Tim's first "guess"
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In-service program/Methods course documentation
Sharing strategies for using instructional resources effectively
(D7.4)
Narrative account of an inquiry-based instructioal experience where the
video "The Theorem of Pythagoras" was used (by J. Fonzi
& R. Borasi)
(5 pages)
The following vignette reports on a 7-hour-long instructional experience that was developed within an unusual high school course entitled "Math Connections". The overall goal of this course can be briefly described as broadening the students' conceptions of mathematics by exploring connections between mathematics and disciplines such as art, music, literature and science. Within this course, the students engaged in a number of inquiry experiences around specific topics they themselves had contributed to identify. The narrative that follows reports on one of these experiences, which developed quite spontaneously towards the end of the course, as students tried to understand the nature and significance of analog and analytic thinking in the context of mathematics - the topic of an essay two students in the class had been assigned to read and present to the rest of the class using reading strategies that had been introduced earlier in the course. (For a detailed narrative of this instructional episode, see Fonzi & Smith [1992].)
The 2-page essay entitled "Analog and analytic mathematics" (from Davis and Hersh's popular book The Mathematical Experience [1981]) introduced these two fundamental ways of thinking by means of the following explanation:
TEXT: "[A]nalog mathematizing is sometimes easy, can be accomplished rapidly, and may make use of none, or very few, of the abstract symbolic structures of "school" mathematics. It can be done by almost everyone who operates in a world of spatial relationships and everyday technology. […] In analytic mathematics, the symbolic material predominates. It is almost always hard to do. It is time consuming. It is fatiguing. It requires special training. […] Analytic mathematics is perfomed only by very few people."
Each approach was further illustrated by a series of examples, such as the following:
TEXT: "Problem: How much liquid is in this beaker? Analog solution: Pour the liquid into a graduated measure and read off the volume directly. Analytic solution: Apply the formula for the volume of a conical frustrum. Measure the relevent linear dimensions and then compute."
Since the two students responsible for reading and reporting on this essay found themselves in disagreement about their interpretation of analog and analytic thinking, the class as a whole tried to resolve this controversy and come to an understanding of the distinctions between these approaches. This was done mainly by looking in depth at some of the examples discussed in the essay, as well as new examples proposed by the students themselves. However, despite all these efforts, the class was not able to come to a satisfactory resolution of what really distinguishes analog and analytic thinking in mathematics.
In the effort to help the students better understand the nature of these complementary approaches and their relevance to learning and doing mathematics, at this point the teacher thought it would be worth having the students themselves use both of these approaches to create and prove their own theorems. In preparation to this challenging activity, the students watched a video, The Theorem of Pythagoras (Project Mathematics!, 1988), which discusses this fundamental theorem by telling the story of the many different times and places in history where the theorem was discovered and/or proved differently. Through the use of computer graphics this video shows, without labelling them as such, both analog and analytic approaches to proving the theorem. The students were told that they were not expected to fully make sense of the technical aspects of the mathematics from this video; rather, the purpose for viewing the video was to get a sense of how a specific theorem was studied from multiple perspectives. They were asked to identify, and jot down, examples of analog and analytic approaches to the theorem and to write down any general comments about the approaches and what they looked like.
In the next class, as the students shared the ideas about the nature of analog and analytic approaches that they recorded while watching the video, the teacher created a list integrating all of these ideas. On the basis of this list and their previous discussions, students were then asked to identify themselves as someone who usually takes an analog approach or someone who usually takes an analytic approach, and then to form heterogeneous small groups with respect to their approaches. The groups were given the task "to create a theorem about two different figures that have the same area and prove it, using both an analog and an analytic approach in the process".
Each group was given plain paper, lined paper, scissors, and a sheet with some formulas for the area of some simple figures. In addition, the teacher reminded the students that the video and all of the classroom resources (e.g. texts, materials) were available. The inquiry took most of two class periods during which each of the groups had complete autonomy over how they completed the task.
