Implementing Oregon's Diploma Requirements
Using the Math Scoring Guide – An Introduction for High School Math Teachers
Information provided by Oregon Department of Education Office of Assessment and Information Services 2011-12
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Design for Mathematics Essential Skill Workshops Training for Trainers & Materials Provided by ODE The workshops described below are designed to be delivered by school district personnel who have received training and materials from ODE through “Training of Trainer” WebEx sessions.
Level 1
Level 2
Overview of the Essential Skill of Mathematics (30 – 45 minutes) General audiences – posted on ODE Website at http://www.ode.state.or.us/search/page/?id=2666 Introductory Training – Using the Mathematics Scoring Guide Content Area Teachers with math Mathematics Teachers (3 hours) emphasis Introduction to new Math Scoring (2 hours) Introduces Math Scoring Guide and Guide with practice scoring student provides practice scoring student papers; emphasis is on updating papers; emphasis is on classroom use scoring accuracy for Essential Skills work samples and classroom use In-Depth Training – Using the Mathematics Scoring Guide for Essential Skills Work Samples Content Area Teachers
Level 3
Level 4
In-depth Training on Scoring Guide – Expand understanding of scoring guide and increase accuracy in scoring papers with examples from content areas classes (3 to 3 ½ hours)
Mathematics Teachers In-depth Training on Scoring Guide – Expand understanding of scoring guide and increase accuracy in scoring papers with emphasis on Essential Skills proficiency (3 ½ to 4 hours)
In-Depth Training – Creating Mathematics Work Samples (3 ½ to 4 hours) Hands-on workshop showing characteristics of effective Mathematics Problem Solving Tasks for both Content Teachers and Mathematics Teachers; review of Guidelines for Work Samples, and opportunity for participants to draft a work sample for use in their classrooms.
*Estimated time needed for trainer to deliver the workshop to district/school participants Training of Trainer WebEx Sessions Level 1 training for presenters is provided in a one hour WebEx session which includes reading, writing and mathematics. It is designed to be delivered to general audiences by anyone with a basic understanding of the Essential Skills. No content expertise is required. Level 1 workshop materials are also available on the ODE website at http://www.ode.state.or.us/search/page/?id=219. Select the desired Essential Skill and go to Resources and Promising Practices. Levels 2 – 4 provides training for presenters with expertise in high school mathematics. Level 2 Training of Trainers is delivered in one 2-hour WebEx session, Level 3 in another 2-hour WebEx session and Level 4 in a final 2-hour WebEx session. All workshop materials, including ready-to-print handouts, are provided to attendees following each WebEx Training of Trainers session.
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11/10/2011
The Essential Skill of Mathematics Using the New Math Scoring Guide: An Introduction for High School Math Teachers
Goals for this workshop 1. Review
¾ Oregon’s Math Problem Solving Scoring Guide ¾ Classroom uses of the Math Scoring Guide ¾ Supporting colleagues in using math work samples in content classes
2. Understand
¾ Options for Demonstrating Proficiency in the Essential Skill of Mathematics for the Oregon Diploma
3. Score student papers and calibrate to scoring standards 4. Set the stage for follow-up training
OAR: 581-22-0615 For students first enrolled in grade 9 during the 2010-2011 school year [and subsequent years], school districts and public charter schools shall require students to demonstrate proficiency in the Essential Skills listed • (A) Read and comprehend a variety of text; and • (B) Write clearly and accurately • (C) Apply mathematics
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Essential Skill Definition Apply Mathematics in a variety of settings ¾Interpret a situation and apply workable mathematical concepts and strategies, using appropriate technologies where applicable. ¾Produce evidence, such as graphs, data, or mathematical models, to obtain and verify a solution. ¾Communicate and defend the verified process
The Common Core State Standards For Mathematics describe varieties of expertise... that rest on important “processes and proficiencies” …[including the] NCTM process standards of –problem solving –reasoning and proof –communication –representation –and connections
Make sense of problems and persevere in solving them “Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution….
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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.”
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning g of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
FromCommonCoreState StandardsforMathematics
Essential Skill Proficiency Three options for diploma requirement 1. OAKS Statewide Mathematics Assessment • Score of 236 or higher
2. Other approved standardized assessments Test ACT or PLAN
Score 19/19
WorkKeys
5
Compass
66 (College Alg. test)
Asset
41 (Int. Alg. test)
SAT/PSAT
450/45
AP & IB
various
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Option 3 Math Work Samples Mathematics Work Sample scored using Official State Scoring Guide Two Mathematics Work Samples Required: algebra, geometry, statistics Students must earn a score of 4 or higher in each dimension for each work sample
Level of Rigor Work samples must meet the
level of rigor required on the OAKS assessment. Work samples provide an optional ti l means tto d demonstrate t t proficiency not an easier means.
LET’S REVIEW THE SCORING GUIDE !
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The Math Problem Solving Scoring Guide
Background In use since 1988 (minor revisions in 2000) 2009-2010 new version based on Oregon Mathematics Content Standards 2010-11 aligned to the Common Core State Standards Adopted by Oregon State Board of Education May 2011
Mathematics Problem Solving Scoring Guide Making Sense of the Problem
Representing and Solving the Problem Communicating Reasoning Accuracy Reflecting and Evaluating
Making sense of the problem Interpret the concepts of the task and translate them into mathematics
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Representing and solving the problem
Use models, pictures, diagrams, and/or symbols to represent the problem and select an effective strategy to solve the problem.
Communicating Reasoning Communicate mathematical reasoning coherently and clearly use the language of mathematics.
Accuracy Clearly identify and support the solution.
