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CFN 609Professional Development | March 8, 2012

RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services

Elementary SchoolPerformance Tasks for Mathematics

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Make 25¢ with:

1 coin 2 coins3 coins 4 coins…

…23 coins 24 coins25 coins

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Focus on Performance Tasks

AGENDA• Tasks: Criteria for Judging Tasks• Metacognition• Aligning Tasks to the CCLS• Cognitive Demand• Task Implementation• Resources

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Some Criteria for Considering Tasks

• Level of challenge: accessible to the struggling, challenging enough for the advanced

• Multiple points of entry• Various solution pathways• Identifying the math concept

involved with, and strengthened by, working on the task

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Fraction Value

A and B are two different numbers selected from the first twenty counting numbers, 1 through 20 inclusive. What is the largest value that fraction A × B can have? A – B

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Some Criteria for Considering Tasks

• Opportunities to exercise the standards for mathematical practice

• Opportunities to bring out student misconceptions, which can be identified and addressed

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What number can replace the square to make the statement true?

5 × 11 = + 12

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Research on Retention of Learning: Shell Center: Swan et al

Misconception Learning verses Remedial Learning: Test Scores

10.4

17.819.1

7.9

15.8

12.7

0

5

10

15

20

25

Pre-test Post-test Delayed Test

Students who weretaught by addressingmisconceptions

Students who weretaught using remedialmeasures

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Three Responses to a Math Problem

1. Answer getting2. Making sense of the problem

situation3. Making sense of the

mathematics you can learn from working on the problem

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Bob, Jim and Cathy each have some money. The sum of Bob's and Jim's money is $18.00. The sum of Jim's and Cathy's money is $21.00. The sum of Bob's and Cathy's money is $23.00.

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Standards for Mathematical Practice

1 Make sense of problems and persevere in solving them.

2 Reason abstractly and quantitatively.3 Construct viable arguments and critique the

reasoning 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.

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Math Olympiad for Elementary and Middle School

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Math Olympiad 76-2

HE HE+HE AH

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Different letters represent different digits, and E is twice H. What two-digit number does AH, the sum, represent?

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Working on a Task: Four Phases

1.Understand2.Plan3.Try it4.Look back

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Some Strategies for Approaching a Task

• Make an organized list• Work backward• Look for a pattern• Make a diagram• Make a table• Use trial-and-error• Consider a related but simpler

problem

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Pizza PartyTwelve friends were having a party and

ordered seven small pizza pies. How can the pies be divided so that each friend gets exactly the same amount of pizza? No pie can be cut into more than four pieces.

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And Some More Strategies

• Consider extreme cases• Adopt a different point of view• Estimate• Look for hidden assumptions• Carry out a simulation

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Metacognition

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A Problem (DO NOT SOLVE)

Make as many rectangles as you can with an area of 24 square units. Use only whole numbers for the length and width. Sketch the rectangles, and write the dimensions on the diagrams. Write the perimeter of each one next to the sketch.

What questions do you ask yourself as you encounter this problem?

How do these questions help you to develop a solution approach?

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Meta-Cognition

• Thinking about thinking.• The usually-unconscious process

of cognition.• Habit of mind: taking the time to

look back and reflect

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Meta-cognition implications for lessons.

• Make thinking public• Use multiple representations• Offer different approaches to solution• Ask questions about the problem posed.• Set a context, define the why of the

problem• Focus students on their thinking, not the

solution• Solve problems with partners• Prepare to present strategies

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Aligning Tasks to the Common Core Learning Standards

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Comparing Two Mathematical TasksTASK AMAKING CONJECTURES Complete the conjecture based

on the pattern you observe in the specific cases.

1. Conjecture: The sum of any two odd numbers is ______?

1 + 1 = 2 7 + 11 = 181 + 3 = 4 13 + 19 = 323 + 5 = 8 201 + 305 = 506

2. Conjecture: The product of any two odd numbers is ____?

1 x 1 = 1 7 x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = 15 201 x 305 = 61,305

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Comparing Two Mathematical TasksTASK BMAKING CONJECTURES Complete the conjecture based on

the pattern you observe in the specific cases.Then explain why the conjecture is always true or

show a case in which it is not true.1. Conjecture: The sum of any two odd numbers is

______?1 + 1 = 2 7 + 11 = 181 + 3 = 4 13 + 19 = 323 + 5 = 8 201 + 305 = 506

2. Conjecture: The product of any two odd numbers is ____? 1 x 1 = 1 7 x 11 = 77

1 x 3 = 3 13 x 19 = 2473 x 5 = 15 201 x 305 = 61,305

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Draw a Picture

Every odd number (like 11 and 13) has one loner number. Add the two loner numbers and you will get an even number (24). Now add all together the loner numbers and the other two (now even) numbers.

