Yesterday, I was working with a fantastic group of primary elementary school teachers and our topic of discussion was building number sense. Many defined number sense as understanding what a number is and also knowing how it is relates to other numbers. I was thrilled to hear that their answers centered around deeply “knowing” a number. I then shared with them this quote from the Arithemtic Teacher:
“Number sense is good intuition about numbers and their relationships that develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Howden, 1989)
Feeling good about their working definition of number sense, I asked them to use a ten frame to build 7 using two color counters. Here is what all but one teacher in the room created:
I was intrigued by this observation. After our discussion about number sense, I anticipated that there would be many different ways of making seven in the room. Yet only one teacher made seven differently:
At this point, I had enough formative assessment data to know that these teachers were ready to build some flexibility with this tool. I first asked one of the teachers who filled the top of the ten frame first to share her strategy with the group. As she described how she built her seven she stated that she “followed the rule” of filling in the top of the ten frame first before moving to the bottom row. As many of the other teachers in the room nodded their heads in agreement, one replied in jest, “Did you get a rule book with your ten frames?” Laughter ensued…
I then asked the teacher who had filled the ten frame in differently present her solution. It was evident that “breaking the rule” made several teachers in the room nervous. We began to talk about the importance of students “seeing” the 5 in a ten frame, as that is a valuable benchmark for students in this base ten math system. I then expressed that students can use this same ten frame tool to “see” all the hidden partners within a number. In connection to this, I shared this quote from Standards for Math Practice (SMP) 5:
“Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.” (CCSS-M, 2013)
The spirit of the standard indicates that we require mathematically proficient students to make decisions about which tool to use and how to use it. With this new information in the forefront, I proceeded to ask the teachers to build 7 again on their ten frames, pushing them away from the 5 frame structure and into a different composition. It was amazing how quickly they became comfortable with using this tool flexibly. They then walked around the room comparing their composition to others, determining how they were the same and how they were different from each others compositions.
As we debriefed our learning from this task, we determined that there were three strategies for using the ten frame: Building 5-Wise, Pair-Wise, and Otherwise.
We also determined that their is benefit to teaching students multiple ways to use a tool in order to build greater number sense. Walking away from this day of learning with these teachers, I found myself thinking, “SMP 5 for the win!”