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Reflecting and Evaluating State the solution in the context of the problem. p Defend the process. Evaluate and interpret the reasonableness of the solution
Simplified Mathematics Scoring Guide Exemplary Strong Proficient
3 2
1
6
5
4
Developing
Emerging
Beginning
6 íEnhanced or connected to other mathematics 5 – Thoroughly developed 4 – Work is proficient (not perfect) 3 – Work is partially effective or partially complete 2 – Work is underdeveloped or sketchy 1 – Work is ineffective, minimal, or not-evident
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What does a Math Work Sample l llook k like?
Mathematics Problem Solving Work Samples •Present complex, multi-step tasks •Are designed to judge student abilities to apply specific knowledge & skills •Allow a variety of problem-solving approaches •May simulate real-word mathematics problems
The Task: Farmer John has a rectangular holding pen that measures 10 yards long and 5 yards wide to contain his cattle. He is acquiring more cattle from the neighboring farmer and wants to add the same amount of fencing to each side to create a new holding pen that encloses 176 square yards. How much should Farmer John add on to each side of his existing holding pen to achieve his goal?
Farmer John
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• Share your solution
Share your solution!
Scoring the First Anchor Paper This anchor paper met the achievement standard in each trait. Why did this paper earn these scores?
Scoring the 2nd Anchor Paper This anchor paper did not meet the achievement standard. What scores did this paper earn?
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Scoring Within the Traits . . . What differentiates a 3 and a 4?
Scoring Papers 3, 4, & 5 Use the scoring guide to rate each paper. What scores did these papers earn?
Bike Rental A local bicycle rental company charges $12 to rent a bicycle. They normally have 300 rentals per month. The company owner has determined that each increase in price of $2 will decrease the number of rentals by 15. What price will maximize the revenue?
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Share your solution!
Scoring the First Anchor Paper for Bike Rental This anchor paper met the achievement standard in each trait trait. Why did this paper earn these scores?
Scoring the 2nd Anchor Paper For Bike Rental This anchor paper also met the achievement standard. What scores did this paper earn?
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Scoring Papers 8, 9, & 10 Use the scoring guide to rate each paper. What scores did these papers earn?
The Mathematics Scoring Guide Purposes 1. Instructional Tool 2 Formative Assessment 2. 3. Summative Assessment 4. Demonstrate Proficiency in the Essential Skill of Apply Math to earn an Oregon Diploma
Building Consensus on Definitions of Assessments Assessment
Screening g
Formative
Purpose
When Administered?
Identify students at risk of mathematics difficulties & provides info to target instruction for all students
Beginning of year or semester;; when new students arrive
Supports learning and informs instruction
Embedded directly in instruction to inform teacher decisions
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Multiple Uses for the Scoring Guide Instructional Tool • Makes targets explicit to students • Opportunities to show students models from website or other examples
Screening Tool
• Help determine likelihood of reaching proficiency – on target, need assistance, at risk • Help determine which students need additional instruction and coaching
Building Consensus on Definitions of Assessments Assessment Interim and Predictive
Summative
Purpose
When Administered?
Determine the progress of individuals or groups of students based on focused elements of content
Occasional, based on curriculum & other instructional milestones
Determine how much knowledge and skills individuals or groups of students (e.g. programs, schools, districts and states) have acquired.
Periodically after a substantial period of time (e.g. end of the year and end of course).
Multiple Uses for the Scoring Guide
Formative & Interim Assessments – Inform instructional strategies – Provide data on student progress
Classroom/ Summative Assessment – End of unit, course, etc. or Essential Skills
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Formative Assessment The Scoring Guide can help to identify math strengths and weaknesses.
Students learn where to focus to improve math skills.
Teachers learn where additional instruction is needed.
Assessment
Does your school have an assessment plan? Explain
Does your school have a data analysis & use plan? Explain
Benchmark
Formative
Interim and Predictive
Summative
Requirements for Essential Skill Proficiency Using Math Work Samples 2 work samples í Algebra, geometry, or statistics Score of 4 or higher in all dimensions on Official Scoring Guide
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Work Sample Design • Math Problem Solving Tasks must be carefully designed to provide opportunities for students to demonstrate skills i all in ll dimensions di i off th the scoring guide. • Math tasks must be at the appropriate difficulty level and address high school content standards.
Work Sample Implementation Administration Work samples must be the product of an individual Work samples must be supervised by an authorized adult; Students may not work on work samples outside a supervised setting.
Work Sample Implementation Scoring All work samples must be scored using Oregon’s Official Math Scoring g Guide. All raters must have been trained to use the Scoring Guide. Only one set of scores is required for a work sample. (Districts may want more than one rater for borderline papers.)
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Work Sample Implementation Feedback and Revision FEEDBACK: Only 2 options 1. Oregon’s Official Scoring Form 2. Oregon’s Scoring Guide (highlight/underline) STUDENT REVISION: 1. Students are allowed to revise and resubmit their work samples following scoring/feedback. 2. Most papers should be revised only once.
Resources & Coming Attractions
ODE Website: www.ode.state.or.us/go/worksamples OCTM Website: http://www octm org/ http://www.octm.org/ Follow-up workshops (List any scheduled) Contact information (List your information here)
A Parting Thought
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To meet the graduation requirement for the essential skill of “apply mathematics” follow these steps. 1. Student takes the OAKS Mathematics Assessment.
