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Making Student Thinking Visible: A Close Reading of Online Conversations

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... some different way than I am accustomed to, you weigh less than your height at birth. ... so confused that I have no idea what I am supposed to do anymore. ... – PowerPoint PPT presentation

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Title: Making Student Thinking Visible: A Close Reading of Online Conversations


1
Making Student Thinking Visible A Close
Reading of Online Conversations
  • University of Maryland
  • September 29, 2004
  • Anita Salem
  • Professor of Mathematics
  • Rockhurst University

2
Online Calculus Conversations Making Student
Thinking Visible
A Carnegie Foundation Scholarship of Teaching
Learning Project
  • Anita Salem (Mathematics)
  • Renee Michael (Psychology)

3
Problem
  • Inability of most students to apply methods and
    concepts used in a practiced problem to a new
    situation

4
Literature Review
  • Types of Intelligence (Robert Sternberg)
  • Book Smarts (skills, methods, procedures)
  • Street Smarts (using common sense to find new
    strategies for solving problems)

5
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

6
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.

7
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.

8
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.

9
The Main Idea
10
The Conversation
11
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.

12
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.

13
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?

14
One graph or two graphs?
15
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.

16
Get rid of the age factor !
17
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.

18
Bingo!
  • We have the first conceptual response.
  • However it is very poorly incompletely
    expressed.

19
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!!!!!!

20
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.

21
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.

22
What about those squiggles?
23
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.

24
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.

25
Lets dump the graph idea!
26
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.

27
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.

28
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.

29
Quantitative Analysis
Exam Question Average
  • TABLE 2

30
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

31
Quantitative ResultsExam Question Averages by
Activity Scores
32
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.

33
Coding the Responses
  • Practical Response (P)
  • Conceptual Response (C)
  • Intercommunication (I)
  • Language Extremes (L)
  • Pose Questions (Q)

34
Qualitative Results
35
Real Results
  • Activity provided a window into students
    understandings and misunderstandings
  • Observation that students struggle to rise above
    the details of a problem

36
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.

37
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.

38
Acknowledgements
  • Participating Calculus Instructors
  • John Koelzer
  • Paula Shorter
  • Keith Brandt
  • Julie Prewitt Kramschuster
  • Research Assistance
  • Craig Sasse
  • Tom Jones

39
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.

40
Student Attitudes
  • 51 approved
  • 30 disapproved
  • 13 no comment
  • Typical Positive Responses
  • Typical Negative Responses

41
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.

42
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.

43
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

44
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.

45
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.

46
The Collaboration
  • Collaborator Colleagues
  • My reflections on the collaboration
  • Reflections of my collaborator
  • By-products of collaboration

47
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.

48
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

49
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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