5.1 Homework Solutions

Amy Jones LewisNovember 2010Green River Regional Educational CooperativeMathPLUSContent Day 1: Student-Centered Problem Solving1Day 1: Student-Centered Problem SolvingSolve, analyze, and discuss mathematical tasks.Consider the effects of different levels of mathematical tasks on students achievement. Work in a student-centered environment to complete a mathematical task.2There 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, 1995Analyzing Mathematical Tasks3What are Mathematical Tasks?We define mathematical tasks as a set of problems or a single complex problem the purpose of which is to focus students attention on a particular mathematical idea.

4A mathematical task can be defined as one large problem (US Shirts) or as a set of smaller problems (Page 49: 1-37 odd). Regardless of the type of curriculum that a teacher uses, s/he is using some sort of mathematical task.Why Focus on Mathematical Tasks?Tasks form the basis for students opportunities to learn what mathematics is and how one does it

Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information

The level and kind of thinking required by mathematical instructional tasks influences what students learn5This sets the stage for the Task Sort activity that will take place over the next few hours.Professional Practice NormsListening to and using others ideas.

Adopting a tentative stance toward practice -- wondering about the rationale/outcome for others professional decisions instead of espousing certainty and being judgmental about what the teacher was thinking or what you believed should have happened.

Backing up statements with evidence and providing reasoning.

Talking with respect yet engaging in critical analysis of teachers and students portrayed.

6As a facilitator, you will be continually asking participants to defend their solutions. Encourage participants to start asking the same of their group members.

Because of your participants are adults, proper behavior is not the issue in the groups. Willingness to be critical, to demand explanations and evidence, and to push each other to use alternate methods are the biggest challenges. Put this on the table from the start to set the appropriate expectations for their interactions as a group.Comparing Two Mathematical Tasks

7Goals:Raise awareness of how mathematical tasks differ with respect to their levels of cognitive demandHighlight the importance of analyzing and discussing tasks in order to determine the level of thinking required to solve themThe point is not that one type of task is better than another; rather, it is important to know the potential of a task so that it can be appropriately mapped on to the goals for students learning. This activity as raises teachers awareness of worthwhile mathematical tasks as defined by Professional Standards for Teaching Mathematics (NCTM 1991).

Debriefing notes are provided in Characterizing the Cognitive Demands of Mathematical Tasks: A Task Sorting Activity by Smith, Stein, Arbaugh, Brown and Mossgrove in Professional Development Guidebook for Perspectives on the Teaching of Mathematics, Bright, G.W. & Rubenstein, R. (Eds.), Reston, VA: NCTM, 2004.

Comparing Two Mathematical Tasks Solve Two Tasks:

Marthas Carpeting Task

The Fencing Task

8The next four slides show what participants are to do:1. Marthas Carpeting Task2. The Fencing Task3. When participants have completed both tasks, they should discuss the similarities and differences between the two tasks. There is a recording sheet for this activity.

The blank slide can be left up while participants work on the task. The tasks slides are repeated after the blank slide in case you want to use them to begin the debriefing for each task.

Implementation:Begin by having teachers work individually on Marthas Carpeting and the Fencing tasks for five minutes and then continue to work on the tasks with a partner. This gives teachers both a first-hand experience in solving a high-level (Fencing Task) and low-level (Marthas Carpeting task) task and a basis for distinguishing between the levels of tasks during the sorting activity.Comparing Two Mathematical TasksMartha was re-carpeting her bedroom, which was 15.25 feet long and 10.5 feet wide. How many square feet of carpeting will she need to purchase?

Stein, Smith, Henningsen, & Silver, 2000, p. 1Marthas Carpeting Task9Read this together with your group, then go on to the next two slides to set the stage for this first activity.Comparing Two Mathematical TasksAdapted from Stein, Smith, Henningsen, & Silver, 2000, p. 2The (Modified) Fencing TaskMs. Olson's 7th grade class at Roosevelt MS will raise rabbits for their spring science fair. The class will use someportionof the school building as one ofthesides of its rectangular rabbit pen and will use the fencing that was left over from the school play to enclose the other three sides of the pen.

