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Additional essays on teaching mathematics through inquiry (C1)
Teaching practices that support an inquiry approach to mathematics instruction (Borasi & Fonzi, 1998) -- Main text of the essay

(unpublished manuscript prepared for inclusion in the multi-media package "Introducing math teachers to inquiry: Framework and supporting materials to design professional development" [Borasi & Fonzi, 1998])

Introduction

Elsewhere (see Section C.1 of these materials) we have identified "facilitating students' inquiries and learning in the classroom through the use of appropriate teaching practices and techniques" as a key element of teaching mathematics through inquiry. In this essay we have attempted to further articulate what this means by providing a rich list of practices that we believe are crucial for orchestrating successful inquiry experiences.

The teaching practices discussed in this essay were all derived from the analysis of various examples of inquiry-based mathematics instruction in a variety of instructional settings, including "experiences as learners" conducted within a professional development program. Although none of the practices we have identified are necessarily unique to an inquiry approach, they are nonetheless an important component of teaching mathematics through inquiry. Also note that we have purposefully chosen to focus our attention in this essay only on teaching practices that play out in classroom activities. Therefore, in what follows we will not discuss what it takes to plan successful inquiry units for mathematics students, nor how to develop appropriate assessment for the learning taking place as a result of engaging in inquiry experiences.

We have organized our list of teaching practices that support an inquiry approach to mathematics instruction into eight major categories, as summarized in Figure 1 below. A list of all the recommended teaching practices within each of these categories will be provided at the end of the essay (see Figure 2), after we have introduced and briefly discussed each practice.

Figure 1.
Main categories of teaching practices
discussed in this essay

Modelling Making explicit the purpose of an inquiry learning experience Synthesizing and reflecting on the results of an inquiry experience Orchestrating and facilitating students' inquiry when working in small groups Orchestrating and facilitating students' inquiry when working individually Orchestrating and facilitating "sharing"sessions where students communicate the results of their inquiries Orchestrating and facilitating students' inquiry when working as a whole class Responding to the learning needs of diverse students in inquiry math classes

Within the first category -- "modeling" -- we suggest a number of teaching practices that have proven effective in scaffolding students inquiry, especially when it takes place within a technical subject like mathematics. The next two categories -- "making explicit the purpose of a learning activity" and "synthesizing and reflecting on the results of a learning activity" -- highlight what the teacher can do to help students appreciate the nature and value of the unusual learning experiences taking place in an inquiry-based mathematics classroom. The next four categories illustrate what it takes to orchestrate and facilitate student inquiry in different situations -- i.e., when students engage in inquiry in small groups, individually or as a whole class; note that we also chose to discuss separately how the sharing of individual and/or small group results can be supported, given the importance of this kind of communication for successful inquiries. Finally, we will point out some strategies to respond to the needs of students with different learning styles and backgrounds while implementing inquiry experiences.

Most of the teaching practices we are going to identify and discuss were purposefully modeled by the instructors in the "experiences as learners" featured in the professional development videos included in the package -- i.e., "Exploring tessellations: A professional development experience" (Video #2) and "Developing area formulas: A professional development experience" (Video #3). Thus, throughout the essay we illustrate each teaching practice with references to examples from these videos whenever available. For the sake of brevity, these references are limited to identifying one, or at most two, representative examples of each teaching practice -- although the videos usually included many more. To complement these illustrations, therefore, we have also included as Appendices the more detailed commentary of selected segments of the videos that we thought well illustrated the most complex of the practices discussed in the essay -- that is, the practices within the categories of "Modeling," "Orchestrating and facilitating students' inquiry when working in small groups," "Orchestrating and facilitating sharing sessions where students communicate the results of of their inquiries," and "Orchestrating and facilitating students' inquiry when working as a whole class" (these files can be accessed from Section C Menu or through the links at the end of the main text of this essay).

Note that, in order for our illustrations to make sense, you should have watched the two videos in their entirety before reading further. If you have not yet done so, please watch these videos now.

