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

Return to Section C Menu
Return to List of additional essays on teaching mathematics through inquiry

Additional essays on teaching mathematics through inquiry (C1)
Teaching practices that support an inquiry approach to mathematics instruction (Borasi & Fonzi, 1998) -- Appendix D

Commented video index for the class discussion of "what should count as a tessellation," highlighting the teaching practices modeled within the category of "Orchestrating and facilitating students' inquiry when working as a whole class."

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

Orchestrating and facilitating students' inquiry when working as whole class:
* Choosing engaging topics for whole class activities
* Preparing the ground to generate genuine debate
* Moderating class discussions
* Helping the class stay focused on the topic of a discussion
* Dealing with unexpected questions within a class discussion
* Helping students articulate and elaborate upon their contributions to a class discussion
* Building on students' contributions in a class discussion
* Highlighting controversy and the need to resolve it
-- Providing opportunities for individual thinking within a class activity
* Providing feedback without taking away from the class' inquiry
* Recording key points raised during the class activity

Time Description of the video segment Teaching practice being modeled

Events leading up to this whole class discussion:

This "experience as learners" is a whole class inquiry that involves several small group inquiries within it. The segment we are focussing on here is the whole class discussion of the results of their small group work examining potential examples of tessellations brought in by the participants to determine if everyone agrees that they are examples of tessellations. We selected the topic of "tessellations" because we thought it would be "new" to most of the participants and thus could generate genuine interest in exploring it. The assignment to bring in 2 examples which met the given definition of tessellation was intended to develop some personal investment in the examples as well as to give the participants individual time to interpret the meaning of the definition. The small group work was intended to elicit participants different interpretations of the definition by having them carefully examine each example. The facilitators purposefully tried to insure that all examples were examined by at least 2 groups so as to elicit any differing interpretations during the whole group sharing, thus highlighting the role the definition plays in determining a class of figures.

choosing engaging topics for whole class activities
preparing the ground to generate genuine debate

07:45
Video segment:

Whole class sharing of results of small group examination of potential examples of tessellation

Facilitator calls the group together and sets up the discussion so that it is very clear that she is not expecting everyone to agree on the decisions. She describes the work they have done in such a way as to highlight the fact that the participants themselves are the ones making the decisions. The facilitator asks them to share the "easy" ones first to either establish some agreed upon examples or make clear very quickly that even "easy" one are controversial since ultimately she is looking to generate controversy.

Notice, throughout the discussion that the facilitators do not confirm nor deny the decisions rather they elicit the decisions made by the other groups.

preparing the ground to generate genuine debate
providing feedback without taking away from the class' inquiry

08:40 Sheila is asked to holds her example as she shares that her group decided it was a tessellation. Other groups confirm that they made the same decision and Sheila is asked to post the example in the column for "tessellations".

09:35
Ceiling / floor example

The facilitator makes it a point to "hear" the concerns Sheila is expressing in order to value them and bring the ideas out in the open for discussion. As Sheila expresses her concerns about the "boarder" being a different size and color the contributing group is having a private discussion. The facilitator insists that the let the whole class be "let in on" the discussion hoping to nurture a deeper examination and discussion of the examples. When the group says they were only looking at the "middle" of the ceiling /floor the other facilitator makes public the comment she overheard from another participant, "you have to define your limits", in order to demonstrate that there is some agreement about the need to know precisely you are looking at and to generalize the issue.

highlighting controversy and the need to resolve it
moderating class discussion
helping students articulate and elaborate on their contributions to a class discussion

10:45

11:50

13:35

Participants begin to grapple with the issue of boarders/boundaries/limits of a pattern. The facilitator makes public the potentially controversial decisions by asking "who defines the limits?" and she gets the group to talk about what could be the key factors in this decision making process. Notice that the participants are bringing other examples into the discussion as a way to make their points. Here they begin to make public the fact that they are making decisions on their intuitive instincts but actually question whether they fit the given definition.

The other facilitator asks if they are saying that the definition is not adequate in order to help them realize that they may be able to settle this controversy by identifying what is causing it and how it can be dealt with. She records their issues on newsprint.