1a: IfIfthe 2a. thestudent studentreceives receivesaascore of 236, score ofhe 236, hashe/she met thehas met
2b. If the student receives a score lower than 236, he/she can meet the graduation requirement standard in one of three ways:
the graduation graduation requirement requirement g. standard for readin mathematics. 3c. Complete 2 mathematics work samples that are: 3a. The student studies and retakes the OAKS Mathematics Assessment and receives a score of 236. 3b. Take one of a number of approved standardized
x scored using the Official State Mathematics Scoring
Guide; x receive a score of 4 or higher in 5 traits for each work
sample. In addition the work samples will be drawn from: x Content Strands: Algebra, Geometry and Statistics x Traits: Making Sense of the Problem, Representing
& Solving the Task, Communicating & Reasoning, Accuracy, Reflecting & Evaluating.
tests and receive the following scores: ACT: 19 PLAN: 19 SAT: 450 PSAT: 45 ASSET: 41 Compass: 66 Work Keys: 5 AP or IB: varies
4a. If the student attains a score
4b. If the student attains a score
of 4 or higher for each trait
of 3 or lower on any trait,
on each work sample,
he/she does not meet the
he/she has met the
graduation requirement
graduation requirement
standard for mathematics.
standard for mathematics. REVISION IS POSSIBLE: Work samples that nearly meet the standard (scoring a mix of 4s and 3s) may be returned to students for revision. Teachers may mark areas on the scoring guide or Official Scoring Form to show students in what areas they need to work (no other instructions are allowed). The work sample is then rescored.
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From the Introduction to the Common Core State Standards for Mathematics “These Standards define what students should understand and be able to do in their study of mathematics. Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b) (x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.” “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).”
“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. 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.” i Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. i Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. i Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”
The Common Core Standards for Mathematics can be found at the following link: http://www.ode.state.or.us/search/page/?=1527 There are a variety of resources to assist Oregon educators at this link and additional resources will be added over time.
Oregon Department of Education
2011-12
Office of Assessment and Information Services
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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.
Process Dimensions Making Sense of the Task Interpret the concepts of the task and translate them into mathematics.
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.
Communicating Reasoning Coherently communicate mathematical reasoning and clearly use mathematical language.
Accuracy Support the solution/outcome.
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.
**6/ 5
4
3
*2 / 1
The interpretation and/or translation of the task are x thoroughly developed and/or x enhanced through connections and/or extensions to other mathematical ideas or other contexts. The strategy and representations used are x elegant (insightful), x complex, x enhanced through comparisons to other representations and/or generalizations.
The interpretation and translation of the task are x adequately developed and x adequately displayed.
The interpretation and/or translation of the task are x partially developed, and/or x partially displayed.
The strategy that has been selected and applied and the representations used are x effective and x complete.
The strategy that has been selected and applied and the representations used are x partially effective and/or x partially complete.
The interpretation and/or translation of the task are x underdeveloped, x sketchy, x using inappropriate concepts, x minimal, and/or x not evident. The strategy selected and representations used are x underdeveloped, x sketchy, x not useful, x minimal, x not evident, and/or x in conflict with the solution/outcome.
The use of mathematical language and communication of the reasoning are x elegant (insightful) and/or x enhanced with graphics or examples to allow the reader to move easily from one thought to another. The solution/outcome is correct and enhanced by x extensions, x connections, x generalizations, and/or x asking new questions leading to new problems. Justifying the solution/outcome completely, the student reflection also includes x reworking the task using a different method, x evaluating the relative effectiveness and/or efficiency of different approaches taken, and/or x providing evidence of considering other possible solution/outcomes and/or interpretations.
The use of mathematical language and communication of the reasoning x follow a clear and coherent path throughout the entire work sample and x lead to a clearly identified solution/outcome.
The use of mathematical language and communication of the reasoning x are partially displayed with significant gaps and/or x do not clearly lead to a solution/outcome.
The solution/outcome given is x correct, x mathematically justified, and x supported by the work.
The solution/outcome given is x incorrect due to minor error(s), or x a correct answer but work contains minor error(s) x partially complete, and/or x partially correct
The solution/outcome is stated within the context of the task, and the reflection justifies the solution/outcome completely by reviewing x the interpretation of the task x concepts, x strategies, x calculations, and x 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 x the task situation, x concepts, x strategies, x calculations, and/or x reasonableness.
The use of mathematical language and communication of the reasoning are x underdeveloped, x sketchy, x inappropriate, x minimal, and/or x not evident. The solution/outcome given is x incorrect and/or x incomplete, or x correct, but o conflicts with the work, or o not supported by the work. The solution/outcome is not clearly identified and/or the justification is x underdeveloped, x sketchy, x ineffective, x minimal, x not evident, and/or x 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.
For use beginning with 2011-2012 Assessments Oregon Department of Education
Office of Assessment and Evaluation Adopted May 19, 2011
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Introduction to the Scoring Guide The scoring guide maintains vertical consistency at each number through each of the dimensions. 6: 5: 4: 3: 2: 1:
Work is insightful Work is thoroughly developed Work is complete and effective (not necessarily perfect) Work is partially effective or partially complete Work is underdeveloped or sketchy Work is ineffective, minimal, or not evident
Making Sense of the Task (MS) Interpret the concepts of the task and translate them into mathematics
The student translates the words from the problem into appropriate mathematics. The key concepts are addressed. Evidence that makes a paper more thoroughly developed or insightful may include extending their thinking to other mathematical ideas or making connections to other contexts. Representing and Solving the Task (RS) Use models, pictures, diagrams, and/or symbols to represent and solve the task situation and select an effective strategy to solve the task
The strategies chosen by the student are effective and complete for this task. Evidence that makes a paper more thoroughly developed or insightful may include generalizing a strategy using an algebraic representation versus a numeric or tabular representation. Communicating Reasoning (CR) Coherently communicate mathematical reasoning and clearly use mathematical language.
