# Mathematics Instruction: Planning, Teaching, and Reflecting

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Mathematics Instruction: Planning, Teaching, and Reflecting
Modifying Tasks to Increase the Cognitive Demand Tennessee Department of Education Elementary School Mathematics Grade 4 Overview of the Module: 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). Equally important to tending to students’ mathematical thinking is understanding the instructional goals of the lesson and aligning these goals with student thinking. By engaging in setting goals, anticipating student responses, and considering the Selection and Sequencing with the intent of advancing student understanding of the goals, teachers will learn about the relationship between these parts of lesson planning. No Prior Knowledge Necessary. Materials: Slides with note pages Mathematics Common Core State Standards (CSSS) (the Standards for Mathematical Practice and the grade-level Standards for Mathematical Content) Participant handouts Chart paper and markers

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. Directions: (SAY) Take a minute and read the rationale for the lesson. As you can see from the rationale, “understand” is really important. The Common Core State Standards include standards that focus on understanding of mathematical concepts and the development of skills. We will engage in the lesson with the goal of deepening our understanding of concepts related to the task.

Rationale 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. Directions: Read or paraphrase the rationale.

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. Directions: Give participants a minute to read the slide.

Overview of Activities

Mathematical Tasks: A Critical Starting Point for Instruction
All tasks are not created equal−different tasks require different levels and kinds of student thinking. (SAY) We know the level of thinking the CCSS for Mathematical Practice and Mathematical Content require, so we must be cognizant of the types of tasks we are using to develop that kind of 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.

Mathematical Tasks: A Critical Starting Point for Instruction
The level and kind of thinking in which students engage determines what they will learn. Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997 (SAY) The task is a critical point for instruction. We have seen in earlier modules what students can learn when the tasks we use are, in fact, high level.

Mathematical Tasks: A Critical Starting Point for Instruction
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 Directions: Read the quote.

Revisiting the Characteristics of Cognitively Demanding
Tasks (SAY) Now we will analyze some textbook tasks that have been modified to increase the cognitive demand of the task. Our goal is to identify the type of changes that have been made to the task so that we in turn can make similar modifications to textbook tasks.

Comparing the Cognitive Demand of Two Tasks
Compare the two tasks. How are the tasks similar? How are the tasks different? Directions: Compare Task #1 and Task #2. How are they similar and how are they different from each other?

Solve each equation = ___ = ___ = ___ = ___ = ___ = ___ 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? (ASK) What is similar? What is the difference between this task and the previous task?

Linking to Research/Literature: The QUASAR Project
Low-Level tasks Memorization Procedures without Connections High-Level tasks Procedures with Connections Doing Mathematics (SAY) The characteristics of both low- and high-level tasks appear in the TAG on page 10 in your handout. The QUASAR Project further identifies two types of low-level tasks. A “Memorization” and a “Procedures Without Connections”, both of which appear on the left-hand side of the document. Also identified are two types of high-level tasks, a “Doing Mathematics” category and a “Procedures With Connections” category. The “Doing Mathematics” task is an open-ended task, whereas the “Procedures With Connections” task is one in which procedures are given to students, but they must still make connections or discuss mathematical relationships. Both appear on the right hand side of the document. Take two minutes to re-familiarize yourself with the characteristics of both high- and low-level tasks as described on the TAG.

Directions: Suppose I told you that the first task is a “doing mathematics task,” do the characteristics describe the task? If so in what ways? (Students are not given a pathway for solving the task, they aren’t told how to solve it. Some anxiety might exists because they have two problems and they must compare the solutions, draw a picture and write an equation. Students will notice mathematical relationships. They might discover the commutative property of addition because of the numbers in the tasks. Students must draw on prior knowledge because some experience with put together problems is required.) Problem #2 might initially when students first learn to solve story problems a problem like this could be considered a challenge because students would make connections between a drawing, an equation and the context but as time goes on and students solve more and more of these problem they will not have to think deeply about relationships between the drawing, equation and the context. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p New York: Teachers College Press.

