School Leaders, Pre-K - 5

School Leaders, Pre-K - 5

Guiding Principles to Support Effective Teaching and Learning:

Professionalism Diane J. Briars

President National Council of Teachers of Mathematics

[email protected]

Goals • Increase your understanding of effective teaching

practices to promote high quality mathematics learning by all students and how to support teachers successful implementation of these practices.

• Prepare you to use the principles and practices in Principles to Actions: Ensuring Mathematical Success for All as a framework for improving mathematics teaching and learning in your school or district.

Our Work

Session 1: Teaching and Learning, Professionalism

Sessions 2, 3, & 4: Eight Effective Mathematics Teaching Practices

Session 5: Action planning

Introductions • Name • District • Role • 1 challenge/need you’re trying to address

Effective Instructional Leaders:

• Make high achievement by all students a priority.

• Recognize and value high quality content, instruction and assessment.

• Focus: Focus strongly where the standards focus.

• Coherence: Think across grades, and link to major topics

• Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application

Key Features of CCSS-M

Key Features of CCSS-M • Focus: Focus strongly where the

standards focus.

• Coherence: Think across grades, and link to major topics

• Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application

Curriculum Standards, Not Assessment Standards

Understand and apply properties of operations and the relationship between addition and subtraction. (1.OA) 3. Apply properties of operations as strategies to add and

subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

Skill-algorithm understanding from the rote application of an algorithm through the selection and comparison of algorithms to

the invention of new algorithms (calculators and computers included)

Property-proof understanding from the rote justification of a property through the derivation of properties to the proofs of new

properties

Use-application understanding from the rote application of mathematics in the real world through the use of mathematical

models to the invention of new models

Representation-metaphor understanding from the rote representations of mathematical ideas through the analysis of such representations

to the invention of new representations

Vocabulary

Problem Solving Zalman Usiskin, 2012

Dimensions of Mathematical Understanding in CCSS

Phil Daro, 2010

Phil Daro, 2010

Other “Butterflies”?

Key Instructional Shift

From emphasis on: How to get answers

To emphasis on:

Understanding mathematics

Guiding Principles for School Mathematics

1. Teaching and Learning

Effective teaching is the non-negotiable core that ensures that all students learn

mathematics at high levels.

We Must Focus on Instruction

Student learning of mathematics “depends fundamentally on what happens inside the classroom as teachers and learners interact over the curriculum.”

(Ball & Forzani, 2011, p. 17) “Teaching has 6 to 10 times as much impact on achievement as all other factors combined ... Just three years of effective teaching accounts on average for an improvement of 35 to 50 percentile points.”

Schmoker (2006, p.9)

Beliefs about Teaching and Learning?

Complete the survey (p. 1): • Rate the extent to which you agree

with each statement. • Compare with others sitting near you. • How might these beliefs affect your

work? • Who else’s beliefs about teaching and

learning affects your work?

Unproductive vs Productive Beliefs about Teaching and Learning

• Beliefs should not be viewed as good or bad.

• Beliefs are unproductive when they hinder implementation of effective instructional practice or limit student access to important mathematics content and practices.

p. 4

Common Core State Standards for Mathematics

Two type of standards:

• Standards for Mathematical Practice

• Standards for Mathematical Content

Common Core State Standards for Mathematics

Two type of standards:

• Standards for Mathematical Practice

• Standards for Mathematical Content

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.

Standards for Mathematical Practice K-5 Elaborations

• Read your assigned practice • What additional information/clarification

does it provide about that practice that would be important for K-5 mathematics teachers?

• What would you “look for” to see this in the classroom—teacher actions/students actions.


• Describe the thinking processes, habits of mind and dispositions that students need to develop a deep, flexible, and enduring understanding of mathematics; in this sense they are also a means to an end.

SP1. Make sense of problems “….they [students] analyze givens, constraints,

relationships and goals. ….they monitor and evaluate their progress and change course if necessary. …. and they continually ask themselves “Does this make sense?”


