Chicago Public Schools Bureau of Student Assessment
201 Math Problem Solving
Source: Charles, Randall, Lester, Frank and O'Daffer, Phares. How to Evaluate Progress in Problem
Solving. Reston, VA: National Council of Teachers of Mathematics, 1987. In Stenmark, Jean, Mathematics
Assessment: Myths, Models, Good Questions and Practical Suggestions. Reston, VA: National Council of
Teachers of Mathematics, 1991.
Subjects: Mathematics # of scales 3
Grade(s) Not specified Scale length 3
Scale I: Understanding the Problem
2 Complete understanding of the problem
1 Part of the problem misunderstood or misinterpreted
0 Complete misunderstanding of the problem
Scale II: Planning a Solution
2 Plan could have led to a correct solution if implemented properly
1 Partially correct plan based on part of the problem being interpreted correctly
0 No attempt, or totally inappropriate plan
Scale III: Getting an Answer
2 Correct answer and correct label for the answer
1 Copying error; computational error; partial answer for a problem with multiple
answers
0 No answer, or wrong answer based on an inappropriate plan
Chicago Public Schools Bureau of Student Assessment
202 California Generalized Rubric for Math
Source: California State Department of Education, A Question of Thinking. Sacramento, CA: California
State Department of Education, 1989
Subjects: Mathematics # of scales 1
Grade(s) Not specified Scale length 6
Holistic Scale
6 Exemplary response. Gives a complete response with a clear, coherent,
unambiguous and elegant explanation; includes a clear and simplified diagram;
communicates effectively to the identified audience; shows understanding of the
open-ended problem's mathematical ideas and processes; identifies all the important
elements of the problem; may include examples and counterexamples; presents
strong supporting arguments.
5 Competent response. Gives a fairly complete response with reasonably clear
explanations; may include an appropriate diagram; communicates effectively to the
identified audience; shows understanding of the problem's mathematical ideas and
processes; identifies the most important elements of the problem; presents solid
supporting arguments.
4 Minor Flaws But Satisfactory. Completes the problem satisfactorily, but the
explanation may be muddled; argumentation may be incomplete; diagram may be
inappropriate or unclear; understands the underlying mathematical ideas; uses
mathematical ideas effectively.
3 Serious Flaws But Nearly Satisfactory. Begins the problem appropriately but may
fail to complete or may omit significant parts of the problem; may fail to show full
understanding of mathematical ideas and processes; may make major
computational errors; may misuse or fail to use mathematical terms; response may
reflect an inappropriate strategy for solving the problem.
2 Begins, But Fails to Complete Problem. Explanation is not understandable;
diagram may be unclear; shows no understanding of the problem situation; may
make major computational errors.
1 Unable to Begin Effectively. Words do not reflect the problem; drawings
misrepresent the problem situation; copies parts of the problem but without
attempting a solution; fails to indicate which information is appropriate to the
problem.
Chicago Public Schools Bureau of Student Assessment
203 Analytical Scale for Problem Solving
Source: Szetela, Walter and Nicol, Cynthia. Evaluating Problem Solving in Mathematics. Educational
Leadership, May 1992, pp. 42-45.
Subjects: Mathematics # of scales 3
Grade(s) Not specified Scale length 3-5
Scale I: Understanding the Problem
4 Complete understanding of the problem
3 Misinterprets minor part of the problem
2 Misinterprets major part of the problem
1 Completely misinterprets the problem
0 No attempt
Scale II: Solving the Problem
4 A plan that could lead to a correct solution with no arithmetic errors
3 Substantially correct procedure with minor omission or procedural error
2 Partially correct procedure but with major fault
1 Totally inappropriate plan
0 No attempt
Scale III: Answering the Problem
2 Correct solution
1 Copying error; computational error, partial answer for problem with multiple
answers; no answer statement; answer labeled incorrectly
0 No answer or wrong answer based upon an inappropriate plan
Note: This rubric is based on rubric 201.
Chicago Public Schools Bureau of Student Assessment
204 North Carolina Math Rubric I
Source: North Carolina Department of Public Instruction
Subjects: Mathematics # of scales 1
Grade(s) Elementary Scale length 4
Holistic Scale
3 All parts of the question are answered accurately and completely. All directions are
followed.
