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School of Education

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  1. Freudenthal institute US School of Education Beyond rise over run: Contexts, representations, and a learning trajectory for slope RME 4 Boulder CO USA Sept. 29, 2013 Frederick Peck (Frederick.Peck@Colorado.edu) Freudenthal Institute US University of Colorado Boulder

  2. Freudenthal institute US School of Education What are some characteristics of the function represented by this graph? Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  3. Freudenthal institute US School of Education • The dependent variable increases by two units for every unit increase in the independent variable • In the U.S., 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” Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  4. Freudenthal institute US School of Education • The dependent variable increases by two units for every unit increase in the independent variable Geometric • In the U.S., 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 Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  5. Freudenthal institute US School of Education http://www.youtube.com/watch?v=R948Tsyq4vA Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  6. Freudenthal institute US 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) Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  7. 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

  8. Freudenthal institute US School of Education Summary: In the U.S.: Slope • The dependent variable increases by two units for every unit increase in the independent variable Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  9. Freudenthal institute US School of Education Summary: In the U.S.: Slope measures steepness is procedural Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  10. Freudenthal institute US School of Education Summary: In the U.S.: y = mx + b Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  11. Freudenthal institute US School of Education A Design Experiment (Cobb, 2000)on Slope Goal: To develop versatile and adaptable mathematical realities around slope. • 19 students, 4 weeks • High school Algebra 1 (9th grade, ages 14-15) • Combination of lecture, whole class discussion, small group work, and individual work. • RME principles Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  12. Freudenthal institute US 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) Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  13. Freudenthal institute US 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) Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  14. Freudenthal institute US 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) Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  15. Freudenthal institute US 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) Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  16. Freudenthal institute US School of Education Realistic Mathematics Education (RME) (Treffers, 1987) • Activity • Reality • Reinvention • Intertwinement • Social Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  17. Iceberg metaphor; Emergent modeling (Webb, et al., 2008) (Gravemeijer, 1999) Formal The Algebraic Ratio Preformal “models for” Informal “models of”

  18. Iceberg metaphor; Emergent modeling (Webb, et al., 2008) (Gravemeijer, 1999) Formal The Algebraic Ratio Context Making predictions Preformal “models for” Informal “models of”

  19. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  20. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  21. The Ms. Moeller Running Problem Freudenthal institute US 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? Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  22. The Ms. Moeller Problem: Students making predictions Freudenthal institute US School of Education Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  23. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  24. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  25. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  26. The Xbox Shipping Problem Freudenthal institute US School of Education • The table shows the cost of shipping Xbox games • Predict the cost of shipping 12 games Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  27. The Xbox Shipping Problem: Student strategies for predicting Freudenthal institute US School of Education Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  28. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  29. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  30. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  31. The Window Problem Freudenthal institute US 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? Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  32. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  33. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  34. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  35. The Window Problem II: Excel Freudenthal institute US School of Education Write a formula to calculate the rate of change Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  36. Lessons learned Freudenthal institute US School of Education Situations that involve “making predictions” can be powerful contexts for ensembles to invent progressively more formal productions involving slope Reinvention is distributed: • Contexts and representations are active participants in the invention process. • Changing the context and representations changes the way invention is distributed. Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  37. Lessons learned: Reinvention is distributed Freudenthal institute US School of Education What work is the context and/or representation doing in this problem? Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  38. Lessons learned: Reinvention is distributed Freudenthal institute US School of Education What work is the context doing here? Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  39. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  40. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) (Coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  41. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) (Coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  42. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) (Coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  43. Lessons learned: Reinvention is distributed Freudenthal institute US School of Education Contexts that do work to make students distinguish between change and value: • Dynamic experiences • Negative rates of changein situations where negative values are impossible. • “Clock time” (e.g. 11:00 pm) for values when time is the independent variable Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  44. Lessons learned: Reinvention is distributed Freudenthal institute US School of Education Consider the work that the “find one strategy” and the ratio table do to structure this solution strategy Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  45. Lessons learned: Reinvention is distributed Freudenthal institute US School of Education Consider the work that the “find one strategy” and the ratio table do to structure this solution strategy Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  46. Lessons learned: Reinvention is distributed Freudenthal institute US School of Education Ratio table and “find one strategy” work to promote a within unit (scale factor) strategy. Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  47. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) (Coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  48. Functional property (rate of change) Algebraic ratio Formal Many-to-one (coordination of two quantities changing together; intensive) (Coordination of two quantities changing together; intensive) Preformal “models for” Informal “models of” Many-as-one (measure of one quantity; extensive)

  49. Lessons learned: Reinvention is distributed Freudenthal institute US School of Education Ratio table and “find one strategy” work to promote a within unit (scale factor) strategy. To create the unit rate as a measure of covariation (an intensive quantity), students need to consider the between unit factor Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013

  50. Lessons learned (again) Freudenthal institute US School of Education Situations that involve “making predictions” can be powerful contexts for ensembles to invent progressively more formal productions involving slope Reinvention is distributed: • Contexts and representations are active participants in the invention process. • Changing the context and representations changes the way invention is distributed. Frederick Peck Beyond rise over run: Contexts, representations, and a learning trajectory for slope Freudenthal Institute US, School of Education, University of Colorado Boulder RME 4, Boulder CO, 2013