Math Tasks

Allegheny Valley Curriculum WritingDay 1Rigorous/High Level TaskMath & Science Collaborative at the Allegheny Intermediate Unit1

GoalsUnderstand the importance of engaging students in rigorous and relevant tasks and activities.Recognize the characteristics of rigorous, high-level tasks in mathematicsUnderstand the importance of engaging students in high-level tasks in order to more deeply learn the content and see relevance to the real world.

Math & Science Collaborative at the Allegheny Intermediate Unit2GoalsDevelop a deep understanding of the PA Core Standards, Keystone Assessment Anchors and Eligible Content, Understand the importance of engaging students in the Standards for Mathematical Practice as the means of learning important content.Understand the focus and coherence of the PA Core standards. Become familiar with Learning Progressions as narrative documents describing the progression of a topic across grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics

Math & Science Collaborative at the Allegheny Intermediate Unit33GoalsWrite Curriculum and Plan Lessons/UnitsUtilize the different Curriculum Maps from PA and other states to organize curriculum writingUtilize the Learning Progressions as touchstone documents to assist with curriculum writing Select high-level tasks to include in lessons/units from a variety of vetted resourcesMath & Science Collaborative at the Allegheny Intermediate Unit4Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003

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 and Briars, 1995Analyzing Mathematical Tasks5At the heart of teaching well is the core challenge of getting learners engaged in productive work. Mathematical tasks are the sites for engaging students in core mathematical activity. Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003What are mathematical tasks?We define mathematical tasks as a set of problems or single complex problem the purpose of which is to focus students attention on a particular mathematical idea.

6Mathematical tasks can be examined from a variety of perspectives including the number and kinds of representations evoked, the variety of ways in which they can be solved, and their requirements for student communication.Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003Why 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; andThe level and kind of thinking required by mathematical instructional tasks influences what students learn.7*The day-in and day-out cumulative effect of classroom-based tasks leads to the development of students implicit ideas about the nature of mathematics - about whether mathematics is something about which they can personally make sense and about how long and how hard they should have to work to do so. * If we choose to give students well defined problems in which they will utilize previously learned procedures, this is what they will come to understand as doing mathematics. When they find themselves in a problem situation which is somewhat ambiguous, they will quickly give up and come to believe that they cannot deal with mathematically ambiguous problems. Will this approach to mathematics serve well our 21st century students of mathematics?Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003

Comparing Two Mathematical Tasks8Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003

Comparing Two Mathematical Tasks Solve Two Tasks:

Hundreds, Tens and OnesMuffles Truffles

910Here is an example of a place value chart that you get when you search for place value worksheets online. It is also a non-example of work that would elicit conceptual understanding. As you can see, it would not be possible to assess whether your students had a conceptual understanding of place value by them completing this worksheet. It would be fairly obvious to a student who does not understand place value that the first number goes with hundreds, the 2nd number with tens and so on. Even on problem letter h, where it could have asked for deeper understanding, the worksheet places a 0 for tens to eliminate any need for thinking.

Muffles TrufflesHere are the truffles that Muffles assistant Patricio needs to package: 218 raspberry truffles 132 strawberry truffles 174 dark chocolate truffles 83 vanilla truffles with cinnamon and nutmeg 126 green truffles with pistachios 308 truffles with pecans and caramel 97 butterscotch crunch truffles covered in milk chocolate 22 truffles with white and dark chocolate swirls 44 chocolate-covered cherry truffles 46 almond and raisin trufflesTell story before clicking to get the list11Muffles Truffles How many boxes does Patricio need for each flavor? How many leftovers of each kind will there be?

Is there a shortcut way to know how many boxes of each kind he needs to pack and how many leftovers there will be for the assortment boxes?

How many assortment boxes can he make?

Muffles sells his fancy truffles for $1.00 each so his boxes of truffles cost $10 each. How much money will he collect if he sells them all?Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003Comparing Two Mathematical TasksHow are Hundreds, Tens and Ones and Muffles Truffles the same and how are they different?

13Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003

Comparing Two Mathematical TasksDo the differences between Muffles Truffles and Hundreds, Tens and Ones matter?

Why or Why not?14Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003

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, 200015This is the math classroom version of You reap what you sow. Do we want our students to be mathematical problem solvers ? What types of mathematical tasks will build problem solving skills?Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003Comparing Two Mathematical TasksThe level and kind of thinking in which students engage determines what they will learn.

