Unit 2 jcs mid


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Unit 2 jcs mid

  1. 1. What the Research Says---
  2. 2. <ul><li>According to the 2007 National Assessment of Educational Progress (NAEP) only 39% of fourth-grade students and only 32% of eighth-grade students scored at the proficient level in mathematics. </li></ul><ul><li>(National Center for Educational Statistics, 2007) </li></ul><ul><li>Feedback? </li></ul>
  3. 3. <ul><li>Schmidt, Houang, and Cogan (2002) reported that by the end of high school, US students performed near the bottom of the international distribution in the Third International Mathematics and Science Study (TIMSS). </li></ul><ul><li>Feedback? </li></ul>
  4. 4. <ul><li>The National Council for Teachers of Mathematics (NCTM) </li></ul><ul><li>highlights the need for a well designed curriculum and quality teacher preparation. </li></ul>
  5. 5. Federal Recognition of Lack of Research Studies <ul><li>IDEIA 2004 and NCLB clearly define a high standard for research-based reading practices </li></ul><ul><li>IDEIA 2004 did not clearly define a high standard for research-based math practices, because we did not have the same research for math as we did for reading. </li></ul><ul><li>What Works clearinghouse: </li></ul><ul><li>http://www.whatworks.ed.gov </li></ul>
  6. 6. National Mathematics Panel Report 2008 What do students need for success in Algebra? <ul><li>Major Findings: </li></ul><ul><li>Proficiency with whole numbers, fractions and certain aspects of geometry and measurement are the critical foundations of algebra </li></ul><ul><li>Explicit instruction for students with disabilities shows positive effects. </li></ul><ul><li>Students need both explicit instruction and conceptual development to succeed in math. </li></ul><ul><li>http://www.ed.gov/about/bdscomm/list/mathpanel/index.html </li></ul>
  7. 7. Math Basics <ul><li>International Research </li></ul>
  8. 8. TIMSS from Improving Mathematics Instruction (Ed Leadership 2/2004) <ul><li>1995 Video Study </li></ul><ul><ul><ul><li>Japan, Germany, US </li></ul></ul></ul><ul><ul><ul><li>Teaching Style Implicated </li></ul></ul></ul><ul><li>1999 Video Study </li></ul><ul><ul><ul><li>US, Japan, Netherlands, Hong Kong, Australia, Czech Rep. </li></ul></ul></ul><ul><ul><ul><li>Implementation Implicated </li></ul></ul></ul>
  9. 9. Style vs. Implementation <ul><li>High Achieving countries use a variety of styles to teach (calculator vs. no calculator, ‘real-life’ problems vs. ‘naked’ problems) </li></ul><ul><li>High Achieving countries all implement connections problems as connections problems </li></ul><ul><li>U.S. implements connection problems as a set of procedures </li></ul>
  10. 10. Exponents, Geometry, Measurement <ul><li>What is 4 2 ? </li></ul><ul><li>Why is it 4 x 4 when it looks like 4 x 2? </li></ul><ul><li>It means ‘make a square out of your 4 unit side’ </li></ul>
  11. 11. Exponents, Geometry, Measurement <ul><li>What is 4 2 ? </li></ul><ul><li> --4 units-- </li></ul><ul><li>1 </li></ul><ul><li>1 </li></ul><ul><li>1 </li></ul><ul><li>1 </li></ul><ul><li>You’d get how many little </li></ul><ul><li>1 by 1 inch squares? 4 2 = 16 </li></ul>
  12. 13. Are these the same?
