moving ahead with the common core learning standards for mathematics

Copyright © 2010 Pearson Education, Inc.or its affi

liate(s). All rights reserved.

CFN 602Professional Development | February 17, 2012

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

Moving Ahead with the Common Core Learning Standards for Mathematics



Block Stack25 layers of blocks are stacked; the top four layers are shown. Each layer has two fewer blocks than the layer below it. How many blocks are in all 25 layers?

Math Olympiad for Elementary and Middle Schools



AGENDA• Standards for Math Content:

Conceptual Shifts• What’s Different• Math Performance Tasks• Formative assessment• Resources

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liate(s). All rights reserved.4

What are Standards?•Standards define what students should understand and be able to do.

•The US has been a jumble of 50 different state standards. Race to the bottom or the top?

•Any country’s standards are subject to periodic revision.

•But math is more than a list of topics.



NAEP & NY STATE TEST RESULTSNYC MATH PERFORMANCE

PERCENT AT OR ABOVE PROFICIENT

NAEP NY State Test NAEP NY State Test

2003 2009 2003 2009 2003 2009 2003 2009

4th Grade 8th Grade

DESPITE GAINS, ONLY 39% OF NYC 4TH GRADERS AND 26% OF 8TH GRADERS ARE PROFICIENT ON NATIONAL MATH TESTS



What Does “Higher Standards” Mean?

•More Topics? But the U.S. curriculum is already cluttered with too many topics.

•Earlier grades? But this does not follow from the evidence. In Singapore, division of fractions: grade 6 whereas in the U.S.: grade 5 (or 4)



Lessons Learned• TIMSS: math performance is being compromised

by a lack of focus and coherence in the “mile wide. Inch deep” curriculum

• Hong Kong students outscore US students in the grade 4 TIMSS, even though Hong Kong only teaches about half the tested topics. US covers over 80% of the tested topics.

• High-performing countries spend more time on mathematically central concepts: greater depth and coherence. Singapore: “Teach less, learn more.”

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Common Core State Standards Evidence Base

8

For example: Standards from individual high-performing countries and provinces were used to inform content, structure, and language.

Mathematics1. Belgium (Flemish)2. Canada (Alberta)3. China4. Chinese Taipei5. England6. Finland7. Hong Kong8. India9. Ireland10.Japan11.Korea12.Singapore

English language arts1. Australia

• New South Wales• Victoria

2. Canada• Alberta• British Columbia• Ontario

3. England4. Finland5. Hong Kong6. Ireland7. Singapore



Why do students have to do math problems?1.To get answers because

Homeland Security needs them, pronto

2.I had to, why shouldn’t they?

3.So they will listen in class4.To learn mathematics



Answer Getting vs. Learning Mathematics

United StatesHow can I teach my kids to get the

answer to this problem?Use mathematics they already know. Easy, reliable,

works with bottom half, good for classroom management.

JapanHow can I use this problem to teach

mathematics they don’t already know?

10



Three Responses to a Math Problem1. Answer getting2. Making sense of the problem

situation3. Making sense of the

mathematics you can learn from working on the problem



Answer Getting

Getting the answer one way or another and then stopping

Learning a specific method for solving a specific kind of problem (100 kinds a year)



Butterfly method



Use butterflies on this TIMSS item

1/2 + 1/3 +1/4 =



Foil FOIL• (a + b)(c +d) = ac + bc + ad + bd• Use the distributive property• This IS the distributive property when a

is a sum: a(x + y) = ax + ay• Sum of products = product of sums• It works for trinomials and polynomials

in general

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Answers are a black hole:hard to escape the pull• Answer getting short circuits

mathematics, especially making mathematical sense

• High-achieving countries devise methods for slowing down, postponing answer getting



A dragonfly can fly 50 meters in 2 seconds.

What question can we ask?

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Rate × Time = Distance

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Posing the problem• Whole class: pose problem, make sure

students understand the language, no hints at solution

• Focus students on the problem situation, not the question/answer game. Hide question and ask them to formulate questions that make the situation into a word problem

• Ask 3-6 questions about the same problem situation; ramp questions up toward key mathematics that transfers to other problems



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.



What problem to use?• Problems that draw thinking toward the

mathematics you want to teach. NOT too routine, right after learning how to solve

• Ask about a chapter: what is the most important mathematics students should take with them? Find problems that draw attention to this math

• Near end of chapter, external problems needed, e.g. Shell Centre



What do we mean by conceptual coherence?Apply one important concept in 100 situations

rather than memorizing 100 procedures that do not transfer to other situations: – Typical practice is to opt for short-term

efficiencies, rather than teach for general application throughout mathematics.

