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					                                                                                       Teacher: ________________________

1 Identifying the Purpose                                                              Subject Area: ____________________

   List the standards from National Core Curriculum or Arkansas Framework that will be the focus of the activity.

Ask Yourself These Questions
   1. What kinds of essential tasks, achievements, or other valued competencies am I missing with traditional
      instruction and paper and pencil exercises? What accomplishments of those who practice my discipline
      (mathematicians) are valued but left unmeasured by conventional instruction?

Intellectual Skills
   Place a check mark next to the skills needed to use the knowledge or content.

           COMMUNICATING                                    ORGANIZING
              Explaining               Programming           Classifying              Arranging
              Modeling                 Proposing             Categorizing             _____________
              Demonstrating            Drawing               Sorting                  _____________
              Graphing                 _____________         Ordering                 _____________
              Displaying               _____________         Ranking                  _____________
              Writing                  _____________
              Advising                 _____________

           MEASURING                                        PROBLEM SOLVING
            Counting                   Estimating          Stating questions          Assessing risks
            Calibrating                Forecasting         Identifying                Monitoring
            Rationing                  Defending             problems                  _____________
            Appraising                 _____________       Developing                 _____________
            Weighing                   _____________         hypotheses                _____________
            Balancing                  _____________       Interpreting               _____________
            Guessing                   _____________

           INVESTIGATING                                    DECISION MAKING
            Gathering                  Hypothesizing       Weighing                   Assessing risks
              references                _____________         alternatives              Monitoring
            Interviewing               _____________       Evaluating                 _____________
            Using                      _____________       Choosing                   _____________
              references                _____________       Supporting                 _____________
            Experimenting                                   Electing                   _____________
                                                             Adopting
         * For further information:                                                                 Figure 1.1
           Educational Testing and Measurement. Page 168.

                                                                                                                 Page 1 of 7
1 Identifying the Purpose
Mathematical Practices
Identify the mathematical practices the student will use and choose the specific behaviors.

    Make sense of problems and persevere in solving                       Construct viable arguments and critique the
     them.                                                                  reasoning of others # 3
          Explain to themselves the meaning of a problem                        Understand and use stated assumptions, definitions,
          Look for entry points to the solution.                                 and previously established results in constructing
          Analyze givens, constraints, relationships and goals.                  arguments.
          Make conjectures about the form and meaning of the                    Make conjectures and build a logical progression of
           solution.                                                              statements to explore the truth of conjectures.
          Plan a pathway to a solution rather than just jumping into a          Analyze situations by breaking them into cases.
           solution attempt.                                                     Recognize and use counterexamples.
          Consider analogous problems.                                          Justify conclusions and communicate to others.
          Try special cases and simpler forms of the original problem           Respond to arguments of others.
           in order to gain insight into the solution.                           Reason inductively about data, making plausible
          Monitor and evaluate progress and change course if                     arguments that take into account the context from
           necessary.                                                             which the data arose.
          Transform algebraic expressions or change the view                    Compare the effectiveness of two plausible arguments.
           window on the graphing calculator to get the information              Distinguish correct logic or reasoning from that which is
           they need.                                                             flawed, and explain the flaw.
          Explain correspondences between equations, verbal                     Elementary students can construct arguments using
           descriptions, tables, and graphs, or draw diagrams of                  concrete referents such as objects, drawings, diagrams,
           important features and relationships, graph data, and                  and actions. Such arguments can make sense and be
           search for regularity or trends.                                       correct without being formal or generalized until a later
          Younger students might rely on using concrete objects or               grade.
           pictures to help conceptualize and solve a problem.                   Determine domains to which an argument applies.
          Check answers using a different method, and continually               Listen or read the arguments of others and decide
           ask, “Does this make sense?”                                           whether they make sense.
          Understand the approaches of others to solving complex                Ask useful questions to clarify or improve the
           problems.                                                              arguments.
          Identify correspondences between different approaches.
                                                                           Model with mathematics # 4
    Reason abstractly and 1uantitatively # 2                                    Apply mathematics to solve problems arising in
          Make sense of quantities and their relationships in                    everyday life, society, and the workplace.
           problem situations.                                                   In early grades, write an addition equation to describe a
          Have the ability to de-contextualize or abstract a given               situation.
           situation, represent it symbolically and manipulate the               Middle grades might apply proportional reasoning to
           representing symbols as if they have a life of their own               plan a school event or analyze a problem in the
           without necessarily attending to referents.                            community.
          Have the ability to contextualize, to pause as needed                 High school might use geometry to solve a design
           during the manipulation process in order to probe into the             problem or use a function to describe how one quantity
           referents for the symbols involved.                                    of interest depends on another.
          Create a coherent representation of the problem at hand.              Make assumptions and approximations to simplify a
          Consider the units involved.                                           complicated situation, realizing that these may need
          Attend to the meaning of quantities, not just how to                   revision later.
           compute them.                                                         Identify important quantities in a practical situation and
          Know and flexibly use different properties of operations               map their relationships using such tools as diagrams,
           and objects.                                                           two-way tables, graphs, flowcharts, and formulas.
                                                                                 Analyze those relationships mathematically to draw
                                                                                 Interpret mathematical results in the context of the
                                                                                 Reflect on whether the results make sense, possibly
                                                                                  improving the model if it has not served its purpose.

