In October and November 2017, we conducted a design study on percents with our seventh-grade student and teacher partners at Highland Middle School in Harnett County, NC. Following the study, we analyzed the data we collected, which took the form of assessment results, student work, and classroom observations, to refine our learning trajectories on percent and improve the curriculum we developed. However, many of our findings were broadly applicable to teachers everywhere.
General Instructional Suggestions
- With any curriculum, it can be very valuable to look through the materials as a grade-level team and think through the goals of the lessons. The “Thinking Through a Lesson Protocol” (Smith & Bill 2004; Hughes & Smith 2004) can be a valuable tool for this, but the important thing is to recognize the goals of the lesson, decide on how to launch the lesson, and come up with plans for supporting students to work through the lesson without lowering the cognitive demand of the tasks.
- It can be challenging to determine how much time and direction to give students before releasing them to work on the task. A helpful expectation is that not all students will complete all the components of any activity, otherwise many students will be finished and off-task by the time the last student finishes. Knowing the goal of the lesson, and keeping it in mind can help to identify the essential challenge or idea that ALL students need to encounter in order for the math content to be meaningful. Then, when the essential content is known, it can help to clarify the content that must be established at the beginning of the lesson.
- Through a lot of persistence on the part of our teacher partners, we saw many students adopt and value ratio boxes as a tool for solving percent problems. Requiring students to label the columns of the ratio boxes and identify the meaning of the values that they are entering are key for ensuring ratio boxes are a meaningful sense-making tool .
- Percent bars were less frequently used, but appeared to be a helpful model, particularly in solving percent change problems for the original amount. Using percent bars throughout a unit on percents can help to keep this representation familiar and increase its utility in describing percent change contexts, such as percent error and percent change.
- Solving percent problems for the original amount proved to be one of the more challenging skills. While continued use with ratio boxes may help, students should encounter these kinds of problems again after working with equations. A student who is struggling may find equations to be a helpful tool in solving these problems.
Interpreting and Discussing Diagnostic Assessment Results:
- Spend more time in student-centered discussion of the items, especially the common items that all students have worked on. Discussions should be student-centered, having them explain how they thought about them both rightly and wrongly. Let students volunteer to explain them to others. We recently added features to support this in MathMapper. Questions in the class reports now appear with response lists for students in the class, information about misconceptions, and answer keys. For more information about using these new features, see this discussion guide.
- Refer students to the learning trajectories in the test results page. Focus on one construct at a time and talk about how the progress levels are related to the items. This will build their metacognition about their studying.
- Remind students that these are diagnostic items, designed to bring up underlying conceptual issues that can be confusing. Therefore, they are diagnosing issues—not checking to see if they can parrot routine problems. As a result, students may score lower, but with persistence, discussions, revising, and retesting, they can improve significantly. These assessments are not intended to produce the typical school grading scale.
- Use information presented in the heat maps to inform decisions regarding reviewing assessment items. For example, levels that contain a significant amount of orange may warrant whole-class discussion, items with relatively little orange may be better addressed by pulling out those students for a small group discussion. Items that were answered incorrectly by approximately half of the class could be addressed through collaborative group work. Additionally, use the heat maps to identify students that were able to answer items at certain levels correctly and allow them to lead the discussion of those items.
- Remind students that they can retake assessments if they are unhappy with or want to improve on their results. Students may also complete practice sessions to review the content of a particular construct.