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Build a Model

If I take the numbers 5 and 11 and organize the counters as shown, you can see the pattern.

You can see that when you put the sets together (add the numbers), the two extra blocks will form a pair and the answer is always even. This is because any odd number will have an extra block and the two extra blocks for any set of two odd numbers will always form a pair.

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Use a Pattern

When we count, odd numbers alternate with even: 1, 2, 3, 4, 5… or odd, even, odd, even, odd…. If we start counting on an odd number and we count an even number of spaces forward, we land on another odd number, for example, start on 5 and count forward by 2 or 4: we get 7 or 9. But if we start on an odd number and count an odd number of spaces forward we always land on an even number, for example start on 5 and count forward by 3 or 5: we get 8 or 10.

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Logical Argument

An odd number = [an] even number + 1. e.g. 9 = 8 + 1

So when you add two odd numbers you are adding an even no. + an even no. + 1 + 1. So you get an even number. This is because it has already been proved that an even number + an even number = an even number.

Therefore as an odd number = an even number + 1, if you add two of them together, you get an even number + 2, which is still an even number.

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Use Algebra

  If a and b are odd integers, then a and b can be written a = 2m + 1 and b = 2n + 1, where m and n are other integers.

If a = 2m + 1 and b = 2n + 1, then a + b = 2m + 2n + 2.

If a + b = 2m + 2n + 2, then a + b = 2(m + n + 1).

If a + b = 2(m + n + 1), then a + b is an even integer.

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Comparing Two Mathematical Tasks

How are the two versions of the task the same and how are they different?

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Tasks A and B

SameBoth ask

students to complete a conjecture about odd numbers based on a set of finite examples that are provided

Different Task B asks students to develop an

argument that explains why the conjecture is always true (or not)

Task A can be completed with limited effort; Task B requires considerable effort – students need to figure out WHY this conjecture holds up

The amount of thinking and reasoning required

The number of ways the problem can be solved

The range of ways to enter the problem

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Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively3. Construct viable arguments and critique the

reasoning of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated

reasoning

Common Core State Standards for Mathematics, 2010, pp.6-7

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Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively3. Construct viable arguments and critique the

reasoning of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated

reasoning

Common Core State Standards for Mathematics, 2010, pp.6-7

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Button Pattern

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Analysis of Tasks

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Levels of Cognitive DemandLower-level• Memorization• Procedures without

connectionsHigher-level• Procedures with connections• Doing mathematics

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Norman Webb’s Depth of Knowledge

• Level 1: Recall and Reproduction

• Level 2: Skills and Concepts• Level 3: Strategic Thinking• Level 4: Extended Thinking

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Slide 39

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Cognitive ComplexityBLOOMS TAXONOMY WEBB’S DEPTH OF KNOWLEDGE

KNOWLEDGE- the recall of specifics and universals, involving more than bringing to mind the appropriate material

RECALL- recall of fact, information, or procedure

COMPREHENSION- Ability to process knowledge on a low level such that the knowledge can be reproduced or communicated without a verbatim repetition

APPLICATION- the use of abstractions in concrete situations

APPLICATION of SKILL / CONCEPT- use of information, conceptual knowledge, procedures of two or more steps

ANALYSIS- the breakdown of a situation into its component parts

STRATEGIC THINKING- requires reasoning, developing a plan or sequence of steps; has some complexity; more than one possible answer

SYNTHESIS & EVALUATION- putting together elements and parts to form a while, then making value judgments about the method

EXTENDED THINKING- requires an investigation; time to think and process multiple conditions of the problem / task; non-routine manipulations

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What is Depth of Knowledge?

• A language system used to describe different levels of complexity

• A framework for evaluating curriculum, objectives, and assessments so they can be studied for alignment

• Focuses on content and cognitive demand of test items, instructional strategies, and performance objectives

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DOK Levels

Level 1 measures Recall at a literal level.Level 2 measures a Skill or Concept at an interpretive level.Level 3 measures Strategic Thinking at an evaluative level.Level 4 measures Extended Thinking and Reasoning

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DOK Level 1: Mathematics

• Recall and recognize information such as facts, definitions, theorems, terms, formulas or procedures

• Solve one-step problems, apply formulas, and perform well-defined algorithms

• Demonstrate an understanding of fundamental math concepts

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DOK 1

What is the place value of 9 in the number 74.295?

A. hundreds

B. tenths

C. hundredths

D. thousandths

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DOK Level 2: Mathematics

The cognitive demands are more complex than inLevel 1.