– Result: typical students do OK on unit tests, but don’t remember what they ‘learned’ later when they need to learn more mathematics

– Use basic “rules of arithmetic” (same as algebra) instead of clutter of specific named methods



Teaching against the test

3 + 5 = [ ]3 + [ ] = 8[ ] + 5 = 8

8 - 3 = 58 - 5 = 3



Anna bought 3 bags of red gumballs and 5 bags of white gumballs. Each bag of gumballs had 7 pieces in it. Which expression could Anna use to find the total number of gumballs she bought?

A. (7 × 3) + 5 = B. (7 × 5) + 3 = C. 7 × (5 + 3) = D. 7 + (5 × 3) =



Math StandardsMathematical Practice: varieties of expertise

that math educators should seek to develop in their students.

Mathematical Content:Mathematical Performance: what kids should

be able to do.Mathematical Understanding: what kids

need to understand.

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Standards for Mathematical Content

Organization by Grade Bands and Domains

K–5 6–8 High SchoolCounting and Cardinality

Operations and Algebraic Thinking

Number and Operations in Base Ten

Number and Operations—Fractions

Measurement and Data

Geometry

Ratios and Proportional Relationships

The Number System

Expressions and Equations

Geometry

Statistics and Probability

Functions

Number and Quantity

Algebra

Functions

Modeling

Geometry

Statistics and Probability

(Common Core State Standards Initiative 2010)



Progressions within and across Domains

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Daro, 2010



Math ContentGreater focus – in elementary school, on

whole number operations and the quantities they measure, specifically:

Grades K-2 Addition and subtractionGrades 3-5 Multiplication and division and

manipulation and understanding of fractions (best predictor algebraic performance)

Grades 6-8 Proportional reasoning, geometric measurement and introducing expressions, equations, linear algebra

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Why begin with unit fractions?

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Unit Fractions

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Units are things that you count

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•Objects•Groups of objects•1•10•100•¼ unit fractions•Numbers represented as expressions

Daro, 2010



Units add up• 3 apples + 5 apples = 8 apples• 3 ones + 5 ones = 8 ones• 3 tens + 5 tens = 8 tens• 3 inches + 5 inches = 8 inches• 3 tenths + 5 tenths = 8 tenths• 3(¼) + 5(¼) = 8(¼)• 3(x + 1) + 5(x+1) = 8(x+1)

Daro, 201034



There are 125 sheep and

5 dogs in a flock.How old is the shepherd?



A Student’s Response

There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

125 x 5 = 625 extremely big125 + 5 = 130 too big125 - 5 = 120 still big125 5 = 25 That works!



How CCLS support changeThe new standards support improved

curriculum and instruction due to increased:– FOCUS, via critical areas at each grade

level– COHERENCE, through carefully developed

connections within and across grades– RIGOR, including a focus on College and

Career Readiness and Standards for Mathematical Practice throughout Pre-K through 12

(Massachusetts State Education Department)



Critical Areas

• There are two to four critical areas for instruction in the introduction for each grade level, model course or integrated pathway.

• They bring focus to the standards at each grade by providing the big ideas that educators can use to build their curriculum and to guide instruction.

Grade level

PK K 1 2 3 4 5 6 7 8

# of Critical Areas

2 2 4 4 4 3 3 4 4 3



()



Critical Areas: Grade 6Ratio and Rate:• Connecting to whole number

multiplication and division.• Equivalent ratios derive from, and

extend, pairs of rows in the multiplication table.

Number:• Dividing fractions in general.• Extending rational number system to

negative integers (order, absolute value).



Critical Areas: Grade 6Expression and Equations:• Use of variables, equivalent

expressions.• Solve simple one-step equations.Statistics:• Different ways to measure center of

data.Geometry:• Find areas of shapes by decomposing,

rearranging or removing pieces, and relating shapes to rectangles.



(Page 46)



Critical Areas: Grade 7Proportional Relationships:• Use to solve variety of percent

problems.• Graph and understand unit rate as

the steepness of the line, or slope.Unified Understanding of Number:• Fraction, decimal and percent are

different representations of rational numbers.

• Same properties and operations apply to negative numbers.

