Response to: Investigating Mathematics with Technology: Lesson Structures That Encourage a Range of Methods and Solutions

The article Investigating Mathematics with Technology: Lesson Structures That Encourage a Range of Methods and Solutions by Rebecca McGraw and Maureen Grant compares "two kinds of lessons (we call these Type 1 and Type 2) according to the extent to which they engage students in (1) identifying patterns and searching for relationships, (2) making and investigating mathematical conjectures, and (3) developing and evaluating mathematical arguments" (McGraw & Grant, pg. 303).  The authors describe Type 1 lessons by how the instructions "are written to focus students' attention on particular mathematical relationships. Technology is used to help make these relationships transparent to students" (McGraw & Grant, pg. 304).  McGraw and Grant claim the difference between Type 1 and Type 2 is that the instruction is different and "the roles of the teachers and students during implementation" (McGraw & Grant, pg. 304).

This difference is immediately apparent when reading through dialogue between the teacher and students for each of the lesson types.  In Type 1, the instructor Mr. Jeffries appears to be pulling teeth and really stretching the students for answers about the activity, where students examined equations using their graphing calculators.  The lesson seemed to be centered around the teacher and as a result only engaged students who understood the material and had a firm grasp on the concepts being examined.

Type 2 is comparable to lessons that remove the scaffolding and encourage students to arrive at their own conjectures through exploration, which is facilitated by the graphing calculator.  In this activity, the students were first presented with graphs and are instructed to group the graphs.  The lesson then slowly unravels details about the subject matter by providing supplemental materials in a "reverse" order.   Students receive a handout with the equations and are asked to match graphs to equations which leads the students to organize their own thoughts through communication of their conjectures.

While it would be fantastic as an instructor to have every lesson and class period follow Type 2, I do not believe it is practical. However, instead of presenting students with the information up front (i.e. terminology) students can investigate concepts on their own and the teacher can utilize recaps and discussions from Type 2 lessons to incorporate terminology that is necessary for mathematics communication.

McGraw, R., & Grant, M. (2005). Investigating mathematics with technology: Lesson structures that encourage a range of methods and solutions. In W. J. Masalski (Ed.), Technology-supported mathematics learning environments (pp. 303-317). Reston, VA: National Council of Teachers of Mathematics.