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Mathematics Instruction Best Practices: A Formula for Success How Many Triangles? Pair off with another person, count the number of triangles, explain the process, and record the number. How Many Triangles? Math Inventory What are our beliefs about Math? • Results from Survey Monkey: Why Math? Increasing recognition of the importance of mathematical knowledge “For people to participate fully in society, they must know basic mathematics. Citizens who cannot reason mathematically are cut off from whole realms of human endeavor. Innumeracy deprives them not only of opportunity but also of competence in everyday tasks.” (Adding it Up, 2001) State of Mathematics • Achievement on the NAEP trending upward for 4th/8th grade and steady for 12th grade – Large numbers of students still lacking proficient skills – Persistent income and ethnicity gaps – Drop in achievement at the time algebra instruction begins • TIMSS data indicate significant lower levels of achievement between the US and other nations • Jobs requiring intensive mathematics and science knowledge will outpace job growth 3:1 and everyday work will require greater mathematical understanding (STEM) Meaningful Differences in Math Readiness • Long term trajectories are established as early as kindergarten (Morgan & Farkas, 2009) – 70% of students exiting K below the 10th %ile remain below the 10th %ile at the end of 5th grade • Middle and high SES children come to school with much more informal instruction in numbers and quantitative concepts (Griffin, 1994) • Children lacking these opportunities require formal explicit instruction to develop early numeracy skills The National Mathematics Advisory Panel report offers recommendations for how we can best prepare elementary and middle school students for success in algebra, a gateway to mathematics in high school and beyond. Critical Benchmarks for Algebra Success Fuzzy Math? We need BALANCE in mathematics… CONCEPT/APPLICATION + COMPUTATION = STUDENT ACHIEVEMENT At the Elementary Level, Most Students that Struggle in Math Have Difficulty with: • Solving problems (Montague, 1997; Xin Yan & Jitendra, 1999) • • • • • Visually representing problems (Montague, 2005) Processing problem information (Montague, 2005) Memory (Kroesbergen & Van Luit, 2003) Self-monitoring (Montague, 2005) Fluency of math facts (Fuchs, 2005) What should we do for these students? What to do? • Implement an effective core CURRICULUM based on – Critical mathematics content (Common Core State Standards) – Research-based instructional design principles (EnVision Math) • • Ensure student understanding through high-quality instruction using both student-centered and teacher-centered strategies. – Procedural Understanding/Skill Acquisition (Instructional Focus Continuum) – Conceptual Understanding (CRA model) – Scaffolded Instruction (I do, We do, Ya’all do, You do) Use reliable assessment tools – Regular formative assessment (District CFA’s) – Benchmark screening and progress monitoring (M-CBM’s) ~ Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008). Types of CURRICULUM • Recommended Curriculum = recommended by experts in the field (NCTM, National Standards, IRA) • Written Curriculum = state, district, school, and teacher documents specifying what is to be taught (math maps) • Supported Curriculum = what is in instructional materials (textbooks, media) • Tested Curriculum = what is embodied in state tests, school tests, and teacher tests Taught Curriculum = what teachers actually deliver Learned Curriculum = what students learn Hidden Curriculum = unintended content learned from school culture and climate Excluded Curriculum = what has been left out, intentionally or not Common Core State Standards • All students need to develop mathematical practices such as solving problems, making connections, understanding multiple representations of mathematical ideas, communicating their thought processes, and justifying their reasoning. • All students need both conceptual and procedural knowledge related to a mathematical topic, and they need to understand how the two types of knowledge are connected. • Curriculum documents should organize learning expectations in ways that reflect research on how children learn mathematics. • All students need opportunities for reasoning and sense making across the mathematics curriculum—and they need to believe that mathematics is sensible, worthwhile, and doable. EnVision Instructional Content EnVision focuses on key strands rather than a broad array of mathematical content – Numbers and Operations – Geometry – Measurement – Data Analysis – Algebra NCTM Curriculum Focal Points (2006) What to do? • Implement an effective core curriculum based on – Critical mathematics content (Common Core State Standards) – Research-based instructional design principles (EnVision Math) • Ensure student understanding through high-quality INSTRUCTION using both student-centered and teachercentered strategies. – Procedural Understanding/Skill Acquisition (Instructional Focus Continuum) – Conceptual Understanding (CRA model) – Scaffolded Instruction (I do, We do, Ya’all do, You do) • Use reliable assessment tools – Regular formative assessment (District CFA’s) – Benchmark screening and progress monitoring (M-CBM’s) ~ Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008). INSTRUCTION HOW do I teach . . . Research on Effective Instruction indicates: Quality of Instruction - reflects quality of curriculum, lesson preparation, and teaching skill Appropriate Level - lesson is neither too easy nor too difficult Effective Pacing – time is used efficiently, the pace is “perky” Incentive - students are engaged and motivated to learn Instructional Focus Continuum Instructional Strategies for Building Skill Accuracy • Explicit Teaching – Teacher modeling, guided practice, independent practice • Teacher Feedback – Specific positive confirmations – Corrective feedback on errors • “Cover, copy, compare” Explicit Instruction • What is Explicit? – Precise and consistent language – Clear, accurate, and unambiguous teaching – I do it, we do it, Ya’all do it, You do it – Why is it important? • Often when students encounter improper fractions (e.g. 5/4) the strategies they were taught using a single unit don’t work. • Many commercially developed programs suggest that students generate a number of alternative problem solving strategies. Teachers need to select only the most generalizable, useful, and explicit strategies (Stein, 2006). Explicit Instruction • High Achieving countries all implement connections problems as connections problems • U.S. implements connection problems as a set of procedures Exponents and Geometry? What is 42 ? Why is it 4 x 4 when it • looks like 4 x 2 and • sounds like a geometry term? It means ‘make a square out of your 4 unit side’ Exponents and Geometry? What is 42 ? --4 units-1 1 1 1 You’d get how many little 1 by 1 inch squares? 