levels should also align with students’ learning progression for a particular mathematical idea at a particular grade level. For example, rigorous state standards suggest providing opportunities consistent with level 3 or 4 tasks while students are in the process of learning to make sense of multiplication in grades 2 through 4, as expressed in the grade 4 Common Core standard:
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (NGA & CCSSO, 2010; 4.NBT.B.5)
By grade 5, we expect students to demonstrate mastery of the standard algorithm for multidigit multiplication, which would be supported by engaging in tasks at level 2 that provide practice and promote automation and mastery.
Note that in the progression of learning multiplication from grades 2 to 5, opportunities for students to understand, unpack, and develop strategies and procedures, using problems, contexts, and representations that make sense, precede the memorization of multiplication facts and mastery of standard algorithms. Too often, students’ learning of a mathematical topic or procedure begins with the teacher telling or showing students everything they need to know, modeling procedures, or providing definitions, while students’ mathematical activity is limited to practicing procedures (level 2) or taking notes and memorizing (level 1). Typically, teachers wait until students have mastered procedures or memorized the appropriate facts or properties before providing opportunities for real-world applications or problem solving. After all, how would students know how to solve problems if they were not shown how to solve them first? On the contrary, students can develop mathematical procedures given contextual problems prior to any direct instruction or modeling (Carpenter, Fennema, Franke, Levi, & Empson, 2014). Students can discover many mathematical properties through investigation, such as the congruent angles formed when parallel lines are cut by a transversal (figure 1.6, page 22), the formula for area of a trapezoid (figure 1.2, page 13), or the rules for adding integers (figure 1.2, page 13). The Making Sense of Mathematics for Teaching series provides many suggestions for supporting students’ learning of specific mathematical topics at each grade band.
Summary
It is important to rate the level of instructional tasks to be aware of the type of thinking and access a task can provide for each and every student. A task at level 1 or 2 does not provide much space for discussion, as the focus is on the correctness of memorized knowledge or rote procedures. Additionally, a task at level 1 or 2 often does not provide access for students unless they know the mathematics to be recalled or the specific procedure requested. A level 3 or 4 task is often necessary to support quality mathematical discourse and teacher questioning, as we will discuss in upcoming chapters. We provide additional support for rating tasks in appendix D (page 141).
Even though reform efforts call for mathematics learning for each and every student (NCTM, 2014), learners who struggle in mathematics or who have special education placements often have less access to demanding mathematics (Weiss, Pasley, Smith, Banilower, & Heck, 2003). To successfully include all learners in the mathematics classroom, we need to design instruction that is accessible to all.
Chapter 1 Transition Activity: Moving From Tasks to Implementation
Before moving on to chapter 2, engage in the transition activity with your collaborative team. The transition activity will enable you to build on ideas about tasks from chapter 1 to begin to explore implementation in chapter 2.
• Select a chapter, unit, or any set of two to three consecutive lessons in the mathematics curriculum materials you use in your school or classroom. Rate the tasks that appear in a set of lessons over two to three days of instruction.
• Identify a task at level 3 or 4 to use as the main instructional task to teach a mathematics lesson. Indicate what features make the task a level 3 or 4.
• Implement the task in your class. Collect sets of student work (at least four samples). Select samples that show a variety of strategies, thinking, and reasoning.
• Analyze students’ responses. Did students actually engage in or produce the level and type of thinking you identified when considering the potential of the task?
Save the sets of student work, your ratings, and notes or any written reflections from the transition activity, as you will refer to them in chapter 2.
CHAPTER 2
Implementation of the Task
To ensure that students have the opportunity to engage in high-level thinking, teachers must regularly select and implement tasks that promote reasoning and problem solving.
—National Council of Teachers of Mathematics
In this chapter, you will explore how teachers can implement mathematical tasks during mathematics lessons in ways that support or possibly diminish students’ opportunities to engage in thinking and reasoning. By the end of the chapter, you will be able to answer the following questions.
■ What happens when teachers enact high-level tasks in the classroom? How do teachers maintain (or limit) opportunities for thinking and reasoning during the lesson?
■ What teacher actions seem to support (or diminish) students’ opportunities for thinking and reasoning as well as provide (or take away) access for students to engage in the task?
■ What does student work indicate about students’ engagement in thinking and reasoning during the lesson and their level of access to the lesson?
■ What