Communication of the reasoning refers to the connections among all of the dimensions, and the identifiable solution – allowing the flow of the paper to help the reader understand the path from one part to another. A clear path does not require a linear sequence of thoughts or communication. The student uses math vocabulary and labels appropriately. A significant gap is when the reader is using his/her own knowledge about the problem and mathematics to infer why a student might have moved from one part of the work to another. Evidence that makes a paper more thoroughly developed or insightful may include additional graphics or examples to help the reader move easily through the student work. Accuracy (AC) Support the solution/outcome.
The student’s solution is correct, mathematically justified, and supported by the work. It is critical that students who are “close” to having a proficient response, with minor errors or partial answers, be given an opportunity to rework the problem given the scoring feedback, but no further instruction. Evidence that makes a paper more thoroughly developed or insightful may include extending the solution by asking new questions leading to new problems. Although possible, it is a rare occurrence to get a 5 or 6 in accuracy.
Reflecting and Evaluating (RE) State the solution/outcome in the context of the task. Defend the process, evaluate and interpret the reasonableness of the solution
The student states the solution within the context of the problem. This requires the student to review the task and reflect on what was asked. There should be evidence on the student has reviewed ALL the dimensions in solving the task. The reflection (a second look) could be embedded in the original work or after arriving at a solution and/or a combination of both. Evidence that makes a paper more thoroughly developed or insightful may include solving the task from a different perspective. Students evaluating their approaches taken may include addressing the efficiency of an approach or the relative use of a procedure. Additional considerations: It is important to consider each of the dimensions as a separate entity. The weakness in the work should only reduce the score for the dimension in which the weakness occurred. On the other hand, strength in one dimension may improve a score in another dimension. An answer that is not correct may still have strong work in some or all other dimensions. Likewise, a paper with a correct solution still needs careful consideration for success in each dimension. Because a single scoring guide is used for a variety of tasks, the student work is not always expected to exactly match the criteria described at each numbered level. It will, however, have characteristics similar to those described in the criteria. Guidelines for using the scoring guide: It is important to prepare for the scoring process by doing the following; x x x x
Work the task yourself State the answer within the context of the problem List the key concepts necessary to complete the task Anticipate alternative solutions and strategies
Although different scorers may use different styles, here is a typical process for using the scoring guide: x Scan the student papers and find a sample of papers that represent the spectrum of student work x Possibly sort the student work so that you will score all work with similar approaches one after another x Carefully read all through one student’s work. x Read the criteria for a score of 3 on one dimension. x Review the student work again. If it seems stronger than a 3, read the 4 through 6 criteria. If is weaker than a 3, consider the 1 and 2 criteria. x Assign a score for that dimension x Repeat the process for the other 4 dimensions
Additional scoring hints or considerations include the following: x x x
Any scores that you are uncertain about should be set aside to look at again after scoring the rest of the papers (these papers might also be good candidates for collegial discussions). A score of 4 meets the standard, and a 3 nearly meets – so this is a critical distinction As a way of validating your scores, it is very helpful to have a colleague score a few of your papers without seeing the scores you gave, then having a discussion about any differences in your scores. When double scoring, as a general rule, scores that are within 1 point in any given dimension are considered “aligned” but a discussion to agree on the same score can help to “calibrate” your scoring.
Reworking “official” work samples When time allows and if no discussion of the task has taken place in class, it is encouraged that students with 4’s or better in some dimensions, and a few scores of three, or possibly lower, should be given an opportunity to revise their work (no further instruction is allowed if this response is to be used as evidence of proficiency for the purpose of the Essential Skill of Mathematics)
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Mathematics Work Sample High School 2011 – 2012 – Farmer John Use the information provided to solve the problem listed below. Be sure to show your work at all phases of problem-solving. Refer to the Student Problem Solving Tips to receive the highest score in each of the five areas. Algebra
Geometry
Statistics
Content Standard: CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.A.REI.4 Solve quadratic equations in one variable.
Achievement level descriptor: Determine area, surface area, and/or volume. Solve for missing dimensions. Solve related context- based problems.
Name: ______________________________________ School: _____________________________________ Teacher: ____________________________________
Farmer John has a rectangular holding pen that measures 10 yards long and 5 yards wide to contain his cattle. He is acquiring more cattle from the neighboring famer and wants to add the same amount of fending to each side to create a new holding pen that encloses 176 square yards. How much should Farmer John add on to each side of his existing holding pen to achieve his goal?
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Mathematics: Essential Skills Scores and Commentary Algebra ;Geometry Statistics High School Content Standard: H.1G.5 Determine the missing dimensions, angles, or area of regular polygons, quadrilaterals, triangles, circles, composite shapes, and shaded regions.
Work Sample Title: Farmer John
MS 4
Paper Number: R1
RS 4
CR 4
AC 4
RE 4
Making Sense of the Problem: The interpretation of the task into increasing the dimensions by the same amount until reaching the desired area was adequate and complete. The mathematics is not connected or extended to other mathematical ideas. Representing and Solving the Problem: The strategy of making a systematic list adding 1 square yard to each original dimension and checking the area until reaching 176 was effective and complete. The strategy is not complex nor is the student able to generalize the situation. Communicating Reason: The student clearly represents the two rectangular pens and labels their dimensions. Although the student does not explain what he/she is specifically doing with the table, it is clear that he/she is adding 1 unit to each dimension and then multiplying to find area. However, using a systematic list or guess and check table are not considered elegant or insightful at high school level.
Accuracy: The solution of adding 6 yards to each side is correct and supported by the work. The work does not include extensions, connections, or generalizations.
Reflecting and Evaluating: The student summarizes the answer in a sentence at the bottom of the page, indicating they have answered the question asked and did not just give the dimensions. The review is complete, since the student basically re-did the problem again on the second page (adding an additional set of numbers), although they did not restate the solution of adding 6 yards to each side. **Note: This is a good example of a case where a student earns a "4" in all dimensions for problem solving but does not demonstrate proficiency for Essential Skills.
