Exploring the Rule of 3 in Elementary School Math Teaching and Learning
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Transcript of Exploring the Rule of 3 in Elementary School Math Teaching and Learning
Exploring the Rule of 3 in Elementary School Math Teaching and Learning
Timothy BoerstJane Addams Elementary School, South Redford
andThe Center for Proficiency in Teaching Mathematics,
University of Michigan
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/Graphic Algebraic
Defining the Rule of 3Numerical- Representation focuses on specific values within
algorithms, equations, lists, tables and the like.
Defining the Rule of 3Algebraic- Representation focuses on verbal and symbolic
notation to generalize, formalize, model and extend.
Defining the Rule of 3Graphic- Representation focuses on spatial/pictorial/
geometric/visual displays.
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
Classroom Action Research
Classroom tasks Student work products Student surveys Student interviews Instructional video Case partner observation
Case Construction Description of rationale for inquiry, classroom context, and instructional practices Reflection upon inquiry Relevant research and literature Questions for group members Classroom
Context Artifacts
Teacher Reflection Group Meeting
Case based group discourse and video
A Group Member’s Written Case
Classroom Context
Video
Case Partner Input
TRG Video (used for next version of the
case)
Content Sources Classroom context Individual case context TRG meeting context
Teacher Reflection Group: Year Long Process of a Contemporary Professional
Development Approach
represents daily effort to take what is learned from inquiry to construct and enact improved instructional action
Beginning of the year wonderings (individual and group)
End of the year document sharing overview of “findings” and next
steps (individual and group)
Rough Case Phase
Revised Case Phase
Summative Case Phase
Rule of 3 Rationale National standards State measures Reformed texts Student learning strengths Subject matter rigor Professional growth
Applying the Rule of 3Try solving or communicating a solution for the following
problem using numerical, algebraic, and graphic representations.
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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
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 ≥ $2.95Where W=number of weeks
.50 + .50 + .50 + .50 +.50 + .50 > $2.95
0
0.5
1
1.5
2
2.5
3
Week1
Week3
Week5
Money Saved
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?
Students Use of the Rule of 3
Numerical
50 › 0
50+50 › 75
50+50+50 = 75+75
Students Use of the Rule of 3
Graphic
Students Use of the Rule of 3
Algebraic
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…)
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