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List of Benchmarks for Mathematics

Standard 1.Uses a variety of strategies in the problem-solving process
  Level Pre-K (Grade Pre-K)
   1. Not appropriate at this level
  Level I (Grade K-2)
   1. Draws pictures to represent problems
   2. Uses discussions with teachers and other students to understand problems
   3. Explains to others how she or he went about solving a numerical problem
   4. Makes organized lists or tables of information necessary for solving a problem
   5. Uses whole number models (e.g., pattern blocks, tiles, or other manipulative materials) to represent problems
  Level II (Grade 3-5)
   1. Uses a variety of strategies to understand problem situations (e.g., discussing with peers, stating problems in own words, modeling problem with diagrams or physical objects, identifying a pattern)  A 
   2. Represents problems situations in a variety of forms (e.g., translates from a diagram to a number or symbolic expression)
   3. Understands that some ways of representing a problem are more helpful than others
   4. Uses trial and error and the process of elimination to solve problems
   5. Knows the difference between pertinent and irrelevant information when solving problems  A 
   6. Understands the basic language of logic in mathematical situations (e.g., "and," "or," "not")  A 
   7. Uses explanations of the methods and reasoning behind the problem solution to determine reasonableness of and to verify results with respect to the original problem  A 
   8. Understands basic valid and invalid arguments (e.g., counter examples, irrelevant approaches)
  Level III (Grade 6-8)
   1. Understands how to break a complex problem into simpler parts or use a similar problem type to solve a problem
   2. Uses a variety of strategies to understand problem-solving situations and processes (e.g., considers different strategies and approaches to a problem, restates problem from various perspectives)  A 
   3. Understands that there is no one right way to solve mathematical problems but that different methods (e.g., working backward from a solution, using a similar problem type, identifying a pattern) have different advantages and disadvantages
   4. Formulates a problem, determines information required to solve the problem, chooses methods for obtaining this information, and sets limits for acceptable solutions  A 
   5. Represents problem situations in and translates among oral, written, concrete, pictorial, and graphical forms  A 
   6. Generalizes from a pattern of observations made in particular cases, makes conjectures, and provides supporting arguments for these conjectures (i.e., uses inductive reasoning)  A 
   7. Constructs informal logical arguments to justify reasoning processes and methods of solutions to problems (i.e., uses informal deductive methods)  A 
   8. Understands the role of written symbols in representing mathematical ideas and the use of precise language in conjunction with the special symbols of mathematics
   9. Uses a variety of reasoning processes (e.g., reasoning from a counter example, using proportionality) to model and to solve problems  A 
  Level IV (Grade 9-12)
   1. Uses a variety of strategies (e.g., identify a pattern, use equivalent representations) to understand new mathematical content and to develop more efficient solution methods or problem extensions
   2. Constructs algorithms for multi-step and non-routine problems
   3. Understands the concept of a mathematical proof
   4. Constructs logical verifications or counter examples to test conjectures and to justify algorithms and solutions to problems (i.e., uses deductive reasoning)
   5. Uses formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations
   6. Understands the difference between a statement that is verified by mathematical proof (i.e., a theorem) and one that is verified empirically using examples or data
   7. Understands connections between equivalent representations and corresponding procedures of the same problem situation or mathematical concept (e.g., a zero of a function corresponds to an x-intercept of the graph of the function, the correspondence of binary multiplication to a series electrical circuit and the logical operation "and")
   8. Understands the components of mathematical modeling (i.e., problem formulation, mathematical model, solution within the model, interpretation of solution within the model, validation in original real-world problem situation)
  Level V (Grade (College Readiness))
   1. Translates from specific instances to generalizations and identifies generalizations that follow from specific cases
   2. Understands the structure of an axiomatic system (e.g., axioms, definitions, theorems)

 A  = Assessment items available