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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