AND….

• Describe mathematical content students need to learn. SP1. Make sense of problems

“……. students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.”

Developing Proficiency in the Standards for Mathematical Practice Requires

Explicit attention to development of students’ proficiency in the Standards for Mathematical

Practice.

Diane J. Briars, January 2014

Developing Proficiency in the Standards for Mathematical Practice Requires

Instructional practices that promote students’ development of

conceptual understanding and proficiency in the Standards for

Mathematical Practice.



Teaching and Learning An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their

ability to make sense of mathematical ideas and reason mathematically.


Effective Mathematics Teaching Practices • Establish mathematics goals to focus learning. • Implement tasks that promote reasoning and problem

solving. • Use and connect mathematical representations. • Facilitate meaningful mathematical discourse. • Pose purposeful questions. • Build procedural fluency from conceptual understanding. • Support productive struggle in learning mathematics. • Elicit and use evidence of student thinking.

(formative assessment)



Professionalism In an excellent mathematics program, educators hold themselves and their

colleagues accountable for the mathematical success of every student

and for their personal and collective professional growth toward effective

teaching and learning of mathematics.


Professionalism In an excellent mathematics program, educators hold themselves and their

colleagues accountable for the mathematical success of every student

and for their personal and collective professional growth toward effective

teaching and learning of mathematics.

Professionalism Obstacle

In too many schools, professional isolation severely undermines attempts to significantly

increase professional collaboration … some teachers actually embrace the norms of

isolation and autonomy. A danger in isolation is that it can lead to teachers developing

inconsistencies in their practice that in turn can create inequities in student learning.

Principles to Actions, p. 100

Principles to Actions, pp. 103-104

Collaboration Team Work

• An examination and prioritization of the mathematics content and mathematics practices students are to learn.

• The development and use of common assessments to determine if students have learned the agreed-on content and related mathematical practices.

• The use of data to drive continuous reflection and instructional decisions.

• The setting of both long-term and short-term instructional goals. • Development of action plans to implement when students

demonstrate they have or have not attained the standards. • Discussion, selection, and implementation of common research-

informed instructional strategies and plans.

Rate Your Teams

Critical Issues for Team Collaboration

Principles to Actions, pp. 103-104

Collaboration Team Work

• An examination and prioritization of the mathematics content and mathematics practices students are to learn.

• The development and use of common assessments to determine if students have learned the agreed-on content and related mathematical practices.

• The use of data to drive continuous reflection and instructional decisions.

• The setting of both long-term and short-term instructional goals. • Development of action plans to implement when students

demonstrate they have or have not attained the standards. • Discussion, selection, and implementation of common research-

informed instructional strategies and plans.

Collaborative Team Work

Principal’s Tasks 1. Create a schedule that includes regular

grade-level and cross-level collaborative planning

2. Set explicit expectations about how to use this planning time, as well as what products should result

3. Monitor the collaborative work




Effective Instructional Leaders

Expect and support all teachers in implementing effective

mathematics teaching practices.

How Many Students Will Have High Quality Mathematics Instruction K-5?

K 1 2 3 4 5

Ms. A Ms. C Mr. E Ms. G Ms. I Ms. K

Ms. B Mr. D Ms. F Ms. H Mr. J Ms. L

64 Students


K 1 2 3 4 5



64 Students

32


K 1 2 3 4 5



64 Students

32 16


K 1 2 3 4 5



64 Students

32 16 8


K 1 2 3 4 5



64 Students

32 16 8 4


K 1 2 3 4 5



64 Students

32 16 8 4 2


K 1 2 3 4 5



64 Students

32 16 8 4 2 1




• Expect all faculty to implement effective mathematical teaching practices.

• Create a learning community that supports teachers and administrators as they work to improve instructional practices, including teachers working in collaborative teams.