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

16Components of a Math TaskDeveloping the ContextSupporting the InvestigationPreparing for the math congressFacilitating the math congressIntegrating mini-lessons, games and routinesMath & Science Collaborative at the Allegheny Intermediate UnitToday, we are going to examine a set of lessons that use math tasks as their basis. The components are listed here. Well talk more about each component as we work through the tasks or investigations.17Components of a Math TaskDeveloping the ContextCan use stories, situations (realistic or fictional), contexts, modelsChildren work to explore and make sense of the situationsThey try out strategies to solve and make sense of the use of the strategiesThey explore and generate patternsThey generalizeAnd mathematizeSupporting the InvestigationPreparing for the math congressFacilitating the math congressIntegrating mini-lessons, games and routines

Math & Science Collaborative at the Allegheny Intermediate UnitThe context is established at the beginning. Often, this portion is called the LAUNCH of the task or the lesson.

Our context is Muffles Truffles task, as taken from the Investigating Multiplication and Division series from Contexts for Learning Mathematics by Catherine Fosnot and colleagues at City College of New York. Developed through the Math in the City project mitcccny.org18Muffles TrufflesUse blank paper to record your own thinking about solving the task. Actually do the work to solve the problem yourself. Think as a learner.Share your solution strategies with your small groupMath & Science Collaborative at the Allegheny Intermediate Unit19AnticipatingActively envision how students might mathematically approach the instructional task or tasks that they will work on Involves developing considered expectations about how students might mathematically interpret a problem, the array of strategies both correct and incorrect that they might useand how those strategies relate to the mathematical concepts, representation, procedures and practices5 Practices for Orchestrating Productive Mathematics Discussions by Margaret Smith and Mary Kay SteinMath & Science Collaborative at the Allegheny Intermediate UnitYou have to do the task before you give it to kidsYou have to think about it as a learner and as a facilitator20Muffles TrufflesDo some AnticipatingTogether, think as teachers. Strategize about other solution strategies that students in your grade 3 or 4 classroom might use.Record these strategies for yourself.

Now, examine the student work samples.Were you able to anticipate these strategies? If not, no worries. Just by doing the anticipating, you are more prepared to deal with unanticipated strategies.Math & Science Collaborative at the Allegheny Intermediate UnitStrategies that students might use to determine the number of boxes:Drawing boxes and filling them one by oneDrawing boxes and labeling them as ten (doing same for each type of truffle) [there is a student work sample showing this]Using additive reasoning or expanded notation [student work sample 2 shows this]Combing the individual leftover truffles for assortment boxes, but not realizing to do this for the full boxes as wellAdd up all the individual numbers and see how many tens and leftoversUnitize with tens for all the individual numbers, so you do not have to find the total number of truffles [student work sample 3 shows this]

To determine money, students might:Count by onesSkip count by tensCount number of boxes and multiply by ten (either by kind or as a total of all the boxes)

21Components of a Math TaskDeveloping the ContextSupporting the Investigation AKA: Monitoring Facilitator observes strategiesListens to discussionsConfers with pairs or small groupsAsk questions and make comments (not leading ones)Help me understand your methodWhat made you decide to use that strategy?Preparing for the math congressFacilitating the math congressIntegrating mini-lessons, games and routinesMath & Science Collaborative at the Allegheny Intermediate UnitMake sure student work spaces are conducive for pairs or small groups (tables, floor spaces)Make sure students have or can easily get the needed materials so they do not have to rely on you at every turnMay need to get easily-distracted students started first

Make sure to ask questions of both partnersWork with the mathematician Dont just try to fix the mathematics

Help students stay connected to the context while mathematizing. Connection to math practice 2. 22Sample student dialogueExamine the snip-it of conversation on the Conferring with Students at Work handout. This is from Tonis monitoring of the students while they work.

What is Toni doing to assist his/her students?What isnt Toni doing to assist his/her students?Math & Science Collaborative at the Allegheny Intermediate UnitComponents of a Math TaskDeveloping the ContextSupporting the InvestigationPreparing for the math congressHave children talk with partners about what they want to share (can use posters or paper with doc camera)Might do a gallery walkFacilitator decide what ideas to have shared and what order to have the ideas shared AKA: Selecting and SequencingFacilitating the math congressIntegrating mini-lessons, games and routinesMath & Science Collaborative at the Allegheny Intermediate UnitHave participants actually do this now.

Preparing for the congress allows for further reflection and some metacognition to take place about what students' thought processes were, what is important and what is not. They refine their thinking. They develop their argument.For gallery walk, students can write comments or questions about the math and / or strategy and stick those on poster (sticky notes) They get better at this with experience. At first, the comments will be trivial, but will become more relevant and helpful.

