Making Sense of Mathematics for Teaching to Inform Instructional Quality. Juli K. Dixon
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. In completing the task, students actually learn mathematics.
Tasks in row 4 contain all of the features of tasks in row 3, with the added feature that the task directions explicitly require students to provide an explanation or justification. In addition to completing the mathematics necessary to solve the task, the task includes a prompt for students to reflect on, explain, or justify some aspect of their work.
Source: Questions adapted from Dixon, Nolan, Adams, Tobias, & Barmoha, 2016, p. 60; Nolan, Dixon, Roy, & Andreasen, 2016, pp. 36, 105.
Figure 1.2: Benchmark Tasks grid.
Visit go.SolutionTree.com/mathematics for a free reproducible version of this figure.
Figure 1.3: Benchmark Tasks recording sheet.
Visit go.SolutionTree.com/mathematics for a free reproducible version of this figure.
Students can use prior knowledge to help solve the problems in rows 3 and 4, which allows more access to these types of problems. For example, in problem 2C, if students do not know the formula for a trapezoid, they most likely would be unsuccessful in solving and would not attempt the problem. However, in problems 3C and 4C, students could use their knowledge of other formulas to break up the trapezoid into other shapes, find the area of those shapes, and then find the area of the figure. While this may not be the most efficient strategy, it allows access to the problem and students can find a solution. Then, through discussion with peers, students can connect their solution to others and engage in the lesson. It is interesting that tasks that teachers often consider to be more difficult actually provide more access to students. Additionally, if students are not able to complete the mathematics necessary to solve the task, the explanations and justifications the students provide can assist the teacher in diagnosing gaps in students’ understanding so that the teacher can then address those gaps. (Note: In this book, we use the terms demanding and challenging to mean stimulating and thought provoking, rather than difficult. A difficult task—for example, multidigit long division—may be difficult but not necessarily cognitively challenging.)
How did your responses on the Benchmark Tasks recording sheet compare to these descriptions? Take a moment to consider or discuss any new ideas introduced in this section using your recording sheet from activity 1.2. Then, proceed to the following section, The IQA Potential of the Task rubric, where we present a framework from the IQA Toolkit to assist you and your collaborative team in assessing the potential cognitive demand of mathematical tasks.
The IQA Potential of the Task Rubric
The level and type of thinking required by a task is referred to as the level of cognitive demand (Stein et al., 2009). Teachers can use the IQA Potential of the Task rubric (figure 1.4) to rate the level of cognitive demand a task can elicit from students. In activity 1.2, you identified examples and features of tasks at different levels of cognitive demand. The identification of these task features will help you to rate a task using the Potential of the Task rubric, with which you will consider the level of thinking required for students to successfully complete the task as initially written, without any changes, extensions, resources, or other expectations that may arise as students work on the task.
Source: Adapted from Boston, 2017.
Figure 1.4: IQA Potential of the Task rubric.
The IQA Potential of the Task rubric is intended to align with our previous ideas about tasks in rows 1 through 4 of the Benchmark Tasks recording sheet (page 14)—which we hereafter refer to as levels in the IQA Toolkit—and to provide additional detail and support for rating tasks. Look back through your task ratings and rationales for activity 1.2 and consider the following questions.
■ In what ways does the Potential of the Task rubric appear to be consistent with ideas on your recording sheet from activity 1.2? What words or phrases on the rubric do you find helpful?
■ In what ways does the Potential of the Task rubric appear to be inconsistent with or different from ideas on your recording sheet from activity 1.2?