Making Sense of Mathematics for Teaching to Inform Instructional Quality. Juli K. Dixon

if students already know the properties, procedure, or definitions at the heart of a discovery task, there would not be anything for them to discover. For example, the adapted Angles task can help students discover relationships between angles only if students do not already know those relationships.

      Additionally, be aware of the implications of the wording of the task and how this can impact the work students produce. While a task may ask students to explain how or show your steps, there is a difference between explaining a procedure and explaining your thinking. For example, if a task requires students to solve fraction division problems such as ¾ ÷ ¼ using the traditional algorithm, asking students to explain might generate responses such as, “I used invert and multiply and computed ¾ × .” While this explanation of how students solved the problem indicates knowledge of an algorithm, it does not indicate conceptual understanding, reasoning, or sense making. Adding a prompt to explain onto a procedural task does not raise the cognitive demand; the task itself must first elicit some thinking, reasoning, problem solving, or understanding for the student to have something worth explaining.

Aligning With Learning Goals

      While we categorize procedural tasks at a level 2, note that there are many important mathematical procedures that students should be able to apply fluently and with automation after having established an appropriate level of conceptual understanding (NCTM, 2014). There are also appropriate occasions when you would use or assign a level 2 task to provide students the opportunity to practice or demonstrate their ability to perform a procedure, or a level 1 task when the goal is memorizing rules, properties, or definitions. Task levels (as well as mathematical content) should align with the goals for students’ learning.

      Task 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.

       What opportunities would students have to engage in thinking and reasoning?

       What is the balance of levels across the lessons?

      • 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.

       What thinking, reasoning, or sense making would the task potentially elicit from students?

       What products or processes would serve as evidence that students actually engaged in this thinking, reasoning, or sense making?

       How does the task provide access for all students?

      • 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.