Teaching practices that support an inquiry approach to mathematics instruction (by Raffaella Borasi and Judith Fonzi)

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Teaching practices that support an inquiry approach to mathematics instruction (Borasi & Fonzi, 1998) -- Appendix A

Commented video index for the Diamond activity that took place within the Area inquiry experience, highlighting the teaching practices modeled within the category of "Modeling."

Below is a list of the teaching practices we have identified within the category of "Modeling"; the practices noted with an * have been identified within this video segment:

Modeling:
-- Genuinely engaging in an activity as a learner and making one's thought process explicit
* Directing the class through a new process and then articulating key steps
* Making explicit a process that someone in the class has illustrated
* Articulating the key steps of the modeled process in writing

Time Description of the video segment Teaching practices(s) being modeled

Events leading up to the modeling activity:

This session is the entire second component of the Area "experience as learner" activity. During the first component participants were introduced to the entire activity and then engaged individually and in pairs in finding the area of a complex figure, the "fish". The first component set the stage for appreciating the complexity of the concept of area and the value of learning to develop area formulas. This modeling activity is designed to identify a general process for developing area formulas for complex figures by engaging the whole group in developing a formula for "diamonds".

The facilitator purposefully did not define a priori a class of figures called "diamonds" but rather begins by proposing examples commonly thought of as diamonds and later introduces some which do not "fit" this common image.

25:40
Video segment:

The facilitator introduced this segment of the Area experience, articulating its rationale and scope, and making connections with the previous "fish" activity. She describes the task, creating an area formulas for "diamonds", and presents an overhead of 2 rhombi. She asks the participants for ideas about what it means to develop an area formula for figures like this. The facilitator waits patiently for a participant to offer an idea.

Finally, someone suggests finding the area of the specific figures and then try to find a general procedure. After the facilitator rephrases the suggestion she asks participants to work in pairs on the task for a few minutes.

The entire activity was specifically designed to model the development of area formulas by directing the class through a new process and then articulating the key steps so we will only identify it here at the beginning of the segment. For the remainder on the segment we will only identify the additional "modeling" teaching practices being employed.

making explicit a process that someone in the class has illustrated

28:10
The facilitator asks participants to share the results of their work. She creates the drawings as each of the volunteers describes her process.

A participant, Meghan, explains how she tried to transform the diamond into a rectangle so they could use the formula A=bxh. The facilitator asks questions to help Meghan better explain and justify her strategy. She also makes connections with some strategies identified in the "fish" activity to get the participants to realize that these strategies may be generalizable.

making explicit a process that someone in the class has illustrated

29:20

30:50

32:15

Another participant, Heather, shares her approach which was to "see" the diamond as two equal triangles formed by one of the diagonals and then use the formula A = 2 (1/2 bh) or A=bh.

At this point participants begin to grapple with issues about variables when the facilitator has some difficulty fully understanding what this participant is trying to communicate, and other participants intervene to help. The facilitator restates the process Heather used and reminds the group that they ended up with A=bh using the approach previously shared.

Using a clean overhead of the 2 diamonds, the facilitator, with the participants' input, demonstrates that the two formulas actually measure different parts of the diamond. At this point a participant suggests they identify the parts being measured in the second approach as the diagonals and the formula is rewritten as A = d₁ x ½ d₂. A third formula is suggested, A = d₁². This discussion helped the participants appreciate that the symbols used in area formulas made have different meanings.

- making explicit a process that someone in the class has illustrated

34:00 Another participant raises the question of why another formula was needed when we already had one that worked. The facilitator used this as an opportunity to review how they came to the additional formulas and help the participant herself provide reasons for having more than one formula.

36:10 The facilitator now suggests that the 3 formulas proposed by the participants need to be tested to see if they work for all diamonds. She records the 3 formulas and puts on an overhead with some additional examples of "diamond". These new examples challenge the commonly held image of "diamond" and some participants say they do not like all of these as diamonds. The facilitator shares her definition, "a quadrilateral with perpendicular diagonals", and the participants agree to use this definition for now.

38:40 Working as a whole group they attempt to test the 3 formulas. Two of the formulas are quickly eliminated, as they do not work for all of the new examples. To check the third formula, A = ½ (d₁ x d₂₎, the facilitator draws the diagonals in each of the examples but the group is still not sure if this formula works for all of the examples.

41:45 Some time is given to the participants to work on their own on testing the formulas on the new set of example, so that they can individually "make sense" of the results proposed.

42:10 The facilitator calls the participants back for a whole group discussion to come to some final resolution about which area formula(s) can be used with diamonds. As a result of the individual work the facilitator is now able to get all the participants to confirm that the formula, A = ½ (d₁ x d₂₎, works for all diamonds.

