Before going to the big topic of “why teach math?” I wanted to write a post about my most recent focus, mathematical fluency in relation to basic arithmetic. At first glance, one might think Montessori manipulatives used for arithmetic such as the Stamp Game build this fluency. However, their role is more about enabling the child to do the algorithms abstractly, i.e., on paper without memorization. They are not geared towards promoting the type of fluency that allows a child to quickly add, say 5+7. (I think there are other great benefits to these manipulatives as well and I hope to dig into them in the future).
So what, within Montessori, addresses this? With my background in a Children’s House, I knew of the Snake Game, Addition Strips, Multiplication Board, etc that are geared towards memorization. I think these materials, along with the bead chains and bead bars can start to really build the foundations of a basic numerical fluency. However, I was working with children coming from both a non-Montessori background and a Children’s House and both were struggling. For instance, a Montessori child had trouble quickly adding two to a number. This was a child who had counted the long 8 chain, finished the Addition Strip Boards, and more. I saw patterns like this repeated with non-Montessori children as well.
This isn’t saying something is missing from the Children’s House necessarily. My guess is that it will typically be rare to see the sort of arithmetical fluency I think is desirable developed at such a young age regardless of the work they do. The question I’m asking is: considering these Children House materials are not typically in an elementary and the manipulatives that are aren’t geared towards memorization like, say, the Addition Strip Board is, what is the best path forward? Should I work to get the Children’s House material or will it not be appealing to the 2nd plane child? Can some of the material typically in an elementary, such as the bead chain, be used instead? I’m just now starting to experiment.
Initially, I tried flash cards. However, I think that even with children with great sequential counting skills, proficiency with 4-digit+ addition, and lots of skip counting experience, approaching fluency as a rote memorization drill seems inappropriate after my initial observations. It’s too easy for the child to just memorize something like 9+5 as an isolated fact. If she does, even if she can answer what 9+5 is immediately, if you then ask her what 9+6 is she would falter unless she had memorized that fact as well.
A different approach I’m now looking into would be to teach simple strategies. In this case, one might show how when adding single-digit numbers to 9, you can take “1” from the other number to make “9” a “10” making it a simpler problem. The child learns one strategy and can now quickly compute many math facts. He or she might even be able to apply this same strategy to adding 8. There are many strategies like this that provide the child with a more fundamental approach to math facts. I also think being able to quickly and creatively apply these strategies is more in line with what “arithmetical fluency” actually means. Further, seeing how these strategies are useful algorithms that can be generalized in new ways, I could also see this approach help the child begin to think more creatively algorithmically, one key outcome of learning math in my opinion. Rote memorizing is appropriate in some cases but it can rob the child of learning important strategies to derive the answer if used prematurely.
Do materials need to be used to teach strategies? For some of these strategies, being able to get some perceptual representation of the problem is important. The so-called “Ten Frame” is what I’ve come across so far to aid in this. It is similar to the Addition Strip Board but can more easily demonstrate certain strategies.
Along with this approach, I’m also looking at how to experiment with some of the Children’s House material when I can get my hands on some. At some point, I hope to provide an update on how the different approaches are working.