If I gave you a pile of cards and asked you to divide the cards into three equal piles, you could count the cards, divide that number by three, then make up three piles with that final number of cards.
In fact, I suspect most students would do exactly that.
But you could also just deal the cards one-at-a-time to three piles.
This is what I will call a "natural operation". You are not dividing in the sense that it is taught in school. But this still could be called division. In fact, if you made up a realistic story problem involving division, I think it would be difficult for you to avoid having the problem be almost exactly like this.
(Technically, your typical story problem would be: "John has 100 pieces of candy that he wants to divide equally among four children. How many pieces would each child get?" But the real life analog to this problem would be that John has a pieces of candy that he wants to divide equally among four children. John does not care how many pieces of candy he started with or how many pieces each child ends up with -- as stated in the problem, all John cares about is that the candy is divided equally.)
I have written elsewhere that there is something fundamentally wrong if a student can add 2 + 3 but cannot say how many apples you would have if you started with two and then someone gave you three. There is a real-world operation, called "combining", which corresponds to the mathematically operation of addition. (Or more than one -- combining lengths is a little different from combining apples.)
Students can learn to add without ever thinking about the connection between addition and combination. And they can do fine on addition. But it's not right, and they will then have trouble with story problems. (Story problems should come first, and students should never be taught the key-word method of doing story problems.)
So this is teaching the underlying operations that correspond to the mathematical operations of addition, subtraction, multiplication, and division.
I came at this from a no-counting point of view. Perhaps for that reason, I use that wrapper. I start out saying that I am going to teach the students how to not count. Then I wonder aloud at where "not" is supposed to be there -- aren't I supposed to be teaching the students to count? Oh no! I prepared to teach them to not count. Then I worry that the "not" is really a squished bug, or someone added it. Then I think of a solution -- I find out from the class that they already know how to count.
Of course, if they can't count, then they can't know the numbers. If they don't know the numbers, they have to use "natural" methods. So this could have been approached from a "no-numbers" point of view.
After my initial exercise, which is a counting exercise, I say that they can't count any more. If they need to count, they should raise their hand and ask me for permission. When they do, I tell them no.
The Initial Counting Exercise
I start out with a problem that is naturally solved with counting. Everyone can solve it, minus the occasional counting error. At this point, you can have them try to define counting. THIS IS AN EXCELLENT EXERCISE.
Of course, they are likely to gush an answer, then be satisfied with that answer. You can spend a lot of productive time working on this, because counting is a sophisticated concept. For my class, I put some objects on the board then counted them, getting an obviously wrong answer, but using a procedure consisten with what they had on the board.
This is also useful later if you get into techniques that are in between counting and not counting. Essentially, "matching" is a natural procedure, but counting relies on matching.
Which is Greater?
The simplest operation is equality/inequality -- there is a collection of A's and B's. Are there more A's then B's, more B's than A's, or are they the same?
This is my first exercise, which of course is easily and naturally counted.
A A A B B B
A A A B B
B B B
A A A A B B B B
A A B B B
A A B B
Sorry if this doesn't fit on your computer screen. When I print it out, all the A's are on one side and all the B's are on the other side.
Then I give them this:
This of course is easily solved without counting, and most students will do that. If you have let students choose how to do this, you now have to deal with the students who thought they had to count. I didn't do this, which was wrong. Probably the best way is to slowly run them through the problem row by row, until they see that A and B are going to always be tied until they get to the last row.
I then asked how they solved this. This is again the standard exercise, and it too is very worthwhile.
Then I have four problems on determining equality/inequality. Depending on age, some or all of your students can jump right to the hardest problem.
A B A B
A B A
B A B
A B A B
A B A B
B A A B
B A B
B A A
B A B
B A A B
A B A B
A B B A A
A A B B
B B A
A B A B
A B B B
B A B A
I also asked the class the question if there were more people or noses in the classroom. I then had a contest -- counting versus not counting. One student was given the task of answering my next question with counting. The other students had to answer without counting. The question was whether there were more ears or eyes in the classroom. It was no contest.
The contest idea will work for any of the other operations described below, though it might not always win.
The Other Operations
I start with division -- subtraction and multiplication are harder, and addition is so simple a student might not think to ever realize that the correct answer is correct.
Division. They are given a pile of cards. They need to divide that pile into three equal piles. Cover story: They are passing out apples to 3 needy families.
A correct method is to deal the cards into three piles, one at a time. For any of these, there may be other correct methods, at least in theory, but I suspect for at least this one, everything is going to look roughly like this.
This was pretty difficult for third-graders -- only two out of 16 bright third-graders could solve this problem. Sixth graders who had done similar problems did better, though I don't know how much better.
Addition. They are given two (or more) piles of cards. They have to make a pile of cards equal to the sum of the two piles they are given.
The correct method is to combine the two piles.
Multiplication. The problem is, given a pile of cards, produce a pile of cards with three times as many cards. One solution is to produce a 3xn rectangular array. Another is to deal the cards in to n piles, and do this three times. (This requires counting to three, but since three is a part of the problem, I think this should be allowed.)
Subtraction. Given a large pile of cards and a small pile of cards, produce a pile of cards equal to the large pile minus the small pile.
By this point, this problem might be easy. From the large pile, produce a small pile equal to the given small pile. The remaining cards in the large pile is the answer.
I did not get into it, but a related problem with a little counting is this. There is a pile of cards. Someone takes five cards from the pile. How can you restore the pile to it's starting value, if you don't know what the starting value was?
How many A's are there? Determine without counting.
A A A A A A A A A A
There's a trick. If you put each finger on each A, you will get a perfect one-to-one match between the A's and your fingers (including your thumbs as fingers). Given that you alreay know you have 10 fingers, you can know there are 10 A's.
As an aside, lot's of people used to use the finger trick. Enough that this problem
A A A A A A A A A A A
Could be identified as "leave one" and this problem.
A A A A A A A A A A A A
Could be identified as "leave two." These phrases have evolved into our modern words "eleven" and "twelve".
What about this?
There are two A's, and people can perceive this without consciously counting. Psychologists call this ability subitizing. Is this counting? I don't know. I called it counting and didn't get into the issue.
Advanced Determining Number
I had a pile of pennies. The problem was to determine how many pennies there were without counting them. I also gave a student a piece of paper with the numbers 1 to 30 written out in large letters. He put each penny on the numbers 1 to 12 and told me there were 12 pennies.
I don't think this is counting. But it is very close to being like counting. The "closeness" occurs because counting itself relies (I am pretty sure) on the one-to-one matching principle.
I think it would be profitable to have the student/class describe the procedure for using this method. Also, the method changes if the paper contains the numbers 0 to 30, or if the numbers on the page are not in order.
The same problems can be given for length (or presumably any other physical measurement that has the operations of addition, subtraction, multiplication, and division).
One of the really interesting things about length is this. If you have a ruler, and count up the number of feet in an object, you are "counting" -- this is analogous to counting the A's. But you line up the ruler against an object and proclaim that the object is 8 inches long, you have not counted. You have used a method analogous to putting pennies on numbers (I think).
Relationships to other Topics
I think this is a pre-requisite to doing the Base-10 Operation problems.
The skills used here are probably the same as doing operations in the tally system of representing numbers.