High-level math tasks provide critical opportunities for students to develop mathematical understanding as well as learn about the nature of mathematics and how one engages in it. Such tasks, however, are difficult for teachers to implement effectively. Through thoughtful pre-planning, purposeful implementation, and deliberate reflection on one's practice, teachers are better able to implement such tasks and support student learning. This module discusses various components of this process: (a) the characteristics of high-level tasks, (b) effective implementation of lessons around high-level tasks.

In order to assist teachers as they work to transform their practice, a set of four tasks as well as their Lesson Guides (LG's) -- which are detailed suggestions for how each task might be enacted – are provided. The detailed lesson guides permit teachers to study, use and internalize instructional practices long before they may feel prepared to design similar lessons themselves. Specific questions and strategies that can support the learning of ELLs are also highlighted on the lesson guides. It is intended that, over time, teachers will generalize these practices to their instruction more broadly and design and enact similar lessons.

This module provides four high-level tasks, two of which are suitable for use in Algebra 1 and two are suitable for use in Geometry.

This module will guide the reader through Custom T-Shirts and its Lesson Guide in order to ground the discussion and illustrate the features of the resources available.

To determine whether or not a task is high-level it is necessary to consider both the kind of thinking and the level of thinking that is required of students as they engage with and solve the task (Stein, Smith, Henningsen and Silver, 2000; Stein and Smith, 1998). It is important to keep in mind that a "high-level" mathematical task is not the same as a "difficult" task. A high-level task is one that requires students to put forth some cognitive effort as they work to understand, make connections to, and build upon, mathematical concepts and ideas. Students are required to represent the mathematical ideas in multiple ways and make connections between the representation of underlying mathematical ideas in tables, graphs, verbal explanations, equations, and both real-world and mathematical contexts. High-level tasks should also allow for multiple entry points, opportunities for students to begin working on the task using a variety of approaches and techniques.

A specific mathematical procedure might be suggested or implied, however a high-level task would require students to make sense of the procedure by making connections to underlying mathematical ideas and structures. Students might also be required to develop a unique pathway or procedure for solving the task as they explore underlying mathematical concepts, processes, or relationships. In short, high-level tasks promote thinking, reasoning, and mathematical sense-making (Stein, Smith, Henningsen, Silver, 2000). Conversely, tasks that can be solved by simply reproducing memorized facts or applying learned procedures would not be categorized as high-level, even if the facts or procedures are complex or difficult for a particular student.

Why is it important for students to have an opportunity to engage in high-level tasks? The tasks in which students engage determines what they learn. Lappan and Briars contend that "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, 1995, p. 138). Providing opportunities for students to build new understanding by drawing upon their prior knowledge as they work to solve rich tasks helps students build the connections between mathematical ideas that are important for developing deep understanding.

High-level tasks also address important socio-cognitive needs in a heterogeneous classroom as they "enable more students to contribute ideas and feel valued" (Boaler, 2006, p. 42). It is important to keep in mind, however, that for a task to be a high-level task for a particular group of students, it must be able to form a bridge between their prior knowledge and experiences and the mathematical understanding that the teacher wants them to construct. Therefore a task that might be high-level for one group of students might be routine for another group, and impossible for yet another group. In addition, although low-level tasks may provide an opportunity for students to develop procedural fluency, this is only one strand of mathematical proficiency that today's students must develop – the other strands are conceptual understanding, strategic competence, adaptive reasoning, and a productive disposition (Kilpatrick, et al., 2001). High-level tasks work on all five strands as an interconnected whole.

The Custom T-Shirts task, which can be used to introduce Pre-Algebra or Algebra 1 students to linear functions, will be used to illustrate our discussion.

Although high-level tasks provide important opportunities for student learning, merely selecting a high-level task and presenting it to students does not guarantee that students will engage with the challenging aspects of the task (Stein and Smith, 1998). The characteristics of a high-level task are often not lived out in classrooms because the practices that students need to engage in and the instructional practices that teachers must be able to use are challenging for both students and teachers (Stein, Grover and Henningsen, 1996). As a result, time that teachers spend thinking through the lesson beforehand, both individually and with other teachers, can help ensure that the cognitive demands of the task will be maintained as the lesson unfolds.

Smith, Bill and Hughes (in press) have developed the Thinking Through the Lesson Protocol (TTLP) that provides prompts that promote detailed and thoughtful planning by teachers. It "provides a framework for developing or reflecting on lessons that use students' mathematical thinking as the critical ingredient in developing their understanding of key disciplinary ideas" (Smith, Bill and Hughes, in press). The TTLP divides the planning process into three phases: selecting and setting up the task, supporting student's exploration of the task, and sharing and discussing the task. The Lesson Guides (LG) presented in this module were developed using this protocol and provide examples of ways that the TTLP can help structure your planning.

Boaler, J. 2006. How a detracked mathematics approach promoted respect, responsibility, and high achievement. Theory into Practice, 45(1), 40-46.

Boaler, J. and Brodie, K. (2004). The importance, nature, and impact of teacher questions. In McDougall, D.E. & Ross, J. A. (Eds.), Proceedings of the Twenty-Sixth Annual Meeting of the North American Chapter of the International Group for Psychology of Mathematics Education, Vol. 2, pp. 773-782. Toronto, Ontario.

Hiebert, J., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth: Heinemann.

Horn, I.S., 2005. Learning on the job: A situated account of teacher learning in high school mathematics departments. Cognition and Instruction, 23(2), 207-236.

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

Kilpatrick, J., J. Swafford, and B. Findell (Eds.), 2001. Adding it up: Helping children learn mathematics. Washington DC: National Academy Press.

NCTM Commission on Teaching Standards for School Mathematics, (1991). Professional standards for teaching mathematics. Reston: National Council of Teachers of Mathematics.

Sherin, M. G. (2000). Facilitating meaningful discussion of mathematics. Mathematics Teaching in the Middle School 6(2), 122-125.

Smith, M.S., Bill, V., and Hughes, E. K. (in press). Thinking through a lesson: A key to successfully implementing high-level tasks.

Stein, M.K., Engle, R.A., Hughes, E.K., and Smith, M.S. (in press) Orchestrating productive ...

Stein, M. K., Grover, B., and 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), pp. 455-488.

Stein, M.K. and Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School 3(4), 268-275.