Linking to Research/Literature: The QUASAR Project

Linking to Research/Literature: The QUASAR Project
The Mathematical Tasks Framework TASKS as set up by the teachers TASKS as implemented by students TASKS as they appear in curricular/ instructional materials Student Learning We know that the task matters a lot. And that, in fact, if the task chosen does not have a high level of cognitive demand and does not align to the goals set, it is difficult, if not impossible, to orchestrate an Accountable Talk discussion around the task. Stein, Smith, Henningsen, & Silver, 2000 Setting Goals Selecting Tasks Anticipating 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 Instruction and Select an Appropriate Task

The Structure and Routines of a Lesson
The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small Group Problem Solving Generate and Compare Solutions Assess and Advance Student Learning Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 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 PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT: Engage students in a Quick Write or a discussion of the process. Set Up the Task Set Up of the Task Directions: (SAY) This diagram represents the structures and routines of a lesson. In this session we will be focusing on the Share, Discuss, Analyze Phase of the Lesson (point to SDA box), specifically selecting and sequencing student work and orchestrating a discussion around that work. And

The CCSS for Mathematics: Grade 4
Number and Operations – Fractions 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. Directions: (Say) These are the standards that were identified by the teacher as the goals of the lesson. The fractions standards are familiar to us from our work together on previous modules. Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

The CCSS for Mathematics: Grade 4
Number and Operations – Fractions NF Understand decimal notation for fractions, and compare decimal fractions. 4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, 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 decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Directions: (Say) These standards are from the fractions domain. Read the standards. Probing Facilitator Questions and Possible Participant Responses (Italics): What do you notice? These standards link fractions to decimals but in a very narrow way. So there is deep understanding of the relationship. What are the implications for instruction? There is the ability to use manipulatives and demonstrate with the numbers involved. There is a lot of prior knowledge necessary with comparisons. Is every part of each of these standards addressed by the task? No. The students would need more numbers and more experiences decomposing fractional place values. Common Core State Standards, 2010, p. 31, NGA Center/CCSSO

Mathematical Practice Standards Related to the Task
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Directions: (Say) The bolded standards are the Standards for Mathematical Practice identified by the teacher as goals of the lesson. Probing Facilitator Questions and Possible Participant Responses (Italics): What does it mean to reason abstractly and quantitatively? Abstract values from the problem, perform calculations with them, and interpret the results in the context of the problem. What does it mean to attend to precision? Attending to precision means using precise mathematical notation and language, explaining the meaning of symbols, and clearly articulating the steps in a mathematical process, as well as not making errors and rounding values appropriately. Give an example of what it means for a student to look for and make use of structure. If a student recognizes a need for and writes an equivalent fraction/decimal to solve a problem, (s)he is looking for and making use of the structure of place value, as well as the meaning of the numerator vs. the denominator. 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. Directions: Give participants time to solve the task and discuss solution paths. The purpose at this time is to give participants an opportunity to think about ways that students might enter into and respond to the task. Do not facilitate a true Share, Discuss, and Analyze Phase by selecting and sequencing participant responses. Don’t work too hard to make connections between solution paths. This will come later.

The Pizza Task Jolla has 1 4 of a pizza. Sarah has 30 100 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. Show each of the student’s amount of pizza. Compare the students’ amounts of pizza. Explain with words and use the >, <, or = symbols to show who has the most pizza. Explain with words and use the >, <, or = symbols to show who has the least amount of pizza.

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? Probing Facilitator Questions and Possible Participant Responses (Italics): How does this task address 4.NF.A.2? Students are tasked with comparing numbers less than one; because the numbers are given in fractional form. They are comparing fractions; they will have the opportunity to compare them as decimals when they use the 10x10 grid to show tenths or hundredths. How does this task address 4.NF.C.5? It is likely that many of the students will immediately write the fractional amounts as decimals. If they do not, however, the teacher can ask students to use both forms. What structures are students looking for and making use of in this task? Students will make use of the structure of fractions (meaning of the numerator and of the denominator) as well as the structure of place value with decimals.

Modifying Textbook Tasks to Increase the Cognitive Demand
Giving it a Go: Modifying Textbook Tasks to Increase the Cognitive Demand of 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. Directions: Read or paraphrase the directions. Ask you enter groups listen to make sure they know the goal of the original textbook page. Some possible questions that you might ask when guiding a group include: (ASK) What is the math goal of the original textbook page? (visual images can help us determine the number of items in a set. The items are arranged like the dots on a dice.) Guiding participants to think about a prompt that will engage students in using one problem to solve another problem.

