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

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      While 4⅚ ÷ ⅔ and the Leftover Pizza task both require the same mathematical operation and perhaps address similar content standards (dividing fractions), they provide much different opportunities for students’ thinking and reasoning. The Leftover Pizza task engages students in interpreting, modeling, and making sense of a context that requires the division of fractions, the process of dividing fractions, and the meaning of the quotient. In this way, the Leftover Pizza task elicits the types of mathematical thinking identified in the Process Standards of NCTM’s (2000) Principles and Standards for School Mathematics, such as representations and connections. It also engages students in the Mathematical Practices called for by the Common Core’s Standards for Mathematical Practice, such as Mathematical Practice 1, “Make sense of problems and persevere in solving them,” and Mathematical Practice 4, “Model with mathematics” (NGA & CCSSO, 2010).

       How might this task provide access for each and every learner?

      This task has many features that allow access for each and every student. It can be described as having a low threshold and a high ceiling (McClure, 2011). There are numerous ways to solve the pizza task, ensuring multiple entry points for all learners. Students can use fraction tiles, pattern blocks, or other manipulatives to model the situation and utilize those models when communicating how they solved the problem to their peers and teachers. The pizza task allows students access because they are able to model the serving sizes and determine that they can make at least seven servings. Then, using the models or through discussion with peers or the teacher, students can get to the ¼ of a serving that remains. By allowing students entry into the problem, teachers provide them with something to discuss with the class to further their understanding. If a student was just given the problem 4⅚ divided by ⅔ and did not have the means to perform the operation or solve the problem, he or she would not have access or the ability to solve the problem and would then be left out of the conversation regarding the solution. Multiple entry points and solution methods, as well as a meaningful context, allow students to interact with the task on at least some level so that when there is a discussion, all students have ideas to bring to the table and then can use those ideas to make sense of the mathematics and build a better understanding of the division problem.

      In activity 1.2, you will continue to explore how different types of tasks provide different opportunities for students’ thinking.

Activity 1.2: Considering Different Types of Tasks

      It is valuable to engage with tasks as learners to make sense of what those tasks have to offer students. Be sure to devote attention to this experience. Explore the tasks on your own before engaging in the activity.

       Engage

      For activity 1.2, you may want to print figure 1.2 (page 13) and figure 1.3 (page 14) from this book or from the online resources (see go.SolutionTree.com/mathematics). Look over the tasks in figure 1.2 and use figure 1.3 to record your responses to the following questions.

      ■ What is similar about the tasks in each column? How do the tasks change as you move up (or down) a column?

      ■ What is similar about the tasks across each row? Identify phrases that characterize the nature of tasks in each row of the grid, and write these phrases on your recording sheet.

      Compare your work and ideas in your collaborative team before moving on to the activity 1.2 discussion. Keep your recording sheet to use as your rubric in activity 1.3 (page 16).

       Discuss

      How do your responses compare with those in your collaborative team? What themes emerged during your discussion? In this section, we present ideas for you to consider.

       What is similar about the tasks in each column? How do the tasks change as you move up (or down) a column?

      The columns of the Benchmark Tasks grid each contain tasks that address related mathematical ideas. In column A, the tasks address division with remainders for students in grade 4. Tasks in column B relate to addition and subtraction of integers in grade 7. The tasks in column C all involve the area of trapezoids in grade 6. However, as you move up or down a column, the tasks provide different opportunities for students’ thinking and reasoning about each mathematical topic. Research shows that attending to the level and type of thinking that a task can elicit from students is equally as important as considering the mathematical ideas in the task, and different types of tasks provide different opportunities for thinking and reasoning (Stein et al., 2009) and impact students’ learning in different ways.

       What is similar about the tasks across each row? Identify phrases that characterize the nature of tasks in each row of the grid, and write these phrases on your recording sheet.

      Tasks across the rows of the Benchmark Tasks grid elicit similar types and levels of thinking from students. Tasks in row 1 mainly draw on students’ memorized knowledge or recall of mathematics facts, rules, formulas, or vocabulary. Teachers have described tasks in this row as “You either know it or you don’t”—nothing in the task helps students to learn what it is asking, and there is no procedure they can apply to determine an answer. Students only have access to the task if they are able to recall what it is asking. Taking notes would also belong in row 1 of this grid, as taking notes engages students in writing down or reproducing mathematics rather than doing any mathematical thinking on their own. Tasks such as these are appropriate when the goal for student learning is recall and memorization.

      In row 2, students can solve the tasks by applying a procedure, computation, or algorithm. The goal of these tasks is to perform a procedure or computation and arrive at a correct answer. While students may use a conceptually based strategy to solve the task, nothing in the task requires or supports students to make sense of the mathematics or demonstrate their understanding of the mathematics. This denies access to the task if students are not fluent with the procedure or computation used to solve the problem. Successfully completing the task only requires that students perform a procedure and produce an answer.

      In row 3, tasks may ask students to engage in problem solving, though they may also ask them to apply specific procedures or use specific representations. The main difference between tasks in row 2 and row 3 is that tasks in row 3 provide opportunities for mathematical connections, reasoning, and sense making. The questions, representations, and contexts in the task support students to develop an understanding of a mathematical concept or procedure or to engage in complex and non-algorithmic thinking. СКАЧАТЬ