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Introduction to Issues in Teaching and Learning: 11.124
Understanding Math and Science
Bldg 38-153
M-W 3:00-4:30
Jeanne Bamberger
4-082
x3-7374
jbamb@mit.edu
Maurice Page
Cambridge Rindge and Latin High School
876-7759
mopage544@aol.com
R. Feynman's Last Blackboard:
"What I cannot create, I do not understand"
Note: the Course Reader can be purchased at Graphic Arts (11-004).
Some Basic Questions To Be Addressed
1. What does it mean to "understand" X ( motion, angle, ratio, velocity, force; matter, gears, pulleys)?
2. What, how, do you understand X?
Probing your own understanding:
o How did you acquire it?
o What can you do with it?
o What are the assumptions implicit (or explicit) in your explanations?
o What features and relations are given names; what are the units of measure; what assumptions are built into these?
o How would you explain X to another?
o What would constitute a different representation of X; what are the significant differences? What kinds of relations are accounted for in each representation; what can you do with each representation? Are the several representations basically incommensurable? --which are, which are not?
3. How is another understanding X?
0 What features is the other attending to and which is he/she ignoring?
o What is the intuitive theory (IT) behind the other's explanation?
o How can you find out; how can he or she find out? What is your
evidence? How is the other's representation/explanation different from
yours?
o What are the "prototypes" (from other experiences) that are being
brought to bear on the other's explanations/representations? How do you
know?
4. What kinds of transformations would be involved in helping the other to move between IT and the conventional theory (CT)? Can the other (or you) hold more than one theory of X each of which is behind an explanation? Is this useful?
5. When and why are these questions useful in teaching?
o How can you probe for these issues on-the-fly in a classroom?
o How can you use your own understanding to facilitate (rather than inhibit) the process of helping another to learn?
o Why is it important in teaching to have a broad understanding of X, including a wide repertoire of different representations and explanations?
6. How do you know that you have succeeded in understanding another's understanding?
7. What are standard curricula for teaching X (texts, software)?
o What are the assumptions built into a given approach to teaching? For instance, what does the text assume that students know and understand already--as reflected in the representations/explanations used in the text (including such things as graphs, formulas, math involved, etc.)? Does the material and the approach take into account what is already known about ITs?
o Why do you think each curriculum succeeds or fails? Would you do it differently or the same? Why?
CALENDAR
Weeks 1-4 (9/9-9/30)
SIMPLE MACHINES
Pulleys, gears, levers
Fractions, ratio, angle, units and measurement
Multiple representations
Guest: Crispin Miller, PhD: Mechanical Engineering, MIT '93
Week 1: Sept. 9-16
Readings
#1: Clement, "Students' preconceptions in introductory mechanics"
#2: Smith, et al. "Misconceptions Reconceived: a constructivist analysis of knowledge in transition."
Due in class, September 21
Note: Reader may be picked up at Graphic Arts
Wednesday (Sept 9): Introduction:
o Questions as backdrop
o Overview of course
o Classroom observations and tutoring
Introduction to Pulley Mechanisms (Design and sketch)
You will be given a specific problem in class that must be solved by constructing a working pulley mechanism. You will begin by sketching a mechanism that you think will solve the problem, and also describe the theory you believe it to embody--i.e., why it will work.
Monday (Sept. 14): Building Pulleys
You will work in groups of three with two people building and the other observing and keeping notes. At the mid-point in the hour, you will switch roles--i.e., starting from wherever the builders are in the process, the observer will become one of the builders and the one of the builders will become an observer.
Observing
The observer will be primarily responsible for keeping track of the design and construction process. The observer can ask questions concerning what participants are thinking and/or what they are trying to do, etc.. However, the effect of asking a question should be noted.
The following are suggested aspects to watch:
1. How is the task entered?
What do participants see as "the task?"
Where do the participants begin their thinking/acting?
What is the problem they set to begin with?
Are there differences among the builders about where to begin; if so, what are they?
2. What are the elements (objects, relations) of the phenomena that initially come into play; what things are named; what are the names?
3. Are differences among builders confronted, left hidden, ignored; made explicit; accounted for? Are they resolved? What is the basis for the differences?
4. What events trigger shifts:
When/why does the problem-situation change?
What new questions emerge; what new features are "liberated?"
What role does measuring play? What is measured; how is it measured?
5. What puzzles are encountered? Does the group get stuck? How do they get
un-stuck?
6. Are there specifically different explanations? If so, what are the
underlying conceptual differences? How are they resolved?
7. What kinds of interpersonal relations develop: e.g., leader/follower;
concept-maker/doer, etc.?
Wednesday (Sept.16): Discussing Pulleys
Reports from each group.
