supporting rigorous mathematics teaching and learning

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Constructing an Argument and Critiquing the Reasoning of Others

Tennessee Department of EducationMiddle School MathematicsGrade 6-8

Mathematical Understandings

[In the TIMSS report the fact] that 89% of the U.S. lessons’ content received the lowest quality rating suggests a general lack of attention among teachers to the ideas students develop. Instead, U.S. lessons tended to focus on having students do things and remember what they have done. Little emphasis was placed on having students develop robust ideas that could be generalized. The emergence of conversations about goals of instruction – understandings we intend that students develop – is an important catalyst for changing the present situation. Thompson and Saldanha (2003). Fractions and Multiplicative Reasoning. In Kilpatrick et al. (Eds.), Research companion

to the principles and standards for school mathematics, Reston: NCTM. P. 96.

In this module, we will analyze student reasoning to determine attributes of student responses and then we will consider how teachers can scaffold student reasoning.

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 3

Session Goals

Participants will learn about:• elements of Mathematical Practice Standard 3;

• students’ mathematical reasoning that is clear, faulty, or unclear;

• teachers’ questioning focused on mathematical reasoning; and

• strategies for supporting writing.


Overview of Activities

Participants will:• make sense of Mathematical Practice Standard 3;

• analyze student work to differentiate between writing about process versus writing about mathematical reasoning; and

• review strategies for supporting writing.


Making Sense of Mathematical Practice Standard 3

Study Mathematical Practice Standard 3: Construct a

viable argument and critique the reasoning of others,

and summarize the authors’ key messages.

Common Core State Standards:Mathematical Practice Standard 3

The Common Core State Standards recommend that students:

• construct viable arguments and critique the reasoning of others;

• 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;

• recognize and use counterexamples;

• justify 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; and

• 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.

Modified from the Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO


Talk is NOT GOOD ENOUGH.

Writing is NEEDED!

The Writing Process

In the writing process, students begin to gather, formulate, and organize old and new knowledge, concepts, and strategies, to synthesize this information as a new structure that becomes a part of their own knowledge network.

Nahrgang & Petersen, 1998

When writing, students feel empowered as learners because they learn to take charge of their learning by increasing their access to and control of their thoughts.

Weissglass, Mumme, & Cronin, 1990

Talk Alone is NOT GOOD ENOUGH!Several researchers have reported that students tend to process information on a surface level when they only use talk as a learning tool in the context of science education.

(Hogan, 1999; Kelly, Druker, & Chen, 1998; McNeill & Pimentel, 2010)

After examining all classroom discussions without writing support, they concluded that persuasive interactions only occurred regularly in one teacher’s classroom. In the other two classes, the students rarely responded to their peers by using their claims, evidence, and reasoning. Most of the time, students were simply seeking the correct answers to respond to teachers’ or peers’ questions. Current research also suggests that students have a great deal of difficulty revising ideas through argumentative discourse alone.

(Berland & Reiser, 2011; D. Kuhn, Black, Keselman, & Kaplan, 2000)

Writing involves understanding the processes involved in producing and evaluating thoughts rather than the processes involved in translating thoughts into language.

(Galbraith, Waes, and Torrance (2007, p. 3).(Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through argumentation.)


What’s the Difference? Task

Points A and B are the same distance from 0 on the number line.

Explain how you can find the following differences:

A – B = B – A =

What is the relationship between these differences?


Analyzing Student Work

• Analyze the student work.

• Sort the work into two groups—work that shows mathematical reasoning and work that does not show sound mathematical reasoning.

What can be learned about student thinking in each of these groups, the group showing reasoning and the group that does not show sound reasoning?


Student 1


Student 2


Student 3


Student 4


Student 5


Student 6


Student 7


Essential UnderstandingThe Sum of Two Rational Numbers Can be Located on a Number LineThe sum of two numbers p and q is located q units from p on the number line, because p and q represent linear distances with direction from zero. When q is a positive number, p + q is to the right of p. When q is a negative number, p + q is to the left of p. Subtraction is Adding the InverseThe difference of two numbers p – q is equal to p + (-q) because both can be modeled by the same point on the number line. That is, if q is positive, p – q and p + (-q) are both located at the point q units to the left of p and if q is negative, both are located at a point q units to the right of p. The Results of the Expressions b – a and a – b are Opposite ValuesThe results of the expressions b - a and a - b are opposite values because, since each represents the distance between the same two points, their absolute values must be the same while their direction must be opposite. Addition is Commutative, but Subtraction is NotThe order of the values being added does not affect the sum, but the order of the values being subtracted does affect the difference because movement along the number line is in opposite directions.

Essential Understandings

Common Core State Standards:Mathematical Practice Standard 3The Common Core State Standards recommend that students:

• construct viable arguments and critique the reasoning of others; • 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; • recognize and use counterexamples;• justify 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; and • 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.



Two Forms of Writing

Consider the forms of writing below. What is the purpose of each form of writing? How do they differ from each other?

• Writing about your problem solving process/steps when solving a problem

• Writing about the meaning of a mathematical concept/idea or relationships

A Balance: Writing About Process Versus Writing About Reasoning

Students and groups who seemed preoccupied with “doing” typically did not do well compared with their peers. Beneficial considerations tended to be conceptual in nature, focusing on thinking about ways to think about the situations (e.g., relationships among “givens” or interpretations of “givens” or “goals” rather than ways to get from “givens” to “goals”).

This conceptual versus procedural distinction was especially important during the early stages of solution attempts when students’ conceptual models were more unstable. Lesh & Zawojewski, 1983


Strategies for Supporting Writing


Reflecting on the Benefit of Using Supports for Writing

How might use of these processes or strategies assist students in writing about mathematics? Record your responses on the recording sheet on page 29 of your participant handout.

Reflect on the potential benefit of using strategies to support writing.

1. Make Time for the Think-Talk-Reflect-Write Process2. The Use of Multiple Representations3. Construct a Concept Web with Students4. Co-Construct Criteria for Quality Math Work5. Engage Students in Doing Quick Writes6. Encourage Pattern Finding and Formulating and Testing

Conjectures

Checking In: Construct Viable Arguments and Critique the Reasoning of Others

The Common Core State Standards recommend that students:• construct viable arguments and critique the reasoning of others; • 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; • recognize and use counterexamples;• justify 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; and • 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.



Bridge to Practice – Module 7

• Journal and Reflect:• How and where can you effectively incorporate writing in

your math classroom? What strategies for student writing will you use?

• What is an effective way to get your students to understand MP3 and incorporate that learning in their work?

supporting rigorous mathematics teaching and learning

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