Accessing Meaning Through Function Stories in College

Algebra

Lauretta Garrettgarrettl@mytu.tuskegee.edu

Kristen Miller

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

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

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)

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

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?

Classroom Practice: Analogy• Commonly experienced easily

understood settings providing “groundedness”

• Groundedness can facilitate “meaning-making and self-monitoring processes”

(Koedinger & Nathan, 2004, p. 158)

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)

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

Pictorial Mapping

Simple drawing

Connects to a mapping representation

Demonstrates characteristics of a function

Whole Group DiscussionConnect idea to other representations

–Mappings–Tables–Sets of coordinate points –Equations–Graphs

Add Notation to Analogy

Domain {x1, x2, x3}

Range {y1, y2}

• 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

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

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.”

Characteristics• Emotion• Engaging – want to know more• Possibly part of student’s lived

experience• Personal connection to mathematical

idea

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

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.”

Louise’s Example

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

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.

Emotional Connections• Contexts appear to come from

something the student might have personally experienced or about which the student feels strongly.

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)

Unexpected Topics

• Connection to another topic or discipline

• Setting not provided or anticipated, • Appears to be of interest to the

student.

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.

One Possible Tabular Representation of Shequita’s Example

Input OutputBaby boy 1 Sister and Son-in-law 1Baby girl Adopted sister and Son-in-

law 2Baby boy 2 Adopted sister and Son-in-

law 2

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?

Complex Situations• The context is complex• It provides elements unnecessary

to the goal that add to the richness

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

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

Three Emergent Categories• Complex situations

–Go beyond minimal mathematical structure

• Emotional connections–Very personal to the student

• Unexpected topics–Multi-cultural or cross disciplinary

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)

Starr’s Work

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

What’s a Story?• Burke (1969) included the following

– Actor, Action, Goal/Intention, Scene, Instrument (Burke, 1969)

• Labov (1973) included– A complicating action

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)

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)

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?

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)

Trenicia’s Table of Values

• 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

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

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

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 . . .

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?

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

students!

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.

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.

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, 1973.

• Sand, Mark. (1996). A function is a mail carrier. Mathematics Teacher, 89(6), 468-469.