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High-Level Algebra and Geometry Tasks: Resources for Both Student and Teacher Learning

Module by: Institute for Learning

Summary: High-level mathematical tasks provide opportunities for students to develop deep mathematical understanding, however, such tasks are difficult for teachers to implement effectively. This module offers practical suggestions as well as four algebra and geometry tasks, and detailed lesson guides, that can support teachers' efforts to provide such opportunities for their students.

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.

High-Level Mathematical Tasks

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.
  • The pricing plan of a t-shirt company embodies the important mathematical concepts of constant (y-intercept) and rate of change (slope) that are foundational to students' understanding of linear functions.
  • The four prompts provide entry points for students as they build upon their understanding of arithmetic processes and generalize from these procedures to build their algebraic representations.
  • While the prompts help students organize their work, they do not give away the solution paths -- student still must make sense of the mathematical ideas for themselves.
  • Students construct tables, graphs and equations, and will work to connect the various representations as they construct their explanations and listen to their peer' ideas.
  • The task is at the appropriate level of difficulty for students who are in the process of transition between arithmetic and algebra.
  • The homework provides an opportunity for students to continue to think about the mathematical ideas that were the focus of this task and to begin to think about the roles of the constant and the coefficient in different representations of a linear function.

Implementing High-level Tasks

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.

Phase One: Selecting and Setting up the Task

One of the key components of planning is identifying the mathematical goals of the lesson - what students are to learn or be in the process of learning as they engage in the task, not what they will do. The TTLP also prompts the teacher to consider: how the task builds on students' prior knowledge, the ways that students might solve the task, the errors and misconceptions that might surface, and the expectations for how students will engage with the task.
The Lesson Guide that accompanies Custom T-Shirts will be used to illustrate how these are taken into consideration when planning to implement the task.
The Lesson Overview and Set-Up Phase of the LG (pages 1 and 2) discusses:
  • The overall flow of the lesson
  • NCTM Standards that the task addresses
  • Mathematical and Academic Language Goals of the lesson
  • Assumption of prior knowledge.
  • How the task can be introduced and expectations set so that students understand what they are expected to do, however the cognitive challenge of the task has not been removed.

Phase Two: Supporting Students' Exploration of the Task

It is important to carefully think about how you will support student learning during the Explore phase of the lesson. The TTLP underscores the central role of questioning in this process, however it also recognizes that constructing good questions as one is also reacting and responding to students is not easy. Considering possible questions that you could ask as you plan the lesson is crucial (Smith, Bill and Hughes, in press). The TTLP suggests that a teacher's carefully crafted questioning can serve many purposes: questions can help floundering students get started, focus student thinking on key mathematical ideas of the lesson, support the sharing and building of ideas by small groups, and challenge and assist students in clarifying misconceptions or errors.
In planning for the questions that you might ask, beginning with clearly stated mathematical goals is a key starting point. Solving the task yourself, or with peers, in order to anticipate possible ways your students might solve the task, as well as the likely misconceptions and errors that might surface as students engage in the lesson is a critical next step. One technique that we have found to be very useful in helping teachers to construct questions that move students forward during the Explore phase of the lesson is to construct questions that you might ask to:
  • assess student understanding of key mathematical ideas, problem solving strategies and representations and
  • advance student thinking and understanding towards the mathematical goals of the lesson.
In order for this to occur the teacher must circulate and monitor what is occurring in all of the groups. This also provides an opportunity for the teacher to begin to construct a plan for how the whole-class discussion can be built upon the emerging ideas.
The Lesson Guide that accompanies Custom T-Shirts will be used to illustrate these points.
The Explore Phase of the LG provides ideas for supporting students' individual and small group work and monitoring students as they work on the task by suggesting:
  • How the private think time might be conducted - approximately how much time to allow and what you should - and should not do - as students work individually on the task
  • General reminders for interacting with students during small-group work:
    • Ask questions related to mathematical ideas, problem solving strategies and connections between representations;
    • Ask students to explain their thinking and reasoning to you and to their peers;
    • Ask students to listen to each other's explanations and to rephrase what each other has said;
    • Support students' use of mathematical language.
  • What you could look for as you monitor small-group work so that you can orchestrate the whole group discussion to build on student ideas, and move towards the mathematical goals of the lesson.
  • Different ways that students might approach the task and possible questions that you can ask them to assess their understanding and to advance their thinking towards the mathematical goals of the lesson (questioning strategies are discussed in more detail below).
  • Possible misconceptions and errors that you can anticipate, and possible questions that prompt them to revisit their notions and begin to correct their own thinking.
As you examine the Explore Phase of the LG you will also note that:
  • it is suggested that you begin with an assessing question. As teachers we often assume that we understand the thinking that has gone into a particular piece of student work. However, students frequently approach a problem in a novel way, and it is important to determine this before you try to move a student's thinking forward. Also, having students put their thinking into words is important in its own right.
  • advancing questions should consider not only mathematical concepts, but also strategies, representations, and connections among representations. Advancing questions should also keep the goals of the lesson in mind.
IT IS IMPORTANT TO NOTE THAT THE LG IS NOT A SCRIPT, but provides a way to consider the type of planning that will help support task implementation, and student learning (see Smith, Bill and Hughes, in press, for a more detailed discussion of the TTLP as a tool for planning).

