So that I don't have to repeat my introduction, you may want to read this first.
One of the nice things about blogging is the ability to choose one's subject at will. Today's subject is about my experiences with juvenile math. I've come to the conclusion that it is easier to teach very young children math that it is to teach them to read. Even so, my essay is not intended as a "curriculum" for small children, but only a set of observations based on a very small sample of children (two).
This is being written in the first person because it is a first person account. What has worked for me might not work for someone else.
I am fortunate enough to have three grandsons. Equally fortunately, all of them are pretty bright. None of them are geniuses, or if they are, it is not obvious. If you don't think your child or grandchild can begin learning math at an early age, you might be wrong. I will explain how I taught math concepts to a now six year old first grader and how I am now teaching them to a four year old.
I started the program after observing my middle grandson's interaction with his older brother as I began to teach the oldest grandson math concepts at age six. I suddenly had a four year old competing with the six year old to shout out the answers to the questions I was asking, and the four year old was right more often the six year old.
Every bit of what I write about will be concept math. For example, I don't teach multiplication tables. I teach how those tables work. If they understand how the operation works, they can figure out the answer without resorting to memory. That is not to suggest that they don't eventually need to memorize the tables, because they do.
The first thing they need to learn is one to one correspondence (the number one corresponds to one finger, one penny, etc.), and that can easily happen at about age three. Some people don't like to teach children to use their fingers when they are doing math. I have discovered that if you teach a three year old to use his fingers, he will have no need of them by the time he is five. Kids and grownups carry around ten fingers, so why not put them to use?
Dice are great manipulatives to teach one on one correspondence.
The best place to teach math is when a child is in a car seat. They get bored, and math relieves the boredom. Have them count while the adult in the front passenger seat holds up fingers. That teaches one to one correspondence. Once they seem to have it down, hold up fingers and ask them to tell you how many you have up. They will likely count them in the beginning, but that is exactly what you want them to do.
The next step is addition. Keep it simple in the beginning by adding 1 to numbers. Again hold up the fingers so they can count the answer, for example, two fingers on one hand and one on the other. Here is where it gets tricky. You want them to learn that 1 plus 2 is the same as 2 plus 1, so tack that skill on once they understand that adding one to a number produces the next number. Develop a habit of asking the questions just that way. Once they understand that 5 plus 3 is the same as 3 plus 5 you will discover that while they may have to count on their fingers for 5 plus 3, they won't for 3 plus 5, which helps them memorize their addition tables. It becomes a game.
Don't forget that zero is a number and is represented by a closed fist. What is 3 plus 0?
After they have simple addition down, keeping the product numbers to 10 and below, you can move on to modified two digit addition. That is, adding something like 23 plus 3.
Before you can do that, you need to teach them two digit numbers. Teach them to count by tens to 100, again holding up a finger for each ten spot. I also taught them to count by hundreds to a thousand at the same time and in the same way. Fingers are wonderful. Sometime later, teach them to count to a million the same way, first by thousands to 10 thousand, then by 10 thousands to a hundred thousand, and finally by hundred thousands.
Do not try to teach them to add in situations where either the problem or the answer is within the teens because those numbers are not intuitive. Both grandsons knew that 29 plus 4 is 33 long before they could deal with 9 plus 4. The youngest at 4 and a half still has problems with teens. The six year old grew out of it. As an exception, do teach them 2 plus 2 through 10 plus 10. They don't seem to have a problem with those "teen" numbers.
To teach this modified two digit addition, first ask what 3 plus 3 is. They should know that the answer is six. Then ask what they think 23 plus 3 is. 33 plus 3 etc. After you have done this for a while, start working with them in the tens place. 10 plus 23. 20 plus 23, etc. When they have that down, it is time for 23 plus 23. Then move on to problems that involve carrying. While I described this last part in one paragraph, you probably will find that the whole process takes six months to teach. Again, there is no hurry.
Four big hints: Don't wear this out. I only see the grandkids about once a week, so this wasn't an every day thing. When a child starts giving wrong answers to problems you know he should know, stop immediately. Keep the pressure off. You are teaching them things that other kids won't learn for years, so there is no hurry. One of my grandchildren asked me one day if math was a game, and that was at a time when he could do multiplication at age 5.
The second is always start each session with the exact same problem. Currently, the four year old is getting 23 plus 3 as the first problem. He knows the answer, and he knows he knows the answer. That gives him confidence to work with the next problems.
The third is always ask them if they want a hard problem or an easy problem and then give them what they ask for according to their level.
The forth is to encourage them to use what they learn. If you see them using a calculator to do operations that you know they know how to do, tell them to do it in their head. Most games are based on math, so playing games is really good for reinforcing their skills.
You will discover that there will be other opportunities to teach math. We attended a party last week and about 9 pm I asked the 4 year old if he would like to go outside and do math. We worked for about 15 minutes on it and he only missed two problems. As we left, I told him that I wanted him to think about what 3 plus 3 plus 3 was for next time He said immediately, it's nine! His mother later told me that on the way back to the party he announced "and 3 plus 3 plus 4 is 10."
Once a child understands the concept of addition, teaching subtraction is very easy. Again, avoid answers in the teens in the beginning. Do teach negative numbers at this point. The language that works best is what is 3 take away 1?
None of this is done with pencil and paper. All of it is mental. You do have to work with pencil and paper eventually. Not long ago, I started using pencil and paper with the four year old. I wrote down the number 33. Before I could ask him what it was, he said "six." It drove home the point that he hadn't yet learned to recognize the written form of two digit numbers, and that he had never seen a plus sign. Since he is a year from kindergarten, we probably have time to correct this deficiency.
This system works, at least for me. By the time our middle grandson was in kindergarten he could do addition, subtraction, one digit multiplication and modifications of it, division, squares, square roots of perfect squares up to 144, simple prime numbers, and word problems involving all of the above. He also had a full grasp of the negative part of the number line. More importantly, he understood the underlying concepts and though he didn't have the high part of his multiplication tables memorized, he could compute the answer to the part he didn't have memorized.
To put what I have written in context, schools expect children entering kindergarten to be able to count to five. He just entered first grade and tested at the 3.9 grade level in math overall and 4.5 in some areas.
I want to really emphasize that I taught concepts without worrying about whether he had his tables memorized. In fact, I felt free to move on to the next concept long before they had their tables memorized. Memorization is drudgery and it comes with time and usage anyway. Math concepts can be, and really should be a game.
Next time I go off subject, I'll write about teaching multiplication. Again, I teach concepts, so a child doesn't need to know all of his addition facts before we start thinking about multiplication.
If you come to this site only to see essays about this project and want to skip the political stuff, I created an "education" category for it. Just look in there.