Differentiating Math Instruction: Project EQUAL Small Group Instruction Responding to Learners’ Needs

Big Ideas of Mathematics ~Number & Operations ~Algebra ~Geometry ~Measurement ~Data analysis & probability Processes for Doing Mathematics ~Problem Solving ~Reasoning & Proof ~Connections ~Communications ~Representation Responsive Teaching Framework for Differentiating Mathematics Instruction

How programs are designed is critical! - Spiral vs. Strand -Traditional vs. Explicit -Scaffolding to increase mastery & generalization of skills/strategies vs. demonstrate & replicate -Prior knowledge: Instruction vs. Assumption -Examples & non-examples -Sequencing of skills -Progress monitoring vs. “wait and see”

Adapted from: Allsopp, D., Teaching Mathematics Meaningfully, 2007 Making mathematics accessible through responsive teaching Understanding & teaching The big ideas in math AND The big ideas for DOING math Understanding learning characteristics/ barriers for students with difficulties In mathematics Continuously assessing learning To make informed instructional decisions Model for Meaningful Mathematics Instruction Pre- Assessment Formative Assessment Summative Assessment Making mathematics accessible through responsive teaching

Educators must create meaningful learning experiences for students with persistent math difficulties. This is accomplished by considering learner profile/characteristics and creating a match for the student through careful selection/use of: Methods Practices Procedures Making Mathematics Accessible Through Responsive Teaching

ReadinessInterestLearning preferences Differentiation of Instruction based on students’ teachers can differentiate Tomlinson, The Common Sense of Differentiation, ASCD, 2005 OPTIONS, FDLRS Action Resource Center Differentiated Instruction is A teacher’s response to a learner’s needs clear learning goals respectful tasks flexible grouping ongoing assessment and adjustment positive learning environment ContentProcessProduct guided by general principles of differentiation, such as

Forming Flexible Small Groups Teachers may group students in order to: oProvide more explicit, intensive, well-scaffolded instruction oIncrease a students’ stage of learning from initial acquisition to advanced acquisition oProvide independent learner experiences to build proficiency oProvide learning extension experiences

Using data to flexibly group students What math data do you currently gather on: ALL students? Some students? Few students?

Small Group Responsive Teaching Framework Step 1: Revisit Summary of MDA Results SA ZD JD AD RF FJ RJ ? SK NM JM XM TR JT TW FFFMIMMFIIIFMMFFFMIMMFIIIFMM IIIMMMMIMMMIMMIIIMMMMIMMMIMM IIIIMIIIMMMIIIIIIIMIIIMMMIII MMMMMMMMMMMIMMMMMMMMMMMMMIMM IIIMMMMIMMMIMMIIIMMMMIMMMIMM MMMMMMMMMMMMMMMMMMMMMMMMMMMM Name Abstract Expressive Receptive Representational Expressive Receptive Concrete Key: M=Mastery 95%, I=Instructional 70/75%-95% F=Frustrational below 70%

Step 2: Consider additional Flexible Student Interviews Who might we want to have an additional conversation with? What would we want to know? SA, ZD, JD, TR Ask them to look at & create concrete & representational examples of 2 fractions, and explain how they know which one is greater… What evidence do they show re: fractions = area? Do they understand that fractions represent area between 0 and 1 on the number line?

Step 2: Consider additional Flexible Student Interviews Ideas for your “Conversation”: o“Listen” for student’s mathematical thinking oAsk them to describe to you how they solved the problem oAsk them to “teach” you how to solve the problem oAsk them to watch you solve the problem and generate question(s) oUse concrete objects or drawings to further explore student understandings

So… What if I have a small group problem or an individual student problem… What do I do to increase student achievement? How do I meet the needs of students who are struggling with mathematics? Make Instruction Explicit!

Step 3: Develop Hypothesis for Small Group(s) A group of my students can… demonstrate receptive concrete and representational understanding when comparing fractions with like denominators Using… manipulatives and drawings However, they do not demonstrate an understanding of this concept… At the expressive concrete or representational levels I think this is because… They do not have conceptual understanding what fractions represent (area) or the part : whole relationship.

Step 4: Form Flexible Small Group(s) for Mathematics Instruction Who would be in our small, skill-based group?

SA, ZD, JD, TR A group of my students can…demonstrate receptive concrete and representational understanding when comparing fractions with like denominators Using… manipulatives and drawings However, they do not demonstrate an understanding of this concept…At the expressive concrete or representational levels I think this is because…They do not have conceptual understanding what fractions represent (area) or the part : whole relationship.

