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 C

Commented video index for the "sharing" session that concluded the Fish activity in the Area inquiry experience, highlighting the teaching practices modeled within the category of "Orchestrating and facilitating 'sharing' sessions where students communicate the results of their inquiries."

Below is a list of the teaching practices we have identified within the category of "Orchestrating and facilitating "sharing" sessions where students communicate the results of their inquiries"; the practices noted with an * have been identified within this video segment:

Orchestrating and facilitating "sharing" sessions where students communicate the results of their inquiries:
* Developing a common interest in the results to be shared
* Encouraging the use of artifacts to support one's presentation
* Providing the opportunity to first share with a partner
* Orchestrating a good sequence of presentations
* Helping students articulate and elaborate upon their results in a presentation
-- Encouraging audience participation in the presentation
* Building upon what students have shared

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

Events leading up to the sharing session:

This session took place within the first component of the Area "experience as learner" activity.

The facilitator first articulated the rationale and nature of the whole "Area" segment and then introduced the first task, "to find the area of the fish to be painted on the aquarium floor to find out how much paint would be needed."

She passed out a copy of the "fish" to each participant and explained the procedure they would follow, i.e., work individually for a few minutes, then talk to a partner to share your approach, finally share your approach with the whole group. Participants were also told of the "tools" that were available for their use.

While the participants were working on the task the facilitator publicly accepted the suggestion "to measure in square units" when another participant asked what she really meant by find the area. The facilitator walked around and observed throughout the individual and pairs work to get an idea of the variety of approaches that were being used.

developing a common interest in the results to be shared
providing the opportunity to first share with a partner

01:35
Video segment:

Participants' sharing

The facilitator begins by soliciting and recording the numerical results of calculating the area of the "fish." At this point she has accepted all of the answers and has not given any indication that any one is better than another. She suggests that their differences make it important to see how people got to their results. She mentions that as she walked around the room she saw almost as many solution approaches as there were people.

She invites volunteers to share their solution approaches and provides them with a clean overhead of the "fish" so that they can demonstrate their approach as well as describe it.

developing a common interest in the results to be shared
encouraging the use of artifacts to support one's presentation

02:45

05:20

The first participant, Heather, shares her strategy, consisting of "breaking up" the fish into rectangles and triangles and computed the area.

After Heather shared her strategy, the facilitator summarizes her approach in general terms always asking Heather for confirmation of her summary and asking specific questions about how she computed, "did you use formulas?".

building upon what students have shared
helping students articulate and elaborate on their results in a presentation

When soliciting a next volunteer the facilitator asks "did anyone did it differently?"

A second participant, Nancy, shares her solution based on the idea of "boxing" the "fish" and then subtracting the extra pieces which she subdivided into rectangles and triangles. She acknowledges that she made some arithmetic mistakes but did not feel they affected the validity of her approach.

After Nancy has completed her explanation the facilitator pointed out the significance of this different approach even if she did not get the "correct" answer. Then, once again the facilitator summarized the approach in general terms and confirmed the specific decisions and processes with Nancy.

orchestrating a good sequence of presentations
building upon what students have shared

08:00

08:05

Once again the facilitator solicits the next volunteer by asking for someone who did it differently. At this point, informed by her observations during the pairs work, she is prepared to call on specific people whose unique solutions can add to the discussion if they do not volunteer by themselves.

A third participant, Brenda, shares her solution. She recognized the symmetry within the "fish" and folded the paper to identify half of the figure to be calculated. Brenda is not providing enough information in her explanation so the facilitator intervenes to ask occasional questions to prompt her to say more. The facilitator gets her to explain that she transformed a trapezoid into a rectangle because she did not remember the formula.

orchestrating a good sequence of presentations
helping students articulate and elaborate on their results in a presentation

10:00 A fourth participant, Marie, shares her solution consisting of "boxing" the fish, but then computing the area to be subtracted by simply counting squares. Again the facilitator intervenes to ask for more information. This time her question is intended to raise what could be a controversial issue - "how did you count pieces of the square?". The facilitator knew that Marie's process was valid but her result was quite high.

helping students articulate and elaborate on their results in a presentation

11:15 Facilitator concludes the sharing without having all the participants share. She then explicitly says "let me try to pull together or summarize what we have done". She generalizes the approaches and points out that even the simplest strategies got pretty close to the "exact" answer (though she has not made this answer public).

building upon what students' have shared

The facilitator then invites the participants to reflect on what is really necessary to solve this problem so that they will be the ones to state that only a couple of simple area formulas are all that is necessary. She pushes them further to describe what the critical skills are for solving these complex area problems. After stating the key strategies she returns to the early question of "what it means to compute area" to demonstrate that even this one activity can raise several of the current curricular issues and some very thought-provoking questions about the math of area. She further articulates and records these issues on newsprint to return to at a later time.

