Basic Concepts of Number
I was talking to my niece, when she was very young, and she said her father had just turned 34. Trying to be a good uncle, I feigned interest. She then added that he had been 33 before that.
I did not have to pretend to be interested in that. She thought it was informative to tell me he had been 33 before 34. As far as she knew, maybe he could have been some other age.
Seizing the teaching moment, or just wanting to know more, I asked if her father had ever been 32. She thought for a long time, then answered uncertainly, "Yes."
Had he ever been 31? More long thought, and a little more confidence, "Yes".
Had he ever been 17? "Of course", she said quickly, as if she was saying the most obvious thing in the world and I was the idiot for asking.
There is a principle here. It's hard to put into words, but to get from 0 to 34, you have to do all the numbers in between. I like to think that she learned that principle from my questions. So that it was obvious by the time we got to 17. Or maybe it was just obvious to her that 17 was less than 34 and not as obvious that 32 was less than 34.
Density of the Number Line
Are there any numbers between 4 and 5? I suppose it depends on what you mean by a number, but clearly there are values between 4 and 5, and for the sake of discussion I will say there are numbers between 4 and 5, even if they don't have names. One day I was working with a math-phobic high schooler. I was trying to explain fractions, and I realized he did think of the number line as having numbers between 4 and 5. He knows that if I am 53 now, I was once 27 years old and once 28 years old. So far so good. Now, are there any ages between 27 and 28?
I am sympathetic to the answer of no. When someone is 27 years old, they say they are 27 for a whole year, then they suddenly start telling people they are 28. On the other hand, we don't go a year saying that a baby is 0 years old. The space between 0 and 1 is divided into months, and if the child is young enough, weeks, or days, or even hours or minutes.
At older ages, it is a little trickier. I taught my daughters to say that they were 3 and a half. I think they conceptualized 3 1/2 as a new integer lying between 3 and 4, so one might count 1, 2, 3, 3 1/2, 4, 5. I also naturally used modifiers for their, such as almost 4 years old, and just-turned 3. These imply ages that are not integers.
Back to the math-phobic high schooler. I introduced months, and he agreed I was once 27 years and 8 months, and I was once 27 years and 9 months. Then I asked if there were any ages in between these two. There might have been a language problem, but he said no, definitely not. So then I introduced days. Going through the same process, he agreed that I was once 27 years, 8 months, 11 days, and I was once 27 years, 8 months, 12 days, but there was definitely nothing in between. As you might expect, hours came next. He kept saying there was nothing in between until we got to minutes. Then he could think of seconds by himself. However, he then said there was definitely nothing between 27 years, 8 months, 12 days, 13 hours, 5 minutes, 37 seconds and 27 years, 8 months, 12 days, 13 hours, 5 minutes, 38 seconds. But he immediately recognized his error when I added a fraction (27 years, 8 months, 12 days, 13 hours, 5 minutes, 37 1/2 seconds), and then we went into decimal places.
The goal here is to help the student understand the nature of the number line. In mathematical terms, I think the concept is that numbers are dense. These words make no sense of course until you first understand the concept, so it is very difficult to teach by explanation. Put another way, I am trying to teach the existence of the real numbers. We assume that students know this, because it seems so obvious. But apparently it is not. You could do the same exercise with lengths, and it would work especially well if the student was comfortable with the metric system.
When my daughter was 3 or 4, I let her climb the ladder. After a while of doing this, she was on the ladder and asked my how many more steps she could climb. (We had established that there was a limit to how high I would let her climb.) I said 3 steps. She went up a step and asked how many more steps she could climb. This was a little exasperating, but it was math, and if she was asking, it must not have been boring to her. So she must be learning something, though she was probably just solidifying a concept that was already pretty solid. So I said 2 steps, she climbed a step and asked how many more steps she could climb, I said 1 step, and she climbed one more step.
Then she asked how many more steps she could climb. In a sense, this question was mathematical. Having established a pattern, it is mathematical to try to continue a pattern past the point it makes sense. For example, we teach people what it means to divide by 17, and then we divide something by 17.1. Tell me that makes any sense. Well, I can continue a pattern as well as anyone. I said "Zero steps."
She gave me a look as if to say sarcastically, "Right dad. What a joker." She knew she wasn't supposed to climb any more steps, and she didn't, even though I had just told her there were more steps to climb.
The very deep concept is this. The simplest numbers are the counting numbers -- 1,2,3,4,5,7,8, etc. They do not include 0. Zero is a difficult concept, even though none and nothing are simple concepts. I try to explain it this way. If you have one apple, I can see that you have 1 apple. If you have zero apples, I cannot see that. How do I know that it's apples that you have zero of? How do I look at you and say you have zero apples? For all I know, you might have zero oranges, or 0 giraffes. Or, put another way, you have zero of so many things, why focus on apples?
So, to my daughter, zero was a number, and if it was a number, then I was saying she had more steps to climb. I don't know what sense she then made of it -- maybe she was trying to imagine climbing zero steps. (Really, I assume that she ignored my instruction, but if she did act it out, how would I know?)
I don't really know how you teach the concept of zero. Everyone seems to get it, so my guess is that you don't have to teach it. My older daughter explained to my younger daughter that zero means nothing or none. But if you have to understand it that way, I am not sure you get it.
I did think of one exercise for teaching about zero. My daughter came in to show me how high she could jump. I feigned interest. Then I asked her, how low can you jump. She made a small jump, and I asked her if she could jump even smaller. Well, this exercise has one of two possible outcomes. The first is that finally they are jumping so small that they really are jumping as small as they can and still be jumping off the ground. You can keep asking, but maybe that's the best they can do. The other ending is that they have the insight to not jump at all. Perhaps not jumping at all doesn't count as a jump -- like I said, zero isn't as good of number as the other numbers. So, while you would hope a child does not jump at all, you have to be sympathetic to a child who doesn't count that as a jump.
You can play this game with almost any activity. It naturally generalizes to other quantities, but it can also be for smiles. We actually developed this game, so that I would ask my daugher to show me how high she could jump, then how high her older sister could jump (with the answer being a very poor jump.) Or how big they could smile, or whatever -- show me how polite you can be; show me how polite your sisten can be. It's a fun game, and great for creativity, and I suppose things like smiles and politeness allow exploration of negative numbers (a frown or being rude).