• Use to formulate equations, solve problems



Critical Areas: Grade 7Geometry:• Area and circumference of a circle.• Surface area of solids.• Scale drawing and informal

constructions.• Relationships among plane figuresData:• Compare two data distributions, to

see differences between populations.• Informal work with random sampling.



(Page 52)



Critical Areas: Grade 8Proportional Reasoning:• Equations for proportions as special

linear equations: y = mx or y/x = m• Constant of proportionality is the

slope, graphs are lines through the origin.

Functions:• Function as a rule that assigns to

each input exactly one output.• Functions describe situations where

one quantity determines another.



Critical Areas: Grade 8

Geometry:Ideas about distance and angles, how

they behave under translations, rotations, reflections and dilations and ideas about congruence and similarity



A Coherent Curriculum• Is organized around the big

ideas of mathematics• Clearly shows how standards

are connected within each grade

• Builds concepts through logical progressions across grades that reflect the discipline itself.



International Comparison

The mathematics curriculum of top-achieving countries on international assessments looks different from the U.S. in terms of topic placement

Charts in the next three slides are taken from:Schmidt, W.H., Houang, R., & Cougan, L. (2002). A coherent curriculum: The case of Mathematics. American educator, 26(2), 10-26, 47-48.

As cited by Massachusetts State Education Department



Topic Placement in Top Achieving Countries



Topic Placement in the U.S.



International Comparison

In what ways do the curricula of the top-achieving countries exhibit coherence?



Domain Progression in the New Standards




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.

57



Take the number apart?Tina, Emma and Jen discuss this

expression:6 × 5⅓

Tina: I know a way to multiply with a mixed number that is different from what we learned in class. I call my way ‘take the number apart.’ I’ll show you. First, I multiply the 5 by the 6 and get 30. Then I multiply the ⅓ by the 6 and get 2. Finally, I add the 30 and the 2 to get my answer, which is 32.

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Take the number apart?Tina: It works whenever I have to multiply a

mixed number by a whole number.Emma: Sorry Tina, but that answer is wrong!Jan: No, Tina’s answer is right for this one

problem, but ‘take the number apart’ doesn’t work for other fraction problems.

Which of the three girls do you think is right?Justify your answer mathematically.

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Distributive Property5⅓ = 5 + ⅓6 × 5⅓ = 6(5 + ⅓)6(5 + ⅓)= 6 × 5 + 6 × ⅓Since a(b + c) = ab + acCould illustrate with area of rectangle 6

by 5⅓

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NCTM process standards

• Problem Solving• Reasoning and Proof• Communication• Representation• Connections

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National Research Council’s reportAdding It Up:• Conceptual Understanding (comprehension of

mathematical concepts, operations and relations)

• Procedural Fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately)

• Adaptive Reasoning• Strategic Competence• Productive Disposition (habitual inclination to

see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy)

62



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New York State Assessment Transition PlanELA & Math Revised October 20, 2011

1 New ELA assessments in grades 9 and 10 will begin during the 2012-13 school year and will be aligned to the Common Core, pending funding.

2 The PARCC assessments are scheduled to be operational in 2014-15 and are subject to adoption by the New York State Board of Regents. The PARCC assessments are still in development and the role of PARCC assessments as Regents assessments will be determined. All PARCC assessments will be aligned to the Common Core.

3 The names of New York State’s Mathematics Regents exams are expected to change to reflect the new alignment of these assessments to the Common Core. For additional information about the upper-level mathematics course sequence and related standards, see the “Traditional Pathway” section of Common Core Mathematics Appendix A.

4 The timeline for Regents Math roll-out is under discussion.

5 New York State is a member of the NCSC national alternate assessments consortium that is engaged in research and development of new alternate assessments for alternate achievement standards. The NCSC assessments are scheduled to be operational in 2014-15 and are subject to adoption by the New York State Board of Regents.

Assessment – Grade / Subject 2011-12 2012-13 2013-14 2014-15

ELA Grades 3-8 Aligned to 2005 Standards Grade 91 Grade 101

Aligned to the Common Core

Grade 11 Regents Aligned to 2005 Standards

PARCC2

Math

Grades 3-8 Aligned to 2005 Standards Aligned to the Common Core Algebra I3 Aligned to 2005 Standards Aligned to the Common Core Geometry3 Aligned to 2005 Standards Aligned to the Common Core4 Algebra II3 Aligned to 2005 Standards

PARCC2

Additional State Assessments NYSAA Aligned to the Common Core NCSC5 NYSESLAT Aligned to 2005 Standards Aligned to the Common Core

