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Responsiveness To Instruction: Mathematical Understanding Rowan-Salisbury Schools RtI Foundations Training August 2010 Erin Banks & Sarah Brown, School.

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Presentation on theme: "Responsiveness To Instruction: Mathematical Understanding Rowan-Salisbury Schools RtI Foundations Training August 2010 Erin Banks & Sarah Brown, School."— Presentation transcript:


2 Responsiveness To Instruction: Mathematical Understanding Rowan-Salisbury Schools RtI Foundations Training August 2010 Erin Banks & Sarah Brown, School Psychologist and Program Specialist

3 When you need help with math…… Call Somebody!!! y=9A7EF76E53A5E67C y=9A7EF76E53A5E67C

4 Its not MY Fault!!!

5 How Our Students Feel About Numbers.....

6 The Ultimate Test Anxiety...

7 And Why Its All Worth It.....

8 Whats Going on With Math? NAEP 2009 results were released in March 2010 How do you think NC math results compare to: United States norms? Reading results?

9 NAEP (2009) Math Results:

10 In 2009, the average score in North Carolina was lower than those in 4 states/jurisdictions higher than those in 29 states/jurisdictions not significantly different from those in 18 states/jurisdictions (

11 Test Yourself! Sample 4th grade NAEP Math test question It takes Mrs. Wylie 15 minutes to drive from her house to the store. What is the best estimate of the distance from her house to the store? 1. 5 feet 2. 5 miles 3. 20 feet 4. 200 miles

12 Are You Smarter than an 8th Grader? Which point is the solution to both equations in the graph? 1.(0,0) 2.(0,4) 3.(1,1) 4.(2,2) 5.(4,0)

13 National Mathematics Panel Report 2008: What do students need for success in Algebra? Major Findings: Proficiency with whole numbers, fractions and certain aspects of geometry and measurement are the critical foundations of algebra Explicit instruction for students with disabilities shows positive effects. Students need both explicit instruction and conceptual development to succeed in math.

14 Math Basics International Research

15 TIMSS from Improving Mathematics Instruction (Ed Leadership 2/2004) 1995 Video Study Largest and most carefully designed study of teacher instruction and mathematics More than 500,000 students (33,000 in US) Grades 4, 8, 12 41 nations participated Results = 4th grade students scored ABOVE international average; BELOW average in measurement, estimation, and number sense! 8th grade students scored BELOW international average overall, including: Measurement, geometry, and proportionality Outperformed by 20 countries 12th grade students scored WELL BELOW international average (outperformed Cyprus and South Africa)

16 TIMSS from Improving Mathematics Instruction (Ed Leadership 2/2004) 1999 Follow-up Study Examined lessons in Grade 8 in United States, Germany, and Japan US teachers teach student HOW to get answers while Japanese teachers teach for UNDERSTANDING US teachers usually dont develop math concepts and ideas US teachers taught material with low-level math content vs. other teachers using high-level math content

17 What Do We Need to Do Differently? Help students develop understanding of math concepts Teach lessons to challenge students (using high level content)

18 TIMSS (2007) Study U.S. students' average mathematics score was 529 for 4th-graders (500 is average) 508 for 8th-graders (500 is average) Fourth-graders in 8 of the 35 other countries that participated in 2007 (Hong Kong, Singapore, Chinese Taipei, Japan, Kazakhstan, Russian Federation, England, and Latvia) scored ABOVE their U.S. peers Among the 16 countries that participated in both the first TIMSS in 1995 and the most recent TIMSS in 2007, at grade 4, the average mathematics score increased in 8 countries, including in the United States, and decreased in 4 countries. Higher than 1995!

19 Style vs.. Implementation High Achieving countries use a variety of styles to teach (calculator vs.. no calculator, real-life problems vs.. naked problems) High Achieving countries all implement connections problems as connections problems U.S. implements connection problems as a set of procedures NCDPI RtI Foundations Training

20 Exponents and Geometry What is 4 2 ? Why is it 4 x 4 when it looks like 4 x 2? It means make a square out of your 4 unit side Making Connections... NCDPI RtI Foundations Training

21 Exponents and Geometry What is 4 2 ? --4 units-- 1 Youd get how many little 1 by 1 inch squares?4 2 = 16 NCDPI RtI Foundations Training

22 Some words about Key Words They dont work… NCDPI RtI Foundations Training

23 We tell themmore means add Erin has 46 comic books. She has 18 more comic books than Jason has. How many comic books does Jason have. But is our answer really 64 which is 46 + 18? NCDPI RtI Foundations Training

24 Sense-Making We need to notice if we are making sense of the math for our students, or if our discussion of the math contributes to --- The suspension of sense-making Schoenfeld (1991) NCDPI RtI Foundations Training

