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DRAFT--August 23, 1993

Note: This document is an adaptation of an earlier version of Chapter 10 in Borasi, R. (1996). Reconceiving mathematics instruction: A focus on errors. Norwood, NJ: Ablex

The goal of this document is to articulate some fundamental principles and beliefs about mathematics education that have informed this project -- and, more specifically, the illustrative units that we have designed and you are now asked to refine and implement.

More specifically, in this document I have tried to explicitly address what characterizes a learning environment supportive of student inquiry and how mathematics teachers could be prepared and supported in their efforts to establish such an environment in their classes. This will involve first of all a discussion of how the proposed approach assumes views of mathematics, learning and teaching that are quite different from those informing traditional mathematics instruction. Secondly, I will highlight some major changes in terms of curriculum choices, evaluation, classroom dynamics and social norms that seem necessary in order to create a learning environment supportive of student inquiry. Finally, I will make explicit some of the principles that have informed the design of the professional development initiative in which you are now involved.

1) The assumptions informing an "inquiry approach" to mathematics instruction

1.1) Rethinking the nature of mathematics

Most people perceive mathematics as the "discipline of certainty" par-excellence and, consequently, associate the ideals of objectivity, absolute truth and rigor to mathematics. An inquiry approach, on the contrary, is informed by the belief that mathematics, as all other products of human activity, is a humanistic discipline. That is, mathematical results are not pre-determined but rather socially constructed. Thus, their "truth" will depend on a number of factors including, besides logical coherence, the context of application, the criteria established by the mathematical community, and even to a certain extent personal values and judgements. Once it is accepted that mathematicians strive to reduce uncertainty without the expectation of ever totally eliminating it, ambiguity and limitations become integral and dynamic components of mathematical activity.

Let me expand on some of the above points in light of the Tesselation activities that you have experienced during the Summer Institute:

• Mathematical results are not pre-determined but rather constructed: Your own experience developing, testing and refining conjectures about Tessellation is likely to have make you doubt that mathematical results are actually "discovered" in a straightforward manner, as the "neat" and organized way they are now reported in most textbooks or lectures may lead many students to believe. Indeed, a critical look at the history of mathematics would confirm such an impression (see Kline's Mathematics: The Loss of Certainty (1980), for a wealth of interesting historical examples supporting this claim). The troubled histories of some fundamental mathematical topics (such as the calculus and infinity, for example) have revealed the centuries of intellectual struggle that were sometimes needed to produce even fundamental mathematical results. The debates and controversies that characterized the development of topics such as infinity and non-Euclidean geometries have also shown that mathematicians have occasionally proposed alternative yet legitimate solutions to mathematical problems (as we have just experienced when finding alternative definitions of "tessellation" in different mathematical texts!). Thus, one has to realize that there is not just one unique way in which mathematics could develop. Rather, the mathematics community has occasionally to make choices among possible alternatives. These choices, while not pre-determined, are far from being random either, as they are always guided by the consideration of potential coherence with the existing system as well as of the potential benefits that could be derived from each alternative (for example, what definition of "tessellation" seems more rich in inviting interesting mathematical conjectures to explore).
• Mathematical truth is not absolute but may change in time: Once one realizes that mathematical results are the product of human construction rather than of the gradual discovery of a pre-determined system, one has also to accept that they are fallible just like any other product of human activity. Indeed, the construction of mathematical results often consists of an iterative process producing increasingly refined results, each of which can be assumed as "true for the time being", i.e. until it is disproved and revised. If one abandons the hope of determining mathematical truth absolutely, it necessarily follows that mathematical results can only be sanctioned by a community of practice (the mathematical community of the time) on the basis of agreed upon criteria as well as of the existing mathematical knowledge.
• Mathematical truth depends on the context of application: Whenever alternative systems or solutions are logically possible and plausible in mathematics, their use in a specific situation will require the consideration of the context of application. For example, we ourselves have experienced this in the context of tessellation, when we realized that the decision of whether a specific example should or not be considered a tessellation depended essentially on the definition of tessellation assumed.
• Ambiguity and limitations are an integral part of mathematics: The examples developed in the above points have provided supporting evidence to this regard in various ways.

The more relativistic, contextualized and socially constructed view of mathematics that results from the above considerations has the potential to affect not only the work of a small elite of professional mathematicians and philosophers, but also the everyday mathematical experience of non-specialists. Indeed, the humanistic elements of mathematics discussed in this section affect all areas of mathematics, including "elementary" ones such as arithmetic and geometry. Appreciating these humanistic aspects of mathematics, in turn, could affect mathematics students at all levels in a number of complementary ways.