Attempt to use an order of sharing that will enable students to make connections and ask questions to bring these connections outand perhaps even make generalizationsDO share error-prone strategies and misconceptions! Dont necessarily fix these, allow the students to see them and have them comment (ask them to reflect on inconsistencies and answers that are not reasonable.)24Components of a Math TaskDeveloping the ContextSupporting the InvestigationPreparing for the math congressFacilitating the math congressNOT a whole class shareDesigned to push the math development of studentsMany possible structures to use Make sure to ask questionsHelp students make connections among ideas, among strategies, among representations, etc.Integrating mini-lessons, games and routinesMath & Science Collaborative at the Allegheny Intermediate UnitMath congress is NOT a whole class share too much time, too repetive

Possible structures include:From least to most efficientProgressively focusing more and more on the relevant big idea (example multiplication with turkey task)Use of different representations and how they are connected to one another (concrete, maybe representational, maybe abstract) but do NOT necessarily have one in mind that you want all students to use.25Math CongressExamine the sample excerpt from the Math Congress in Tonis class, entitled A Portion of the Math Congress

What is Toni doing to assist his/her students?What isnt Toni doing to assist his/her students?

Math & Science Collaborative at the Allegheny Intermediate UnitWhat is the math in this task?CONTENTPlace Value patterns, especially with groups of tenUnitizingQuotative division Finding how many groupsMath & Science Collaborative at the Allegheny Intermediate UnitBy nature, quotative division problems are easier for students than partitive (fair-share, how many in a group) problems because they know what is in the group and can use this quantity to reason with.27Math CongressWhat would you focus on in the Math Congress for this task?UnitizingPlace valueConnection in this task is the fact that the number of tens (the unit)is the number of full boxes

Might move from least to most efficient strategiesMath & Science Collaborative at the Allegheny Intermediate UnitWhat is the math in this task?PRACTICESRead the elementary elaborations (draft) of the Standards for Mathematical Practices.Which Practices were STRONGLY exhibited in the Muffles Truffles task and debrief? How? Some of them will NOT be strongly exhibited.Math & Science Collaborative at the Allegheny Intermediate UnitDivide the practices so that everyone is reading 2 or 3 of them29Some general questionsWho can put what Sarah just said into your own words?Who has a question or comment for Daniel?Who agrees with Ashley, but used a different strategy?Who still needs convincing that Carmines strategy will work?Will Baileys strategy always work? How do you know for sure?Math & Science Collaborative at the Allegheny Intermediate UnitTalk in general at this point about the facilitation of a task.

Here are some general questions to consider while monitoring and sharing strategies

Keeping these general questions as well as the three variations on Muffles Truffles that we have encountered, What math practices might we be working on here?

Who can think of some questions that could focus on SMP 2? SMP 8?30Components of a Math TaskDeveloping the ContextSupporting the InvestigationPreparing for the math congressFacilitating the math congressIntegrating mini-lessons, games and routinesCan be used at the start of a lesson for 10-15 minutesDesigned to highlight a particular computational strategyDesigned to help build fluency Might help with mental mathMake sure to structure the games so that they actually support strategies and discussions, not just fact practiceMath & Science Collaborative at the Allegheny Intermediate UnitTodays mini-lesson is students sitting around a circle and counting. See handout describing the mini-lesson and with a dialogue included.

As you can see, after this mini-lesson, students are introduced to a variation on Muffles original task. Now, his boxes are of varying sizes. Read that part of the handout, if you have not done so already.31Multiplication and Division Word Problem Types

Rigor What it isMath & Science Collaborative at the Allegheny Intermediate Unit33Rigor refers to academic rigor learning in which students demonstrate a thorough, in-depth mastery of challenging tasks to develop cognitive skills through reflective thought, analysis, problem-solving, evaluation, or creativity. Rigorous learning can occur at any school grade and in any subject.(Rigor/Relevance Framework- International Center for Leadership in Education, Dr. Bill Daggett) 3 aspects of Rigor, defined by CCSSM, are:Conceptual understandingProcedural skill and fluencyApplications

3 aspects of Rigor as defined by CCSSM : conceptual understanding, procedural skill and fluency, applications.To help students meet the expectations of the Standards, educators will need to pursue, with equal intensity, three aspects of rigor in the major work of each grade33Characterizing TasksMath & Science CollaborativeSAS Secondary Mathematics Teacher Leadership Academy, Year 134We would like to engage you in an activity which is about characterizing mathematical tasks. The goal of the activity is for you to participate in a thoughtful analysis of the tasks. Sort the Tasks into two categories [high level and low level]