Discussion will focus on the observation questions, and on educational issues raised by the pulley exercise in the light of the Clement and Smith, &al. papers.
Assignment:
In a paper of not more than five pages, make a "post mortem analysis" of the pulley and gear sessions as an exercise in "reflective thinking." The paper should include:
o An accounting of your view of the process as both builder and observer;
o A description of specific problems and surprises you and/or your observee encountered along the way, including the situation at the time of the encounters and what strategies were invoked in dealing with them.
o A description of the role played by modes of representation: e.g., as you moved from sketch to implementation, transformations in your initial understanding of the problem, the mechanism, and/or the theory.
o An explanation of how and why your finished pulley/gear mechanism actually works.
o In the light of your own experience and the readings, discuss what intuitive theories relevant to pulleys/gears you would expect to find among your future high school students? How could you find out what intuitive theories they hold? What kinds of conceptual transformations imight be involved in helping them understand the canonical physics principles?
Due in class: September 30
Week 2: September 21-23
Reading:
#3: Klaassen & Lijnse, "Interpreting students'and teachers' discourse in
science classes: an underestimated problem."
#4: Hammer, D., "More than misconceptions: Multiple perspectives on student
knowledge and reasoning..."
Due in class: September 28
Monday (Sept 21): Building gear mechanisms
The session will be similar to the one on building pulley mechanisms.
Wednesday (Sept. 23): Discussion of gear mechanisms.
Week 3: Sept. 28-30
Monday (Sept 28): Discussion of readings and reflections on Weeks 1 and 2
In class: Choose a short passage from two of the readings that you believe present significantly different, even conflicting views. Be prepared to discuss the following: What is the reasoning behind each; what are the implications for teaching; which do you find more convincing and why?
Question: In the light of the sessions on pulleys/gears and your readings,what issues are raised when moving between your initial sketches, the formalisms associated with relevant physics as you learned them, your intuitive know-how, and your experience actually working with materials?
Wednesday (September 30): Exploring interactive physics software
Guest: John Samp. Physics teacher, CRLS
Pully/gears paper due today
Week 4: Oct. 5-7
Monday (Oct. 5): Continuation of software explore
Guest: John Samp
Wednesday (Oct. 7): Reports from classroom observations and tutoring
You should now be scheduled into classes at CRLS for observations. You should also have made at least initial arrangements for working with a student tutee.
Reading:
#5: Schifter, Deborah, "Learning Mathematics for Teaching: Lessons in/from
the domain of fractions."
Due in class : Oct. 7
WEEKS 5-8
MATHEMATICS
Problems, Puzzles, and Student Work
Question: What makes a hard problem hard?
Week 5: October13 (Tuesday) -14
This section of the course will involve actually doing some mathematics together while watching the varied strategies, attitudes, representations (internal and external) that you collectively and individually bring to bear on the work. There will also be an emphasis on curricula, National Standards, and work in school classrooms including close study of specific examples of teaching strategies and the responses of younger students's to problems similar to those we will do in class. We will circle back on the background questions (pp. 1-2) focusing especially on questions 5-7. The specific content of each session will evolve in response to the concerns raised by the examples and by your reflections on your own understanding and your own strategies.
Reading: NCTM Standards (excerpts) and selected mathematics curricula
Due in class: Oct. 14
DiSessa et al, "Inventing Graphing"
Due in class: Oct. 21
Assignment:
One question will wend its way through all of our discussions: Is there a class of hard problems and, if so, what are their shared conceptual issues? Each group of two or three students, will choose one such problem to focus on. Your task will be to design a lesson in response to the results of your analysis of the problem and then to present the lesson to our class (see Week 8)..
Tuesday: Oct. 13 Introduction: Some puzzles; where is the math?
A first look at your list of hard math problems; why is a hard problem hard?
Wednesday (Oct. 14): .Discussion of NCTM (Math) Standards.
Be prepared to discuss: What are the big goals of the NCTM? What teaching strategies are advocated for achieving these goals? Compare the approaches proposed, here, to your own learning strategies.
Week 6: October 19-21
Monday (Oct. 19): Approaches to teaching and learning mathematical functions
Wednesday (Oct. 21): Approaches to teaching and learning graphing
Discussion of "Inventing Graphing."
Questions: What and how does a graph represent? Compare and contrast an equation, a graph, and a table of values all for the same relationship. Bring some examples of when and why one kind of representation would be more useful than another.
Week 7: October 26-28
Monday (Oct 26): Examples and critique of math software related to multiple representations
Wednesday (Oct. 28): Integrating technology into the curriculum
Week 8: November 2-4
Monday (Nov. 2): Presenting math lessons
Wednesday (Nov. 4): Presenting math lessons.
