Effective Mathematics Instruction: The Role of Mathematical Tasks* *Based on research that undergirds the cases found in Implementing Standards-Based Mathematics Instruction (Stein, Smith, Henningsen, & Silver, 2000). This presentation accompanies “Implementing Standards-Based Mathematics Instruction,” (Teachers College Press: 2009). For more information or to order, please visit: http://store.tcpress.com/0807749575.shtml

Why Instructional Tasks are Important

Comparing Two Mathematical Tasks Martha’s Carpeting Task The Fencing Task NOTE TO SPEAKER: Have the two mathematical tasks printed on paper and ready to hand out to your participants.

Martha’s Carpeting Task Martha was recarpeting her bedroom, which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase?

The Fencing Task Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen to keep the rabbits. If Ms. Brown’s students want their rabbits to have as much room as possible, how long would each of the sides of the pen be? How long would each of the sides of the pen be if they had only 16 feet of fencing? How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.

Comparing Two Mathematical Tasks Think privately about how you would go about solving each task (solve them if you have time) Talk with you neighbor about how you did or could solve the task Martha’s Carpeting The Fencing Task NOTE TO SPEAKER: If there is time, you may want to have participants share their solution strategies on a blackboard or on large pieces of chart paper before showing the solutions that appear on the upcoming slides.

Solution Strategies: Martha’s Carpeting Task

Martha’s Carpeting Task Using the Area Formula A = l x w A = 15 x 10 A = 150 square feet

Martha’s Carpeting Task Drawing a Picture 10 15

Solution Strategies: The Fencing Task

The Fencing Task Diagrams on Grid Paper

The Fencing Task Using a Table Length Width Perimeter Area 1 11 24 2 10 20 3 9 27 4 8 32 5 7 35 6 36 The table shows that all the configurations have a perimeter of 24, but different areas. The area for the 6 x 6 pen is the largest; both before and after that, the areas are smaller than 36 square feet.

The Fencing Task Graph of Length and Area

Comparing Two Mathematical Tasks How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? NOTE TO SPEAKER: If there is time, you may want participants to generate similarities and differences before sharing the upcoming slide.

Similarities and Differences Both are “area” problems Both require prior knowledge of area Differences The amount of thinking and reasoning required The number of ways the problem can be solved Way in which the area formula is used The need to generalize The range of ways to enter the problem Way in which area formula is used: Martha’s Carpeting can be solved by knowing and using the area formula but this formula alone is not sufficient to solve the Fencing Task The need to generalize: Martha’s carpeting does not lead to a generalization but the Fencing Task does The range of ways to enter the problem: Martha’s Carpeting Task cannot be started by a student who does not know the formula for area; the Fencing Task can be started in other ways, such as sketches on graph paper.

Mathematical Tasks: A Critical Starting Point for Instruction Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking. Stein, Smith, Henningsen, & Silver, 2000

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

Mathematical Tasks: 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

Mathematical Tasks: 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

Levels of Cognitive Demand & The Mathematical Tasks Framework

Linking to Research: The QUASAR Project Low-Level Tasks High-Level Tasks

Linking to Research: The QUASAR Project Low-Level Tasks memorization procedures without connections to meaning High-Level Tasks procedures with connections to meaning doing mathematics For more information on what constitutes the various levels of task, see Chapter 1 of Stein, Smith, Henningsen, & Silver, 2000.

Linking to Research: The QUASAR Project Low-Level Tasks memorization procedures without connections to meaning (e.g., Martha’s Carpeting Task) High-Level Tasks procedures with connections to meaning doing mathematics (e.g., The Fencing Task)

The Mathematical Tasks Framework as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students For more information regarding the Mathematical Tasks Framework, see Chapter 2 of Stein, Smith, Henningsen, & Silver, 2000. Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

The Mathematical Tasks Framework as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

The Mathematical Tasks Framework as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

The Mathematical Tasks Framework as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

The Mathematical Tasks Framework as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

Cognitive Demands at Set Up In a research study, QUASAR researchers coded the instructional tasks used in 144 lessons across four high-poverty middle schools. These bars represent the percentage of tasks set up at each of the levels of cognitive demand. Stein, Grover, & Henningsen, 1996

The Fate of Tasks Set Up as Doing Mathematics This slide shows what happened to the “doing-mathematics” tasks represented on the prior slide during the implementation phase. • 37% were implemented as doing-mathematics tasks • 22% declined into what the researchers labeled as “unsystematic exploration” (tasks in which students explored, discussed and attempted to engage with the task, but missed the important and central mathematical substance at the heart of the task.) • 17% declined into no mathematical activity—students were off-task. • 14% declined into proceduralized routines • 10% declined into other forms of thinking and activity that could not be coded. Stein, Grover, & Henningsen, 1996

The Fate of Tasks Set Up as Procedures WITH Connections to Meaning This slide shows what happened to the “procedures-with-connections” tasks represented on the set up slide during the implementation phase. Stein, Grover, & Henningsen, 1996

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes This slide identifies the classroom factors that researchers observed when high-level tasks declined during a lesson. Stein, Grover & Henningsen, 1996

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore By contrast, this slide identifies the classroom factors that researchers observed when the cognitive demands of high-level tasks were maintained during a lesson. Stein, Grover & Henningsen, 1996

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Decline Maintenance Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore

Does Maintaining Cognitive Demand Matter? YES

Research shows . . . That maintaining the cognitive complexity of instructional tasks through the task enactment phase is associated with higher student achievement.

The QUASAR Project 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 enacted at high levels of cognitive demand (Stein & Lane, 1996).

Patterns of Set up, Implementation, and Student Learning Task Set Up Task Implementation Student Learning A. High High High B. Low Low Low Evidence 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). C. High Low Moderate Stein & Lane, 1996

TIMSS Video Study Higher-achieving countries implemented a greater percentage of high level tasks in ways that maintained the demands of the task (Stigler & Hiebert, 2004). 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).

TIMSS Video Study 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). TIMSS Video Mathematics Research Group, 2003 Making connections is defined as constructing relationships among mathematical ideas, facts, or procedures and possibly engaging in special forms of mathematical reasoning such as conjecturing, generalizing, and verifying.

Boaler & Staples (2008) The success of students in the high-achieving school was due in part to the high cognitive demand of the curriculum and the teachers’ ability to maintain the level of demand during enactment through questioning. In a recent study Boaler and Staples report the results of a five-year longitudinal study of 700 students in three high schools. Students at one high school, Railside, used a standards-based curriculum designed by teachers around key concepts (e.g., What is a linear function?) and featuring groupworthy tasks drawn from curricula such as College Preparatory Mathematics (CPM) and the Interactive Mathematics Program (IMP) and a textbook of activities that use algebra manipulatives. The students at the other two high schools used conventional curricula. The researchers report that students at Railside achieved at higher levels than those at other schools. In particular by the end of the second year Railside students significantly outperformed all other students in a test of algebra and geometry. Boaler and Staples indicate that one factor contributing to the success of students at Railside was the high cognitive demand of the curriculum and the teachers’ ability to maintain the level of demand during enactment through questioning.

Conclusion Not 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.

Additional Articles and Books about the Mathematical Tasks Framework Research 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.

Additional Articles and Books about the Mathematical Tasks Framework Practitioner Articles Stein, 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.

Additional Books about the Mathematical Tasks Framework Books Stein, 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.

Additional References Cited in This Slide Show   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. 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.