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School of Education

School of Education. Beyond over . Rise. Run. Developing robust understandings of slope. Michael Matassa ( Michael.Matassa@Colorado.edu) Frederick Peck ( Frederick.Peck@Colorado.edu) University of Colorado Boulder. ICME 2012 Seoul July 11, 2012 17:00 – 18:30 pm.

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School of Education

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  1. School of Education Beyond over Rise Run Developing robust understandings of slope Michael Matassa (Michael.Matassa@Colorado.edu) Frederick Peck (Frederick.Peck@Colorado.edu) University of Colorado Boulder ICME 2012 Seoul July 11, 2012 17:00 – 18:30 pm

  2. School of Education Objective Participants will further develop their perspectives on linear functions by participating in discussion and activity. Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  3. School of Education Overview • Discuss diverse international perspectives on linear functions • Explain the traditional American approach • Explore a learning trajectory for linear functions that: • Places “rate of change” as the key concept • Enables students to construct formal mathematics • Helps students develop versatile and adaptable understandings of linear functions • Present lessons learned, including some important considerations for context Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  4. School of Education What are some characteristics of this linear function? Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  5. School of Education • The dependent variable increases by two units for every unit increase in the independent variable How would you describe this behavior in your country? Draw a schematic or diagram of how this concept is developed or taught in your country Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  6. School of Education Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  7. School of Education • The dependent variable increases by two units for every unit increase in the independent variable • In America, we would say that the slope of this line is 2 • In English, slope means “steepness” • Students are taught to calculate slope as “rise over run” or “change in y over change in x” Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  8. School of Education • The dependent variable increases by two units for every unit increase in the independent variable Geometric • In America, we would say that the slope of this line is 2 • In English, slope means “steepness” • Students are taught to calculate slope as “rise over run” or “change in y over change in x” Procedural Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  9. School of Education http://www.youtube.com/watch?v=R948Tsyq4vA Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  10. School of Education Sub-constructs of slope (Stump, 1999) • Algebraic ratio (i.e., ) • Parametric coefficient (i.e., the “a” in y = ax + b) • Geometric ratio (i.e., “rise over run”) • Physical Property (i.e., steepness) • Functional Property (i.e., rate of change) Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  11. Physical property Functional property Geometric ratio Algebraic ratio The number one result when searching for “rate of change” in Google! http://www.regentsprep.org/regents/math/algebra/AC1/Rate.htm

  12. School of Education Summary: In America: Slope • The dependent variable increases by two units for every unit increase in the independent variable Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  13. School of Education Summary: In America: Slope measures steepness is procedural Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  14. School of Education Summary: In America: y = mx + b Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  15. School of Education “Mathematics should be thought of as the human activity of mathematizing- not as a discipline of structures to be transmitted, discovered, or even constructed, but as schematizing, structuring, and modeling the world mathematically.”Hans Freudenthal (as quoted in Fosnot & Jacob, 2010) Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  16. School of Education “Mathematics should be thought of as the human activity of mathematizing- not as a discipline of structures to be transmitted, discovered, or even constructed, but as schematizing, structuring, and modeling the world mathematically.”Hans Freudenthal (as quoted in Fosnot & Jacob, 2010) Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  17. School of Education “Mathematics should be thought of as the human activity of mathematizing- not as a discipline of structures to be transmitted, discovered, or even constructed, but as schematizing, structuring, and modeling the world mathematically.”Hans Freudenthal (as quoted in Fosnot & Jacob, 2010) Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  18. School of Education Realistic Mathematics Education (RME) (Treffers, 1987) • Mathematical exploration should take place within a context that is recognizable to the student. • Models and tools should be used to bridge the gap between informal problem-solving and formal mathematics • Students should create their own procedures and algorithms • Learning should be social. • Learning strands should be intertwined Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  19. The Iceberg Metaphor(Webb, et al., 2008) Algebraic Ratio Formal Notationsand Algorithms The Algebraic Ratio Preformal Structure, Tools, Models Informal Contexts that are experientially real Delivery Costs Snowfall Accumulation Running Pace Scaling triangles Context How do we use math to make predictions?

  20. The Iceberg Metaphor(Webb, et al., 2008) Algebraic Ratio Formal The Algebraic Ratio Preformal Informal Guided re-invention Delivery Costs Snowfall Accumulation Running Pace Scaling triangles Context How do we use math to make predictions?

