Student Engagement through the use of High-Level Tasks Presented by: Raven Hawes iZone Mathematics

Rationale Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true….…Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

High-Level Tasks Cognitively challenging tasks that promote thinking reasoning problem solving

Structures and Routines of a Lesson

Set-Up of the Task 0 What are the mathematical goals for the lesson (i.e., what is it that you want students to know and understand about mathematics as a result of this lesson)? 0 What questions will you ask to help students access their prior knowledge? 0 What are all the ways the task can be solved? 0 What misconceptions might students have? Errors? 0 What are the expectations for students as they work on and complete this task? 0 How will you introduce students to the activity so as not to reduce the demands of the task and provide access to all students? What will you hear that lets you know students understand the task?

Explore Phase What will you do or say if no progress has been made on the task? What will you see or hear that lets you know how students are thinking about the mathematical ideas? What questions will you ask to assess students’ understanding of key mathematical ideas, problem solving strategies, or the representations? What questions will you ask to advance students’ understanding of the mathematical ideas? What questions will you ask to encourage all students to share their thinking with others or to assess their understanding of their peer’s ideas? How will you ensure that students remain engaged in the task?

Share, Discuss, and Analyze Phase 0 How will you orchestrate the class discussion so that you accomplish your mathematical goals? Specifically: Which solution paths do you want to have shared during the class discussion? In what order will the solutions be presented? Why? In what ways will the order in which solutions are presented help develop students’ understanding of the mathematical ideas that are the focus of your lesson? 0 What specific questions will you ask so that students will: make sense of the mathematical ideas that you want them to learn? expand on, debate, and question the solutions being shared? make connections between the different strategies that are presented? look for patterns? begin to form generalizations? 0 How will you ensure that, over time, all students will have the opportunity to participate and be recognized as competent?

Solve the Task Work privately on the Task. (5 minutes) Work with others at your table. (15 minutes) Compare your solution paths.

Shares of Fudge Four friends each have a fraction of a bar of fudge. All of the bars of fudge are the same size. Who has the most fudge and how do you know? You may choose your own tools and models for solving this problem. Jill: 2/4 of a bar of peanut butter fudge James: 5/6 of a bar of peanut butter fudge Fred: 6/10 of a bar of peanut butter fudge Juan: 9/10 of a bar of peanut butter fudge Their friend Russell claims that he has the most fudge because he bought two different kinds of fudge. He bought 13 of a bar of cherry fudge and 23 of a bar of vanilla fudge. Do you agree or disagree with Russell? Why or why not?

How Many are Equivalent? Ms. Smith asked students to generate equivalent expressions for 25x + 15x + 20 Students gave the following responses: 20x + 20x (5x + 3x + 4) 60x 5(8x + 20) 40x + 20 Which of the expressions above are equivalent? Why? Prove your thinking with diagrams and/or substitutions for the variable.

Buying Tools

Expectations for Group Discussions Share solution paths. Listen with the goals of: 0 putting the ideas into your own words; 0 adding on to the ideas of others; 0 making connections between solution paths; and 0 asking questions about the ideas shared. The goal is to understand the mathematics and to make connections among the various solution paths.

Whole Group Discussion 0 What relationships can be discovered between the solution paths? 0 How can one solution help you think about another?

Linking to Research/Literature Connections between Representations Pictures Written Symbols Manipulative Models Real-world Situations Oral Language

MathMath – Online Resources for High Level Tasks