2 Answer deals correctly with most aspects of the question, but something is missing.
May deal with all aspects but have minor errors.
1 Addresses item but only partially correct; something correct related to the question.
0 Does not address task, unresponsive, unrelated or inappropriate. Nothing correct.
Chicago Public Schools Bureau of Student Assessment
205 North Carolina Math Rubric II
Source: North Carolina Department of Public Instruction
Subjects: Mathematics # of scales 1
Grade(s) Elementary Scale length 53
Holistic Scale
2 Answer is complete and correct; all parts of the question are addressed.
1 Student gives a partially correct answer, or task is incomplete (i.e., one of two parts
answered correctly.
0 Does not address task, unresponsive, unrelated or inappropriate.
Chicago Public Schools Bureau of Student Assessment
206 Maine Holistic Rubric for Mathematics Open-Ended Items
Source: Maine Department of Education
Subjects: Mathematics # of scales 1
Grade(s) Not specified Scale length 5
Holistic Scale
4 A correct solution and an appropriate strategy are shown or explained and the
solution is shown with correct label or description if necessary.
3 A complete, appropriate strategy is show or explained but:
nan incorrect solution is given due to a simple computational or other error or
nno solution is given.
A correct solution is given with no solution strategy or explanation shown.
A correct solution and appropriate strategy is shown or explained, but not labeled
correctly when necessary.
2 Some parts of an appropriate strategy are shown or explained, but some key
elements are missing.
Some parts of an appropriate strategy are shown or explained, along with some
inappropriate parts.
Appropriate strategy shown or explained, but implemented incorrectly.
1 Some work or explanation beyond re-copying data, but work would not lead to a
correct solution.
One or more incorrect approaches attempted or explained.
0 No work or solution shown or explained.
Incorrect solution and no work shown or explained.
Some data from the problem copied over, but no evidence of any strategy is shown
or explained.
Chicago Public Schools Bureau of Student Assessment
207 Vermont Math Problem Solving Criteria
Source: Vermont Department of Education
Subjects: Mathematics # of scales 4
Grade(s) 8 Scale length 4
Scale I: Understanding the Problem
4 Identified special factors that influenced the approach before starting the problem.
3 Understood the problem.
2 Understood enough to solve part of the problem or to get part of the solution.
1 Didn't understand enough to get started or make progress.
Scale II: How Student Solved Problem
4 Approach was efficient or sophisticated.
3 Approach would work for the problem.
2 Approach would only lead to solving part of the problem.
1 Approach didn't work.
Scale III: Decisions Along the Way
4 Clearly explained the reasons for the correct decisions made throughout the
problem.
3 Didn't clearly explain the reasons for decisions, but work suggests correct reasoning
used for only part of the problem.
2 Only partly correct reasoning, or correct reasoning used for only part of the
problem.
1 No reasoning is evident from the work or reasoning is incorrect.
Scale IV: Outcomes of Activities
4 Solved the problem and made general rule about the solution or extended the
solution to a more complicated situation.
3 Solved the problem and connected the solution to other math or described a use for
what was learned in the "real world."
2 Only partly correct reasoning, or correct reasoning used for only part of the
problem.
1 Solved the problem and stopped.
Chicago Public Schools Bureau of Student Assessment
208 QUASAR General Rubric (page 1 of 2)
Source: Lane, Suzanne, The Conceptual Framework for the Development of a Mathematics Performance
Assessment Instrument. Educational Measurement: Issues and Practice, Summer 1993, 16-23.
Subjects: Mathematics # of scales 1
Grade(s) Not specified Scale length 5
Holistic Scale
4 Mathematical knowledge: Shows understanding of the problem's mathematical
concepts and principles; uses appropriate mathematical terminology and notations;
and executes algorithms completely and correctly.
Strategic knowledge: May use relevant outside information of a formal or informal
nature; identifies all the important elements of the problem and shows
understanding of the relationships between them; reflects an appropriate and
systematic strategy for solving the problem; and gives clear evidence of a solution
process, and solution process is complete and systematic.
Communication: Gives a complete response with a clear, unambiguous explanation
and/or description; may include an appropriate and complete diagram;
communicates effectively to the identified audience; presents strong supporting
arguments which are logically sound and complete; may include examples and
counter-examples.