Brief description of suggested teaching practices

1. Modelling

Doing inquiry does not mean that mathematics students always have to discover on their own tools and procedures that may have taken mathematicians centuries to develop. Yet, this also does not imply that they need to be "told" upfront all that may become useful as they later engage in their own inquiry. Modelling often provides a viable alternative between these two extremes. It can be especially valuable when students are asked to engage in learning activities that are very new to them, and as a means to help them "learn how to learn" from what they are doing. At the same time, the modeling a teacher does can take on complementary forms in a mathematics classroom informed by an inquiry approach, including:

• Genuinely engaging in an activity as a learner and making one's thought process explicit. Having an expert perform the process that one is trying to learn is probably the image of "modeling" that first comes to mind, as we have all experienced the power of this kind of demonstration in every-day life. Yet, we may be less aware that the effectiveness of this kind of modelling greatly increases when the expert not only shows what needs to be done, but also articulates the key steps followed and their rationale. It is also worth mentioning that seeing the teacher genuinely engage as a mathematics learner may have considerable affective value for students, and contribute to developing a sense of community in the classroom. In the Tessellation and Area inquiries featured in our professional development videos, this kind of modelling occurred as facilitators other than the one conducting the experience, as well as more experienced teachers, joined with the participants during the group activities and become co-learners with them; unfortunately, the edited video does not capture any of these episodes.
• Directing the class through a new process and then articulating key steps. Modelling does not always mean that the teacher performs while the students are watching. Sometimes students get a better feeling for the process under study if they are asked to take a more active role at carefully selected points -- although always under the guidance of the instructor, who "walks them through" the process at this initial stage. Similarly, students may be asked to take part in identifying and describing the key steps they have just gone through. A good illustration of this kind of modelling is offered in the area unit, when the whole group engaged in developing an area formula for diamond in preparation for developing area formulas for their assigned stars -- the experience we chose to examine in more detail in Appendix A.
• Making explicit a process that someone in the class has illustrated. Sometimes, students may unwittingly model a new process for the rest of the class, as they share an unusual solution approach. In this case, the teacher's role is still very important, as she can validate such an approach by calling everybody's attention to it, and also fully articulate it so enable other students to understand and use it. This is demonstrated multiple times in the Diamond activity, as indicated in the commented video index of this experience included in Appendix A.
• Articulating the key steps of the modelled process in writing. Regardless of how the process under consideration has been experienced and discussed, it is critical that its main elements are recorded in writing, so as to make sure that they are fully articulated and understood, and also to provide a record that students can refer to in their independent work. This practice was an important component of the modeling of how to develop area formulas, as at the end of this experience the instructor carefully articulated and recorded the key steps of the process on large pieces of newsprint that were then posted around the room as references.

2. Making explicit the purpose of an inquiry learning experience

As mathematics students are asked to engage in unusual learning activities as part of their inquiry experiences, it is important that the teacher makes explicit the goals and rationale of such activities, so as to set the right expectations and gain students' full cooperation. More specifically, this may involve:

• Articulating the goals of the forthcoming inquiry experience. This is important at the beginning of any major inquiry experience, and especially the first few times when this is likely to present a contrast to what students have come to expect from mathematics lessons. Note that this does not mean "giving away" any of the discoveries that we hope the learners themselves will make during the inquiry, but rather involves setting up appropriate expectations about the kind of learning experience that is going to take place and explaining its significance. For example, as the Tessellation experience was introduced to the teachers participating in our Summer Institute, the instructor was very explicit about its nature, role and rationale, so as to prepare the participants for an experience likely to challenge their expectations about the role of learners of mathematics.
• Articulating the goals of specific activities and their role within the inquiry. As specific activities within the same unit are introduced and/or concluded, it is also important to make explicit their specific goals and rationale, as well as their connections with previous and/or forthcoming activities in the same unit. The instructors in both the Tessellation and Area experiences often did this as they moved to new segments within the unit, using this as an opportunity to both "recap" what had been done up to that point and motivate the introduction of new tasks.
• Reviewing the scope of an activity when needed. Students may need reminding of the scope of an activity in the middle of it, when the group needs help to focus and direct their efforts. For example, in the whole group discussion about what should count as a tessellation that took place towards the beginning of that inquiry experience, after a while the participants seemed to have lost track of what they were doing as they struggled with conflicting interpretations of the given definition; it was crucial for the instructor at that point to remind them of why reaching consensus on an interpretation of the definition of Tessellation was necessary so that they could proceed with developing and testing conjectures about tessellations and be able to share the results of their investigations in a meaningful way.

