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Implementing Oregon's Diploma Requirements
The Essential Skill of Mathematics An Overview
Information provided by Oregon Department of Education Office of Assessment and Information Services
Updated Aug 2011
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THE ESSENTIAL SKILL OF MATHEMATICS
Using the Mathematics Problem SolvingScoring Guide
An Overview
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Updated Aug 2011
GOALS FOR THIS WORKSHOP
Participants will know:Requirements for demonstrating proficiency in the Essential Skill of Apply Mathematics
Updated Math Problem Solving Scoring Guide and Traits
Resources and professionaldevelopment available
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Apply Mathematics Requirements
OAR: 581-22-0615Students who entered high school as 9th
graders in 2010-11 will be required to demonstrate proficiency in three essential skills:
read & comprehend a variety of textwrite clearly and accuratelyapply mathematics
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APPLY MATHEMATICS IN A VARIETY OF SETTINGS
This Essential Skill includes all of the following:
Interpret a situation and apply workable mathematical concepts and strategies, using appropriate technologies where applicable.
Produce evidence such as graphs data or Produce evidence, such as graphs, data, or mathematical models, to obtain and verify a solution.
Communicate and defend the verified process
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3 W3 WAYSAYS TOTO DDEMONSTRATEEMONSTRATEPPROFICIENCYROFICIENCY ININ MMATHEMATICSATHEMATICS
1. OAKS Mathematics Assessment
Score of 236
2. Other Approved Test Options
ACT or PLAN 19
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ACT or PLAN 19WorkKeys 5Compass* 66
Asset* 41SAT/PSAT 450/45AP & IB varies
*Intermediate Algebra Test
3. L3. LOCALOCAL WWORKORK SSAMPLESAMPLES
Mathematics Work Sample scoredusing Official State Scoring Guide
Two Mathematics Work Samples Requiredone each for two of the following:
geometry, algebra or statistics
Score: 4 or higher on each of the five traits 6
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WHAT IS THE STATUS OF THE
MATHEMATICS SCORING GUIDE?
In use since 1988 (minor revisions in 2000)
2009-2010 new version based on Oregon Mathematics Content StandardsMathematics Content Standards
2010-11 aligned to the Common Core State Standards
Adopted by Oregon State Board of Education May 2011
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WHAT IS THE STATUS OF THE MATHEMATICS
SCORING GUIDE?
Districts should use the new scoring guide when training teachers for classroom activities and Essential Skills work samples.
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Making sense of the task(MS)
Representing and solving the task(RS)
C i i R i (CR)
OOFFICIALFFICIAL MMATHEMATICSATHEMATICS SSCORINGCORINGGGUIDEUIDE TTRAITSRAITS
Communicating Reasoning (CR)
Accuracy (Acc)
Reflecting and evaluating (RE)
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LLEVELEVEL OFOF RRIGORIGORWork samples must meet the level of rigor required on the OAKS assessment.Work samples provide an Work samples provide an optional means to demonstrate proficiency not an easiermeans.
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SSIMPLIFIEDIMPLIFIED MMATHEMATICSATHEMATICS SSCORINGCORING GGUIDEUIDE
4
5
Proficient
Strong
6Exemplary
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Beginning1
2
3
Emerging
Developing
FFORMATIVEORMATIVE AASSESSMENTSSESSMENTand the Scoring Guideand the Scoring Guide
The Scoring Guide is intended to be more than a final assessment tool.
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Both teachers and students can use the Scoring Guide to improve math skills.
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Districts must administer one local performance
Districts do not need to assess
RUMOR VERSUS REALITY
Rumor Reality
performance assessment in mathematical problem-solving during high school … (continued)
math until the class of 2014
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REALITY, CONTINUED
Districts should offer students opportunities to demonstrate proficiency in the Essential Skill p yof Mathematics beginning with students who entered 9th grade in 2010 - 11.
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RUMOR VERSUS REALITY
Rumor Reality
Work Samples must be scored by 2 raters
Only one rater is required
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In the case of a borderline passing paper, districts may wish to have more than 1 rater.
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RUMOR VERSUS REALITY
Rumor Reality
Students may not revise after a work sample has been
Students may revise and resubmit work samples to be
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sample has been scored
samples to be rescored
Revisions must remain the product of the student’s independent efforts
RUMOR VERSUS REALITY
Rumor
Teachers may not provide any feedback
Reality
Feedback is allowed using ONLY the Scoring Guide and/or the Official Scoring Form
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Official Scoring Form
A student must score a 4 in each required trait for each individual
A student may combine scores from multiple work samples to meet the achievement
RUMOR VERSUS REALITY
Rumor Reality
each individual work sample.
the achievement standard
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RUMOR VERSUS REALITY
Rumor Reality
Only trained raters can score work
TRUE – raters must be trained to use the
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samples scoring guide accurately
“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.