3 Mathematical knowledge: Shows nearly complete understanding of the problem's
mathematical concepts and principles; uses nearly correct mathematical
terminology and notations; executes algorithms completely; and computations are
generally correct but may contain minor errors.
Strategic knowledge: May use relevant outside information of a formal or informal
nature; identifies the most important elements of the problems and shows general
understanding of the relationships between them; and gives clear evidence of a
solution process, and solution process is complete or nearly complete, and
systematic.
Communication: Gives a fairly complete response with reasonably clear
explanations or descriptions; may include a nearly complete, appropriate diagram;
generally communicates effectively to the identified audience; presents supporting
arguments which are logically sound but may contain some minor gaps.
(cont'd.)
Chicago Public Schools Bureau of Student Assessment
208 QUASAR General Rubric (page 2 of 2)
2 Mathematical knowledge: Shows understanding of some of the problem's
mathematical concepts, and principles; and may contain serious computational
errors.
Strategic knowledge: Identifies some important elements of the problems, but
shows only limited understanding of the relationships between them; and gives
some evidence of a solution process, but solution process may be incomplete or
somewhat unsystematic.
Communication: Makes significant progress towards completion of the problem,
but the explanation or description may be somewhat ambiguous or unclear; may
include a diagram which is flawed or unclear; communication may be somewhat
vague or difficult to interpret; and arguments may be incomplete or may be based
on a logically unsound premise.
1 Mathematical knowledge: Shows very limited understanding of the problem's
mathematical concepts, and principles; may misuse or fail to use mathematical
terms; and may make major computational errors.
Strategic knowledge: May attempt to use irrelevant outside information; fails to
identify important elements or places too much emphasis on unimportant elements;
may reflect an inappropriate strategy for solving the problem; gives incomplete
evidence of a solution process; solution process may be missing, difficult to
identify, or completely unsystematic.
Communication: Has some satisfactory elements but may fail to complete or may
omit significant parts of the problem; explanation or description may be missing or
difficult to follow; may include a diagram which incorrectly represents the problem
situation, or diagram may be unclear an difficult to interpret.
0 Mathematical knowledge: Shows no understanding of the problem's mathematical
concepts and principles.
Strategic knowledge: May attempt to use irrelevant outside information; fails to
indicate which elements of the problem are appropriate; copies part of the problem,
but without attempting a solution.
Communication: Communicates ineffectively; words do not reflect the problem;
may include drawings which completely misrepresent the problem situation.
Chicago Public Schools Bureau of Student Assessment
209 Maryland Math Communication Rubric
Source: Maryland State Department of Education, Sample activities, student responses and Maryland teachers' comments
on a sample task: Mathematics Grade 8, February 1991.
Subjects: Mathematics # of scales 1
Grade(s) 8 Scale length 5
Holistic Scale
4 Uses mathematical language (terms, symbols, signs, and/or representations) that is
highly effective, accurate, and thorough, to describe operations, concepts, and
processes.
3 Uses mathematical language (terms, symbols, signs, and/or representations) that is
partially effective, accurate, and thorough to describe operations, concepts and
processes.
2 Uses mathematical language (terms, symbols, signs and/or representations) that is
minimally effective and accurate, to describe operations, concepts, and processes.
1 An incorrect response— attempt is made.
0 Off task, off topic, illegible, blank or insufficient to score.
Chicago Public Schools Bureau of Student Assessment
210 California Performance Standards for Student Work
Source: California Department of Education, A Sampler of Mathematics Assessment, 1991.
Subjects: Mathematics # of scales 1
Grade(s) Not specified Scale length 6
Holistic Scale
6 Fully achieves the purpose of the task, while insightfully interpreting, extending
beyond the task, or raising provocative questions.
Demonstrates an in-depth understanding of concepts and content.
Communicates effectively and clearly to various audiences, using dynamic and
diverse means.
5 Accomplishes the purposes of the task.
Shows clear understandings of concepts.
Communicates effectively.
4 Substantially completes purposes of the task.
Displays understanding of major concepts, even though some less important
ideas may be missing.
Communicates successfully.