3. Synthesizing and reflecting on the results of an inquiry experience

We believe that students may miss the significance of what they learned as a result of engaging in inquiry experiences if they are not explicitly helped to synthesize and reflect on what they have done and accomplished. This, however, can be achieved in several complementary ways:

• Articulating key results and their significance. Occasionally, the instructor may choose to explicitly identify some of the key results the students achieved, and point to some of the implications/significance of such results. Having the instructor do so may be especially effective at the end of the learners' first inquiry experiences, so as to provide a model for this kind of synthesis. Alternatively, the instructor may want to take on this role in situations where she wishes to highlight the significance of results or connections the learners are not likely to realize on their own. This was the case, for example, in the Area experience; after the participants had shared the area formulas they had created for various stars, the instructor chose to highlight some interesting mathematical results that had been used and/or derived in these presentations, and point out their significance in mathematics.
• Providing structures to engage in systematic reflections on the inquiry. It is also important that the students learn to identify and analyze what they learned. Since this is not an easy process, learners at all levels need some structure to support them in these reflections -- especially at the beginning. There are two practices that we have found especially effective in this regard. The first one, which was named by some of our students as "a walk down memory lane," consists of asking the learners to identify all that took place in the course of an inquiry, and then examine what each event contributed to their learning -- an activity the teachers who engaged in the inquiry on Tessellation also participated in, although it is not featured in the video. This activity is often followed by the creation of a "what did I learn" list, as each learner is asked to identify what s/he learned from the experience and then share it with the rest of the group, with the goal of creating a comprehensive list of the learning opportunities offered by that experience.
• Providing prompts to help students reflect on what was done from new perspectives. Appropriate reading and writing prompts can be very effective in helping learners focus their reflections on a learning experience and look at what they have done from a new perspectives. For example, at the end of the first day of the Tessellation experience, the essay entitled "What does it mean to study a geometric figure?" was assigned to help the participants appreciate that their struggle with the definition of Tessellation had not been a waste of time, on the contrary, it illustrated the process many mathematical definitions have gone through before being "accepted" by the mathematics community. The power of such a prompt is confirmed by the participants' comments in the discussion that followed this assignment, as captured in the Tessellation video.
• Creating a written record of key results. Regardless of how the reflection/synthesis on a learning experience is generated, it is important to publicly record its results in class, so as to further articulate and validate these results and also to have a record the class can go back to later as needed. This practice is illustrated in the multiple uses made of such recordings on newsprint throughout the two professional development videos.
• Providing opportunities for individual synthesis. While we think it is crucial that the class as a whole engages in some reflection and synthesis on what they have done and learned, it is also important that individual students are given the opportunity to make personal sense of their learning experiences. Writing assignments that require each learner to synthesize what s/he learned from a group or class activity are invaluable to help them further understand the outcomes of group work or class discussions and enable them to make use of what was learned in future activities -- as illustrated once again by the writing assignment given after the discussion of "what should count as a tessellation."

4. Orchestrating and facilitating students' inquiry when working in small groups

Inquiry experiences often involve students working in small groups. How can this group work be made most engaging and productive? In this, as well as the next three categories, we have found it helpful to further divide the teaching practices we recommend into two major groups. Practices within the first group are suggested to help teachers orchestrate the learning experience, so that it is most conducive to inquiry and learning. The second group of practices, instead, provides ideas about how teachers can facilitate the student inquiry as it takes place in the classroom, so that it is most productive.

Strategies to orchestrate group work conducive to inquiry and learning:

• Designing engaging tasks for group work. First of all, we believe it is crucial that the task the groups are asked to work on be sufficiently well defined, and yet open-ended and challenging enough to stimulate genuine inquiry and engage everyone's interest and participation. For example, the first group activity in the Tessellation experience, where participants examined the examples of Tessellation they had collected in order to decide whether they did or did not meet the given definition of tessellation, was very engaging; this was due to the fact that the task was very concrete, and the participants' generated artifacts were more controversial than they expected, thus raising considerable debate within each group.
• Providing adequate directions for group work. Students' engagement in the task as intended will also depend on their understanding of the task itself. Thus, the assigned task will always need to be clearly articulated and explained by the teacher. An oral explanation at the beginning of the activity, however, is usually not enough -- as some students may forget the details by the time they get involved in the group work. Therefore, we recommend that teachers also articulate the task in writing, and make this description accessible to students throughout the group activity -- either by posting it on the board or through a hand-out distributed to all students (as we did, for example, in the case of the group project to develop the area of a given star).
• Providing group members with opportunities for individual thinking. Often group work is most effective when its members are given the opportunity to think independently about the task, and thus produce individual results and ideas that can then be shared and discussed with the rest of the group. The success of the activity of examining "potential" examples of tessellations, for instance, can also be attributed in part to the fact that each participant had already given some thought to what should count as a tessellation, as a result of engaging in the preliminary homework assignment of finding at least two examples satisfying the given definition. It is important to note, however, that opportunities for individual thinking do not always have to happen prior to the group work. For example, in the Area experience participants worked individually on the task of developing an area formula for their assigned star only after the group had a chance to brainstorm ideas and possible approaches; these preliminary conversations helped the group set productive tasks for each group member and minimized frustration.
• Assigning students to appropriate groups. The decision of how to group students is also an important one, and it is closely linked with the nature of the task as well as the teacher's instructional goals for that specific activity. Teachers will need to decide both the ideal number of people in each group, and criteria for group composition. For example, in the Tessellation experience we grouped the participants heterogeneously, because we expected everyone to be rather unfamiliar with this topic and we wanted them to realize the contributions that could be provided by people with different learning strengths, regardless of their mathematical background. Conversely, more homogeneous groups comprised of only mathematics teachers, or special education and elementary teachers, were formed for the task of creating an area formula for the various stars. This decision was motivated by the realization that elementary and special education teachers might be intimidated by math specialists working in the same group and "defer the thinking" to them -- when we knew they were equally capable of reaching very interesting and creative results! Also note that both of these situations involved groups of 3-5 people, as we felt that each group would benefit from different perspectives and ideas, and sharing would be made more manageable with only 5-6 groups presenting.

