Common Core Math: How do I teach that to my students with disabilities and a history of low performance?

Brad Witzel, Ph.D.

Winthrop University

1

Why Math? Why Now?

How does the US fare internationally? http://www.youtube.com/watch?v=niU1E3SSTAM

What does the future hold? http://dotsub.com/view/f3f91861-f623-4614-8c1e-c24a9a53b4cf

Should improvements in math education be aimed at the elite? Why or why not?

2

U.S. Math Performance

The 2011 National Assessment of Educational Progress (NAEP) reported US math achievement as:

– 18% of Grade 4 students scored below the basic level – 28% of Grade 8 students scored below the basic level *Grade 12 students scores could not be compared

2011 NAEP data revealed that: - 21% of 4th grade SC students scored below basic - 30% of 8th grade SC students scored below basic

3

U.S. Math Performance Students with Disabilities

2011 NAEP report on students with disabilities National percent scoring below basic

• 45% of Grade 4 • 65% of Grade 8

South Carolina

• 53% of 4th grade • 71% of 8th grade

4

U.S. Math Performance Students learning English

2011 NAEP report on ELL population National percent scoring below basic

• 42% of Grade 4 • 72% of Grade 8

South Carolina

• 21% of 4th grade • 43% of 8th grade

5

6

Characteristics of those who struggle the most

• Lack of strategic approaches to mathematics • Unorganized, impulsive, unaware of where to begin an

assignment • Unaware of possible steps to break the problem into a

manageable task, possibly due to the magnitude of the task • Exhibit problems with memory (working memory) • Unable to focus on a task • Experience feelings of frustration, failure, or anxiety • Lack persistence to solve longer problems or personal

struggles • Attribute failure to uncontrollable factors (e.g., luck,

teacher's instructional style)

7

Learner Characteristics (from Riccomini)

• Strategic Learners – Able to analyze a problem and develop a plan – Able to organize multiple goals and switch flexibly from

simple to more complicated goals – Access their background knowledge and apply it to novel

tasks – Develop new organizational or procedural strategies as the

task becomes more complex – Use effective self-regulated strategies while completing a

task – Attribute high grades to their hard work and good study

habits – Review the task-oriented-goals and determine whether they

have been met http://iris.peabody.vanderbilt.edu/srs/chalcycle.htm

8

Learner Characteristics (from Riccomini)

• Non-Strategic Learners – Unorganized, impulsive, unaware of where to begin an

assignment

– Unaware of possible steps to break the problem into a manageable task, possibly due to the magnitude of the task

– Exhibit problems with memory

– Unable to focus on a task

– Lack persistence

– Experience feelings of frustration, failure, or anxiety

– Attribute failure to uncontrollable factors (e.g., luck, teacher's instructional style)

http://iris.peabody.vanderbilt.edu/srs/chalcycle.htm

9

What have we done to infuse strategic learning in low-scoring students?

1) Multi-modal; teaching the concepts and reasons to why math procedures work a certain way; academic and social relevance to the mathematics; 2) Technology (KhanAcademy) to stop and pause the lesson; 3) Station teaching to address needs and instructional approaches (differentiation) 4) Summer academic enrichment (school happens everyday) 5) Scripted programs; PD for teachers

10

A change of math ed is in the air

• National Study of Algebra Teachers

• Common Core

• Arithmetic to Algebra Gap

• Proactive Ideas

• Reactive Ideas

11

What Math Knowledge is Needed to Solve these Equations?

2x + 5 = 18

- 5 - 5

2x = 13

2x = 13

2 2

1x = 6 ½

(y – 5)(y + 2)

(y)(y) + (y)(2) – (5)(y) - (5)(2)

y2 + 2y – 5y – 10

y2 – 3y – 10

12

Nationally, what do algebra teachers say? (NMP, 2008)

743 algebra teachers in 310 schools nationally responded to a survey on algebra instruction and student learning in 2007. Findings:

• The teachers generally rated their students’ background preparation for Algebra I as weak. The three skill areas in which teachers reported their students have the poorest preparation are rational numbers, word problems, and study habits.

• Regarding the best means of preparing students, 578 suggested a greater focus on mastery of elementary mathematical concepts and skills.

• Teachers were less excited about how current textbook approaches meet the needs of diverse student populations.

13

More findings from the NSAT • Use of calculators was quite mixed with 33% saying they

never use them and 31% use them frequently (more than once a week).

• 60% use physical tools less than once a week and only 9% use them frequently.

