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Methods course documentation
Developing a mathematical inquiry independently (D7.2)
Facilitator's plan (by Raffaella Borasi)
ALTERNATIVE GEOMETRIES
INQUIRY EXPERIENCE
Key decisions and their rationale: (about 1 page)
- I conceived of this experience as another inquiry experience, more
open-ended and less "teacher-directed" than the area experience,
that would enable the students to experience a more "independent inquiry"
where they would be in more control (and have more responsibility) for
determining the direction and structure of their investigations -- in preparation
for developing inquiries of their own as the prerequisite for orchestrating
similar inquiries for their students;
- I still wanted, however, to provide some stimulus for the students'
inquiry (as I wondered if they would still not know what to do if I gave
them an open assignment like "do your own inquiry on a math related
topic") while at the same time being open for them to pursue various
directions determined by their own interests;
- I envisioned that, in an inquiry that is more independent and student
centered (in contrast to a more "teacher orchestrated" inquiry
like the one experienced in the Area unit), it would be unrealistic (and
even not "genuine") to impose that the students use only themselves
as "resources"; instead, I wanted to encourage them to use texts
and other resources as well to both generate new questions and help them
make sense/respond to the ones they had already generated;
- I initially thought of doing this in the context of taxi-geometry,
and then expand the "theme" to that of alternative geometries
for a number of reasons:
- most importantly, I wanted to have a wide enough theme to allow more
opportunities for students to find something they would really be interested
in exploring; (in fact, I want to be very clear that, if after the first
experiences intended to generate interest in alternative geometries they
are still not intrigued with the topic and would rather develop a mathematical
inquiry of their own on a different topic, they should feel free to do
so -- after consulting with me to help them define the scope of their inquiry
in a productive and manageable way)
- I realized that I had already assigned as a reading the chapter on
Taxigeometry on my book Learning Mathematics through Inquiry, which
may have given away some of the more "intuitive" directions for
exploration in taxigeometry
- I thought that students could find more resources/texts on the more
general topic of alternative geometries -- if they wanted to consult them
- I did not think about it in time, but I wish now that I had assigned
the reading of John Sheedy's essay "Beyond Straight Lines" prior
to the beginning of the unit, as a way to have students start thinking
and becoming intrigued with the topic of alternative geometries
- I also decided that I would use this experience as the basis for introducing
and discussing more explicitly the important roles that reading and writing
can play in mathematics instruction, and math inquiries in particular
Overview Plan (about 2 pages)
LESSON 1:
1. Introduction. (5')
2. "Set the stage" for an inquiry on alternative geometries
through the reading of John Sheedy's story "Moving around the city"
in class: (30-45')
- introduce and model "say something" strategy as a class
- read the text using "say something" strategy in pairs
BREAK
3. Large group sharing of the reading experience (20')
- share and record on newsprint on what each pair learned from reading
the story about geometry (and taxigeometry in particular) and what new
questions this reading raised
4. Reflecting on the process (and the reading specifically): (20-30')
- students report on how they read and on the perceived benefits/drawbacks
of this way of reading texts
- generate and discuss some possible modifications of the say something
strategy
- synthesize key points about reading raised, and invite to keep reading
and writing in mind as they proceed in their inquiry
5. Open to the more general theme of alternative geometries, and begin
to generate possible directions for inquiry: (15-20')
- open discussion on the whole issue of alternative geometries (of which
taxigeometry is one example), starting from the questions recorded on newsprint
- point out historical significance of alternative geometries and show
books I brought in as a possible resource
- as a class or in small groups, generate further questions and possible
directions for inquiry
HW. 1:
- (MAYBE) Reading on history of non-Euclidean geometries (from Kline)
- Begin to develop your own inquiry on an alternative geometry OR another
topic of your choice -- be ready to report on it in small groups in class
- Remind of deadline for presentation and final report
LESSON 2 (11/15/95):
5. Brief sharing and discussion on their inquiry process: (15')
- have people briefly share with the large group the "direction"
of their inquiry, and how they have proceeded
- initiate some discussion, as appropriate, on some of the strategies
and resources used
6. Sharing and further elaboration in small groups of their preliminary
inquiries: (60'-90')
- divide in small groups of 2-3 (on the basis of similar interests OR
to make sure groups will be "productive");
- have students in small groups share and further elaborate their preliminary
results (going around to help push the inquiry further)
HW. 2:
- Conclude your inquiry on an alternative geometry OR an aspect of alternative
geometries; prepare a brief presentation (15') of your results to the class
+ first draft of written report (to be turned in for feedback)
- Read some articles on "writing to learn" in math
LESSON 3:
- Interactive presentation of results from individual inquiries (120')
HW. 3:
- Write final draft of report
- Read some more articles on reading and writing in math instruction
Detailed Plan for Lesson 1 (about
5 pages)
LESSON 1:
1. Introduction: (5')
- make explicit the scope and nature of this new inquiry experience,
as a complement to the "area" experience we already went through:
- point out how the area experience, though it had many elements of an
inquiry approach, was also quite structured and teacher directed -- partly
because it was the first inquiry experience for them (and so it was designed
so as to "model" the process), partly because it had different
instructional goals and roles within this course
- now I would like you to engage in a more open-ended inquiry experience,
where you are in more control of the scope and direction of the inquiry;
I think it is important to do so not only to experience the educational
potential of this type of inquiry, but also because I think that as math
teachers you need to be able to develop inquiries of this kind as a prerequisite
to organizing "inquiries units" for your students
- at the same time, I thought it would be useful to give a first stimulus/push
to these more independent inquiries by providing a possible theme (alternative
geometry), chosen "wide" enough so as to aloow for many different
directions within it, and some initial thought-provoking activities that
may invite questions and "doubt" to be generated; BUT, feel free
to chose another topic if at the end of this class you feel that that would
be more interesting for you
- point out how I am going to use these "setting the stage"
activities also as an opportunity for experiencing some unusual ways to
use reading and writing in the math class -- a theme that we will continue
to pursue in a parallel way throughout the next few lessons.