Strategies to facilitate group work as it takes place in the classroom:

• Monitoring group work. While the previous examples highlighted important elements to take into consideration when planning for small group work, the role of the teacher does not stop here. First of all, a teacher needs to know what the students are doing. Thus, moving around the room -- eavesdropping and occasionally asking students what they are doing -- is very important as it can help the teacher get a sense of whether some clarifications of the task are needed, what different groups are doing and thus also which groups may need some help or additional prompts. In our professional development videos you can often see the facilitators doing just that as the groups are working.
• Helping groups articulate where they are. While moving around the groups, the instructor may choose to use her request for an update on the group work as an opportunity to engage the group in better articulating and clarifying for themselves what they have been doing, We believe that doing this can enable some members to "catch up" with the rest of the group, if needed, and also help the whole group move forward with their task. This is what happened, for example, in one of the groups exploring their own conjectures about tessellations featured in the video segment discussed in the commentary reported in Appendix B.
• Providing feedback without taking away from the group's inquiry. The information gathered while moving around the groups may enable the teacher to identify difficulties, or even mistakes, encountered by specific groups. In each of these occasions, the teacher will need to decide whether to intervene somehow or let the group recognize the problem and address it on their own -- in the interest of letting them develop strategies to deal with this kind of problem and thus eventually increasing their confidence and ability to work independently. An example of choosing the latter option is offered in the Area video, when the instructor stopped by the group working on developing an area formula for the 6-pointed star and realized they had developed an incorrect formula; since the group had just begun to work on this task, here the instructor chose to show her doubts about the formula they had developed, but then left the group without further explanations or suggestions. Further example of this kind of practice are offered in the video segment discussed in Appendix B.
• Intervening as necessary to support the group work. Although the instructor will generally try to let the learners work independently in their small groups, there are occasions when it may be useful, and even necessary, for the instructor to take a more active role -- so as to help the participants get over an impasse they do not seem to be able to resolve on their own, move in more productive directions, or consider alternative approaches. Depending on the situation, this kind of intervention can take on different forms, as illustrated once again throughout the video segment discussed in Appendix B.

5. Orchestrating and facilitating students' inquiry when working individually

Although the inquiries featured in our Tessellation and Area videos were all conducted within small or large group settings (due to time constraints as well as the main goals of the experiences as learners within the professional development program), mathematical inquiries are also conducted by students in the context of individual projects. Making such projects most conducive to inquiry and learning requires variations of some of the strategies identified earlier for small group settings, as discussed below:

Strategies to orchestrate individual projects conducive to inquiry and learning:

• Designing engaging tasks for individual work. Once again, students' engagement in individual inquiries, and what they will learn from these experiences, will depend to a great extent on the nature of the task assigned to them. As discussed in the case of tasks undertaken by small groups, individual projects, too, will need to be challenging and open-ended enough to stimulate genuine interest and exploration. In addition, we believe that teachers should take advantage of the flexibility offered by individual projects to provide each student with choices and the opportunity to "customize" the project to their own interests and background. Individual projects also offer teachers the opportunity to easily make adjustments in the assigned task, so as to respond to the specific level of ability, learning needs or gaps in mathematical background of different students.
• Providing adequate directions for individual work. Since students are expected to work independently on their projects, and the projects are likely to be individualized, a clear written description of each project is crucial to direct each student's work. At the same time, it is important to remember that some students -- especially young ones -- may have difficulties understanding written directions, and may need additional oral explanations and support from the teacher in order to make sense of them.
• Providing opportunities for peer sharing and feedback. Working independently on a project does not mean that students need to conduct their inquiries in total isolation. To the contrary, we recommend that students be given opportunities in class to occasionally share their progress and problems with a partner or in small groups, so that they can benefit from feedback and ideas.

