1. Planning Effective Mathematics Instruction in a Variety of Educational Environments David Allsopp, Ph.D. University of South Florida Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

2. Essential Questions What is the target mathematics concept? What did I learn from my mathematics dynamic assessment (or other pre-unit assessment) What is my instructional hypothesis (what students know, don’t know, and why) How do students think about these ideas differently from adults and how can I use this information to inform instruction? How will I differentiate the instructional needs of my students? Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

3. Essential Questions What authentic contexts will I use? How will I introduce/model the target concept to the whole class? How will I differentiate the instructional/scaffolding/extension (generalization and adaption) needs of my students? How will I provide practice opportunities that promote proficiency/maintenance? How will I evaluate my students’ learning and determine the effectiveness of my instruction? Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

4. Planning for Responsive Mathematics Instruction Consider what students know, don’t know, and why Consider stages of learning Determine differentiated instruction needs/objectives Select authentic contexts Plan whole class instruction Plan differentiated instruction Plan practice opportunities Plan how you will evaluate learning and your instruction Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

5. 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

6. 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

7. Core Instruction: The areas to be studied in mathematics from pre-kindergarten through eighth grade should be streamlined Proficiency with whole numbers, fractions, and certain aspects of geometry and measurement are the foundations for algebra. Conceptual understanding, computational and procedural fluency, and problem solving skills are equally important and mutually reinforce each other. Students should develop immediate recall of arithmetic facts to free the "working memory" for solving more complex problems. What we know… National Math Panel

8. 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.

9. How programs are designed is critical! - Spiral vs. Strand design -Traditional vs. Explicit -Use of scaffolding to increase mastery & generalization of skills/strategies vs. demonstrate & replicate -Prior knowledge: Instruction in related vocabulary and review of mastered prerequisite skills vs. assumption of prior knowledge -Examples & non-examples -Sequencing of skills (macro/micro) -Progress monitoring vs. “wait and see”

10. Time to Reflect… Individual Think-Write What are some of the characteristics of my core math program? Mix and Match 2

12. Summarizing CRA Results Based on C-R-A Assessment Mrs. Carsen concludes students: oHave difficulty representing fractions that are >, <, or = using unlike denominators oHave difficulty determining >, <, or = using symbols between fractions with unlike denominators (abstract, representational) oHave some ability to do this with fractions that have natural relationships (1/2 and 2/4, 4/6 and 2/3-abstract) oHave difficulty relating written fractions to drawings (Lack concept of the meaning of a fraction) oDifficulty with concept of “equivalent area” of whole to part when drawing

13. MDA Results Hypothesis SA ZD JD AD RF FJ RJ SK NM JM XM TR JT TW FFFMIMMFIIIFMMFFFMIMMFIIIFMM IIIMMMMIMMMIMMIIIMMMMIMMMIMM IIIIMIIIMMMIIIIIIIMIIIMMMIII MMMMMMMMMMMIMMMMMMMMMMMMMIMM IIIMMMMIMMMIMMIIIMMMMIMMMIMM MMMMMMMMMMMMMMMMMMMMMMMMMMMM

14. Instructional Hypothesis What’s it all about? Provides you with a focused approach to teaching that specifically addresses the needs of your students based on the results of a mathematics dynamic assessment.

15. Instructional Hypothesis Context: What Students Can Do: What Students Can’t Do: Reason: “Given _____________________, Students can _______________ Students cannot _____________ Because ____________________” Keep It Simple!

16. Given two fractions… Students are able to… determine >, <, and = when fractions have like denominators at concrete, representational and abstract levels. Students are unable to… determine >, <, and = when fractions have unlike denominators at concrete, representational and abstract levels. …because they lack understanding of the area that fractions represent Instructional Hypothesis

17. MDA Results Hypothesis SA ZD JD AD RF FJ RJ SK NM JM XM TR JT TW FFFMIMMFIIIFMMFFFMIMMFIIIFMM IIIMMMMIMMMIMMIIIMMMMIMMMIMM IIIIMIIIMMMIIIIIIIMIIIMMMIII MMMMMMMMMMMIMMMMMMMMMMMMMIMM IIIMMMMIMMMIMMIIIMMMMIMMMIMM MMMMMMMMMMMMMMMMMMMMMMMMMMMM Given a set of 2 fractions, the majority of my students can determine >, <, and = when fractions have like denominators at the C R & A levels; however they cannot determine >, <, and = when fractions have unlike denominators because they lack understanding of the area that fractions represent.

18. Consider Stages of Learning: A Framework for Understanding How Struggling Learners Learn Entry level AcquisitionProficiencyMaintenanceGeneralizationAdaption InitialAdvanced Accuracy Rate Retention Stages of Learning Extension To focus instruction, it is important to know at what stage of learning students are with respect to the target math concept/skill…

19. Planning for Responsive Mathematics Instruction Determine differentiated instructional needs/objectives Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

20. Planning for Responsive Mathematics Instruction Determine differentiated instructional needs/objectives Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

21. Planning for Responsive Mathematics Instruction Select Authentic Context Plan whole class instruction Select Authentic Context Plan whole class instruction Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

22. Planning for Responsive Mathematics Instruction Plan differentiated instruction Plan practice opportunities Plan differentiated instruction Plan practice opportunities Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

23. Planning for Responsive Mathematics Instruction Plan evaluation of learning and instruction Plan practice opportunities Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

24. Planning for Responsive Mathematics Instruction Plan evaluation of learning and instruction Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

25. Planning for Responsive Mathematics Instruction Plan differentiated instruction Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.

26. Planning for Specific Learning Barriers of Students Allsopp, Kyger, and Lovin (2007). Teaching Mathematics Meaningfully. Paul H. Brookes Publishing.