© 2013 university of pittsburgh mathematics instruction: planning, teaching, and reflecting...

© 2013 UNIVERSITY OF PITTSBURGH

Mathematics Instruction: Planning, Teaching, and ReflectingModifying Tasks to Increase the Cognitive Demand

Tennessee Department of Education

Elementary School Mathematics

Grade 4

Rationale

There is wide agreement regarding the value of

teachers attending to and basing their instructional

decisions on the mathematical thinking of their students

(Warfield, 2001).

By engaging in an analysis of a lesson-planning

process, teachers will have the opportunity to consider

the ways in which the process can be used to help them

plan and reflect, both individually and collectively, on

instructional activities that are based on student thinking

and understanding.

© 2013 UNIVERSITY OF PITTSBURGH 3

There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.

Lappan & Briars, 1995

By determining the cognitive demands of tasks and being cognizant of the features of tasks that make them high-level or low-level tasks, teachers will be prepared to select or modify tasks that create opportunities for students to engage with more tasks that are high-level tasks.

Rationale

4© 2013 UNIVERSITY OF PITTSBURGH

Session Goals

Participants will:

• deepen understanding of the cognitive demand of a task;

• analyze a set of original and modified tasks to learn strategies for increasing the cognitive demand of a task; and

• recognize how increasing the cognitive demand of a task gives students access to the Common Core State Standards (CCSS) for Mathematical Practice.


Overview of Activities

Participants will:

• discuss and compare the cognitive demand of mathematical tasks;

• identify strategies for modifying tasks; and

• modify tasks to increase the cognitive demand of the tasks.


Mathematical Tasks:A Critical Starting Point for Instruction

All tasks are not created equal−different tasks require

different levels and kinds of student thinking.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 3.

New York: Teachers College Press.



The level and kind of thinking in which students engage

determines what they will learn.

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997



If we want students to develop the capacity to think,

reason, and problem-solve, then we need to start with

high-level, cognitively complex tasks.

Stein & Lane, 1996


Revisiting the Characteristics of Cognitively Demanding

Tasks


Comparing the Cognitive Demand of Two Tasks

• Compare the two tasks.

• How are the tasks similar? How are the tasks different?


Task #1: A Place Value Task

Identify the place value for each of the underlined digits.

a. 351

b. 76

c. 4,789


Task #2: What is Changing? Solve each equation.

234 + 10 = ___

379 + 10 = ___

389 + 10 = ___

399 + 10 = ___

489 + 10 = ___

499 + 10 = ___

When ten is added to each of the numbers above, how is the sum changing from one equation to the next?

Sometimes the tens place changes and sometimes the hundreds place change when ten is added to the number. Why does this happen and when does it happen?

Look at the number 2,399. Which numbers will change when ten is added to this number and why?

12


Linking to Research/Literature:The QUASAR Project

• Low-Level tasks

– Memorization

– Procedures without Connections

• High-Level tasks

– Procedures with Connections

– Doing Mathematics


The Mathematical Task Analysis Guide

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction:

A casebook for professional development, p. 16. New York: Teachers College Press.

TASKS

as they appear in curricular/ instructional materials

TASKS

as set up by

the teachers

TASKS

as implemented by students

Student Learning

The Mathematical Tasks Framework

Stein, Smith, Henningsen, & Silver, 2000

Linking to Research/Literature: The QUASAR Project

TASKS

as they appear in curricular/ instructional materials

TASKS

as set up by

the teachers

TASKS

as implemented by students

Student Learning

The Mathematical Tasks Framework

Stein, Smith, Henningsen, & Silver, 2000

Linking to Research/Literature: The QUASAR Project

Setting GoalsSelecting TasksAnticipating Student Responses

Orchestrating Productive Discussion• Monitoring students as they work• Asking assessing and advancing questions• Selecting solution paths• Sequencing student responses• Connecting student responses via Accountable

Talk discussions


Identify Goals for Instructionand Select an Appropriate Task


The Structure and Routines of a Lesson

The Explore Phase/Private Work TimeGenerate Solutions

The Explore Phase/Small Group Problem Solving

1. Generate and Compare Solutions2. Assess and Advance Student Learning

Share, Discuss, and Analyze Phase of the Lesson

1. Share and Model

2. Compare Solutions

3. Focus the Discussion on Key

Mathematical Ideas

4. Engage in a Quick Write

MONITOR: Teacher selects examples for the Share, Discuss,and Analyze Phase based on:• Different solution paths to the same task• Different representations• Errors • Misconceptions

SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification.REPEAT THE CYCLE FOR EACH

SOLUTION PATHCOMPARE: Students discuss similarities and difference between solution paths.FOCUS: Discuss the meaning of mathematical ideas in each representationREFLECT: Engage students in a Quick Write or a discussion of the process.