Disclaimer The National Council of Teachers of Mathematics is a public voice

of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

56

57



Effective Mathematics Teaching Practices Session 2

Diane J. Briars President

National Council of Teachers of Mathematics [email protected]

Principle on Teaching and Learning

An excellent mathematics program requires

effective teaching that engages students in

meaningful learning through individual and

collaborative experiences that promote their

ability to make sense of mathematical ideas

and reason mathematically.







Promoting Proficiency in the Standards for Mathematical Practice

“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

What Research Tells Us

Learner’s should: • Acquire conceptual knowledge as well as

skills to enable them to organize their knowledge, transfer knowledge to new situations, and acquire new knowledge.

• Engage with challenging tasks that involve active meaning-making to build conceptual knowledge. Heibert and Grouws, 2007

What Are Mathematical Tasks?

Mathematical tasks are a set of problems or a single complex problem the purpose of which is to focus students’ attention on a particular mathematical idea.

Why Focus on Mathematical Tasks?

• Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it;

• Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information;

• The level and kind of thinking required by mathematical instructional tasks influences what students learn; and

• Differences in the level and kind of thinking of tasks used by different teachers, schools, and districts, is a major source of inequity in students’ opportunities to learn mathematics.

Why Focus on Mathematical Tasks?

• Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it;

• Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information;

• The level and kind of thinking required by mathematical instructional tasks influences what students learn; and

• Differences in the level and kind of thinking of tasks used by different teachers, schools, and districts, is a major source of inequity in students’ opportunities to learn mathematics.

The QUASAR Project

• Assisted schools in economically disadvantaged communities to develop instructional programs that emphasize thinking, reasoning and problem solving in mathematics.

• Worked with lowest achieving middle schools in six urban sites.

• Studied the impact of high quality curricula and professional development upon student achievement.

Edward Silver, Margaret S. Smith, Mary Kay Stein

Comparing Two Mathematical Tasks

Martha was re-carpeting her bedroom which was 15 feet long and 10 feet wide. How many

square feet of carpeting will she need to purchase?

Smith, Stein, Arbaugh, Brown, and Mossgrove, 2004

Comparing Two Mathematical Tasks

Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen in which to keep the rabbits.

1. If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be?

2. How long would each of the sides of the pen be if they had only 16 feet of fencing?

3. How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.

Smith, Stein, Arbaugh, Brown, and Mossgrove, 2004

Compare the Two Tasks • Work each task. • Share solution strategies. • Discuss:

How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?

Solution Strategies: Martha’s Carpeting Task

Martha’s Carpeting Task Using the Area Formula

A = l x w A = 15 x 10 A = 150 square feet

Martha’s Carpeting Task Drawing a Picture

10

15

Solution Strategies: The Fencing Task

The Fencing Task Diagrams on Grid Paper

The Fencing Task Using a Table

Length Width Perimeter Area

1 11 24 11

2 10 24 20

3 9 24 27

4 8 24 32

5 7 24 35

6 6 24 36

7 5 24 35

The Fencing Task Graph of Length and Area

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Length

Are

a

The Fencing Task Using Calculus

A = lw A = w(12-w) A = 12w – w2

A’ = 12 – 2w 0 = 12 – 2w 2w = 12 w = 6

Cognitive Level of Tasks

• Lower-Level Tasks (e.g., Martha’s Carpeting Task)

• Higher-Level Tasks

(e.g., The Fencing Task) The Quasar Project

With a Partner: • Categorize Tasks A – L into two

categories: high level cognitive demand and low level cognitive demand.

• Develop a list of criteria that describe the tasks in each category.

Characterizing Tasks

Lower-Level Tasks • Memorization

– What are the decimal equivalents for the fractions ½ and ¼?

• Procedures without connections

– Convert the fraction 3/8 to a decimal.

Higher-Level Tasks • Procedures with connections

– Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5.

• Doing mathematics

– Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine:

a) The decimal part of area that is shaded;

b) The fractional part of area that is shaded.

Principles to Actions, pp 18-19







27

28



Effective Mathematics Teaching Practices Session 3



Goals • Increase your understanding of effective teaching

practices to promote high quality mathematics learning by all students and how to support teachers successful implementation of these practices.

