Introduction
Knowledge in any field exists in the dynamic interaction of the individuals that make up that field. In other words knowledge in mathematics consists of the accepted theorems and postulates at any given time. New ideas are always being considered for acceptance or rejection to that body of knowledge. Accepted ideas are being tested to see if they need to be altered or rejected. Knowledge is not static, nor does it exist in the mind of any one individual.
In my high school, those of us that did well in math would meet in the hallway before school to discuss our math homework. We had developed a mathematics community that supported the growth of each of our individual conceptual foundations.
I propose the development of knowledge within the classroom using the same process. The teacher builds upon the current body of knowledge in the classroom through peer discussion and by providing experiences that allow the challenging of misconceptions and provides the parameters to develop new conceptions for dealing with new problems. This method has four distinct advantages over direct instruction for the development of concepts. First, the students develop and learn to trust their own ability to reason. Second, they learn that knowledge is our best effort at explaining the relationships within the world rather than a discrete set of facts. Third, the method does not emphasize one mode of learning over another so that most students are able to participate more fully than with more narrow practices. Finally, students develop a balance between individual and cooperative effort that is more conducive to learning than total independence or total dependence on the teacher.
This paper addresses the elements that are necessary to achieve a community of mathematicians in an elementary classroom. The critical elements consist of the social relationships between the students, the students’ perceived role of the teacher, the students’ concepts about what learning is and how they relate to that process, and the teachers’ understanding of mathematical processes. Before looking at these elements, I would like to share an example of a discussion in a second grade mathematics community.
New concepts are not presented by the teacher. A new concept is presented by solving a problem that demands a new concept in order to solve it effectively. For instance,
Students take one minute to think, and then begin working in pairs to find a method of solution. Once most of the pairs have an idea, they share it with another pair, explaining their method. Finally students take turns sharing different solution methods with the class. Examples of different student solutions are:
32
14
22

Thirty
minus ten is twenty;
four minus two is two; so the answer is twentytwo. 
32,
31, 30,
29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18

Start
with thirtytwo;
count back fourteen; you get eighteen. 
32 22
10
 4
22 18

Thirtytwo
minus ten is twentytwo;
twentytwo minus four more is eighteen. 
24 =
2 30
10
20
20 2
=18

Two
minus four is negative two;
thirty minus ten is twenty; twenty and negative two is two less than twenty; that is eighteen. 
Through the process of students challenging each others solutions and defending their own, they each come up with a workable solution for themselves over the course of a couple of weeks. In a normal classroom there will be several workable ways to solve the problem.
Peer Relationships
Having classroom discussions where students share their beliefs and risk exposing errors in their beliefs, relies on positive relationships between students in the classroom. They must develop a community that trusts each other enough to take the risk of looking foolish.
The necessary relationships between the members of the community will not develop spontaneously. Direct guidance is necessary to establish and maintain these relationships. First, the teacher must set a high standard for the way that students treat each other. Any name calling, teasing, putdowns, or lack of respect for each other will undermine the ability of the group to share their understanding in an open manner that is necessary for a community of knowledge to operate. Fortunately, just as within the scientific community, this relationship does not have to be perfect. However, the standard is critical. The closer the class comes to meeting the standard the better the classroom will operate.
Next, an atmosphere of cooperation toward a common goal, rather than competition for limited resources, needs to be established. Individual testing needs to be viewed as a measure of progress and guidance for future instruction. Testing should never be used to rank students. On the other hand, it is important that students do not feel that their level of understanding of a concept at a given moment of time is any more secret than who can run the fastest on the playground. Students know anyway, and the secrecy undermines the acceptance of the fact that individual differences are all right.
Students are encouraged
to challenge all ideas, at least in their own minds. No idea is to be accepted
or rejected based upon the advocates reputation. This is done in the spirit
of understanding and clarifying ideas, not of attacking the individual.
No student should back down from an idea they believe in until they understand
why that idea is incorrect. Peer pressure is not acceptable as a form of
argument in the classroom.