Set Up the TaskSet Up of the Task

The CCSS for Mathematics: Grade 4

Number and Operations – Fractions 4.NF

Extend understanding of fraction equivalence and ordering.

4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Common Core State Standards, 2010, p. 30, NGA Center/CCSSO



Understand decimal notation for fractions, and compare decimal fractions.

4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction withdenominator 100, and use this technique to add two fractions withrespective denominators 10 and 100. For example, express 3/10 as30/100, and add 3/10 + 4/100 = 34/100.

4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. Forexample, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size.Recognize that comparisons are valid only when the two decimalsrefer to the same whole. Record the results of comparisons with thesymbols >, =, or <, and justify the conclusions, e.g., by using a visualmodel.


Mathematical Practice Standards Related to the Task

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


Identify Goals: Solving the Task(Small Group Discussion)

Revisit the Pizza Task.

Solve the task.

Discuss the possible solution paths to the task.


The Pizza TaskJolla has of a pizza.

Sarah has of a pizza.

Maria has of a pizza.

Tim’s pizza is shaded on the pizza. How much pizza is Tim’s share?

Jake has of a pizza.

Juan has of a pizza.

1. Show each of the student’s amount of pizza.

2. Compare the students’ amounts of pizza. Explain with words and use the >, <, or = symbols to show who has the most pizza.

3. Explain with words and use the >, <, or = symbols to show who has the least amount of pizza.


The Pizza Task (continued)

Jolla’s Pizza Tim’s Pizza Juan’s Pizza

Sarah’s Pizza Maria’s Pizza Jake’s Pizza


Identify Goals Related to the Task(Whole Group Discussion)

Does the task provide opportunities for students to access the Mathematical Content Standards and Practice Standards that we have identified for student learning?


Giving it a Go:Modifying Textbook Tasks to

Increase the Cognitive Demand of Tasks


Your Turn to Modify Tasks

• Form groups of no more than three people.

• Discuss briefly important NEW mathematical concepts, processes, or relationships you will want students to uncover by the textbook page. Consult the CCSS.

• Determine the current demand of the task.

• Modify the textbook task by using one or more of the Textbook Modification Strategies.

• You will be posting your modified task for others to

analyze and offer comments.


Strategies for Modifying Textbook Tasks

Increasing the cognitive demands of tasks: • Ask students to create real-world stories for “naked

number” problems. • Include a prompt that asks students to represent

the information another way (with a picture, in a table, a graph, an equation, with a context).

• Include a prompt that requires students to make a generalization.

• Use a task “out of sequence” before students have memorized a rule or have practiced a procedure that can be routinely applied.

• Eliminate components of the task that provide too much scaffolding.


Strategies for Modifying Textbook Tasks (continued)

Increasing the cognitive demands of tasks:

• Adapt a task so as to provide more opportunities for students to think and reason—let students figure things out for themselves.

• Create a prompt that asks students to write about the meaning of the mathematics concept.

• Add a prompt that asks students to make note of a pattern or to make a mathematical conjecture and to test their conjecture.

• Include a prompt that requires students to compare solution paths or mathematical relationships and write about the relationship between strategies or concepts.

• Select numbers carefully so students are more inclined to note relationships between quantities (e.g., two tables can be used to think about the solutions to the four, six, or eight tables).


Gallery Walk

• Post the modified tasks.

• Circulate, analyzing the modified tasks. On a “Post-It” Note,” describe ways in which the tasks were modified and the benefit to students.

• If the task was not modified to increase the cognitive demand of the task, then ask a wondering about a way the task might be modified.


The Cognitive Demand of Tasks

• Does the demand of the task matter?

• What are you now wondering about with respect to the task demands?



Extend understanding of fraction equivalence and ordering.

4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.




Understand decimal notation for fractions, and compare decimal fractions.

4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction withdenominator 100, and use this technique to add two fractions withrespective denominators 10 and 100. For example, express 3/10 as30/100, and add 3/10 + 4/100 = 34/100.

4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. Forexample, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size.Recognize that comparisons are valid only when the two decimalsrefer to the same whole. Record the results of comparisons with thesymbols >, =, or <, and justify the conclusions, e.g., by using a visualmodel.



Accountable Talk Discussions

Recall what you know about the Accountable Talk features and indicators. In order to recall what you know:

• Study the chart with the Accountable Talk moves. You are already familiar with the Accountable Talk moves that can be used to Ensure Purposeful, Coherent, and Productive Group Discussion.

