I have many friends with school age children who know that I work in the field of math education and are always eager to pick my brain on what their children are doing in class during math instruction. Since the implementation of the common core standards, a question I get asked many times is, “Why are they teaching them all these weird strategies for addition and subtraction?” (i.e. decomposing numbers, making tens.). A few years ago I would have answered that question with a response like, “Teachers want to make sure that students have a conceptual understanding of addition and subtraction before they take it to the abstract level of the standard algorithm.” But as I get further into my work as a Research Associate and read the work of Dr. Les Steffe on how children develop their understanding of number and looking at their understanding through the perspective of radical constructivism, I feel I need to revise my answer.
If we look at one common core standard for first grade in the domain of Operations and Algebraic Thinking it states:
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 -1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g knowing that 8 + 4 =12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 +1 = 12 + 1 = 13).
What I find interesting is that we in education have interpreted the words “such as” to “must teach” and have started teaching these strategies to students in such a step by step manner that they become empty procedures that have no meaning for students. Much like learning standard algorithms, students learn these new strategies as procedures and still have no better conceptual understanding of addition and cannot explain why these strategies work.
What if we look at the standard from a different point of view? What if the strategies called out were ones that students constructed on their own with just the right tasks and prompts posed by their teacher? The research that we are studying suggests that you cannot directly teach the strategies called out in the standards to students until they are at a certain point in their development of understanding number. Even at this point it is more meaningful if students construct these strategies on their own.
So my newly revised response to the question posed above actually is, “Does the strategy they are using make sense to them or is it just a step-by-step procedure they are plugging into? In other words, “Is it my strategy they are learning or their own strategy they are constructing?” What do you think? I would love to hear your thoughts!