Reflections of the Rule of 3
Timothy
Boerst 
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Seeing the math within (pdf)
Understanding new curriculum materials
Roles of representations
All students 
Seeing the math within
There has been a consistent call for teachers to become more knowledgeable about mathematics. Using the Rule of 3 as a way of thinking about instruction has pushed me to more deeply explore the range of mathematics within the problems that I assign and to begin perceiving the relative strengths of different representation in relation to particular problems.
At the beginning of the year I worked through all of the math menu problems from my first unit looking for ways in which numerical, graphic, and algebraic representations could be used in solving and discussing important mathematical issues. It did not take long to realize that a number of the problems would be difficult to represent in certain ways and/or that I would have trouble even conceptualizing the connection between some of the problems and certain representations (1). It was not initially apparent whether this was: a reason to retool the problems that were on the menus; an instance when it would be necessary to marshal other resources to determine the representations; or the first sign that the Rule of 3 was not going to be consistently useful in working on fifth grade mathematics. It was however an initial hurdle to enacting instruction using the Rule of 3 and a reason to question the extent to which such instruction would advance student learning.
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Understanding new curriculum materials
The Rule of 3 is one way of understanding the structure of new, standardsbased curriculum materials, such as those that my district piloted last year (Everyday Mathematics, Investigations, and Mathematics Trailblazers). Finding ways for students and teachers to make sense of new curricular materials is important, particularly when traditional materials stand in stark contrast to standardsbased materials.
It was a coincidence that late in the first year of studying the Rule of 3 in the classroom my district decided to pilot several of the NSF funded elementary math texts. As I worked through selected units in these texts it became apparent that the closely structured spiraled organization of content in the texts had a connection to my Rule of 3 work. Often within any particular unit, the topics under study would be developed through numerical, algebraic, and multiple graphic representations. As I listened to other teachers talk about how "scattered" (many topics in short spans of the text) and "repetitive" (the same topics covered in multiple units) the texts seemed to be, it occurred to me that other users of the text were interpreting the use of multiple representations within particular math strands as a jumbling of multiple mathematical topics. In this section I will show how knowledge of the Rule of 3 can help teachers perceive "unity" from the apparent variety in the organization of standardsbased texts.
Teachers are called upon to help students accomplish an amazing variety of mathematical objectives over the course of a school year. This often leads to a characterization of the curriculum as a mile wide and an inch deep. With or without a text that closely interrelates representations, the Rule of 3 can serve as a way of interconnecting content and representations, thereby increasing the mathematical mileage of the work that I assign. In a sense the Rule of 3 can provide direction for instruction and learning, but also structure economy of engagement and interconnection of ideas.


Roles of Representation
The ways in which students take up the Rule of 3 as they learn highlights a distinction between use of multiple representations to communicate problem solving solutions and the use of multiple representations to actually solve problems. This distinction is significant in terms of learning in mathematics (Polya, 1954) and in terms of how lessons are conducted (NCTM, 2001).
Mathematical representations can be used in many different ways for many different purposes. When I first began encouraging students to use a variety of representations, they most often solved problems using numerical means and then a few students would work to develop subsequent solutions that involved other representations to convey their (already determined) answers to others. This distinction between representations as tools for determining solutions and as tools for mathematical communication has long been recognized by mathematicians and mathematics education researchers alike. In this section I will share the literature that describes these uses of representation. I will also share narratives from my classroom teaching and work samples that show the ways in which students used representations and their movement toward the use of multiple representations to both solve and communicate. These will also illustrate how I worked to encourage students to see multiple representations as useful in different facets of their engagement in mathematics.


All Students
Borrowing an idea from reform calculus may sound as though it would at best serve only highly skilled students. In practice, using the Rule of 3 provided new routes to help students at a variety of skill levels to gain access to mathematics problems. In this section I address these ideas by highlighting two disjunctions between perceptions of the Rule of 3 and what I learned about the Rule of 3 in practice.
Disjunction 1: The Rule of 3 as pushing "higher level math" in a fifth grade context vs. Rule of 3 as a tool to access more the sophisticated understandings and skills dictated for elementary students in state benchmarks
Readers of this research might question the appropriateness of intensive and routine work with multiple representations. In this section I will use standards from NCTM and state of Michigan benchmarks to address these questions by illustrating their connection to Rule of 3 work. Even though some may question the developmental grounding of these documents, they comprise the guiding framework for the high stakes tests upon which students are judged as "proficient" and schools/teachers are judged as "passing/failing". In such an environment it would be unwise not to attempt understanding the ways in which teaching and learning can incorporate the use of multiple representations.
Disjunction 2: The Rule of 3 as a diversion for highly able students vs. Rule of 3 as a scaffold for learners at various levels of proficiency
While this is not unrelated to the first disjunction, this disjunction originates in a far different place. Teachers are often asked what they will do for students who are perceived as high skilled or intelligent. My work on the Rule of 3 actually began as a way of thinking about this issue. However as I worked with the Rule of 3 in my classroom and deliberated with peers about this approach to teaching math content it became clear that it could be a way of reaching a wide variety of students. In this section I show student work samples and video from TRG meetings to illustrate a journey from Rule of 3 "for some" to "for all."


