Accessing Meaning Through Function Stories in College A lgebra

1. Accessing Meaning Through Function Stories in College Algebra Lauretta Garrett garrettl@mytu.tuskegee.edu Kristen Miller

2. Agenda • Description of the study • Classroom practices used • Student work • Emerging themes • What’s a story? • Teacher Practice

3. What was done • Fall 2011, Spring 2013 • Data collected from college algebra/pre-calculus algebra • Methodology • Student work examined for emerging theory about student thinking and learning

4. Standards for College Algebra • Modeling • Learn through modeling real-life situations • Connecting with other disciplines • View mathematics as interrelated with culture • Linking multiple representations • Select, use, and translate among numerical, graphical, symbolic, and verbal Blair, 2006, “Beyond Crossroads” (p. 5)

5. Mathematical Topic: Functions • “[A] clear description of how one thing depends on another”(Crauder et al., 2003, p. 88) • Each domain element paired with one and only one range element • Different representations • Map, Graph, Table, Equation, Words

6. Research Questions • When students are encouraged to reset the concept of function in settings of their own choosing, what representations do they produce? • What qualities do those representations have?

7. Classroom Practice: Analogy • Commonly experienced easily understood settings providing “groundedness” • Groundedness can facilitate “meaning-making and self-monitoring processes” (Koedinger & Nathan, 2004, p. 158)

8. Analogy: Delivering the Mail • Mail carrier places each letter in a mailbox • Could place more than one letter in one mailbox • One letter cannot be placed in more than one mailbox (Sand, 1996)

9. Groundedness • Most students have gotten mail in a mailbox or sent mail • Common experience connecting the class • Acommunity of learners who can all understand and relate to this idea

10. Pictorial Mapping Simple drawing Connects to a mapping representation Demonstrates characteristics of a function

11. Whole Group Discussion Connect idea to other representations • Mappings • Tables • Sets of coordinate points • Equations • Graphs

12. Add Notation to Analogy Domain {x1, x2, x3} Range {y1, y2}

13. Student Work Source • Students were asked to . . . • Describe a real life example of a functional relationship • Tell why the example demonstrates the mathematical properties of a function

14. Student Work • Jonelle • Malia • Louise • Tremaine • Shequita • Starr • Trenicia • Tawana

15. Jonelle’s Example “When in a relationship it should be 2 people and not extra people. That would be called cheating. . . . A boy is not supposed to date 2 girls at a time.”

16. Characteristics • Emotion • Engaging – want to know more • Possibly part of student’s lived experience • Personal connection to mathematical idea

17. A Story . . . A larger narrative could be easily built upon this • Characters • Conflict

18. Malia’s Example • “You have 4 children and 3 cars headed to the fair. More than two of the children can fit into one car but one child cannot be in more than one car at the same time.”

19. Louise’s Example “Each child receive one piece of candy. You can’t promise ‘ONE’ piece of candy to ‘two’ kids.”

20. Malia and Louise • Show emotion on the faces of characters in their “stories.” • Malia • The children who can all go to the fair are happy. • Louise • The children who get candy are happy. The children who have to split it are sad.

21. Emotional Connections • Contexts appear to come from something the student might have personally experienced or about which the student feels strongly.

22. Tremaine’s Example • “In some religions a man can have more than one wife, but a wife can’t have more than one husband. A woman can’t have more than one husband. A husband can have more than one woman. An x-value can’t have more than one y-value. A y-value can have more than one x-value.” • (illustration to follow)

24. Unexpected Topics • Connection to another topic or discipline • Setting not provided or anticipated, • Appears to be of interest to the student.

25. Shequita’s Example • In a family of five, a mother, a father, sister, brother, and adopted sister, each child married and one sister had a baby boy, another had a baby girl, and the last had a baby boy also. The input is different, which is the children, because they are not alike, but two have the same output.

26. One Possible Tabular Representation of Shequita’s Example

27. Characteristics • Does not parrot typical textbook examples • Characters in the story that are not part of the function. • Three generations • One wants to know more about the story • Why did they adopt one of the sisters? • Was this situation part of the student’s lived experiences?