  21. School of Education A Teaching Experiment (Cobb, 2000)on Slope Goal: To develop a robust understanding of slope. • Versatility and adaptability • 60 students • 4 weeks • Algebra 1 • 9thgrade (ages 14-15) • Combination of lecture, whole class discussion, small group work, and individual work. Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  22. School of Education Our focus today • Algebraic ratio (i.e., ) • Parametric coefficient (i.e., the “a” in y = ax + b) • Geometric ratio (i.e., “rise over run”) • Physical Property (i.e., steepness) • Functional Property (i.e., rate of change) Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  23. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Using rate and start to make predictions Learning Outcome:Seeing rate in a table and reasoning backwards to find start Covariation Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  24. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Learning Outcome:Seeing rate in a table and reasoning backwards to find start Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  25. The Apple iPhone Problem School of Education Apple already building iPhones at rate of 40 million a year? By Slash Lane Published: 10:00 AM EST Monday, August 4, 2008 http://www.appleinsider.com/articles/08/08/04/apple_already_building_iphones_at_rate_of_40_million_a_year.html Apple is reportedly testing the limits of its overseas manufacturing facilities in order to keep up with demand for the new iPhone 3G, with production already cranked nearly sevenfold compared to the first-generation model. Foxconn, the company's Taiwanese handset and iPod manufacturer, has recently ramped production of the new iPhone to 800,000 units per week, says TechCrunch, citing a person "close to Apple with direct knowledge of the numbers." The build rate is said to be "above current full capacity" for the Foxconn facilities allotted to Apple's handset business, which has led to concerns that quality control may suffer. At the current rate, Apple stands to produce more than 40 million iPhone 3Gs over the course of twelve months. Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  26. The Apple iPhone Problem School of Education Underline the prediction that the author makes. • “Apple will produce more than 40 million iPhone 3Gs over the course of 12 months.” • Imagine that there were already 15,000,000 iPhone 3Gs when the article was written. • Based on the prediction above, how many iPhone 3Gs will there be 12 months later? Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  27. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Learning Outcome:Seeing rate in a table and reasoning backwards to find start Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  28. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Learning Outcome: Finding rate in a table and reasoning backwards to find start Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  29. The Xbox Shipping Problem School of Education • The table shows the cost of shipping Xbox games • Find: • Rate of change: • Start: Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  30. The X-Box Problem: Students talking about rate School of Education Video not available online Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  31. The X-Box Problem: Students talking about start School of Education Video not available online Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  32. Video Discussion: School of Education • What strategies did students have to invent/reinvent in order to solve this problem? • What other reactions do you have to this video? Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  33. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Finding rate in a table and reasoning backwards to find start Learning Outcome: Subtracting to find change Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  34. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Learning Outcome:Seeing rate in a table and reasoning backwards to find start Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  35. The Ms. Moeller Running Problem School of Education Ms Moeller runs 6 miles every day. On average, she can run six miles in 54 minutes. At this rate, how long would it take Ms. Moeller to run an 11-mile race? Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  36. The Ms. Moeller Problem: Student strategies for making predictions School of Education Strategy #1 Strategy #2 Video not available online Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  37. Video Discussion: School of Education • What strategies did students have to invent/reinvent in order to solve this problem? • What other reactions do you have to this video? Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  38. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Finding rate in a table and reasoning backwards to find start Learning Outcome: Subtracting to find change Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  39. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Learning Outcome:Seeing rate in a table and reasoning backwards to find start Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  40. The Window Problem School of Education Leslie is a window installer. On Friday, she installed two windows, and charged 402 dollars. Last week, on another job, she charged 517 dollars to install seven windows. A new customer has asked Leslie to install five windows. How much will this cost? Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  41. The Window Problem: Students reinventing strategies School of Education Video not available online Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  42. Video Discussion: School of Education • What strategies did students have to invent/reinvent in order to solve this problem? • What other reactions do you have to this video? Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  43. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Learning Outcome: Subtracting to find change Xbox Shipping Problem Learning Outcome:Seeing rate in a table and reasoning backwards to find start Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  44. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Learning Outcome:Seeing rate in a table and reasoning backwards to find start Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  45. The Window Problem II: Excel School of Education Write a formula to calculate the rate of change Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  46. The Iceberg Metaphor(Webb, et al., 2008) Algebraic Ratio The Algebraic Ratio Delivery Costs Snowfall Accumulation Running Pace Scaling triangles Context How do we use math to make predictions?

  47. Lessons learned School of Education Students can mathematize situations involving rate of change to arrive at the other sub-constructs of slope. Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  48. School of Education Realized Learning Trajectories Hypothetical Learning Trajectory (Simon 1995) Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Using rate and start to make predictions Learning Outcome:Seeing rate in a table and reasoning backwards to find start Covariation Change and covariation Ms. Moeller Running Problem Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  49. School of Education Realized Learning Trajectories Window Problem II: Excel Learning Outcome: Formalizing strategies into the algebraic ratio Window Problem I Xbox Shipping Problem Learning Outcome: Subtracting to find change Using rate and start to make predictions Learning Outcome:Seeing rate in a table and reasoning backwards to find start Ms. Moeller Running Problem Change and Covariation Apple iPhone Problem Learning Outcome: Dividing to find a unit rate Learning outcome: Using rate and start to make a prediction Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

  50. Lessons learned:Confusion between change and value School of Education • In their discourse, students didnot distinguish between change and value Ambiguity Resolution Montage Video not available online Michael Matassa & Frederick Peck Beyond rise over run: Developing robust understandings of slope University of Colorado Boulder School of Education ICME 2012, Seoul

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