Phase Three: Sharing and Discussing the Task

After students have had the opportunity to explore the mathematical ideas underlying the task, it is crucial for the teacher to bring the class together to examine, analyze, and connect the various ideas that surface and to focus the discussion on the key mathematical ideas (Sherin, 2000; Smith, Bill and Hughes, in press). By thinking ahead of time about how you could orchestrate the class discussion around the ideas that might surface during small-group work, you will be able to make decisions about: the student work that you will select for presentation as you monitor small-group work; the order in which you will have the students present their solutions; and the questions you might ask so that students will make connections between different strategies and representations and make sense of the key mathematical ideas. It is also important that you consider what you will expect to see and hear in the student discussion of the mathematics that will indicate that students understand those mathematical ideas, and what your next steps will be.
Note: It is not expected that each student, or each group, will have completely solved the task prior to the Sharing and Discussing of the task. By carefully selecting and sequencing the work, and orchestrating the discussion around the key ideas, a story can be built and the class as a whole will have an opportunity to continue to work to construct and analyze the mathematical ideas. The teacher plays a crucial role in this process (Sherin, 2000; Smith, Bill and Hughes, in press).
The Lesson Guide that accompanies Custom T-Shirts will be used to illustrate these points.
The Share, Discuss, and Analyze Phase of the LG provides three critical aspects related to the lesson:
  • A possible sequence for presenting student work
    • By allowing different groups to chart and present different segments of their solution, you can ensure that a wider variety of students' input is honored and incorporated into the discussion. This is also a time saving device, since it is not always necessary for each group to construct a chart that contains all parts of the solution.
  • Mathematical ideas and rationale
    • The LG is not a script to be followed, but a discussion of possible ways in which a lesson might unfold. The LG presents a rationale for one particular set of decisions, and the mathematical ideas that are linked to that decision. In your practice you may make different decisions – the lesson to be learned here, however, is that all decisions should keep the mathematical goals of the lesson in mind.
  • Possible questions and student responses
    • The questions are provided as a starting point for discussion, with the idea of asking students to explain their thinking, model their solution processes so that they are made explicit for other students to see, and to make connections among various representations.
    • Possible student responses allow the "student voice" to come alive and to illustrate how mathematical language might be incorporated in their explanations. These are meant only as an ideal to which you might aim over the course of the year. Initially students should be allowed to express their thinking in ways that make sense to them. It is then your job to explicitly link mathematical vocabulary to their ideas, to create charts in the classroom that display the vocabulary for students to refer to, and that you press students to use, and take ownership of, this language.
Homework is also provided so that students will continue to think about, and build upon, the mathematical ideas explored in the lesson, and to form a link to the following day's exploration and continued learning
Development supported by The James Irvine Foundation.

References

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.

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