Transition Mathematics Project

Jon Hasenbank, PhD, facilitated a workshop on "Framework for Procedural Understanding, Theory into Practice" He kicked his session off by highlighting the promising work of the TMP College Readiness Standards as being aligned with understanding. He reviewed student work that addressed the question, How well do you think your students ‘understand’ mathematics? One answer is that we cannot infer understanding from skilled execution of procedures. Dr. Hasenbank proceeded to outline dimensions of knowledge:

• Type:
  Concepts vs. Procedures
  (Ideas vs. Algorithms)
• Aptitude:
  Novice vs Practiced Novice vs. Practiced
  (Arduous vs. Automatic)
• Depth:
  Shallow vs. Deep
  (Disconnected vs. Connected)

Dr. Hasenbank described a study comparing two understanding frameworks:
The “Framework for Procedural Understanding” (NCTM, 2001) which can be used to: define procedural understanding, deepen lesson content, deepen lesson content, select and develop homework tasks, and assess understanding through homework, quizzes, tests, journals, and interviews. It is a student-centered guide to understanding procedures.

The “Teaching for Understanding" (TfU) Framework which consists of 4 parts:

1. generative topics
2. understanding goals
3. performances of understanding
4. ongoing assessment

TfU is intended to guide teachers in designing units that elicit understanding in all subject areas. Be patient!

Key lessons learned to this work include the heightened value of performance: It’s important for students to perform their understanding, and faculty will learn more about what students understand (or misunderstand) and they will be able to sort out what students really think through these performances. Performances include:

• Daily homework presentations (rotate through students).
• Small group tasks, thoughtfully selected
• Journal reflections
• Individual projects
• Finding connections (in math, to life, to history…) and explain it (to me, or to the class.)
• Quiz prompts (formative)
• Exam prompts (summative)

As students engage in understanding performances, we should help students make connections by asking them to explain their answers, give reasons for their choices, offer supporting evidence, give reasons for their choices, make predictions, listen for common questions or sources of confusion that should be addressed in whole-group discussion or lecture.

Last, students value what we assess so our assessments must reflect our understanding goals. Further, we must provide students with clear responses to their performances of understanding in a way that helps them improve their next performance.

Discussion handout for Dr. Hasenbank’s session