Students’ Perceived Role of the Teacher
In lessons designed to get to students to construct concepts, the teachers must suspend their authority concerning correctness of reasoning so that the students see reasoned discourse as the authority. In other words, the students must learn not to look to the teacher for the correct answer, or evaluation of their thinking. The teacher is following the course of reasoning, but it is for the purpose of guiding the discussion, not to give direct feedback to the students about the correct way of thinking.
The students must see themselves as responsible for constructing the solution to the challenges presented by the teacher. Rather than giving the students problems and then showing them how to solve the problem, the teacher guides the students in finding a solution that works. Students often work in small groups first and then share their solutions with the class. The members of the class challenge any ideas that they think will not work. Through this process the class will often come up with several solutions that will work. Sometimes the class does not produce a working solution. Then the teacher suggests situations that demonstrate the error in their solutions. Once in a while the class becomes satisfied with a single solution, such as "counting on" in an addition problem. Although the solution works, it is not practical with large numbers, so the teacher needs to lead the class to see the difficulty through the selection of problems given to the class.
Teachers often find difficulty in relinquishing authority concerning students' statements. The first hurdle to overcome is that students expect teachers to be the authority, to judge their own and their peers' work, so that they will know what is true. When the teacher refuses to give this feedback, young students will become indignant, but older students may become quite angry, because the teacher has upset the status quo that they worked so long to master. Teachers may need to convert slowly and carefully to this type of teaching so as to prevent the creation of too many behavior problems.
Starting with young students has an advantage because it is easier to deal with their indignation than older students’ anger. The worst experience I have had with a third grader was having him come in and inform me that I was not such a great teacher. His mom had shown him how to work the problem and now he understood how to do it. On the other hand, I have had fifth graders that will constantly pull the discussion off track in order to avoid having to think about what they know and how it relates to a new problem. By sixth grade, teachers have had open rebellion in the classroom because the students resist having to figure out the processes themselves. One sixth grade teacher had students complain to the principal that the teacher would not answer enough questions. They expected to be lead by the hand through the assignment, and resisted making any decisions on their own. In older students the patterns of dependency are well established, and hard to change.
The second hurdle is getting the students to stop trying to "read the teacher" during a challenge discourse lesson and start thinking for themselves. The teachers need to learn to control their own facial expressions and voice tones. Although some teachers can send no facial expression, many find that it is easier to purposefully send the wrong message part of the time. Once the students learn not to trust the teacher’s facial and tonal messages, then they will have to rely on their own thinking processes.
There is one caution; let the students in on what you are doing. Make it clear that your goal in using the wrong facial expressions is for them to come up with what they think and NOT what they think you think. If students are not aware of the goal of your false facial expressions, they may feel betrayed by your signals. After all they have learned to read people as a legitimate method of social interaction. We expect students to read our facial and body expressions as to when to back off of teasing or when to go along with the game the teacher is playing. Body language is an important part of communication. It just happens to get in the way in a challenge discourse lesson.
At the same time the
teacher needs to begin challenging students both when they are right and
when they are wrong. Teachers must intervene when other students don't
correct their peers' errors, but to avoid getting automatic "oh I did that
wrong" responses, the teacher needs to match those challenges with challenges
where the students are correct. This also gives the student the opportunity
to defend their views.
Students’ Concepts About What Learning Is
Students often think about learning in behavioral terms. They see learning as memorizing the "correct" response. In order to develop a mathematics community in the classroom, the students must come to understand that learning can, and in fact will, involve making mistakes. Certainly, being right feels good, and in indeed being right is the goal when memorizing needed facts, but students do not develop new concepts by being right. Mistakes help define the parameters of a concept. They are the most efficient method of finding those parameters because exact definitions of any concept are almost impossible. If you doubt this, try giving a clear, concise definition that defines the differences between a chair, bench, and a stool. Yet you all have a workable concept of each. Young children have already learned to set the parameters for words to classify things by labeling things incorrectly, such as calling the first sheep they see a dog, and then having someone correct them. The same applies to the refining of mathematical concepts.
Unlike younger children, elementary age children have developed social skills that can interfere with the learning process. When an authority points out a mistake, the student will often try to memorize the new information, because they know the teacher knows more than they do. Instead, students need to evaluate the new information so that they can challenge misconceptions, either their own, or in the ideas being presented. Memorized information will not replace misconceptions, so the misconceptions will be used in trying to solve problems in the future. In fact memorized information, parroted by the student, will hide their thought process so that the teacher does not know what foundation to build upon or what misconceptions to confront.