• 51% consider “mixed-ability” grouping to be a moderate or serious problem with instruction.

• The greatest challenge to teachers was #1 – “working with unmotivated students.” This was chosen by 58% of the middle school teachers and 65% of the high school teachers. The next most frequent response was “making mathematics accessible and comprehensible to all my students,” selected by 14% of the middle school teachers and 9% of the high school teachers.

14

What is Common core

Developed by the National Governor’s Association and the Council of Chief State School Officers Scope and sequence were informed by standards in higher-performing countries. These standards must make up at least 85% of each state’s standards in English language arts and mathematics. Sheehy, US News, July 5, 2012 • “The Common Core State Standards set a consistent bar for math

and English achievement. With an emphasis on building and expressing logical arguments, and applying math to real-world issues, the initiative aims to align high school lessons in those subjects with college and work expectations.”

• “60 percent of registered voters surveyed said they know nothing about the new academic standards.”

15

Standards for Mathematical Content

Grade Level Domains

High School Conceptual Categories

K – 5 Counting and Cardinality Operations and Algebraic Thinking Number and Operations in Base Ten Number and Operations – Fractions Measurement and Data Geometry

6-8 Ratios and Proportional Relationships The Number System Expressions and Equations Functions Geometry Statistics and Probability

• Number and Quantity

• Algebra

• Functions

• Modeling

• Geometry

• Statistics and Probability

16

From RI CCSS presentation

CCSS vocabulary

Domains: group of related standards

Cluster Headings: overview of the mathematical ideas within a domain

Clusters: groups of related standards

Standards: what students should understand and be able to do

17

From RI CCSS presentation

Pieces within the CCSS

18

CCSSM, p.5

CONCEPTUAL CATEGORY DOMAIN

Standards within the CLUSTER

CLUSTER HEADING

19

From RI CCSS presentation

Common core “process standards”.. Oops, I mean math practices

Practice Standards (NCM called these process standards) 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.

20

A new focus within the CCSSM

1. More Instructional Time: fewer topics covered in greater depth 2. Planned Progressions: instruction is connected within and across grades 3. Proficiency: perform mathematics procedures with speed and accuracy 4. Application: applying math to solve a problem 5. Balanced Learning: achieve fluency and conceptual understanding

21

Proactive measures

• Plan approaches across grade levels and courses.

Develop algebra as a benchmark not as the only outcome in mathematics.

• Horizontally plan instructional approaches for the most challenging content.

Combine procedural facility and sense-making.

22

Vertical planning needs One teacher’s success is connected to that of each preceding teacher.

23

Successful transition into no intervention

abstract algebra concepts necessary

Successful in divisible and

sequential learning associated

with arithmetic

Unsuccessful transition need for more

associated with overgeneralization concrete learning of

Student of arithmetic abstract algebra

Performance

Unsuccessful in divisible and Compounded difficulty trying to need for continued

sequential learning associated attach limited arithmetic knowledge arithmetic assistance

with arithmetic to algebra and more concrete

learning of abstract

algebra

(Current Study)

Arithmetic Computation Learning Abstract Learning Associated with Algebra

Figure 1. Flowchart of Algebraic Needs for Students Who Experience Difficulty in Math

Arithm

etic to Algebra G

ap

Arithmetic to Algebra Gap (Witzel, Smith, & Brownell, 2001)

What should be covered before formal algebra? (Gersten, Clarke, & Witzel, 2008)

• Fluency with standard algorithms

• Understanding properties

– Commutative

– Associative

– Distributive

• Basic measurement concepts and operations of 2 and 3 dimensional objects

• Word problem translations into symbols

24

How do students tackle problems?

3x – 6y – 9 3 they might see the immediate possibility that:

13x – 26y – 39 = 1x – 2y - 3 13 But what happens when computation isn’t as straight

forward? 3x – 6y – 9 x they might see the immediate possibility that:

3x – 6y – 9 = 3 – 6y – 9 = -6y - 6 1x

25

Horizontal planning needs

• Teachers must agree to instructional process on challenging materials

• A consistent delivery of effective instruction

• Focus on explicit instruction for the most troublesome concepts

• Reasoning first; shortcuts after mastery

26

Math Instruction Consistency

• Steps to Teaching at College Park ES in Berkeley Co.