If Ms. Olson's class wants its rabbits to have as much room as possible, what would the dimensions of the pen be? Try to organize your work so that someone else who reads it will understand.10Read aloudHow are Marthas Carpeting Task and the Fencing Task the same and how are they different?

(Consider your own experience in solving the tasks and the mathematical possibilities of the tasks.)

Comparing Two Mathematical Tasks11Use this to set the stage for what participants should be accomplishing during this time. Allow them to begin work now.

When participants have solved each task, they can discuss the similarities and differences with members of their group.

Teachers will have a recording sheet Venn diagram of Marthas Carpeting and Fencing Task.

12You may wish to project this blank slide while participants work on the tasks. Model the kind of facilitation that we expect the teachers to do with their students.

Things to look for as participants work the Fencing Task: Individuals who think squares are not rectangles, so dont consider the 6 x 6 square as a possibility. (Do not clarify this during the set upmuch more effective when groups discuss this . . . This should definitely be discussed during debriefing.)Groups that randomly try rectanglesGroups that systematically try rectangles with different dimensions, e.g., increasing the length by one unit each time, etc.Groups that make a table to work on the problem.Groups that make diagrams to work on the problem. Groups that make a graph to solve the problem (e.g., length as independent variable; area as the dependent variable).Have groups with different strategies present their solutions. Might want them to record their solutions on poster paper.Martha was re-carpeting her bedroom which was 15.25 feet long and 10.5 feet wide. How many square feet of carpeting will she need to purchase?

Stein, Smith, Henningsen, & Silver, 2000, p. 1Marthas Carpeting Task

Comparing Two Mathematical Tasks13Read this together with your group, then go on to the next two slides to set the stage for this first activity.Comparing Two Mathematical TasksAdapted from Stein, Smith, Henningsen, & Silver, 2000, p. 2The Fencing TaskMs. Olson's 7th grade class at Roosevelt MS will raise rabbits for their spring science fair. The class will use someportionof the school building as one ofthesides of its rectangular rabbit pen and will use the fencing that was left over from the school play to enclose the other three sides of the pen.

If Ms. Olson's class wants its rabbits to have as much room as possible, what would the dimensions of the pen be? Try to organize your work so that someone else who reads it will understand.14Read aloudHow are Marthas Carpeting Task and the Fencing Task the same and how are they different?

(Consider your own experience in solving the tasks and the mathematical possibilities of the tasks.)

Comparing Two Mathematical Tasks15Use this to set the stage for what participants should be accomplishing during this time. Allow them to begin work now.

When participants have solved each task, they can discuss the similarities and differences with members of their group.

Teachers will have a recording sheet Venn diagram of Marthas Carpeting and Fencing Task.

Comparing Two Mathematical TasksNot all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.

Stein, Smith, Henningsen, & Silver, 200016Comparing Two Mathematical Tasks

The level and kind of thinking in which students engage determines what they will learn.