  13. 14. Concept Development & Connections <ul><li>“ Research has shown a mutually beneficial pairing of procedures and concepts” </li></ul><ul><li>David Chard </li></ul>
  14. 15. Characteristics of Students with Learning Difficulties in Mathematics <ul><li>Slow or inaccurate retrieval of basic arithmetic facts </li></ul><ul><li>Impulsivity </li></ul><ul><li>Problems forming mental representations of mathematic concepts (number line, visual means to represent subtraction as a change process) (Geary 2004) </li></ul><ul><li>Weak ability to access numerical meaning from mathematical symbols (i.e. poorly developed number sense) (Gersten and Chard 1999 and Noel 2006) </li></ul><ul><li>Problems keeping information in working memory. (Passolunghi and Siegel 2004; Swanson and Beebe-Frankenberger 2004) </li></ul>Research Brief NCTM 2007 “What are the Characteristics of Students with Learning Difficulties in Mathematics?”( adapted from Gersten and Clarke)
  15. 16. Concrete to Representational to Abstract
  16. 17. C-R-A Cecil Mercer <ul><li>The student moves through stages. </li></ul><ul><li>The teacher has the responsibility to explicitly and directly instruct students through these stages. </li></ul><ul><li>Make connections for the students! </li></ul>7 + 5 = 12 Concrete  Representational  Abstract
  17. 18. Sharon Griffin Core Image of Mathematics 1 2 Symbols Counting Numbers Quantity “ one” “ two” “ three” + - 1 2 3 X =
  18. 19. Different Forms of a Number Number Worlds Griffin
  19. 20. Prototype for lesson construction Touchable Visual Discussion: Makes sense of concept 1 2 Learn to record these ideas V. Faulkner and DPI Task Force adapted from Griffin Symbols Simply record keeping! Mathematical Structure Discussion of the concrete Quantity Concrete display of concept
  20. 21. <ul><li>CRA instructional model </li></ul>Abstract Representational Concrete 8 + 5 = 13
  21. 22. Synthesizing CRA & Griffin Model <ul><li>CRA (instructional model) </li></ul>Quantity Math Structure Symbolic Abstract Representational Concrete Griffin (cognitive development model) Connection 1 Connection 2
  22. 23. Synthesizing CRA & Griffin Model <ul><li>CRA (instructional model) </li></ul>Quantity Math Structure Symbolic Abstract Representational Concrete Griffin (cognitive development model) Connection 1 Connection 2 representational Structural/ Verbal
  23. 24. Concrete Reality <ul><li>8 - 5 = 8 </li></ul><ul><li>7 - 4 = 7 </li></ul>
  24. 25. = -
  25. 26. Use Math Research <ul><li>60th percentile and above on the EOG </li></ul><ul><li>What should instruction look like? </li></ul><ul><ul><li>TIMSS studies </li></ul></ul>
  26. 27. 25 th percentile to 60 th percentile What should instruction look like?
  27. 28. Language, Reading and Mathematics Connections and Disconnections
  28. 29. <ul><li>Oral Language </li></ul><ul><li>Reading and Writing </li></ul><ul><li>Mathematics </li></ul>
  29. 30. Mathematically Authentic Instruction <ul><li>Research indicates that teachers who have a more robust understanding of the math present math in a more accurate and effective manner. </li></ul><ul><li>Deeper understanding on the part of teacher positively affects student performance. </li></ul><ul><li>Deborah Ball and Liping Ma </li></ul>
  30. 31. Targeted Instruction is <ul><li>Explicit </li></ul><ul><ul><li>Focus on making connections </li></ul></ul><ul><ul><li>Explanation of concepts </li></ul></ul><ul><li>Systematic </li></ul><ul><ul><li>Teaches skills in their naturally acquired order </li></ul></ul><ul><li>Multi-sensory </li></ul><ul><li>Cumulative </li></ul><ul><ul><li>Connecting to prior knowledge and learning </li></ul></ul><ul><li>Direct </li></ul><ul><ul><li>Small group based on targeted skills </li></ul></ul><ul><ul><li>Progress monitored </li></ul></ul>
  31. 32. Make sure Instruction is Explicit <ul><li>What is Explicit? </li></ul><ul><ul><li>Clear, accurate, and unambiguous </li></ul></ul><ul><ul><li>Why is it important? </li></ul></ul><ul><ul><ul><li>Often when students encounter improper fractions (e.g. 5/4) the strategies (or tricks) they were taught for proper fractions don’t work. </li></ul></ul></ul><ul><ul><ul><li>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). </li></ul></ul></ul>
  32. 33. Direct Instruction for all? <ul><li>Why would this not be necessary? </li></ul><ul><li>All students can learn math!!! </li></ul>
  33. 34. Instructional Design Concepts <ul><li>Sequence of Skills and Concepts </li></ul><ul><li>Knowledge of Pre-skills </li></ul><ul><li>Example Selection </li></ul><ul><li>Practice and Review </li></ul>
  34. 35. Sequencing Guidelines <ul><li>Pre-skills of the mathematical concept or skill are taught before a strategy </li></ul><ul><ul><li>4008 </li></ul></ul><ul><ul><ul><li>- 9 </li></ul></ul></ul><ul><li>Information presented in concrete  representational  abstract sequence </li></ul>