42 = 16 Precise and Consistent Language Kids are Sponges! Ambiguous Language Examples 4<7 8>2 What NOT to teach… • Math strategies that do not support – Future learning • EX: “When subtracting, the larger number ALWAYS goes on top.” – Accurate conceptual understanding • EX: “When subtracting, borrow a 1 from the tens place.” Treatment Integrity for Cover, Copy, & Compare _____ Provided worksheet with problems and solutions on the lefts side of the page and the same problems without answers to the left of the page. _____ Provided checklist of the steps to complete. _____ Watched each student to be sure they are doing steps correctly. _____ Provided error correction and praise as needed. Student Directions for Cover, Copy, Compare _____ Look at the problem and the answer. _____ Cover the problem and answer. _____ Write the answer. _____ Uncover the problem and check if you wrote the answer correctly. _____ If your answer is not the same, try again until it is. Cover, Copy, Compare Skill Probe Generator: http://www.lefthandlogic.com/mathprobe_ol d/allmult.php Share with your partner any additional ideas you have for building accuracy in a skill. Instructional Focus Continuum Instructional Strategies for Building Skill Fluency • • • • Multiple exposures to skill or concept Goal setting for increased automaticity Computer games Peer games Computer Resources for Math Skill Fluency • http://www.internet4classrooms.com/grade_level_h elp.htm • http://nlvm.usu.edu/en/nav/index.html • http://www.gamequarium.org/dir/Gamequarium/M ath/ • http://cte.jhu.edu/techacademy/web/2000/heal/ma thsites.htm Peer Math Games • Math Fact WAR w/Flash cards • Dot Game w/Flash cards or Dice • Math Fact Bingo Share with your partner any additional ideas you have for building fluency in a skill. Instructional Focus Continuum Instructional Strategies for Building Skill Application • Word Problem Solving • Use questioning strategies that require learners to go deeper • Peer tutoring Share with your partner any additional ideas you have for building application in a skill. What to do? • Implement an effective core curriculum based on – Critical mathematics content (Common Core State Standards) – Research-based instructional design principles (EnVision Math) • Ensure student understanding through high-quality instruction using both student-centered and teachercentered strategies. – Procedural Understanding/Skill Acquisition (Instructional Focus Continuum) – Conceptual Understanding (CRA model) – Scaffolded Instruction (I do, We do, Ya’all do, You do) • Use reliable assessment tools – Regular formative assessment (District CFA’s) – Benchmark screening and progress monitoring (M-CBM’s) ~ Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008). Concrete-Representational-Abstract Instructional Approach (C-R-A) • CONCRETE: Uses hands-on physical models or manipulatives to represent numbers and unknowns. • REPRESENTATIONAL: Draws or uses pictorial representations of the models. • ABSTRACT: Involves numbers as abstract symbols of pictorial displays. 49 Concrete Level • Definition: A teaching method that uses actual objects such as people, shoes, toys, fruits, cubes, base-ten blocks, or fraction tiles. • What concrete items have you used in your classroom to teach math concepts? Representational Level • Definition: A teaching method that uses pictures, tally marks, diagrams, and drawings. These pictorial representations relate directly to the manipulatives and set up the student to solve numeric problems without pictures. • From your experiences, what have you used that is representational in your math classroom? Abstract Level • Definition: A teaching method that uses written words, symbols (such as variables or numerals), or verbal expressions. Amber has 3 toy cars. If there are 4 wheels on each car, how many wheels are there on her toy cars? Concrete Representational Abstract Amber has 3 toy cars. If there are 4 wheels on each car, how many wheels are there on her toy cars? Concrete Representational Abstract With your partner, come up with 2- 3 different ways you would teach this problem using CONCRETE objects With your partner, come up with 2- 3 different ways you would teach this problem using REPRESENTATION With your partner, come up with 2- 3 different ways you would teach this problem using ABSTRACT symbols What to do? • Implement an effective core curriculum based on – Critical mathematics content (Common Core State Standards) – Research-based instructional design principles (EnVision Math) • Ensure student understanding through high-quality instruction using both student-centered and teachercentered strategies. – Procedural Understanding/Skill Acquisition (Instructional Focus Continuum) – Conceptual Understanding (CRA model) – Scaffolded Instruction (I do, We do, Ya’all do, You do) • Use reliable assessment tools – Regular formative assessment (District CFA’s) – Benchmark screening and progress monitoring (M-CBM’s) ~ Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008). Scaffolded Instruction Gradual Release of Responsibility “I do - We do –Ya’all do- You do” Explicit Instruction and Modeling Guided Practice Independent Practice A Day in the Life of EnVision Math The CORE and More Lesson Checklist • INSERT Core and more document here What to do? • Implement an effective core curriculum based on – Critical mathematics content (Common Core State Standards) – Research-based instructional design principles (EnVision Math) • Ensure student understanding through high-quality instruction using both student-centered and teacher-centered strategies. – Procedural Understanding/Skill Acquisition (Instructional Focus Continuum) – Conceptual Understanding (CRA model) – Scaffolded Instruction (I do, We do, Ya’all do, You do) • Use reliable ASSESSMENT tools – Regular formative assessment (District CFA’s) – Benchmark screening and progress monitoring (M-CBM’s) ~ Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008). ASSESSMENT Where do I START instruction? How do I EVALUATE student learning? Screening Assessment: Foundational skills Identify children at risk Diagnostic Assessment: What deficits are impeding the development of proficiency? Multi-faceted approach Progress Monitoring (Formative) Assessment: Curriculum-based On-going (students at risk assessment more frequently) Outcome (Summative) Assessment