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

2. 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

3. 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”

4. 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

5. 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

6. 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

7. 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

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

11. 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%

12. 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?

13. 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

14. 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!

15. 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.

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

17. 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.

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

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

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

22. Make Instruction Explicit!

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

24. Sequence of Skills and Concepts: Strategic Integration 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

25. Instructional Strategies: Explicit & Conspicuous 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

26. Prior Knowledge: 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!

27. 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.

28. Systematic Practice & Judicious Review 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

29. Make Instruction Explicit!

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

31. 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

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

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

34. 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!

35. Make Instruction Explicit!

36. 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

37. What evidence did you observe of explicit & intensive instruction?

38. 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

39. Guided Practice! Math Lab: Explicit Instructional Routine

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

41. SA, ZD, JD, TR