What is
the Rule of 3? A Provisional Definition for Dayto Day Use Timothy Boerst It is important to get a sense of the daytoday meanings of representations in the Rule of 3, so that they can be purposefully incorporated into/connected within instruction, consciously observed as students interact/create mathematical products, and fairly/openly assessed. The algebraic, numerical, and graphic representations that comprise the Rule of 3 may seem like fairly distinct entities that could be easily understood by teachers. Indeed, my TRG peers, with whom I interacted a great deal concerning the Rule of 3, never seemed confused or in doubt about what I was referring to when I talked about the representations. Perhaps one seemed to involve letters, another numbers, and the last diagrams. However in the course of writing about my research it has proven difficult to pin down the particular/unique meaning of each representation within the Rule of 3. This is perhaps due to the interconnected nature of mathematical representations (NCTM, 2000; Golden, 2001). For instance stem and leaf plots display specific numbers in graphic form and equations often combine specific numbers and algebraic variables. With these qualifications in mind, I share here a sort of baseline for the focus and components of each representation, knowing that each is certainly more nuanced and interconnected with other representations than expressed here. I have also included linguistic representation since it is a tool used by students and teachers to understand and explain the use of Rule of 3 representations. 

Representation focuses on specific values within algorithms, equations, lists, tables and the like. 

Representation focuses on verbal and symbolic notation (such as variables) to generalize, formalize, model and extend.. 

Representation focuses on spatial, pictorial, geometric, visual displays. 

Oral and written language used to understand, describe, analyze, explain, or reflect upon numerical, algebraic, or graphic representations. 


All Rule of 3 representations can be used individually or in concert to solve mathematical problems, support exploration, and share the results of mathematical work. In other words they can support the plausible reasoning necessary to work through a problem (a means of solving a problem) and also the demonstrative reasoning necessary to clearly communicate and prove the validity of one’s solution (the ends of problem solving activity) (Polya, 1954). Links to electronic sources with more information about the Rule of 3

1. Originally the Rule of 3 was delineated in reformed calculus materials as encompassing geometric, numerical, and algebraic representations (HughesHallett et al, 1994). However subsequent editions of these texts and interpretations of the Rule of 3 by others in the field have made it clear that attention would not be as much upon geometric representations (at least not as most teachers would interpret that title) as upon graphical/spatial representations of mathematical topics. In fact most current references to the Rule of 3 replace “geometrically” with “graphically”. Furthermore, HughesHallett and others have amended the Rule of 3 to include verbal representation as well. For these reasons I chose, early in my TRG work, to define the Rule of 3 as comprised of numerical, algebraic, and graphic representations. I also have attended closely to verbal representations as I have conducted my research, but did not want to change terminology to the “Rule of 4” to avoid confusion among TRG members (and myself for that matter). 2. Elsewhere in this snapshot I discuss the importance and utility of interconnected representations. 