Strategies to facilitate productive individual projects:

• Monitoring individual progress. Just as we argued in the case of group work, teachers need to monitor what each student is doing in their independent projects, so as to determine whether the student is making sufficient progress and/or needs some support to redirect his/her work in more productive directions. Logistically, this may not be easy to do with an entire class, especially when each student is working on a somewhat different project! Providing some class time when students can work on their project, and/or discuss it with peers, can give the teacher the opportunity to get a sense of what everyone is doing while circulating in the room. Asking students to turn in their written work for feedback at critical points in the project is also a possible solution, although its effectiveness will depend on the nature of the task and the students' facility with writing.
• Providing feedback without taking away from the student's inquiry. Once again, as teachers monitor individual work they may realize difficulties and mistakes that require feedback. In each case they will need to decide if and how to provide this feedback, so as to maximize each student's learning without generating too much frustration.
• Intervening as needed to maximize individual learning. There may also be cases when students need additional support from the teacher in order to proceed with their project. Once again, teachers will have to decide in each case how much intervention is needed and how to provide it so that it does not take away from the student's own inquiry and learning. Occasionally, the support may also be provided by a peer or through some suggested readings. Logistically, this intervention could take place either during the class time assigned to individual work or out of class.

6. Orchestrating and facilitating "sharing" sessions where students communicate the results of their inquiries

We have chosen to discuss as a separate category how teachers can orchestrate and facilitate the sharing of individual and small group results with the rest of the class, given the importance of this kind of communication for successful inquiries and the concerns many teachers have voiced regarding their students' reluctance to listen to each other and/or their inability to communicate effectively.

Strategies to orchestrate productive sharing sessions:

• Developing a common interest in the results to be shared. A first prerequisite for successful sharing is for all the participants to genuinely want to hear from each other and believe that they can gain from such an exchange. This can naturally happen when the learners have all been working on the same open-ended and challenging task -- as illustrated in the presentations where participants shared the alternative approaches they had used to find the area of the "fish" (the video segment we chose to examine in more depth in Appendix C). Note that the interest, in this case, was generated by the fact that there were many possible solutions to the task, so every participant was intrigued by seeing strategies they had not thought about -- in contrast, think how boring it would be (for students and teacher alike) to listen to the same kind of explanation over and over again! Developing a common interest, however, does not necessarily mean giving every individual or group the same task. The same result can also be achieved whenever the class is working towards a common goal, with individuals and/or groups pursuing complementary aspects of the investigation. In this case, individual and/or group results are perceived as potential contributions to the class inquiry, to be carefully and critically examined by everyone in the community to decide whether they can move the class, as a learning community, towards their shared goal. This situation is well illustrated by the activity of developing an area formula for "regular stars." Here each group worked on developing the formula for a star with a specific number of points but, based on the presentations of the results of this work, the class was then able to propose an area formula for a generic "n-point regular star." As evident in the video, the audience was extremely interested and participated quite actively in these presentations!
• Encouraging the use of artifacts to support one's presentation. Presenting results in a clear and concise manner is not easy for school students and adults alike, especially when dealing with technical content. Unclear or unfocused presentations, in turn, are likely to cause the audience to get distracted, and even disruptive. We have found that asking students to prepare some kind of visual support to accompany their presentations can be very helpful for both presenters and audience. First, the task of creating the artifact forces the presenter(s) to identify their key results and to articulate them in a clear and concise manner. This may be especially important in group presentations, where the various group members may need to synthesize individual results in a coherent way before they can even decide on the role each member is going to play in the presentation. Second, having a visual to refer to may both support the presenter and help students in the audience keep their attention focused during the presentation. These multiple benefits of using artifacts to support presentations are demonstrated by the use of "posters" made in all the major presentations featured in the Tessellation and Area videos, and even by the instructor's choice in the Fish activity to ask each presenter to draw his/her solution approach on the overhead projector for everyone to see.
• Providing the opportunity to first share with a partner. Some students may feel intimidated (at least initially) when asked to share the tentative results of their inquiries in front of the class. To address this problem, we found it effective to occasionally ask each student, before s/he is called to report to the rest of the class, to share, verify and elaborate her/his results with a partner. This set up -- often refered to as "think-pair-share" -- also has the additional advantage of allowing every student the chance to share and elaborate on their thinking, even when time constraints and concerns about repetition may make it impossible for every pair to present to the class. The activity of computing the area of the fish provides a good illustration of how this "think-pair-share" structure can work.