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 CCSS for Mathematics: Grade 4
Number and Operations – Fractions 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

The CCSS for Mathematics: Grade 4
Number and Operations – Fractions NF Understand decimal notation for fractions, and compare decimal fractions. 4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, 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 decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Common Core State Standards, 2010, p. 31, NGA Center/CCSSO

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. Directions: Direct participants to read the Accountable Talk Moves in their participant packet. Probing Facilitator Questions and Possible Participant Responses (Italics): How are the AT moves associated with accountability to community, knowledge, and rigorous thinking similar? How are they different? Do you disagree with how any of the moves are classified? If so, why? The community moves are all designed to make sure that everyone is involved in the discussion and understands the ideas being discussed. The knowledge moves are used to make sure that we are discussing mathematics that is accurate and that we are attending to precision of language. The rigorous thinking moves are made to press for reasoning and meaning. Let’s think more about Academic Rigor… How do the AT moves support accountability to rigorous thinking relate to the characteristics of academically rigorous lessons we identified? If students are not prompted to articulate and expand their reasoning, there are no opportunities for “aha” moments. Pressing students for reasoning ensures that they are carrying the cognitive load instead of us doing the thinking for them. That is where the rigor lies in the lesson. Expanding reasoning moves prompts students to make connections. “Aha” moments often occur when we see how ideas are related to one another.

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. Directions: Direct participants’ attention to the Accountable Talk Features and Indicators in their participant packets. Give them an opportunity to independently read the Accountable Talk Features and Indicators. Ask them to talk in small groups about how these features and indicators support the implementation of academically rigorous lessons. Probing Facilitator Questions and Possible Participant Responses (Italics): What is the benefit of asking students to elaborate or build on each others’ ideas? Elaborations on ideas is a means by which the group co-constructs a solution path that they might not have been able to do independently. They are part of the struggle that students must engage in before they can make a connection triggering an “aha” moment. Why do you think the authors claim that evidence of all three features of AT must be present? Whoever talks the most learns the most. Students need to be the ones talking so that the teacher can assess what they know or don’t know. It is the students’ knowledge and reasoning that we need to hear, not just any kind of talk. What does it mean when you press students for an explanation, but they can’t explain the reasoning underlying the concept? Students often don’t understand a concept and they can’t share their mathematical reasoning because it is hard work. Students need to have a deep enough understanding of why the mathematical ideas are working the way they are working. How might you scaffold student learning in order to make it possible for students to share their mathematical reasoning? Make a table so they see a repeating pattern, link to the context, ask them if the problem reminds them of other similar problems, provide students with manipulatives, share your reasoning so they have opportunities to hear what it sounds like, invite students to “try to share their reasoning” and permit others to add on. All of these things will help students make connections between representations and to prior learning. These connections are characteristic of what it means to move thinking and to have an academically rigorous experience.

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: 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? Directions: These AT moves slides are to be used as a reference for the discussions that will occur around the questions on the previous slides.

To Support Accountability to Knowledge To Support Accountability to
Accountable Talk Moves (continued) 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 to Rigorous 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? Directions: These AT moves slides are to be used as a reference for the discussions that will occur around the questions on the previous slides.

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) (Say) This is an example of an Accountable Talk discussion script. Let’s read through it together. Directions: It may be helpful to role-play this discussion. The facilitator asks the questions on the slide and participants can predict what students might say in response to the questions. Keep this brief. It is meant to give the participants an idea of what they are expected to produce.

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 𝟑 𝟏𝟎 is equal to 𝟑𝟎 𝟏𝟎𝟎 . 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) (SAY) Here is the same AT discussion we looked at before, but this time with a hook at the beginning of the discussion. (Ask) How does adding this hook impact the discussion that will follow? Possible Participant Responses: It gives the students something specific to consider as they listen to the others in the group share their reasoning. The hook draws students’ attentions to a particular mathematical relationship.

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.” Directions: These observations are being shared at the individual tables so that all participants will be able to participate in the whole group discussion that follows.

Step Back and Application to Our Work
What have you learned today that you will apply when planning or teaching in your classroom? Directions: Give participants a few minutes to write down their thoughts in response to this question. Ask for volunteers to share. Do not discuss participant responses. Simply allow 3–5 people to share and thank them for sharing. The purpose is for participants to hear some of what their colleagues are taking away from this module.