Structuring Numbers
How to use these materials
The following videos are NOT short clips that introduce a specific instructional activity that a teacher can then quickly implement. Rather, they are dense and require time and thought to understand and implement. We recommend that viewers watch each video in its entirety and then go back to study it in more detail before making use of the information in their classrooms. In our experience, teachers often work in teams or with a teacher partner to incorporate the instructional sequences into their instruction. In doing so they create a community in which they can share ideas and explore questions and issues that arise when they implement the sequences with their students. Such a community can be particularly important for teachers that have not previously provided mathematics instruction that makes heavy use of class discussions as an essential feature of children’s mathematical learning.
The list of videos below form the instructional sequence Structuring Numbers (SN) that is designed to follow the P&P sequence in the first grade year. Aspects of this instructional sequence are also appropriate for grades K and 2.
The accompanying print materials include templates for visual and print materials needed to implement the Structuring Numbers instructional sequence. In addition, we provide detailed notes to accompany each video. These sets of notes include essentially the same information as the videos and can be read independently of the videos. They can also be used as hands-on print materials to easily review the information provided in the video.
The instructional materials described in these videos and the accompanying print materials form sequences and, as such, should be used in the order listed below. Accordingly, the videos should be viewed in that same order. Videos later in a sequence assume the viewer is familiar with information provided in the earlier videos.
Videos
Video Group 1: Sequence Overview and Structuring Numbers Part I – Becoming Efficient With Reading the Rack
This video group provides an overview of the entire instructional sequence and a detailed description of Part I of the sequence, “Becoming Efficient With Reading the Rack.” There are three segments that comprise this set of videos.
Video Segment 1a
This segment begins with an overview of the entire Structuring Numbers instructional sequence. This overview is followed by a detailed description of the first two activities of part I of the sequence. The first of these is the introductory activity that includes introducing the arithmetic rack, introducing the scenario that underlies the sequence, and establishing the conventions of using the arithmetic rack. The next activity is the first one in which children use their racks. We describe the activity, its purpose, and how children engage in it.
Video Segment 1b
This segment is devoted to a detailed description of an activity that is used many times in Part I, “Figure out how many people are on the bus when the configuration is specified.” It also includes details about what children do. A partner version of the activity is described as well.
Video Segment 1c
This segment describes activities designed to foster mental imagery. Children use visual images of the rack that the teacher flashes to figure out how many people are on the bus, without having their own racks to use.
Video Group 2: Structuring Numbers Part II – Becoming Efficient with Adding and Subtracting Using Grouping Strategies Prompted by the Rack
In Part II of the sequence children solve problems that involve people getting on or getting off the bus.
Video Segment 2a
This segment includes the initial activities of Part II. It includes extensive details about various ways children use the beads to solve the tasks posed and explains how these ways form the basis from which children develop grouping strategies.
Video Segment 2b
This segment describes an anticipation activity that forms a bridge between children’s use of beads to solve the tasks posed and an activity in which they solve the tasks entirely mentally while viewing an image of the teacher’s rack. It also includes a description of the latter activity. The segment ends with a discussion of several pragmatic issues including how the teacher chooses which activities to include within a lesson and how she selects which specific tasks to pose for those activities.
Video Segment 2c
This segment is devoted entirely to the pragmatic issue of recording children’s solutions. Specific recommendations are given for productive ways to record solutions, using the examples of children’s solutions first described in video segment 2a. Special attention is given to how the notation links grouping strategies implicit in the movement of the beads to numerical notation. This segment ends with a timeline for the entire sequence with emphasis on a detailed timeline for Part II of the sequence.
Video Group 3: Structuring Numbers Part III – Becoming Proficient With Using Grouping Strategies to Solve Additive Tasks Without the Rack
This video group describes activities that are designed to further develop children’s proficiency with using grouping strategies to solve additive tasks independent of the support of the rack. The first activity involves posing tasks in story contexts. In the second activity tasks are posed purely numerically. The discussions of these activities include attention to how the teacher’s use of notation serves to link children’s reasoning with grouping and thinking strategies to numerical notation, without reference to the beads on the rack. The final activity described in this segment is designed to focus entirely on numerical notation. This video segment ends with a detailed timeline for the entire Structuring Numbers sequence.