  35. 36. Preskills <ul><li>Component skills of any strategy should always be taught before the strategy itself. </li></ul><ul><li>We can call component skills the pre-skills. </li></ul><ul><ul><li>i.e. what is 45% of 80? </li></ul></ul><ul><ul><ul><li>What are the pre-skills? </li></ul></ul></ul><ul><ul><ul><li>Is there another way to look at what might constitute the pre-skills for another student? </li></ul></ul></ul><ul><ul><ul><li>Can we teach a systematic, mathematically accurate and authentic method that does not involve multiplication, or multiplication of decimals? </li></ul></ul></ul>
  36. 37. Selecting Examples <ul><li>First only use problems with the new material in practice exercises. </li></ul><ul><ul><li>Few programs provide a sufficient number of examples for new material – and do not emphasize mathematical connections. </li></ul></ul><ul><ul><li>The programs rarely present a discussion of examples versus non-examples to help frame students’ thinking. </li></ul></ul>
  37. 38. <ul><li>Secondly, we integrate problems with both old and new material. </li></ul><ul><ul><li>The purpose of this: </li></ul></ul><ul><ul><ul><li>is to allow discrimination between using the new strategy and not using the strategy. </li></ul></ul></ul><ul><ul><ul><li>is to allow practice of previously learned skills (cumulative review). </li></ul></ul></ul><ul><ul><ul><ul><li>Research suggests a strong relationship between retention and practice & review (Stein, 2006) </li></ul></ul></ul></ul>
  38. 39. What are the components of a practice set? <ul><li>Mastery level of understanding </li></ul><ul><ul><li>What if you made a practice of sending home two or three different sets of HW? (93-100% accuracy) </li></ul></ul><ul><ul><li>This practice can be used to build automaticity. </li></ul></ul><ul><ul><li>Homework should not be seen as an opportunity for student to learn a skill, simply to correctly practice that skill. </li></ul></ul>
  39. 40. Consider both of these problems <ul><li>3002 </li></ul><ul><li>- 89 </li></ul><ul><li>364 </li></ul><ul><li>-128 </li></ul>
  40. 41. Sequencing of skills <ul><li>The problem of 3002-89 is much more difficult than 364-128. </li></ul><ul><ul><li>Do they both require renaming? </li></ul></ul><ul><ul><li>Then why is the first more difficult –Work both problems out to find the answer. </li></ul></ul>
  41. 42. Practice and Review <ul><li>1. Mastery is reached when a student is able to work problems with automaticity and fluency (fact learning). </li></ul><ul><li>2. Provide systematic review of previously learned skills. </li></ul>
  42. 44. Anxiety, Performance and Instructional Considerations Testing Grading opportunities Self-concept
  43. 45. “ Understanding Math Anxiety” <ul><li>What implications does the article make with respect to math anxiety? </li></ul><ul><li>What can we do as instructors to ease the trauma of math anxiety? </li></ul>
  44. 46. “Guess and Check” <ul><li>Analogy with “maintaining meaning” </li></ul><ul><li>We’re actually reinforcing the wrong thing. We are creating a ceiling for them. </li></ul><ul><li>“ 4th grade slumps” </li></ul><ul><li>Larger numbers, negative numbers improper fractions, fractional parts (decimals, ratios) make guessing ineffective and counter-productive, and stressful. </li></ul>
  45. 47. Some words about “Key Words” <ul><li>They don’t work… </li></ul>
  46. 48. We tell them— more means add <ul><li>The average high temperature in January was 9 degrees F. The average low temperature was -5 degrees F. How much more was the average high temperature than the average low temperature? </li></ul>
  47. 49. Sense-Making <ul><li>We need to notice if we are making sense of the math for our students, or if our discussion of the math contributes to --- </li></ul><ul><li>“The suspension of sense-making…” </li></ul><ul><li>Schoenfeld (1991) </li></ul>
  48. 50. Mayer <ul><li>Translating </li></ul><ul><ul><li>Relational sentences and reversal errors </li></ul></ul><ul><ul><ul><li>There are 6 students for every principal </li></ul></ul></ul>Students Principals
  49. 51. Characteristics of Math Disabilities
  50. 52. Geary and Hoard, Learning D isabilities in Basic Mathematics from Mathematical Cognition , Royer, Ed. <ul><li>1-1 Correspondence </li></ul><ul><li>Stable Order </li></ul><ul><li>Cardinality </li></ul><ul><li>Abstraction </li></ul><ul><li>Order-Irrelevance </li></ul>Gellman and Gallistel’s (1978) Counting Principles
  51. 53. <ul><li>Standard Direction </li></ul><ul><li>Adjacency </li></ul><ul><li>Pointing </li></ul><ul><li>Start at an End </li></ul>Briars and Siegler (1984) Unessential features of counting Geary and Hoard, Learning D isabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed.
