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  1. Class 3 September 6, 2012 Please turn in RAP #1 Review Ch. 2 p. 24 Relational Understanding by Skemp Review/Skim Ch. 3 problem solving and classroom discourse.

  2. Quiz! What would Skemp say about Holly’s lesson? Out of all of the suggestions in the book which three do you want to remember when it comes to selecting worthwhile problems? What is classroom discourse? How can effective classroom discourse be encouraged during math time? What is an example of “productive talk”?

  3. What would Skemp say about Holly’s lesson?

  4. Today I will be able to… • Describe teaching for, about, and through problems • Evaluate tasks • Connect lesson planning to problem solving • Apply the before, during, after lesson plan to (a) actual teaching and (b) planning

  5. What is problem solving in math?

  6. Problem 1: How did you solve it? The Adorable Alabama Athletes are creating red hair ribbons for the first football game. Each Alabama Athlete has 12 feet of ribbon and each hair ribbon takes 2/3 foot of a ribbon. How many will each Athlete be able to make?

  7. Nine Toys Problem 2 • Some toys are in the container and some are out of the container. • Leo has nine toys. • How many might be in the container and how many might be out of the container?

  8. Problem 3: Camel Dilemma 35 Camels • 1/ 2 to my oldest child • 1/3 to my youngest child • 1/9 to my middle child

  9. Problem 4: What is the Question? from Barlow & Cates, Teaching Children Mathematics, December 2006/January 2007 • The answer is 20 cookies. What is the question?

  10. Three Problem Solving Approaches Schroeder & Lester, 1989 • Teaching for problem solving. • Teaching about problem solving. • Teaching via problem solving.

  11. Teaching for Problem Solving • Uses real-life problems as a setting in which students can apply and practice recently taught concepts and skills. • Janalea has 2 dogs. Landree has 5 dogs. How many more dogs does Landree have than Janalea? • Traditional problem-solving experiences familiar to most adults.

  12. Teaching About Problem Solving • Refers to instruction that focuses on strategies for solving problems • Polya (1954) Four Step Method • Heuristics • Focuses on process not just on procedure • Critical Thinking • Examples

  13. Teaching About Problem Solving • Polya (1954) Four Step Method • Understand the Problem • Design a strategy • Implement the strategy • Look back

  14. Teaching About Problem Solving • Hueristics • Draw a Picture • Write an Equation • Can you come up with five more?

  15. Unifix Names (class one) Ribbon Making Nine Toys Camel Dilemma 20 Cookies In what ways did the 5 Problems Lend to Teaching About Problem Solving?

  16. Criteria for Problems • There is a perplexing situation that the student understands. • The student is interested in finding a solution. • The student is unable to proceed directly toward a solution. • The solution requires the use of previously acquired knowledge, skills, and understanding.

  17. Teaching via Problem Solving Uses a problem as a means of learning new ideas and for connecting new and already existing constructs. • What math content was learned in these tasks? • Unifix Names • Ribbon Making • Nine Toys • Camel Dilemma • 20 Cookies

  18. Diverse Learners

  19. Student Profiles:Who is in our classes? • Gender • Mild disabilities • Significant disabilities • English language learners • Performance levels • Variety of Socio- economic levels • Gifted • Talented • Advanced learners • Strugglers • Hard workers • Unmotivated • Ethnicity

  20. The question now becomes . . . How can we focus on ALL students EVERY day?

  21. What does it mean to accommodate and modify instruction?

  22. Accommodations • Accommodations are changes to the environment or circumstances to meet the needs of particular students. • The teacher makes adaptations to his or her typical practices to meet the needs of particular students. • Accommodations do not change the task or what the students are expected to do.

  23. Example of an Accommodation The teacher writes instructions instead of just saying them.

  24. Modifications • Modifications are changes to the problem or task itself. • Modifications do change what the students are expected to do but do not lower the expectations.

  25. Modifications • Content • Product • Process

  26. Example of Modification: Original Task Eduardo had 9 toy cars. Erica came over to play and brought 8 cars. Can you figure out how many cars Eduardo and Erica have together? Explain how you know. The teacher distributed cubes to students to model the problem and paper and pencil to illustrate and record how they solved the problem. He asked students to model the problem and be prepared to explain their solution.