DRAFT



Instructional Expectations 2011-12 • Strengthening student work

CurriculumAssessmentClassroom instruction

• Strengthening teacher practice

Feedback64



Instructional Expectations 2011-12: Core DocumentsFramework for Teaching, Charlotte

DanielsonDepth of Knowledge, Norman WebbUnderstanding by Design, Grant

WigginsUniversal Design for LearningCurriculum Mapping, Heidi Hayes

Jacobs66



Curriculum Mapping: Heidi Hayes Jacobs

A subject or course’s Essential Map is developed by identifying:

The core curriculum concepts The critical focal skills Benchmark assessmentsCommon essential questionsEssential learnings / power

standards67



Specifics of Math Task:

• Will be administered to all students in spring 2012

• Based on Standards for Math Practice # 3 and 4 and selected domains

• Sample tasks, curriculum and student work available at DoE website

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Spring 2012 Task – Domains Chosen

69

Grade Domain

1-2 Number and Operations in Base Ten3 Operations and Algebraic Thinking4-5 Number and Operations—Fraction6-7 Ratios and Proportional Relationships8 Expressions and EquationsAlgebra Reasoning with Equations and InequalitiesGeometry Congruence



Math Tasks

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Some Criteria for Choice of 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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How many 'C' balls does it take to balance one 'A' ball?



Some Criteria for Choice of Tasks• Opportunities to exercise the

standards for mathematical practice

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

75



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 addressingmisconceptionsStudents who weretaught using remedialmeasures



Pedagogy• Make conceptions and

misconceptions visible to the student

• Students need to be listened to and responded to

• Partner work• Revise conceptions• Debug processes• Meta-cognitive skills



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?



Meta-Cognition• Thinking about thinking.• The unconscious process of

cognition.• Meaning-makingIt is hard to articulate how you

think about thinking. It is even harder to model



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



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?



Aligning Tasks to the Common Core Learning Standards



Comparing Two Mathematical TasksTASK AMAKING CONJECTURES Complete the conjecture based

on the pattern you observe in the specific cases.29. 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

30. 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



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.29. 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

30. 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



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.



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.



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.



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.



Comparing Two Mathematical Tasks

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



Tasks A and B

SameBoth ask

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

DifferentTask 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




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



What is the value of 17 × 13 + 61 × 13 + 22 × 13?

94



Analysis of Tasks

95



Levels of Cognitive DemandLower-level• Memorization• Procedures without

connectionsHigher-level• Procedures with connections• Doing mathematics

96



Norman Webb’s Depth of Knowledge

• Level 1: Recall and Reproduction

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

97



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



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



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



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



DOK 1

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

A. hundredsB. tenthsC. hundredthsD. thousandths




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



DOK 2

Draw the next figure in the following pattern:



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?




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



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…



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.




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.



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.



DOK 4George 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.



Implementationof Tasks




115

“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




116

The eight Standards for Mathematical Practice – place an emphasis on student demonstrations of learning…

Equity begins with an understanding of how the selection of tasks, the assessment of tasks, the student learning environment create great inequity in our schools…



Opportunities for all students to engage in challenging tasks?

117

• 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?



The nature of tasks used in the classroom…

118

will impact student learning!



But WHAT TEACHERS DO with the tasks matters too!

119

The Mathematical Tasks Framework



Students’ beliefs about their intelligence

120

• 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



Students can develop growth mindsets

121

• 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



Changing View of Assessment:

Assessment for Learning

124



Formative Assessment Strategies1. 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

125



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.



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…”



Ambiguities inherent in mathematics

6½6x61

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



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



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



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?



Identify where students might use and extend their existing knowledge• You’ve used substitution to solve

all these simultaneous equations. Can you use elimination?



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?



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.



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”



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



Assessment for LearningTeachers use assessment,

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

137

Dylan Wiliam



Assessment for Learning

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

138



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

139



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

Responsiveness140



SampleTasks



Animals



Giantburgers



Security Camera



Pythagorean Triples



Multiplying Cells



Circle Pattern



Skeleton Tower



Resources

150



Web Links

151

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

http://www.corestandards.org/

http://commoncoretools.wordpress.com/

http://www.engageny.org/

http://www.parcconline.org/

http://www.insidemathematics.org/index.php/home

http://map.mathshell.org/materials/index.php

http://schools.nyc.gov/Academics/CommonCoreLibrary/default.htm

http://www.moems.org/



Pearson Professional Development

pearsonpd.com

RONALD SCHWARZ, [email protected]