*Educational Leadership*, experienced researchers at the American Institutes for Research will discuss research-based practices, providing educators on the front lines of school improvement efforts with the information they need to make the best instructional decisions. Steve Fleischman, a principal research scientist at AIR, will be series editor, identifying the effective practices featured here each month. Send questions or topic suggestions to Steve at editorair@air.org. We also welcome your comments at el@ascd.org.

*border*and

*surrounding*, and is a special case of measuring length. Unfortunately, many people will blame this situation on the “mathematical weaknesses” of the students, or even of the teacher, rather than on instructional sequencing that flies in the face of research.

## What We Know

*instrumental practices*and

*relational practices*to differentiate two approaches to teaching and learning. Instrumental practices involve memorizing and routinely applying procedures and formulas. These practices focus on what to do and how to get answers. In contrast, relational practices emphasize the

*why*of learning. These practices involve explaining, reasoning, and relying on multiple representations—that is, on teaching for meaning and helping students develop their own understanding of content.

*x*= 18. One group was taught procedures (subtract 6 from both sides); the other was not. Both groups then received instruction about the meaning of variables and equations. Next, they used trial and error to balance an equation. On post-tests, the students who received only meaningful, or relational, instruction performed better in applying the procedure and solving the equations. In contrast, the students who first received procedural instruction on how to solve an equation tended to resist new ideas and appeared to apply procedures without understanding.

Wearne and Hiebert (1988) investigated the effectiveness of different approaches for teaching decimal concepts. They suggested thatstudents who have already routinized rules without establishing connections between symbols [and what they mean] will be less likely to engage in the [conceptual] processes than students who are encountering decimals for the first time. (p. 374)

*yours is not to reason why, just invert and multiply*may not enhance the performance of many students. Alternatively, instruction that places a premium from the start on meaning and conceptual understanding may improve classroom productivity.

## What You Can Do

- Promote students' discussion of making meaning by posing open-ended questions:
*Why do you think that? Can you explain your reasoning? How do you know that?* - Make explicit connections and incorporate pictures, concrete materials, and role playing as part of instruction so that students have multiple representations of concepts and alternative paths to developing understanding.
- Avoid instruction focused on teaching a single correct approach to arrive at a single correct answer.