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