background

Understanding the influence of perceptual cues is critical for educators.

Perceptual groupings are consistent across languages and cultures and thus may offer a scaffolding pathway that does not depend on students’ backgrounds. 

Leveraging perceptual cues in instruction and problem-solving helps students notice important systematic structures of math notation that afford efficient strategies and lead to long-term learning by changing the way students perceive, encode, and interpret math information.

project goals

We aim to understand how middle school students’ mathematics learning could be improved by using perceptual scaffolding (e.g., color and spacing of problem features) while solving math problems. Specifically, we are examining the individual effects of cues in mathematical notation on sixth-grade students’ performance on order-of-operations problems. We are examining the following conditions:

• congruent spacing (e.g. 1 + 12×3 - 6, where numbers involved in higher-order operations are grouped more closely together), 

• congruent color (e.g. 1 + 12 × 3 - 6, where numbers involved in higher-order operations are colored), 

• incongruent spacing (e.g. 1 + 12 × 3-6, where numbers involved in lower-order operations are grouped more closely together), 

• incongruent color (e.g. 1 + 12 × 3 - 6, where numbers involved in lower-order operations are colored), 

• and no perceptual cues (e.g. 1 + 12 × 3 - 6).

To address the aims of this project, we will conduct three experimental studies: 

• Study 1 focused on the effects of congruent cues on math learning.
  Research activities occured between October 2023 and May 2024. 

• Study 2 focused on the effects of both congruent and incongruent cues on math learning.
  Research activities occured between August 2024 and February 2025. 

• Study 3 will focus on the effects of Incongruent cues as a DESIRABLE DIFFICULTY. This is the idea that harder problems that initially make students struggle lead to stronger long term learning. Research activities will occur between September 2026 and December 2026. 

Study 1 

Participants 

800 6th graders and their 6th grade teachers were invited to participate in this study. This study was implemented from October 2023 to May 2024.

Study 1 Design and procedure

Participating teachers assigned a link to and supervise students while they completed a 45-minute online session. The session had sections as follows:

• First, students were asked to complete an assessment of their prior order-of-operations knowledge and perceptual processing skills. 

• Then, students were randomly assigned to one of the three conditions to solve a set of 16 order-of-operations problems, in either a congruent spacing, congruent color, or no perceptual cues condition. 

• Finally, students took a posttest of 12 order-of-operations problems without any perceptual cues, and a questionnaire regarding their feelings about math.

Study 2 

Participants 

We will recruited 15 teachers and approximately 1200 sixth-grade students from 12 middle schools.

Study 2 Design and procedure

Similar to Study 1, students solved order-of-operations problems in an online platform under their teachers' supervision. Study 2 will be conducted on 4 school days across the span of approximately 8 weeks. Every 2 weeks, teachers will receive instructions for the next session.

• Session 1. Students completed an assessment of their prior order-of-operations knowledge, metacognitive monitoring and confidence, perceptual processing skills, math anxiety, and math value. 

• Session 2. Students were randomly assigned to one of nine conditions and solved a series of 21 problems specific to their condition. Each condition had a specific spacing setting (Congruent spacing, Incongruent spacing, or Neutral spacing) and color setting (Congruent color, Incongruent color, or Neutral color). Students will receive immediate post-problem feedback and worked examples.

• Session 3. First, students completed a series of 9 problems, again specific to their condition, with immediate post-problem feedback and worked examples. Then, students completed a mirrored posttest of 10 order-of-operations problems without any perceptual cues. Students also answered questions about their metacognitive monitoring and confidence.

• Session 4. Students completed a delayed posttest, which will include a 10-item problem set akin to Session 3 posttest, as well as an assessment of metacognitive monitoring and confidence and perceptual processing skills.

Study 3 

Participants 

We are recruiting 15 teachers and approximately 1500 sixth-and seventh grade students from middle schools in Massachusetts. 

Study 3 Design and procedure

Similar to Study 3, students will solve order-of-operations problems online in their math class. Study 3 will be conducted on 4 school days across the span of approximately 8 weeks. Every 2 weeks, teachers will receive a link for the next activity. In each activity, students will either receive Incongurent Color Cues  or No Cues. For each problem, they will receive feedback and be provided with an example that helps them learn the correct order of operations. Their learning will be assesed by changes in math performance from Days 1 and 4 of the activities.