Each group was also expected to share the theorems they had created to the rest of the class, and to explicitly identify the analog and analytic approaches they had used to prove them. Along with the product of their activity (i.e., their theorem) each group was also asked to report on their process by writing a "story" of the experience. A number of "historical snippets" about the Pythagorean theorem, rpeviously collected by the instructor, were offered as models and inspiration for this final writing assignment.
In the remaining part of this narrative, we will report on what happened within one of the groups, so as to illustrate what took place in this unusual kind of activity.
A more in-depth look at what happened in Group 2
Several members of this group immediately offered their individual ideas in response to the given problem. One girl drew two semi-circles and a circle and said, "Take a circle and cut it in half and then one half plus the other is the circle." Another student said that they could find the dimensions and then use the formula to find the area. She identified that approach as analytic. She then described an analog approach as reading the information on a bag of sand to see that it covers so many square feet. Another student shared yet a different image of the problem suggesting that "two Silver Stadiums = one War Memorial" (two local sports arenas), based on the number of people they could each hold.
The group then began to struggle with understanding the conditions for the theorem that was required: A theorem about two different figures that have the same area. Someone offered, "Maybe like a 2 by 3 square equals a 6 by 4 triangle?". Another person wondered what an analog approach to finding the area could be and someone else continued to talk about the ideas of separating a figure into two parts and then adding them together.
Finally, the group decided to take a 4 by 4 square and make a rectangle with the same area. As one person began to measure and draw the square the following interchange took place:
Tim: I've got to measure it, 4 inches by 4 inches.
Shellie: What can I do?
Eric:You can make the rectangle.
Shellie: Okay, what size?
Tim: We're not sure yet.
Shellie: How you gonna make a square fit in a rectangle? (jokingly) Two squares equals one rectangle?
Tim: I'm not sure.
Eric: No, one square equals one rectangle.
Shellie: It can't. A rectangle's big. … Oh, the area.
Eric and Margie: Yeah !
Tim finally said that "Maybe a 2 by 6 would be adequate, yeah, a 2 by 6 rectangle." And when asked how he decided he said he was guessing and then went on to explain how he was guessing.
Tim: Take two inches off the top and that equals 6, and this is down to 2, so then it would be equal area.
Tim: (He elaborates and revises his original idea when Shellie says she doesn't understand it at all) Actually, I think cutting the square in half would be analytic and analog is just adding these two together; cuz, go take two off here and put it over here, see... Wait a minute... If you take this top and put it over here... Ohhh... That'll be eight. Should have been 8, a 2 by 8...
They proceeded to carefully measure, draw and cut out their figures, while continuing to talk about whether the rectangle was going to fit into the square or vice versa. Ultimately they decided it didn't matter and followed through with their plan.
Tim: We're gonna make this rectangle fit inside this square.
Shellie: It's gonna cuz this little piece can go right next to it. (To Eric) Don't cut it!
Eric: Have to... (He works on the cutting.) Now, hopefully, this little... (He places it on top of the square.)
The group: (Joyously) It fits !
Now this group was ready to create their theorem. They spent a great deal of time trying to determine just what a theorem is and ultimately decided to read a few theorems from a textbook to get a feel for what they are like. They tried to verbalize their work using the same tone as the textbook. They began with: The area of a square equals the area of a rectangle. As they continued to discuss the nature of theorems, i.e., that they seem to apply in general and that they have to be understood by an outsider, they created several iterations of their own theorem. Each time they identified a weakness in their statement they tried to improve on it, finally settling on the following:
If the length of side (a) on the square multiplied by the length of (b) at the top of the square, equals the length of side (c) on the rectangle multiplied by the length of side (d) at the top of the rectangle, then the rectangle and the square have the same area.
It is interesting to note that in the group we have chosen to portray here no one person knew how to complete the entire task. Yet as they listened to, and questioned, each other they were able to construct a process for completing the task and produce an appropriate product. Though this product may, on the surface, look rather unsophisticated the mathematical concepts they reconstructed, i.e., the conservation of area, the nature and role of theorems, and the development of a process to calculate the area of a square and a rectangle, are quite sophisticated.
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