• Prepare you to use the principles and practices in Principles to Actions: Ensuring Mathematical Success for All as a framework for improving mathematics teaching and learning in your school or district.

Our Work

Session 1: Teaching and Learning, Professionalism

Sessions 2, 3, & 4: Eight Effective Mathematics Teaching Practices

Session 5: Action planning




Implement Tasks that Promote Reasoning and Problem Solving

Mathematical tasks should:

• Provide opportunities for students to engage in exploration or encourage students to use procedures in ways that are connected to concepts and understanding;

• Build on students’ current understanding; and

• Have multiple entry points.

Lower-Level Tasks • Memorization

– What are the decimal equivalents for the fractions ½ and ¼?

• Procedures without connections

– Convert the fraction 3/8 to a decimal.

Higher-Level Tasks • Procedures with connections

– Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5.

• Doing mathematics

– Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine:

a) The decimal part of area that is shaded;

b) The fractional part of area that is shaded.

Principles to Actions, pp 18-19

Level of Cognitive Demand

Is different than:

• Difficulty

• Importance

Implementation Issue

Do all students have the opportunity to engage in mathematical tasks that

promote reasoning and problem solving on a regular basis?

Opportunities for all students to engage in challenging tasks?

• 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?

Classroom “Look Fors”

• What is the teacher doing?

• What are students doing? • What happens to the demand during the lesson? • To what extent are all students engaged in the

mathematics learning?

p. 24

The Mathematical Tasks Framework

Stein, Grover & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000)

Tasks as set up by teachers

Tasks as they appear in curricular

materials

Tasks as enacted by

teachers and

students Student learning

LSC Evaluation Study

While teachers were using the materials more extensively in their classrooms, there was a wide variation in how well they were implementing these materials. Teachers were often content to omit rich activities, skip over steps and jump to higher level concepts, or leave little time for students to ‘make sense’ of the lessons.

Weiss, et al, 2006

LSC Evaluation Study

In fact, classroom observations indicated that the lessons taught as the developers intended were more likely to provide students with learning opportunities than those that were “adapted.”

Weiss, et al, 2006

Types of Math Problems Presented 1999 TIMSS Video Study

61

7784

41

57

69

15 16 13

54

2417

0102030405060708090

Australia CzechRepublic

Hong Kong Japan Netherlands US

Using procedures Making connections

How Teachers Implemented Making Connections Math Problems

31

16 18 20 19

59

8

5247 48

37

00

10

20

30

40

50

60

70

80

Australia CzechRepublic

Hong Kong Japan Netherlands US

Using procedures Making connections

Classroom “Look Fors”

• Who is doing the mathematical thinking? • What is the instructional goal—understanding

mathematics or getting answers? • What is the cognitive demand of the tasks? • What happens to the demand during the lesson? • To what extent are all students engaged in the

mathematics learning?

Stein & Lane, 1996

A.

C.

Maintaining Demand Matters!

High High

Low Low

High Low Moderate

High

Low

Task Set-Up Task Implementation Student Learning

Teacher Actions that Affect Cognitive Demand

• Task set-up • Supporting students’ exploration of

the task • Orchestrating debriefing discussion

A LOOK INTO A CLASSROOM

Sharing Brownies

Four friends want to share 7 brownies equally.

How many brownies, or what fractional part of a brownie, will each person get?

Note: NCTM does not currently have permission to share the Brownie Task video. As an alternative, you might consider the Fractions with Geoboards lesson, from the Annenberg K-4 video library, http://www.learner.org/vod/vod_window.html?pid=905

Sharing Brownies

1. Individually do the task. 2. Then compare your work with others. 3. Consider each of the following questions and be

prepared to share your thinking with the group: a) What mathematics content does the task involve? b) Which mathematical practices are needed to

complete the task? c) What is the level of demand of the task?

About the Lesson

• Grade 4 classroom.