Strategies to facilitate productive sharing sessions:

• Orchestrating a good sequence of presentations. The success of a sharing session may also depend on the order in which specific results are shared -- for example, in many cases it is advantageous to discuss the most "obvious" solutions first, so that students who came up with them do not feel discouraged from sharing after more sophisticated solutions have been presented. The instructor may gain a sense of whether this may be the case earlier on, when circulating around the groups, and then decide whether groups should be asked to share in a specific order, or volunteers could be called upon at least at the beginning. Some of these decisions may need to take place during the presentation itself, as illustrated in the case of the sharing that took place in the context of the Fish activity.
• Helping students articulate and elaborate upon their results in a presentation. As mentioned earlier, presenting mathematical results and processes is quite challenging, and students need support in order to learn to do so effectively. At the same time, teachers have the responsibility of making the presentations understandable so that all the other students in the audience can benefit from them. There are multiple ways teachers can try to achieve these goals while the presentations are taking place: by asking clarifying questions that can help the presenter better explain some confusing point; by rephrasing what the presenter has just said in a way that can be more easily understood by the audience, and could also model a better way to express the same point (ex: by using some example, or more precise terminology); by pointing out some weaknesses or contradictions in what was said and inviting the presenter to think about how they could be resolved; by asking questions to push the presenter's thinking further by suggesting a different perspective to look at the problem, or highlighting implications or connections s/he may not have thought about. Illustrations of all of these complementary strategies can be found in the Area video, as the instructor facilitated the sharing of participants' approaches to computing the area of the fish -- as identified in the commented video index in Appendix C.
• Encouraging audience participation in the presentation. Presentations are most successful for both presenter and audience when they involve the audience in actively making sense of what is presented. Having a common interest in the content presented is certainly a prerequisite for high audience participation, and the use of artifacts as a visual support can also contribute to keeping the audience engaged during the presentations and inviting them to ask questions. In addition, the teacher can try to encourage audience participation as the presentations are taking place, by: reminding the audience at the beginning that they are encouraged to ask questions and make public possible connections with their own work; turning some questions to the audience (rather than always addressing them to the presenter); inviting someone in the audience to contribute insights from their own results that may shed light on the issue presented. Illustrations of how the instructor tried to take advantage of these strategies are especially evident in the group presentations on the area formulas developed for given stars.
• Building upon what students have shared. Although teachers have to be careful that the presenters' voice remains central, this does not mean that they have to abstain from contributing to the conversation. On the contrary, it is important that the teacher occasionally validates what is shared by commenting on the significance of some contributions and their implications. The teacher can also help students expand on what they learned from the presentations by elaborating on some results and by making explicit connections between different contributions, or between some of the results presented and what they learned in other parts of the course. The instructor's interventions during and after the sharing in the Fish activity are good illustrations to this regard -- as indicated in the commented video index.

7. Orchestrating and facilitating students' inquiry when working as a whole class

Mathematics students will often engage in inquiry activities that involve the whole class. Worthwhile discussions and other kinds of whole class learning experiences will need to be carefully planned and facilitated by the teacher -- and this may require skills and strategies that are quite different from those used to manage more traditional classroom activities. Some of the teaching practices we suggested to facilitate group work and sharing sessions apply in this context as well, although with some variations, while others are more unique to this setting. We believe that the discussion of "what should count as a tessellation" that took place at the beginning of the Tessellation video can be considered a prototypical example of "whole class activities informed by inquiry;" therefore, we chose to comment on this episode in more detail in Appendix D and to draw from it most of our examples for the following discussion.

Strategies to orchestrate whole class activities conducive to inquiry and learning:

• Choosing engaging topics for whole class activities. In order to have a class genuinely engage in mathematical discussions, the topic they are working on needs to be sufficiently controversial to generate different viewpoints that can then be shared, examined and debated -- as was obviously the case in the discussion of "what should count as a tessellation," when the participants found themselves in disagreement about the interpretation of the definition of Tessellation originally given to them.
• Preparing the ground to raise genuine debate. At the same time, it is important to realize that, given the way mathematics is usually experienced in school, most students are not ready to immediately see mathematical topics as debatable. In most cases, preliminary experiences will need to be developed so as to generate doubt and alternative solutions that can then create the need to engage in discussions and come to some resolution as a whole class. This, in turn, will also provide individual students the opportunity to do some "pre-thinking" and thus have more to contribute in the class discussion later on. In the case of the discussion of "what should count as a tessellation," both the preliminary assignment of finding examples based on a given definition of Tessellation, and then the small group activity where these examples were first examined, were carefully planned so as to generate the controversy that made the following discussion so lively.

