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Accessing Meaning Through Function Stories in College Algebra Lauretta Garrett Kristen Miller

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

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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

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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)

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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

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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?

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

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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)

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

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

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

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

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

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Student Work Jonelle Malia Louise Tremaine Shequita Starr Trenicia Tawana

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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.”

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

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

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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.”

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

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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.

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

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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)

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

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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.

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One Possible Tabular Representation of Shequita’s Example InputOutput Baby boy 1Sister and Son-in-law 1 Baby girlAdopted sister and Son-in- law 2 Baby boy 2Adopted sister and Son-in- law 2

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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?

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

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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

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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

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

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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)

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Starr’s Work

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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

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

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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)

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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)

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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?

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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)

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Trenicia’s Table of Values

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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 Teacher Practice: Using Stories

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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

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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

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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...

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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?

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

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

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References Darby, Linda. (2008). Making Mathematics and Science Relevant through Story. Australian Mathematics Teacher, 64(1), Franz, Dana Pomykal, & Pope, Margaret. (2005). Using children's stories in secondary mathematics. American Secondary Education, 33(2), 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),

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References Labov, William. “The Transformation of Experience in Narrative Syntax.” Language in the Inner City: Studies in the Black English Vernacular. Philadelphia: University of Pennsylvania Press, Sand, Mark. (1996). A function is a mail carrier. Mathematics Teacher, 89(6),

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