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G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
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Mathematics: Essential Skills Scores and Commentary
Work Sample Title: Farmer John
Revised Paper Number: R2
Algebra ;Geometry Statistics
High School Content Standard: H.1G.5 Determine the missing dimensions, angles, or area of regular polygons, quadrilaterals, triangles, circles, composite shapes, and shaded regions. G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Achievement Level Descriptor: Determine area, surface area, and/or volume. Solve for missing dimensions. Solve related context-based problems.
MS 6
RS 6
CR 5
AC 6
RE 6
Making Sense of the Problem: The interpretation of the task into generalizing the amount being added to both dimensions as “x”, and using a quadratic equation to represent the situation is enhanced. The student thoroughly develops the meaning of the x verbally and in the diagram and also looks at the mathematics thoroughly (he/she gets two answers: -21 and 6 but comments why 6 is the only one that makes sense). He/she also describes how to use a graphing calculator to solve the problem, indicating that the student has made sense of this problem in several ways. Representing and Solving the Problem: The strategy of assigning “x” to the additional amount needed and writing, solving, and interpreting the solution for a quadratic equation makes this paper complex. The additional information on the back about how to use the graphing calculator to solve the problem adds elegance and insightfulness, and connects the problem to another representation. Communicating Reason: The work follows a clear and coherent path and the student uses mathematical language precisely (I will solve this equation algebraically, Let y1 represent the product of the, set the window so the domain….). The reader is able to move easily from one thought to another through the use of words, symbols and the diagram to the right. The diagram of the two pens helps explain the origin of the equation (10 + x)(5 + x) = 176 but clearly shows that there would be “x” added to both sides of the pen according to the way it is labeled. There is a slight gap between the diagram and the equation (why is it 5 + x instead of 5 + 2x?). The student also does a good job describing the process used with the graphing calculator, but a sketch of the graph and more details in the description would have helped this paper become a 6 in this dimension Accuracy: The solution of adding 6 yards to each side is correct and justified. It is also extended through the use of technology, since the student uses the technology to check if the intersection is truly the solution they arrived at using the system. The student also thought about the two solutions obtained from solving the system and commented that “6 is the only answer that makes sense”, although they do not say why. Reflecting and Evaluating: The solution is clearly stated within the context of the task. He/she reworks the task using a graphing calculator (and if you include the original method of making a systematic list, the student has worked the problem 3 ways). He/she also considers the solution of -21 and dismisses this as possible. The approach used with the graphing calculator shows that the student reflected on his/her solution and realized that if 6 is the solution, then it should be one coordinate of the intersection point of the two lines. This evidence moves the score for this dimension to a 6.
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Mathematics Work Sample High School 2011 – 2012 – Bike Rental Use the information provided to solve the problem listed below. Be sure to show your work at all phases of problem-solving. Refer to the Student Problem Solving Tips to receive the highest score in each of the five areas. ; Algebra
Geometry
Content Standard:
Statistics 2
H.3A.5 Given a quadratic equation of the form x + bx + c = 0 with integral roots, determine and interpret 2 the roots, the vertex of the parabola that is the graph of y = x + bx + c, and an equation of its axis of symmetry graphically and algebraically. CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CC.9-12.F.BF.1 Write a function that describes a relationship between two quantities.
Achievement level descriptor: Name: ______________________________________ School: _____________________________________ Teacher: ____________________________________
A local bicycle rental company charges $12 to rent a bicycle. They normally have 300 rentals per month. The company owner has determined that each increase in price of $2 will decrease the number of rentals by 15. What price will maximize the revenue?
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TR10
ʹͲͳͳǦͳʹ
ǣ 6/5**
Process Dimensions Making Sense of the Task
·
3
·
·
·
Only a small amount of the problem is understood OR
·A complete plan is used that contains ·A plan using pictures, charts, words,
·The plan could solve some parts of the
Choose the strategy that works best for this problem.
pictures, charts, words, graphs or numbers and may contain more than one step.
graphs or numbers is used to solve the problem.
problem or the plan has a few missing parts.
The plan ·has many missing parts, ·cannot work OR ·No work is shown
HOW?
A superior strategy is used to solve the problem.
Communicating Reasoning
WHAT? Representing and Solving the Task
The problem is changed into math ideas that work.
1/2*
Parts of the problem are changed into math ideas that can work or parts of the problem are understood.
Understand the ideas and change them into a math task
The problem is changed into complete ideas that work.
4
ʹͲͳͳǦͳʹ
·
The ideas are connected to other math ideas that.
·
Use the language of math (words, equations, graphs, charts) to make your ideas clear to others.
·The steps to complete the work are
·The path through the work can be
·The path is not clear OR doesn’t show
very clear.
followed to a clearly identified solution.
much of the work
WHY?
An explanation of the why for each part is given.
Some attempt is made to explain why each step was used.
·The solution is correct and may be
·The answer given is correct.
Accuracy The answer is…
·No understanding is shown.
·
The steps to complete the work are ·just started OR ·There are no steps shown.
·
extended or shown another way.
·The answer given may have a small error but the important parts work fine.
The answer given · is not correct, · is not finished OR · doesn’t match the work.
IS IT RIGHT? Reflecting and Evaluating Check your answer and explain why it makes sense.
CHECK?
·A second look has been taken to
·The problem is solved a second time ·Some, but not all of the work is checked.
completely check the work.
to check the work and method.
·
The explanation cleary shows why the solution makes sense.
·
·
The student makes an attempt to explain why the answer makes sense.
A new way may be used to check the work.