3 Purpose of the task not fully achieved; needs elaboration; some strategies may be
ineffectual or not appropriate; assumptions about the purposes may be flawed.
Gaps in conceptual understanding are evident.
Limits communication to some important ideas; results may be incomplete or
not clearly presented.
2 Important purposes of the task not achieved; work may need redirection; approach
to task may lead away from its completion.
Presents fragmented understanding of concepts; results may be incomplete or
arguments may be weak.
Attempts communication.
1 Purpose of the task not accomplished.
Shows little evidence of appropriate reasoning.
Does not successfully communicate relevant ideas; presents extraneous
information.
Chicago Public Schools Bureau of Student Assessment
211 Kentucky Holistic Scoring Rubric for Grade 12 Math
Source: Kentucky Department of Education Open-Response Released Items and Scoring Rubrics: Grade 12
1991-92
Subjects: Mathematics # of scales 1
Grade(s) 12 Scale length 5
Holistic Scale
5 The student completes all important components of the task and communicates
ideas clearly.
The student demonstrates in-depth understanding of the relevant concepts and/or
processes.
Where appropriate, the student chooses more efficient and/or sophisticated
processes.
Where appropriate, the student offers insightful interpretations or extensions
(generalizations, applications, analogies).
4 The student completes most important components of the task and communicates
clearly.
The student demonstrates understanding of major concepts even though she/he
overlooks or misunderstands some less important ideas or details.
3 The student completes some important components of the task and communicates
those clearly.
The student demonstrates that there are gaps in his/her conceptual understanding.
2 Student shows minimal understanding.
Student unable to generate strategy or answer may display only recall effect.
Answer lacks clear communication.
Answer may be totally incorrect or irrelevant.
1 Blank/no response.
Chicago Public Schools Bureau of Student Assessment
212 California General Rubric for Mathematics (page 1 of 2)
Source: California Department of Education, A Sampler of Mathematics Assessment: Addendum,
Preliminary Edition, 1993.
Subjects: Mathematics # of scales 1
Grade(s) 4, 8, 10 Scale length 6
Holistic Scale
6 Solid work that may go beyond the requirements of the task(s), showing for
example:
ncomplete understanding of the task's mathematical concepts and processes.
nclear identification of all of the important elements of the task(s).
nwhere appropriate, clear evidence of doing purposeful mathematics, including
investigating, experimenting, modeling, designing, interpreting, analyzing, or
solving.
nexcellent prose and mathematical supporting arguments that may include
examples or counter-examples.
ncreativity and thoughtfulness in communicating the results and the
interpretations of those results, to an identified audience, using dynamic and
diverse means.
nmultiple solutions based upon different assumptions about or interpretations of
the task(s).
nunusual insights into the nature of and the resolution of problems encountered
in the task(s).
na high level of mathematical thinking that includes, where appropriate, making
comparisons, conjectures, interpretations, predictions, or gen- eralizations.
nexceptional skill in choosing appropriate mathematical tools and techniques in
the resolution of problems in task(s).
5 Fully achieves the requirements of the task(s), showing for example:
ngood understanding of the task's mathematical concepts and processes.
nidentification of most, if not all, of the important elements of the task(s).
nevidence of doing purposeful mathematics, including where appropriate,
investigating, experimenting, modeling, designing, interpreting, analyzing, or
solving.
nclear, successful communications with an identified audience.
none solution and interpretation of those results.
nevidence of mathematical thinking that includes, where appropriate, making
comparisons, conjectures, interpretations, predictions, or gen-eralizations.
nuse of variety of tools and techniques appropriate to the form of the task(s) and
the requirements of the task.
4 Substantially completes the requirements of the task(s), showing for example:
nan understanding of most of the task's mathematical concepts and processes.
nidentification of the important elements of the task(s), but some less important
ideas are missing.
(cont'd.)
Chicago Public Schools Bureau of Student Assessment
212 California General Rubric for Mathematics (page 2 of 2)
Source: California Department of Education, A Sampler of Mathematics Assessment: Addendum, Preliminary
Edition, 1993.
nsome aspects of investigations, experiments, model building, designs,
interpretations, analysis, solutions require by the task(s) may be missing,
nadequate communication with an identified audience, but with limited clarity
and variety.
noccasional evidence of mathematical thinking involving comparisons,
conjectures, interpretations, predictions, or generalizations.
na limited variety of tools and techniques used to resolve the situation presented
in the task(s).
