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 EducationElementary School MathematicsGrade 3

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:• analyze a video and discuss students’

mathematical reasoning that is clear, faulty, or unclear;

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

• study 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

NCTM Focal Points:

Reasoning and sense making are of particular importance, but historically “reasoning” has been limited to very selected areas of the high school curriculum, and sense making is in many instances not present at all. However, an emphasis on student reasoning and sense making can help students organize their knowledge in ways that enhance the development of number sense, algebraic fluency, functional relationships, geometric reasoning, and statistical thinking.

NCTM, 2008, Focus in High School Mathematics: Reasoning and Sense Making

The Relationship Between Talk and UnderstandingWe come to an understanding in the course of communicating it. That is to say, we set out by offering an understanding and that understanding takes shape as we work on it to share it. And finally we may arrive cooperatively at a joint understanding as we talk or in some other way interact with someone else (p. 115).

This view is supported by Chin and Osborne‘s (2008) study. They state that when students engage socially in talk activities about shared ideas or problems, students must be given ample opportunities for formulating their own ideas about science concepts, for inferring relationships between and among these concepts, and for combining them into an increasingly more complex network of theoretical propositions. For Hand (2008), the oral language component is heavily emphasized in the social negotiated processes in which students exchange, challenge, and debate arguments in order to reach a consensus. (Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through argumentation.)


Determining Student Understanding

What will you need to see and hear to know that students understand the concepts of a lesson?

Watch the video. Be prepared to say what students know or do not know. Cite evidence from the lesson.


Context for the Lesson

The visiting teacher is a TN Common Core Coach. She is demonstrating a lesson for the classroom teacher, who is interested in gaining a better understanding of ways of encouraging classroom talk.

Teacher: Cindy ClichéPrincipal: Roseanne BartonSchool: John Pittard Elementary SchoolSchool District: Murfreesboro, TennesseeGrade: 4Date: February 4, 2013


A Fraction of the Whole TaskIn each case below, the area of the whole rectangle is 1. Shade an area equal to the fraction underneath each rectangle. Compare the two amounts.

1. Which is more and how do you know? Use the “greater than” or “less than” symbol to compare the amounts.

2. Name another fraction that describes the shaded portion.

The CCSS for Mathematics: Grade 3Number and Operations – Fractions 3.NFDevelop understanding of fractions as numbers.

3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Common Core State Standards, 2010, p. 24, NGA Center/CCSSO


Essential UnderstandingEqual Size PiecesA fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.Comparing Fractions with Unlike DenominatorsIf two fractions have the same numerator, the one with the lesser denominator is farther to the right on the number line or has more; it is the greater fraction because there are fewer cuts or pieces and, therefore, they are larger. (modified NCTM Essential Understandings, 2011) Comparing Fractions with Like DenominatorsFractions with the same size pieces, or common denominators, can be compared with each other because the size of the pieces is the same. Recognizing Equivalent FractionsWhen creating equivalent fractions, all of the pieces in a whole are subdivided or partitioned, thus the amount of pieces named in the numerator are automatically partitioned in the same way. What is created is an equivalent fraction. Continuous and Discrete FiguresA fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but, regardless of the model, what remains true about all of the models is that they represent equal parts of a whole. A Whole Can be Represented as a FractionIf all of the equal pieces in the denominator are represented in the numerator, then the whole figure is represented.

Essential Understandings (Small Group Discussion)

The CCSS for Mathematical Practice

1. 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.

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


Determining Student Understanding(Small Group Work)

What did students know and what is your evidence?

Where in the lesson do you need additional information to know if students understood the mathematics or the model?

Cite evidence from the lesson.


Determining Student Understanding(Whole Group Discussion)

In what ways did students make use of the third Mathematical Practice Standard?

Let’s step back now and identify ways in which student understanding shifted or changed during the lesson.

Did student understanding evolve over the course of the lesson? If so, what ideas did you see changing over time?

What do you think was causing the shifts?

Common Core State Standards: Mathematical Practice Standard 3(Whole Group Discussion)How many of the MP3 elements did we observe and if not, what are we wondering about since this was just a short segment?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.