Develop a list of criteria that describe the tasks in each category

Math & Science CollaborativeCharacterizing TasksSAS Secondary Mathematics Teacher Leadership Academy, Year 135Categorizing 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

Math & Science CollaborativeSAS Secondary Mathematics Teacher Leadership Academy, Year 136Tasks that require students to perform a memorized procedure in a routine manner lead to one type of opportunity for student thinking; tasks that demand engagement with concepts and that stimulate students to make purposeful connections to meaning or relevant mathematical ideas lead to a different set of opportunities for student thinking.Categorizing TasksAre all high-level tasks the same? [Is there an important difference between Tasks H and I?]

Are all low-level tasks the same? [Is there an important difference between Tasks E and O?]

37Levels of Cognitive Demand &The Mathematical Tasks FrameworkMath & Science CollaborativeLinking to Literature/Research: The QUASAR ProjectLow-Level Tasks

High-Level Tasks

Math & Science CollaborativeSAS Secondary Mathematics Teacher Leadership Academy, Year 139When determining the level of cognitive demand provided by a mathematical task, it is important not to become distracted by superficial features of the task and to keep in mind the students for whom the task is intended.

Low-level tasks, for example, can appear to be high-level when they have characteristics of reform-oriented instructional tasks such as requiring the use of manipulatives; using real-world contexts; involving multiple steps, actions or judgments; an/or making use of diagrams. ( use 2 elementary tasks as an example)

It is also possible for tasks to be designated low-level when in fact they should be considered high level. (use 2 elementary tasks as an example)

Another consideration when deciding the level of challenge provided be a task is the students and the norms and expectations for work in their classroom. The age, grade-level, prior knowledge and experiences need to be taken into consideration when deciding whether the task is likely to provide an appropriate level of challenge for their students. Linking to Literature/ Research: The QUASAR ProjectLow-Level Tasksmemorizationprocedures without connectionsHigh-Level Tasksprocedures with connectionsdoing mathematics Math & Science CollaborativeSAS Secondary Mathematics Teacher Leadership Academy, Year 140Talk through the characteristics given in the task analysis guide. A copy of this guide should probably be given to participants.

Discuss how the high level tasks constitute rich materials and resources which would support student learning. Also discuss how these types of tasks should not be the add-ons but rather the content of core instruction (RTI Tier I)

Cite which of the previous tasks were at which levelsMemorizationWhat are the decimal equivalents for the fractions and ?Procedures without connectionsConvert the fraction 3/8 to a decimal.Lower-Level TasksThere is a place for lower-level tasks, but they should not be the main portion of the curriculum41Procedures with connectionsUsing a 10 x 10 grid, identify the decimal and percent equivalents of 3/5.Doing mathematicsShade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine:The decimal part of area that is shaded;The fractional part of area that is shaded.Higher-Level TasksMath PracticesHow does using High Level Tasks allow you to engage students in the Math Practices?Allow participants to give you reasons as to why the high level tasks support students engagement with the SMPs. They might say things like, 43Linking to Literature/ Research: The QUASAR ProjectThe Mathematical Tasks Framework

Math & Science CollaborativeTASKS as they appear in curricular/ instructional materialsTASKS as set up by the teachersTASKS as implementedby studentsStudent LearningStein, Smith, Henningsen, & Silver, 2000, p. 4SAS Secondary Mathematics Teacher Leadership Academy, Year 144The Math Tasks Framework is designed to consider the evolution of tasks during a lesson. The fact that tasks take on lives of their own after being introduced into classroom settings has been noted by a variety of classroom researchers.

As mathematical tasks are enacted in classroom settings, they become intertwined with the goals, intentions, actions, and interactions of teachers and students.

We will now consider each phase of the math task framework.Math & Science CollaborativeTIMSS Video Study

FIGURE 1 Types of Math Problems PresentedFIGURE 2 How Teachers ImplementedMaking Connections Math ProblemsSAS Secondary Mathematics Teacher Leadership Academy, Year 145To highlight how the math task framework plays out in the United States this TIMSS research shows what types of task we use in the United States as well as how they are implemented. Although we are in line with other high achieving countries in terms of the number of high level tasks we use, we do not implement any of them at a high level. We tend to take the struggle out of the mathematics in our country.