3 Limited completion of the requirements of the task(s), showing for example:
nan understanding of some of the task's mathematical concepts and processes,
but with evidence of gaps in those understanding.
nidentification of some of the important elements of the task(s), but assumptions
about some of the elements may be flawed.
ncommunication of some ideas, but generally makes inadequate attempts to
communicate, often failing to address the identified audience, and difficulty in
expressing mathematical ideas.
ninadequate mathematical thinking that includes ineffective analysis procedures,
limited solution strategies, unclear mathematical arguments, and inappropriate
interpretation of results.
na selection of some inappropriate tools and techniques used to resolve the
situation presented in the task(s).
2 Requirements of the task(s) not completed, showing for example:
nonly fragmented understanding of the task's mathematical concepts and
processes, accompanied by disorganized, incomplete results.
nidentification of only a few, usually superficial elements of the task(s).
nattempts to address the intended audience that may be incoherent, muddled, or
incomplete.
nattempts to explain or justify results that are convoluted, illogical, circular, or
unrelated to the results shown.
1 Does not achieve any requirements of the task(s), showing for example:
nan irrelevant, nonsensical, or illegible response that has no valid relationship to
the task(s).
nno understanding of the task's mathematical concepts and processes.
nunsuccessful attempt, if any, to communicate with the intended audience.
Usually communication is not attempted.
nno attempt to explain or justify results. If attempt is made, it is often unrelated
to the task, illegible, or incoherent.
Chicago Public Schools Bureau of Student Assessment
213 Temple ISD Math Rubric
Source: Temple Independent School District, Temple, Texas
Subjects: Mathematics # of scales 1
Grade(s) Not specified Scale length 4
Holistic Scale
3 Response is exemplary, detailed and clear
2 Response is generally correct
1 Response is partially correct, but lacks clarity
0 No response or response is incorrect
Chicago Public Schools Bureau of Student Assessment
214 Illinois Rubric for Mathematics (page 1 of 2)
Source: Illinois State Board of Education
Subjects: Mathematics # of scales 3
Grade(s) 3-12 Scale length 5
Scale 1 Mathematical Knowledge
4 Shows complete understanding of the problem's mathematical concepts and
principles. Uses appropriate mathematical terminology and notation (e.g.,
labels as appropriate*). Executes algorithms completely and correctly.
3 Shows nearly complete understanding of the problem's mathematical
concepts and principles. Uses nearly correct mathematical terminology and
notation. Executes algorithms completely. Computations are generally
correct, but may contain minor errors.
2 Shows understanding of some of the problem's mathematical concepts and
principles. May contain serious computational errors.
1 Shows very limited understanding of the problem's mathematical concepts,
and principles. May misuse or fail to use mathematical terms. May contain
major computational errors.
0 Shows no understanding of the problem's mathematical concepts and
principles.
Scale II: Strategic Knowledge
4 Identifies all important elements of the problem and shows understanding the
relationship between them. Reflects and appropriate and systematic strategy
for solving the problem. Gives clear evidence of a solution process, and
solution process is complete and systematic.
3 May use relevant outside information of a formal or informal nature.
Identifies the most important elements of problem and shows general
understanding of the relationships between them. Solution process is nearly
complete.
2 Identifies some important elements of the problem but shows only limited
understanding of the relationships between them. Gives some evidence of a
solution process.
1 May attempt to use irrelevant outside information. Fails to identify important
elements or places too much emphasis on unimportant elements. May reflect
an inappropriate strategy for solving the problem. Gives minimal evidence of
a solution process. Process may be difficult to identify.
0 Attempts to use irrelevant outside information. Fails to indicate elements of
the problem. Copies part of the problem, but without attempting a solution.
*:"As appropriate" or "if appropriate" relate to whether or not the specific element is called
for in the stem of the item.
Chicago Public Schools Bureau of Student Assessment
214 Illinois Rubric for Mathematics (page 2 of 2)
Source: Illinois State Board of Education
4 Gives a complete written explanation of the solution process employed.
Includes appropriate and complete diagram with explanation of elements. May
provide examples and counter examples if appropriate.
