Facilitator’s Packet for The Essential Skill of

Facilitator’s Packet for

The Essential Skill of Mathematics:

An Overview

This packet contains the following:Facilitator’s Agenda PowerPoint Slides with Facilitator’s notes Sample Student Work Commentary with Scores for Student Work

Updated for 2012-12

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Essential Skill of Applying Mathematics Overview Session Facilitator’s AGENDA: 30 – 45 MINUTES

30 minutes without sample student papers; 45 minutes with discussion of student papers

Updated for 2012-13

5 minutes

1. Welcome and Introductions May be done by the host or by the presenter. Focus on making participants feel welcome and let them know what

to expect Take care of any housekeeping details

Handout: The Essential Skill of Mathematics – An Overview booklet which contains all handouts. References to Handouts are marked in the facilitator notes in the PowerPoint presentation.

PowerPoint 12 - 15 minutes

2. Applying Mathematics Work Samples PowerPoint Materials: Laptop with PowerPoint & projector

At the end of the PowerPoint presentation, turn off the projector and have participants refer to the materials in their handout packet.

10 -12 minutes

3. First examine the scoring guide (if not done during PowerPoint presentation) (Facilitator option – focus on 2-3 traits rather than all 6)

4. Have participants read the problem presented to students, Tetra Dice. This is a problem involving probability. (Facilitator Option: Have participants try to solve problem.)

Note: Papers are in the correct order in the handout, but the numbers are not sequential! 5. Have participants read Sample Student Paper #16. Discuss general strengths & weaknesses. (Refer to Commentary 1)7. Tell participants that this paper scored all 4’s ( with a 5 in MS) – it meets the

standard.

5 – 7 minutes 10.

8. Have participants read Sample Student Paper# 2.9. Discuss strengths & weaknesses. (Refer to Commentary 2)

Tell participants that this paper scored 3’s and 2’s – and did not meet the standard.

5 – 7 minutes 11.Have participants read Sample Student Paper # 3.12.Discuss strengths & weaknesses. (Refer to Commentary 3)13.This paper is an example of student work that clearly exceeds the standard

and earned scores of 5’s & 6’s.

(Optional)5 – 7 minutes

14.Optional Question & Answer or Summary

Total = 30 – 45 minutes

Participant Packet Contains Printed copy of PowerPoint for note-taking if desired; Oregon’s Mathematics Problem-Solving Scoring Guide; Blank Problem-Solving Task: Tetra Dice Sample student papers & commentary: Tetra Dice

Sample Paper # 1 Sample Paper # 2 Sample Paper # 3

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Intro slideREFER TO HANDOUT: Participants PowerPoint Notes Pages

1

This slide sets goals for the presentation. These are general awareness and introductory knowledge goals.

This list shows essential skills required of students who entered grade 9 in the 2010-11 school year. 2014 is their projected graduation year. Students who graduate before 2014 or who take additional time to meet diploma requirements must still meet these essential skill requirements.

3

These bullets are part of the Essential Skill Definition, which can be found on the ODE website http://www.ode.state.or.us/teachlearn/certificates/diploma/essential-skills-definitions.pdf

4

This and next slide explain 3 ways students can show proficiency in Essential Skill of Mathematics for diploma.

The State Board adopted specific Advanced Placement and International Baccalaureate tests in March 2011. These are the tests and scores required to use AP or IB tests to demonstrate proficiency in Apply Mathematics

AP Statistics -- 3AP Calculus AB -- 3AP Calculus BC -- 3

IB Mathematics SL -- 4IB Mathematics HL -- 4IB Math Studies -- 4

5

This slide explains the third option for demonstrating proficiency in the Essential Skills. Students may complete 2 math work samples in any two of three mathematics strands: geometry, algebra, or statistics. For a work sample to be used to demonstrate proficiency in the Essential Skill of Mathematics, all 5 traits must receive a score of 4 or higher.

6

This slide explains the history and recent revision process for the Mathematics Scoring Guide.

7

This is now the Official Mathematics Scoring Guide and should be used for classroom and Essential Skills scoring.

8

REFER TO HANDOUT: Official Scoring Guide. Oregon scores 5 traits of Mathematics.

Presenter choose: briefly review scoring guide here or review it after PowerPoint presentation.

9

Explanation that work samples require equal rigor but provide a different format to demonstrate proficiency.

10

This shows the continuum of scores students may achieve. Point out that the Official Scoring Guide contains detailed descriptions of each score level for each trait.

11

Teachers use the Scoring guide to give students feedback during instruction and as formative assessment. When students learn the language and expectations of the scoring guide, their performance often improves.

12

The information presented in the Rumor Versus Reality section of the workshop can be found in Appendices L, M, and N of the 2010-11 Test Administration Manual. You can find the TAM on the ODE website at www.ode.state.or.us/go/tam

13

This slide and the next explain what is allowed in student revisions of work samples and teacher feedback.

14

Emphasize that teachers may provide feedback for revisions by highlighting on the Official Scoring Guide, and/or checking boxes on the Official Scoring Form.

15

16

It is important to stress that each work sample must “stand on its own.” That is --each must earn a score of 4 or higher in all traits. Scores cannot be combined from different work samples to meet the proficiency requirement.

Resources for assessing the Essential Skill of Mathematics will be posted as they become available at www.ode.state.or.us/go/worksamples Mathematics.Training in using the scoring guide and developing math work samples is available through the Professional Development Cadre of the Oregon Council of Teachers of Mathematics at www.octm.org.ODE will offer “Training of Trainer” WebEx sessions throughout the 2011-12 school year to help local district/school personnel deliver quality training. The training schedule and a calendar may be found at http://www.ode.state.or.us/search/page/?=2042

17

The Oregon State Board of Education has adopted the Common Core State Standards (CCSS) for Mathematics and English Language Arts. All Essential Skills resources will be aligned to the CCSS.

This quote from the CCSS provides a clear picture of how mathematically proficient students attack problem solving. The Oregon mathematics problem-solving model and scoring guide support this approach.

18

Resources are being developed and will continue to be added to these websites.

19

Move into the student work samples provided in the participants’ packets.

Facilitator option: You may wish to focus on just the first 2 traits from the scoring guide rather than having participants try to understand all 5 in a short time.

Let participants know that additional training sessions will be provided for more in-depth understanding of the scoring guide and development of work samples.

20

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 EvaluationOregon 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.


partially developed, and/or partially displayed.


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.

CommunicatingReasoningCoherently 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 andlead 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.

AccuracySupport 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 EvaluatingState 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.

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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?

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Sample 1 - Overview

Scores and Commentary: Sample Paper #1: Tetra Dice

MakingSense of the Task

5

Representingand Solving

the Task 4

Communicatingand

Reasoning4

Accuracy

4

Reflecting and

Evaluating4

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.

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Sample 2 - Overview



3


the Task 3

Communicatingand

Reasoning3

Accuracy

2

Reflecting and

Evaluating2

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.

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Sample 3 - Overview



5


the Task 6

Communicatingand

Reasoning6

Accuracy

5

Reflecting and

Evaluating5

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.

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Documents

Facilitator’s Packet for The Essential Skill of