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 nonroutine 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 prespecified 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. 
The student demonstrates problem solving
by using mathematical concepts and skills to solve nonroutine 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 sensemaking
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. 
The student demonstrates problem solving
by using mathematical concepts and skills to solve nonroutine 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 nonroutine and multistep 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. 
