A mathematics program that revisits fundamental problems throughout a student's academic career, while encouraging the use of problem-solving strategies that match his or her current level of conceptual understanding, has the potential to enhance understanding of variable, representation, and mathematical models in a manner not possible with curricular materials that view mathematics as skills obtained in a disjoint collection of courses. This is consistent with the NCTM Problem Solving Standard that suggests that "students be given the opportunity to learn important content through their exploration of the problems and to learn and practice a wide range of heuristic strategies" (NCTM, 2000, p. 341).
Here is a list of general strategies that are available to our students as they solve mathematics problems using technology. For convenience sake, we've divided these strategies into four levels. The levels correspond roughly to courses taken by high school mathematics students or developmental students at the post-secondary level.
- Level 1: Data Analysis-based Strategies (~ introductory algebra)
- Level 2: Dynamic Geometry-based Strategies (~ introductory geometry)
- Level 3: Functional Model-based Strategies (~ advanced algebra or trigonometry)
- Level 4: Calculus-based Strategies (~ precalculus or calculus)
Figure 2 provides a graphical depiction of these strategies.
Figure 2. Strategies for solving mathematics problems with technology
The following aspects of the graphical model are worth examining in greater detail.
A given level within the figure wholly encompasses all levels that precede it. For instance, data analysis and geometry-based strategies (Levels 1 and 2, respectively) lie within Level 3. This suggests that students in our courses who have access to function-model problem-solving strategies may also examine problems from either a geometric or data analysis-based perspective. As our students progress mathematically, they have more choices associated with solution strategies. Recognizing this, we strive to build activities that are accessible to as many of the levels as possible.
Although it is tempting to consider higher levels to be mathematically more rigorous, this is not necessarily the case. For instance, students using dynamic geometry tools (a Level 2 technique) may ultimately write proofs to generalize their findings. Such an activity may be more "rigorous" than calculator-based regression (a Level 3 approach).