The divison board is one of Maria Montessori's mathematical masterpieces. I say that because she has made division work so absolutely understandable that it questions any other method of initially learning this work. The board has a green colored heading. The box is divided into two sections - one is for skittle-like "people" and one is for counters. Both are also green. There are grooved spaces for up to nine skittle-like "people" and 81 counters.
Initially, during the first lesson, I introduce the child to the division board and the skittle-like "people" and the counter beads. Also, I give the child a divison tables booklet which can easily be made on the computer. Examples of these equations are as follows: 2 divided by 2, 4 divided by 2, 6 divided by 2, the last equation being 18 divided by 2. I read several equations re-introducing / reminding the child what the division sign looks like (they would have already done division with the golden bead materials and with the stamp game) and I give them the name of the division symbol - obelus.
I make it a point to tell the children that mathematical notation is read like sentences are read. I also tell them this so as to remind them to show their work so that at any time I may stop by their table and read the specific problem they are currently working on. I feel very strongly that children should learn earlier on to show their work and that it should become routine for them.
At this stage of the work there are no remainders. But, the children quickly figure out the pattern in the division tables booklet and either fill in the answers without using the board or don't want to do the work as it is either too easy or too boring. So I have blank sheets that I use for division on which I write a list of equations. I must remember not to give an equation that can not be answered on the board. Ex. 99 divided by 9.
The equations that I write have no pattern and therefore the children must be very focused on completing the one at hand and to read the next one so as to do the work correctly. I find creating these individual sheets works the best. The children will often say that the work is hard or challenging which is what older five year olds and 6 year olds want. They want the other children in the classroom to see that they are doing difficult work. This is part of the profile of a child about to graduate from the primary classroom.
After a child has worked on this material once or twice, I introduce the concept of a remainder. That may be sooner than my album states but again I find that it works the best. I make an extra box next to the one in which the child writes the answer to the equation. Notation in this box is always done with a red pencil. At the top of the page, just above the placement of this extra box I write a red r. This r is for remainder.
I really didn't catch onto the concept of a remainder when I was first learning this as a child in second or third grade. But the language is the key! In the picture below note that there is what looks like a brown circle - the limitation of this photo - this is actually a small wooden bowl like the one used with the golden beads bank game. If the child is given the equation 9 divided by 4, they place four skittle-like "people"at the top of the board in their specific places and then take nine counter beads and place them in the bowl. Carefully they give each of the "people" the same amount, no more - no less. Therefore each of the four "people" would have been given two counter beads. The answer in division is what one gets. That is what we tell the children.
Then they note that there is one bead left in the bowl - now what?! Here is the beauty of Maria Montessori's mathematical mind -she didn't just give teachers this great board material for lessons on division, but she (because she listened so carefully to the dialogue of her students and of their questions regarding math that she knew that how the method of the work is explained, the very phraseology that the teachers used, was intrinsicly important to the child's ability to master the work) gave us the very language to use for the child to master divison - so we too look in the bowl and see the one bead left there and then look at the child and say so clearly - "What remains in the bowl is the remainder - r." The child generally lets out a little laugh. They find that explaination a relief and that it simply makes sense. Next I show the child how to use the red pencil to write 1 in the box to the right of the answer box. The equation now reads 9 divided by 4 equals 2 r 1
This child is solving the equation four divided by two. The child initially placed two skittle-like people at the top in the designated area of the heading. Then they put four counter beads in the bowl. Now they give the beads to each skittle remembering the first rules: In division the answer is found by giving each "skittle" or "person" the same amount, no more no less. The answer is what one gets.
After reading several of their completed equations, I introduce the language for division - if I have not done so earlier with other materials - I take a singular problem and name the parts as I would in giving a three period lesson:
64 divided by 8 = 8
64 is the Dividen and 8 is the Divisor.
Also after several difficult equations, I show them that they may check their work using the division boards (the ones with the problems and answers listed - they are rectangle in shape and larger than the above spoken board), but that multiplication is also the opposite of division and therefore may be used to check division problem answers. The above equation of 64 divided by 8 = 8 can be easily checked using the multiplication bead bars or any material that will allow the child to take eight eight times.
I will post additional photos to give a larger overall demonstration of the work next week - we are on Spring break all this week. Good Luck!
PS - Materials purchasing note - The division board and the multiplication board are two of the least expensive math materials - My school purchased my classroom's through Appleseed Montessori Materials. Each board was under $60. Save your pennies and buy them. They are worth it!