                                                                                                                                  Page 2 of 7
1 Identifying the Purpose
  Use appropriate tools strategically # 5                                Look for and make use of structure # 7
         Consider the available tools when solving a mathematical              Look closely and discern a pattern for structure.
          problem.                                                              Young students realize that three and seven more is the
         Tools might include paper, pencil, concrete models, ruler,             same amount as seven and three more.
          protractor, calculator, spreadsheet, computer algebra                 Young students may sort a collection of shapes
          system, statistical package, or dynamic geometry software.             according to how many sides the shapes have.
         Make sound decisions about when each of these tools                   Older students will see 7 X 8 equals the well
          might be helpful, recognizing both the functions and                   remembered (7 x 5) + (7 X 3), in preparation for learning
          solutions generated using a graphing calculator.                       about the distributive property.
         Detect possible errors by strategically using estimation and          Older students can see that in the expression
          other mathematical knowledge.                                         x + 9x + 14, the 14 as 2 x 7 and the 9 as 2 + 7.
         Use technology to enable them to visualize the results of             Recognize significance of an existing line in a geometric
          varying assumptions, explore consequences, and compare                 figure and can use the strategy of drawing an auxiliary
          predictions with data.                                                 line for solving problems.
         Identify relevant external mathematical resources, such as            Can step back for an overview and shift perspective.
          digital content located on a website, and use them to pose            Can see complicated things, such as some algebraic
          or solve problems.                                                     expressions as single objects or as being composed of
         Use technological tools to explore and deepen their                    several objects. For example, they can see 5 - 3(x - y)
          understanding of concepts.                                             as 5 - a positive number times a square and use that to
                                                                                 realize that its value cannot be more than 5 for any real
  Attend to precision # 6                                                       numbers x and y.
    Try to communicate precisely with others.
    Try to use clear definitions in discussion with others and in        Look for and express regularity in repeated
          their own reasoning.                                             reasoning # 8
         State the meaning of the symbols they choose, including               Notice if calculations are repeated.
          using the equal sign consistently and appropriately.                  Look for general methods and shortcuts.
         Specify units of measure, and labeling axes to clarify the            Upper elementary might notice that when dividing 25
          correspondence with quantities in a problem.                           by 11 that they are repeating the same calculations
         Calculate accurately and efficiently, express numerical                over and over again, and conclude they have a
          answers with a degree of precision appropriate for the                 repeating decimal.
          problem context.                                                      Middle school students might notice that if they check
         Young students give carefully formulated explanations to               to see whether points are on the line through (1, 2) with
          each other.                                                            slope 3, they might abstract the equation (y - 2)/(x - 1) =
         Older students have learned to examine claims and make
                                                                                Noticing the regularity in the way terms cancel when
          explicit use of definitions.
                                                                                 expanding might lead older students to the general
                                                                                 formula for the sum of a geometric series.
                                                                                Maintain oversight of a process, while attending to
                                                                                Continually evaluate the reasonableness of
                                                                                 intermediate results.

 The Standards for Mathematical Content

 are a balanced combination of procedure and understanding.              In this respect, those content standards, which set an
 Expectations that begin with the word “understand” are often            expectation of understanding, are potential “points of
 especially good opportunities to connect the practices to the           intersection” between the Standards for Mathematical
                                                                         Content and the Standards for Mathematical Practice. These
 content. Students who lack understanding of a topic may rely
                                                                         points of intersection are intended to be weighted toward
 on procedures too heavily. Without a flexible base from which to        central and generative concepts in the school mathematics
 work, they may be less likely to consider analogous problems,           curriculum that most merit the time, resources, innovative
 represent problems coherently, justify conclusions, apply the           energies, and focus necessary to qualitatively improve the
 mathematics to practical situations, use technology mindfully to        curriculum, instruction, assessment, professional
 work with the mathematics, explain the mathematics accurately           development, and student achievement in mathematics.
 to other students, step back for an overview, or deviate from a
 known procedure to find a shortcut. In short, a lack of
                                                                                                                                 Figure 1.3
 understanding effectively prevents a student from engaging in
 the mathematical practices.