• The teacher, Ms. Katherine Casey, has been working several years on standards-based mathematics instruction.

• Ms. Lucy West, a mathematics coach, is working with Ms. Casey during this lesson.

Norms for Watching Video

• Video clips are examples, not exemplars. – To spur discussion not criticism

• Video clips are for investigation of teaching and learning, not evaluation of the teacher. – To spur inquiry not judgment

• Video clips are snapshots of teaching, not an entire lesson. – To focus attention on a particular moment not what came before

or after

• Video clips are for examination of a particular interaction. – Cite specific examples (evidence) from the video clip, transcript

and/or lesson graph.

A Look into a Classroom… Look for: • What happened to the cognitive

demand when the task was implemented?

• What specific teacher actions affected the cognitive demand and supported/hindered students’ engagement reasoning and problem solving?

A Look into a Classroom…

• Debriefing Activity – In pairs…

• One person is the teacher • The other person is the coach/administrator • The coach engages the teacher in reflection and

moves the person toward one or two short-term goals tied to implementing tasks that promote reasoning and problem solving.

Sharing Brownies

..\..\..\..\Diane Files\Brownie lesson\Yesvideo\VIDEO_TS\VIDEO_TS.IFO

A Look into a Classroom…




The Mathematical Tasks Framework

Stein, Grover & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000)

Tasks as set up by teachers

Tasks as they appear in curricular

materials

Tasks as enacted by

teachers and

students Student learning







Leaves and Caterpillars

A fourth-grade class needs five leaves each day to feed its 2 caterpillars.

How many leaves would they need each day for 12 caterpillars?

David Crane and the Caterpillar Task

Read David Crane and the Caterpillar Task • What were the strengths of this lesson? • How might this lesson be revised to

increase student learning?

Five Practices for Orchestrating Effective Discussions

• Anticipating likely student responses • Monitoring students’ actual responses • Selecting particular students to present their

mathematical work during the whole class discussion

• Sequencing the student responses • Connecting different students’ responses—to

each other and to key mathematical ideas. Smith & Stein, 2011

Smith & Stein, 2011

Planning with the Student in Mind

• Anticipate solutions, thoughts, and responses that students might develop as they struggle with the problem

• Generate questions that could be asked to promote

student thinking during the lesson, and consider the kinds of guidance that could be given to students who showed one or another types of misconception in their thinking

• Determine how to end the lesson so as to advance

students’ understanding Stigler & Hiebert, 1997

Leaves and Caterpillars

Make a poster showing: • What is the mathematical goal of the lesson? • Which students would you have present their

solutions? • In what order would you have them present? • Why? • What questions would you want to be sure to

ask?






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48

School Leaders, Pre-K - 5 Guiding Principles to Support Effective

Teaching and Learning: Effective Mathematics Teaching

Practices Session 4






Adding 1 to an Addend

What generalization is suggested by these problems?

• State the generalization in common language. • Which mathematical practices are involved in this task? • Construct an argument to show that your generalization is

true.

Ms. Kaye’s 3rd Grade • Explored the question. • Created the conjecture:

“In addition, when one of the addends goes up by one, the sum always has to go up by one.”

• Task: – Use a story context, model, picture, or number line to

convince someone who wasn’t here that this is always true. – How can you make your story, model, picture or number line

general enough to convince someone that this is always true no matter what the numbers are?

Ms. Kaye’s 3rd Grade

As you watch the video, consider: • How do these students use their representations

to create a general argument for their rule, an argument that doesn‘t depend on particular examples?

• What does the teacher do to support/inhibit students engagement in reasoning and problem solving?

A Look into a Classroom… #2




6

Ms. Kaye’s 3rd Grade Will It Always Work?

http://www.mathedleadership.org/ccss/itp/operations.html

Part 3 of video; starting about 5:50.

http://www.mathedleadership.org/ccss/itp/operations.html

A Look into a Classroom… #2




8




Adding 1 to an Addend Maureen’s 2nd Grade Class

• Read the transcript of second graders investigating this problem.