– Engagement (word problems and challenges)

– Explore (high engagement)

– Explain (teach how)

– Extend (Apply and practice with formative assessment)

– End with closure

27

10 instructional principles linked to effective instruction of new material

1. Begin a lesson with a short review of previous learning. 2. Present new information in small steps with student practice

after each step. 3. Ask a large number of questions and check the responses of

all students. 4. Provide models. 5. Guide student practice. 6. Check for student understanding. 7. Obtain a high success rate. 8. Provide scaffolds for difficult tasks. 9. Require and monitor independent practice. 10. Engage students in weekly and monthly review . (Rosenshine, 2012, p.12)

28

Scaffolding Instruction

The National Mathematics Advisory Panel stated that “Explicit systematic instruction typically entails teachers explaining and demonstrating specific strategies and allowing students many opportunities to ask and answer questions and to think about the decisions they make while solving problems” (p.48).

29

Avoid unguided instruction

• “Decades of research clearly demonstrate that for novices (comprising virtually all students), direct, explicit instruction is more effective and more efficient than partial guidance” (Clark, Kirschner, & Sweller, 2012, p. 6).

• “..teachers are more effective when they provide explicit guidance accompanied by practice and feedback, not when they require students to discover many aspects of what they must learn” (Clark, Kirschner, & Sweller, 2012, p. 6).

30

Division of Fractions example

• I do it (1/2)÷(3/4) = _____ (2/3) ÷(4/3) = _____

• We do it

(1/5)÷(1/4) = _____ (1/3) ÷(2/3) = _____

(3/8)÷(1/2) = _____ (1/2) ÷(2/4) = _____

• You do it (1/3)÷(2/5) = _____ (2/3) ÷(1/5) = _____

(3/5)÷(1/8) = _____ (3/4) ÷(2/3) = _____

31

2012-2013 Project: Student-created Video Modeling

1. Teach a difficult skill to mastery; 2. Present a problem for student to independently

solve; 3. Video students solving the problem and

explaining their reasoning; 4. Show the problem solving to others in the same

class; 5. Use these videos for future classes. What are the potential effects?

© Witzel, 2012 32

Sense-making: converting fractions

Convert this mixed fraction into an improper fraction.

4 2/5 How did you know how to do it? Did you… a. 4x5 b. + 2 = 22 c. 22/5 Why? Say, “Four and two – fifths” 4/1 + 2/5 or 20/5 + 2/5 = 22/5. 33

Sense-making: Division of Fractions

• Why is it that when you divide fractions, the answer is larger? Also, why do you invert and multiply?

• 2/3 divided by 1/4 = 2/3 (4/1) = 8/3

2/3 (4/1) 8/3 8/3 8/3 1/4 (4/1) 4/4 1/1

34

Why teach the basics correctly

35

Adding with unlike denominators

Division of fractions

Simple graphing: Y = 2/3x - 2

36

1) Plot the y-intercept

2) Use the coefficient of x as the slope and go rise (numerator) then run (denominator)

3) Make a point 4) Connect and

continue to make a line

5) But wait!

Take a closer look: Y = 2/3x - 2

37

The rise over run is actually 2/3 over 1

Planning

a) List some of the ways to improve instruction for students with math disabilities and early difficulties?

b) How can we ensure these practices consistently? Provide some steps or ideas.

38

RtI- Tiered Instruction and Intervention

Tier 1: Benchmark

School-wide (not Title I) research-supported instruction available to ALL

students including standards-aligned concepts and competencies, and

instruction.

Tier 2: Strategic

Academic and behavioral strategies, methodologies and practices designed

for students not making expected progress in the standards-aligned system.

These students are at risk for academic failure.

Tier 3: Intensive Interventions

Academic and behavioral strategies, methodologies and practices designed

for students significantly lagging behind established grade-level

benchmarks in the standards aligned system. 39

Behavioral Systems

Intensive, Individual

Interventions

•Individual Students

•Assessment-based

•High Intensity

•Of longer duration

Targeted Group

Interventions

•Some students (at-

risk)

•High efficiency

•Rapid response

Universal

Interventions

•All students

•Preventive,

proactive

Academic Systems

1-5% 1-5%

5-10% 5-10%

80-90% 80-90%

Intensive, Individual

Interventions

•Individual Students

•Assessment-based

•Intense, durable

procedures

Targeted Group

Interventions

•Some students (at-

risk)

•High efficiency

•Rapid response

Universal

Interventions

•All settings, all

students

•Preventive,

proactive

RtI

40

From Tilley, W. D. (in press). The evolution of school psychology to science-based practice. In A. Thomas & J. Grimes (Eds.) Best practices in school psychology (5th ed.). Silver Springs, MD: National Association of School Psychologists.