3 Gives a fairly complete written explanation of the solution process employed.
May contain some minor gaps. May include a nearly complete diagram with
some explanation.
2 Gives some explanation of the solution process employed, but communication
is vague or difficult to interpret. May include diagram that is flawed, unclear, or
not explained.
1 Provides minimal explanations of solution process. May fail to complete or
may omit significant parts of the problem. Explanation missing or difficult to
follow. May include a diagram which incorrectly represents the problem
situation or diagram may be unclear and difficult to interpret.
0 Words do not reflect the problem or no written explanation given. May include
drawings which completely misrepresent the problem situation
Note: This rubric was adapted from the QUASAR mathematics rubric (#208)
Chicago Public Schools Bureau of Student Assessment
215 Ann Arbor Kindergarten Report Card -- Math
Source: Ann Arbor Public Schools, Ann Arbor, Michigan
Subjects: Mathematics # of scales 1
Grade(s) Kindergarten Scale length 5
Holistic Scale
Note: Scale points 2 and 4 are not explicitly defined. A score of 2 would be assigned to
work that exceeded criteria for a score of 1, but did not meet criteria for a score of
2. Similarly a score of 4 would be assigned to work that exceeded criteria for a
score of 3, but did not meet criteria for a score of 5.
5 Achieving
3 Developing
1 Not yet
Note: Scale points are defined in more detail for each outcome. For example, the
outcome "Count to 20" is scored as follows:
5 Achieving. Counts from 1 to 20.
3 Developing. Counts from 1 to 10. Cannot count from 11 to 19.
1 Not yet. Counts to 5 or less.
The outcome "Extending patterns" is scored this way:
5 Extends pattern 3 times.
3 Same attributes but not correct order.
1 Randomly adds on.
For a copy of the entire Ann Arbor report card, contact the Chicago Public Schools Bureau
of Student Assessment.
Chicago Public Schools Bureau of Student Assessment
216 Ann Arbor First Grade Report Card -- Math
Source: Ann Arbor Public Schools, Ann Arbor, Michigan
Subjects: Mathematics # of scales 1
Grade(s) 1 Scale length 5
Holistic Scale
Note: Scale points 2 and 4 are not explicitly defined. A score of 2 would be assigned to
work that exceeded criteria for a score of 1, but did not meet criteria for a score of
2. Similarly a score of 4 would be assigned to work that exceeded criteria for a
score of 3, but did not meet criteria for a score of 5.
5 Achieving
3 Developing
1 Emerging
Note: Scale points are defined in more detail for each outcome. For example, the
outcome "Tells time to the nearest hour and half hour" is scored as follows:
5 Achieving. Tells time correctly for all six hour and half hour times given.
3 Developing. Misses one to two of the six times given. Tells time to hour,
but not half hour. Needs prompt to get times correct.
1 Not yet. Provides no response or gives three or more incorrect responses.
For a copy of the entire Ann Arbor report card, contact the Chicago Public Schools Bureau
of Student Assessment.
Chicago Public Schools Bureau of Student Assessment
217 Kentucky Open-Ended Scoring Guide for Grade 8
Mathematics, Social Studies and Science
Source: Kentucky Department of Education
Subjects: Science, mathematics,
social studies
# of scales 1
Grade(s) 8 Scale length 5
Holistic Scale
4 n The student completes all important components of the task and communicates
ideas clearly.
n The student demonstrates in-depth understanding of the relevant concepts and/or
processes.
n Where appropriate, the student chooses more efficient and/or sophisticated
processes.
n Where appropriate, the student offers interpretations or extensions
(generalizations, applications, analogies).
3 n The student completes most important components of the task and
communicates clearly.
n The student demonstrates understanding of major concepts even though he/she
overlooks or misunderstands less important ideas or details.
2 n The student completes some important components of the task and
communicates those clearly.
n The student demonstrates that there are gaps in his/her conceptual understanding.
1 n Student shows minimal understanding.
n Student unable to generate strategy or answer may display only recall. Answer
lacks clear communication.
n Answer may be totally incorrect or irrelevant.
0 n Blank/no response
Note: Scale points are defined in greater detail for each test question.