Children also need to come to respect the development of their own reasoning process. They need to discover that they reason pretty well and can improve that process. Solid reasoning combined with a challenging peer group will give them the power to learn new concepts on their own, without always needing a teacher around. That should be the ultimate goal in mathematics, and any other, education.
In addition, students
need to be able to discern which learning involves concepts, and needs
to be constructed by them; and which learning involves memorizing and should
be learned by using memory techniques. Students also need to learn to respect
the power of their minds to solve problems without conscious effort. They
need to know when to walk away from a problem that has them stumped, knowing
that when they return to it, their mind will have made progress in a way
that could not occur while they are putting forth conscious effort.
Teachers Understanding Of Mathematical Processes
Perhaps the teacher’s biggest challenge in developing mathematics communities in the classroom is developing an in depth understanding of the mathematics that they are teaching. Most teacher preparation programs have mathematics classes that are based upon memorizing algorithms. These classes do not challenge the misconceptions of the aspiring teachers. Thus the teachers try desperately to memorize what is being told to them. However, it usually does not make sense to them, so not only do they not have the concepts that underlie our mathematical system, but they end up misremembering much of what they are taught. Rather than trying to improve elementary mathematics by requiring higher order math classes for elementary teachers, colleges should be teaching for understanding of our basic number system. Developing the basic concepts of our number system, elementary algebra, geometry, and informal logical reasoning will serve teachers better than trying to memorize the quadratic formula or formal methods of geometric proof.
Only when teachers have basic math concepts firmly in mind, can they judge if a child’s reasoning is sound or if it needs intervention. Understanding the process of mathematics will also help the teacher come up with counter examples when a student is starting to accept a fallacious generalization. Also, teachers should not be teaching in a vacuum; they should be relying on discourse among themselves to come up with ideas about how to challenge students’ misconceptions. By engaging in mathematical discourse with their peers, teachers would be experiencing what they are trying to get their classes to do. In addition, I would like to see resources for teachers to reach out to one another electronically when they cannot find a workable solution within their school.
Once the teacher begins to understand the mathematical concepts being taught, they need to start developing the skills of posing problems that challenge students' thinking. The goal is to ask questions that will create a conflict between the students' views and their past experience. Then when the students try to apply their faulty logic, they should find the conflict. However, the teacher may need to help the students through this process by following through on the problem posed, applying the students’ logic, and possibly even pointing out the conflict. There will be times when the teacher will guess wrong about the students views, then a new example must be found that will help the student see the conflict. The conflict must involve the student's viewpoint for real changes in their belief system to occur.
One of the most serious
problems that teachers will run into is when students memorize words that
sound right, but they do not understand the concept behind what they are
saying. Therefore, teachers must be very careful to avoid giving students
verbiage that will be used as a smoke screen to keep the teacher from evaluating
the students’ misunderstandings.
Conclusion
In order to make a real difference in mathematics education that reaches a broad range of students we must begin by offering mathematical understanding to elementary teachers. We must then recognize the difference between what needs to be memorized, what needs to taught as a concept, and when we need to allow students to learn on their own. When teaching concepts, we need to teach our students ground rules of courtesy and how to engage in intellectual discourse. This will allow them to construct valid concepts which will serve as a firm foundation for higher mathematical understanding.
One of the advantages
I have found in this type of teaching is that all students become engaged
in the process. Those need to speak in order to think will get the chance.
Introspective students have the chance to get started thinking during the
minute of silence that I give them at the beginning of any discussion.
Girls, who traditionally drop out of mathematics, learn to assertively
present their position. Students that were presumed to be dull, often end
up giving clear descriptions of the process in question. More adept students
learn to listen carefully so that they can present arguments to which their
fellow classmates will listen. All students get the opportunity to maximize
their learning styles. In other words, I have seen mathematics move from
a limited resource for the lucky few, to a subject that is accepted with
energy and enthusiasm by the vast majority.