Making Student Thinking Visible: A Close Reading of Online Conversations

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Making Student Thinking Visible A Close
Reading of Online Conversations

• University of Maryland
• September 29, 2004
• Anita Salem
• Professor of Mathematics
• Rockhurst University

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Online Calculus Conversations Making Student
Thinking Visible
A Carnegie Foundation Scholarship of Teaching
Learning Project

• Anita Salem (Mathematics)
• Renee Michael (Psychology)

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Problem

• Inability of most students to apply methods and
  concepts used in a practiced problem to a new
  situation

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Literature Review

• Types of Intelligence (Robert Sternberg)
• Book Smarts (skills, methods, procedures)
• Street Smarts (using common sense to find new
  strategies for solving problems)

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Reasoning Modes (Brown, Collins Duguid)

• Students reason with
• laws acting on symbols
• resolving well-defined problems
• producing fixed meaning
• Practitioners Just Plain Folks reason with
• casual stories acting on situations
• resolving emergent problems
• producing negotiable meaning

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Project Goal

• Increase students conceptual understanding
• of first principles in calculus
• by creating an activity where students could
  practice solving problems
  using a Just Plain Folk approach.

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Project Description

• Student-to-Student web-based threaded discussion
  (participation required graded)
• Three questions each focusing on key concepts in
  Calculus
• Follow up in-class student-to-student discussion
• Exam question conceptually related to the key
  concept but contextually varied from the practice
  problems.

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Calculus Conversations Question 1

• Was there ever a time since you were born that
  your weight in pounds was equal to your height in
  inches ?
• Provide a mathematical explanation for your
  answer.

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The Main Idea
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The Conversation
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Tuesday, 1224 pm

• I think we should probably pretend that this
  takes the form of a graph.
• Maybe making two lines.
• One would represent a persons height and the
  other line representing a persons weight.

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Tuesday, 541 pm

• The graph idea is great. When the two line
  graphs are shown together in the same graph, the
  intersections would show the age when the height
  and the weight are the same.
• The y-axis would have to have the same
  calibrations for the height weight.
• The x-axis would have the age of the person.

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Tuesday, 905 pm

• Can you really do this make it clear to the
  outside observer what our graph represents or
  does it have to be done in two graphs and find a
  common point?
• Do you need to specify inches pounds?

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One graph or two graphs?
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Tuesday, 905 pm

• We might want to start out with a scatter plot at
  first for a basis to start with. Label one side
  weight and the other height.
• Then youd need to take it to the next level and
  add the age factor.

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Get rid of the age factor !
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Tuesday, 926 pm

• Unless people were born in some different way
  than I am accustomed to, you weigh less than your
  height at birth.
• I was 11 inches tall and weighed 6 pounds.
• This should take care of the question of yes or
  no.
• But the answer to where that point takes place, a
  scatter plot seems to be the best answer.

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Bingo!

• We have the first conceptual response.
• However it is very poorly incompletely
  expressed.

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Tuesday, 927 pm

• I think the idea of having age on the horizontal
  axis is a great idea.
• For the vertical axis, I think that maybe it
  should just be a listing of numbers you know
  from 0 to maybe 200.
• Then just draw two different graphs in two
  different colors to depict between the two of
  them.
• It would be really easy to tell when the two
  (height and weight) are the same because they
  will be the same point!!!!!!

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Tuesday, 1118 pm

• I feel that you should make two graphs.
• One showing the results of the comparison between
  age and height and
• the other showing the comparison between age and
  weight.
• Then you should lay one graph on top of the other
  and see if there is a point when the height and
  weight equal each other.

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Wednesday, 950 am

• The lines for weight height arent functions
  since they arent continuous.
• Weight tends to fluctuate in a range of 10 to 15
  pounds on most people weekly.
• This creates quite a bumpy line that cannot be
  simulated by a function.

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What about those squiggles?
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Wednesday, 321 pm

• Just because a line is wavy or bumpy does not
  mean that the line is not a function.
• It cannot be a function if you assign two values
  of height to one age which I agree is hard if you
  measure age in years.
• But why not measure age in months? Or days?
• By zooming in on the graph, and calculating age
  in days, I believe it is possible to find the
  point of intersection.

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Wednesday, 1102 pm

• Really, one would not have to draw a graph at
  all.
• If one were to make a table with three columns,
  the same idea could be captured.
• For instance the first column would be labeled
  Time, the second column Height and the third
  column Weight
• Time should start with birth. Then with every
  input of time, there should be a height and
  weight to correspond.
• Of course, this would take extensive time and a
  really, really long piece of paper.

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Lets dump the graph idea!
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Thursday, 638 pm

• I believe that yes there will be a time in our
  lives that we do weigh in pounds as much as we
  are tall in inches.
• Now as far as a graph, I think you would have to
  weigh yourself monthly or bi-monthly until you
  begin to get very close to the barrier.
• Then when you got closer you would have to start
  weighing yourself at least once a day.
• Also, then maybe you get sick and lose some
  weight in which you could have crossed back over
  the barrier.
• It is possible that you can cross it more than
  once.

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Response Continued

• There could be a legend used for clarification as
  to whether the increments on the y-axis represent
  pounds or inches (depending on which line the
  observer was observing.)
• We know that babies length in inches will exceed
  its weight given normal circumstances.
• Therefore the weight line will start below the
  height line.
• But at our current age we know that our weight
  in pounds exceeds our height in inches.
• So, at some point in time the weight line grew
  above the height line, and this is where the two
  lines crossed.