**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 inadequate in some of the attributes; while a 1 would be inadequate in most of the attributes
The check ·doesn’t work, ·is barely started OR · is not there at all.
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Student Problem Solving Tips You may use manipulatives or a calculator to work on your problem. To receive the highest score in each of the five areas, you will want to be certain your work SHOWS each of these parts of a successful solution. 1. Making Sense of the Task
I turned important information into numbers, symbols and/or diagrams.
The mathematics I used fits the problem.
If possible, I showed connections and/or extended my work to other math ideas.
2. Representing and Solving the Task
I used strategies that fit the problem.
I showed all my work (diagrams, pictures, models, numbers, symbols and/or words).
If possible, I was able to make generalizations and/or compare my work to other ways the task could have been completed.
3. Communicating Reasoning
The path leading through my complete solution has no gaps for the reader to fill in. I used mathematical language/labels appropriately. My answer is clearly identified. If possible, I used graphics and/or examples to enhance my work.
4. Accuracy
My final answer is complete and justified.
The work shown on my paper leads to my answer.
If possible, I extended my solution by asking new questions leading to new problems. . 5. Reflecting and Evaluating
My solution matches what the problem was asking.
I reviewed ALL of my work (interpretation of the problem, concepts, strategies, and calculations) to show that my answer makes sense and it is correct.
If possible, I worked the entire problem a second way.
RECAP: Show your answer and all of your work so everything is clear to the reader
Oregon Department of Education
2011-12
Office of Assessment and Information Services
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Participant Score Recording Sheet PAPER #
Title
1
Farmer John
2
Farmer John
3
Farmer John
4
Farmer John
MS
RS
CR
AC
Space is provided in this table to allow you to record your original score, the expert score and any comments you wish.
RE
5
Farmer John
6
Bike Rental
7
Bike Rental
8
Bike Rental
9
Bike Rental
10
Bike Rental
Space is provided in this table to allow you to record your original score, the expert score and any comments you wish.
School Math Problem Solving Assessment & Data Analysis Plan Assessment
Describe your school’s math problem solving assessment plan?
Describe your school’s data analysis & use plan?
Benchmark
Formative
Interim and Predictive
Summative
Introduction to the Math Problem Solving Scoring Guide Workshop – Level 2
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Guidelines for High School Mathematics Work Samples Definition: A Mathematics Work Sample is individual student work used to provide students with the opportunity to demonstrate problem-solving skills and/or to demonstrate proficiency in the Essential Skill of Applying Mathematics. Purposes: 1. To meet requirements for one local performance assessment in high school in mathematics 2. To demonstrate proficiency in the Essential Skill of Mathematics in order to earn an Oregon High School Diploma Required assessment instrument: Oregon’s Official Mathematics Scoring Guide Requirements for mathematics work samples Number One work sample for local performance assessment (any of three math subjects: algebra, 1. geometry, statistics ) 2. Two work samples for Mathematics Essential Skill proficiency (any two of three math subjects above). One math work sample may also count as the local performance assessment. Required scores and traits i For local performance assessment, there are no required scores. i For Mathematics Essential Skill Proficiency, a minimum score of 4 out of 6 in each of the five scoring guide dimensions is required. Making Sense of the Problem Representing and Solving the Task Communicating Reasoning Accuracy Reflecting and Evaluating i For Mathematics Essential Skill Proficiency, student work must demonstrate proficient application of high school level mathematics knowledge and skills.
Individual work i must represent what the individual student can do with no outside assistance, teacher or peer feedback i no collaborative group projects or products are allowed i Appendices L & M of the 2011-12 Test Administration Manual contain more information (http://www.ode.state.or.us/search/page/?=486) Opportunities for revision: i work samples that nearly meet the achievement standard (scoring a mix of 4s and 3s) may be returned to students for revision i In addition to scores, the only allowable feedback to students is highlighting phrases on the Official Mathematics Scoring Guide and/or using the Official Mathematics Scoring Form provided by ODE (http://www.ode.state.or.us/search/page/?id=2704)
Oregon Department of Education
Guidelines for Mathematics Work Samples
Page 1 of 2
2011-12
Guidelines for High School Mathematics Work Samples (continued) Who should complete mathematics work samples? iLocal Performance Assessments: All students must have the opportunity to complete at least one mathematics work sample during high school.
iEssential Skill Proficiency: Students who have not demonstrated proficiency by meeting the mathematics standards with a score of 236 on the OAKS Mathematics Assessment may use work samples as evidence of their proficiency in the Essential Skill of Mathematics. (Typically, these would be students in the “nearly meets” category: students whose assessment scores or classroom work indicate that they may have the necessary mathematics skills, but are not demonstrating those skills on the OAKS assessment. Students who need significant additional instruction to reach a high school level of mathematics proficiency are not likely to benefit from the work sample option until their skills have improved.) Who should score mathematics work samples? One trained classroom teacher or other district employee trained on Oregon’s Official Mathematics Scoring Guide. (Some schools may choose to use more than one rater or to score work samples in a group setting for anonymity and to facilitate discussion of close scores.)
Recommendations for Developing Mathematics Work Samples Problem-Solving Tasks: Complex problems requiring multi-step solutions and reflecting content from the mathematics standards are appropriate for work samples. These work samples may be “stand-alone” tasks that provide students with opportunities to practice and demonstrate their problem- solving skills or they may arise naturally in the curriculum as part of a particular unit of study in a math or other content class. Mathematics tasks released by the Oregon Department of Education may be used as practice activities and as models to develop local math problem solving tasks http://www.ode.state.or.us/search/page/?id=281 Choice: Whenever possible, work samples should be designed to offer student choices, among several different problem-solving situations.
