Abstract:

Perseverance, or initiating and sustaining productive struggle in the face of setbacks, is imperative for the conceptual learning of mathematics. When students engage with challenging tasks, such setbacks often manifest as problem-solving errors, which can be devastating for students both cognitively and emotionally and often lead to giving up. This outcome is even more prevalent when students are working remotely, without a teacher present to offer guidance. Thus, to expect students to persevere with a challenging task, we must support them to learn from their errors to help re-initiate their effort. One way to support this kind of resilient engagement is through personalized mathematical feedback that responds directly to a student’s specific error. This type of attentive support, however, is impossible for teachers to provide to all students simultaneously, and therefore it should be investigated how digital technology could be leveraged to efficiently support students in these ways.

The purpose of this exploratory study was to show how a newly designed digital application called Drawn2Math (D2M) could anticipate students’ errors with sketching-based fractions tasks and offer corresponding mathematical feedback to help support their perseverance. A key feature of D2M is that students draw solutions to fractions tasks, such as area models that represent rational numbers. D2M’s grading algorithm can identify students sketched anticipated errors based on common misconceptions informed by the mathematics education literature and our own pilot research. We designed specific textual and visual mathematical feedback, at various scaffolding levels, to be delivered to the student in response to their errors. The intent of this feedback was to meet the student at their current stage of thinking and support them to persevere with their problem-solving after making an error.

We completed problem-solving interviews with 10 fourth-grade students who were learning about fractions for the first time. Each participant thought-aloud as they completed about 15 conceptual fraction tasks that required freehand sketching a visual representation. To test the validity of our error anticipation, we employed a Wizard of Oz usability experiment in which students sketched on an iPad and received predefined feedback when they made errors we anticipated ahead of time. The predefined feedback was administered by a remotely observing researcher simulating the grading algorithm.

Our findings showed that a large majority of participants’ sketching errors were anticipated. These errors ranged from procedural (e.g., miscalculations, not following directions) to conceptual (e.g., numerator misconceptions, denominator misconceptions, partitioning and/or iterating misconceptions). Further, participants were able to leverage the personalized feedback in response to their anticipated error to persevere with the task, even at moments when they were most challenged and frustrated. Participants were persevering on these tasks despite working on them remotely without a teacher present. These findings suggest that mathematics education technology can help students learn from their mathematical errors and persevere with challenging mathematics, even when working alone. The next steps for this research will iterate on D2M to improve its sketch-recognition algorithm and feedback effectiveness, and will conduct a classroom trial to show D2M’s effect at a larger scale.

Keywords:

Mathematics education, digital technology, perseverance, feedback.