Essential Elements of RtI

1. Strong “Tier 1” programming/screening

2. Problem definition (initial assessment)

3. Research evidenced programs and implementation

4. Referral to intervention (identification of students who struggle not necessarily special education)

5. Intervention selection

6. Intervention implementation (fidelity)

7. Decision-making/outcome evaluation

41

42

Shortcut of RtI Ideas

Tier 1: Explicit instruction; teach the how and why;

concrete prompts; teacher think alouds; differentiation; PALS; whole class

Tier 2: CRA; student think alouds; strategy

instruction; extra practice; small group

Tier 3: Individually designed curricula per student

needs with a slower pace; completely interactive

Research and support provided by: The National Mathematics Panel and Gersten, Baker, and Chard with the Center on Instruction in Math

What have you implemented so far? (based on the IES Math RtI Practice Guide, Gersten, et al, 2009)

Category Recommendation Yes No How

Overall plan Assessment

Screening all students to identify those who need interventions

Intervention content

Interventions that focus on whole numbers (K-5) and rational numbers (4-8)

Intervention instruction

Interventions are taught explicitly

Intervention content

Structural word problem instruction

Intervention instruction

Interventions include visual representations

Intervention content

Interventions include at least 10 minutes on fluent fact retrieval

Assessment Progress monitoring for those receiving interventions as well as those at-risk

Intervention instruction

Motivational strategies for those in interventions

43

44

Instruction and Intervention

• Focus on Early Numeracy in math (ES) • Whole Number Operations (ES and MS) • Fractions (ES, MS, HS) • Integers (MS and HS) • Equations (MS and HS) • Applications of Whole and Rational Numbers (MS and MS) • Word Problems and Problem Solving approaches (ES, MS,

HS)

• Although an oversimplified approach, it is an important start

45

“Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.”

Albert Einstein

46

Assessment Cycle extras – see Riverside Publishing

47

Screening

Progress Monitoring

Review of Student Records

These assessment extras do not eliminate the need for daily error pattern analysis, chapter quizzes and tests, HW, or even diagnostics.

Q1 Q2 Q3 Q4

48

Set up a Screening Plan

• Implement screening at least twice / year.

• Examine each tool for its content to match to the CCSS.

• Use the screening tool in connection with SBAC.

• Use the same screening tool across your district.

• Warning – Computational Fluency alone has had low predictive validity as a screener..

• Options: MAP; STAR; Aimsweb; Voyager; EasyCBM; MBSP; iSTEEPetc…

49

50

Column A Column B

Name _______________________________ Date ________________________ Test 4 Page 1

Applications 4

• N

• M

(B) L K • •

• Z (A)

(C)

(D)

point

ray

line segment

line

Write the letter in each blank.

(1)

Look at this numbers.:

356.17

Which number is in the hundredths place?

(2)

(3)

Jeff wheels his wheelchair for 33 hours a week at school and for 28 hours a week in his neighborhood. About how many hours does Jeff spend each week wheeling his wheelchair?

Solve the problem by estimating the sum or difference to the nearest ten.

(4)

Write the number in each blank.

3 ten thousands, 6 hundreds, 8 ones

2 thousands, 8 hundreds, 4 tens, 6 ones

(5)

Write a number in the blank.

1 week = _____ days

(6) Vacation Plans for Summit School Students

Summer School

Camp

Travel

Stay home

0 20 10 30 40 50 60 70 80 90 100

Number of Students

The P.T.A. will buy a Summit School T-Shirt for each student who goes to summer school. Each shirt costs $4.00. How much money will the P.T.A. spend on these T shirts?

How many students are planning to travel during the summer?

How many fewer students are planning to go to summer school than planning to stay home?

Use the bar graph to answer the questions.

(A) meters

(B) centimeters

(C) kilometers

To measure the distance of the bus ride from school to your house you would use

(7)

$ .00

One page of a three-page measure for mathematics concepts and applications (24 problems total)

Measure taken from Monitoring Basic Skills Progress: Basic Math Concepts and Applications (1999)

SBAC – Assessment www.smarterbalanced.org

Smarter Balanced is a state-led consortium developing assessments aligned to the Common Core State Standards in English language arts/literacy and mathematics that are designed to help prepare all students to graduate high school college- and career-ready. Uses computer adaptive testing Timeline: 2012-13 pilots conducted 2013-14 – broad field tests Full implementation by 2014-15 SC is a governing state

“South Carolina has been an active participant in the Consortium with representatives serving on three of the 10 state-led Smarter Balanced work groups: Accessibility and Accommodations, Formative Assessment Practices and Professional Learning, and Test Design,” said Joe Willhoft, Ph.D., executive director of Smarter Balanced. “I know South Carolina’s decision to become a Governing State went through a thorough and thoughtful process. We are excited about continuing to learn from the Palmetto State’s expertise with high-quality assessments as we develop and implement the Smarter Balanced system.”