Recommendations for Administering and Scoring Mathematics Work Samples Allow adequate time for students to show their best work. If students need more than one session to complete a work sample, all student materials in progress must be collected and kept secure between sessions. Provide access to appropriate tools such as calculators and formula tables from the OAKS Mathematics Assessment.
Oregon Department of Education
Guidelines for Mathematics Work Samples
Page 2 of 2
2011-12
ȋǯ
Ȍ Student Name: ___________________________________________ Date: ________________ Task Title: ____________________________________________ Grade Level:_____________ The Student Work Demonstrates: _____Algebra
_____Geometry ______Statistics/Probability
Essential Skills Requirement for Oregon Diploma _____ Uses High School or Advanced High School level mathematicsRU&&66 and _____ Meets at “4” level or above in all scoring dimensions Standard(s) Addressed: ___________________________________________ Bullets describe a score of 4. ; indicates areas that meet the standard. No other feedback beyond the Official Scoring Guide may be provided. Making Sense of the Task
6
5
4
3
2
1
2
1
Important information was changed into mathematical ideas.
The way the problem is changed into mathematics fits what was asked. Representing and Solving the Task
6
5
4
3
The strategies used fit the problem.
All pictures, models, diagrams, and/or symbols used to solve the problem are shown. Communicating Reasoning
6
5
4
3
2
1
The path leading to a complete solution is shown with no gaps for the reader to fill in.
The work connects all the parts (i.e. concepts, strategies, reflection, answer and reasoning).
Mathematical language/labels are used appropriately throughout. Accuracy
6
5
4
3
2
1
6
5
4
3
2
1
The final answer is complete and justified.
The answer is supported by the work.
The solution/outcome is correct. Reflecting and Evaluating
The solution/outcome matches what the problem was asking.
The defense of the solution reviews the interpretation of the task, concepts, strategies, calculations and reasonableness. Raters may mark the boxes and circle specific words to explain reasons for the current scores. Rater ID Number, Initials, or Name: Date of revision: ________________ Revised scores: MS___ RS___ CR____AC____RE___
OregonDepartmentofEducation
MathWorkSampleFeedbackForm
OfficeofAssessmentandInformationServices
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Suggestions for using the Oregon Department of Education Feedback form. The feedback form was designed as a consistent tool for teachers to use with the state scoring guide to give students more specific feedback than dimension scores alone can provide. Although the form can be used for all students with all work samples, it isn’t often feasible to do so. This is meant to provide feedback to students in two categories: x Those who have earned scores of 3 in some dimensions and have a good chance of editing their work in order to earn scores of 4 in all dimensions. x
Those who have produced successful work samples (scores of 4 in all dimensions) but not at the level of high school or advanced standards. This information is critical when work samples are used to provide evidence of meeting the essential skills requirement for the Oregon Diploma in lieu of a passing score on the OAKS test.
1. 2. 3. 4.
Score the student work using the official state scoring guide. Circle the scores in each dimension on the feedback form. Check the boxes corresponding to what the student did well and need no revision. For areas that need revision, circle key words that will give the student more specific guidance. 5. Determine the strand (Algebra, Geometry, or Statistics) used in the solution and indicate it in the box near the top of the form. 6. Indicate whether the student has addressed a high school or advanced standard and identify the standard (more than one standard may be addressed). Teachers should be familiar with the high school and advanced standards.
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Resources for Supporting Mathematics Instruction WEBSITES The following websites contain information specific to Oregon: Assessment of Essential Skills Toolkit: http://assessment.oregonk-12.net/ A web-based interactive planning system that will help districts and high schools develop a local plan for assessing the essential skills by guiding them through a series of 10 Steps. Common Core State Standards Crosswalk (ODE) http://www.ode.state.or.us/search/page/?id=3211 Identifies matches between individual Common Core standards with Oregon standards in mathematics and includes rater comments that highlight the similarities and differences between the standards. Mathematics Achievement Standards: http://www.ode.state.or.us/search/page/?id=3182 A detailed explanation of the standard setting process and rationale for the new cut scores is provided along with links for additional resources, tools, and materials. Moving Math Education Forward (MMEF): http://www.ode.state.or.us/search/page/?id=2702 Workshop components and resources focus on elements of effective instruction, analyzing cognitive demand, and formative assessment. Access to materials from professional development workshops provided during the summer of 2009 to mathematics educators across Oregon. OAKS Online Practice Test: http://www.ode.state.or.us/search/page/?id=441 Helps students prepare for OAKS Online by providing sample tests for grades 3-8 and high school with the same look and feel as the actual test. OAKS Test Specifications and Blueprint Documents: http://www.ode.state.or.us/search/page/?id=496 These documents explain how the Oregon mathematics standards will be assessed. Included are the content standards, accessible content and vocabulary, achievement level descriptors, and scoring guides. Oregon Council of Teachers of Mathematics (OCTM) Math Links: http://www.octm.org/mathlinks.html Links to numerous k-12 math resources on a variety of math topics including; problem solving, task banks, and practice tests. Oregon Diploma: http://www.ode.state.or.us/search/results/?id=368 Find up to date information about Oregon requirements needed to earn a diploma, including credit requirements, Essential Skills, and the personalized learning requirements. Portland Public Schools Mathematics Curriculum: http://www.pps.k12.or.us/departments/curriculum/1476.htm From here you can select
elementary, middle, or high school resources developed by Portland Public Schools, including assessments. Salem-Keizer Curriculum, Instruction, and Assessment Online Resources: https://salkeiz-cia.orvsd.org/math Includes a number of resources for the current Oregon math standards, including deconstructed standards and pacing guides. Test Administration Manual: http://www.ode.state.or.us/search/page/?=486 Contains specific procedures and guidelines for assessment of OAKS, local performance assessments, and work samples. See Appendices L,M, and N. Work Sample Resource Page (ODE) http://www.ode.state.or.us/search/page/?id=219 News releases and work sample resources by subject area. Work Sample Tasks for Problem Solving: http://www.ode.state.or.us/search/page/?id=281 Links to tasks that have been aligned to the 2011-2012 scoring guide and the 2007/2009 math standards.
