Affordances

1
Affordances Constraints in Abstract
AlgebraAre Prospective Teachers Connecting the
Mathematics?

• Dr. Cindy S. Henning
• Columbus State University
• Columbus, Georgia
• AMTE January 2008

2
Theoretical PerspectiveSituative Research

• Knowledge Construction is situated (Greeno, 2003
  Cobb Bowers, 1999).
• Knowledge is always created in a context and
  cannot be separated from that context
• Affordances and constraints exist in this context
  
• Individuals impact are impacted by the
  environment
• Diverse personal histories experiences
• Perceptions of self-efficacy, confidence and
  motives Learning is becoming attuned to the
  affordances and constraints of the social system
• Patterns of participation are the unit of
  analysis

3
Research Questions

• What affordances and constraints existed in the
  abstract algebra classroom?
• To what extent did the affordances and
  constraints impact undergraduates learning
  orientations?
• Are undergraduates able to recontextualize their
  knowledge of algebra or of mathematics, in
  general? (i.e. To what extent were undergraduates
  able to make connections between secondary level
  algebra and abstract algebra?

4
Learning Motivation Attribution Theory (Weiner,
1984)

• Locus of Cause Stability of Attribute
• Success hard work
• Internal, controllable factors
• Success talent
• Internal, uncontrollable (stable) factors
• Success luck, teacher, etc.
• External factors, uncontrollable factors

5
Learning Motivation Goal Theory (Seifert
OKeefe, 2001)

• Student behaviors are functions of goals they
  wish to achieve.
• Learning Goals (Mastery)
• Typically attribute high level of efforts with
  success
• Prefer challenges, use more strategies, make more
  positive self-statement
• Performance Goals (Ego-oriented)
• Pre-occupied with ability, ability is the cause
  of success, intelligence is fixed.
• Prefer to compare themselves with others
• More likely to develop perceptions of poor
  self-efficacy
• Social Goals
• Seek approval from peers
• May be motivated in non-academic pursuits

6
Situative Perspective Learning Orientations
(Greeno, 2003)

• Skill-oriented
• Learners focus upon acquiring skills and attend
  to learning the steps to procedures and when to
  apply them.
• Understanding-oriented
• Learners focus upon explanations of solutions,
  relationships of concepts, and purposes or
  meanings of activities.
• Participation-oriented
• Learners focus on participating in a mathematics
  community by proposing problems, evaluating
  solutions, and communicating answers, examples
  and arguments.

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The Classroom

• First course in Abstract Algebra
• Introductory group theory
• Groups, subgroups, cyclic groups, permutations,
  orbits, abelian groups, semi-groups
• Not Rings Fields
• Fifteen week semester
• Text
• Fraleigh, J. (2003) A first course in Abstract
  Algebra, Seventh Ed., Pearson Education, Inc.,
  Boston.
• Eight students enrolled
• Five prospective teachers one mathematics major
  two dual enrollment students
• One participant was 29, the others were 19-24
  years old

8
Data Collection

• Pre- Post Course Interviews with undergraduates
  and instructor
• 70 of course session observed
• Field notes, course artifacts (e.g. handouts,
  assignments, text, exams)
• Data analysis utilized open and axial coding
  NVivo qualitative software

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Instructor Expectations

• New material so most students have a clean
  slate
• To engage students in creating mathematics
• I want to get the students to try and start
  using their intuition and developing the
  willingness to make conjectures.
• To see student gains in personal
  confidencebecause theyve experienced the
  excitement and joy of doing mathematics
• He held the same expectations for both math and
  secondary math majors

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Affordances and Constraints

• First Six Weeks
• Students were instructed to not to peek at
  their texts and to set them aside
• Instructor provided tasks
• Students were allowed to explore and discuss
  observations and conjectures
• Instructor would refine their language, provide
  new tools for analysis
• Instructor would provide summaries of
  discussions, including definitions and theorems
• Remaining Nine Weeks
• Text reading assignments and exercises with
  students working problems at the board

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Example Introduction to the Course Content

• When we look at abstract, we are looking at the
  essence of the thing, or the minimal
  characteristics or propertiesWere going to
  study numbers like this and try to minimum and
  eliminate things to get down to the essence.
• How many choices do I have for filling in this?
• Ex a b a

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Commonly observed classroom interactions

• Instructor acted as transcriber
• Proofs completed with one student contributing
  one line of a proof
• Students randomly chosen to do problems at board,
  but could pass the chalk
• Students verified accuracy of their work with the
  instructor or the class.

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Instructors invitations to research discovery

• Shared an undergraduate research project
• Distributed handouts relating abstract algebra to
  cryptography
• Instructor feign ignorance at discoveries
• Instructor would allow divergent directions

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How the professor reacted to the class struggling
with a problem

• This reminds me of some research I was doing
  with my students who had trouble. I suggested
  that we assume what they wanted to prove was true
  and find results from that, and then what
  happened, she eventually proved what she wanted.
  This is a trick mathematicians use.