The facilitator listens with interest allowing the participants to discuss this issue without her input. After some time, hoping to determine how technical this group was going to be about determining limits, she picks up on the concrete suggestion to "cut on the line". She asks Diane to go further and justify why she believes this would make the example fit the definition, and then invites the groups reaction.

Again as several people talk at once the facilitator is silent to let them try to work through this issue by themselves.

highlighting controversy and the need to resolve it
helping the class stay focussed on the topic of the discussion
recording key points raised during the class activity
moderating class discussions
helping students articulate and elaborate upon their contributions to a class discussion
moderating class discussions

15:00

15:15

The other facilitator, feeling that the group has come to a decision regarding limits they are all comfortable with, explicitly asks if there is a way they would want to modify the definition to try to respond to some of their concerns.

A participant (Laurie) suggests they add something about color which takes the facilitators completely by surprise ! The facilitators immediately work to understand what she is talking about so as to validate the question but then get Laurie's agreement to take on the issue later. The issue is recorded on newsprint as a reminder to come back to it.

helping the class stay focussed on the topic of discussion
building on students' contributions in a class discussion
dealing with unexpected questions within a class discussion
helping the class stay focused on the topic of a discussion
recording key points raised during the class activity

16:35

18:25

19:25

The other facilitator offers the learners the opportunity to modify the definition to make it more clear.

The other facilitator helps the learners recognize their process and predicament and invites them to think about how to proceed mathematically.

In an effort to keep a participant's question of common goals from derailing the forward movement the facilitator reminds the group of the scope of the activity pointing out that they do all have the same goal. The facilitator goes on to recap what seems to be the roadblocks to reaching that goal and why the suggestion was made to consider modifying the definition.

building on students' contributions in a class discussion
building on students' contributions in a class discussion
dealing with unexpected questions within a class discussion
helping the class stay focused on the topic of a discussion

21:15
Several people offer ideas that sound similar as they hone in on their interpretation of the "limits" implied by the phrase "covers a flat surface" in the definition. The facilitator is quite surprised when a participant asks "if it can [the pattern] go on for a while wouldn't it go on longer?". Feeling a need to be sure that the participants understand the significance of this misconception the facilitator slowly and carefully articulates the points to help them see the subtle but important distinctions. She lets the group discuss their different ideas for a few minutes and then articulates the 3 different situations that have been suggested. She once again reminds the group that they must come to consensus on this issue in order to decide which examples are tessellations and which are not and points out that the suggestion is to make the decision clear in the definition by changing it to reflect their decision.

(This seems to be a turning point in this inquiry.)

highlighting controversy and the need to resolve it
providing feedback without taking away from the class' inquiry

26:30 Several people offer ideas for modifying the definition so that it better conveys their understanding. One participant, Sharon, argues against the suggestion that the space can be "finite" because it challenges her intuitive understanding of tessellating figures. As she describes the case of 2 pentagons she is asked to draw the figure on newsprint so everyone can see what she is describing. Sharon claims that 2 pentagons do not represent a tessellation and several participants challenge her. Ultimately it becomes clear that some participants do not want to allow for figures that only tessellate in a finite space to "count".

helping students articulate and elaborate upon their contributions to a class discussion

28:15
The other facilitator synthesizes the discussion by making concrete suggestions for how to change the definition and asks if this will provide them with a tool that will allow them to make precise decisions.

A participant, Ed, resists altering a mathematical definition trying to deny its lack of precision or the controversy it has raised. The facilitator once again points out the considerable controversy that has been raised.

highlighting controversy and the need to resolve it
building on students' contributions in class discussion

30:00 At this point some people have become very invested in the results of this discussion. As a participant, Beth, points out the need for the community to develop a common language another participant, Sharon, challenged her even before she could finish ! The facilitator had to work hard to keep the issues alive yet keep the tone civil and collaborative. - moderating class discussions

31:10 After the facilitator has again summarized the issues and reminded the participants about why they needed to come to some decisions about these issues a participant, Laurie, was still willing to believe that the controversy was the result of their different math backgrounds rather than some inherent "fuzziness" in the definition !

helping the class stay focused on the topic of a discussion
dealing with unexpected questions within the class discussion
highlighting the controversy and the need to resolve it