• What did the teacher do to support her students learning? Cite evidence from the lesson to support any claims that you make.

Adding 1 to an Addend Maureen’s 2nd Grade Class

• Identify a question in the lesson that you think is good.

• Discuss the questions the members of your group have identified and see if you can come to consensus on 2 or 3 good questions. Highlight these on your transcripts in some way.

• Make a list of the characteristics of a good question.

Boaler and Brodie’s Framework Question Type Description

1 Gathering information, leading students through a procedure

Requires immediate answer

Rehearses known facts/procedures

Enables students to state facts/procedures

2 Inserting terminology Once ideas are under discussion, enables correct mathematical language to be used to talk about them

3 Exploring mathematical meanings and/or relationships

Points to underlying mathematical relationships and meanings. Makes links between mathematical ideas and representations.

4 Probing, getting students to explain their thinking Asks student to articulate, elaborate or clarify ideas.

5 Generating discussion Solicits contributions from other members of class.

6 Linking and applying Points to relationships among mathematical ideas and mathematics and other areas of study/life.

7 Extending thinking Extends the situation under discussion to other situations where similar ideas may be used.

8 Orienting and focusing Helps students to focus on key elements or aspects of the situation in order to enable problem-solving.

9 Establishing context Talks about issues outside of math in order to enable links to be made with mathematics.

Funneling vs. Focusing

Funneling pattern of questioning: Teacher uses a set of questions to lead students to a desired procedure or conclusion, while giving limited attention to student responses that veer from the desired path. The teacher has decided on a particular path for the discussion to follow and leads the students along that path, not allowing students to make their own connections or build their own understanding of the targeted mathematical concepts.

Typical Teacher Questioning-IRE

• Teacher initiates a question. • Student responds (usually in one or two words). • Teacher evaluates the response as right or wrong. Authority for deciding whether answer is right or wrong lies with the teacher—instead of with discipline-based reasoning

Funneling vs. Focusing

Focusing pattern of questioning: The teacher attends to what the students are thinking, pressing them to communicate their thoughts clearly, and expecting them to reflect on their thoughts and those of their classmates. The teacher is open to a task being investigated in multiple ways. On the basis of content knowledge related to the topic and knowledge of student learning, the teacher plans questions and outlines key points that should become salient in the lesson.

Productive Discourse

• How did you get that? • How do you know that? • Can you explain your

idea? • Why? • Can you convince us? • Did anyone get

something else?

• Can someone tell me or share with me another way?

• Do you think that means the same things?

• Is there another opinion about this?

• Why did you say that, Justin?

Boaler, J., & Brodie, K. (2004).









Promoting Productive Struggle

https://www.youtube.com/watch?v=TTXrV0_3UjY

https://www.youtube.com/watch?v=TTXrV0_3UjY

Students’ Beliefs about Their Intelligence Affect Their Academic Achievement

• 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


When confronted with challenging school transitions or courses, students

with growth mindsets outperform those with fixed mindsets, even when they

enter with equal skills and knowledge.

Dweck, 2007

Students Can Develop Growth Mindsets

• Teacher praise influences mindsets

– Fixed: Praise refers to intelligence – Growth: Praise refers to effort, engagement,

perseverance

• 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.

• Reading stories of struggle by successful individuals can promote a growth mindset

“Effort Praise” Promotes Growth Mindsets

“You really studied for your English test, and your improvement shows it. You read the material over several times, outlined it, and tested yourself on it. That really worked!”

“I like the way you tried all kinds of strategies on that math problem until you finally got it.”

“It was a long, hard assignment, but you stuck to it and got it done. You stayed at your desk, kept up your concentration, and kept working. That's great!”

“I like that you took on that challenging project for your science class. It will take a lot of work—doing the research, designing the machine, buying the parts, and building it. You're going to learn a lot of great things.”

Dweck, 2007

“Effort Praise” Promotes Growth Mindsets

What about a student who gets an A without trying?

