Beyond rise over run: Contexts, representations, and a learning trajectory for slope

Transcription

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

2 y x What are some characteristics of the function represented by this graph? Freudenthal Institute US,, University of Colorado Boulder

3 7 y x 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 Freudenthal Institute US,, University of Colorado Boulder

4 x y 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 Geometric Students are taught to calculate slope as rise over run or change in y over change in x Freudenthal Institute US,, University of Colorado Boulder Procedural

5 Freudenthal Institute US,, University of Colorado Boulder

6 Sub-constructs of slope (Stump, 1999) y 2 y 1 x 2 x 1 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) Freudenthal Institute US,, University of Colorado Boulder

7 Physical property Geometric ratio Functional property Algebraic ratio The number one result when searching for rate of change in Google!

8 Summary: In the U.S.: Slope 7 y x The dependent variable increases by two units for every unit increase in the independent variable Freudenthal Institute US,, University of Colorado Boulder

9 Summary: In the U.S.: Slope measures steepness is procedural Freudenthal Institute US,, University of Colorado Boulder

10 Summary: In the U.S.: y = mx + b Freudenthal Institute US,, University of Colorado Boulder

11 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 (9 th grade, ages 14-15) Combination of lecture, whole class discussion, small group work, and individual work. RME principles Freudenthal Institute US,, University of Colorado Boulder

12 Our focus today y 2 y 1 x 2 x 1 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) Freudenthal Institute US,, University of Colorado Boulder

13 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) Freudenthal Institute US,, University of Colorado Boulder

14 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) Freudenthal Institute US,, University of Colorado Boulder

15 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) Freudenthal Institute US,, University of Colorado Boulder

16 Realistic Mathematics Education (RME) (Treffers, 1987) Activity Reality Reinvention Intertwinement Social Freudenthal Institute US,, University of Colorado Boulder

17 Iceberg metaphor; Emergent modeling (Webb, et al., 2008) (Gravemeijer, 1999) Formal The Algebraic Ratio Δx = y 2 y 1 x 2 x 1 x y Preformal models for Δx Informal models of

18 Iceberg metaphor; Emergent modeling (Webb, et al., 2008) (Gravemeijer, 1999) Formal The Algebraic Ratio Δx = y 2 y 1 x 2 x 1 Context Making predictions x y Preformal models for Δx Informal models of

19 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

20 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

21 The Ms. Moeller Running Problem 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? Freudenthal Institute US,, University of Colorado Boulder

22 The Ms. Moeller Problem: Students making predictions Strategy #1 Strategy #2 Freudenthal Institute US,, University of Colorado Boulder

23 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

24 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

25 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

26 The Xbox Shipping Problem The table shows the cost of shipping Xbox games Number of games Total cost Predict the cost of 0 shipping 12 games Freudenthal Institute US,, University of Colorado Boulder

27 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

28 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

29 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

30 The Window Problem 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? Freudenthal Institute US,, University of Colorado Boulder

31 The Window Problem: Students inventing a strategy Freudenthal Institute US,, University of Colorado Boulder

32 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

33 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

34 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

35 The Window Problem II: Excel Write a formula to calculate the rate of change Freudenthal Institute US,, University of Colorado Boulder

36 Lessons learned 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. Freudenthal Institute US,, University of Colorado Boulder

37 Lessons learned: Reinvention is distributed What work is the context and/or representation doing in this problem? Freudenthal Institute US,, University of Colorado Boulder

38 Lessons learned: Reinvention is distributed What work is the context doing here? Ambiguity Resolution Freudenthal Institute US,, University of Colorado Boulder

39 Formal Algebraic ratio Δx = y 2 y 1 x 2 x 1 Functional property (rate of change) x y Preformal models for Δx Many-toone (coordination of two quantities changing together; intensive) Informal models of Many-asone (measure of one quantity; extensive)

40 Formal Preformal models for Algebraic ratio Δx = y 2 y 1 x 2 x 1 Δx (Coordination of two quantities changing together; x y Functional property (rate of change) Many-toone (coordination of two quantities changing together; intensive) intensive) Informal models of Many-asone (measure of one quantity; extensive)

41 Formal Preformal models for Algebraic ratio Δx = y 2 y 1 x 2 x 1 Δx (Coordination of two quantities changing together; x y Functional property (rate of change) Many-toone (coordination of two quantities changing together; intensive) intensive) Informal models of Many-asone (measure of one quantity; extensive)

42 Formal Preformal models for Algebraic ratio Δx = y 2 y 1 x 2 x 1 Δx (Coordination of two quantities changing together; x y Functional property (rate of change) Many-toone (coordination of two quantities changing together; intensive) intensive) Informal models of Many-asone (measure of one quantity; extensive)

43 Lessons learned: Reinvention is distributed Contexts that do work to make students distinguish between change and value: Dynamic experiences Negative rates of change in situations where negative values are impossible. Clock time (e.g. 11:00 pm) for values when time is the independent variable Freudenthal Institute US,, University of Colorado Boulder

44 Lessons learned: Reinvention is distributed Consider the work that the find one strategy and the ratio table do to structure this solution strategy Freudenthal Institute US,, University of Colorado Boulder

45 Lessons learned: Reinvention is distributed Consider the work that the find one strategy and the ratio table do to structure this solution strategy Freudenthal Institute US,, University of Colorado Boulder

46 Lessons learned: Reinvention is distributed Ratio table and find one strategy work to promote a within unit (scale factor) strategy. Freudenthal Institute US,, University of Colorado Boulder

47 Formal Preformal models for Algebraic ratio Δx = y 2 y 1 x 2 x 1 Δx (Coordination of two quantities changing together; x y Functional property (rate of change) Many-toone (coordination of two quantities changing together; intensive) intensive) Informal models of Many-asone (measure of one quantity; extensive)

48 Formal Preformal models for Algebraic ratio Δx = y 2 y 1 x 2 x 1 Δx (Coordination of two quantities changing together; x y Functional property (rate of change) Many-toone (coordination of two quantities changing together; intensive) intensive) Informal models of Many-asone (measure of one quantity; extensive)

49 Lessons learned: Reinvention is distributed 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 Freudenthal Institute US,, University of Colorado Boulder

50 Lessons learned (again) 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. Freudenthal Institute US,, University of Colorado Boulder

51 Discussion! Web: (for slides and complete unit) Freudenthal Institute US,, University of Colorado Boulder