Engage in mental processing beyond recall or habitual response:

• Determine how to approach a problem• Solve routine multi-step problems• Estimate quantities, amounts, etc.• Use and manipulate multiple formulas, definitions,

theorems, or a combination of these• Collect, organize, classify, display, and compare

data• Extend a pattern

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DOK 2

Draw the next figure in the following pattern:

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DOK 2

On a road trip from Georgia to Oklahoma, Maria determined that she would cover about 918 miles. What speed would she need to average to complete the trip in no more than 15 hours of driving time?

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DOK Level 3: Mathematics

Engage in abstract, complex thinking• Determine which concepts to use in solving complex

problems• Use multiple concepts to solve a problem• Reason, plan, and use evidence to explain and justify

thinking• Make conjectures• Interpret information from complex graphs• Draw conclusions from logical arguments

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DOK 3

Find the next three items in the pattern and give the rule for following the pattern of numbers:

1, 4, 3, 6, 5, 8, 7, 10…

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DOK 3A local bakery celebrated its one year anniversary on

Saturday. On that day, every 4th customer received a free cookie. Every 6th customer received a free muffin.

A. Did the 30th customer receive a free cookie, free muffin, both, or neither? Show or explain how you got your answer.

B. Casey was the first customer to receive both a free cookie and a free muffin. What number customer was Casey? Show or explain how you got your answer.

C. Tom entered the bakery after Casey. He received a free cookie only. What number customer could Tom have been? Show or explain how you got your answer.

D. On that day the bakery gave away a total of 29 free cookies. What was the total number of free muffins the bakery gave away on that day? Show or explain how you got your answer.

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DOK Level 4: Mathematics

Extended Thinking/Reasoning requires complex reasoning, planning, developing, and thinking most likely over an extended period of time.

The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking.

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DOK 4

Take the bakery problem and ask students to write equations. Solve the system of equations and explain why it satisfies the conditions. Determine what any customer might receive, i.e. the 1000th customer.

Refer back to the task on equations in the form y=mx + b.

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DOK 4

George Smith charges $4.00 an hour for his services to walk/feed/water outdoor pets when his clients take weekend trips. Charles Wood charges $45.00 for weekly lawn care – mowing, weeding, raking. Marty Rogers cleans and organizes items in sheds/garages at the rate of $6.50 per hour. If each of these boys’ families needs the services of the other two boys, determine a fair way (as fair as possible) to arrange for services to be rendered among the three families without the exchange of money.

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Working on Tasks and Unit Plans

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Implementationof Tasks

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Standards for Mathematical Practice

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“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.”

Stein, Smith, Henningsen, & Silver, 2000

“The level and kind of thinking in which students engage determines what they will learn.”

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

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Opportunities for all students to engage in challenging tasks?

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• Examine tasks in your instructional materials:

– Higher cognitive demand? – Lower cognitive demand?• Where are the challenging tasks?• Do all students have the opportunity

to grapple with challenging tasks?• Examine the tasks in your

assessments: – Higher cognitive demand? – Lower cognitive demand?

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The nature of tasks used in the classroom…

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will impact student learning!

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But WHAT TEACHERS DO with the tasks matters too!

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The Mathematical Tasks Framework

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Ideas on Implementation

• Focus on the mathematics, not just getting the answer

• Maintain the mathematical rigor• Habits of mind before: understanding the situation,

deciding on a plan or strategy• Habits of mind after: reflecting on the thinking, does

the answer make sense, was this the best strategy?• Explaining the thinking• Comparing and considering other solution pathways

and strategies

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Students’ beliefs about their intelligence

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• Fixed mindset: – Avoid learning situations if they might

make mistakes – Try to hide, rather than fix, mistakes or

deficiencies – Decrease effort when confronted with

challenge• Growth mindset: – Work to correct mistakes and deficiencies – View effort as positive; increase effort

when challenged Dweck, 2007

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Students can develop growth mindsets

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• Explicit instruction about the brain, its function, and that intellectual development is the result of effort and learning has increased students’ achievement in middle school mathematics.

• Teacher praise influences mindsets – Fixed: Praise refers to intelligence – Growth: Praise refers to effort,

engagement, perseverance

NCSM Position Paper #7Promoting Positive Self-Beliefs

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Changing View of Assessment:

Assessment for Learning

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Formative Assessment Strategies

1. Clarifying, sharing and understanding goals for learning and criteria for success with learners

2. Engineering effective classroom discussions, questions, activities and tasks that elicit evidence of students’ learning

3. Providing feedback that moves learning forward

4. Activating students as owners of their own learning

5. Activating students as learning resources for one another

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When teachers start from what it is they want students to know and design their instruction backward from that goal, then instruction is far more likely to be effective.

