Understanding a mathematics topic consists of having the ability to operate successfully in three cognitive domains. The first domain, knowing, covers the facts, procedures, and concepts students need to know, while the second, applying, focuses on the ability of students to make use of this knowledge to select or create models and solve problems. The third domain, reasoning, goes beyond the solution of routine problems to encompass the ability to use analytical skills, generalize, and apply mathematics to unfamiliar or complex contexts.
Without access to a knowledge base that enables easy recall of the language and basic facts and conventions of number, symbolic representation, and spatial relations, students would find purposeful mathematical thinking impossible. Facts encompass the factual knowledge that provides the basic language of mathematics, and the essential mathematical facts and properties that form the foundation for mathematical thought.
Procedures form a bridge between more basic knowledge and the use of mathematics for solving routine problems, especially those encountered by many people in their daily lives. In essence, a fluent use of procedures entails recall of sets of actions and how to carry them out. Students need to be efficient and accurate in using a variety of computational procedures and tools. They need to see that particu- lar procedures can be used to solve entire classes of problems, not just individual problems.
Knowledge of concepts enables students to make connections between elements of knowledge that, at best, would otherwise be retained as isolated facts. It allows them to make extensions beyond their existing knowledge, judge the validity of mathematical statements and methods, and create mathematical representations.
Behaviors Included in the Knowing Domain
- Recall – Recall definitions, terminology, notation, mathematical conventions, number properties, geometric properties.
- Recognize – Recognize entities that are mathematically equivalent (e.g., different representations of the same function or relation).
- Compute – Carry out algorithmic procedures (e.g., determining derivatives of polynomial functions, solving a simple equation).
- Retrieve – Retrieve information from graphs, tables, or other sources.
In items aligned with this domain, students need to apply knowledge of mathematical facts, skills, procedures, and concepts to create represen- tations and solve problems. Representation of ideas forms the core of mathematical thinking and communication, and the ability to create equivalent representations is fundamental to success in the subject.
Behaviors Included in the Applying Domain
- Select – Select an efficient/appropriate method or strategy for solving a problem where there is a commonly used method of solution.
- Represent – Generate alternative equivalent representations for a given mathematical entity, relationship, or set of information.
- Model – Generate an appropriate model such as an equation or diagram for solving a routine problem.
- Solve Routine Problems – Solve routine problems, (i.e., problems similar to those students are likely to have encountered in class). For example, differentiate a polynomial function, use geometric properties to solve problems.
Reasoning mathematically involves the capacity for logical, system- atic thinking. It includes intuitive and inductive reasoning based on patterns and regularities that can be used to arrive at solutions to non- routine problems. Non-routine problems are problems that are very likely to be unfamiliar to students.
Behaviors Included in the Reasoning Domain
- Analyze – Investigate given information, and select the mathematical facts necessary to solve a particular problem. Determine and describe or use relationships between variables or objects in mathematical situations. Make valid inferences from given information.
- Generalize – Extend the domain to which the result of mathematical thinking and problem solving is applicable by restating results in more general and more widely applicable terms.
- Synthesize/ Integrate – Combine (various) mathematical procedures to establish results, and combine results to produce a further result. Make connections between different elements of knowledge and related representations, and make linkages between related mathematical ideas.
- Justify – Provide a justification for the truth or falsity of a statement by reference to mathematical results or properties.
- Solve Non-routine Problems – Solve problems set in mathematical or real-life contexts where students are unlikely to have encountered similar items, and apply mathematical procedures in unfamiliar or complex contexts.