  52. 54. Geary and Hoard, Learning D isabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed. <ul><li>Pseudo-Error: Adjacency. </li></ul><ul><ul><li>Students with MD/RD did not understand Order-Irrelevance </li></ul></ul><ul><li>Error: Double Counts </li></ul><ul><ul><li>Students with MD/RD saw double counts as errors at end of sequence, but missed them at beginning of sequence. </li></ul></ul><ul><ul><ul><li>Working memory </li></ul></ul></ul>Students who are identified with a math disability
  53. 55. Geary and Hoard, Learning D isabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed. <ul><li>Retrieval </li></ul><ul><ul><li>MD/RD, MD consistently differ from peers in retrieval ability </li></ul></ul><ul><ul><li>Retrieval ability DOES NOT improve substantively through elementary schooling. </li></ul></ul><ul><ul><li>Consistent with left-hemi lesions: This suggests that the issue is cognitive/organically based and NOT due to lack of exposure/motivation/IQ </li></ul></ul>
  54. 56. Geary and Hoard, Learning D isabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed. <ul><li>Memory and Cognition </li></ul><ul><ul><li>Working Memory </li></ul></ul><ul><ul><li>Executive deficit: difficulty monitoring the act of counting. </li></ul></ul><ul><ul><li>Children with MD(only) exhibited deficits in span tasks only if they involved counting (not other forms of language) </li></ul></ul>
  55. 57. Geary and Hoard, Learning D isabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed. <ul><li>Procedures: sum, min, max </li></ul><ul><ul><li>Children with MD/RD and MD do not tend to shift into memory based procedures for simple arithmetic problems until 1 st -2 nd grade </li></ul></ul><ul><ul><li>Procedural sophistication with the basics does tend to develop later in elementary years. </li></ul></ul>
  56. 58. SPED Research <ul><li>Concrete-Representational-Abstract </li></ul><ul><li>Meta-cognitive support </li></ul><ul><li>Routine and Structure </li></ul><ul><li>Teach the BIG IDEAS </li></ul><ul><li>Direct Instruction: Systematic, Explicit, Cumulative and Multi-Sensory </li></ul>
  57. 59. 25th percentile and below What should instruction look like?
  58. 60. Just look at this wall young man! You’ve got some explaining to do . Einstein as a Boy
  59. 61. References <ul><li>Chard, D. (2004) Council for Exceptional Children Conference. New Orleans, LA </li></ul><ul><li>Geary and Hoard, Learning Disabilities in Basic Mathematics from Mathematical Cognition, Royer, Ed. </li></ul><ul><li>Griffin, S Number Worlds (2004) </li></ul><ul><li>National Center for Educational Statistics, 2007 </li></ul><ul><li>Schmidt, Houang, and Cogan (2002) </li></ul><ul><li>TIMSS, Improving Mathematics Instruction (Ed Leadership 2/2004) </li></ul>
  60. 62. Required Reading for Days 1 <ul><li>“ Reform by the Book” (Deborah Loewenberg Ball / David K. Cohen) </li></ul><ul><li>“ Understanding Math Anxiety” (Sean Cavanaugh) </li></ul>
  61. 63. Assignments from Units 1 and 2 <ul><li>Learning Task 1 </li></ul><ul><ul><li>Due Day 1 </li></ul></ul><ul><ul><ul><li>Discussion of Math Program </li></ul></ul></ul><ul><ul><ul><li>Initial Lesson Plan </li></ul></ul></ul><ul><li>Readings for Units 3 and 4 </li></ul><ul><ul><li>Assigned reading from Liping Ma </li></ul></ul>
  62. 64. Knowing and Teaching Elementary Mathematics Liping Ma
  63. 65. Assignments for Liping Ma <ul><li>Chapter 1- Subtraction </li></ul><ul><li>Chapter 2- Multiplication </li></ul><ul><li>Chapter 3- Division of Fractions </li></ul><ul><li>Chapter 4- Area and Perimeter </li></ul>
  64. 66. Be prepared to discuss in Expert Groups <ul><li>The approach of the U.S. Teachers to the topic </li></ul><ul><li>The approach of the Chinese Teachers to the topic </li></ul><ul><li>Biggest pedagogical shift you may need to take regarding this topic and/or the most interesting idea you learned. </li></ul>