Perhaps your students’ mathematical thinking rivals the discussion in this Abbott and Costello skit. I always enjoy that one, and it’s funny in part because it speaks to our befuddlement with math algorithms. I don’t remember learning long division so I can’t comment on how well I understood it at the time. However, if I did understand the meaning behind the mechanics, it was lost over the years such that this Montessori lesson seemed exciting and new. I’m guessing that textbooks include some good graphics and descriptions to conceptually illustrate the algorithm (since I’ve not taught such young grades in a traditional setting I’m not sure what form these take), but I also wanted to share this series of pictures from my training.

A note that in Montessori lessons units, tens, and hundreds are represented using green, blue, and red respectively. Before this lesson children learn place value (including the colors) and work with division more concretely (think Cuisenaire rods — I still remember that day-glo orange). They also do a “first passage” with the board and beads in which they do all the mechanical distributing and exchanging but record only the quotient. I’ve included the complete recording so you can see how the materials lead into each step of the algorithm. Let me know what you think!

One day all children in this nation will have the opportunity to attain an excellent education.

Have a great week!

I don’t know about this, but I had a great time teaching fourth graders to do basic division 9/2, 15/5, 16/3, etc, using toothpicks. I taught lower grades, but I volunteered for a Saturday “test prep” session for fourth grade. When they started to pull out their calculators, the other volunteer teacher and I drew a line, and brought out the manipulatives. They loved it! Turns out, teachers are no longer using manipulatives to teach math after second grade, even though fourth graders still need and love it. The reason is simple: standardized testing starts in 3rd grade, and calculators are allowed on the tests. It’s very sad that teachers are allowing kids as young as 8 to use calculators instead of really understanding math. Even if the “Common Core” changes this, which I doubt it will, teachers like you (middle school) will still be dealing with the fallout from this (kids unable to do basic arithmetic without a calculator) for many years to come.

Also, our principal walked in on our tutoring session and said we didn’t have time to use manipulatives, to pull out the calculators so they could really get ready for the big test.