First of all, the recognition of limitations and "human" elements in the discipline could make it more attractive to people who have so far been intimidated by the absolute and authoritarian image of mathematics currently presented in schools -- especially women and some ethnic minorities. Furthermore, such a recognition could challenge some common yet dysfunctional expectations about learning mathematics that have proved unsuccessful and invite students, instead, to realize that doing mathematics requires not only good technical knowledge, but also the ability to take into account the context in which one is operating, the purpose of the activity, the possibility of alternative solutions, and also personal values and opinions that can affect one's decisions.

An appreciation that mathematics as a discipline is not totally objective and pre-determined, but rather is influenced by economic, cultural and even political agendas -- just like any other human domain -- should also make any mathematics educator question the choices made so far about what mathematics should be covered in the pre-college curriculum and open to consider the possibility and legitimacy of curriculum choices alternative to the pre-established syllabus of their courses.

Most importantly, once we accept that mathematics is a social construct, the view of "knowing as inquiry" proposed by philosophers such as Dewey and Peirce can help us appreciate that the uncertainty that permeates the discipline should be perceived as a positive element rather than a limitation. The presence of ambiguity, limitations and unavoidable errors in mathematics, revealed by the previous analysis, should be thus recognized as a major force for that can stimulate doubt and, consequently, inquiry, on the part not only of mathematics researchers, but users and students as well (as we ourselves experienced in our first tessellation activity).

1.2) Rethinking the nature of learning mathematics

Another fundamental assumption of an inquiry approach to mathematics education is that learning should be seen as a generative process of meaning-making from the part of each student (as articulated in more depth in the chapter we read by Schoenfeld [1988]). It is also important to realize that such a process is often stimulated by some perceived disequilibrium and involves making sense of situations and problems in lights of the available data as well as one's previous knowledge and by building on social interactions.

Let me comment more specifically on the some fundamental aspects of such a view of learning mathematics:

• Learning mathematics as making-sense: A characterizing element of our exploration of tessellation was that throughout the activity we, as the students, were engaged in trying to make sense of some mathematical phenomenon as we had to make use of what we already knew about mathematics, even in areas that may not have seemed immediately related to the issue in question, as well as information specific to the situation under study, to come to some satisfactory resolution. It is also worth pointing out that in several cases our "making sense" involved generating and pursuing new questions.
• Learning mathematics as stimulated by anomalies: If you think back at the discussion that occurred in your small group as you were generating and exploring some conjectures about tessellation, you will realize that in most cases the "catalyst" for the activity was created by something puzzling, that did not make sense -- in other words, an anomaly. It seems indeed crucial that inquiry-based mathematics lessons involve situations which are sufficiently "problematic" and open-ended to stimulate curiosity and plausible alternative resolutions and, thus, engage students in genuine debate, meaning- making and learning.
• Learning mathematics as a social activity: While each student learning mathematics has to make-sense and construct his/her own understanding of mathematical concepts, problems or situations, this activity should not be conceived as occurring in isolation. Rather, our experiences with tessellation during the Summer Institute have shown how social interaction is a crucial component of this process. Remember how each of us seemed to benefit considerably from the other group members as we were forced to articulate our solutions and procedures, listen to other people's results as well as feedback on our own work, provide justifications for and/or revise our results when challenged by a peer, compare and evaluate alternative solutions proposed by different people, put together and/or elaborate partial individual contributions, and even reflect on the process as well as the product of such an activity and its significance.

I would also like to point out that the "mathematical inquiries" we engaged in as we explored tessellation were characterized by the fact that:

• the issue(s) addressed were sufficiently open-ended and controversial to allow for the generation of plausible alternative solutions and of genuine debate around them;
• the problems or issues discussed were not always set by the teacher and, at the very least, the students had a role in determining the scope of the inquiry (i.e., the specific questions and/or conjectures to be explored), how the inquiry should develop and when it could be considered satisfactorily concluded;
• "digressions" from the original "planned activity" were welcomed and encouraged;
• the students were expected to monitor and justify their mathematical activity and results;
• the students were expected to communicate their results convincingly to an interested audience (consisting at the very least of the other members of the class).

