Problem Solving in the Quantitative Reasoning Pathway

A problem is generally considered to be a mathematical task for which one does not know a solution method in advance. Problem solving therefore generally involves the mental activity in which one must engage to identify, select, perform, and evaluate solution methods. Developing productive problem-solving habits and a disposition for critical thinking are essential components of mathematical proficiency. Problem-solving skills are necessary to aid the critical thinking on which quantitative reasoning relies. A goal of entry-level mathematics instruction is to enhance students’ problem-solving ability while leveraging it as a foundation for their learning of central ideas.

Faculty at the MIP Quantitative Reasoning Initiation Workshop in May, 2021 recommended that instructional resources developed by CoRDs and ARCs addressing Problem Solving for success in the Oklahoma Quantitative Reasoning Pathway should:

1. Use contextual problems, affording students an opportunity to address the social context of the data (e.g. including multiple values/perspectives/needs, attending to implicit biases). Because Quantitative Reasoning problems are posed in real world context, the context plays a significant role in decision making.

2. Identify the cognitive habits and characteristics of effective problem solvers.

3. Attend specifically to ways to help students explore and make sense of problems (e.g., focus on understanding the question, organizing and making sense of the information given, processing, asking what other information is needed and what information, if any, is unnecessary, thinking about what form an answer to the given question could take (e.g. a number, set of numbers, equation, graph, explanation, etc).

Addressing Components of Inquiry

Participants of the MIP Workshop on Quantitative Reasoning suggested the following ways resources about Problem Solving could address the three MIP components of mathematical inquiry:

Active Learning: Because problem solving involves identifying and outlining methods of approaching mathematical situations (for which no method is known ahead of time), problem solving tasks inherently afford opportunities for students to engage in cyclic, iterative processes of identifying, selecting, performing, and evaluating actions. CoRD members could potentially explore the relationship of this process to problem solving frameworks that are outlined in the literature.

Meaningful Applications: Problem solving tasks afford an opportunity for students to identify mathematical relationships, make and justify claims, and generalize across contexts because the nature of problem solving are inherently not context-specific. CoRD members are therefore encouraged to identify context-invariant aspects of problem solving that students themselves might be supported in attending to and developing an awareness of as they engage in various tasks.

Academic Success Skills: Problem solving – that is, when one is devising a solution to a situation in which the method for obtaining a is not known in advance – is related to a number of key academic success skills, including perseverance (often the first few methods attempted are not successful), mathematical identity and confidence (seeing a difficult, multifaceted problem through to its solution can support positive self-images related to mathematics), and classroom community building (collaborating and sharing ideas to help devise a solution method). CoRD members are encouraged to provide some specific guidance to how such academic success skills might be fostered.

References

Carlson, M. P., & Bloom, I. (2005) The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58(1), 45-75.

Silver, E. A. (Ed.). (2013). Teaching and learning mathematical problem solving: Multiple research perspectives. Routledge.

Thompson, P. W. (1985). Experience, problem solving, and learning mathematics: Considerations in developing mathematics curricula. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 189-243).

Wilkersen-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?: Resources and acts for constructing and understanding mathematics. Educational Studies in Mathematics, 78(1), 21-43.