25 Just Do It! NCDPI RtI Foundations Training

26 Content Standards Number & Operations Algebra Geometry Measurement Data Analysis & Probability Process Standards Problem Solving Reasoning & Proof Communication Connections Representation

27 What does NC SCOS tell us? Number and Operations Measurement Data Analysis & Probability Algebra Problem-Solving (uses the same 5 strands as NCTM Principles)

28 What do researchers tell us? Use Direct and Explicit Instruction to focus upon: Number Sense Basic math operations Problem-solving skills (Kroesbergen & Van Luit, 2003)

29 Why Intervention in Math? Students have difficulty with: Mastering math computation skills or Application of math Speed and Accuracy are IMPORTANT! Students who cannot master basic computational skills are very unlikely to succeed at application of those skills. (Shapiro, 2004)

30 Number Sense Computation Prerequisite ……in the same way as……… Phonemic Awareness Prerequisite Reading Fluency

31 What is Number Sense? Manipulate Numbers Adding on Counting up Skip counting One-to-One correspondence Counting steps as they walk down/up Understanding of how number systems work

32 NCTM Strands Without Number Sense Number & Operations Measurement Geometry Data Analysis and Probability Algebra

33 NCTM Strands With Number Sense Number & Operations Measurement Geometry Data Analysis and Probability Algebra

34 Fluency (Automaticity) Strong correlation (relationship) between poor retrieval of arithmetic combinations (math facts) and global math delays Automatic recall of arithmetic combinations frees up student cognitive capacity to allow for understanding of higher-level problem-solving Working memory!!!!

35 How to Increase Fluency PRACTICE!!!!!!!! However, practice alone may not be sufficient Area of concernInterventions Speed and Fluency 1. Immediate feedback 2. Contingent free time 3. Interspersal of easy and difficult material on worksheets (folding-in technique) Speech, Fluency, AND accuracy 1. Using simple reminders 2. Reminding students of steps 3. Visual representations of problem

36 Language in Mathematics NCDPI RtI Foundations Training

37 Are these the same? NCDPI RtI Foundations Training

38 Break During the break, read the article: Early Identification and Interventions for Students With Mathematics Difficulties (Gersten, Jordan & Flojo, 2005) Discussion upon returning to group

39 Early Identification 1. Describe the nature of math difficulties. 2. How are mathematical difficulties and reading difficulties related? 3. What is number sense as operationalized by Kalchman, Moss and Case in this article? 4. What are the two distinct factors in mathematics proficiency in kindergartners? 5. What is the role of socioeconomic status as stated by Griffin, Case and Sigler in 1994? 6. What are the instructional implications for the findings? NCDPI RtI Foundations Training

40 Tier I Consultation Between Teachers- Parents Tier II Consultation With Other Resources AMOUNT OF RESOURCES REQUIRED TO MEET THE STUDENTS NEEDS INTENSITY OF NEEDS Nds-circles-pub Tier IV IEP Consideration Tier III Student Study Team Intensive Interventions 1-7% Strategic Interventions 5-15% Core Curriculum 80-90% How Does Math Fit into the RtI Model? NCDPI RtI Foundations Training

41 What Works Clearinghouse InterventionEvidenceExtent of Evidence Odyssey Math +Small Everyday Math +Medium to Large Progress in Math 0Small Scott Foresman-Addison Wesley Elementary Math (enVision) 0Small Houghton Mifflin Math 0Medium to Large Saxon Elementary Math 0Small


43 Tier I - Math Characteristics: 1. Classroom teacher 2. Core Math instructional time (60 minutes recommended) 3. Benchmarks - Fall, Winter, Spring Instruction: 1. Quality Lesson Design 2. Research-Based Strategies 3. Differentiated Instruction 4. Explicit Instruction 5. Questioning 6. Connections

44 Assessment At Tier I Universal Screening/Benchmarking Fall, Winter, Spring Investigating Further.... Number Knowledge test (handout) Money test (handout) Use Data to: Develop groups for differentiation Groups for supplemental intervention/instruction Groups for enriched instruction

45 Differentiated Instruction in Math

46 What is Differentiated Instruction? A teaching theory based on the premise that instructional approaches…. …should vary and be adapted… …in relation to individual and diverse students in classrooms.


48 Ways to Differentiate in Math: Manipulate Content/Topic Manipulate Process/Activities Manipulate Product Manipulate Environment How have you been successful using these techniques in your classroom?