The following websites contain information that is NOT specific to Oregon: AAAMath: http://aaamath.com/ a comprehensive set of interactive arithmetic lessons and unlimited practice is available on each topic, K-8. Common Core State Standards Webpage (ODE): http://www.ode.state.or.us/search/page/?id=2860 This site has links to Common Core State Standards for Math and English Language Arts, Resources for teachers, Communication Tools, Timelines and Transitions, Assessment and more. Mathematics Assessment Resource Service (MARS): http://www.nottingham.ac.uk/education/MARS/ Site contains samples of balanced assessment tasks including solutions, student work, and commentaries. NCTM Illuminations: http://illuminations.nctm.org/ Direct link to NCTM’s lessons and activities. National Council of Teachers of Mathematics (NCTM): http://www.nctm.org/ Contains information about standards, conferences, professional development, journals, research, and lessons and activities. PBS Teachers - Math: http://www.pbs.org/teacherline/catalog/browse/?sa=1 Multimedia resources and professional development are made available for Pre-K through 12 educators. Silicon Valley Mathematics Initiative: http://www.svmimac.org/ Resources include coaching materials, performance assessment, lesson study, reports, and presentations.
Texas Assessment of Knowledge and Skills: http://www.tea.state.tx.us/student.assessment/taks/ This site contains released assessments and answer keys. Some of the materials include a Spanish version. TI Education Technology: http://education.ti.com/us/home/down/ae This Texas Instruments site has calculator and technology-related lessons and activities for math and science. WEBINARS A Closer Look at the Common Core State Standards for Mathematics (Education Northwest) http://educationnorthwest.org/event/1346 Archived webinar from December 13, 2010 featuring Dr. William McCallum an focusing on the progression and connection of key concepts across grades. Common Core Standards Initiative: Preparing America’s Students for College and Career (Carnegie Learning) http://www.carnegielearning.com/webinars/commoncore-standards-initiative-preparing-americas-students-for-college-and-career/ Archived webinar from October 27, 2010 featuring Chris Minnich. Chris leads the standards and assessment work at CCSSO, where they are currently working on implementing common standards across states. National Council of Supervisors of Mathematics (NCSM) Webinar Series http://ncsmonline.org/events/webinars.html has two archived webinars, one from November 30, 2010 and one from February 23, 2011. The first is a getting started webinar and the other is a diving deeper webinar - both focusing on Common Core State Standards. Includes pdfs of the presentation and handouts. WORKSHOPS, TRAININGS, AND CONFERENCES Annenberg Learner http://www.learner.org/index.html Annenberg Learner's multimedia resources help teachers increase their expertise in their fields and assist them in improving their teaching methods. Many programs are also intended for students in the classroom and viewers at home. All Annenberg Learner videos exemplify excellent teaching. Resources can be accessed for free at learner.org.
Essential Skills Work Samples Training of Trainers (ODE) http://www.ode.state.or.us/search/page/?id=2042 ODE has scheduled Training of Trainer WebEx sessions to assist districts in implementing work samples for the Essential Skills of Reading, Writing and Mathematics. These sessions are designed to help district staff prepare to conduct workshops training teachers to use the Official Scoring Guides to assess work samples for the purpose of determining proficiency in the Essential Skills. Mathematics and Science Services (Education Northwest) http://educationnorthwest.org/services/math-science Provides educators with top-
quality professional development, technical assistance, evaluation, and research services. National Council of Supervisors of Mathematics (NCSM) http://ncsmonline.org/events/index.html NCSM offers various seminars, workshops, and conferences throughout the school year. Please check the website for current offerings. National Council of Teachers of Mathematics (NCTM) Conference Page: http://www.nctm.org/conferences/default.aspx?id=52 Lists current, available conferences and workshops available, including NCTM’s annual meeting and exposition. Northwest Mathematics Conference October 13-15, 2011 http://www.northwestmathconf.org/NWMC2011/ Strands will include best practices in instruction and assessment, standards and NCTM Focal Points, historical perspectives in mathematics, and future trends in mathematics. OCTM Professional Development Cadre: Math Workshops http://www.octm.org/pdc.html Workshops targeting effective instruction, including: oregon math standards, Higher order thinking, questioning strategies, lesson leverage, assessment for learning, problem solving, and common core state standards. PRISM (Preparation for Instruction of Science and Math) Courses http://www.pdx.edu/ceed/prism a collaborative effort of seven Oregon universities to offer graduate level courses and professional development modules in math and science that are available online, in weekend workshops, at summer institutes, or combinations of these formats. Teacher’s Development Group http://www.teachersdg.org/ A non-profit organization dedicated to increasing all students’ mathematical understanding and achievement through meaningful, effective professional development. The Math Learning Center http://www.mathlearningcenter.org/development/workshops Each year hundreds of educators attend Math Learning Center workshops for practical teaching strategies and insights into how children learn. Whether it focuses on a broad topic or a specific set of materials, an MLC workshop offers a rich learning experience.
TIPS (Teachers Inspiring Problem Solvers) http://www.math-tips.com/ Works with teachers and students to help deepen their understanding of mathematics while developing the habits and characteristics of a successful problem solver.
Problem Solvers (TIPS) is to work with teachers and students, helping deepen their understanding of mathematics while developing the habits and characteristics of a successful problem