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

17Characterizing Tasks

18Sorting Activity: Teachers are now ready to begin work on the sorting activity. Working in pairs or triads, have teachers sort the cards into two groups those they consider to be high-level, and those they consider to be low-level and develop the criteria for each category. As groups finish sorting, have them record their decisions on an overhead.Discussion of Sorting Activity:Note: refer to Smith and Stein (1998) prior to this activity.Orchestrate a whole-group discussion of the sort. Begin by focusing on a task that the majority of the groups characterized as low level. Ask teachers to describe the characteristics of this task while you record their statements on chart paper. Move on to consideration of other tasks that were characterized as low level, adding to and clarifying the criteria as needed. You should then repeat this process for high-level tasks.Once the group has come to some agreement regarding a subset of tasks at each level, you are ready to focus on the tasks for which there is some disagreement. For each task, ask teachers whether it is more like the high-level or the low-level tasks, comparing the characteristics on your list. If teachers indicate that the task is low level, as questions such as What is the rule or procedure you would use to solve the task? or What have you memorized that you are being asked to recall? If teacher indicated that the task is high level, as questions such as What is it you have to think about in order to solve the task? Or What decisions or judgments do you have to make?Participants Anticipated Responses:Results from the entire group will most likely show that the teachers disagree about the placement of a number of tasks. Those differences should prompt rich discussions with regard to analysis and characteristics of different levels of tasks. Disagreements of ten result from making assumptions regarding the cognitive level of a task based on surface features of the task or from equating high level with difficult. For example, some teacher might assume that the feature requires an explanation is always associated with tasks with high-level demands. Although many tasks in the sorts are consistent with this view, others can serve as counterexamples to this assertion. For example, in the middle school sort task A (high-level) and task N (low-level), both require an explanation. The point is to dig beneath the surface in determining the level of thinking required to complete a task. The following questions should foster lively discussion about these issues.Does a particular feature (e.g., writing an explanation as part of your answer, drawing a picture to explain what you did, using manipulatives to solve the task) indicate that the task has a certain level of cognitive demand?Is there a difference between level of cognitive demand and difficulty?What effect does context (e.g., setting in which the task is used, students prior experience, grade level) have on the level of cognitive demand required by a task?Answers to the Task-Sorting Activity:Note: these answers are for the use of the facilitator in orchestrating the discussion. The goal of this activity is for teachers to participate in a thoughtful analysis of the tasks, not to come to a consensus about the placement of tasks. Also note that the characteristics of the tasks have been identified so as to facilitate the identification of counterexamples (e.g., if a teacher claims that manipulatives are only used in high-level tasks, you can use the matrix to identify a low-level task that also uses manipulatives).Characterizing TasksGoals: Identify characteristics of high and low level mathematical tasks.

Marthas Carpet: Low levelThe Fencing Task: High level

19Dont push for a complete understanding of High vs. Low. You are looking for teachers to begin to think about the similarities and differences. You will add to this discussion after the Task Sort activity.

Sort Tasks A P into two categories: high level and low level.

Develop a list of criteria that describe in general the characteristics of low level and high level tasks.

Characterizing Tasks20In this activity, participants will identify characteristics of high/low level tasks inductively: first decide whether each task is high or low level, then develop a set of criteria for each type based on the tasks identified as high level or low level.All the materials to actually DO a sorting task with teachers appears in the NCTM 2004 yearbook companion --Professional Development Guidebook for Perspectives on Teaching of Mathematics. CHAPTER 6. Also contains detailed facilitator notes.There are three different sets of tasks: elementary, middle, and high. The middle school tasks are most appropriate for this course. Participants dont have to completely solve problem before categorizing it; however, they may want to try some problems to better understand them.Participants to should work in pairsnot table groupsto categorize tasks. Reason: the more classifications, the more interesting the discussion. Each pair records their classification on the overhead recording sheet, or poster paper recording sheet.Tip: Helpful to keep a copy of results from a larger group for future use if needed. The sort works best with a large group; if you have less than 10 participants (i.e.., fewer than 5 groups), helps to augment their sort with results from larger group.

You can have pairs vote by using dots on a poster, using a transparency, or using a document camera. Either way, you must ensure that the pairs who are still working cannot see the votes of the pairs who have already finished. 21Debrief task sort:

Start with task that majority characterized as low level. (Ideally, there will be at least one task for which there is consensus.) Ask to describe characteristicsrecord on chart paper.Move on to other low level tasks for which there is consensus or strong majority.Repeat process for high level tasks for which there is consensus or near consensus.Then move to tasks upon which there is disagreement. For each task, ask why task was classified as high/low (give groups that did each a chance to respond). Then ask if the task is more like the high or low tasks for which there is consensus . . . If participants think task is low, ask What rule or procedure would you use to solve the task? Or What have to memorized that you are being asked to recall?If high, ask What is it you have to think about in order to solve the task? or What decisions or judgments did you have to make?This discussion can be longer or shorter depending on time available. See Smith et al for important points to make explicit during the discussion.Categorizing TasksAre all high-level tasks the same? [Is there an important difference between Tasks J and M?]