• “All right, that was too easy for you. Let‘s do something more challenging that you can learn from.”

• What about a student who works hard and doesn't do well?

• “I liked the effort you put in. Let's work together some more and figure out what you don't understand.”

Dweck, 2007


• What are the implications of this research on praise and mindsets?

• How might you use this research?




Monitoring and Observation Guide



29

30



Assessment and Planning Diane J. Briars

President National Council of Teachers of Mathematics

[email protected]

Beliefs about Assessment?

Complete the survey: • Rate the extent to which you agree

with each statement. • Compare with others sitting near you. • How might these beliefs affect your

work? • Who else’s beliefs about assessment

affect your work?


Assessment An excellent mathematics program ensures that

assessment is an integral part of instruction, provides evidence of proficiency with important mathematics content and practices, includes a

variety of strategies and data sources, and informs feedback to students, instructional

decisions and program improvement.

5

What is Assessment?

Assessment is the process of

gathering evidence about

student’s knowledge of, ability

to use, and disposition towards

mathematics and of making

inferences based on that

evidence for a variety of

purposes.

NCTM Assessment Standards for School Mathematics, 1995.

Formats That Support Valid Inferences

Learning target: Understanding the definition of a triangle.

Performance task: Draw a triangle.

Grade 2

2.G.3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Grade 2

Grade 2

Understanding a Concept

• Explain it to someone else • Represent it in multiple ways • Apply it to solve simple and complex problems • Reverse givens and unknowns • Compare and contrast it to other concepts • Use it as the foundation for learning other

concepts

Assessment Purposes

• Monitoring students’ progress to promote student learning

• Making instructional decisions to modify instruction to facilitate student learning

• Evaluating students’ achievement to summarize and report students’ demonstrated understanding at a particular moment in time

• Evaluating programs to make decisions about instructional programs









Assessment for Learning

“ . . .is any assessment for which the first priority in its design and practice is to serve the purpose of promoting students’ learning. . . An assessment activity can help learning if it provides information that teachers and their students can use as feedback in assessing themselves and one another and in modifying the teaching and learning activities in which they are engaged.”

Black and Wiliam, 2004

Few interventions have the same level of impact as assessment for learning.

The most intriguing result is that while all students show achievement gains,

the largest gains accrue to the lowest achievers.

Stiggins, et al. (2006, p. 37)

Formative Assessment

“Assessment becomes “formative assessment” when the evidence is actually used to adapt the teaching work to meet learning needs.”

Black and Wiliam, 2004

Formative Assessment is:

Students and teachers Using evidence of learning To adapt teaching and learning To meet immediate learning needs Minute-to-minute and day to day Dylan Wiliam, University of London

What Research Says

• Read Five “Key Strategies” for Effective Formative Assessment.

• Individually identify three key ideas. • Share with table members-First Word/Last Word:

– Person 1 shares one key idea without comment – Other table members comment – Person 1 comments and summarizes – Person 2 shares a different key idea . . . Etc. – Other table members comment – Continue until each table member has shared one of

their ideas.

Formative Assessment Strategies

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

• Providing feedback that moves learning forward.

To do this requires

• Task analysis • Gathering and analyzing evidence of students

thinking, understandings, and misconceptions.

Analyze a Task

Grades K-2: Subtraction Task

Grades 3-4: Baker Task

Grades 4-5: Fractions Task

Analyze a Task

• Work the task individually, and in more than one way if possible.

• Discuss solution strategies with others at your table.

What are the Core Elements of Performance?

• What does the task require? • What does the task have the potential to reveal

about: – The mathematical content (concepts; skills) – The mathematical practices (CCSS)

• Anticipating student solutions: – On what parts will students have success? – On what parts will students struggle or be unsuccessful?

Analyzing Students’ Work

Goal: Diagnose what students thought, could do, where they had errors or incomplete understanding.