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Time	Description of the video segment	Teaching practice(s) being modeled
	Events leading up to the sharing session: This session took place within the first component of the Area "experience as learner" activity. The facilitator first articulated the rationale and nature of the whole "Area" segment and then introduced the first task, "to find the area of the fish to be painted on the aquarium floor to find out how much paint would be needed." She passed out a copy of the "fish" to each participant and explained the procedure they would follow, i.e., work individually for a few minutes, then talk to a partner to share your approach, finally share your approach with the whole group. Participants were also told of the "tools" that were available for their use. While the participants were working on the task the facilitator publicly accepted the suggestion "to measure in square units" when another participant asked what she really meant by find the area. The facilitator walked around and observed throughout the individual and pairs work to get an idea of the variety of approaches that were being used.	developing a common interest in the results to be shared providing the opportunity to first share with a partner
01:35	Video segment: Participants' sharing The facilitator begins by soliciting and recording the numerical results of calculating the area of the "fish." At this point she has accepted all of the answers and has not given any indication that any one is better than another. She suggests that their differences make it important to see how people got to their results. She mentions that as she walked around the room she saw almost as many solution approaches as there were people. She invites volunteers to share their solution approaches and provides them with a clean overhead of the "fish" so that they can demonstrate their approach as well as describe it.	developing a common interest in the results to be shared encouraging the use of artifacts to support one's presentation
02:45 05:20	The first participant, Heather, shares her strategy, consisting of "breaking up" the fish into rectangles and triangles and computed the area. After Heather shared her strategy, the facilitator summarizes her approach in general terms always asking Heather for confirmation of her summary and asking specific questions about how she computed, "did you use formulas?".	building upon what students have shared helping students articulate and elaborate on their results in a presentation
	When soliciting a next volunteer the facilitator asks "did anyone did it differently?" A second participant, Nancy, shares her solution based on the idea of "boxing" the "fish" and then subtracting the extra pieces which she subdivided into rectangles and triangles. She acknowledges that she made some arithmetic mistakes but did not feel they affected the validity of her approach. After Nancy has completed her explanation the facilitator pointed out the significance of this different approach even if she did not get the "correct" answer. Then, once again the facilitator summarized the approach in general terms and confirmed the specific decisions and processes with Nancy.	orchestrating a good sequence of presentations building upon what students have shared
08:00 08:05	Once again the facilitator solicits the next volunteer by asking for someone who did it differently. At this point, informed by her observations during the pairs work, she is prepared to call on specific people whose unique solutions can add to the discussion if they do not volunteer by themselves. A third participant, Brenda, shares her solution. She recognized the symmetry within the "fish" and folded the paper to identify half of the figure to be calculated. Brenda is not providing enough information in her explanation so the facilitator intervenes to ask occasional questions to prompt her to say more. The facilitator gets her to explain that she transformed a trapezoid into a rectangle because she did not remember the formula.	orchestrating a good sequence of presentations helping students articulate and elaborate on their results in a presentation
10:00	A fourth participant, Marie, shares her solution consisting of "boxing" the fish, but then computing the area to be subtracted by simply counting squares. Again the facilitator intervenes to ask for more information. This time her question is intended to raise what could be a controversial issue - "how did you count pieces of the square?". The facilitator knew that Marie's process was valid but her result was quite high.	helping students articulate and elaborate on their results in a presentation
11:15	Facilitator concludes the sharing without having all the participants share. She then explicitly says "let me try to pull together or summarize what we have done". She generalizes the approaches and points out that even the simplest strategies got pretty close to the "exact" answer (though she has not made this answer public).	building upon what students' have shared
	The facilitator then invites the participants to reflect on what is really necessary to solve this problem so that they will be the ones to state that only a couple of simple area formulas are all that is necessary. She pushes them further to describe what the critical skills are for solving these complex area problems. After stating the key strategies she returns to the early question of "what it means to compute area" to demonstrate that even this one activity can raise several of the current curricular issues and some very thought-provoking questions about the math of area. She further articulates and records these issues on newsprint to return to at a later time.