every group
has a number, some of these numbers are the same. Two groups have the same number if we can
take the objects in the two groups and pair them up completely, with nothing left over in either
group. One way to pair objects up is to connect them with lines. For example, the circles and
squares above have the same number, because we can pair them up as follows:

The circles and triangles, however, have different numbers because however we try to
pair them up, there will always be one circle left over:

The number five, thus, is not the shape of the symbol "5," but a property shared by all the groups
of objects which can be paired up exactly with the group of circles pictured above.

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If we were to take a picture of a group of objects, rearrange them, and then take another
picture, there is a natural way to pair up the objects in the two pictures: draw a line from where
each object started to where it ended up. For example, suppose the bottom two circles of the
above group were moved to the right as illustrated:

We could then draw lines to indicate where the circles began and ended:

This is a natural way to pair up the objects in the two groups. We can conclude that moving
objects around does not change their number. This might sound obvious, but it is an important
aspect of the concept of number. There are some children, for example, who feel that the squares
below have the number 5, but the circles do not. Such children must be taught to separate the
idea of arrangement from that of number.

The third and final aspect of numbers is that they come in an order. The smallest natural
number is the number 1. You cannot have a group of fewer objects (it may be debatable if 1
object really constitutes a group, but 0 things are certainly not a group). Whenever you add one
more object to a group, you end up with the next larger number. In this sense, we develop the
natural numbers:

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It is curious that this last concept of a number is often the very first which is taught.
Before teaching a child what a number is (a property of a group) or what it means for two groups
to have the same number (they can be paired up), most parents teach their children how to count.
It is very valuable for a child to learn how to count, because it teaches the names of the numbers
and reinforces the concepts of order and sequence. However it is also important for a child to
learn the other two aspects of the concept of number.
In order to teach the idea that a number is a property of a group, have the child count
objects. This is a learning exercise that can occur anywhere – in a car, in a room, while reading a
story, or anywhere. How many birds are in this picture? How many people are in this room?
How many cars can you see? By counting things like this, the child will be able to connect the
names of numbers with groups of objects.
When a child is good at counting the number of objects in a group, try reversing the
process. Have the child separate out a given number of objects from a pile. For example, have
the child give you 6 pennies from a large pile of coins. Similarly, you can have the child ask you
for a certain number of objects (give me four buttons) and then correct you if you are wrong
("no, that is five buttons. Here, let me show you four..."). You could also show a child a number
of different groups of items (some pencils, some paper clips, some buttons, etc.) and then ask the
child to tell you which was a group of eight. As an extra challenge, you could set up the piles so
that there were several correct answers, working to show the child that it is possible for several
groups of objects to have the number eight.
There are a number of tricks to teaching a child about the pairing concept of numbers.
One is to have the child count a number of objects, then move the objects around and have the
child count them again. Similarly, you can have a child count a row of objects, first from left-to-
right, and then from right-to-left. Ideally, after a few tries the child will announce that there is no
need to keep counting the items, that the number will remain the same. Such a child has just
made a fundamental step forward in understanding numbers.
Another way to develop the pairing concept of number is to give have a child compare
two groups of items, each one too numerous for him or her to count. For example, give a child a
large box with empty bottles and bottle caps, then ask whether the number of bottles and caps are
the same. Ideally, the child will think to screw the caps on the bottles. If there are screw caps
left over, then there are more caps. If there are bottles left over, then there are more bottles.
Otherwise, the numbers are the same. A simlar exercise would be to show a child a large picture
with many people on it, most of them wearing hats, and ask the child if there are more hats or
more people. Even if there are too many people to count, a child ought to be able to compare the
numbers of hats and people. These are ways to reinforce the pairing concept of number. While
it initially might help the child if the items go together (bottles and bottlecaps, hats and heads),
the child should eventually be able to compare two large groups of unrelated items (for example:
pennies and popsicle sticks) by pairing them up.

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