Math Relearning/Number Sense
The ancient Greeks focused on geometry, but mathematics began before them with the basic numeric calculations the Babylonians and Egyptians needed for their societies to function. Math education for children today begins with teaching them what numbers are and how they generally work. If numbers are only one type of mathematical object, then there might be more than one starting point for learning about math. But since a lot of research has been done on starting math education with numbers and since numbers pervade so much of math and may even form its basis, I thought I'd start at the same place. But there will be some differences between this project and elementary school education. I'm going to start with the most basic number-related ideas I can find and build up from there, but I'll discuss these things abstractly rather than having us all do the kinds of concrete exercises children have to do to get used to the concept.
Properties of number
What are numbers, and what are their basic features that shape how we work with them? A helpful way to understand numbers is to examine the ways we use them, the way humanity developed its knowledge of them, and the ways we learn about them now. I'll refer to these factors as I go along. From my observations so far I believe the attributes of number can be broken out along several lines: the ways people represent numbers; the skills people need to work with them; the capacities humans have that enable those skills; the functions numbers serve; the methods people use to carry them out; and the properties of numbers that enable those representations, functions, and methods.
First I'll offer a preliminary, operational definition of numbers that incorporates the properties I'd like to highlight, and then I'll break it down and talk about each property and how it relates to others. Here's my definition: Numbers are terms representing stable, interrelated abstractions of quantity that can be associated with objects to identify their quantity-related attributes. I suspect this definition isn't complete or broad enough, but I think it captures the characteristics that children are taught about numbers, and so it's a convenient and adequate entry point for thinking about them and about math. The features I'll discuss are representation, quantity, association, abstraction, stability, and relatedness.
One caveat to these first few chapters is that even though the topics of number sense, measurement, geometry, and the basic operations are treated separately in education and I'll be treating them separately here, they're so closely interrelated that I have trouble talking about one concept without bringing in the rest. I think this is because although math is a highly abstract activity, our knowledge of it is rooted in our interactions with the physical world, and those interactions use all these concepts at once.
At face value, numbers are words, and at least in English, the written number takes two forms, symbolic and alphabetical. The spoken form of each written form is the same. Sometimes there's more than one symbolic way to represent the same number, and each symbolic form can be spelled out alphabetically.
As words, numbers represent things. What a number represents varies depending on how you're using it. We use numbers in three ways: quantifying, ordering, and naming. In the first case, we count and measure objects and their attributes to determine their amounts or magnitudes. This quantifying use gives us the cardinal numbers, which are the number names we use for counting. In the second case, we arrange things in some kind of order and number them to keep track of that order. This ordering use gives us the ordinal numbers: first, second, and so on. I think of ordinal numbers as a more complex type of measurement than the cardinal numbers, so I'll wait to cover them till the chapter on measurement (Measurement). In the third case, we label things with arbitrary numbers simply as a way to refer to them without having to make up new names, as in the case of player numbers in sports teams. The naming use is a linguistic use for number words rather than a mathematical one, so we'll mostly leave it behind in our discussions.
It seems to me that the fundamental, defining feature of numbers is that they represent quantities, and their other features either arise from that fact or make it useful. Quantity uniquely defines the concept of number because we don't really have another way to represent specific quantities, and most of the time when we use numbers, quantity is at least implied.
Quantity is a value that identifies how much or how many of something there is. To a certain degree our grasp of quantity is inborn. Even infants have some sense of quantity, and we recognize collections of up to three objects without having to count, a process called subitizing. Children gain a fuller sense of quantity by rational counting (see Measurement) and by comparing quantities of different sizes (see the Relatedness section below).
This is one of the first places in math that patterns come into play. A number represents the repeated experience of collections containing a particular quantity.
Numbers can be associated with things. This is a property of number that makes quantities useable. It's certainly not unique to number, though, because associating things is characteristic of the way human minds work in general. It's how we create new words, for example. This property of numbers is a basis for all three number uses--cardinal, ordinal, and nominal. I'll cover its use for cardinal and ordinal numbers in Measurement.
We can think of these things that we assign numbers to as objects, even if it's a collective object or an abstract one like temperature. It's a convenient term for the whole, enormous class of things that have the trait that something else can be associated with them. And it brings out a further feature of association, that we're very flexible in our choice of targets for our associations. For example, you can pick a single product on an assembly line and give it a serial number, or you can look at the whole group of products that were made in one day and identify their total.