The results of the recent TIMSS video study provide additional evidence of the relationship between the cognitive demands of mathematical tasks and student achievement. In this study, a random sample of 100 8th grade mathematics classes from each of six countries (Australia, the Czech Republic, Hong Kong, Japan, the Netherlands, Switzerland) and the United States, were videotaped during the 1999 school year. The six countries were selected because each performed significantly higher than the U.S. on the TIMSS 1995 mathematics achievement test for eighth grade (Stigler & Hiebert, 2004). The study revealed that the higher-achieving countries implemented a greater percentage of making connections tasks in ways that maintained the demands of the task. With the exception of Japan, higher-achieving countries did not use a greater percentage of high-level tasks than in the U.S. All other countries were, however, more successful in not reducing these tasks into procedural exercises. Hence, the key distinguishing feature between instruction in the U.S. and instruction in high achieving countries is that students in U.S. classrooms rarely spend time engaged in the serious study of mathematical concepts (Stigler & Hiebert, 2004, p. 16).

Approximately 17% of the problem statements in the U.S. suggested a focus on mathematical connections or relationships. This percentage is within the range of many higher-achieving countries (i.e., Hong Kong, Czech Republic, Australia).

Virtually none of the making-connections problems in the U.S. were discussed in a way that made the mathematical connections or relationships visible for students. Mostly, they turned into opportunities to apply procedures. Or, they became problems in which even less mathematical content was visible (i.e., only the answer was given).

Other findings from the TIMSS research are addressed on the next slides.\ASK HOW UNDERSTANDING THE PRACTICES WILL HELP TEACHERS KNOW HOW TO KEEP High Level Tasks at a high level when implementedDoes Maintaining Cognitive Demand Matter?YES4646Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003Factors Associated with the Maintenance and Decline of High-Level Cognitive DemandsRoutinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answerProviding insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behaviorEngaging in high-level cognitive activities is prevented due to classroom management problemsSelecting a task that is inappropriate for a given group of studentsFailing to hold students accountable for high-level products or processes

Stein, Grover & Henningsen, 19964747This slide identifies the classroom factors that researchers observed when high-level tasks declined during a lesson. Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003Factors Associated with the Maintenance and Decline of High-Level Cognitive DemandsScaffolding of student thinking and reasoningProviding a means by which students can monitor their own progressModeling of high-level performance by teacher or capable studentsPressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedbackSelecting tasks that build on students prior knowledgeDrawing frequent conceptual connectionsProviding sufficient time to explore Stein, Grover & Henningsen, 1996

4848By contrast, this slide identifies the classroom factors that researchers observed when the cognitive demands of high-level tasks were maintained during a lesson.

Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003Factors Associated with the Maintenance and Decline of High-Level Cognitive DemandsDecline MaintenanceRoutinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answerProviding insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behaviorEngaging in high-level cognitive activities is prevented due to classroom management problemsSelecting a task that is inappropriate for a given group of studentsFailing to hold students accountable for high-level products or processes

Scaffolding of student thinking and reasoningProviding a means by which students can monitor their own progressModeling of high-level performance by teacher or capable studentsPressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedbackSelecting tasks that build on students prior knowledgeDrawing frequent conceptual connectionsProviding sufficient time to explore 4949Developed under the auspices of the NSF-funded ESP Project (ESI - 0301962) -- Directed by Margaret Smith, University of Pittsburgh, 2003Stein & Lane, 1996A.B.C.HighHighLowLowHighLowModerateHighLowTask Set UpTask ImplementationStudent LearningPatterns of Set up, Implementation, and Student Learning5050Evidence gathered across scores of middle school classrooms in four QUASAR middle schools has shown that students who performed the best on project-based measures of reasoning and problem solving were in classrooms in which tasks were more likely to be set up and implemented at high levels of cognitive demand. Results from QUASAR also show that students who had the lowest performance on project assessments were in classrooms where they had limited exposure to tasks that required thinking and reasoning (Stein & Lane, 1996).