51

Accessibility and Accommodations

• An individual SBAC work group

• First draft is currently under review

• Charge: Address accessibility and accommodations for Consortium summative, interim, and formative instruments; provide definitions and background information on target populations; create supporting documents and guidelines for implementation.

• http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/03/Work-Group-Governance.pdf

52

Which of these classroom accommodations will transfer over to SBAC?

• Extra wait time • Procedures clarification • Minimize classroom

distractions • Homework reminders and

planners • Weekly progress report and

home checks • Increased 1:1 assistance • Peer tutoring or reciprocal

teaching • Homework from previous

week

• Classroom signals for attention

• Visual organizer • Scribe or notetaker • Guided notes • Shortened assignments • “Chunked” lesson of brief

assessed activities throughout a lesson

• Frequent praise to teach proper academic and social behaviors

Any more????

53

Why won’t these transfer?

• Altered grading procedures

• Alternate but related standard during lesson

• Different reading assignments

• Different questions

• Alternate assessment content and / or expectations

• Elimination of parts of assignments if they remove a standard

• Calculator during math fluency assignment

54

We must teach differently to prepare our students

55

Concrete introductions

• Manipulatives to explain reasoning

• Visuals and arrays

• Automaticity

• Think alouds

56

Connect Algebra to Arithmetic: Build on what they know

• 8 ÷ 2

• 8 ÷ 3

• 13 ÷ 4

57

• The National Math Panel (2008) concluded that use of visual representations, such as the CRA sequence of instruction is a powerful instructional tool.

• Highest effect sizes with secondary students were from CRA instruction (Gersten et al., 2009;

Witzel, Mercer, & Miller, 2003; Witzel, 2005).

Mathematics

58

CRA

• Concrete to Representational to Abstract Sequence of Instruction (CRA)

• Concrete (expeditious use of manipulatives)

• pictorial Representations

• Abstract procedures

Excellent for teaching accuracy and understanding

• http://www.coedu.usf.edu/main/departments/sped/mathvids/index.html

59

CRA (Gersten et al, p. 35)

60

Let’s try some CRA from Multisensory Algebra and the Middle School

Intervention Series

• From the basic facts

• To variations that often cause trouble

• To crazy things like fractions

• To simple equations

61

62

Witzel (2004) Multisensory Algebra Guide

63

Witzel (2004) Multisensory Algebra Guide

64

Witzel (2004) Multisensory Algebra Guide

List what else can be used with CRA. ex. Trigonometric ratios

65

hyp

opp

adj

Measurement - relates the concepts of measurement to similarity and proportionality in real-world situations.

Use visuals that explain processes: Arrays / Matrices

66

Connecting arrays and matrices Physics department, NYU

Your Turn: Draw an array to show 2/5 x 2/4

• 5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

67

68

(2/5) (2/4)

Your Turn: Make an array to show (1 1/3)(2 1/2)

Total = ?

69

1 1

1

1/3

1/2

Answer Make an array to show (1 1/3)(2 1/2)

Total = 1 + 1 + 1/2 + 1/3 + 1/3 + 1/6 = Refer to the grid above = 6/6 + 6/6 + 3/6 + 2/6 +

2/6 + 1/6 = 20/6

70

1 1

1

1/3

1/2

Your Turn: Make an array using abstract terms

to show (2 1/3)(4 1/2)

71

• Fifth grade “Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.”

• (2 1/3) (4 1/2)

8 + 4/3 + 1 + 1/6 = 8 + 8/6 + 1 + 1/6 = 10 3/6

72

multiply 2 1/3

4 8 4/3

1/2 2/2 1/6

Your Turn: Make an array to show 7.6 x 2.4

• Try using abstract/Arabic terms only

73

• Fifth grade “Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.”

• 7.6 x 2.4 = ?

14+1.2+2.8+0.24 = 18.24

74

multiply 7 .6

2 14 1.2

.4 2.8 .24

Your Turn: Make an array to show (2x + 5)(x – 4)

• Total = ?

