Abstract:

Personalized feedback is integral for effective mathematics learning. An expert teacher looking over the shoulder of a student can provide a personalized hint to a student making a mistake. If the student makes the same mistake again, the teacher may provide the student a different hint, responding to a new detail in the student’s apparent thinking. If the hint is too large, the expert teacher will be reducing the cognitive demand of the task without promoting productive struggle, and if the hint is too small, the student may remain stuck. An automated system to provide personalized feedback within a digital application, like an expert teacher, could allow each student to advance at their own pace. To enable this, a hierarchy of feedback that can support all students at critical moments as they make progress with challenging tasks is needed.

This paper presents the design of a feedback hierarchy and the subsequent effects on engagement for students working with a new application called Drawn2Math (D2M). This application was embedded with a feedback hierarchy designed to keep K-6 students engaged with challenging tasks and support their perseverance to learn mathematics conceptually. A key feature of the app is that students draw solutions to math problems, such as visual representations of area models used in fractions tasks. The app can identify if the student’s sketch is correct, and whether a mistake they make matches a predefined common misconception. Authors of the assignment define a set of common misconceptions with appropriate feedback. Feedback spans different levels of scaffolding, from low-level conceptual reminders, to medium-level restructuring, to high-level environmental provisioning. The approach utilizes the rich amount of information in student sketches, which illuminates their misconceptions, as well as teacher expertise in authoring the assignments, which captures their classroom experience regarding common mistakes. Each feedback message is correlated to a specific sketch that a student draws, and is designated a hierarchy level (e.g, first hint for a given mistake), the number of times the hint can be triggered, and follow-on hints for the same mistake. A feedback hierarchy table was developed to allow the assignment author to designate their feedback approach.

An exploratory study implemented the feedback hierarchy with 10 fourth-grade students who were learning fractions for the first time. To test the validity of this system, a Wizard of Oz usability experiment was conducted in which students sketched on an iPad and received triggered support from the predefined feedback hierarchy on their screen, but the feedback was administered by a remotely observing researcher acting as the grading algorithm. Our findings showed that participants working on tasks within D2M were able to leverage the personalized feedback to persevere with the task despite working on them without a teacher present, even at moments when they were most challenged and frustrated. All 10 participants showed evidence of this kind of success. All participants made a sketching mistake during their initial attempt at solving, but the feedback helped them stay engaged and continue to make progress. These findings suggest that feedback hierarchies embedded within this education technology can be carefully designed to help students stay engaged with challenging mathematics and learn conceptually, even when working independently or even remotely.

Keywords:

Mathematics, app, feedback hierarchy, perseverance, sketching.