Math & Science CollaborativeConclusionNot all tasks are created equal -- they provided different opportunities for students to learn mathematics.High level tasks are the most difficult to carry out in a consistent manner. Engagement in cognitively challenging mathematical tasks leads to the greatest learning gains for students.Professional development is needed to help teachers build the capacity to enact high level tasks in ways that maintain the rigor of the task.Being cognizant of the factors that lead to maintenance of the cognitive demands of the task leads to higher students achievementSAS Secondary Mathematics Teacher Leadership Academy, Year 15151Math StandardsReexamine some of the high level tasks from the card sort and discuss the SMPs and content standards that are there.Math & Science CollaborativeCould use this slide to talk about which tasks best support the implementation or use of a given SMP. Which of these tasks best supports the implementation of SMP 1?...SAS Secondary Mathematics Teacher Leadership Academy, Year 152Math & Science CollaborativeAdditional Articles and Books about the Mathematical Tasks FrameworkResearch Articles

Boston, M.D., & Smith, M.S., (in press). Transforming secondary mathematics teaching: Increasing the cognitive demands of instructional tasks used in teachers classrooms. Journal for Research in Mathematics Education.

Stein, M.K., Grover, B.W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2(1), 50 - 80.

Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.

SAS Secondary Mathematics Teacher Leadership Academy, Year 15353Math & Science CollaborativeAdditional Articles and Books about the Mathematical Tasks FrameworkPractitioner ArticlesStein, M. K., & Smith, M.S. (1998). Mathematical tasks as a framework for reflection. Mathematics Teaching in the Middle School, 3(4), 268-275.

Smith, M.S., & Stein, M.K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344-350.

Henningsen, M., & Stein, M.K. (2002). Supporting students high-level thinking, reasoning, and communication in mathematics. In J. Sowder & B. Schappelle (Eds.), Lessons learned from research (pp. 27 36). Reston VA: National Council of Teachers of Mathematics.

Smith, M.S., Stein, M.K., Arbaugh, F., Brown, C.A., & Mossgrove, J. (2004). Characterizing the cognitive demands of mathematical tasks: A sorting task. In G.W. Bright and R.N. Rubenstein (Eds.), Professional development guidebook for perspectives on the teaching of mathematics (pp. 45-72). Reston, VA: NCTM.

SAS Secondary Mathematics Teacher Leadership Academy, Year 15454Math & Science CollaborativeAdditional Books about the Mathematical Tasks FrameworkBooksStein, M.K., Smith, M.S., Henningsen, M., & Silver, E.A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.

Smith, M.S., Silver, E.A., Stein, M.K., Boston, M., Henningsen, M., & Hillen, A. (2005). Cases of mathematics instruction to enhance teaching (Volume I: Rational Numbers and Proportionality). New York: Teachers College Press.

Smith, M.S., Silver, E.A., Stein, M.K., Henningsen, M., Boston, M., & Hughes,E. (2005). Cases of mathematics instruction to enhance teaching (Volume 2: Algebra as the Study of Patterns and Functions). New York: Teachers College Press.

Smith, M.S., Silver, E.A., Stein, M.K., Boston, M., & Henningsen, M. (2005). Cases of mathematics instruction to enhance teaching (Volume 3: Geometry and Measurement). New York: Teachers College Press.

SAS Secondary Mathematics Teacher Leadership Academy, Year 15555Math & Science CollaborativeAdditional References Cited in This Slide Show

Hiebert, J., Carpenter, T.P., Fennema, D., Fuson, K.C., Wearne, D., Murray, H., Olivier, A., Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Lappan, G., & Briars, D.J. (1995). How should mathematics be taught? In I. Carl (Ed.), 75 years of progress: Prospects for school mathematics (pp. 131-156). Reston, VA: National Council of Teachers of Mathematics.

Stigler, J.W., & Hiebert, J. (2004). Improving mathematics teaching. Educational Leadership, 61(5), 12-16.

TIMSS Video Mathematics Research Group. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 Video Study. Washington, DC: NCES. Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of Railside School. Teachers College Record, 110(3), 608-645.SAS Secondary Mathematics Teacher Leadership Academy, Year 15656Chart3611577168413415457246917

Using ProceduresMaking Connections

Sheet1FIGURE 1 Types of Math Problems PresentedSeriesAustraliaCzech RepublicHong KongJapanNetherlandsUnited StatesUsing Procedures617784415769Making Connections151613542417SeriesAustraliaCzech RepublicHong KongJapanNetherlandsUnited StatesUsing Procedures311618201959Making Connections8524648370

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Using ProceduresMaking Connections

Sheet1FIGURE 1 Types of Math Problems PresentedSeriesAustraliaCzech RepublicHong KongJapanNetherlandsUnited StatesUsing Procedures617784415769Making Connections151613542417SeriesAustraliaCzech RepublicHong KongJapanNetherlandsUnited StatesUsing Procedures311618201959Making Connections8524648370

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Using ProceduresMaking Connections

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Using ProceduresMaking Connections

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