Strategies to facilitate productive whole class activities:

• Moderating class discussions. The first thing inquiry teachers need to learn when facilitating a discussion is to "stand back" and let students talk to each other (rather than address the teacher). This practice is well illustrated by the instructor in the Tessellation video, as many times during the discussion of "what should count as a tessellation" she chose to stay silent and let the students respond to each others' suggestions and arguments, thus enabling them to develop a truly collaborative form of discourse. At the same time, the teacher still plays an important role as the moderator of the discussion, making sure that everybody has a voice and is given a fair chance of contributing -- as illustrated towards the end of the same discussion, when the instructor had to intervene to ensure that some participants were not overpowering others and that everyone's voice was heard.
• Helping the class stay focused on the topic of a discussion. Given the open-ended nature of most inquiries and the possibilities for side-tracking, another important role of the teacher in class discussions is to monitor that the conversation remains focused and productive. This may require the teacher to stop occasionally the on-going conversation and remind the group of the goals that the discussion was intended to achieve -- as illustrated by the instructor in the Tessellation experience when she stopped to summarize the issues the group had raised and pointed out why it was important to resolve those issues before they could meaningfully proceed with their inquiry into tessellations.
• Dealing with unexpected questions within a class discussion. No matter how carefully a discussion has been planned, there will always be some questions or contributions that will take the teacher by surprise. Each time, the teacher will then have to decide whether the question/comment needs to be addressed immediately, or if it should be acknowledged but left aside for a while so as not to lose focus or momentum. We have examples of both kinds of decisions even within the discussion about "what should count as a tessellation." For instance, towards the beginning of the discussion, as the group was debating the "border issue," one of the participants (Laurie) suggested they modify the definition to address the role of color; in this case, the instructor chose to record this issue on newsprint to be discussed later, so that they could resolve the issue under discussion first. Note, however, that the instructor made this decision and its rationale public, to make sure that the participant did not feel slighted and thus discouraged from asking further questions. In contrast, towards the end of the discussion the instructor decided to respond immediately to Laurie's comment that the difficulty the group had encountered in agreeing on a definition of tessellation was due to differences in the participants' math background, since she thought that it would be important to dispel this misconception right away and that discussing this issue would help all participants better appreciate the significance of the activity they had just engaged in.
• Providing feedback without taking away from the class inquiry. The instructor's role to provide feedback to the learners' thinking and inquiry is as important in class discussions as it is when students work individually or in small groups. And in this setting, too, the greatest challenge for teachers is to choose when and how to provide this feedback without interfering with the students' own struggles and yet avoiding too much frustration. The commented video index of the discussion of "what counts as a tessellation" identifies and comments on a few situations, making explicit the rationale for the teacher's decision in each case.
• Helping students articulate and elaborate upon their contributions to a class discussion. As already argued in the context of facilitating sharing sessions, the teacher should also help students learn to clearly articulate and elaborate upon their thoughts during classroom discussions. The same strategies identified earlier to support students' sharing can be effective in classroom discussions as well -- as illustrated by what the teacher did in the discussion on "what should count as a tessellation" to encourage participants to make their arguments or counter-examples clear to others. Especially significant to this regard is the exchange where the instructor asked a participant (Sharon) to go to the board and draw the counterexample she was talking about. The participant's points were much clearer after she was able to refer to her drawing and they provided a turning point in the discussion.
• Building on students' contributions in a class discussion. Similarly, once again we want to highlight the instructor's role in elaborating on students' contributions by rephrasing and/or expanding on them, as well as making connections and deriving implications -- as indicated in various points in the commented video index in Appendix D.
• Highlighting controversy and the need to address/resolve it. Throughout discussions and other whole class activities, the teacher will also need to continuously monitor the conversation to make sure that the students recognize and capitalize on the controversy she has tried to develop. At times, this may require the instructor to make some pointed comments that highlight contradictory positions stated by different students, or even to play "devil's advocate" to what some student said -- a practice illustrated throughout the discussion of "what should count as a tessellation."
• Providing opportunities for individual thinking within a class activity. Teachers also need to recognize times when individual learners may need to work out on their own the meaning and implications of some questions or contributions. In those cases, the teacher may call a halt in the class discussion or activity, and give everyone the opportunity to think through the specific issue under consideration -- either in class or at home. Depending on the nature of the issue examined, this individual thinking could take different forms, such as trying to solve the specifical mathematical problem that has been posed, beginning to brainstorm ideas with a partner, or doing some free-writing as a way to generate and organize one's thoughts on the matter. Although the discussion of "what should count as a tessellation" did not call for this kind of scaffolding (mostly because participants had been given the opportunity to do this individual thinking prior to the class discussion, through the preliminary assignment and the small group activity), a good example of this practice is offered in the Diamond activity, where the instructor stopped the class discussion a few times to allow participants to work out on their own some of the results that had been suggested.
• Recording key points raised during the class activity. We have found it very effective to have the instructor record on newsprint key points made during a whoel class activity. This kind of on-going recording can foster the need to identify and clearly articulate key contributions, helps everyone keep focused, and provides artifacts that can be revisited at various points during the same discussion or later activities -- as illustrated throughout all the discussions taking place in the Area and Tessellation videos. This kind of recording is especially important in discussions that take place over multiple sessions, as it provides the means to revisit what was done earlier and facilitates a final synthesis of the results of the discussion.

