*"The ability to think is the most valuable skill we can pass on to our students, for more important than any particulars of knowledge. If our goal is to let students think for themselves, we must constantly change how we pose problems"*I start by getting them to take a handful of tiles and put them on a pile in front of themselves on the table and count them. Then I ask them to take a second pile, keeping this pile separate from the first pile. Count how many are in that pile.

Push the two piles together and count how many there are all together.

Once we have done this a number of times, they begin recording them.

He now is creating his own addition problems. He is free to create problems of any size that he wishes. He can advance at any rate he feels comfortable. I tell him that if he has created enough two-pile problems, he can begin making up problems that use three handfuls. There is no special praise or no pressure to advance to the next level, so he can form his own contentment at his own level of success.

The suggestions of other things to do are just to keep the interest alive. Adding with three, four of five piles are just variations on the same theme. Suggesting that he predict how many there are all together can also be another variation.

*"The more we allow our students to rely on their own intelligence in finding answers to problems, the less difficult teaching mathematics or any other subject becomes."*

There is no guarantee that he can count accurately enough to produce correct answers to the problems created, but it is more likely that the answers will be correct if the emphasis is on the

*making and doing of the problems*and not the finishing of the page or chapter or book. He begins to own and take pride for his problems and his answers.

After he seems to tire of addition problems, I introduce subtraction problems in a similar way. I get him to take a large handful of tiles and put them in front of him. Count them. This time instead of taking another handful of tiles, I get him to take some of the tiles in front of him into his hand. Count these and put them back in the box of tiles. Count the ones remaining. Once this procedure is learned, he can begin recording.

First list the whole pile, then the ones taken away and held in the hand, a line and then those remaining.

Once he tires of making and recording subtraction problems, then I ask him to make up addition and subtraction problems alternately. If it has not come up before, we talk about the difficulty in telling right away whether they are addition or subtraction problems and use this opportunity to teach the use of plus and minus symbols.

Once he has done this for a period of time, I suggest a game in which we take a number, 9 for example, and add 2 more, and then keep adding 2 more and recording the numbers. We then look at the answers for patterns. Do the patterns repeat themselves? Are they odd or even numbers? How many numbers are in the right-hand column before the pattern repeats itself?

These can be done for a while, choosing different sets of numbers of his choice.

Quentin wanted to add some multiplication and division problems with 5 as well, so we did this, too. |

Once we tire of this, we do another game in which I pick a number and he tells me what the problem could have been to get that particular answer. Can you come up with a different problem that could get this answer? Is 5 plus 3 the same as 3 plus 5? These can also be written down.

Once we tire of these games with the materials we have been using, I bring out different material. He then gets to see that all that he has learned about tiles can transfer to cubes and that it is not tied to the materials themselves. All of this is done at

*his*own pace.

(These activities, of course, are not all done in one sitting, but over several months of sittings. This method of teaching and all quotes are from

*Mathematics...A Way of Thinking*, Robert Baratta-Lorton)

For more advanced work, go to

*Advanced Addition and Subtraction.*

Totally OT, but I passed the Versatile Blogger Award onto you and your blog. If you'd like, you can check the post on my blog.

ReplyDeleteI love this post! It's amazing how you can take the same thing but approach it differently and it's far more engaging. Plus, it uses different thinking skills. Left to think creatively, I've notice children go beyond what is in traditional workbooks. I love all the quotes you've added. It paints a complete picture.

ReplyDeleteI like the use of thinking before memorization in math.

ReplyDeleteI agree with An Almost Unschooling Mom. I love the emphasis on *doing*. He is learning how to figure it out, not just memorizing facts. This is such a fantastic post. I'm going to pin this so I can reference back. Thank you so much for sharing this!

ReplyDeleteI really think the best learning happens when the student is self-motivated. My daughter is extremely interested in math at the moment. That's what she is thinking all day and she teaches herself multiplication for fun :)

ReplyDelete