42:45
Articulating the steps to create area formulas

The facilitator now explicitly states that she is going to summarize the steps that were used to generate the area formula for the diamond so that the process can be used to develop other area formulas. She records on newsprint key steps followed in their process of trying to create an area formula for a new figure. As she synthesizes and reflects on the experience she solicits input from the participants. She takes great care to both elaborate on and organize the steps she is recording. This final discussion takes quite a long time but in the end the "key steps to developing area formulas" are fully rationalized and articulated.

articulating the key steps of the modeled process in writing

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Time	Description of the video segment	Teaching practices(s) being modeled
	Events leading up to the modeling activity: This session is the entire second component of the Area "experience as learner" activity. During the first component participants were introduced to the entire activity and then engaged individually and in pairs in finding the area of a complex figure, the "fish". The first component set the stage for appreciating the complexity of the concept of area and the value of learning to develop area formulas. This modeling activity is designed to identify a general process for developing area formulas for complex figures by engaging the whole group in developing a formula for "diamonds". The facilitator purposefully did not define a priori a class of figures called "diamonds" but rather begins by proposing examples commonly thought of as diamonds and later introduces some which do not "fit" this common image.
25:40	Video segment: The facilitator introduced this segment of the Area experience, articulating its rationale and scope, and making connections with the previous "fish" activity. She describes the task, creating an area formulas for "diamonds", and presents an overhead of 2 rhombi. She asks the participants for ideas about what it means to develop an area formula for figures like this. The facilitator waits patiently for a participant to offer an idea. Finally, someone suggests finding the area of the specific figures and then try to find a general procedure. After the facilitator rephrases the suggestion she asks participants to work in pairs on the task for a few minutes.	The entire activity was specifically designed to model the development of area formulas by directing the class through a new process and then articulating the key steps so we will only identify it here at the beginning of the segment. For the remainder on the segment we will only identify the additional "modeling" teaching practices being employed. making explicit a process that someone in the class has illustrated
28:10	The facilitator asks participants to share the results of their work. She creates the drawings as each of the volunteers describes her process. A participant, Meghan, explains how she tried to transform the diamond into a rectangle so they could use the formula A=bxh. The facilitator asks questions to help Meghan better explain and justify her strategy. She also makes connections with some strategies identified in the "fish" activity to get the participants to realize that these strategies may be generalizable.	making explicit a process that someone in the class has illustrated
29:20 30:50 32:15	Another participant, Heather, shares her approach which was to "see" the diamond as two equal triangles formed by one of the diagonals and then use the formula A = 2 (1/2 bh) or A=bh. At this point participants begin to grapple with issues about variables when the facilitator has some difficulty fully understanding what this participant is trying to communicate, and other participants intervene to help. The facilitator restates the process Heather used and reminds the group that they ended up with A=bh using the approach previously shared. Using a clean overhead of the 2 diamonds, the facilitator, with the participants' input, demonstrates that the two formulas actually measure different parts of the diamond. At this point a participant suggests they identify the parts being measured in the second approach as the diagonals and the formula is rewritten as A = d₁ x ½ d₂. A third formula is suggested, A = d₁². This discussion helped the participants appreciate that the symbols used in area formulas made have different meanings.	- making explicit a process that someone in the class has illustrated
34:00	Another participant raises the question of why another formula was needed when we already had one that worked. The facilitator used this as an opportunity to review how they came to the additional formulas and help the participant herself provide reasons for having more than one formula.
36:10	The facilitator now suggests that the 3 formulas proposed by the participants need to be tested to see if they work for all diamonds. She records the 3 formulas and puts on an overhead with some additional examples of "diamond". These new examples challenge the commonly held image of "diamond" and some participants say they do not like all of these as diamonds. The facilitator shares her definition, "a quadrilateral with perpendicular diagonals", and the participants agree to use this definition for now.
38:40	Working as a whole group they attempt to test the 3 formulas. Two of the formulas are quickly eliminated, as they do not work for all of the new examples. To check the third formula, A = ½ (d₁ x d₂₎, the facilitator draws the diagonals in each of the examples but the group is still not sure if this formula works for all of the examples.
41:45	Some time is given to the participants to work on their own on testing the formulas on the new set of example, so that they can individually "make sense" of the results proposed.
42:10	The facilitator calls the participants back for a whole group discussion to come to some final resolution about which area formula(s) can be used with diamonds. As a result of the individual work the facilitator is now able to get all the participants to confirm that the formula, A = ½ (d₁ x d₂₎, works for all diamonds.
42:45	Articulating the steps to create area formulas The facilitator now explicitly states that she is going to summarize the steps that were used to generate the area formula for the diamond so that the process can be used to develop other area formulas. She records on newsprint key steps followed in their process of trying to create an area formula for a new figure. As she synthesizes and reflects on the experience she solicits input from the participants. She takes great care to both elaborate on and organize the steps she is recording. This final discussion takes quite a long time but in the end the "key steps to developing area formulas" are fully rationalized and articulated.	articulating the key steps of the modeled process in writing