37:25 Finally, the other facilitator begins to bring the discussion to close by suggesting that they come to consensus as a community so that they can move on. She is ultimately able to get the whole group to agree to modify the definition so that it does not allow for patterns that can not cover a plane. She records the changes in the definition and asks the participants to re-examine the examples in light of the modified definition and post the results.

helping the class stay focused on the topic of a discussion
helping students articulate and elaborate on their contributions to a class discussion
recording key points raised during the class activity

TOP

Return to Main Text
Return to List of additional essays on teaching mathematics through inquiry

Time		Description of the video segment	Teaching practice being modeled
		Events leading up to this whole class discussion: This "experience as learners" is a whole class inquiry that involves several small group inquiries within it. The segment we are focussing on here is the whole class discussion of the results of their small group work examining potential examples of tessellations brought in by the participants to determine if everyone agrees that they are examples of tessellations. We selected the topic of "tessellations" because we thought it would be "new" to most of the participants and thus could generate genuine interest in exploring it. The assignment to bring in 2 examples which met the given definition of tessellation was intended to develop some personal investment in the examples as well as to give the participants individual time to interpret the meaning of the definition. The small group work was intended to elicit participants different interpretations of the definition by having them carefully examine each example. The facilitators purposefully tried to insure that all examples were examined by at least 2 groups so as to elicit any differing interpretations during the whole group sharing, thus highlighting the role the definition plays in determining a class of figures.	choosing engaging topics for whole class activities preparing the ground to generate genuine debate
07:45	Video segment: Whole class sharing of results of small group examination of potential examples of tessellation Facilitator calls the group together and sets up the discussion so that it is very clear that she is not expecting everyone to agree on the decisions. She describes the work they have done in such a way as to highlight the fact that the participants themselves are the ones making the decisions. The facilitator asks them to share the "easy" ones first to either establish some agreed upon examples or make clear very quickly that even "easy" one are controversial since ultimately she is looking to generate controversy. Notice, throughout the discussion that the facilitators do not confirm nor deny the decisions rather they elicit the decisions made by the other groups.			preparing the ground to generate genuine debate providing feedback without taking away from the class' inquiry
08:40	Sheila is asked to holds her example as she shares that her group decided it was a tessellation. Other groups confirm that they made the same decision and Sheila is asked to post the example in the column for "tessellations".
09:35	Ceiling / floor example The facilitator makes it a point to "hear" the concerns Sheila is expressing in order to value them and bring the ideas out in the open for discussion. As Sheila expresses her concerns about the "boarder" being a different size and color the contributing group is having a private discussion. The facilitator insists that the let the whole class be "let in on" the discussion hoping to nurture a deeper examination and discussion of the examples. When the group says they were only looking at the "middle" of the ceiling /floor the other facilitator makes public the comment she overheard from another participant, "you have to define your limits", in order to demonstrate that there is some agreement about the need to know precisely you are looking at and to generalize the issue.			highlighting controversy and the need to resolve it moderating class discussion helping students articulate and elaborate on their contributions to a class discussion
10:45 11:50 13:35	Participants begin to grapple with the issue of boarders/boundaries/limits of a pattern. The facilitator makes public the potentially controversial decisions by asking "who defines the limits?" and she gets the group to talk about what could be the key factors in this decision making process. Notice that the participants are bringing other examples into the discussion as a way to make their points. Here they begin to make public the fact that they are making decisions on their intuitive instincts but actually question whether they fit the given definition. The other facilitator asks if they are saying that the definition is not adequate in order to help them realize that they may be able to settle this controversy by identifying what is causing it and how it can be dealt with. She records their issues on newsprint. The facilitator listens with interest allowing the participants to discuss this issue without her input. After some time, hoping to determine how technical this group was going to be about determining limits, she picks up on the concrete suggestion to "cut on the line". She asks Diane to go further and justify why she believes this would make the example fit the definition, and then invites the groups reaction. Again as several people talk at once the facilitator is silent to let them try to work through this issue by themselves.			highlighting controversy and the need to resolve it helping the class stay focussed on the topic of the discussion recording key points raised during the class activity moderating class discussions helping students articulate and elaborate upon their contributions to a class discussion moderating class discussions