How can teachers reach & teach these students? Differentiating Mathematics Instruction: An Explicit, Intensive Small Group Lesson Sequence

How does this instructional sequence differ from the whole group lesson?

Variables that influence students’ acquisition of mathematics: Instructional Design Instructional Delivery Classroom Organization & Management

Make Instruction Explicit!

Instructional Design Decisions affecting What to Teach Sequence of skills and concepts Explicit instructional strategies Pre-skills Example selection Practice and Review

Sequence of Skills and Concepts: The order of which skills and strategies are introduced affects the difficulty students have in learning them. Pre-skills of a strategy are taught before the strategy. Easy skills are taught before more difficult ones. Strategies and information that are likely to be confused are not introduced consecutively. Stein, Kinder, Silbert & Carnine, Designing Effective Mathematics Instruction: A Direct Instruction Approach, 2006

Instructional Strategies: Clear, accurate & unambiguous (Gersten, 2002) Strategies must be well-designed and generalizable Should draw focused attention to the relationship between and among math concepts and skills

Pre-skills: Component skills of a strategy are taught before the strategy itself is introduced Ensure students have mastered the pre-skills before introducing a new instructional strategy Assess preskills! This helps in determining where instruction should begin!

Example Selection: Include only problems that students can solve by using a strategy that has been previously taught Include both examples of the currently introduced type of problem as well as previously introduced problem types that are similar (discrimination) CAUTION! Many commercial programs do not contain sufficient numbers of examples in their initial teaching to reach mastery, and rarely employ and adequate number of discrimination problems.

Systematic Practice is necessary for those with low performance rates; without it, students may confuse or forget earlier taught strategies. Facilitates retention over time As students reach their goal, gradually decrease the amount of practice of that skill Practice should never entirely disappear! Kinder, Silbert, Carnine, 2006

Make Instruction Explicit!

Instructional Delivery: How to Teach Initial assessment and progress monitoring Presentation techniques Error-correction procedures Diagnosis & Remediation

Instructional Delivery: Initial Assessment & Progress Monitoring Initial (Pre) Assessment: Answers the Question: Which skills must be taught? Dynamic Mathematics Assessment Focus: Accuracy Progress Monitoring: Answers the Question: How are my students responding to instruction? On-going / continuous Progress Monitoring Focus: Rate of Learning + Accuracy

Instructional Delivery: Presentation Techniques Adequate Time Appropriate Pacing Frequent Student Responding Responsive Scaffolding Precise Monitoring Positive, Corrective, & Immediate Feedback

When an error occurs: Model/Lead Test Retest Instructional Delivery: Error Correction Procedures

Instructional Delivery: Diagnosis & Remediation Diagnosis: Determining the cause of a pattern of errors “Can’t do… or … won’t do?” If you need deeper diagnostic informationregarding a student’s knowledge of fractions, consider administering a Fraction Inventory Remediation: Process of re-teaching a skill Provide Instruction! Address Motivation!

Make Instruction Explicit!

Classroom Organization & Management Elements of Daily Math Lessons Teacher-Directed Instruction -Introduction of new skills -Remediation of previously taught skills Independent Work -Exercises students complete without assistance -Never assign independent work involving problem types/skills that have NOT been instructed Workcheck -Designed to correct errors students make on independent work -Provides insights into nature of student errors -Enables teacher to make necessary instructional adjustments

What evidence did you observe of explicit & intensive instruction?

I.Teacher explains task II.Teacher models task “I do…” III.Teacher & students practice task together—guided practice “We do…” IV.Students (independently) practice tasks “You do...” Step 6: Plan & Deliver Instructional Routine

Guided Practice! Math Lab: Explicit Instructional Routine

Step 7: Develop & implement student monitoring plan Determine: …who will be monitored? …on what skill? …using which tool? …frequency (how often)?

Precision Teaching or Progress Monitoring? Indicator: Progress Monitoring (Uses Equal Interval Chart) Precision Teaching (Uses semi-logarithmic chart) Uses one-minute timings to analyze data to determine student progress toward specific skills Yes Helpful in making instructional decisions based on the review and analysis of student data Yes One-minute timings always on grade level skills YesNot necessarily. If a student is working below grade level, the timing would be on the student’s instructional level. One minute timings conducted three-four times per year on grade level skills YesNo. The precision teaching fluency- building timings are conducted daily or three to four times per week.

