1. Prospective teachers facilitate Problem-Solving by enabling students to
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Reason Inductively, Conjecture, and Generalize
Problem Solve via a Process (such as that of Polya)
Persevere in a Productive Struggle
Demonstrate Precision and Attention to Detail
Make Connections Among Mathematical and Scientific Phenomena
Represent Mathematical Ideas in Multiple Ways
Use Tools (Technology) Appropriately to Aid Problem Solving
2. Prospective teachers facilitate Abstract Reasoning, quantitatively, efficiently, and effectively by enabling students to
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Reason Inductively, Conjecture, and Generalize
Reason Deductively and Use Logic
Generalize and Specialize Appropriately
Mathematize (Quantify Arguments)
Construct Viable Logical and Quantitative Arguments
Logically Critique the Arguments of Others
Support Conjectures via Logical and Deductive Arguments
Represent Mathematical Ideas in Multiple Ways: Graphically, Numerically, Symbolically, and Verbally
3. Prospective teachers facilitate Modeling with mathematics by enabling students to
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Reason Inductively, Conjecture About, and Construct Viable Models
Use of Appropriate Tools strategically to Construct Viable Models
Algorithms to Construct Models with the Aid of Technology
Represent Models in Multiple Ways: Graphically, Numerically, Symbolically, and Verbally
Affirm the Viability of Models Using Logic and Deductive Reasoning
Affirm the Limitations of Models Using Logic and Deductive Reasoning
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Construct and Refine Mathematical Models via a cycle, as in the flow chart:
4. Prospective teachers facilitate looking for and making use of Mathematical Structure and expressing Regularity in Repeated Reasoning by
enabling students to
Common Prompts for Problem-Solving, Abstract Reasoning, and Mathematical Modeling
Prospective teachers teach Problem-Solving, (Abstract) Reasoning, and Mathematical Modeling by enabling their students to
1. Employ problem-solving, reasoning, or modeling processes (for example, respectively, Polya’s Process, Deductive Reasoning, or the CCS
Modeling Flow Chart).
2. Reason inductively, conjecture, and generalize toward a problem solution or mathematical model.
3. Reason deductively, and use logic to affirm conjectures, arguments, generalities, or mathematical models.
4. Logically affirm the limitations of conjectures, arguments, solutions, and models.
5. Generalize and specialize appropriately in solutions, arguments, and mathematical models.
6. Demonstrate precision and attention to detail.
7. Logically and mathematically connect mathematical and scientific constructs.
8. Mathematize (quantify) and logically construct arguments.
9. Logically and mathematically critique the arguments of others.
10. Represent mathematical ideas in multiple ways.
11. Recognize and make use of mathematical structure in order to simplify, streamline, specify problem solutions, algorithms or
mathematical models, or to generalize mathematical or logical statements.
Reduce the rubric to five outcomes. Six is inauthentic and too many to grade. Here is the rubric that will fit for each dimension.
Not Effective (0pt)
Inconsistent (1 pts)
The supporting
The supporting evidence is
evidence is all or
inconsistent, lacks some
mostly missing,
elements of support, or lacks
vague, or lacks
some professional structure.
professional structure.
Proficient (2 pts)
The evidence is
adequate and affirms
professional
demonstration of the
standard.
Very Effective (3pts)
The evidence is
comprehensively and
redundantly affirms
professional
demonstration of the
standard.
Highly
Accomplished (4 pts)
The evidence
comprehensively,
redundantly, and creatively
affirms professional
demonstration of the standard.
Missing or
Incomplete (0 pt)
The supporting
evidence is missing
or substantially
lacks professional
structure.
Not
Effective (1 pt)
The supporting
evidence is
mostly missing,
vague, or lacks
professional
structure.
Very
Effective (4 pts)
The supporting
The evidence is
The evidence is
evidence is
adequate and affirms comprehensively
inconsistent, lacks
professional
and redundantly
some elements of
demonstration of the affirms
support, or lacks some standard.
professional
professional structure.
demonstration of
the standard.
Inconsistent (2 pts)
Proficient (3 pts)
Highly
Accomplished (5 pts)
The evidence
comprehensively,
redundantly, and
creatively affirms
professional
demonstration of the
standard.
Either the
artifact or
the
reflection
is missing
or
unprofessi
onal.
Missing or Incomplete (0 pt)
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able to achieve a deep
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Not
Effective (
1 pt)
Evidence
Very
Highly
Inconsistent ( Proficient (
Missing or
Effective (4 Accomplis
2 pts)
3 pts)
Not
pts)
hed
Profession
ally
Presented