Chicago Public Schools Bureau of Student Assessment
218 Norwood Park Draft Math Problem Solving Rubric (page 1 of 2)
Source: Faculty of Norwood Park Elementary School, Chicago, Illinois
Subjects: Mathematics # of scales 5
Grade(s) K-8 Scale length 4
Scale I: Shows Evidence That Problem Was Understood
Distinguished Shows rigorous understanding of the problem
Proficient Shows substantial understanding of the problem
Apprentice Shows limited understanding of the problem
Novice Shows little or no understanding of the problem
Scale II: Uses Information Appropriately
Distinguished Explains why certain information is essential to the solution
Proficient Uses all appropriate information correctly
Apprentice Uses some appropriate information correctly
Novice Uses inappropriate information
Scale III: Applies Appropriate Procedures
Distinguished Explains why procedures are appropriate for the problem
Proficient Applies completely appropriate procedures
Apprentice Applies some appropriate procedures
Novice Applies inappropriate procedures
(cont'd.)
Chicago Public Schools Bureau of Student Assessment
218 Norwood Park Draft Math Problem Solving Rubric (page 2 of 2)
Source: Faculty of Norwood Park Elementary School, Chicago, Illinois
Scale IV: Uses Representations, e.g., Diagrams, Graphs, Pictures,
Manipulatives, Equations
Distinguished Uses a representation that is unusual in its aesthetic value or
mathematical precision
Proficient Uses a representation that clearly depicts the problem
Apprentice Uses a representation that gives some important information about the
problem
Novice Uses a representation that gives little or no significant information about
the problem
Scale V: Shows Competent Use of Mathematics
Distinguished Makes a general rule about the solution that can be applied to another
problem
Proficient Shows complete competence in using mathematics
Apprentice Shows some competence in using mathematics, skips some important
steps, or omits some important information
Novice Shows incompetent use of mathematics
Chicago Public Schools Bureau of Student Assessment
219 Oregon Mathematics Problem Solving (page 1 of 3)
Source: Oregon Department of Education
Subjects: Mathematics # of scales 4
Grade(s) 3, 5, 8, 11 Scale length 5
Note: Scale points 2 and 4 are not explicitly defined. A score of 2 would be assigned to
work that exceeded criteria for a score of 1, but did not meet criteria for a score of
2. Similarly, a score of 4 would be assigned to work that exceeded criteria for a
score of 3, but did not meet criteria for a score of 5.
Scale I: Conceptual Understanding
Conceptual Understanding includes the ability to interpret the problem and select
appropriate information to apply a strategy for solution. Evidence is communicated
through making connections between the problem situation, relevant information,
appropriate mathematical concepts and logical/reasonable responses.
5 Full Conceptual Understanding: The student uses all relevant information to solve
the problem.
n The student's answer is consistent with the question/problem.
n The student is able to translate the problem into appropriate mathematical
language.
3 Partial Conceptual Understanding: The student extracts the "essence" of the
problem, but is unable to use this information to solve the problem.
n The student is only partially able to make connections between/among the
concepts.
n The student's solution is not fully related to the question.
n The student understands one portion of the task, but not the complete task.
1 Lack of Conceptual Understanding: The student's solution is inconsistent or
unrelated to the question.
n The student translates the problem into inappropriate mathematical concepts.
n The student uses incorrect procedures without understanding the concepts
related to the task.
(cont'd.)
Chicago Public Schools Bureau of Student Assessment
219 Oregon Mathematics Problem Solving (page 2 of 3)
Source: Oregon Department of Education
Scale II: Procedural Knowledge
Procedural Knowledge deals with the student's ability to demonstrate appropriate use of
concepts. Evidence includes the verifying and justifying of a procedure using concrete
models, or the modifying of procedures to deal with factors inherent in the problem.
5 Full Use of Appropriate Procedures: The student uses principles efficiently while
justifying the solution.
n The student uses appropriate mathematical terms and strategies.
n The student solves and verifies the problem.
n The student uses mathematical principles and language precisely.
3 Partial Use of Appropriate Procedures: The student is not precise in using
mathematical terms, principles, or procedures.
n The student is unable to carry out a procedure completely.
n The process the student uses to verify the solution is incorrect.