Make sense of problems and persevere in solving them.
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They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.”
Common Core State Standards for Mathematics
RRESOURCESESOURCES
ODE website:http://ode/state.or.us/go/worksamples
Some resources are available now and additional resources will be posted here in the
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additional resources will be posted here in the future
OCTM website:http://www.octm.org
A variety of resource material is available here and additional assistance with the Essential Skill of Mathematics will be added.
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LET’S EXPLORE SOME
STUDENT WORK SAMPLES
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2011-2012 Mathematics Problem Solving Official Scoring Guide 2011-2012
Apply mathematics in a variety of settings. Build new mathematical knowledge through problem solving. Solve problems that arise in mathematics and in other contexts. Apply and adapt a variety of appropriate strategies to solve problems. Monitor and reflect on the process of mathematical problem solving.
For use beginning with 2011-2012 Assessments Office of Assessment and Evaluation Oregon Department of Education Adopted May 19, 2011
Process Dimensions **6/ 5 4 3 *2 / 1 Making Sense of the Task Interpret the concepts of the task and translate them into mathematics.
The interpretation and/or translation of the task are • thoroughly developed and/or • enhanced through connections
and/or extensions to other mathematical ideas or other contexts.
The interpretation and translation of the task are • adequately developed and • adequately displayed.
The interpretation and/or translation of the task are • partially developed, and/or • partially displayed.
The interpretation and/or translation of the task are • underdeveloped, • sketchy, • using inappropriate concepts, • minimal, and/or • not evident.
Representing and Solving the Task Use models, pictures, diagrams, and/or symbols to represent and solve the task situation and select an effective strategy to solve the task.
The strategy and representations used are • elegant (insightful), • complex, • enhanced through comparisons to
other representations and/or generalizations.
The strategy that has been selected and applied and the representations used are • effective and • complete.
The strategy that has been selected and applied and the representations used are • partially effective and/or • partially complete.
The strategy selected and representations used are • underdeveloped, • sketchy, • not useful, • minimal, • not evident, and/or • in conflict with the
solution/outcome.
Communicating Reasoning Coherently communicate mathematical reasoning and clearly use mathematical language.
The use of mathematical language and communication of the reasoning are • elegant (insightful) and/or • enhanced with graphics or
examples to allow the reader to move easily from one thought to another.
The use of mathematical language and communication of the reasoning • follow a clear and coherent path
throughout the entire work sample and
• lead to a clearly identified solution/outcome.
The use of mathematical language and communication of the reasoning • are partially displayed with
significant gaps and/or • do not clearly lead to a
solution/outcome.
The use of mathematical language and communication of the reasoning are • underdeveloped, • sketchy, • inappropriate, • minimal, and/or • not evident.
Accuracy Support the solution/outcome.
The solution/outcome is correct and enhanced by • extensions, • connections, • generalizations, and/or • asking new questions leading to
new problems.
The solution/outcome given is • correct, • mathematically justified, and • supported by the work.
The solution/outcome given is • incorrect due to minor error(s), or • a correct answer but work
contains minor error(s) • partially complete, and/or • partially correct
The solution/outcome given is • incorrect and/or • incomplete, or • correct, but
o conflicts with the work, or o not supported by the work.
Reflecting and Evaluating State the solution/outcome in the context of the task. Defend the process, evaluate and interpret the reasonableness of the solution/outcome.
Justifying the solution/outcome completely, the student reflection also includes • reworking the task using a
different method, • evaluating the relative
effectiveness and/or efficiency of different approaches taken, and/or
• providing evidence of considering other possible solution/outcomes and/or interpretations.
The solution/outcome is stated within the context of the task, and the reflection justifies the solution/outcome completely by reviewing • the interpretation of the task • concepts, • strategies, • calculations, and • reasonableness.
The solution/outcome is not stated clearly within the context of the task, and/or the reflection only partially justifies the solution/outcome by reviewing • the task situation, • concepts, • strategies, • calculations, and/or • reasonableness.
The solution/outcome is not clearly identified and/or the justification is • underdeveloped, • sketchy, • ineffective, • minimal, • not evident, and/or • inappropriate.
**6 for a given dimension would have most attributes in the list; 5 would have some of those attributes. *2 for a given dimension would be underdeveloped or sketchy, while a 1 would be minimal or nonexistent.
