“Backing Up and Moving Forward” is an insightful article in this month’s Teaching Children Mathematics. The premise of the article is that carefully worded math problems can assist teachers in first assessing student performance and then determining the next steps: do students move forward or back up? The authors, Barlow et al., share their experiences with 5th graders’ understanding of dividing fractions and how teachers determined what those next steps should be.
In the process of designing their lessons, the authors share some features of effective word problems. In this experiment, the word problem was fashioned around “Chef Frederick” as he made dessert. Do students have familiarity with this kind of cooking experience? Does the problem support varied response styles, such as visual, manipulative, or written?
The authors also established a 4 point scale of student performance, from exceeding expectations to lacking fundamental understanding. From my perspective as a special educator, I can predict that many of my kiddos would fall in the lower end of the proficiency scale. Asking students to demonstrate their understanding visually is an effective indicator of performance. Both teachers and students can better see the reasoning process and where it might have broken down. In one example, a student did not know how to represent a simple fraction; obviously, dividing fractions was introduced before that student had sufficient prior learning.
I’d love to see a follow-up article on how classroom teachers could address major gaps in mathematical understanding. It takes time to replace misconceptions. It takes willingness on both the student’s and teacher’s part to tackle the process. What happens to the struggling learners while the remaining kids move forward? It is encouraging that these educators recognize the need to fill in those gaps, instead of simply pushing forward. If this assessment and repair process occurred routinely at all grade levels, perhaps we wouldn’t see as many kids who are partially memorizing algorithms and procedures without true understanding.