• Study the Accountable Talk moves associated with creating accountability to:

the learning community;

knowledge; and

rigorous thinking.


Accountable Talk Features and Indicators

Accountability to the Learning Community• Active participation in classroom talk.• Listen attentively.• Elaborate and build on each others’ ideas.• Work to clarify or expand a proposition.

Accountability to Knowledge• Specific and accurate knowledge.• Appropriate evidence for claims and arguments.• Commitment to getting it right.

Accountability to Rigorous Thinking• Synthesize several sources of information.• Construct explanations and test understanding of concepts.• Formulate conjectures and hypotheses.• Employ generally accepted standards of reasoning.• Challenge the quality of evidence and reasoning.


Accountable Talk Moves

Function Example

To Ensure Purposeful, Coherent, and Productive Group Discussion

Marking Direct attention to the value and importance of a student’s contribution.

That’s an important point. One factor tells use the number of groups and the other factor tells us how many items in the group.

Challenging Redirect a question back to the students or use students’ contributions as a source for further challenge or query.

Let me challenge you: Is that always true?

Revoicing Align a student’s explanation with content or connect two or more contributions with the goal of advancing the discussion of the content.

S: 4 + 4 + 4.

You said three groups of four.

Recapping Make public in a concise, coherent form, the group’s achievement at creating a shared understanding of the phenomenon under discussion.

Let me put these ideas all together.What have we discovered?

To Support Accountability to Community

Keeping the Channels Open

Ensure that students can hear each other, and remind them that they must hear what others have said.

Say that again and louder.Can someone repeat what was just said?

Keeping Everyone Together

Ensure that everyone not only heard, but also understood, what a speaker said.

Can someone add on to what was said?Did everyone hear that?

Linking Contributions

Make explicit the relationship between a new contribution and what has gone before.

Does anyone have a similar idea?Do you agree or disagree with what was said?Your idea sounds similar to his idea.

Verifying and Clarifying

Revoice a student’s contribution, thereby helping both speakers and listeners to engage more profitably in the conversation.

So are you saying..?Can you say more? Who understood what was said?


To Support Accountability to Knowledge

Pressing for Accuracy

Hold students accountable for the accuracy, credibility, and clarity of their contributions.

Why does that happen?Someone give me the term for that.

Building on Prior Knowledge

Tie a current contribution back to knowledge accumulated by the class at a previous time.

What have we learned in the past that links with this?

To Support Accountability toRigorous Thinking

Pressing for Reasoning

Elicit evidence to establish what contribution a student’s utterance is intended to make within the group’s larger enterprise.

Say why this works.What does this mean?Who can make a claim and then tell us what their claim means?

Expanding Reasoning

Open up extra time and space in the conversation for student reasoning.

Does the idea work if I change the context? Use bigger numbers?

Accountable Talk Moves (continued)


An Example: Accountable Talk Discussion The Focus Essential Understanding

Creating Equivalent Fractions When the denominator is multiplied or divided then the numerator is automatically divided into the same number of pieces because it is a subcomponent of the denominator.Group A Group B

• Explain your set of equivalencies.• Who understood what he said about the 100 and the 10? (Community)• Can you say back what he said how the model shows ? (Community)• Who can add on and talk about the 3 and the 30? (Community)• The denominator tells the number of equal parts in the whole. (Marking) • Do we see in both pieces of work? (Rigor) • Tell us how you found in your picture (Group A). (Rigor)


An Example: Accountable Talk Discussion The Focus Essential Understanding Creating Equivalent Fractions When the denominator is multiplied or divided then the numerator is automatically divided into the same number of pieces because it is a subcomponent of the denominator.Group A Group B

• Both groups say that is equal to . How can this be when the fractions use different numbers? (Hook)

• Can Group B explain why = = ?• Who understood what they said about the denominators? (Community)• Can you say back what they said about the numerator changing?

(Community)• Each group made statements about equivalency. How does the visual

model differ from/support the symbolic model? (Rigor)


Reflecting: The Accountable Talk Discussion

• The observer has 2 minutes to share observations related to the lessons. The observations should be shared as “noticings.”

• Others in the group have 1 minute to share their “noticings.”


Step Back and Application to Our Work

What have you learned today that you will apply when planning or teaching in your classroom?

© 2013 university of pittsburgh mathematics instruction: planning, teaching, and reflecting...

Documents

teachers tasks

highlevel tasks

tasks procedures

tasks different

lowlevel tasks

equaldifferent tasks

modified tasks

features of tasks