Long Division: Lessons Using Montessori Materials
There are a couple of ways that division can be expressed with Montessori's racks and tubes: (1) as sharing equally and (2) as repeated subtraction.
Division as Sharing Equally
Usually, racks and tubes division is done by clearing the boards before adding beads of a lower place value. For example, the division of 9702 by 42 is typically done in three rounds that look like this:
If you choose problems carefully (the sum of the digits of the quotient must be nine or less), you can lay out the total value of the dividend divided into equal parts (remembering that each skittle on the tens board stands for ten unit skittles):
This might be a good way to start to build on the idea of sharing equally.
This method can also be used to give children a preview of the square root work:
In this example, we have 1764 divided by 42 equals 42. We can help the child notice the squares made of red and green beads and the blue rectangles of equal size that occur when we have a two digit divisor equal to the quotient.
Division as Repeated Subtraction
The previous variation on the racks and tubes lesson helps children who might have trouble seeing that equal sharing is happening when they do this work. To help them understand the pencil and paper long division algorithm, it is useful for them to see division as repeated subtraction. For example, 8942 divided by 34 could be shown like this:
Start with the 8942 set out in cups, then when you're ready to do your first clear/dump/shift, you'll have two rows on each board with three green beads and four red beads in each row. You could point to the first row on the board and say, "Here we have thirty-four hundreds. That's the same as one hundred thirty-fours." Point to the second row and say, "Here we have a hundred thirty-fours again. One hundred thirty-fours plus one hundred thirty-fours gives us two hundred thirty-fours." You could use the number cards with this work. Take out the 200 card to keep track of the two hundred thirty-fours you've taken away, then clear/dump/shift. When you're ready to do the next clear/dump/shift, you'll have six rows of thirty-four tens. You point out that each row is ten thirty-fours, so there are sixty thirty-fours all together on the boards. You take out a 60 card and clear/dump/shift. Finally, you end up with three rows of thirty-four (each row has three blue beads and four green beads). You take out the 3 card and put together the 200, 60 and 3 cards to show that you have taken away 263 34's all together.
The next step after this is learn the traditional notation for the long division:
I expect the experience with the Freudenthal notation will help kids better understand why we are doing the subtractions we do and why, e.g., the 2 goes in the hundreds place, the six goes in the tens place and the 3 goes in the units place.
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