8. Responding to the learning needs of diverse students in inquiry math classes

The education community is becoming increasingly aware of the variety of learning styles, as well as mathematical background and ability, that can be found in a mathematics classroom. Rather than try to eliminate such diversity, inquiry teachers should strive to orchestrate learning experiences that can meet different learning needs. The instructors in the inquiry experiences featured in our professional development videos, too, had to deal with considerable diversity in mathematical background and learning styles, as participants included both teachers certified to teach mathematics in grade 7-12 (most of whom had earned the correspondent of a Bachelors degree in mathematics) and special education and elementary teachers (who instead generally had received very limited training in mathematics). This situation, in turn, enabled us to "model" the strategies listed below:

• Assigning tasks that allow for multiple solutions and approaches. As argued in Section C.1 of the main text, tasks that allow for more than one right answer and solution approach are likely to provide more opportunities for success for students with different abilities and learning styles -- as well illustrated in the "fish" activity within the Area unit, where the most creative solutions were offered by non-math specialists!
• Allowing learners to go at their own pace. At the same time, it is also important to realize that students with different learning styles and abilities may require a different amount of time to successfully complete a given task. Although time constraints usually do not allow teachers for much flexibility within a class period, homework assignments and time after school could be judiciously used to overcome these limitations. This is why, for example, we carefully timed the Area inquiry experience over two days, so as to give participants the opportunity to take the necessary time at home to continue to work on developing the area formula for the star assigned to their group.
• Providing opportunities to use different learning modalities during the same lesson. Research on learning has made us aware that students have different strengths and weaknesses with respect to auditory, visual or kinestetic modes of learning. In order to maximize learning for all students in a mathematics class, therefore, teachers should try as much as possible to provide opportunities to use each of these modalities within each lesson. We tried to apply this principle in our own inquiries on Tessellation and Area. Although presentations and discussions were used most extensively thoughout these learning experiences, we also made manipulatives available and encouraged participants to use them as needed to complete their assigned tasks, and consistently recorded directions and key results on newsprint so as to provide a visual support to the on-going activities.
• Modifying tasks as necessary for students with specific learning disabilities. While the previous points have identified strategies that teachers can use to orchestrate their inquiry math classes so that they are more conducive to learning for all students, teachers may also need to modify some tasks and expectations for individual students with specific learning needs or disabilities. These modifications, for example, could take the form of substituting a written report with an oral presentation for students with writing disabilities, or allowing students with attention difficulties to move around the room if that helps them stay focused -- as it was the case for one of the teachers who participated in our Tessellation experience! Since each LD student has different strengths and weaknesses, these modifications will require a good understanding of the students' unique needs and thus would greatly benefit from the input of a special education teacher.
• Scaffolding tasks for students with a more limited math background. While the modifications discussed in the previous point had to do with students' learning styles, students in the same mathematics classroom may also present differences in terms of their mathematics background and abilities. Therefore, there may be cases when the teacher may find it necessary to modify the assigned task to make it more accessible to some students in the class. This can often be done almost "invisibly" by providing those students with modified written directions and/or some additional hints -- as illustrated in the case of the "star" project, where the groups composed of non-math specialists were given figures of their stars with construction lines that suggested possible ways of breaking these figures into simpler ones.

Summary

The teaching practices identified and discussed in this essay (see Figure 2 for a complete list) make clear that teaching mathematics through inquiry requires a lot of planning and careful orchestration from the part of the teacher. Far from "sitting back" and simply letting students go on their own, inquiry math teachers need to think very carefully about how to structure and facilitate activities so that they can maximize students' opportunities for genuine inquiry and learning.

We hope that the list developed in this essay, along with the concrete illustrations we have provided through the selected video segments and their commented indexes (reported in Appendixes A to D), may contribute to teachers' awareness of the importance of these teaching strategies and hopefully enrich their repertoire of such practices.

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."

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."

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."

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."

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