1 Lacks Use of Appropriate Procedures: The student uses unsuitable methods or
simple manipulation of data in his/her attempted solution.
n The student fails to eliminate unsuitable methods or solutions.
n The student misuses principles or translates the problem into inappropriate
procedures.
n The student fails to verify the solution.
Scale III: Problem Solving Skills and Strategies
Problem Solving requires the use of many skills, often in certain combinations, before the
problem is solved. Students demonstrate problem solving strategies with clearly focused,
good reasoning that leads to a successful resolution of the problem.
5 Evidence of Thorough/Insightful Use of Skills/Strategies: The skills and strategies
show some evidence of insightful thinking to explore the problem.
n The student's work is clear and focused.
n The skills/strategies are appropriate and demonstrate some insightful thinking.
n The student gives possible extensions or generalizations to the solution or the
problem.
(cont'd.)
Chicago Public Schools Bureau of Student Assessment
219 Oregon Mathematics Problem Solving (page 3 of 3)
Source: Oregon Department of Education
3 Evidence of Routine or Partial Use of Skills/Strategies: The skills and strategies have
some focus, but clarity is limited.
n The student applies a strategy which is only partially useful.
n The student's strategy is not fully executed.
n The student starts the problem appropriately, but changes to an incorrect focus.
n The student recognizes the pattern or relationship, but expands it incorrectly.
1 Limited Evidence of Skills/Strategies: The skills and strategies lack a central focus and
the details are sketchy or not present.
n The procedures are not recorded (i.e., only the solution is present).
n Strategies are random.
n The student does not fully explore the problem, looking for concepts, patterns or
relationships.
n The student fails to see alternative solutions that the problem requires.
Scale IV: Communication
In assessing the student's ability to communicate, particular attention should be paid to
both the meanings he/she attaches to the concepts and procedures and also to his/her
fluency in explaining, understanding, and evaluating the ideas expressed.
5 Clear, Complete Communication: The student gives a complete response with clear,
coherent, unambiguous, and elegant explanations.
n The student communicates his/her thinking effectively to the audience.
n The details fit and make sense.
n One step flows to the next and shows organization.
n The student presents strong supporting arguments.
3 Partial or Incomplete Communication: The student's explanation is unclear, inconsistent
or not complete.
n The student uses terminology incorrectly or inconsistently.
n The student's visual aids (graphs, tables, diagrams, etc.) are inappropriate or not
directly related.
n The student's explanation centers on his/her solution, not on his/her thinking.
1 Limited or Lack of Communication: The student's explanation is not understandable or
not present.
n The student either does not use or misuses appropriate mathematical terminology.
n The student does not use essential visual aids to enhance or clarify the explanation.
n The student's explanation lacks focus.\
Chicago Public Schools Bureau of Student Assessment
220 Arizona Mathematics Rubric
Source: Arizona Department of Education
Subjects: Mathematics # of scales 3
Grade(s) 3-12 Scale length 5
Holistic Scale
4 A 4 response represents an effective solution. It shows complete understanding of
the problem, thoroughly addresses all points relevant to the solution, shows logical
reasoning and valid conclusions, communicates effectively and clearly through
writing and/or diagrams, and includes adequate and correct computations and/or
setup. It may contain insignificant errors that do not interfere with the
completeness or reasonableness of the student's response.
3 A 3 response contains minor flaws. Although it shows an understanding of the
problem, communicates adequately through writing and/or diagrams, and generally
reaches reasonable conclusions, it shows minor flaws in reasoning and/or
computation or neglects to address some aspect of the problem.
2 A 2 response shows gaps in understanding and/or execution. It shows one or some
combination of the following flaws: an incomplete understanding of the problem,
failure to address some aspects of the problem, faulty reasoning, weak conclusions,
unclear communication in writing and/or diagrams, or a poor understanding of
relevant mathematical procedures or concepts.
1 A 1 response shows some effort beyond restating the problem or copying given
data. It shows some combination of the following flaws: little understanding of the
problem, failure to address most aspects of the problem, major flaws in reasoning
that lead to invalid conclusions, or a lack of understanding of relevant mathematical
procedures or concepts.
0 Response shows no mathematical understanding of the problem or the student has failed
to respond to the item.