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 B

Commented video index for a segment of the small group activity where participants explored their own conjectures about tessellations, highlighting the teaching practices modeled within the category of "Orchestrating and facilitating students' inquiry when working in small groups."

Below is the list of the teaching practices we have identified within the category of "Orchestrating and facilitating students' inquiry when working in small groups; the practices noted with an * have been identified within this video segment:

Orchestrating and facilitating students' inquiry when working in small groups:
* Designing engaging tasks for group work
* Providing adequate directions for group work
* Providing group members with opportunities for individual thinking
* Assigning students to appropriate groups
* Monitoring group work
* Helping groups articulate where they are
* Providing feedback without taking away from the group's inquiry
* Intervening as necessary to support the group work

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

Events leading up to the small group work:

This small group experience took place within the Tessellation "experience as learner" activity. At the end of the previous day the participants had generated and shared a list of conjectures and questions about tessellations based on their new understanding of what "counts" as a tessellation. Participants were told that they would be testing some of these conjectures the following day and were asked to think about what it takes to test a conjecture for homework. This small group activity began with a whole group discussion where participants shared their ideas about how to test conjectures. The facilitator then introduced the activity and carefully explained the process the group would be following. That is, first model the testing of conjectures by having all groups test the conjecture "All triangles tessellate", share the results of that testing to get a better understanding of what is involved in testing a conjecture, and then each small group would select their own conjectures to test. The facilitator also explicitly described the materials that were available for participants' use. The groups, which were arranged at the beginning of the entire Tessellation "experience as learner" activity, were heterogeneous with respect to grade level taught, area of certification, perceived math background, and school team.

Designing engaging tasks for group work
Providing group members with opportunities for individual thinking
Providing adequate directions for group work
Assigning students to appropriate groups

1:04:15
Video segment:

Small group work on the conjecture "All triangles tessellate"

This group is a very good example of a group working together. (The person in the background with the white shirt is a silent observer, not a participant.) The group has been testing the conjecture by making tessellating patterns with specific types of triangles and then determining which "types" tessellate. One of the participants notices, as a result of actually putting triangles together, that "every triangle is ½ a parallelogram". This produces an AHA for some of the group members and they announce to the facilitator that they are done ! One participant states that any 2 triangles can form a parallelogram and all parallelograms tessellate.

The facilitator gets the group to realize that they have more work to do by asking how they know all parallelograms tessellate (which had not previously been proven).

The facilitator then stands back and watches silently to be sure they are moving forward. After a minute or two she quietly intervenes to check her understanding of what they are doing. She does this to both make the point that they need to verbalize the plan so all the groups members know what is being done and why.

Providing feedback without taking away from the group's inquiry
Monitoring group work
Helping groups articulate where they are

1:08:35
Upon hearing a highly technical explanation for why two triangles can be made into a parallelogram, the facilitator suspects that not all the group members know this information. She is able to bring this out by explicitly asking if the other group members (those not giving the explanation) know this information. This immediately prompts the original speaker to explain her statements so that everyone would understand the mathematics involved in their group's work.

Intervening as necessary to support the group work

1:10:15
The facilitator continues to be a silent observer as the group struggles to demonstrate to each other the mathematical thinking they are using. She only becomes involved again when it seems that the ideas are clear to everyone and the struggle for terminology is unnecessarily bogging them down. She states that they seem to be struggling for words and helps them refocus on their larger goals by reminding them of what they were trying to do and how they got to this point. She suggests that they reconfirm their plan and then continue, since as a community they all seem to have an understanding of the process they were struggling to name even if they didn't have "textbook terminology".

As soon as the facilitator is comfortable that they are again working profitably as a group she leaves.

Monitoring group work
Intervening as necessary to support the group work
Providing feedback without taking away from the group's inquiry
Monitoring group work

The small groups shared the results and processes used in their testing of the conjecture "all triangles tessellate" before moving on to test additional conjectures of their own choosing.

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Time	Description of the video segment	Teaching practice(s) being modeled:
	Events leading up to the small group work: This small group experience took place within the Tessellation "experience as learner" activity. At the end of the previous day the participants had generated and shared a list of conjectures and questions about tessellations based on their new understanding of what "counts" as a tessellation. Participants were told that they would be testing some of these conjectures the following day and were asked to think about what it takes to test a conjecture for homework. This small group activity began with a whole group discussion where participants shared their ideas about how to test conjectures. The facilitator then introduced the activity and carefully explained the process the group would be following. That is, first model the testing of conjectures by having all groups test the conjecture "All triangles tessellate", share the results of that testing to get a better understanding of what is involved in testing a conjecture, and then each small group would select their own conjectures to test. The facilitator also explicitly described the materials that were available for participants' use. The groups, which were arranged at the beginning of the entire Tessellation "experience as learner" activity, were heterogeneous with respect to grade level taught, area of certification, perceived math background, and school team.	Designing engaging tasks for group work Providing group members with opportunities for individual thinking Providing adequate directions for group work Assigning students to appropriate groups
1:04:15	Video segment: Small group work on the conjecture "All triangles tessellate" This group is a very good example of a group working together. (The person in the background with the white shirt is a silent observer, not a participant.) The group has been testing the conjecture by making tessellating patterns with specific types of triangles and then determining which "types" tessellate. One of the participants notices, as a result of actually putting triangles together, that "every triangle is ½ a parallelogram". This produces an AHA for some of the group members and they announce to the facilitator that they are done ! One participant states that any 2 triangles can form a parallelogram and all parallelograms tessellate. The facilitator gets the group to realize that they have more work to do by asking how they know all parallelograms tessellate (which had not previously been proven). The facilitator then stands back and watches silently to be sure they are moving forward. After a minute or two she quietly intervenes to check her understanding of what they are doing. She does this to both make the point that they need to verbalize the plan so all the groups members know what is being done and why.	Providing feedback without taking away from the group's inquiry Monitoring group work Helping groups articulate where they are
1:08:35	Upon hearing a highly technical explanation for why two triangles can be made into a parallelogram, the facilitator suspects that not all the group members know this information. She is able to bring this out by explicitly asking if the other group members (those not giving the explanation) know this information. This immediately prompts the original speaker to explain her statements so that everyone would understand the mathematics involved in their group's work.	Intervening as necessary to support the group work
1:10:15	The facilitator continues to be a silent observer as the group struggles to demonstrate to each other the mathematical thinking they are using. She only becomes involved again when it seems that the ideas are clear to everyone and the struggle for terminology is unnecessarily bogging them down. She states that they seem to be struggling for words and helps them refocus on their larger goals by reminding them of what they were trying to do and how they got to this point. She suggests that they reconfirm their plan and then continue, since as a community they all seem to have an understanding of the process they were struggling to name even if they didn't have "textbook terminology". As soon as the facilitator is comfortable that they are again working profitably as a group she leaves.	Monitoring group work Intervening as necessary to support the group work Providing feedback without taking away from the group's inquiry Monitoring group work
	The small groups shared the results and processes used in their testing of the conjecture "all triangles tessellate" before moving on to test additional conjectures of their own choosing.