1.3) Rethinking the nature of teaching mathematics

As it is to be expected, living by the assumptions about mathematics and learning articulated so far will also affect the nature of the teacher's role and behavior. When looked at from the point of view of the teacher, the tessellation experience we shared indeed provide some insights on the meaning of reconceiving teaching mathematics as stimulating and supporting the students' own inquiries within a conducive learning environment. More specifically, in what follows I will try to identify some changes in teacher's role that most distinguish such an approach from traditional mathematics instruction:

• The teacher's role can be described better as "facilitator" rather than "instructor": In all the tessellation activities developed in the Summer Institute, the instructors very rarely "provided" us directly with information, either in the form of explanations or demonstrations. Rather, in this case the instructors' main task has been to design mathematically rich and thought-provoking activities that would raise questions and engage the students actively in inquiry and meaning-making. Yet, while these activities took place, the instructors still played an important albeit non-traditional role, whether in the context of whole class instruction or small group work, as they had to monitor the development of the activity and make decisions about how to best proceed after each stage, provide support to individual students as needed and orchestrate the sharing and discussions of results. It is important to appreciate that assuming such a "facilitator" role imposes much greater demands on the teacher than the traditional one, despite the fact that the teacher takes on a less central position in the classroom.
• Planning is not relinquished though it takes on a very different form: As you can imagine, the tessellation activities you experienced as a student were the result of careful planning on the part of the instructors (even if in some cases the lesson might have deviated somewhat from the original plan!). Thoughtful planning is indeed a necessary prerequisite for success within an inquiry-based classroom, even if here the teacher is much less in control of the class agenda than in traditional classroom instruction, since s/he must always be ready to adjust the original plan in order to respond to her/his students' results and decisions. Within an inquiry approach the teacher is first of all responsible to come up with an initial question, issue, problem or situation that is sufficiently rich and interesting to stimulate student inquiry. Materials and activities that can help structure and stimulate such inquiry also need to be generated and tentatively structured in advance, so as to be available as options when needed to support specific students' explorations. While the development of genuine student inquiry can never be fully predicted, an experienced teacher can do much to foresee possible directions in which inquiry on a certain topic is likely to develop with her students and prepare supporting material accordingly. At the same time, the instructor must be always ready to relinquish some of the activities planned if the students' inquiry moves in different and potentially more productive directions and, more generally, expect to use only a fraction of the ideas and materials developed in advance while continuing to develop new ones in response to the results of the students' work and decisions.
• The teacher's role in establishing compatible beliefs and social norms in the classroom: A successful implementation of an inquiry approach requires students to develop and live by a set of expectations about school mathematics different from the one governing traditional mathematics classes. Teachers should not expect that such a switch will occur spontaneously as inquiry-based activities are first introduced in a mathematics classroom. Rather, explicit attention should be spent, especially at the beginning of the school year, to establish together with the students a new set of social norms that would be compatible and supportive of an inquiry approach. This could involve, first of all, some initial activities especially geared to elicit and discuss the participating students' beliefs about school mathematics. These experiences, however, should also be accompanied by on- going reflections and discussions about the process followed in any non-traditional learning activity, so as to help the students better appreciate their rationale as well as potential benefits.

2) Major implications of adopting an inquiry approach to school mathematics

2.1) Curriculum goals and choices

The emphasis of "process over product" characteristic of an inquiry approach, along with the open-ended nature of the process of inquiry itself, requires first of all a very flexible mathematics curriculum. Ideally, within inquiry- based units students and teachers should feel free to pursue questions till a satisfactory resolution is achieved and to make "digressions" whenever promising new avenues of explorations open up unexpectedly.

Yet, this should not be interpreted as if the mathematics the students encounter as they engage in inquiry could be considered irrelevant. On the contrary, I believe that the educational value of the tessellation activities we experiences lies to a great extent to the fact that the students engaged with some important mathematical concepts such as definition, angles, properties of basic geometric figures, or even geometric transformations. In fact, one could argue that because an inquiry approach requires "more time", teachers must be even more conscious of selecting situations and topics for inquiry that are mathematically sound and valuable. The recommendations provided by the NCTM Evaluation and Curriculum Standards (NCTM, 1989) could provide some valuable guidelines to evaluate such choices, though I would like to warn against the danger of considering them as another "mandated curriculum to be covered". In sum, I suggest that it would pay for teachers to articulate upfront their instructional objectives in terms of both some fundamental mathematical content (always keeping in mind that in this case "more is less") and some processes that one would like the students to have mastered by the end of the course.

Reconceiving the goals of mathematics courses in this way will affect not only the content and organization of the curriculum for the course as a whole, but also how time is distributed among various activities and routines in everyday instruction. Students are very quick to realize which activities the teacher really values based on the time devoted to them in class. Thus, it would be crucial to devote considerable class time to small group as well as whole class explorations, to the sharing and discussion of the results of these activities, as well as to written as well as oral reflections on the process and its outcomes. Once again, since class time is a precious as well as finite commodity, this is likely to imply that the class time devoted to other activities such as teacher explanations, review of homework, quizzes and written exams may necessarily be reduced.

