Analyzing Program Design and Implementation through

Mathematics Learning Theory

Analyzing Program Design and Implementation through Mathematics Learning Theory - Exploring Representations of Addition and Subtraction – Concepts, Algorithms, and Mental Math- Exploring Algebraic Reasoning through Arithmetic, Geometry, and Data Management using manipulatives and graphing calculators - Relating theories of learning and adolescence, lesson design principles , and components of an effective learning environment for effective mathematics teaching of adolescent studentsCreating learning environments to support adolescent students in learning mathematicsCreating learning environments conducive to the intellectual, social, emotional, physical, linguistic, cultural, spiritual and moral development of the adolescent Understanding and implementing Ministry of Education curriculum expectations and Ministry of Education and district school board policies and guidelines related to the adolescent

ABQ Intermediate Mathematics Spring 2010SESSION 13 – June 3, 2010

Sample Bansho Plan

11” 11”

8-1/2”

8-1/2”

AFTERHighlights/Summary-3 or sokey ideas from theDiscussionFor TI grade

AFTERPractice-Problem-2 solutions- focused on TI grade

KnowledgePackageGr 7 to 10-codes anddescription-lessonlearninggoals in recthighlighted

MathVocabularylist

BEFOREActivation-Task orProblem-2 solutionsRelevant to TI grade

DURING-Lesson (bus) Problem-What informationwill WE useto solvethe problem? List info

AFTER Consolidation

Gr7 Gr8 Gr9 Gr10

4 different solutions exemplifyingmathematics from specific grades

labels for each solution that capture the mathematical approach

-Math annotations on and aroundthe solutions (words, mathematical details to make explicit the mathematics in the solutions-Mathematical relationship betweenthe solutions

Is it possible to create two patterns that have the same pattern rule, but look different?

a. Create two growing patterns, using square tiles.

b. Record your growing pattern onsquare grid strip of paper.

c. Explain and record each other’s pattern rule on the paper.

d. Discuss how your patterns are mathematically similar and different.

We usually have studentswrite and compare pattern rules to concrete models; When do we have students re-present a pattern rule in different ways? Such develops a conceptual model of equivalent algebraic expressions

BEFORE - Same Pattern Rule, Looks Different

DURING - Pool Border Problem A

How many one-by-one square tiles are required to surround a 5 x 5 square unit pool?

Show 2 different solutions.

5 x 5 Pool

5 square tiles

5 square tiles

Video : Jo Boaler

Original Pool Border Problem

How many square tiles are needed for the border of any square-shapedpool? a. How many one-by-one tiles required to

surround other square-shaped pools?b. Determine a rule to predict the number of

tiles required to surround a square-shaped pool of any size.

c. How does your rule relate to the number of tiles for the pool and for the pool border?

Pool

5 square tiles

5 square tiles

Same problem -,butdifferent focus - tochoose the arithmeticsolution that has thepotential for generalizeability for anysize square-shaped pool)

• Engage students in the same mathematics (demonstrating achievement of the same expectations) but give them choice about − Strategy − Numbers that they use to do the math

• Allow teachers to ask “Common Questions” of ALL students that make explicit some of the mathematical relationships in the question prompting students to reason and defend their thinking

Parallel Tasks

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• Lisa : $1.65 in quarters and nickels

• Amy : $1.05 with half the quarters and twice the nickels

How many of each coin does Lisa have?

Parallel Tasks

Choice 1

Use equations to model the problem and then solve it.

Choice 2

Solve the problem using only number thinking.

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• Did you need to know how much each coin was worth or just the relationship between them?

• How did you know that Amy had an even number of nickels?

Common Questions

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• Could you be sure that Amy had an even number of quarters?

• How do you know that each girl had fewer than 7 quarters?

Common Questions

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• How did you solve the problem?

• How do you know your solution is correct?

Common Questions

Parallel Tasks

Choice 1: The track is

24 cm long. How long could the pieces be?

Choice 2: The track is 264 cm long. How long could the pieces be?

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Ian put together equal length pieces of train track.

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How do you know that the pieces are not 10 cm long?

How do you know that the pieces are not 25 cm long?

Common Questions for Parallel Tasks

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• Could the pieces be 2 cm long? How do you know?

• Could there be 12 pieces of track? How do you know?

Common Questions for Parallel Tasks

An open question provides valuable information about the range of knowledge in your classroom. The student responses will help you know how to proceed with your lesson.

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Open Questions

Open:

The area of a rectangle is about 400 square units. What could its dimensions be?

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Not open:

The area of a rectangle is 432 square units. The length is 12 units, what is the width?

Contrast

Open:

A pattern begins like this

3, 6, …

How might it continue?

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Open Questions

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• How might such open questions be effective in diagnosing student differences?

• Do you see them more as exposing or evoking thinking?

Purpose Open Questions

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• Assessment for learning is your focus.

• An open question should be accessible to all students.

Open Minds On