                                                                                                                                  Page 3 of 7
2 Designing the Context
Ideas & Brainstorming

   1. What real-world projects and tasks can be adapted to school instruction?

   2. What roles do professional mathematicians acquire that learners can re-create in the classroom?
      (mathematical practices)

Product & Performance:                Examples of how will students might communicate their learning and

                           Products                                                    Performances
          Multi-media presentations                                  Presentations
          Poems                                                      Debates
          Research paper                                             Oral presentation
          Lab report                                                 Perform a construction
          Prototype                                                  Following a recipe
          Model                                                      Job responsibilities
          Business letters                                           Following a set of procedures (i.e. calculator
          Maps                                                        procedures)
          Graphs                                                     Use of specialized equipment
          Essays                                                     Defend a solution
          Chart                                                      _______________________________________
          Exhibit                                                    _______________________________________
          Drawing
          Multimedia product
           (i.e. movie, PowerPoint, animation, graphic, computer
           program, song, video game, interactive spreadsheet)
        ____________________________________
        ____________________________________                                                         Figure 2.2

Define Performance Task
      Describe the chosen performance task in 1-5 sentences.

                                                                                                                   Page 4 of 7
3 Specifying the Scoring Rubrics
Identify Relevant Student Accomplishments & Criteria
List student accomplishments & criteria identified in steps 1 and 2 as intellectual skills (Figure 1.1), mathematical
practices (Figure 1.3), products (Figure 2.2), and performances (Figure 2.2).

  Intellectual Skills – From Step 1                           Mathematical Practices – From Step 1

  Products – From Step 2                                       Performances – From Step 2

                                                                                                                   Figure 3.1

                                                                                                                  Page 5 of 7
3 Specifying the Scoring Rubrics
Choose a Rubric Type
Each of the three scoring systems has strengths and weaknesses.

       Ease of construction: the time involved in coming up with a comprehensive list of the important aspects or traits
       of successful and unsuccessful performance. Checklists, for example, are particularly time-consuming, while
       holistic scoring is not.
       Scoring efficiency: the amount of time required to score various aspects of the performance and calculate these
       scores as an overall score.
       Reliability: the likelihood that two raters will independently come up with a similar score, or the likelihood that
       the same rater will come up with a similar score on two separate occasions.
       Defensibility: the ease with which you can explain your score to a student or parent who challenges it.
       Quality of feedback: the amount of information the scoring system gives to learners or parents about the
       strengths and weaknesses of the performance.
                                                                                                                           Figure 3.2
Circle the relevant considerations for your performance assessment. Use this information in choosing the appropriate
rubric type.

                         Ease of            Scoring
                                                         Reliability   Defensibility   Feedback    More Suitable For
                       Construction        Efficiency

       Checklist           low             moderate         high           high          high          procedures
                                                                                                    attitudes, products,
     Rating Scales       moderate          moderate       moderate      moderate       moderate
                                                                                                         social skills
                                                                                                  products and processes
    Holistic Scoring       high              high           low            low           low
                                                                                                                  Figure 3.3
Choose the scoring system best suited for the type of accomplishments being measured.

    Checklists – List of behaviors, traits or characteristics best suited for complex behaviors or performances divided
     into clearly defined specific actions. Can be scored whether student is present or absent. (i.e. focusing a
     microscope, scientific experiment)
    Rating Scales – Used for aspects of a complex performance that does not lend itself to yes/no or present/absent
     judgments. (i.e. dancing, singing)
    Holistic Scoring – Used when estimating overall quality and assigns a numerical value to that quality rather than
     addition or omission of specifics. Keeps rater focused on purpose of the performance test. (i.e. term papers,
     dance or musical creation, extended essays)
    Combination – Used when students are asked to demonstrate achievements through a variety of primary traits
     and accomplishments (i.e. cooperation + research + delivery)
                                                                                                                           Figure 3.4
Define Cut Score or Benchmark
The minimum level at which you would want most students to perform is your cut score or benchmark.
How well should most students perform?

                                                                                                                    Page 6 of 7
4 Specifying the Constraints

Specify Constraints
Answer the following questions regarding the performance task.

   1. Time. How much time should a learner have to prepare, rethink, revise, and finish a task?

   2. Reference material. Should learners be able to consult dictionaries, textbooks, or notes as they
      perform a task?

   3. Other people. May learners ask for help from peers, teachers, and experts as they perform a
      task or complete a project?

   4. Equipment. May learners use computers or calculators to help them solve problems?

   5. Prior knowledge of the task. How much information about the task situation should learners
      receive in advance?

   6. Scoring criteria. Should learners know the standards by which the teacher will score the

   7. What kinds of constraints authentically replicate the constraints and opportunities facing the
      performer in the real world?

   8. What kinds of constraints tend to bring out the best in apprentice performers and producers?

                                                                                            Page 7 of 7

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