2.2) Classroom discourse and dynamics

The considerations in terms of curriculum and goals articulated in the previous point have obvious consequences on classroom organization as well as teacher's and students' behavior. While traditional classrooms are centered mostly on the teacher (who explains, demonstrates, assigns worksheets, questions students and evaluates their work), in inquiry-based classrooms the focus is shifted on the students' own activities. As mentioned earlier in this chapter, while the teacher maintains an important role in planning and monitoring classroom activities, students also have an increased responsibility and say on the nature and direction of their work, as well as on its evaluation. Furthermore, whether the students work as a whole class, in small groups or even individually on specific tasks, interaction among themselves is always an integral component of their learning experiences, as they act as a community of practice in constructing and critically examining their learning.

This emphasis on interaction and collaboration also highlights the crucial role played by communication in an inquiry-based mathematics classroom. This was well illustrated in our tessellation activities, where we as students were continuously expected to talk and listen to each others, as well as the teacher, so as to share results, provide and receive feed-back, resolve disagreements, verify the validity of procedures and conclusions, or reflect on the process we had engaged in. In sum, both the content and the partipants' role in classroom discourse within inquiry-based classrooms are very different from those characteristic of the traditional classroom, where conversations are essentially dominated by and centered on the teacher.

It is important to appreciate, once again, that the classrooms dynamics and discourses just described imply that teachers assuming an inquiry approach are going to lose the control that is powerfully, although implicitly, provided by a rigid and pre-determined lesson plan combined with an emphasis on student individual seat-work. Thus, mathematics teachers will need to develop new approaches to classroom management so as to respond to the changed relationships and routines established in classrooms informed by a spirit of inquiry.

2.3) Evaluation

The mathematics education community has become increasingly aware that reform in curriculum and teaching practices needs to go hand in hand with a compatible revision of evaluation criteria and tools in order to be really effective (as shown for example in the very conception of the NCTM Curriculum and Evaluation Standards for School Mathematics [1989]). This is especially true in the case of an inquiry approach to mathematics instruction, since currently students' measures of mathematical achievement rely heavily on multiple choice and/or standardized tests, which by their very nature value the production of exact answers to rather mechanical tasks and do not even attempt to measure the kind of learning that students may derive from engaging in genuine inquiries.

In order to support the goals and spirit of an inquiry approach, evaluation should be reconceived so as to address the acquisition not only of procedural knowledge, but also of conceptual knowledge, problem solving and posing heuristics, learning and metacognitive skills, creativity, independence and attitudes towards the discipline, and assessment tools to address all these elements designed and employed. This is an issue that we need to further explore in the context of this porject, with the help of the growing literature on mathematics assessment, based on both research and innovative practice.

2.4) Social norms and expectations

The theoretical assumptions about the nature of mathematics, learning and teaching discussed in section 10.1, along with the instructional changes identified in the previous points, have all practical implications that can challenge considerably the expectations about school mathematics that most students have developed after years of traditional schooling. Consider the following list of typical expectations identified by Schoenfeld in his summary of the results of research on students' beliefs:

• Mathematics problems have one and only one right answer.
• There is only one correct way to solve any mathematical problem -- usually the rule that the teacher has most recently demonstrated to the class.
• Ordinary students cannot expect to understand mathematics; they expect simply to memorize it and apply what they have learned mechanically and without understanding.
• Mathematics is a solitary activity, done by individuals in isolation.
• Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less.
• The mathematics learned in school has nothing to do with the real world.
• Formal proof is irrelevant to processes of discovery or invention.

(Schoenfeld, 1992, p. 359)

It is worth contrasting explicitly these beliefs with a few of the practices and expectations that characterize an inquiry approach to mathematics instruction:

• Students are expected to "make sense" of mathematical problems and situations, to "understand" their own as well as their peer's solutions and their justification;
• Students can find the solution to novel and challenging problems;
• It is worth listening to how other students have approached a problem, even when you have already reached a solution and you are convinced it is the correct one;
• A mathematical problem may take a full lesson or even more in order to be satisfactorily addressed;
• Justifying your results is an integral part of mathematical activity, and the responsibility of the student rather than the teacher;
• Discussing about mathematics as a discipline and about the process of learning mathematics is a legitimate school mathematics activity;
• It is worth to pay explicit attention to recognized errors;
• Confusion is a necessary and valuable component of learning mathematics.