The fact that these associations can be made and broken freely means that numbers are abstract, one of the many abstractions of math. You don't need to find or manufacture a new set of numbers every time you want to quantify a new object or attribute. Numbers are abstract also because there are many kinds of quantifiable attributes, and many kinds of objects have them. We'll especially start to see this in Measurement.
Numbers are made concrete and visible by the ways we represent them. We notate them using symbols and diagrams, and this helps us to work with them.
One of the math learning tasks for children that Piaget identified was number conservation, the understanding that when you rearrange the objects in a collection, it still has the same number of objects. Once you've learned this, of course, it seems obvious, but I think there's a principle to keep in mind that can be consciously applied to new mathematical situations: When you don't operate on a quantity in relevant ways, it remains the same. That is, a change in a situation may affect one quantity and not another, and it's important to distinguish between them. You don't have to re-count the quantity that stayed the same, but you also can't change the quantity with that operation.
The inverse of the stability property is what I'll call relatedness: When you do operate on a quantity in relevant ways, you arrive at another quantity. In this way numbers have relationships with each other. You could also say that individual numbers have properties or behavior and that each number has its own character.
Furthermore, there are many mathematical paths between any number and any other, and this fact makes math a powerful tool for discovery, especially when combined with the many relationships among measurable attributes. Once you know a few quantitative facts about an object, you can perform calculations to learn a lot more about it. Sometimes you'll need to find the path between what you know about the object and what you want to know about it, and that can turn math into a puzzle or a game.
Children learn a few basic number relationships as they're gaining a sense of number. One is that quantities have sizes that can be compared. Even without knowing the specific quantities in two sets of objects, they can match each item in one set with an item in the other and determine which set is larger based on which has items left over. Matching the items in two sets is called one-to-one correspondence.
Comparison is the basis for seriation, the operation of sorting objects into sequences by the size of some attribute, one of the early number tasks children learn. To order a set of objects by ascending size, for example, you compare each to the others and place it in a sequence such that each object is larger than the one before it and smaller than the one after.
Seriation is the basis for the sequence of numbers we use in counting and other operations. Even though we've all practiced the sequence of the counting numbers to the point that they're entirely natural to us, numbers don't occur in a particular sequence out in nature. There's no physical row of numbers we need to observe when we want to learn their order or discover a new number. The number sequence is a result of sorting the numbers by magnitude. We could choose a different sorting order, such as alphabetically by the number's name, but that would be drastically less useful.
Another basic number relationship children learn is what Piaget called number inclusion, that items in a set can be grouped into sets of smaller quantities and that sets can be grouped together into larger sets. Grouping numbers together to form larger numbers is also called composition, and breaking them apart into smaller numbers is called decomposition. Number inclusion is the basis for the arithmetic operations.
Just as numbers often represent sets of objects, numbers themselves can be grouped into sets. This grouping is based on various criteria.
For example, the most basic major set of numbers we work with is the whole numbers. The whole numbers are a number system, not to be confused with the numeration systems in the next section. The whole numbers consist of the natural numbers and 0. The natural, or counting, numbers are 1, 2, 3, and so on. The whole numbers have the characteristic of representing countable quantities of whole units of whatever you're counting. There are numbers in other sets that represent part of a unit, and I'll start talking about those in the fractions chapter.
As another example, whole numbers can be further classified as even or odd. You can find out if a number is even by forming pairs of items in a collection of that quantity. If you have one item left over, the number is odd. If all the items can form pairs, the number is even. So you could say an even number has the trait of being completely pairable within itself.
Our numeration system uses place value to represent a number by means of the regular groupings within it.
- Devlin 1994, 1-2
- Hatfield et al. 2008, 189
- Chapin and Johnson 2006, 2-3; Yolkowski 2013
- Butterworth 1999, 101-105
- Dehaene 2011, 56-57
- Hatfield et al. 2008, 132-135
- Devlin 1994, 9
- Devlin 2000, 74
- Hatfield et al. 2008, 127-128
- Hatfield et al. 2008, 132
- Hatfield et al. 2008, 128
- Hatfield et al. 2008, 127-128
- Chapin and Johnson, 1-2
- Wikipedia, s.v. “Number,” last modified August 4, 2014, http://en.wikipedia.org/wiki/Number
- Chapin and Johnson 2006, 9-10