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Two-Stage Process:

• Clarifying the learning goals• Establishing success criteria

“…discrepancies in beliefs about what it is that counts as learning in mathematics classrooms may be a significant factor in the achievement gaps observed…”

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Ambiguities inherent in mathematics

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Students who do not understand what is important and what is not important will be at a very real disadvantage

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Eliciting evidence of student learning

• By crafting questions that explicitly build in the undergeneralizations and overgeneralizations that students are known to make

• The teacher is able to address students’ confusion during the lesson

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Feedback that moves the learner forward

• Feedback is usually “ego-involving”

• Grades with comments are no more effective than grades alone, and much less effective than comments alone

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Finding errors for themselves

• There are five answers here that are incorrect. Find them and fix them.

• The answer to this question is __ Can you find a way to work it out?

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Identify where students might use and extend their existing knowledge

• You’ve used substitution to solve all these simultaneous equations. Can you use elimination?

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Encourage pupils to reflect

• You used two different methods to solve these problems. What are the advantages and disadvantages of each?

• You have answered ___ well. Can you make up your own more difficult problems?

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Have students discuss their ideas with others

• You seem to be confusing sine and cosine. Talk to Katie about how to work out the difference.

• Compare your work with Ali and write some advice to another student tackling this topic for the first time.

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Activating students as owners of their own learning

• Student motivation and engagement: cost vs. benefits

• “It’s better to be thought lazy than dumb.”

• Focus on personal growth rather than a comparison with others

• Green, yellow red “traffic lights”

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Activating students as learning resources for one another

• Group goals, so that students are working as a group, not just in a group

• Individual accountability• Feedback from a peer: two stars and

a wish• Internalize the learning intentions

and success criteria in the context of someone else’s work

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Assessment for Learning

Teachers use assessment, minute-by-minute and day-by-day, to adjust their instruction to meet their students’ learning needs.

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Dylan Wiliam

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Assessment for Learning

Change of focus from what the teacher is putting into the lesson, to what the learner is getting out of it.

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Danielson’s Framework for TeachingComponents of Professional

PracticeDomain 1: Planning and PreparationComponent 1a: Demonstrating Knowledge of

Content and PedagogyComponent 1b: Demonstrating Knowledge of

StudentsComponent 1c: Selecting Instructional GoalsComponent 1e: Designing Coherent

InstructionComponent 1f: Assessing Student Learning

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Danielson’s Framework for TeachingComponents of Professional PracticeDomain 2: The Classroom EnvironmentComponent 2a: Creating an Environment of Respect and

RapportComponent 2b: Establishing a Culture for LearningComponent 2c: Managing Classroom ProceduresComponent 2d: Managing Student Behavior

 Domain 3: InstructionComponent 3a: Communicating Clearly and AccuratelyComponent 3b: Using Questioning and Discussion

TechniquesComponent 3c: Engaging Students in LearningComponent 3d: Providing Feedback to StudentsComponent 3e: Demonstrating Flexibility and

Responsiveness81

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Math Tasks

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Bananas

If three bananas are worth two oranges, how many oranges are 24 bananas worth?

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Math Olympiad 60-5

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A fast clock gains one minute per hour and a slow clock loses two minutes per hour. You set both clocks to the correct time, and less than 24 hours later, the fast clock shows 9:00 o’clock at the same moment that the slow clock shows 8:00 o’clock. What is the correct time at that moment?

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You were supposed to add A and B. By accident, you subtracted B from A and got 4. This number is different from the correct answer by 12. What is A?

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Tennis Tournament

A tennis tournament has 50 contestants, with these rules: no tie games and the loser of each game is eliminated, the winner goes on to play in the next round. How many games are needed to determine a champion?

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Resources

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Web Links

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Common Core State Standards:http://www.corestandards.org/ Common Core Tools:http://commoncoretools.wordpress.com/ New York State Site for Teaching and Learning Resourceshttp://www.engageny.org/ PARCC:http://www.parcconline.org/ Inside Mathematics:http://www.insidemathematics.org/index.php/home Mathematics Assessment Project:http://map.mathshell.org/materials/index.php Common Core Library:http://schools.nyc.gov/Academics/CommonCoreLibrary/default.htm Mathematical Olympiads for Elementary and Middle Schools:www.moems.org

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Pearson Professional Development

pearsonpd.com

RONALD SCHWARZ, facilitatorRonald.schwarz@yahoo.com