Carnegie Learning Webinar

Teaching for Understanding and Its Impact on Students' Learning in Algebra
Jon Hasenbank & Jennifer Kosiak
Presented March 25, 2010

Abstract: In this webinar we discuss a student-centered Framework used to promote understanding of algebraic procedures. Using student work samples we will demonstrate how traditional mathematics tasks can be enriched to support the development of deeper algebraic knowledge.  We also present the impact of a yearlong teaching for understanding program on students’ learning, including recommendations and resources for enriching algebra in your school.

Resources:

Frequently Asked Questions:

Q:  There just isn’t time to have my students explore all eight "framework questions" every day.

A:  We suggest ‘sprinkling in’ the understanding questions, picking and choosing only those that naturally connect with the topic at hand. The goal is that over time (not every day) questions from all eight Framework categories will be addressed, practiced, and assessed.

Q:  What about homework?

A:  In our college algebra study (Hasenbank, 2006) we reduced the skill question set by 18% to make room for understanding questions in the nightly homework. Overall, even with the added understanding questions, students were assigned 8% fewer problems and yet outperformed the comparison group on both skill items and understanding items. The benefits of a focus on understanding and the law of diminishing returns for repetitive practice help explain these results.

Q:  Writing skills seem to be a barrier to students communicating their understanding. What can we do to work around that?

A:  Students' writing skills can improve, but we need to scaffold that learning. We have used sentence skeletons (“First I…, then I…, and finally I…”) to help them structure their responses. A focus on vocabulary development is another important component, and literacy strategies (e.g. Frayer models and other graphic organizers) can be helpful tools at all levels.

Q:  Can we really infer understanding from written responses?
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Q: If students have been exposed to the answers to these “deep questions” in class, how can we be sure students' responses on tests don’t just reflect answers that are simply being recited back?

A: Due to the complex nature of “understood knowledge,” it is true that inferring deep understanding from responses on a written test is problematic. However, the ability to answer an understanding question is clearly preferable to the inability to do so, even if for some students that knowledge is superficial. Ultimately, the more the learning environment is focused on asking and exploring deep questions, the more likely written responses will reflect true understanding. One-on-one interviews, think-aloud podcasts, and oral presentations or summative projects are other potential mechanisms for assessing the depth of students' understanding.