49 Manipulation of Content Multiple Options for Taking in Information Classroom Objective: All students will subtract using renaming. Manipulation: Some students may learn to subtract two-digit numbers, while others learn to subtract larger numbers in the context of word problems. (Tomlinson, 1999)

50 Manipulation of Process Multiple Options in which a Student Accesses Material One student may explore a learning center, while another student collects information from the web. (Tomlinson, 1999)

51 Manipulation of Project Multiple Options for Expressing What They Know To demonstrate understanding of a geometric concept, one student may solve a problem set, while another builds a model. Tomlinson, 1999

52 Manipulation of Learning Environment Multiple Ways to Create an Environment to Stimulate Learning Create places in the room to work quietly and without distraction Provide materials that reflect a variety of cultures and home settings Set out clear guidelines for independent work that matches individual needs Develop routines that allow students to get help when teachers are busy with other students and cannot help immediately Allow learners to move around if it increases learning (Tomlinson, 1999)

53 Strong Tier I Math Instruction Tiered Assignments Adjusting Questions Compacting Curriculum Learning Centers/Groups Choice Boards Flexible Grouping Acceleration/Deceleration Peer Teaching Independent Study Projects Anchoring Projects

54 Tiered Assignments Focus of Differentiation: Readiness Example: In a unit on measurement, some students are taught basic measurement skills, including using a ruler to measure length. Other students can apply measurement skills to problems involving perimeter.

55 Adjusting Questions Focus of Differentiation: Readiness Example: During large group discussion activities, teachers direct higher level questions to students who can handle them and adjust questions accordingly for students with greater needs. With written quizzes, teacher may assign specific questions for each group of students. They all answer the same number of questions, but complexity required varies from group to group.

56 Compacting Curriculum Focus of Differentiation: Readiness Example: A 3rd grade class is learning to identify the parts of fractions. Diagnostics indicate that two students already know the parts of fractions. These students are excused from completing activities and are taught to add and subtract fractions.

57 Interest Centers/Groups Focus of Differentiation: Readiness, Interest Example: Interest Centers - centers can focus on specific math skills, such as addition, and provide activities that are high interest such as counting jelly beans or adding eyes on aliens. Interest Groups - students work in small groups to research a math topic of interest such as how geometry applies to architecture or how math is used in art.

58 Flexible Grouping Focus of Differentiation: Readiness, Interest, Learning Profiles Example: The teacher may assign groups based on readiness for direct instruction on algebraic concepts and allow students to choose their own groups for projects that investigate famous mathematicians.

59 Learning Contracts Focus of Differentiation: Readiness, Learning Profile Example: A student decides to follow a football team over a 2-month period and make inferences about players performances based on scoring patterns. The student develops a plan for collecting and analyzing data and present to class.

60 Choice Boards Focus of Differentiation: Readiness, Interest, Learning Profile Example: Students can choose to complete an inquiry lesson where they measure volume using various containers, use a textbook to read about measuring volume, or watch a video where steps are explained. The activities are based on visual, auditory, kinesthetic, and tactile learning. Students must complete two activities from the board and choose these activities from two different learning styles.

61 Acceleration/Deceleration Focus of Differentiation: Readiness Example: Students demonstrating a high level of competence can work through the curriculum at a faster pace. Students experiencing difficulties may need adjusted activities that allow for a slower pace in order to experience success.

62 Independent Study Projects Focus of Differentiation: Readiness, Interest, and Learning Profiles Example: Assign a research project where students learn how to develop the skills for independent learning. The degree of help and structure will vary between students and depend on their ability to manage ideas, time, and productivity.

63 Peer Teaching Focus of Differentiation: Readiness, Interest, Learning Profiles Example: Occasionally, a student may have personal needs that require one-on-one instruction that go beyond the needs of peers. After receiving this instruction, the student could be designated as the resident expert for that concept or skill and get practice by teaching others. Both students would benefit.

64 Anchoring Activities Focus of Differentiation: Readiness Example: List of activities that a student can do at any time when they have completed present assignments Short assignments at the beginning of each class as students organize themselves and prepare for work. **These activities must be worth of a students time and appropriate to learning needs.**

65 Tier II/III - Math Instruction Direct/Explicit Instruction Components of Number Sense Use of appropriate LANGUAGE Concrete-Representational-Abstract Approach (CRA) Small group/individualized instruction Frequent monitoring of progress

66 IES Recommendations - Tier II, III Interventions should: Include instruction on how to solve word problems that is based on common underlying structures Intervention materials need to include opportunities for students to work with visual representations of math ideas Interventions at all grade levels should devote about 10 minutes to building fluent retrieval of basic arithmetic facts!!!!!

67 IES Recommendations, cont. Instructional materials should: Focus on in-depth treatment of: Whole numbers in grades K-5 and Rational numbers in grades 4-8 Monitor progress of students receiving supplemental instruction Include motivational strategies in Tier II and Tier III

68 Tier II/III Intervention Instruction D=71 D=71 The multimedia overview discusses characteristics of effective Tier 2 and Tier 3 instruction, offering examples of systematic and explicit instruction, guided practice, and feedback. It also details the use of concrete materials and visual representations to develop abstract concepts as well as the role of motivation in learning math.