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Assessing the Activity

• Quantitative Analysis
• Looked for possible relationships between
  participation in the on-line discussions
  performance on the contextually varied exam
  questions.
• Qualitative Analysis
• Examined student approaches to problem solving.

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Quantitative Analysis
Exam Question Average

• TABLE 2

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Scoring the Activity

• Score 1
• Did Not Participate
• Score 2
• Contribution Confused the Conversation
• Score 3
• Contribution Kept the Conversation Level
• Score 4
• Contribution Moved the Conversation Forward

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Quantitative ResultsExam Question Averages by
Activity Scores
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Statistical Correlation

• A statistically significant relationship
  exists between student performances on the
  Calculus Conversation activities and
  corresponding performances on conceptually
  related but contextually varied exam questions.

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Coding the Responses

• Practical Response (P)
• Conceptual Response (C)
• Intercommunication (I)
• Language Extremes (L)
• Pose Questions (Q)

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Qualitative Results
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Real Results

• Activity provided a window into students
  understandings and misunderstandings
• Observation that students struggle to rise above
  the details of a problem

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Implications for Practice

• Think more carefully about how to get students to
  use Just Plain Folk problem solving approaches.
• Look for ways to capitalize on students comfort
  levels with practical matters to help them move
  into more conceptual practices?
• Be more aware of the effect of our presence in an
  activity.

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Conclusion

• THIS WAS A CASE OF
• An attempt to improve students conceptual
  understanding of fundamental ideas in calculus
• THAT RESULTED IN
• improved teacher understanding of how students
  construct their own meaning of fundamental ideas
  in calculus.

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Acknowledgements

• Participating Calculus Instructors
• John Koelzer
• Paula Shorter
• Keith Brandt
• Julie Prewitt Kramschuster
• Research Assistance
• Craig Sasse
• Tom Jones

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Professional Changes

• Better consumer of the Scholarship of Teaching
  Learning.
• Increased respect for how other disciplines ask
  answer questions that are of interest to me in
  becoming a better teacher.
• Clearly articulate how this work is scholarly in
  every sense of the word.

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Student Attitudes

• 51 approved
• 30 disapproved
• 13 no comment
• Typical Positive Responses
• Typical Negative Responses

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Positive Responses to Calculus Conversations

• Calculus Conversation Questions brought the
  thinking to a whole new level. It was difficult
  to understand what people were thinking because
  it forces us to write about Math. You also have
  to challenge yourself to get your thoughts across
  to the rest of the group. It would have been
  helpful to somehow have learned a way to write
  about Math and get your point and thoughts across
  early. It was quite challenging.
• I found out the importance of weighing in early.
  When I weighed in early, I got more out of it and
  was able to take a more active role in the
  conversation. Though I didnt always get the
  answers, I understood the concepts better.
• The conversations allowed people to think
  independently, but at the same time work to solve
  problems in a group. The conversation allowed us
  to talk out the problem in understandable terms.

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Negative Responses to Calculus Conversations

• This helped me some of the time, but most of the
  time I wished there werent so many people doing
  the same problem. Many times, I would check it
  the second day and there would be so many
  students with postings that I was overwhelmed and
  got it set in my mind that there was no way I
  could do the problem.
• I feel the conversation often starts fast and
  then drags on courtesy of the people at the end
  who have no idea what they are going to write.
  They just repeat what other people have already
  said. This repetition clogs up the discussion
  which detracts from the usefulness of the
  activity.
• I never felt that Calculus Conversations
  contributed to my understanding of a concept.
  There have been numerous times that I have
  totally understood something, then gone to read
  other peoples responses to the topic and gotten
  so confused that I have no idea what I am
  supposed to do anymore.

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Real Results

• Activity provided a window into students
  understandings and misunderstandings
• Observation that students struggle to rise above
  the details of a problem
• Striking link between learning theories, found
  in the literature based on studies from K-12 math
  classes practice, found in the project

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Implications for Practice

• Think more carefully about how to get students to
  use Just Plain Folk problem solving approaches.
• Look for ways to capitalize on students comfort
  levels with practical matters to help them move
  into more conceptual practices?
• Be more aware of the effect of my presence in an
  activity.

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Conclusion

• THIS WAS A CASE OF
• An attempt to improve students conceptual
  understanding of fundamental ideas in calculus
• THAT RESULTED IN
• improved teacher understanding of what students
  know and dont know about those ideas.

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The Collaboration

• Collaborator Colleagues
• My reflections on the collaboration
• Reflections of my collaborator
• By-products of collaboration

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By-Products of Collaboration

• Good model for interdisciplinary work
• Have always heard about the value of
  interdisciplinary work but havent seen much of
  it. This experience gave us one concrete example
  of how it works.
• Helped to create a community for the scholarship
  of teaching
• Provided a strong example of how a teacher
  thinks about and incorporates scholarship into
  her courses.

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My Reflections on the Collaboration

• Provided me with a customized roadmap for
  learning theories and methods of assessment
  applicable to my project
• Allowed me to work more comfortably and
  confidently out-of-discipline
• Sharing responsibility served to keep me on-task
  and it raised the bar for the project

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Collaborator Reflections

• Provided a non-artificial environment for
  mentorship
• Allowed me to practice qualitative research
  skills
• Learned practical ideas for my own course
  development
• Learned about our Calculus courses deepened my
  knowledge as a student advisor