  27. Example of a Modification: Modified Task Eduardo had some toy cars. Erica came over to play and brought her cars. Can you figure out how many cars Eduardo and Erica have together? Explain how you know. The teacher asked students: What is happening in this problem? What task are you going to do? Then he distributed Task Cards that explained how many cars Eduardo and Erica had. He varied the difficulty of the numbers: he gave numbers less than ten to students who are struggling, and he gave numbers greater than ten to students who are more advanced.

  28. Giving Students Choice • Multiple Intelligences • Learning Styles • Content • Process • Product

  29. Common Core Standards 3rd Solve problems involving the four operations, and identify and explain patterns in arithmetic. 9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

  30. Bingo or Tic-Tac-Toe

  31. Giving Students Choice Tania was collecting sea shells. Today she added (8, 20, 27) shells to her collection. Now she has (12, 43, 51) shells. How many did she have to begin with?

  32. Tiered Assignments “the meat and potatoes of differentiated instruction.” Addresses a particular standard, key concept, & generalization Allows several pathways Meets the Needs of ALL

  33. Tiered Assignments Tiered Geometry Lesson • Make a poster showing the three types of angles we discussed: acute, right and obtuse. Draw and label each type. Then look through magazines and catalogs to find at least two examples of each type of angle and add the pictures to your poster. Your poster should teach us about the type of angles. • Compose a story or poem about angles. You must use all three angles that we talked about. Be sure your story or poem includes the distinguishing characteristics of the type of angles and clearly teaches us about angles. Illustrate a cover page for your story or poem. • Design and draw a building that has no right angles. Create a brochure for your building describing the structure and it’s angles. In your description include what the building will be used for. What kind of doors and windows will the building include? Will your design be popular with the general public? (Why?)

  34. Tiering by Outcome: Students all use the same materials, but what they do with the materials is different. Pattern block Math Tier one: Identify all the ways you can group your pattern blocks. Tier two: Identify all the different patterns you can make with your pattern blocks. Tier three: Create a bar graph to show all the different kinds of pattern blocks in your bag.

  35. High Quality Tasks

  36. Why Focus on Tasks? • Classroom instruction is generally organized and orchestrated around mathematical tasks • The tasks with which students engage determines what they learn about mathematics and how they learn it • The inability to enact challenging tasks well is what distinguished teaching in the U. S. from teaching in other countries that had better student performance on TIMSS

  37. The Importance of Mathematical Tasks “There is no decision that teachers make that has a greater impact on students’ opportunities to learn, and on their perceptions about what mathematics is, than the selection or creation of the tasks with which the teacher engages students in studying mathematics.” Lappan and Briars, 1995

  38. The Importance of Mathematical Tasks “Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” Stein, Smith, Henningsen, & Silver, 2000

  39. Task-Focused Activities • Distinguishing between high- and low-level cognitive demand mathematics tasks • Maintaining the cognitive demands of high-level tasks during instruction

  40. Distinguishing • Martha’s Carpeting Task • The Fencing Task

  41. Martha’s Carpeting Task Martha was recarpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase? Stein, Smith, Henningsen, & Silver, 2000, p. 1

  42. The Fencing Task Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen in which to keep the rabbits. • If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be? • How long would each of the sides of the pen be if they had only 16 feet of fencing? • How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it. Stein, Smith, Henningsen, & Silver, 2000, p. 2

  43. Importance of Distinguishing • Low (Cognitive Demand ) Level Tasks • High (Cognitive Demand) Level Tasks

  44. Importance of Distinguishing • Low-Level (Cognitive Demand) Tasks • memorization • procedures without connections • High-Level (Cognitive Demand) Tasks • procedures with connections • doing mathematics

  45. Maintaining TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

  46. Features of Good Problems p.35 It must begin where the students are. The problematic or engaging aspect of the problem must be the mathematics that students are to learn. It must require justifications and explanations for answers and methods.

  47. Problem Solving Pitfalls • Rules often provide the thinking for the children. • If 1 man can jump a stream that is 3 meters wide, how wide a stream can 5 men jump? • 2. Key Words often encourage students to avoid thinking about the problem. • Mary walked 11 meters north. She then turned and walked 7 meters west. Did she turn right or left?

  48. Problem Solving Pitfalls • 3. Unrealistic Problems • Mary’s mother needs three hours to do the laundry. If Mary helps her, they can do the laundry in only two hours. How long would it take Mary to do the laundry by herself? • 4. Non-pertinent Clues • If there are two numbers that are big—Subtract • If there was one large and one small—Divide • If it does not come out even—Multiply

  49. How do we teach in a problem-based manner?