Are all low-level tasks the same? [Is there an important difference between Tasks E and O?]22Participants do not need to differentiate between the four levels this is a guideline for what you are looking for. Keep in mind that easy and hard activities do not mean low or high. You will differentiate the four levels in a few slides. Look for the following language when describing levels:Memorization Definitions to memorizeNot ambiguousNo connection to concept or meaningProcedures without connections Algorithmic Limited cognitive demandLittle ambiguityNo connection to concept or meaningFocused on what and not why no explanations or explanations that just describe proceduresProcedures With Connections Suggest pathways that are broad explicitly or implicitlyRepresented in multiple waysProcedures can not be followed mindlesslyRequires the engagement of the conceptual ideas connections!Focus students attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideasDoing MathematicsNot predictable pathwayRequires students to explore and understand the nature of conceptsRequires students to access knowledge and experiencesAnalyze, justify, Requires considerable cognitive effort

Levels of Cognitive Demand&The Mathematical TasksFramework

23Categorizing TasksIf 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

24Low-Level TasksMemorizationProcedures without Connections (e.g., Marthas Carpeting Task)

High-Level TasksProcedures with ConnectionsDoing Mathematics (e.g., The Fencing Task)Linking to ResearchThe QUASAR Project25Page 269 Stein and Smith -- example of task at each of the four levels.

May also want teachers to review the tasks they sorted and identify a task that fits in each category.

After this discussion has taken place, hand out the one page Task Analysis GuideThe Mathematical Tasks Framework

TASKS as they appear in curricular/ instructional materialsTASKS as set up by the teachersTASKS as implemented by studentsStudent LearningStein, Smith, Henningsen, & Silver, 2000, p. 4Linking to ResearchThe QUASAR Project26Reference Stein and Smith article -- figure 2, page 270The Mathematical Tasks Framework

TASKS as they appear in curricular/ instructional materialsTASKS as set up by the teachersTASKS as implemented by studentsStudent LearningStein, Smith, Henningsen, & Silver, 2000, p. 4Linking to ResearchThe QUASAR Project27Reference Stein and Smith article -- figure 2, page 270The Mathematical Tasks Framework

TASKS as they appear in curricular/ instructional materialsTASKS as set up by the teachersTASKS as implemented by studentsStudent LearningStein, Smith, Henningsen, & Silver, 2000, p. 4Linking to ResearchThe QUASAR Project28Reference Stein and Smith article -- figure 2, page 270

Facilitator might say - Back to the Fencing Task - if I would have started that task by saying remember that squares are rectangles or everyone draw a picture of each rectangle any statement of this kind leads the learner down the pathway without allowing them to struggle and work on their own first.

The Mathematical Tasks Framework

TASKS as they appear in curricular/ instructional materialsTASKS as set up by the teachersTASKS as implemented by studentsStudent LearningStein, Smith, Henningsen, & Silver, 2000, p. 4Linking to ResearchThe QUASAR Project29Reference Stein and Smith article -- figure 2, page 270

The important point here is that it is the teacher that lowers the demand at this level by how he/she responds to the students working on the task.

One example of a teacher move that lowers the demand at this stage would be Facilitator says, alright guys, no one knows how to do number 5! Let me set it up for you and see if you can do the calculations and get an answer.

The Mathematical Tasks Framework

TASKS as they appear in curricular/ instructional materialsTASKS as set up by the teachersTASKS as implemented by studentsStudent LearningStein, Smith, Henningsen, & Silver, 2000, p. 4Linking to ResearchThe QUASAR Project30Reference Stein and Smith article -- figure 2, page 270Stein & Lane, 1996HighLowModerateHighHighHighLowLowLowTask Set-UpTask ImplementationStudent LearningLinking Task Level to Student Achievement31This slide is critical---Its the Big Idea. The reason the task level and implementation levels are important is because they are related to student learning.