Imagine Publicly Marking Student Behavior

In addition to stressing the importance of effort, the teachers were very clear about the particular ways of working in which students needed to engage. D. Cohen and Ball (2001) described ways of working that are needed for learning as learning practices. For example, the teachers would stop the students as they were working and talking to point out valuable ways in which they were working.

(Boaler, (2001) How a Detracked Math Approach Promoted Respect, Responsibility and High Achievement.)


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.)

The Importance of Writing

Yore and Treagust (2006) note that writing plays an important role―to document ownership of these claims, to reveal patterns of events and arguments, and to connect and position claims within canonical science (p.296). That is, the writing undertaken as a critical role of the argumentative process requires students to build connections between the elements of the argument (question, claim, and evidence).

When students write, they reflect on their thinking and come to a better understanding of what they know and what gaps remain in their knowledge (Rivard, 1994).

(Chen, Ying Chih, 2011 Examining the integration of talk and writing for student knowledge construction through argumentation.)

Writing Assists Teachers, TOO

Writing assists the teacher in thinking about the child as learner. It is a glimpse of the child’s reality, allowing the teacher to set up new situations for children to explain and build their mathematics understanding.

Weissglass, Mumme, & Cronin, 1990


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


Essential UnderstandingEqual Size PiecesA fraction describes the division of a whole or unit (region, set, segment) into equal parts. A fraction is relative to the size of the whole or unit.Comparing Fractions with Unlike DenominatorsIf two fractions have the same numerator, the one with the lesser denominator is farther to the right on the number line or has more; it is the greater fraction because there are fewer cuts or pieces and, therefore, they are larger. (modified NCTM Essential Understandings, 2011) Comparing Fractions with Like DenominatorsFractions with the same size pieces, or common denominators, can be compared with each other because the size of the pieces is the same. Recognizing Equivalent FractionsWhen creating equivalent fractions, all of the pieces in a whole are subdivided or partitioned, thus the amount of pieces named in the numerator are automatically partitioned in the same way. What is created is an equivalent fraction. Continuous and Discrete FiguresA fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but, regardless of the model, what remains true about all of the models is that they represent equal parts of a whole. A Whole Can be Represented as a FractionIf all of the equal pieces in the denominator are represented in the numerator, then the whole figure is represented.

Essential Understandings (Small Group Discussion)

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 in 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

1. Make Time for the Think-Talk-Reflect-Write Process

Think: Work privately to prepare a written response to one of the prompts, “What is division?”

Talk: What is division? Keep a written record of the ideas shared.

Reflect: Reflect privately. Consider the ideas raised. How do they connect with one another? Which ideas help you understand the concept better?

Write: Write an explanation for the question, “What is division?” Think about what everyone in your group said, and then use words, pictures, and examples to explain what division means. Go ahead and write.

Hunker & Lauglin, 1996

Pictures

Written Symbols

ManipulativeModels

Real-world Situations

Oral & WrittenLanguage

Modified from Van De Walle, 2004, p. 30

2. Encourage the Use of Multiple Representations of Mathematical Ideas


3. Construct a Concept Web with Students

• Analyze the concept web. Students developed the concept web with the teacher over the course of several months.

• How might developing and referencing a concept web help students when they are asked to write about mathematical ideas?


3. Construct a Concept Web with Students (continued)


4. Co-Construct Criteria for Quality Math Work: 4th Grade

Students worked to solve high-level tasks for several weeks. The teacher asked assessing and advancing questions daily. Throughout the week, the teacher pressed students to do quality work. After several days of work, the teacher showed the students a quality piece of work, told them that the work was “quality work,” and asked them to identify what the characteristics of the work were that made it quality work. Together, they generated this list of criteria.


5. Engage Students in Doing Quick Writes

A Quick Write is a narrow prompt given to students after they have studied a concept and should have gained some understanding of the concept.

Some types of Quick Writes might include:• compare concepts;• use a strategy or compare strategies;• reflect on a misconception; and• write about a generalization.


Brainstorming Quick Writes:

What are some Quick Writes that you can ask students to respond to for your focus concept?


6. Encourage Pattern Finding and Formulating and Testing Conjectures

• How might students benefit from having their conjectures recorded?

• What message are you sending when you honor and record students’ conjectures?

• Why should we make it possible for students to investigate their conjectures?

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

How many of the MP3 elements did we observe?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.


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