PAUSE
WEEK 9: November 9-11
Monday (November 9):
Reflections on observations and tutoring.
In order to keep your reflections focused, you should organize your comments around one incident that stands out in your observations or tutoring. You should choose an incident in which, for example:
you were puzzled about how a student was thinking,
where you encountered a surprise,
where (in tutoring) you made a hunch about the student's "theory"
and you tried an intervention--how do you account for why it worked or did not?
Wednesday (Nov. 11): Holiday
WEEKS 10-13: November 16-December 10
Biology: Experiments with Microbes
The relationship between theory. model, and experiment
GUEST: Professor Brian White, UMass-Boston
In this section of the course, the goal is to produce an explanation of what you see happening as you observe the results of experiments that you will devise. As a class, you will have to develop representations to communicate what you did and what happened. There may be differences among the
representations used by different people. We will examine how these representations can both guide and limit thinking, as well as indicate underlying assumptions.
Reading:
Kevles, "The Assault on David Baltimore"
Question to consider when reading: How does the nature of the scientific controversy in the article relate to the scientific controversy in our discussion of the yeast experiments?
Dewey, "The Child and the Curriculum"
Question: What are the criteria through which to distinguish between what to TELL students (by demonstration, lecture, reading), and what we can ask them to DISCOVER through inquiry, experiment, and group discussion?. How is this question informed by the Dewey reading? How does it relate to the Baltimore case?
Due in class: Nov. 30
Assignment:
In a paper of not more than five pages, reflect on your experiences with the
yeast experiments in the light of the course so far. How does the process of
conceptual change in science correspond to the conceptual change that the
class experienced? How do these, in turn, bear on the process of conceptual
change which you want to encourage among younger students?
What are the interactions between evidence and interpretation in the
following:
o designing experiments
o understanding results
o explaining results?
Which of the following takes precedence at varying moments in the course of arriving at explanations?
o direct observation,
o reasoned argument,
o implicit or explicit theory?
What is the role of domain-specific expertise in the course of designing,interpreting, and explaining your observations? How does your experience with the yeast relate to the D. Baltimore case?
Reflecting back on the semester's work, what is your view on the role of "telling" and the role of "discovery" in moving towards the goal of students' "learning with understanding?" How would you distinguish between matters that need to be "told" and matters that need to be "discovered?"
Due: Dec. 10
Monday (Nov. 16): Introduction & Initial models
You will be presented with a biological phenomenon: a culture plate with red and white yeast growing on it. Your task is to devise experiments to explain this color variation. Your tools will be culture plates and sterile toothpicks.
o Throughout this exercise you will work in groups of 3 or 4. Within each
group:
- some of you will design experiments, interpret results, and develop models to explain the color variation.
- one group member will be an observer. The observer will record the models & experiments as they develop, using these observations to probe for the mental models, assumptions, and thought processes of the experimenters. Then, as a class, we will develop an explanation of the phenomenon.
o Today: we will examine and discuss the phenomenon. We will the break into groups and begin to make models, then discuss these models as a class. and think of experiments. Be prepared to come in on Wednesday (Nov. 18) to carry out the experiments.
Wednesday (Nov. 18): Round one of experiments.
o Carry out experiments in groups. Observers record experiments, models,
assumptions and information required (questions asked) to track the
experimenters1 thought processes,
Week 11: November 23-25
Monday (Nov. 23): Continued reports on observations and tutoring
Wednesday (Nov. 25): Data collection & analysis from round one.
Collect data, discuss & plan next round of experiments.
Week 12: Nov. 30-Dec. 2
Monday (Nov. 30): Discuss readings.
Be prepared to consider the questions posed, above.
Wednesday (Dec. 2): Round 2 of experiments
Week 13: Dec. 7-10
Monday (Dec. 7): Data collection and analysis
Wednesday (Dec. 10): Reflections on Weeks 10-13
Reading List (partial):
Clement, J. "Students preconceptions in elementary mechanics." American
Journal of Physics, 50, 66-71.
DiSessa et al., "Inventing Graphing: Children's meta-representational
expertise." Journal of Mathematical Behavior, 10 (2).
Dewey, J. "The Child and the Curriculum"
Hammer, D. "More than misconceptions..." American Journal of Physics, 64,
(10) October, 1996
Kevles, "The Assault on David Baltimore"
Klaassen & Lijnse, "Interpreting students'and teachers' discourse in science
classes: an underestimated problem."
NCTM Standards
Schifter, Deborah, "Learning Mathematics for Teaching: Lessons in/from the
domain of fractions."
Smith et al. "Misconceptions Reconceived" Journal of the Learning Sciences
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