69 RtI: Tier II: Intervention (Secondary Prevention) Focus Instruction Interventionist Grouping Time Assessment For students identified with mathematics difficulties Personnel determined by the school (e.g., classroom teacher, mathematics interventionist) Small group instruction (no more than 1:6) 30 minutes per day/2-3 days per week in addition to 60 minutes of mathematics instruction 2 times per month using AIMSweb Early Numeracy, MCOMP, or MCAP Standard research-validated intervention protocol Fidelity Observations conducted on fidelity of implementation

70 RtI: Tier III: Intervention (Tertiary Prevention) Focus Instruction Interventionist Grouping Time Assessment For students identified with mathematics difficulties who Have NOT responded to Tier II instruction based on data. Personnel determined by the school (e.g., classroom teacher, mathematics interventionist) Small group instruction (no more than 1:3) 30 minutes per day; 3 days per week in addition to 60 minutes of mathematics instruction 2 times per week using AIMSweb Early Numeracy, MCOMP, or MCAP Standard research-validated intervention protocol Fidelity Observations conducted on fidelity of implementation

71 Concrete-Representational- Abstract (C-R-A) Research-based Tier II instruction for small groups

72 C-R-A Approach CONCRETEREPRESENTATIONALABSTRACT Hands-on physical models Manipulatives to represent numbers Semi-concrete Draws or uses pictures of the models Numbers as abstract symbols of picture displays (Witzell, Smith, and Brownell, 2001)

73 Concrete Phase Students provided with manipulatives and other materials that will provide them opportunity to explore a math concept or process by doing it with the tools. This is the stage of getting their hands dirty with the intent that having an actual experience will enable the construction of the knowledge being targeted.

74 Concrete Modeling Examples

75 Representational Phase Students begin to develop mental images of the manipulatives by drawing on other means for understanding the target knowledge. Students are encouraged at this time to step back from the manipulatives and other concrete tools and focus on the concept involved in performing actions with the tools.

76 Representational Example

77 Abstract Phase Students could manipulate concepts or processes in the absence of the tools that were important in the early phase of learning.

78 Abstract Example

79 Moving from Concrete to Representational Students work on computer to use virtual manipulatives to solve problems Use self-dialogue or think-aloud strategies Provide songs, rhymes, or rhythms to help remember facts Chunk material (i.e. fact families)

80 Moving from Representational to Abstract Work with student individually or in small group to ensure understanding Fade use of guided worksheets to prepare for abstract stage Use cooperative groups and encourage self-talk and talking to others Provide opportunity for practice

81 What Does C-R-A look like? Title: Sharing Pizzas (grades 1-6) 1. Gather appropriate materials 2. Divide students into groups of 4 3. Ask students to show how they would share one pizza using manipulatives 4. Encourage students to describe and draw their thinking to create equal shares 5. Ask students to show how they would share two pizzas with the group 6. Repeat for three, four, and five pizzas (Compiled by Bryan, Horn, Jones, 2008)

82 Direct/Explicit Instruction Teacher-centered instructional approach Focuses on what to teach and how Can be a scripted program Number Worlds Very systematic Step-by-step format Fast-paced instruction (National Council of Teachers of Mathematics)

83 Example of Direct Instruction

84 What does the research say? Students receive immediate feedback Continuous modeling by teachers which fades as mastery is acquired Often used in small groups (appropriate for Tier II and III interventions) Effective for low achieving students and Exceptional Children students

85 Cover, Copy, Compare Practice visualizing and computing through a sequence of easy-to-remember steps Allows for immediate, corrective feedback Modeling Video cas/psyc/rileytillmant/Cover-Copy-Compare- Modeling-Video-1.cfm

86 Measuring Progress Curriculum Based Measurement AIMSweb MCOMP (grades 1-5) MCAP (grades 2-5) Early Numeracy (K-1) Number Identification Oral Counting Quantity Discrimination Missing Number

87 AIMSweb MCAP Domains & NCTM Content Areas

88 DISCLAIMER!!!! We do NOT have the same correlations as we have with reading curriculum based assessments We need to be CAUTIOUS about applying what we know about Reading to the world of Mathematics In Math, we must think about what we are measuring. We are not in the same place (with regard to research) as we are in Reading.

89 Collecting Strong Data MULTIPLICATION: 4-digit number times 2-digit number: with regrouping 7083 x 57 49581 354150 403,731 6 correct digits = 6 points

90 Tier III Positive Response

91 Tier III Negative Response

92 Making sense of Math... Ma and Pa Kettle Math video

93 Comments/Questions? ? ? ?

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