Why are there so many numbers?

Where are they?

In my math tutor practice I constantly answer questions. I usually get a lot of questions from my students. All kinds of math related questions. Some of them are very common, meaning, I get them all the time. For example, Calculus and Pre-Calculus students very often ask: “What is the domain?” “How do I find the domain of a function?” The vast majority of standardized test takers ask questions to the effect of “Why do I have to re-learn all this math stuff?” “When am I ever going to use it?” Sometimes I get questions that make me laugh, like: “How can you stand so much math? My head hurts!” and “Do you also have a real job? Or, is this all you do?”

Recently a student asked me a couple questions I found just fascinating. We were going over some algebra rules. I started giving her some numerical examples to illustrate one of the rules. All of a sudden my student asked: “Why are there so many numbers? Where are they?” These are great questions! They get to the root of the concept of number. Just think about these questions for a moment. “Where are they?” Where are the numbers? It is almost like asking: “Where do numbers come from?” or even “How can I believe numbers really exist? Can I see them?” It is clear numbers are not physical objects but we use them to count physical objects all the time. You believe in the existence of something concrete, like cars, because you can easily see them (often in big numbers), but what about numbers themselves? Do we ever get to see a number? What we usually call numbers, like street addresses, or numbers in license plates, or ID cards, or page numbers in a book, all those are not actually numbers but numerals, the symbols we use to represent numbers. Numbers are in our mind. They are concepts, ideas, thoughts, more than things. My answer to these questions was along the following lines:

Numbers are everywhere. We do not see the numbers but we put numbers on the things we see. Numbers show up as soon as you are able to tell differences and similarities. Think about counting the chairs in this coffee shop, for example. When you count the chairs you do not count the tables, or the bookcases, only the chairs. So you count them because they are equal, they are all chairs. However, you do not keep pointing your finger at the same chair while going 1, 2, 3,.. You count that chair and immediately you go on to the next chair for the next number. So you count them with different numbers because they are all different chairs. What makes counting possible is our ability to identify a set of objects that are all equal, in a sense, yet different, in another sense. So we can “see” numbers when we look at the stars, at the grains of sand in a beach, or the cars in a highway, and so on. As long as our mind sees the world in terms of “equal” and “different,” numbers will be there, everywhere.

Dealing with the difficulty of memorizing products of digits higher than five

Advantages of using Mayan and Roman numerals

When it comes to memorizing the multiplication tables, each child has his or her own pace. Some children find it easier than others to memorize the lists of numerical facts that make up the multiplication tables. Others need longer practice periods, maybe with the help of flash cards. Rote memorization is not everybody’s best act. Some children find these dry memorization exercises burdensome.
Here I offer a suggestion to help third and fourth graders who are not into rote memorization, to gain a better grasp on multiplicative manipulation of the higher digits.
The main idea is using the distributive property of multiplication over addition to calculate the result of a product of two “big” digits by breaking one of the “big” factors into smaller ones, doing two smaller multiplications, and adding up the partial results at the end.
The essential keys for this approach to be successful are:
First of all, do not even mention the phrase “distributive property.” That’s a big no-no. Do not do it. Show by example only. At this stage children do not need to know there is such a thing as a distributive property. It would be a waste of time. Avoid the confusion. Just do it. Show them how it works but do not try to explain why. The best way to understand why it works is for them to see how it works, period. Do not say aloud any abstract names like “property,” much less “distributive.” Stick to the numbers.
Second, use a standard “breaking scheme.” The number five is an excellent stepping-stone in this process. It is very natural for a variety of reasons. Mainly, because five is half of ten, the base of our numerical system, plus we have five fingers in each hand, so it is very easy to break any higher digit as the sum of five plus a lower digit. Children accept this fact very easily. Furthermore, the multiplication table of five is one of the easiest to remember.
Third, –and this is very important– spend enough time (at least half-an-hour) in a preparation period showing them, or reviewing with them, how to write numbers using Roman numerals and Mayan numerals. Do this before getting into any multiplication practice. This specific type of preparation has a dual purpose. On the one hand, it lowers their anxiety level. You have to understand they are under pressure. Their parents are at least concerned, maybe even worried. That is why they hired a private tutor in the first place. The child knows he or she is not doing great in the class when it comes to memorizing the multiplication tables. Some children may be even beginning to have some dents in their self-esteem, thinking that perhaps there is something wrong with them, or that they are not good at math, or whatever. The main idea in their head at this time is “multiplication is hard.” So when you –the expert– come along and start working with them doing lists of Roman and Mayan numerals, they go “Oh! This is really not that hard. This is easy.” Some children actually have fun with these numerals. Some prefer Roman numerals, while some prefer Mayan ones. The main point is now they are relaxed, at ease, and working with something they understand much better than the monolithic multiplication tables. That is the first goal of this preparation. The second purpose is for them to realize, or remember, or reinforce the idea of just how natural is the use of the number five as a breaking point, or a stepping-stone. Both Roman and Mayan numerals make heavy use of the number five, and of multiples of five, in this fashion. They consistently apply the principle of expressing higher digits as sums of five plus a lower digit. After doing this work with Roman and Mayan numerals, children are so much more receptive to the idea of using the number five as a standard "break point."
So this is the ideal moment for you to start practicing multiplication with the higher digits in this additive fashion. Here I give just one example to illustrate the main idea:
7*8 = 7 * (5 + 3) = (7*5) + (7*3) = 35 + 21 = 56
Do not expect them to know what to do. Guide them with questions. You are supposed to pause and ask as you write:
“Eight is five plus what? Three? Is that correct? O.K.”
“Now, how much is seven times five? Yes, thirty-five, perfect! Thank you.”
“And, how much is seven times three? Yes, twenty one, very good!”
“So, now we just add those two numbers. Can you please add 35 + 21 ? Thanks.”
Just by listening to you asking them these questions and watching you as you write down this multiplication process step by step, they get it, they understand it, and they end up empowered by knowing they can get the right result by themselves even if they do not have memorized the result, even if it takes them a little while doing it in steps like above. They are now much closer to self-sufficiency when multiplying higher digits.