Chapter 08

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Fluency through Meaningful PracticeMathematical Routines & Algebraic Thinking

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Mathematical Routine:

What is the rule and how would this look on a graph?

X (in) Y (out)

2 3

4 7

6 11

3 5

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X (in) Y (out)

24 2

12 1

36 3

48 4

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Conversation in Mathematics

• Look at the different solution strategies students came up with for the problem 35-x=36-20. What understanding does each student have?

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Developing Fluency through Mathematical Routines

• Pedagogy

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Mathematically Powerful

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What is a mathematical routine?

• Purposefully structured activities that help children develop procedural fluency as well as reasoning and problem solving skills through meaningful practice.

• They are not just a part of the daily schedule.• They are planned based on the needs of the

children for that day—not commercially produced.

• They take place outside of the regular math time

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Types of Routines

• Routines can address any math concept, but tend to be used mostly for the development and support of early number concepts and the base ten system, as well as computational fluency with whole and rational numbers.

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Types of routines

• Early Number• Algebraic Thinking• Computational Fluency

– Number Talks– Number Strings– Number Lines

• Data and Graphing

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Algebraic Thinking & Reasoning

• Content

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1 - Equality

8 + 4 = __ + 5

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Response / Percent Responding Grade 7 12 17 12 and 17 1 and 2 3 and 4 5 and 6

5 58 13 8 9 49 25 10 2 76 21 2

From: Thinking Mathematically, by Carpenter, T., Franke, M., & Levi, L., 2003

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Have you ever seen or done this?

4 + 5 = 9 - 1 = 8 + 4 = 12

Instead, we should record our thinking like this: 4 + 5 = 9 9 - 1 = 8 8 + 4 = 12

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Introducing the Equal Sign

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4 + 5 = 9

True/False Questions

9 = 4 + 5

9 = 9

4 + 5 = 4 + 5

4 + 5 = 5 + 4

4 + 5 = 6 +3

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Video

• Watch the kindergarten video around equality. How does the teacher make this abstract concept accessible to her students?

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Inequalities

• Kill the alligator story!

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2- Relational Thinking

7 + 6 = ___ + 5

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37 + 56 = 39 + 54

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33 - 27 = 34 - 26

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3-Growing Patterns & Functions

• Task: A contractor is designing square swimming

pools with a square center. He uses blue tiles to represent the water. Around each pool there is a border of yellow tiles. He wants to figure out a way to know how many blue and yellow tiles there are in a pool of any size.

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Pool 1 Pool 2 Pool 3

What will the size (yellow, blue, and total area) of the 4th pool be? The tenth pool? Any pool?

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• Creating conjecture charts• Editing conjectures throughout the year

4-Making Conjectures

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• General forms of Justification– Restating the conjecture– Concrete examples that are

more than examples: Building on basic concepts– Use of counter-examples

5-Justification & Proof

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Questions to Elicit Justification

• How do you know?• Does that always work?• Does that work with all numbers?• How can you be sure?

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Criteria for Representation Based Proof

• 1-The meaning of the operation can be shown in diagrams, manipulatives, or in a story context.

• 2-The representation will work with a class of numbers (whole numbers, etc.)

• 3-The conclusion matches the representation. From: Reasoning Algebraically about operations Casebook,Education Development Center, Inc. and TERC, 2005.

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Task

• With a small group, you will design a routine around one of the following algebraic concepts:

– Equality– Relational thinking– Growing patterns