28. Complex Situations • The context is complex • It provides elements unnecessary to the goal that add to the richness

29. Student Responses Included • Pictorial mappings • Some closely mimicked the mail carrier example • Included emotion, unexpected topics, and complex situations • Student creations were rich and meaningful • Examples that related to the college experience

30. Hillary’s Example A vending machine – If you press a combination of a letter and a number to get one item you can never press a combination and two things come out. [If] you press A2 then you get that piece of candy you won’t get anything else

31. Three Emergent Categories • Complex situations • Go beyond minimal mathematical structure • Emotional connections • Very personal to the student • Unexpected topics • Multi-cultural or cross disciplinary

32. Starr’s Description “A bunch of bowmen must shoot arrows to the specific colored target it was assigned [to]. A target can receive different colored arrows but that one arrow can only go to one target.” • (illustration to follow)

33. Starr’s Work

34. Cognitive Interplay • Occurs when students are thinking back and forth between analogy or real life setting and mathematical representation • They become conceptually connected • In Starr’s the setting connects directly to the idea of mapping • Builds conceptual understanding of abstract representations

35. What’s a Story? • Burke (1969) included the following • Actor, Action, Goal/Intention, Scene, Instrument (Burke, 1969) • Labov (1973) included • A complicating action

37. Stories in Mathematics • A form of representation that brings mathematics and context together (Clark, 2007) • Connecting mathematics to contexts allows “[more] coherent and deeper understanding” (Darby, 2008, p. 9) • Provide frameworks with which students at all levels can better understand how to make mathematical connections (Franz & Pope, 2005)

38. Stories in Mathematics • Open the way for classroom discourse: mathematics is discussed through the medium of the story. • Grounds representations, enhancing the learning of symbolization (Koedinger & Nathan, 2004). • Support the standards of modeling, connecting, and linking (Blair, 2006)

39. Trenicia’s Example • As you consider her work, note: • Is it complex, emotionally engaging, or unique? • How is it an example of a story? • What is her mathematical understanding?

40. Trenicia’s Example • I came home at 12:00 noon but our fridge is broken so my mom told me every 45 minutes I would manually have to change the degrees. This relationship is functional because every degree that is put with the time is different there are no degrees that are the same. Therefore this relationship is functional. • (Table to follow)

41. Trenicia’s Table of Values

42. Teacher Practice: Using Stories • An open-ended assignment requiring connections to a real life • Encourage and allow the three categories to emerge: complex, emotionally engaging, unconventional settings • You may require as much narrative as you would like

43. Teacher Practice: Following up Stories • Investment in a story can motivate • An examination of data related to the topic • The use of software to model that data • An examination of the properties of that model

44. Teacher Practice: Fostering Cognitive Interplay • How can teachers help support and deepen the cognitive interplay? • Connect to standard representations • Questioning for cognitive interplay • Follow up on stories with an examination of data

45. Connecting to Standard Representations • Encourage the use of multiple representations, including • illustrations • idiosyncratic (non-standard, personal to the student) representations • standard mathematical representations • Tawana’s example . . .

47. Questioning for Cognitive Interplay • Question students about their context • How can you quantify this situation? • What will this mapping look like as a table of values, a set of ordered points, a graph? • What is the best mathematical representation for this situation?

48. One More Story: Kevin • See handout • What do you notice about Kevin’s work? • Stories have the power to engage students!

49. References • Burke, Kenneth. A Grammar of Motives. Berkley: University of California Press, 1969. • Clark, Julie. (2007). Mathematics saves the day. Australian Primary Mathematics Classroom, 12(2), 21-24. • Crauder, Bruce, Evans, Benny, & Noell, Alan. (2003). Functions and change: A modeling approach to college algebra (Second ed.). Boston: Houghton Mifflin.

50. References • Darby, Linda. (2008). Making Mathematics and Science Relevant through Story. Australian Mathematics Teacher, 64(1), 6-11. • Franz, Dana Pomykal, & Pope, Margaret. (2005). Using children's stories in secondary mathematics. American Secondary Education, 33(2), 20-28. • Koedinger, Kenneth R., & Nathan, Mitchell J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13(2), 129-164.