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.

Hi, i personally feel that juvenile math teaching is the best way to improve our own math skills..as the doubts they get will make us think a lot...

cheers,

suma valluru

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http://www.esumz.com

Posted by: suma valluru | September 26, 2007 at 05:23 AM

i do have daughter at the age of 4, now she started to learn plus and minus a number, but may be my problem is that i've start it with book of plus and minus. she can do it well up to 7 + 3 = 10, but as time goes she intend to forget that by pulling up 5 fingers add with 0 is becoming 5 again......

am i missing the concept, or am i introducing wrong concept.

another things, we're malay, which our native language is Malay languages but she is expose to english as early as she's born. Question is should i send her to an english school which mentosseri or just a normal school which combine english and malay medium. Reason being, in the normal school they teach more on languages reading but english school they concentrate on math and of course it will be automatically cover english languages as it come along with the english math instrustion.

your kind view is very much appreciated.

Posted by: nina | January 15, 2008 at 09:23 PM

I think it is great, Nina, that you are teaching your child math.

I would put away the book for several reasons. 1) Most 4 year olds are not really ready to write and using a book will slow the process down as the child becomes frustrated with not being able to write the answers; 2) You want to gear your teaching to your child's responses, not to some problem in a book. 3) It sounds like your child may not have a one to one correspondence with numbers down yet and a book will only teach her rote memory, not the concept of addition. 4) Teaching the concept of addition or any function is much more important than teaching by rote memory because once they have the concept down they can figure out the answer to problems they don't and can't have memorized. 5) Small children have problems with addition products above 10 and below 20, so don't even try to teach 8 + 3 etc. for now. You might want to teach 1 + 1 through 20 plus 1 as an exception.

Don't get frustrated with her inability to grasp a concept like adding zero to a number immediately. She is only four and you are teaching her things that most kids don't learn until they are eight. Give her lots of praise for the things that she learns. My Grandkids only got one math lesson a week, so don't over do this.

Math is its own language. I can't imagine that it matters which language you teach her in. Teach her in both. Once she starts thinking about doing higher level math, it might make a difference, but not at four.

I am totally unqualified to recommend schools. You will find that if you do teach your child math, the schools can't keep up, no matter which kind you select. No one expects a four year old to know math. I think that is wrong.

Keep me up on your daughter's progress. If you are patient and make this fun you will be both amazed and pleased.

I hope this helps.

Posted by: A Watcher | January 15, 2008 at 10:23 PM

Hi i just wanted to know what I am doing with my four year old is correct or not. Well at the moment i have started teaching him how to count. I basically use two colors of beans and since he knows how to write his abcs and numbers 1-20 so that's where I am working on with him now. I use a board to write and then he counts the beans first number with one color of bean and then the other number with other color and then he counts all the beans together and goes writes the answer on the board. another thing i am using is that he writes the number + another number on board and then asks me to count the beans and i purposely give him wrong answer and he then corrects me and says mom you don't know anything I am teaching you everything and gets really excited about it that he's teaching me..it's working at the moment..do you think i'm using the right method or is their an easier way?

Posted by: Hina | September 05, 2008 at 06:38 PM

I think that what you are doing is great. A four-old who can write is pretty amazing. It sounds like he thinks that what you are doing is fun, and that is the key.

Posted by: a watcher | September 05, 2008 at 06:56 PM

I am pleased to have come across your clear and enlightening posting. I like that you give actual examples and doable strategies on how to teach math. The square / square root ideas is good and I'll try that on my kids.

My questions:

1. How do you explain to young kids you can't divide by 0?

2. How do you explain the concept of infinity to a 7yo? It started with the question "What's the biggest number?" and I explain there's always 1 number bigger than whatever number he says. So he repeatedly asks "why? how is that possible?"

Thanks!

Posted by: jo | October 06, 2008 at 10:50 PM

Thanks for the complement.

The infinity question is easy. The youngest grandson loves big numbers. What is a million times a million? I taught him to count to a trillion when he asked that question. Then he wanted to count to infinity, convinced that if we could count to a trillion, we could count to infinity. I held out my hands palms inward to show him what a trillion might look like and then moved my one hand farther apart to tell him that no matter how far we counted, infinity was just a little farther on.

The zero question is a bit harder. I think it has to wait until you have taught them multi place decimals. Intuitively, it isn't so much that you can't divide by zero as that the answer is essentially the same no matter what number you divide zero into. As the decimal divisor approaches zero, the answer approaches infinity no matter the numerator.

Posted by: a watcher | October 07, 2008 at 05:40 AM

My daughter is 4 and was surprised to know yesterday that she knows how to count and she showed me how- exactly the way you describe it. We sometimes under estimate the abilities of little children! :)

Posted by: Reena | November 21, 2008 at 03:34 AM

My son is also 4 years old. We have been home schooling for about 2 months. I bought a base ten blue math set. I have a basket of 10 ones, a basket of 10 Tens, A basket of 10 Hundreds, and only three Thousands cubes. After they can count 10 Ones to you.....I will take the basket of 10 tens and have him count in tens to me. At first he will try to count them as 1, 2, 3, but I stop him and say 10, 20, 30..... So he has learned from that excersice to count in 10's.

I also play a game that involves using 10 of the ONES blocks. I count them and lay them out in front of us. He faces me (By the way). We started out by having him grab how ever many of those 10 ONES so he may grab 3. Then when he says ready I open my eyes and count how many I see. 1, 2, 3, 4, 5, 6, 7.......Then I use my fingers and i continue the count holding up fingers. ...8, 9, 10 (Having 3 fingers up) I say you must have 3).

Today we started something new..... If I have 5 on the floor then He must have 5 in his hand. He likes to trick me. because 5+5 is always 10. I am working thru different facts that add up to 10 such as .....7+3, 3+7, 5+5, etc. He really caught onto those combinations. I will introduce others such as 8+2 after a few days. He was having fun and laughing. Which makes if fun for me!

Posted by: Kimberly | March 20, 2009 at 11:52 AM