Analyzing Students’ Work Step 1: Sort student work into piles of successful,

partially successful and little success. Step 2: For each category of work, identify common

understandings, common errors, misunderstandings or misconceptions.

Step 3: Consider strengths and weaknesses: • What are the implications for future

instruction? • What specific instruction or experiences will

you design for students?

Effective Intervention • Is mandatory, not optional (i.e., scheduled during the

school day whenever possible); • Is based on constant monitoring of students’

progress, as determined from the results of formative and summative assessment, ensuring that students get support as quickly as possible;

• Attends to conceptual understanding as well as procedural fluency; and

• Allows for flexible movement in and out of the intervention as students need it.

(Kanold and Larson 2012)

Effective Intervention

• Additional intervention periods • During regular instructional time

–Flexible groups across teachers for intervention; heterogeneous core instruction

–Re-engagment lessons

Traditionally Teacher Choose One of Three Options

• Go back and re-teach the topic with the entire class.

• Identify the students needing remediation and find some time/opportunity to re-teach the topic while the rest of the class moves on.

• Feeling the pressure of the over-packed curriculum, the teacher ventures on to the next topic. David Foster, 2010

David Foster, 2010

Re-teaching vs. Re-engagement • Teach the unit again. • Address basic skills that are

missing • Do the same or similar

problems over • Practice more to make sure

that students learn the procedures

• Focus mostly on underachievers

• Cognitive level is usually lower

• Revisit student thinking • Address conceptual

understanding • Examine task from different

perspective(s) • Critique student

approaches/solutions to make connections

• The entire class is engaged in the math

David Foster, 2010

Analyze Student Work

• Understandings and misconceptions • Select work that merits examination in re-

engagement • How would you structure re-engagement?

– Task – Questions

What about “Test-Prep”?

Too often, teachers are putting regular instruction “on hold” to spend class time

practicing test questions. While on the surface this may appear to make sense, research

indicates just the opposite— test scores are lower in schools where teachers

spend large amounts of time on test prep.

(Allensworth, Correa, & Ponisciak, 2008)

Intensive Test Prep Produces the Same or Lower Scores as Little or No Test Prep

All Test-Prep


Intensive Test Prep Produces the Same or Lower Scores as Little or No Test Prep

ACT Test-Prep Materials


Effective Assessment Practice Ongoing Review and Practice

Providing students with periodic opportunities to practice using concepts and skills, along with

feedback about their performance, helps students solidify their knowledge and promotes retention, reflection, generalization, and transfer

of knowledge and skill.

IES Practice Guide, 2007

Distributed Practice • Openers • Homework • Incorporate into

instructional and/or assessment tasks

Good Instruction is the Best Test-Prep

• Students acquire conceptual knowledge as well as skills to enable them to organize their knowledge, transfer knowledge to new situations, and acquire new knowledge.

• Students engage with challenging tasks that involve active meaning-making.

• Students know what is expected

Hiebert & Grouws, 2007

Effective Assessment Practices

1. Create and use common high-quality assessments 2. Collaboratively analyze SBAC and PARCC prototype

assessment tasks 3. Use tasks that assess conceptual understanding and

mathematical practices 4. Teach students to take responsibility for their learning 5. Use assessment results formatively; i.e., use errors and

misconceptions as instructional opportunities 6. Provide opportunities for on-going review and practice

instead of stopping instruction for “test prep.”


• Teaching and Learning • Access and Equity • Curriculum • Tools and Technology • Assessment • Professionalism

Understanding and Supporting Teachers’ Enactment of the Effective Teaching Practices:

Planning our Work with Others

Charting a Path for Improvement

1. Clearly articulated vision for teacher learning. 2. Time for embedded professional learning. 3. Attention to adult learning needs. 4. Support for teacher collaboration. 5. Make the learning known. 6. Celebrate successes.

Action Planning Worksheet

1. Goals 2. Action Steps

include session planning steps

3. Time frame 4. Success criteria



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Thank You!

Diane J Briars

[email protected]

Documents

School Leaders, Pre-K - 5