75

Answer: Make an array to show (2x + 5)(x – 4)

Total = x2 + x2 + + 5x – 4x – 4x – 20

= 2x2 – 3x – 20

76

x -4

x

+5

x2

-4x

x2

-4x

5x

-20

x

Your Turn: Make an array to show (3x – 1)(4x + 5)

Try another form of array

77

• Algebra “Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.”

• (3x – 1)(4x + 5)

12x2 – 4x + 15x – 5 = 12x2 + 11x - 5

78

multiply 3x -1

4x 12x2 -4x

+5 15x -5

Develop stepwise graphic

organizers

Miller, teacher, Oconee County HS http://moodle.oconee.k12.ga.us/course/view.php?id=810&topic=7

79

Common Core on Automaticity

• CC.3.OA.7 Multiply and divide within 100. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of one-digit numbers.

80

Practice operational facility to gain fluency and automaticity

• The RtI Panel (Gersten, Beckman, Clarke, Foegen, Marsh, Star, and Witzel, 2009) concluded that all students (K-8) receiving interventions should receive at least 10 minutes of practice per day in fact fluency.

• K-5 should focus on whole numbers

• 4-8 should focus on rational numbers

81

Make time for extra practice (Chapman, 2010)

35

I have 35. Who has 6 groups of 8?

82

Language

• Many of the questions and problem solving activities evolve from word problems and narratives.

• Teach as a separate language including explicit instruction of word problems and problem solving approaching.

• Prepare students for the assessment and activities by incorporating appropriate language throughout.

83

Extend Think Alouds to increase Student Think-Alouds

• Encouraging students to verbalize their thinking and talk about the steps they used in solving a problem – was consistently effective.

• Verbalizing steps in problem solving was an important ingredient in addressing students’ impulsivity directly.

(Schunk & Cox, 1986)

84

Cognitively Guided Instruction

• CGI uses each student’s own math thinking as a basis for instruction and guided practice.

• Thus, it is an excellent way to follow teacher think alouds.

http://cognitivelyguidedinstruction.com/index.html

• What can we do to incorporate more guided instructional practice?

85

Increased Student Independence

• Authentic Experiences – For struggling students,

engage them in authentic experiences after mastery of skills

– For students with average to above average success, ensure modeling throughout the authentic experience and include practice afterwards

• Problem Solving – Problem solving is

everywhere; well outside of the math classroom.

– Help students see how to interact with the world using mathematics

– Review word problems as they do not always align accurately with student’s reading levels

86

Improving Mathematical Problem Solving Practice Guide (Woodward et al., 2012)

• Recommendation 1. Prepare problems and use them in whole-class instruction

• Recommendation 2. Assist students in monitoring and reflecting on the problem-solving process

• Recommendation 3. Teach students how to use visual representations

• Recommendation 4. Expose students to multiple problem-solving strategies

• Recommendation 5. Help students recognize and articulate mathematical concepts and notation

87

Recommendation 1

• “Problem solving must be an integral part of each curricular unit, with time allocated for problem-solving activities with the whole class. In this recommendation, the panel provides guidance for thoughtful preparation of problem-solving lessons. Teachers are encouraged to use a variety of problems intentionally and to ensure that students have the language and mathematical content knowledge necessary to solve the problems” (Woodward, et al., 2012, p. 10).

88

Daily: Use problems that are routine and non-routine

89

1. Carlos has a cake recipe that calls for 2 3/4 cups of flour. He wants to make the recipe 3 times. How much flour does he need? This problem is likely routine for a student who has studied and practiced multiplication with mixed numbers. 2. Solve 2y + 15 = 29 This problem is likely routine for a student who has studied and practiced solving linear equations with one variable. 3. Two vertices of a right triangle are located at (0,4) and (0,10). The area of the triangle is 12 square units. Find a point that works as the third vertex. This problem is likely routine for a student who has studied and practiced determining the area of triangles and graphing in coordinate planes.

(Woodward et al., 2012)

Present non-routine problems

90

1. Determine angle x without measuring. Explain your reasoning. parallel 155° 110°x This problem is likely non-routine for a student who has only studied simple geometry problems involving parallel lines and a transversal. 2. There are 20 people in a room. Everybody high-fives with everybody else. How many high-fives occurred? This problem is likely non-routine for students in beginning algebra. 3. Solve for the variables a through f in the equations below, using the digits from 0 through 5. Every digit should be used only once. A variable has the same value everywhere it occurs, and no other variable will have that value. a + a + a = a2 b + c = b d × e = d a – e = b b2 = d d + e = f The problem is likely non-routine for a student who has not solved equations by reasoning about which values can make an equation true. 4. In a leap year, what day and time are exactly in the middle of the year? This problem is likely non-routine for a student who has not studied problems in which quantities are subdivided into unequal groups.

