I received the following e-mail from a parent; my response follows it:
I am a parent of a 7 year old second grader. My child is having trouble with math. He knows how to add, and subtract. When it comes to determining what half is when you use symbols such as circles or objects, he is fine. But when it comes using his fingers to divide (eight) equally he gets so confused. I am almost at my wits end. Please give me some tips on how to help him. If there is anything that you can do to help me, it will be much appreciated.
Dividing "8 fingers equally" is not an easy thing because they are on two different hands, and especially if he is looking at five fingers and three fingers instead of four and four. Also, it is possible he does not know what the word "equally" means in case you happened to use that with him, as you did here. You may have to explain it means "the same amount in both groups" or some such. But my guess is that the three and five combo on the hands is what is difficult AND that the notion of dividing quantities of things into fractional groups is also not clear to him -- notice that when you are talking about dividing quantities into groups, sometimes half of something will be 8, but half of a different group might be 4, or 100, or 2 million. Or half of 16 is 8, but then half of the 8 is 4, and half of the 4 is 2, etc. AND those quantities are also WHOLE things in themselves. E.g., a stack of 4 can be either one stack or a half of a stack or a fourth of a stack -- depending on how you are thinking about it; and that is not an easy concept because it is screwy if you think about it. Kids are thinking about it, because they notice it is screwy since it doesn't make sense to them at first. This is not true with dividing cookies, or other objects, because a "half of a cookie" looks different from a whole cookie. But when you divide quantities you don't get something that looks like "a" half of something else.
I would think the easiest way to teach this is by using stacks of something like poker chips -- poker chips are the best for this and pretty inexpensive, but if you have a problem with the notion of "poker" chips because they could be associated with gambling, you will have to find something else that stacks in a similar way or that is easy to group (such as M&M's, but I would prefer the poker chips to chocolate so that it doesn't become a food thing, rather than a math thing). Last time I looked for poker chips, Wal-Mart had them and they were less expensive there than the few other places I was able to find them.
Have him count out four poker chips and stack them in one column. Then have him divide the column in half to make two equal columns next to each other (i.e. "two columns that are the same -- equal") and see how many poker chips there are in each. If that goes okay, have him do the same thing with eight and then sixteen and then possibly thirty two chips in one column. Then using sixteen or thirty two you can, if you wish, go on to show him about fourths and eighths, though be careful about that. Don't try to make him absorb too much at one time. Remember all this is new and strange and difficult. You are just used to it because you have had years of practice, but it was likely not easy for you at the beginning.
Give him lots of practice with the poker chips. Then show him some that won't work -- like with odd numbers of chips, or with 10 chips, so that he can get one divide but no more without having to break a chip (you might want to go to cookies then or something else breakable).
You might want to look at my website about fractions and how difficult the concept is at http://www.Garlikov.com/math/fractions.html
And I presume you have found me in the first place from my web page about teaching math to young children, where I talk about introducing children to (the concept of) fractions in an easy and fun way: http://www.Garlikov.com/math/TeachingMath.html
By the way, the poker chips will be useful for teaching all kinds of math concepts, particularly one of the most difficult for children: place value. I describe using them to teach place value in a long web page: http://www.garlikov.com/PlaceValue.html When you go to that page, if you just want to see the method, do an "Edit" search for the phrase "poker chips" and read that paragraph and the ensuing ones. But I would recommend reading the whole article if you have the patience because it explains about teaching concepts in math in general.
Let me know whether the poker chip stacks work with the fractions concept.