Teacher resources and professional development across the curriculum

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Private Universe Project in Mathematics



From: Dennis McCowan (mccowan@massed.net)
Date: Thu Nov 09 2000 - 09:25:01 EST

  • Next message: DWBentz@pulaski.k12.wi.us: "[Channel-talkpupmath] Thoughts after the first session"

    Hi all! weston watched the fourth brioadcast on November 8. We bagan
    by discussing student approaches to the pizza problems. One grade ten
    teacher watched students tstruggle with the problem and than ask "Why
    are you giving us these easy problems? My grade 5 sisiter was doing
    this problem yesterday!" (The fifth grade teacher is also in our

    A seventh grade teacher shared a students solution to a similar "project
    problem" from the text: "Make as many different decimals as possible if
    you can use at most one decimal point, one "1", one "2" and one "3".
    Arrange your answers by size. The student made a great poster
    attacking the problem by cases (just a single digit and the the deciaml
    point, just two digits and the decimal point, all three digits and the
    decimal point) and within each case was clearly controlling for
    variation to get a systematic list of possibilities.

    We then asked ourselves, How did students' mathematical thinking and
    problem solving evolve over the first three tapes? As students moved
    from "Shirt and pants" to "Towers" to "Pizza problems" we noticed the
    following progression- students would

    1. make diagrams
    2. accept different definitions of the task
    3. play with the problem
    4. make lists of random possibilities
    5. check for duplicates
    6. recognize there will be a unique answer
    7. make towers by building opposites
    8. use manipulatives- no initial need for diagram
    9. try to become systematic
    10. search for patterns
    11. absorb ideas from others
    12. use pattern of flipping towers
    13. recognize a pattern may create duplicates-
      willing to disregard a pattern
    14. consider similar easier problems
    15. adopt rules without knowing why they work
    16. find a formula
    17. recognize similar problems
    18. organize lists by cases
    19. organize lists by controlling variation (fixing starting items)-
      thus using "levels" of variability

    As groups we then reacted to four statements proposed by the

    "The ways students learn combinotorics is fundamentally different from
    the way they learn other mathematics"

    "Students should spend most of their math class time solving problems."

    "In our classes it is necessary to use more efficient ways of learning
    than are shown on these tapes."

    "The researchers weren't trying to teach the students anything."

    Our responses follow:

    "The ways students learn combinotorics is fundamentally different from
    the way they learn other mathematics"
    ? We disagree. The same stages are progressed through en route to
    ? We think that the process is very similar for different topics;
    hoever, the amount of guidance may vary.
    ? Disagree. No matter what the topic, students must always find
    patterns and come up with generalizations.
    ? Patterns are everywhere and a natural way to learn math.
    ? No way José.
    ? No. Students use similar techniques in all problem solving which is
    "the central focus of mathematics education"
    ""Students should spend most of their math class time solving problems."
    ? Agree. We're not sure what "most" means, but it is best when a
    majority of time is student directed, learning by discovery.
    ? Students must actually solve problems to understand mathematical
    ? Since almost everything in life can be construed as problem solving or
    creation . . ..
    ? They should spend most time "doing" math.
    ? Some time spent in class solving, some time spent sharing strategies,
    solutions, and proving.
    ? Goal is to become independent thinkers.
    "In our classes it is necessary to use more efficient ways of learning
    than are shown on these tapes."
    ? It should be a mixture of teaching techniques.
    ? Time constraints and standards do place limits on time devoted to
    tasks of this sort. There is definitely a place for these activities.
    ? "Efficient ways" is a school of thought that hinders "ways of
    ? Disagree philosophically, but agree in some instances due to the
    reality of covering curriculum in our present structure of education.
    ? This method of introducing concepts and "regular" classroom methods
    have similar efficiency as a starting point.
    ? In order to be efficient students must internalize the process.
    "The researchers weren't trying to teach the students anything."
    ? They were giving the students an opportunity to learn for themselves
    through a well-chosen activity.
    ? By constant questioning the researchers modeled deeper thinking and
    learning. In this process students learned to defend their solutions.
    ? They were trying to access their innate capabilities.
    ? While the primary goal may not have been to teach students, it is
    clear that students were taught problem solving strategies and were
    pushed to really high level thinking.
    ? Agree. However, learning did take place.
    ? They were trying to teach them to justify their thinking, to explore
    intelligently, to realiza that not all learning is teacher generated.

    Watching the tape, we noticed that Professor Davis usually started with
    the Tower of Hanoi disks on the center post which cleverly avoided
    dealing with the additional complication of which post to use for the
    first move. (Starting at one end and identifying the far post as the
    destination makes the problem even more complicated. Interesting
    question- how do you decide where to put the first disk on move one?)
    We ran out of time before seeing the entire tape and will watch the last
    ten minutes next session.

    An isse we are now wrestling with is the different levels of
    mathematical experience within our group. Do we devote time to helping
    members find solutions to each math problem and thus decrease the time
    we have to watch and discuss the learning of the sample students?
    Probably we need to do more of this!

    Since the packet contained no readings for next time, we will all be
    reading Chapter 12 "Relational Understanding and Instrumental
    Understanding" from The Psychology of Learning Mathematics by Richard
    (a reprint of an article from "Mathematics Teaching" No 77 December

    Dennis McCowan
    facilitator and Math Dept head

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