(Woodward et al., 2012)

Clarify Vocabulary and Context of Problems

Example Problem Vocabulary Context

In a factory, 54,650 parts were made. When they were tested, 4% were found to be defective. How many parts were working?

Students need to understand the term defective as being the opposite of working and the symbol % as percent to correctly solve the problem.

What is a factory? What does parts mean in this context?

At a used-car dealership, a car was priced at $7,000. The salesperson then offered a discount of $350. What percent discount, applied to the original price, gives the offered price?

Students need to know what offered and original price mean to understand the goal of the problem, and they need to know what discount and percent discount mean to understand what mathematical operators to use.

What is a used-car dealership?

91

(Woodward et al., 2012)

92

Beware textbook definitions: MS (Witzel, 2009)

• Diameter is a line segment that passes through the center of a circle and has endpoints on the circle, or the length of that segment.

• Compensation: When a number in a problem is close to another number that is easier to calculate with, the easier number is used to find the answer. Then the answer is adjusted by adding to it or subtracting from it.

93

Sample definitions from major math textbooks: Middle School (Witzel, 2009)

• Polyhedron: A polyhedron is a three-dimensional object, or solid figure, with flat surfaces, called faces, that are polygons.

• Upper quartile: the median of the upper half of the data; also called third quartile.

94

Math reading confusion in MS (Witzel, 2009)

6th grade

readability (variables,

expression,

least common

multiple and

stem and leaf

plot )

Directions

grade

level

range

Word

Proble

m

grade

level

range

Popular

Textbook

program 1

4th-10th 5th-7th

Popular

Textbook

program 2

5th-11th 5th-9th

Popular

Textbook

program 3

5th-17th 5th-6th

8th grade

readability (coordinate

plane and

function)

Directions

grade

level

range

Word

Problem

grade

level

range

Popular

Textbook

program 1

7th-10th 6th-7th

Popular

Textbook

program 2

9th-10th 10th-12th

Popular

Textbook

program 3

7th-9th 7th-12th

95

Example: Intercept (sounds like- intersection)

The place where a line, curve or surface crosses an axis

(adapted from Riccomini, 2006)

Recommendation 2

• Assist students in monitoring and reflecting on the problem-solving process.

• “…the panel suggests that teachers help students learn to monitor and reflect on their thought process when they solve math problems. While the ultimate goal is for students to monitor and reflect on their own while solving a problem, teachers may need to support students when a new activity or concept is introduced” (p.17).

96

(Woodward et al., 2012)

Model problem solving prompts

97

• What is the story in this problem about? • What is the problem asking? • What do I know about the problem so far ? What information is given to

me? How can this help me? • Which information in the problem is relevant? • In what way is this problem similar to problems I have previously solved? • What are the various ways I might approach the problem? • Is my approach working? If I am stuck, is there another way I can think

about solving this problem? • Does the solution make sense? How can I verify the solution? • Why did these steps work or not work? • What would I do differently next time? Note: These are examples of the kinds of questions that a teacher can use as prompts to help students monitor and reflect during the problem-solving process. Select those that are applicable for your students, or formulate new questions to help guide your students.

Math Discourse

Dr. Deborah Ball on math discourse

http://www-personal.umich.edu/~dball/multimedia/index.html

98

Recommendation 3

• Teach students how to use visual representations

• Include such visuals as:

– Strip diagrams use rectangles to represent quantities presented in the problem.

– Percent bars are strip diagrams in which each rectangle represents a part of 100 in the problem.

– Schematic diagrams demonstrate the relative sizes and relationships between quantities in the problem.

99

(Woodward et al., 2012)

Ratio problem solving (Adapted from math/about.com)

Manuel is making snowballs during a snowstorm. He can make 40 snowballs in a hour but 5 snowballs melt every 20 minutes. How long will is take to build 220 snowballs?

What trends appear in this problem?

100

minutes 60 80 100 120 140 160 180

Snowballs 40 80 120 160 200 240 280

Lost to melting 5 10 15 20 25 30 35

Total number of snowballs

35 70 105 140 175 210 245

Strip diagrams (Gersten et al., 2009)

101

Recommendation 4

• Expose students to multiple problem-solving strategies.