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Student Attributes Attunements
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Students Histories Psychologies

• All students reported learning skill-oriented
  math in high school
• Few word problems, little or no proof, and not
  expected to verbally justify their work
• All students reported high self-efficacy with
  respect to secondary mathematics
• All students noted growth in their definition of
  what constitutes mathematics

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18
Fay

• Failed Abstract Previously
• Expressed concerns with previous instructor who
  made her feel stupid
• Equated effort with lack of ability
• I would say if I was stuck on a problem I can
  usually, give me time and I can figure it out.
  When it comes to abstract, probability, real
  analysis, that takes me a little bit more time.
  Im not good at it, Im not good at it.

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Fay

• Attuned to detailed requirements (constraints)
  imposed by the professor
• Will this be on the exam?
• Is that how you want the proof written?
• Desired proof writing algorithms
• Key learning strategies memorization, repeated
  practice. Discovery is not my way of learning.
• Observed social goal-orientation
• Often sought social support from peers to view
  the math as not worthwhile

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21
Kay

• Self-described "perfectionist"
• "better than most" at math made comparisons
  between her performance and others.
• After the course she states
• Its important, understanding what to do and
  understanding why you do it. If you dont
  understand why then you are really not
  understanding math
• .

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Kay

• Interviewer If you think about the course, what
  were the big things you learned out of it?
• Kay Um. I guess the biggest thing is just not
  to be afraid to try new things because usually
  the first thing youre going to try isnt going
  to work, but if you just try to find the one
  thing that will work, youre going to be sitting
  there forever. Um, so I guess it kind of gave me
  courage to try new things and not be afraid to
  fail.
• Interviewer Do you think youve been afraid to
  fail before?
• Kay Yes, usually.
• Interviewer In all of your classes?
• Kay Yes.
• Interviewer And this is related to math?
• Kay Yes.

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Ed

• Failed abstract previously
• Both Ability and Effort were attributed to his
  success
• There was little work I had to do in high school
  because it was just so easy for me to understand
  it and maintain it. As Ive gotten along further
  and further in college thoughI just have to work
  on it harder than I had to work before

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EdConsidered Participation in Mathematics

• When he was talking about those people who were
  doing all these things with publishing ..So I
  thought, let me see what I could come up with and
  I started noticing things. The professor said
  why dont you work on this here and see how that
  turns out. It was different from the book and it
  wasnt something that we were working on from the
  book but it seemed kind of newer something that
  could be discovered I guess. Find something that
  other people didnt know about, so it was very
  exciting into it then.
• But then when we started getting into the book,
  it seemed like the same old thing, nothing new or
  exciting to go over. I just kind of lost
  interest.

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Deb Effort not Skill

• I think that for any of the great
  mathematicians of the past, you know they come up
  with some stuff and the rest of us are struggling
  to understand it, it doesnt mean that they
  didnt struggle with it.

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Do you see any connections between Abstract
Algebra High School Algebra?

• Only one student responded Yes
• Deb
• Learning about the groups and subgroups and
  knowing about the integers and real numbers and I
  can remember initially learning about all those
  things and thinking, Thats nice, but theyre
  all number. But now understanding why there is
  that differentiation between the different types
  of numbers and different things went into a group
  and why are these subgroups of this group.

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Recontextualizing the Content

• Task
• Simplify 2(x-3) 3 (5 3x)
• Explain how to solve this as if you were
  tutoring
• What properties, if any, did you use?
• Did taking this course change how you
  understand this problem?
• Define a group. Are any of those properties
  related?
• Post-course interview only

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Results Pre-Abstract Algebra

• All were successful in solving the problem
  accurately
• All discourse was skill-oriented, citing rules
  and algorithms (a negative times a negative
  makes a positive)
• Intermingled negative and subtraction with
  out attending to the conceptualization of
  inverse or operation
• Properties Noted
• Distributive (5)
• Commutative Associative (1)
• Other rules (e.g. adding like coefficients) (3)
• The individual admitted to being able to
  recognize these, but just named them.

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Results Post-Abstract Algebra

• All were still successful in solving the task
• All discourse was skill-oriented, citing rules
  and algorithms (a negative times a negative
  makes a positive)
• Intermingled negative and subtraction with
  out attending to conceptualization of inverse
  or operation
• Initial property identification Distributive (4)
• After additional prompts
• All but one defined a with reasonable accuracy
  group, though not precisely
• Commutative (3) Associativity (2) Identity (1)

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Summary--Affordances

• Inquiry-based approach afforded students the
  opportunity to participate in building
  mathematical knowledge
• Most students perceived a safe environment for
  risk taking
• Some reported increase in confidence in problem
  solving
• All student attuned to this affordance, despite
  diverse motivations for studying mathematics
• All prospective teachers indicated they may use
  some of the professors instructional techniques
• One student reported engaging in external
  mathematical work.

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SummaryImpediments to Attunement

• Perceptions that equate high levels of effort
  with lack of innate ability
• Phrasing of text exercises that discourage
  inquiry
• Inability to connect knowledge of and about
  abstract algebra to high school mathematics
• With respect to concepts related to the
  definition of a group AND the importance of
  developing reasoning