2. "Set the stage" for an inquiry on alternative geometries
through the reading of John Sheedy's story "Moving around the city"
in class: (45')
- distribute John's story (give 1 copy each); introduce it as a "mathematical
story" that was written by another student teacher in a class like
this one as a way to communicate (in a very creative way!) his own exploration
of a situation requiring a geometry other than the usual Euclidean geometry;
I want them to read and try to make sense of it because, in doing so, I
think that they can begin to get a better sense of what it means to operate
in a situation where Euclidean geometry does not work any more, and begin
to raise some questions that may be worth further investigate;
- explain that, because of these goals and because it is a complex text,
we will be reading it in class, in pairs, using the "say something"
reading strategy; that is, I am expecting each pair to read a short section
of the story aloud, then stop and "talk" about it -- sharing
reactions, ideas, questions, etc. that the reading of that section may
have suggested, and trying together to "make sense" of
the text in whichever way seems most appropriate/productive; point out
that this strategy is one that has been suggested by reading specialists
as a vehicle to help readers (especially young ones) make sense of a text
and be encouraged to develop their own meanings and interpretations of
it, and myself and my research group have adapted and used in the context
of secondary school mathematics with some success;
- since I expect this way of reading to be quite new to most of the people
in the class, I would like to "model" it first as a class --
that is, I will read aloud the first paragraph and then we will all try
to "say something" about it
- do the modeling, and a little bit of reflection afterwards so as to
make explicit the process and its goals (e.g., articulate what "kinds"
of things one could say, what purposes the say something could serve, etc.)
- assign students in pairs and have them read the text using the "say
something" strategy thus modeled (give enough time to go through the
text -- tell them that I will give 30 minutes to do so, so as to give them
another clue about the importance to go in-depth with this text) as they
do so, go around to get a sense of the kind of issues that are raised and
help pairs that seem still not clear about the task and/or seem to be reading
too superficially
EARLY BREAK -- this may also give some flexibility, as some pairs
may some extra time and other finish earlier
3. Large group sharing of the reading experience: (15-20')
- have pairs report on the key things they understood and learned from
their reading (about geometry, or keep it general), as well as questions/issues
that the reading raised for them (and not resolved), and record these on
two different sets of newsprint; encourage elaboration and discussion on
each point;
- things I am expecting to come up:
Things learned:
- something about addresses/coordinate geometry
- locus of points equidistant from 2 given points is not
always a straight line
- how distance definition may change depending on the constraints
Questions raised:
- how would other loci look like in plane taxigeometry
(ex: circle, line segment, etc.)
- what other problems (besides finding ideal appartment
location) may be affected by the new distance
- what if we took yet a different definition of distance
- what if we worked in 3 dim.
- what if we were on a sphere
- how did mathematicians react to finding that there is
not just 1 geometry
(The next two activities may need to be switched, depending on how the
previous activity 3 develops)
4. Reflecting on the process (and the reading part of it specifically):
(20'-30')
- ask students to move away from the "content" of the experience
they went through, and reflect a moment on the process -- the reading of
the story specifically;
- ask to report on how they actually read in their groups, and how stopping
to "say something" on what they read helped or hindered their
reading and understanding of the text; maybe record on newsprint (so that
we can come back to it a few lessons later when we will revisit the role
of reading and writing to support math learning in light of their experiences
and readings) the benefits and drawbacks thus identified;
- ask to reflect also on the large group sharing as an integral part
of this reading experience; add benefits and drawbacks identified on the
newsprint;
- ask to generate some valuable "modifications" of this reading
strategy (for example, if you do not have a partner, or can't read in class,
or have a technical math text to read); things that could be suggested:
- write down your "say something" on notes (to
be shared later)
- "read and do" -- for technical texts, theorems,
worked out examples, etc.
