August 26, 2015 12:21 World Scientific Book - 10in x 7in main page 1 Chapter 1 Counting and Natural Numbers The main subject of elementary school mathematics is numbers. Children’s acquaintance with numbers is a long and gradual process. Their first encounter is with natural numbers, which are those numbers used for both counting and ordering. Over the next few years, children discover fractions, which are numbers that represent parts of a whole, and can be represented as quotients of natural numbers. In middle school, the world of numbers is expanded to include negative numbers as well. Together, whole numbers and fractions, both positive and negative, constitute a set of numbers called rational numbers. During their middle school years, children also get acquainted with irrational numbers, which are numbers that cannot be represented as quotients of whole numbers, e.g., 2. This completes the children’s acquaintance with the set of numbers known as real√ numbers, which are the numbers with which we express results of quantitative measurements. In high school, teenagers pursuing advanced mathematics classes will be exposed to an even more expanded set of numbers, known as complex numbers. The focus of this chapter is the central role of natural numbers in quantifying the size of sets of objects. 1.1 The natural numbers 1.1.1 Development of the number concept Counting has become such a basic skill for us that it is hard to appreciate its underlying complexities. The historic process leading to the development of the contemporary number concept is tens of thousands of years old. To the best of our knowledge, human’s first use of numbers occurred about 30,000 years ago. Long before humans invented numbers, they had to develop some manner of abstraction that would allow them to identify and name objects. Indeed, without identifiable objects there is nothing to count. Let’s assume that there had already been a word for describing an object such as a sheep, and prehistoric men wanted to tell their friends that they had seen two sheep. They could repeat the word for sheep twice. But what would they do if they saw five sheep? They could say sheep and sheep and sheep and sheep and sheep,but that’s awkward, and it becomes even more so if one wants to talk about a whole herd of 1 August 26, 2015 12:21 World Scientific Book - 10in x 7in main page 2 2 Elementary School Mathematics for Parents and Teachers sheep. This mundane example clarifies why, from the moment there were nouns, i.e., words that classify objects into di↵erent categories, there was a need to quantitatively describe the presence of more than one object of the same category. 1.1.2 Matching between sets Consider the following figure, which displays two sets of objects, one kind contains bees and another kind, flowers: Imagine that numbers have not yet been invented. What does the set containing bee, bee, and bee have in common with the set containing flower, flower, and flower? Can you say it without using the word three? This is not a trivial question, especially if we look for an answer that is general enough to remain valid if the bees were replaced by rabbits, and three was replaced by twenty-six. The answer is illustrated below: Every object in one set has been matched to a single object in the other set, without any leftover objects in either set. In mathematics, such a matching between sets is called a bijective matching. In this book we will use less formal language: we will either say August 26, 2015 12:21 World Scientific Book - 10in x 7in main page 3 Counting and Natural Numbers 3 that there exists an exact matching between the two sets, or that the two sets can be matched. With this new concept in hand, we may now define what it means for two sets to be of equal size: Definition 1.1. Two sets of objects are said to be equal in size (or have equal size) if there exists between them an exact matching. Please note that this way of determining whether two sets of objects are equal in size does not require counting nor assigning numerical values to the size of those sets. It also does not depend on the nature of the objects in both sets. Let’s return to prehistoric men. It is quite possible that even without being aware of the concept of exact matching, prehistoric men used to perform such a matching between sets of objects and the fingers of their hands. Thus, for example, if they said the word sheep and put up three fingers at the same time, their listeners would infer that they were describing a group of sheep that can be matched with the fingers shown. This practice would eventually pave the way for the invention of a word expressing the quantity three. The word three replaces putting up three fingers. 1.1.3 Counting What do we do when we count objects? What is the meaning of the process in which we point at a collection of objects one after another and mumble: one, two, three,...? August 26, 2015 12:21 World Scientific Book - 10in x 7in main page 4 4 Elementary School Mathematics for Parents and Teachers What we do, without being aware of it, is to create an exact matching between the objects in the set and a set of words—numbers—that we learned to recite as young children. These numbers form a sequence—the number sequence—meaning that they have a fixed order. When we match objects in a set to those numbers, the numbers have to be invoked one after the other in this order, until we have exhausted all the objects in the set. The figure below exhibits a set of objects of the apple kind matched to elements from the number sequence. By the end of the matching process in which every object has been matched to a number, the last number invoked represents the size of the set (the formal mathematical term for size is cardinality). Set of apples: Number sequence: one two three four five six On natural numbers and counting (1) A sophisticated invention: The numbers we use for counting are called natural numbers. German mathematician Leopold Kronecker (1823–1891) is credited for the famous saying that natural numbers are the creation of God, whereas all the other numbers are the creation of mankind. (2) Connecting counting to cardinality: Counting is usually learned at a young age. Yet, children need quite some time to internalize the fact that in order to count correctly, they have to account for all the objects in the set, and that each time they point to an object, they should say one, and only one, word from the number sequence. Young children sometimes count correctly without understanding the meaning of their action, that is, without realizing that by counting they are in fact quantifying the size of a collection of objects. (3) An essential skill: Counting is the first mathematical skill that children should acquire. This skill must be practiced and honed all along the early school years. One shouldn’t underestimate it, for it is the basis for all future mathematical learning. (4) Numerals: Numbers have names as well as symbolic representations: both the name of a number and its symbolic representation are called numerals.In August 26, 2015 12:21 World Scientific Book - 10in x 7in main page 5 Counting and Natural Numbers 5 addition to the skill of counting, children need to make the connection between a set of objects, the numeral that expresses the size of that set, and its symbolic representation: three 3 (5) Counting at-a-glance: I highly recommend the development of the skill of identifying, in the blink of an eye, the size of sets that include up to five or six objects. Beyond enhancing the child’s confidence, which is in itself an important goal, this skill becomes useful when learning to perform mathematical operations within the range of zero to ten. This ability can be developed through games in which children are briefly shown sets of objects, and then asked to determine how many objects are in each set. (6) Order does not a↵ect cardinality: When counting objects, the result is independent of the way the objects are positioned, or of the order in which we choose to count them. This may sound obvious to an adult, but needs to be clearly pointed out to children. (7) Abstract versus concrete: Numbers are abstract entities, independent of the concrete objects that they quantify. One of the central themes in mathematics is the abstraction of concrete situations. Abstraction allows us to reason about reality with better efficiency and clarity. This gap between the concrete and the abstract is one of the main hurdles a child has to overcome. For example, when children are required to find the sum of three plus two, we expect them to know that the outcome is independent of the objects to which these numbers refer. This skill is often taken for granted, but in fact, it takes a high level of intellectual maturity to treat numbers independently of the objects they quantify. 1.1.4 The number zero Zero is a number that describes the size of an empty set—a set that contains no objects. The number zero has an interesting history. Western culture, influenced by Indian mathematicians, adopted zero as a number only in the 12th century CE.