Multiple Problem-Solving Strategies and Representational Equivalence in the College Algebra and Precalculus Pathway

Students will be required to apply the skills learned in College Algebra and Precalculus in an extremely diverse range of ways, meaning that they must also develop flexible problem-solving strategies related to these skills. MIP modules can foster an image of algebraic skills as general tools and an ability to strategically select, apply, and reflect on their application in a wide variety of goal-oriented activity. Additionally, access to multiple approaches enables a broader range of meaningful student participation and enables students to select approaches that work better for them, to apply multiple reinforcing approaches, or to check reasoning through alternate methods.

Faculty at the MIP College Algebra and Precalculus Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Multiple Problem-Solving Strategies and Representational Equivalence for success in the Oklahoma Functions and Modeling Math Pathway should:

1. Develop general problem-solving strategies, such as drawing a diagram of a situation and representing all relevant quantities in the diagram, drawing a graph of a quantitative relationship and representing all relevant quantities in the graph, reviewing underlying concepts and terminology, attempting a numerical or approximate approach, looking for counterexamples, communicating one’s problem-solving process to another person, and looking for equivalent forms or representations.

2. Develop content-specific problem-solving strategies, such as factoring or expanding, applying a Pythagorean identity, multiplying by a clever form of 1 or adding a clever form of 0, rationalizing an expression, and reducing trigonometric expressions to sines and cosines.

3. Develop the ability to apply algebraic skills in novel situations. Modules will need to help students recognize structural equivalence between problem settings and known tools.

4. Develop an understanding of why algebraic procedures work, thus promoting their meaningful interpretation and recall.

5. Develop an image of equivalence of algebraic expressions under various algebraic operations.

6. Develop an image of the invariance of solution sets of equations or a system of equations under various algebraic operations

Addressing Components of Inquiry

Participants of the MIP Workshop on College Algebra and Precalculus suggested the following ways resources about Multiple Problem-Solving Strategies and Representational Equivalence could address the three MIP components of mathematical inquiry:

Active Learning: It is essential to provide students with tasks that are genuine problems, beyond their current repertoire of familiar procedures, to provide them the genuine opportunity to activate problem-solving strategies. Modules should focus on helping students deliberately attend to all phases of problem-solving, including planning and evaluation as well as carrying out solution methods. Having them explicitly reflect on their partial or completed solution processes can also help reify productive strategies for future use.

Meaningful Applications: Applications are beneficial for developing problem-solving strategies and reasoning with representational equivalence, as they provide opportunities to express mathematical problems in multiple, non-routine contexts. As students begin to see the similarity of structure of situations and their algebraic tools across multiple settings, they are able to begin developing a generalization of the mathematics independent of the settings in which they were originally experienced.

Academic Success Skills: Students should become increasingly open to productive struggle. Many students, even those who have been highly successful in previous mathematics classes, view any difficulty as an indication of their lack of ability. Viewing challenging tasks as a natural part of mathematical tasks, and even as a necessary aspect of learning, can help students develop persistence (e.g., see the framework involving steps in the cyclic problem-solving process and the impact of underlying beliefs and affect in Carlson & Bloom, 2005).

References

Carlson, M., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem solving framework. Educational Studies in Mathematics, 58, 45–75.