- write "reader responses" on what read
- help synthesize by pointing out and/or elaborating on key insights
recorded on the newsprint; points that I would like to emphasize and/or
bring up if they have not been spontaneously raised:
- value of verbalizing thoughts and ideas
- active role played by the reader
- possibility (abnd value of ) having different interpretations
of a text (due to different readers and contexts)
- reading as a social (not just individual) activity
- how reading can go beyond "understanding" the
author's message
- invite people to keep in mind these points about reading as they may
use texts as resources for their inquiry in various ways -- and record
these uses, so that we can share and rediscuss the role of reading at the
end of the inquiry experience
- similarly, remind that we made use of writing quite a lot throughout
the course so far (including the "experiences as learners" where
they learned some math); they will continue to use writing in the course
of this inquiry as well, also in a yet different way as this time they
will be asked to prepare a "report" on the results of their inquiry
that should be readable by other people in class; announce that we will
also come back to discuss the role of writing in learning mathematics at
the end of this inquiry experience.
5. Open to the more general theme of alternative geometries, and begin
to generate possible directions for inquiry: (15-20')
- [if nec., bring attention back to the inquiry they are expected to
initiate now and conduct rather independently for the next few weeks]
- point to the open questions about taxigeometry, and perhaps also alternative
geometries in general, that were raised so far; raise the issue of alternative
geometries more generally, if necessary, and how if seems to me a wonderful
topic/context to explore genuinely new mathematics without requiring much
technical background
- point out something about the historical significance of the development
of alternative geometries (i.e., shaking the whole idea of truth in Euclidean
geometry, and mathematics more generally) -- to generate further interest
and possible directions for inquiry
- invite to now consider/generate various directions for inquiry that
could be pursued within the wide theme of alternative geometries (though
reminding once again that they could choose to do their inquiry on a different
topic if something else intrigues them); show books I brought in as a possible
resource
- as a class, begin to generate/discuss some possible directions for
inquiry OR divide in small groups to further generate some questions and
begin to look at the books; whole-group sharing
MATERIALS NEEDED FOR LESSON 1:
- 10-12 copies of John's story
- 10-12 copies of Kline's excerpt on non-Euclidean geometries + NCTM
Yearbook RLM article + stories? OR FLM-RLM (for homework readings)
- books/articles on alternative geometry to show:
- Kline M. (1980). Mathematics: The loss of certainty
(chapter on non-Euclidean geometries)
- (MAYBE) Krause (1986). Taxicab-geometry (but don't
let students use it at this stage, or it could give much of their possible
inquiries away)
- Abbott's Flatland
- (MAYBE) Sphereland
Articulation of homework assignments: (about 1 page)
Assignment #1:
- Begin your inquiry on the theme of your choice (whether within or outside
of the topic of alternative geometries). Try to go as far as you can, since
final presentations and a first draft of your written report will be due
on the following class. Keep a written "trail" of your explorations
and make sure that you bring it to class, since most of our next session
will be devoted to small groups discussions on each individual's work with
the goal of providing feedback and/or new ideas. Also bring to class a
brief statement (a page or so) identifying the topic of your inquiry and
some key questions/directions for inquiry that you are planning to pursue
(to be handed in to the instructor). (Written assignment -- 3 points)
- You may want to begin to do some of the readings on reading and writing
in mathematics and/or geometry due in two weeks, since a lot of things
will be due by that time, and these readings may even give you ideas for
your own inquiry.
Assignment #2:
- Write a first draft of the final report on your inquiry and bring
2 copies to class. (To be handed in to the instructor at the end of
class, and to be picked up by Friday Dec.1 in your mailfile in Dewey 1-217
with the instructor's comments; though no grade will be assigned to this
first draft, not turning it in on time will result in a penalty of 5 points).
- Be prepared to give a 10-15 minute presentation summarizing the main
results of your inquiry. Come prepared with posters, transparencies, hand-outs
or whatever else you think would be helpful for your classmates to understand
your findings and their significance, and also provide you with some feedback
for further elaborations and/or clarifications.
- You may want to begin/continue to do some of the readings on reading
and writing in mathematics and/or geometry due in a week, since a lot of
things will be due by that time, and these readings may even give you ideas
for your own inquiry.
Assignment #3:
- Write the final draft of the report on your inquiry and bring 2
copies to class. (Major assignment worth 15 points).
- In preparation for a discussion on the roles and uses of reading and
writing in the context of mathematics instruction, read the following:
- Borasi, R., Sheedy, J.R., & Siegel, M. (1990). The power of stories
for learning mathematics. Language Arts.
- Siegel. M. et al. (in press). The unexplored role of reading in mathematics
meaning-making. 1996 NCTM Yearbook.
- Chapters 1 and 2 in Connelly & Villardi (1989). Writing in mathematics
and science instruction.
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a mathematical inquiry independently"
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