The third level is different from a high-level task disintegrating to nothing. This is turning a high-level task into a procedural task.

Take a minute to go back and reinforce the goals of the workshop. We want teachers to not only gain a deeper mathematical understanding but see the importance of the Pedagogy (delivery of the lessons) that is also very critical. You as the facilitator will continue to model this type of behavior throughout the workshop.

The instructional tasks teachers select are crucial in helping students make connections and learn important mathematics concepts.

Tasks that engage students in thinking about the defining characteristics of important mathematical concepts help students develop a deep understanding of core mathematical ideas that increase retention and transfer of knowledge to new situations.

Linking to Research32One conclusionEffective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.

NCTM, 2000, p.16Linking to Research33Enacting MathematicalTasks

34In this segment, teachers will watch a video lesson of a high level task and discuss:the extent to which the teacher maintained the task level during implementation and the instructional strategies used during the lesson.

Currently, this section uses a fourth/fifth grade lesson, Geoboard Fractions, because the video is available. Eventually, this lesson will be replaced by a middle grades lesson.

Materials needed for this lesson: Geoboards and/or geoboard recording sheet. Transparency of geoboard recording sheet Videotape of lesson (or digitized video)

Teachers do the task first, then discuss it. After debriefing, they watch the video lesson and discuss.A optional follow-up task is included if there is time for it.

The S-Pattern TaskDetermine the number of tiles in the next 2 figures.

Describe the 20th figure in this pattern, including the number of tiles it contains and how they are arranged. Explain the reasoning that you used to determine this information.35Teachers notes give several solutions with connections between diagrams and expressions. The included notes vary slightly from the task above. In the given notes below, Term 1 in our pattern is Term 2 in their pattern. The mathematics is the same; there is just a shift in the term numbers due to a differing starting shape.

The S-Pattern TaskWrite an equation for the number of tiles needed for any figure in the pattern. Explicitly connect your equation to the diagrams.Generate as many different equations as you can for this relationship.36Teachers notes give several solutions with connections between diagrams and expressions. The included notes vary slightly from the task above. In the given notes below, Term 1 in our pattern is Term 2 in their pattern. The mathematics is the same; there is just a shift in the term numbers due to a differing starting shape.

The S-Pattern TaskMake a graph that shows the relationship between the figure number and the number of tiles in the figure.

Share your solutions with your group.

37

The S-Pattern TaskMake a graph that shows the relationship between the figure number and the number of tiles in the figure.

Share your solutions with your group.

38

Group PosterShow all the expressions generated by your group.

Explicitly connect your expressions to the diagrams.

39Have participants organize their posters to show the connections between their expressions and their diagrams.

Gallery WalkOne person from each group stays with the groups poster to answer questions.

Rest of the group members view other posters. Look for:The most common representationsThe most unusual/surprising representations40

A Graph of the PatternThe S-Pattern Task41Another Graph of the Pattern

The S-Pattern Task42This graph extends the graph of the problem situation to consider the relationship if n can be any rational number (not just a whole number). Use this as an opportunity to discuss why the graph of the problem situation only contains isolated points and this graph shows a curve.

A Graph of All ValuesThe S-Pattern Task43How does this graph compare to the previous graphs?Could this be the graph of the problem situation? (No, because it shows negative values and N cant be negative.)This highlights that the graph of a problem situation may be different from that of the general mathematical relationship that describes the situation.How would you characterize the level of this task: High or low cognitive demand?What mathematical ideas are embedded in the task?What makes this worthwhile mathematics?De-Briefing the S-Pattern Task44De-Briefing the S-Pattern TaskWhat connections do you see to the Common Core Standards for Mathematical Practice?

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