NCTM Research Presession, April 2010

Teaching for Understanding and Its Impact on Learning in Algebra
Jon Hasenbank & Jennifer Kosiak
Presented April 20, 2010

Abstract: The National Council for Teachers of Mathematics calls for students to learn mathematics with understanding (NCTM, 2000). This vision also includes expanding students’ understanding of algebraic procedures. Building on the six essential elements in teaching and learning algebra presented in NCTM’s Navigating through Algebra in Grades 9-12, we have developed the student-centered Framework for Procedural Understanding shown below. Over the past several years it has been used as a model for guiding the teaching and learning of algebra.  

The Framework for Procedural Understanding:

  1.  What is the goal of the procedure?
  2. What sort of answer should I expect?
  3.  Can I execute the procedure? What are the steps?
  4. What other procedures could I use to complete the task?
  5. Why does the procedure work? Why are the steps valid?
  6. How can I verify my answer? Does it make sense?
  7. When is this the most efficient procedure to use?
  8. What are some applications of this procedure?  

We will report on the results of a year-long quasi-experimental teaching study designed to identify the impact of Framework-oriented instruction in Algebra. A series of three algebra tests were administered to measure students’ algebraic skill and understanding. We also developed and used a digital classroom observation instrument to record the minute-by-minute focus on different facets of procedural understanding as defined by our Framework. The results of this experiment demonstrate the impact of teaching for understanding and illuminate the relationship between procedural skill and understanding in algebra.

Session 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?
     AND
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.