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Exploring the Rule of 3 in Elementary School Math Teaching and Learning Timothy Boerst Jane Addams Elementary School, South Redford and The Center for Proficiency in Teaching Mathematics, University of Michigan

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Defining the Rule of 3 “Every topic should be presented geometrically, numerically, and algebraically.” (Hughes-Hallett et al, 1994) Subsequent definitions have tended to emphasize graphic and verbal representations and attend less to geometric forms. Numerical Verbal Geometric/GraphicAlgebraic

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Defining the Rule of 3 Numerical- Representation focuses on specific values within algorithms, equations, lists, tables and the like.

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Defining the Rule of 3 Algebraic- Representation focuses on verbal and symbolic notation to generalize, formalize, model and extend.

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Defining the Rule of 3 Graphic- Representation focuses on spatial/pictorial/ geometric/visual displays.

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Defining the Rule of 3 In practice numerical, algebraic, graphic, and linguistic representations are often closely intertwined.

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Teacher Reflection Group: One Phase of Work in a Contemporary Professional Development Approach Content Sources Classroom context Individual case context TRG meeting context

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Teacher Reflection Group: Year Long Process of a Contemporary Professional Development Approach

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Rule of 3 Rationale National standards State measures Reformed texts Student learning strengths Subject matter rigor Professional growth

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Applying the Rule of 3 Try solving or communicating a solution for the following problem using numerical, algebraic, and graphic representations. a +.50 = B Tom wants to buy a book that costs $2.95. He can save 50 cents a week. How many weeks will he need to save enough money for the book?.50 +.50 +.50…= $2.95

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Applying the Rule of 3 Tom wants to buy a book that costs $2.95. He can save 50 cents a week. How many weeks will he need to save enough money for the book?.50 W ≥ $2.95 Where W=number of weeks.50 +.50 +.50 +.50 +.50 +.50 > $2.95

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Students Use of the Rule of 3 Examine the student generated representations related to the following problem. A student left Redford with her family for a well earned vacation. They traveled 50 miles per hour heading toward California. One hour after the student left, his teacher remembered an important homework assignment. She raced along the identical route on her motorcycle at a speed of 75 miles per hour to catch up. How long would it take the teacher to catch the student?

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Students Use of the Rule of 3 Numerical 50 › 0 50+50 › 75 50+50+50 = 75+75

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Students Use of the Rule of 3 Graphic

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Students Use of the Rule of 3 Algebraic

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Potential of the Rule of 3: Seeing more by looking through different lenses Geometric perception of prisms and pyramids is enhanced by: Numerical examination (edges (E), faces (F), vertices (V)) Graphic organization (tables where E, F, and V are organized and also sorted by 3D shape type) Algebraic generalization (prisms F=B +2, F+V-2=E, pyramid Bx2=E…)

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Rule of 3 Challenges Determining representations and meshing them with knowledge of student learning and mathematical objectives New instructional territory (translation, refinement, comparative utility) New territory for learners (leading to new sorts of instructional needs) Depth vs. coverage

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