High School Mathematics Problem-Solving Task
Tetra Dice
Strand: Probability & Statistics A game requires each player to roll three specially shaped dice. Each die is a regular tetrahedron (four congruent, triangular faces). One face contains the number 1; one face contains the number 2; on another face appears the number 3; the remaining face shows the number 4. After a player rolls, the player records the numbers on the underneath sides of all three dice, and then calculates their sum. You win the game if the sum divides evenly by three. What is the probability of winning this game?
Scores and Commentary: Sample Paper #2: Tetra Dice
Making Sense of the Task
5
Representing and Solving
the Task 4
Communicating and
Reasoning 4
Accuracy
4
Reflecting and
Evaluating 4
Making Sense of the Task: The interpretation and translation of sample space related to the shape of the dice, sums divisible by three, and theoretical probability is thoroughly developed. The table is systematic and complete and the student connects the outcomes to the sample space to create the probability. The student was also able to take what made sense to them for two dice and extend it to make sense of a problem with three dice. Representing and Solving the Task: The strategy of creating an organized list of all possible sums making up the sample space is complete. Crossing out those with a sum not divisible by three and finding the number not crossed out and comparing it to the size of the sample space, is effective. Communicating and Reasoning: The communication follows a clear and coherent path throughout and leads to a clearly identified solution. It is not a 5 because very little mathematical language is used and the reasoning is not enhanced or elegant. Accuracy: The correct solution is given and is mathematically justified and supported by the work. Reflecting and Evaluating: The solution is stated within the context of the problem and the reflection (under the line) reviews the interpretation of the problem, concepts, strategies, and calculations. The student defends the process by completely reworking the problem, therefore the review is complete. Enough elements from the descriptors of a score level “4” are evident to make this a complete review.
Scores and Commentary: Sample Paper #3: Tetra Dice
Making Sense of the Task
3
Representing and Solving
the Task 3
Communicating and
Reasoning 3
Accuracy
2
Reflecting and
Evaluating 2
Making Sense of the Task: The interpretation of the key concepts for the task (sample space related to the shape and number of the dice, sums divisible by three, and theoretical probability) is present. It appears that the student correctly interprets the results for a first roll of “1” or “4” (6/16 or 3/8) and incorrectly assumes that the results from a first roll of “2” or “3” will be the same. Therefore, the translation is partially developed. Representing and Solving the Task: The strategy of assuming a first roll and going through the different permutations of rolls 2 and 3 is partially complete. Communicating and Reasoning: The reasoning is partially displayed with significant gaps. The sample space of 64 is not connected to the solution and the reader has to infer that the student assumed all 4 situations would have the same result (because of the “x 4”). Accuracy: The solution is incomplete. If the student has continued with the same strategy for all 4 “first rolls” s/he might have solved it correctly. As a result, the error is more than minor. Reflecting and Evaluating: The justification is ineffective because the choice of a first roll of 4 as the defense supported the student’s answer, but using a first roll of 2 or 3 would have exposed the error in the student’s thinking.
Scores and Commentary: Sample Paper #1: Tetra Dice
Making Sense of the Task
5
Representing and Solving
the Task 6
Communicating and
Reasoning 6
Accuracy
5
Reflecting and
Evaluating 5
Making Sense of the Task: The translation of the key concepts (sample space related to the shape and number of dice, sums divisible by three, and theoretical probability) is thoroughly developed in both translations (tree and matrix). Then s/he shows the sample space in three ways (tree, calculations, and matrix) and connects the pieces in the “tree” solution to those in the “matrix”. This is not a 6 because it is not extended or connected to other mathematical ideas. Representing and Solving the Task: The process of identifying each of the possible sums when starting with each possible roll of the dice, finding the sums divisible by three, comparing the number of successful sums to the total number of sums is elegant and insightful in the original approach. The reflection provides further evidence of the key concepts and the original representation is strengthened through its comparison to the second representation. Communicating and Reasoning: The use of mathematical language (sample space, “tree”, outcomes, sum, divisible) is enhanced by the use of clearly executed graphics which allow the reader to move easily from one thought to another. The insightful way the student chose to break the tree diagram apart further enhances the flow. Accuracy: 22/64 is a mathematically justifiable solution to the task and is supported by the work. The solution is enhanced by the two slightly different approaches and connection between the sample space and the outcomes to produce a probability. Reflecting and Evaluating: The student completely justifies the solution by reviewing the interpretation, concepts, strategies, calculations, and reasonableness (starting with the words after the tree diagrams that mention sample space). The reflection is strengthened by reworking the problem using a different recording method.