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Teacher: ________________________ 1 Identifying the Purpose Subject Area: ____________________ Standards 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. ACQUIRING INFORMATION ORGANIZING & USING INFORMATION 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 conclusions. Interpret mathematical results in the context of the situation. 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 2 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 2 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) = 3. 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 details. 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 accomplishments? 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 assessment? 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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posted: | 2/8/2013 |

language: | English |

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