SA, ZD, JD, TR

Differentiating Math Instruction: Project EQUAL Ongoing Assessment

What teachers need to know… How are my students responding to mathematics instruction?

Adding it Up, National Research Council, p. 117, 2007 In order for students to be successful in Mathematics, each of these intertwined strands must work together, so… What do we measure to determine if they are making progress?

Continuously Assessing Learning Educators should evaluate what students know and can do before, during and after instruction Before Instruction: evaluate student prerequisite knowledge, skills, experiences and interests that relate to the target concept. Know where to begin instruction. During Instruction: A re students gaining understanding? Are they able to use knowledge and skills proficiently? Results allow for immediate adjustment, if necessary. After Instruction: Where are students in terms of their conceptual learning and skill proficiency? Assessing for the purpose of answering this question will help in planning future instruction.

Types of Assessment: Mathematics Before: Screening During: Formative Diagnostic (if necessary) After: Summative Pre- Assessment Formative Assessment Summative Assessment

Adapted from: Allsopp, D., Teaching Mathematics Meaningfully, 2007 Making mathematics accessible through responsive teaching Understanding & teaching The big ideas in math AND The big ideas for DOING math Understanding learning characteristics/ barriers for students with difficulties In mathematics Continuously assessing learning To make informed instructional decisions Model for Meaningful Mathematics Instruction Pre- Assessment Formative Assessment Summative Assessment

Big Ideas of Mathematics ~Number & Operations ~Algebra ~Geometry ~Measurement ~Data analysis & probability Processes for Doing Mathematics ~Problem Solving ~Reasoning & Proof ~Connections ~Communications ~Representation Responsive Teaching Framework for Differentiating Mathematics Instruction

Most Intensive Few Some ALL Daily or Weekly, precise Graphically represented EX. : Precision Teaching Great Leaps CBM probes Least Intensive Weekly or Bi-weekly, more precise Graphically represented EX: CBM Probes The more intensive the instruction… the more intensive the monitoring should be! District math benchmark assessments (as indicated) Curriculum embedded assessments Flexible student interviews Error Analysis Student Work Samples Rubrics for Problem Solving (Applied Math) Monitoring Progress or Progress Monitoring? Progress Monitoring Progress for

How can I measure student knowledge levels? Thinking Differently… CRA Level of Understanding MethodCriterion Abstract 1-5 minute timings (depends on nature of target concept) Fluency (Rate & Accuracy) Representational (Drawing) 8-10 tasks Accuracy % 3 times Concrete3 tasks Accuracy 100% 3 times Start

INVESTIGATE! Review the continuous, formative assessment tools you have at your table. As a Table Group, prepare to explain what it is, & how it might be used. Think in terms of the intensity equalizer. Is it more intensive, or less intensive?

Progress Monitoring Probe Abstract Level Procedural: See/Write 2 digit addition without regrouping (sums < 20) Measure: # of digits correct

Examples of Concrete and Representational/ Drawing Probe Tasks Use circle pieces and string to solve the following equations x 4 = 12 Concrete Representational/ Drawing

Making Instructional Decisions Create a visual display of student performance data Chart Graph Think of this visual display as a “picture” of your students’ learning Evaluate what the learning picture reveals about student learning

“goal line” Visual Display “corrects” “incorrects” What does this learning picture show? Allsopp, 2008

Problem Solving Learning Picture Example: Student Use of Strategies Allsopp, 2008

Algebraic Thinking Standard: Represent, describe, and analyze patterns and relationships using tables, graphs, verbal rules, and standard algebraic notation. NCTM Process: Representation

NCTM Process: Communication 0123 Algebraic Thinking Standard: Represent, describe, and analyze patterns and relationships using tables, graphs, verbal rules, and standard algebraic notation. Allsopp, 2008

Some General Guidelines for PM Incorporate at concrete, drawing & abstract levels Use short, easy to evaluate “probes” Pinpoint key concepts for monitoring Teach students to chart their learning Use as a way to engage students in setting learning goals At least 2-3 times weekly, more often if needed CELEBRATE SUCCESS!!

Learn More About Continuous Progress Monitoring at the MathVIDS Website: Other Ongoing Monitoring Resources include:

A Word About Follow Up Choose an instructional practice you learned about in Project EQUAL Implement in your Classroom Join our Project Equal Wiki Post a reflection about your action research.

With Gratitude… Dr. David Allsopp University of South Florida For his Insightfulness and Collaborative Spirit Contact Information: Donna Crocker Karen Geisel Marcia Levy