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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
  Online Calculus Conversations:
   Making Student Thinking Visible




A Carnegie Foundation
Scholarship of Teaching & Learning Project



           Anita Salem (Mathematics)
           Renee Michael (Psychology)
Problem

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


Types of Intelligence (Robert Sternberg)

   Book Smarts (skills, methods, procedures)


   Street Smarts (using common sense to find
    new strategies for solving problems)
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
 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.
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.
    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.
The Main Idea

        Weight (pounds)




                          Height (inches)




            Age
The
Conversation
Tuesday, 12:24 pm

   I think we should probably pretend that this
    takes the form of a graph.

   Maybe making two lines.

   One would represent a person’s height and the
    other line representing a person’s weight.
Tuesday, 5:41 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.
Tuesday, 9:05 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?
One graph or two graphs?
   Weight (pounds)




                           Height (inches)

       Age




                     Age
Tuesday, 9:05 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 you’d need to take it to the
    next level and add the age factor.
Get rid of the “age factor” !
Tuesday, 9:26 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.
Bingo!

   We have the first “conceptual”
    response.

   However – it is very poorly &
    incompletely expressed.
Tuesday, 9:27 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!!!!!!
Tuesday, 11:18 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.
Wednesday, 9:50 am

   The lines for weight & height aren’t functions
    since they aren’t 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.
What about those squiggles?
Wednesday, 3:21 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.
Wednesday, 11:02 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.
Let’s dump the graph idea!
  Age           Height        Weight

  Birth         20   inches   7   pounds


  9   months    27   inches   18   pounds


  48   months   36   inches   35   pounds


  60   months   38   inches   40   pounds
Thursday, 6:38 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.
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.
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.
Quantitative Analysis

            Exam Question Average


                        TABLE 2
Exam 1 Question Avg   Exam 2 Question Avg   Exam 3 Question Avg
         64 %                  86 %                 60 %
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
Quantitative Results
Exam Question Averages by Activity Scores

 100%
  90%
  80%
  70%
  60%                              1
  50%                              2
  40%                              3
  30%                              4
  20%
  10%
   0%
        Exam 1   Exam 2   Exam 3
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.
Coding the Responses
    Practical Response (P)

    Conceptual Response (C)

    Intercommunication (I)

    Language Extremes (L)

    Pose Questions (Q)
Qualitative Results
100%
 90%
 80%
 70%
 60%                             P
 50%                             C
 40%                             I
 30%                             L
 20%                             Q
 10%
  0%
       Problem Problem Problem
          1       2       3
Real Results

   Activity provided a window into students’
    understandings and misunderstandings

   Observation that students struggle to
    rise above the details of a problem
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.
 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.
Acknowledgements
   Participating Calculus Instructors
       John Koelzer
       Paula Shorter
       Keith Brandt
       Julie Prewitt Kramschuster
   Research Assistance
       Craig Sasse
       Tom Jones
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.
Student Attitudes
          51 % approved
         30 % disapproved
         13 % no comment

     Typical Positive Responses

     Typical Negative Responses
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
    didn’t 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.
    Negative Responses to
    Calculus Conversations
   This helped me some of the time, but most of the time I wished
    there weren’t 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.
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
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.
            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 don’t
 know about those ideas.
The Collaboration

   Collaborator & Colleagues

   My reflections on the collaboration

   Reflections of my collaborator

   By-products of collaboration
By-Products of
Collaboration
 Good model for interdisciplinary work
“Have always heard about the value of
  interdisciplinary work but haven’t 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.”
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
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

				
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