Jon R. Star is a professor of education and educational psychologist at the Harvard Graduate School of Education. He studies how children learn mathematics, the development of flexibility in problem solving, and the effectiveness of instructional and curricular interventions. He is an expert in middle school math curricula, and a former middle and high school math teacher. See full bio
By Jon R. Star
This is a familiar challenge for many math teachers. Topics in mathematics are usually taught in a purposeful sequence, from topic A to topic B to topic C, etc. Ideally, students would master topic A before progressing to topic B, then topic B before progressing to topic C. But too often, students do not master those prerequisites. Due to time constraints and curricular expectations, teachers have to finish one unit and move on to the next before every student in the classroom has mastered the former. Unit by unit and year by year, this problem can snowball. Teachers then must cover the grade level material (for example, topic C) while at the same time filling in the gaps (for example, topics B and A).
While there is no easy solution to this challenging situation, my advice would be to keep these two goals — filling gaps, and teaching current material — separate.
Devote instructional time daily to filling gaps. Expose students to mathematics problems that include tasks from prior years and units. You can do this through “do now” or warm-up exercises, additions to homework assignments, or even test problems. The point is to give students opportunities to revisit past content and to refine their understandings of this “old” material.
When it comes to the current material, recognize that it may be necessary (in the short term, at least) to modify the complexity of new content so that it’s approachable for all students, especially those with a weaker knowledge of “old” material. Students will have an easier time learning new content if the assumption is not that each of them has a complete understanding of the prerequisite content. So when presenting new material, consider using “easier” numbers, fewer fractions, and generally more straightforward problems. This way, struggling students can begin to grasp the important ideas of the new material without being handicapped by their fragile understanding of the “old.”
Too often, struggling students fall further and further behind because their lack of understanding of prior content prevents them from learning new material. This is a very difficult cycle to break. But by devoting regular instructional time to reviewing and remediating past material, and by altering the ways that the new material is presented so that it’s more approachable for struggling students, we can increase the chances that these students will gradually find a more stable foothold in math and begin the difficult climb back to full understanding.
This answer was developed in partnership with Usable Knowledge at the Harvard Graduate School of Education.