• Guess and check vs. Schema-based

102

(Woodward et al., 2012)

From Herlong, 2010

103

From Herlong, 2010

104

Schema-based strategies (see the work of Jitendra and Montague)

• Vary Size of groups unknown: – In school, there are 12 calculators for 24 students to share. How many

students will share each calculator?

• Whole unknown: – Isabelle earned $30 for each day that she worked at the

church store. She worked for 5 days. How much money did she earn?

• Referent unknown (compared is part of referent): Laura

and Isha went running. Isha ran 8 laps. She ran 1/2 as many laps as Laura. How many laps did Laura run?

105

Sample Word Problems

1. The map shows that you have traveled 4 out of 10 miles on your trip. Your friend tells you that you are 3/5 of the way there. Are you 3/5 there? Show why or why not using abstract notation.

2. The temperature was 4 degrees below zero. Recently, the temperature rose up 5 degrees. What is the temperature as it relates to zero now? (1) Set up the equation; (2) Explain your reasoning; and (3) Solve the problem.

106

Change Problem (Gersten et al, p.27)

107

Compare problem (Gersten et al, p. 28)

108

Recommendation 5

• Help students recognize and articulate mathematical concepts and notation.

• Increase verbalization in class, particularly on troublesome content.

109

(Woodward et al., 2012)

Incorporate Frequent Formative Assessment

• Formative assessment is the process of collecting data on samples of curriculum or robust indicators at regular intervals (e.g. biweekly or even daily).

• Evidence has shown that this approach is superior to use of weekly or biweekly textbook tests.

110

111

Random numerals within problems (considering specifications of problem types)

Random placement of problem types on page

Sample CBM Math Computation Probe

How do speducators spend their time? (Wasburn-Moses, 2009)

Greater than 2 hours

1. Reading and writing = 42.3%

2. Teaching content = 36.6%

3. Consulting with caseload = 32.3%

4. Completing paperwork = 27.3%

5. Working with gen ed = 19.1%

6. Teaching study skills = 10.3%

7. Working with admin = 3.1%

8. Working with parents = 0.8%

None

1. Working with parents = 13.0%

2. Reading and writing = 11.5%

3. Teaching study skills = 11.1%

4. Working with admin = 10.7%

5. Teaching content = 8.4%

6. Working with gen ed = 8.4%

7. Consulting with caseload = 7.7%

8. Completing paperwork = 0.0%

112

Collaboration: Whose job is it? (Speducator or Geneducator?)

Item Who has authority here? Rationale

Referrals to special education

Special education screening and identification

Research-supported instructional delivery

Meeting students’ needs in the general education class

Meeting students’ needs in intervention setting

Meeting students’ needs in special education

113

Team developer • Lead and Support

One teacher leads and another offers assistance and support to individuals or small groups. In this role, planning must occur by both teachers, but typically one teacher plans for the lesson content, while the other does specific planning for students' individual learning or behavioral needs.

• Station Teaching Students are divided into heterogeneous groups and work at classroom stations with each teacher. Then, in the middle of the period or the next day, the students switch to the other station. In this model, both teachers individually develop the content of their stations.

• Parallel Teaching Teachers jointly plan instruction, but each may deliver it to half the class or small groups. This type of model typically requires joint planning time to ensure that as teachers work in their separate groups, they are delivering content in the same way.

• Alternative Teaching One teacher works with a small group of students to pre-teach, re-teach, supplement, or enrich instruction, while the other teacher instructs the large group. In this type of co-teaching, more planning time is needed to ensure that the logistics of pre-teaching or re-teaching can be completed; also, the teachers must have similar content knowledge for one teacher to take a group and re-teach or pre-teach.

• Team Teaching Both teachers share the planning and instruction of students in a coordinated fashion. In this type of joint planning time, equal knowledge of the content, a shared philosophy, and commitment to all students in the class are critical. Many times teams may not start with this type of format, but over time they can effectively move to this type of co-teaching, if they have continuity in working together across 2-3 years.

• http://www.specialconnections.ku.edu/cgi-bin/cgiwrap/specconn/main.php?cat=collaboration&section=main&subsection=coteaching/types#top

114

Summation

Common Core is sure to shake things up

Start preparations a) instructionally and b) curricular

a) Instructional: horizontal planning; approaches; concrete and visual representations; word problems; formative assessment

b) Curricular: RtI; vertical planning; coteaching; screening

115