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
workshop!)
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
facilitator:
"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
understanding.
› 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 JosÍ.
› 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
concepts.
› 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
learning."
› 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
Skemp
(a reprint of an article from "Mathematics Teaching" No 77 December
1976.)
Dennis McCowan
facilitator and Math Dept head
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