15:00 15:15	The other facilitator, feeling that the group has come to a decision regarding limits they are all comfortable with, explicitly asks if there is a way they would want to modify the definition to try to respond to some of their concerns. A participant (Laurie) suggests they add something about color which takes the facilitators completely by surprise ! The facilitators immediately work to understand what she is talking about so as to validate the question but then get Laurie's agreement to take on the issue later. The issue is recorded on newsprint as a reminder to come back to it.			helping the class stay focussed on the topic of discussion building on students' contributions in a class discussion dealing with unexpected questions within a class discussion helping the class stay focused on the topic of a discussion recording key points raised during the class activity
16:35 18:25 19:25	The other facilitator offers the learners the opportunity to modify the definition to make it more clear. The other facilitator helps the learners recognize their process and predicament and invites them to think about how to proceed mathematically. In an effort to keep a participant's question of common goals from derailing the forward movement the facilitator reminds the group of the scope of the activity pointing out that they do all have the same goal. The facilitator goes on to recap what seems to be the roadblocks to reaching that goal and why the suggestion was made to consider modifying the definition.			building on students' contributions in a class discussion building on students' contributions in a class discussion dealing with unexpected questions within a class discussion helping the class stay focused on the topic of a discussion
21:15	Several people offer ideas that sound similar as they hone in on their interpretation of the "limits" implied by the phrase "covers a flat surface" in the definition. The facilitator is quite surprised when a participant asks "if it can [the pattern] go on for a while wouldn't it go on longer?". Feeling a need to be sure that the participants understand the significance of this misconception the facilitator slowly and carefully articulates the points to help them see the subtle but important distinctions. She lets the group discuss their different ideas for a few minutes and then articulates the 3 different situations that have been suggested. She once again reminds the group that they must come to consensus on this issue in order to decide which examples are tessellations and which are not and points out that the suggestion is to make the decision clear in the definition by changing it to reflect their decision. (This seems to be a turning point in this inquiry.)			highlighting controversy and the need to resolve it providing feedback without taking away from the class' inquiry
26:30	Several people offer ideas for modifying the definition so that it better conveys their understanding. One participant, Sharon, argues against the suggestion that the space can be "finite" because it challenges her intuitive understanding of tessellating figures. As she describes the case of 2 pentagons she is asked to draw the figure on newsprint so everyone can see what she is describing. Sharon claims that 2 pentagons do not represent a tessellation and several participants challenge her. Ultimately it becomes clear that some participants do not want to allow for figures that only tessellate in a finite space to "count".			helping students articulate and elaborate upon their contributions to a class discussion
28:15	The other facilitator synthesizes the discussion by making concrete suggestions for how to change the definition and asks if this will provide them with a tool that will allow them to make precise decisions. A participant, Ed, resists altering a mathematical definition trying to deny its lack of precision or the controversy it has raised. The facilitator once again points out the considerable controversy that has been raised.			highlighting controversy and the need to resolve it building on students' contributions in class discussion
30:00	At this point some people have become very invested in the results of this discussion. As a participant, Beth, points out the need for the community to develop a common language another participant, Sharon, challenged her even before she could finish ! The facilitator had to work hard to keep the issues alive yet keep the tone civil and collaborative.			- moderating class discussions
31:10	After the facilitator has again summarized the issues and reminded the participants about why they needed to come to some decisions about these issues a participant, Laurie, was still willing to believe that the controversy was the result of their different math backgrounds rather than some inherent "fuzziness" in the definition !			helping the class stay focused on the topic of a discussion dealing with unexpected questions within the class discussion highlighting the controversy and the need to resolve it
37:25	Finally, the other facilitator begins to bring the discussion to close by suggesting that they come to consensus as a community so that they can move on. She is ultimately able to get the whole group to agree to modify the definition so that it does not allow for patterns that can not cover a plane. She records the changes in the definition and asks the participants to re-examine the examples in light of the modified definition and post the results.			helping the class stay focused on the topic of a discussion helping students articulate and elaborate on their contributions to a class discussion recording key points raised during the class activity