Algebraic Reasoning Institute

Math & Science Collaborative at the Allegheny Intermediate Unit

Algebraic Reasoning Institute

July, 2011


Overview of the Institute

Goals Overview

Conceptual Flow Graphic

Reasoning Algebraically about Operations from the DMI series

Common Core State Standards for Mathematics (CCSSM)

Thinking Mathematically (CGI)

The work of Dr Margaret Smith from UPitt


Overview of the Institute

Content Assessment

Will serve as a pre and post-test to the course

Not a timed assessment


Overview

Insert conceptual flow and goals overview.

Common Core State Standards Design


Building on the strength of current state standards, the CCSS are designed to be:

Focused, coherent, clear and rigorous

Internationally benchmarked

Anchored in college and career readiness*

Evidence and research based

Common Core Standards for Mathematics

Grade-Level Standards

K-8 grade-by-grade standards organized by domain

9-12 high school standards organized by conceptual categories

The Common Core proposes a set of Mathematical Practices that all teachers should develop in their students.

Math & Science Collaborative at the Allegheny

Intermediate Unit

Standards for Mathematical Practice

Math & Science Collaborative

at the Allegheny Intermediate Unit

• Carry across all grade levels

• Describe habits of mind of a mathematically proficient student


CCSS Mathematical Practices1. Make sense of problems and persevere in solving

them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.


Underlying Frameworks

5 Process Standards

• Problem Solving•Reasoning and Proof•Communication•Connections•Representations

Underlying Frameworks




Strands of Mathematical Proficiency • Conceptual Understanding – comprehension of

mathematical concepts, operations, and relations

• Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently and appropriately

• Strategic Competence – ability to formulate, represent,, and solve mathematical problems

• Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification

• Productive Disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy

The Standards for Mathematical Practice




CCSS Mathematical Practices1. Make sense of problems and persevere in solving

them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.


Math & Science Collaborative at the Allegheny

Intermediate Unit

Read the practice assigned to your group.

Think about what this practice might look like in action

What would this practice sound like in action

Make a poster of the main ideas.

The poster may be a bulleted list, picture, diagram, or any other method that conveys the main ideas of the practice.




Take a moment to examine the first three words of each of the 8 Mathematical Practices.What do you notice?

Mathematically Proficient Students….



Consider the verbs that illustrate the student actions each practice.

For example, examine Practice #3: Construct viable arguments and critique the reasoning of others.

Highlight each of the verbs in the description of that standard.

Discuss with a partner: What jumps out at you?


Mathematical Practice #3: Construct viable arguments and critique the reasoning of othersMathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.


Mathematical Practice #3: Construct viable arguments and critique the reasoning of others

Mathematically proficient students:

understand and use stated assumptions, definitions, and previously established results in constructing arguments.

make conjectures and build a logical progression of statements to explore the truth of their conjectures.

analyze situations by breaking them into cases, and can recognize and use counterexamples.

justify their conclusions, communicate them to others, and respond to the arguments of others.

reason inductively about data, making plausible arguments that take into account the context from which the data arose.

compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is.

construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.

determine domains to which an argument applies.

listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.


Mathematical Practice #3: Construct viable arguments and critique the reasoning of others

On a scale of 1 (low) to 6 (high), to what extent is your school/district promoting students’ proficiency in Practice 3?

Evidence for your rating?


Standards for Mathematical Practice

SMP1: Explain and make conjectures…

SMP2: Make sense of…

SMP3: Understand and use…

SMP4: Apply and interpret…

SMP5: Consider and detect…

SMP6: Communicate precisely to others…

SMP7: Discern and recognize…

SMP8: Notice and pay attention to…

Implementation Issue




React to the Following Statements in Writing• When you hear the word algebra

what kinds of mathematical ideas come to mind?

• What, if anything, does algebra have to do with the content you teach?

• What might it mean to engage with children on algebraic ideas?


Mathematical Themes • Are two different definitions of

even numbers equivalent?

• What comprises an argument that a statement is always true when you cannot check every number?

• What are generalizations about adding and multiplying odd and even numbers and how can they be proved?


Hue and Julio scenario

In a second-grade classroom the teacher commented

there were an even number of children in class that

day.

• Hue said, “I knew it was even because when we lined up for lunch everyone had a partner.”

• Julio said, “I knew it was even because when we split into two groups, the two groups were equal and no one was left over.”


How can you show that these definitions describe the same set of numbers; how can you show these definitions are equivalent?


Conjecture: The sum of two odd numbers is even.

Work with a partner to make an argument for how you know the conjecture is true.

Math Practices in The Classroom



1. Which mathematical practice standards are evident ?

2. How does the teacher support students with the Mathematical Practice Standards?


Acknowledgement

This material is based on work supported by the SW PA MSP 2010 funds administered through the USDOE under Grant No. Project #: RA-075-10-0603. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the granting agency.

Algebraic Reasoning Institute

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