As a follow-up to our design study of a sixth-grade ratio unit at Highland Middle School in Harnett County, NC, we analyzed student responses to several assessments that teachers administered over the course of the unit. The assessments included:
- A pretest administered before the beginning work on the SUDDS ratio curriculum that included items from cluster 4, Finding Key Ratio Relationships, and cluster 5, Comparing Ratios and Finding Missing Values.
- A Cluster Assessment of Progress (CAP) was administered for cluster 4 after completing the cluster 4 curriculum.
- A CAP was administered for cluster 5 after completing the cluster 5 curriculum.
- A SUDDS-designed posttest containing items from both cluster 4 and cluster 5 was administered after completing and reviewing the cluster 5 curriculum.
- An independent assessment containing ratio items, was administered on paper at the end of the study.
The analysis revealed common thought patterns, strategies, and misconceptions that will improve the learning trajectories in Math-Mapper. However, much of what we learned is broadly applicable to math teachers, so we share some of the primary takeaways here.
General Instructional Suggestions:
- Continue providing opportunities to allow students to work on a curricular task while engaging in productive struggle. Teachers can help students by asking questions and relating the ideas to prior learning. Students were frequently able to contribute appropriate and creative ideas to classroom discussion. Allowing students the freedom to grapple with a difficult or messy task with minimal teacher support will increase students’ independence, promote persistence, develop an inclusive classroom culture, and encourage students to take ownership of the concepts they are learning.
- Support students to use sense-making strategies when solving problems or answering assessment items. Using sense-making strategies involves not only applying procedures and rules accurately but also keeping track of units and relating the math to the students’ own experience.
- Continue to encourage students to use ratio tables and ratio boxes, while paying close attention to the units and quantities represented by the columns. Additionally, encourage students to use notation (e.g. arrows along the sides of the ratio table) to represent the multiplicative relationships between rows in the table and quantities in a ratio. Using tables and identifying the multiplicative relationships may support students in making connections between the constructs in the two clusters, from identifying ratio equivalence (construct 9) to finding missing values in proportions (construct 13).
- We observed lower performance on assessments and in class on the finding unit ratio construct. To address these issues, modifications and additions to the curriculum were added in consultation with teachers. However, intentional use of tables could support students to connect their understanding of equivalent ratios and base ratios to unit ratios. Additionally, consider using manipulatives to support students in applying fair-sharing as a visual model for finding unit ratios before focusing on computation using long division. Relating this visual model to long division will provide the conceptual basis for the computation.
- Notice that sometimes the problems request students to reason with their own experience about rates informally and to apply them to a ratio context. Be sure to have students try to make sense of the ideas and not just blindly apply operations or procedures.
- Allow enough time to thoroughly explore representing ratios on the coordinate plane, specifically the relationship between points representing equivalent ratios. Graphing ratios on the coordinate plane is newly introduced in the sixth grade standards, so teachers should take time to help the students see how to relate graphs to the ideas of ratio equivalence.
Interpreting and Discussing Diagnostic Assessment Results:
- Encourage students to use the language of the constructs (e.g. identifying unit ratios as an area for growth). Students also need to understand that they can explore, revise, and reveal answers to specific items from the assessment using the item matrix at the bottom of the student report page. Refer students back to the map and show them how to get to the learning trajectories. Take one construct and talk about how the progress levels are related to the items. This will build their metacognition about their studying.
- 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.
- Remind students that these are diagnostic items, designed to bring up underlying conceptual issues that can be confusing. As a result, students may score lower than on typical school tests, but with persistence, discussions, revising, and retesting, they can improve significantly.
- 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. Very soon, they will also be able to practice on individual constructs.