Though far from complete, this list further supports the claim that an inquiry approach requires establishing a new set of social norms in the mathematics classroom. As argued earlier, the success of introducing any activity or strategy informed by an inquiry approach will be doomed to fail if the teacher does not devote explicit attention and class time to this issue.

3) Supporting teachers in implementing an inquiry approach in their classrooms

It is to be expected that implementing the radical instructional changes outlined in the previous sections will be no easy task for mathematics teachers. Formal as well as informal professional development initiatives may be needed to support interested teachers in such a challenging enterprise and to help them make it a success. The porject you are engaging in is an example.

The principles and experiences used in designing the project have been based on the results of the growing literature on educational reform and teachers' change, as well as on our previous experiences in pre-service and in-service teacher education. In what follows, I will try to make these elements and their rationale more explicit.

We believe that teachers interested in implementing innovation in their classrooms should be supported through long-term professional development initiatives that would enable them to: (1) experience as learners the kind of learning experiences that they would like to provide to their students, so as to better appreciate what these innovative learning approaches may involve and how students may react to them; (2) reflect on these experiences so as to become aware of the potential benefits and drawbacks of the proposed instructional innovation, as well as of its main instructional implications; (3) develop rich images of how the proposed instructional innovation could play out in specific instructional contexts, to be used as a concrete reference and model as they try to create similar experiences for their students; (4) find support, feedback and inspiration as they actually begin to implement the proposed instructional innovation in their classes. Let me examine each of these components more closely:

• Enabling teachers to experience the proposed strategy as learners: The experiences developed around tessellation in our Summer Institute were intended to play this role. I believe that these experiences were a crucial component of our Summer Institute, as they enabled many of the participating teachers better realize the possibility of engaging in genuine mathematical inquiry without requiring a lot of technical background or ability, as well as to recognize the educational value of these activities for mathematics students (as illustrated by the unanimous choice of starting their school year with their own version of the tessellation unit).
• Encouraging teachers reflections on the educational potential and implications of the proposed strategy: As already argued earlier in the case of students, even teachers participating as learners in mathematical inquiries may not fully recognize the potential values and implications of these experiences, unless they explicitly reflect on and discuss the process they went through, what they learned from it and their reactions to the activity. In order to achieve this goal, such reflections should be encouraged by means of written assignments as well as class discussions -- as modeled in our Summer Institute.
• Providing images of how the proposed strategy can play out in practice: While engaging as learners in mathematical inquiries may already be an important step to visualize what such activities could actually look like in the context of mathematics instruction, it may not be sufficient for many teachers. As a prerequisite to implementing an inquiry approach in their own classes, many mathematics teachers may like to "see" how specific units informed by such an approach have been designed for students of age and ability similar to those of their own students, how these students may have reacted to such experiences, how other classroom teachers actually dealt with elements such as curriculum choices and classroom discourse in these situations, and so on. Ideally, this information would be best acquired by means of direct observation, for a sustained period of time, in classrooms where such experiences are taking place. This option, however, may not be available to interested teachers in most cases. As a substitute, we have tried to provide the participants with teh detailed report of a number of experiences implemented with middle school students in a variety of instructional contexts and dealing with different mathematical topics.
• Supporting teachers as they try to implement the proposed strategy in their classes: The failure of many past attempts at educational reform (such as the "New Math" of the 1960's in the U.S.) have made teacher educators and reformers aware that in-service courses and workshops, however well designed and carried out, are rarely sufficient to enable the participating teachers to go back to their classrooms and implement the principles and techniques learned there on their own. Teachers especially need feedback and assistance as this stage, since it is just as they start incorporating new ideas into their teaching practice that they may fully realize the meaning and implications of these ideas and, at the same time, begin to seriously question some of their assumptions and prerequisites. Thus, to be really effective, a professional development initiative aiming at promoting an inquiry approach to school mathematics should include, as an essential and integral component, also some structured and supported field experiences. in our project, we have attempted to do so first of all with the creation of school-based "support groups" that would meet on a regular basis to both provide ideas and constructive criticisms on the teachers' initial plans for innovative classroom experiences and, later, share and discuss the results of implementing such plans. This initiative has been accompanied with the suggestion of a sequence of activities that could help the participant introduce the innovation in question in their classes and evaluate its outcomes (as described in the plan for the field component of the professional development program, and including supporting material to plan some of the beginning experiences -- i.e., our "unit plans" for the units on Tessellation, Remodelling and Area.)

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