Problem Solving and Mathematical Reasoning
 Elementary School The student demonstrates logical reasoning throughout work in mathematics, i.e., concepts and skills, problem solving, and projects; demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow; and solves problems that make demands on all three aspects of the solution process—formulation, implementation, and conclusion. Formulation a Given the basic statement of a problem situation, the student: • makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps; • uses previously learned strategies, skills, knowledge, and concepts to make decisions; • uses strategies, such as using manipulatives or drawing sketches, to model problems. Implementation b The student makes the basic choices involved in planning and carrying out a solution; that is, the student: • makes up and uses a variety of strategies and approaches to solving problems and uses or learns approaches that other people use, as appropriate; • makes connections among concepts in order to solve problems; • solves problems in ways that make sense and explains why these ways make sense, e.g., defends the reasoning, explains the solution. Conclusion c The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: • explains a pattern that can be used in similar situations; • explains how the problem is similar to other problems he or she has solved; • explains how the mathematics used in the problem is like other concepts in mathematics; • explains how the problem solution can be applied to other school subjects and in real world situations; • makes the solution into a general rule that applies to other circumstances.

 Middle School The student demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow, and solves problems that make demands on all three aspects of the solution process—formulation, implementation, and conclusion. Formulation a The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: • formulates and solves a variety of meaningful problems; • extracts pertinent information from situations and figures out what additional information is needed. Implementation b The student makes the basic choices involved in planning and carrying out a solution; that is, the student: • uses and invents a variety of approaches and understands and evaluates those of others; • invokes problem solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table; • determines, where helpful, how to break a problem into simpler parts; • solves for unknown or undecided quantities using algebra, graphing, sound reasoning, and other strategies; • integrates concepts and techniques from different areas of mathematics; • works effectively in teams when the nature of the task or the allotted time makes this an appropriate strategy. Conclusion c The student provides closure to the solution process through summary statements and general conclusions; that is, the student: • verifies and interprets results with respect to the original problem situation; • generalizes solutions and strategies to new problem situations. Mathematical reasoning d The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: • formulates conjectures and argues why they must be or seem true; • makes sensible, reasonable estimates; • makes justified, logical statements.

 High School The student demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow, and solves problems that make demands on all three aspects of the solution process—formulation, implementation, and conclusion. Formulation a The student participates in the formulation of problems; that is, given the statement of a problem situation, the student: • fills out the formulation of a definite problem that is to be solved; • extracts pertinent information from the situation as a basis for working on the problem; • asks and answers a series of appropriate questions in pursuit of a solution and does so with minimal “scaffolding” in the form of detailed guiding questions. Implementation b The student makes the basic choices involved in planning and carrying out a solution; that is, the student: • chooses and employs effective problem solving strategies in dealing with non-routine and multi-step problems; • selects appropriate mathematical concepts and techniques from different areas of mathematics and applies them to the solution of the problem; • applies mathematical concepts to new situations within mathematics and uses mathematics to model real world situations involving basic applications of mathematics in the physical and biological sciences, the social sciences, and business. Conclusion c The student provides closure to the solution process through summary statements and general conclusions; that is, the student: • concludes a solution process with a useful summary of results; • evaluates the degree to which the results obtained represent a good response to the initial problem; • formulates generalizations of the results obtained; • carries out extensions of the given problem to related problems. Mathematical reasoning d The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student: • employs forms of mathematical reasoning and proof appropriate to the solution of the problem at hand, including deductive and inductive reasoning, making and testing conjectures, and using counterexamples and indirect proof; • differentiates clearly between giving examples that support a conjecture and giving a proof of the conjecture.