Throughout my research, the students were rarely given a verbal introduction to the tasks they faced. Instead, they had to rely on their intuition, pre-conceptions, and any previous knowledge to find solutions to the tasks at hand. They were given time and resources to explore their already existing understanding. This balance of time to think and distance from the teacher’s guidance is key, I think, to encouraging students to use their intuition and past experience to solve problems. According to Duckworth (Bransford, 2000), “Accomplished teachers ‘give learners reason’ by respecting and understanding learners’ prior experiences and understandings, assuming that these can serve as a foundation on which to build bridges to new understandings.” When teachers intrigue students with interesting problems to consider, students will naturally respond by relying on their intuition and previous knowledge. If teachers simply pump their students with new information, the new information may quickly be lost. New information will be better retained, I think, if it can be associated with other, already familiar information.
Perhaps one of the most important elements of encouraging students to think on their own may be to give them ample time to do that thinking well. Most students will not get things right the first time around. Teachers must, I think, invest sufficient time so that the student can come up even with incorrect theories, test their theories, and then realize on their own whether or not their theories are correct. Further, Carolyn Maher (1999), in her own research, found that “Students need a classroom environment that allows them time for exploration and reinvention.” Hence I think the classroom environment should be receptive to bold thinkers, who, like my main research subjects, invent and reshape theories and procedures to match the issues at hand.
Through my research, I saw how careful task design can encourage students to use their intuition and their past experience to solve challenging problems. In all the tasks given to the students, they were not told how to solve the problem. One of the probability tasks given them, for example, asked for several different probabilities without introducing a method for calculating probability, and without indicating any sample space. The students knew generally what the word probability meant and had some background upon which to build. However, since they had no formulae to turn to, and no sample space provided for them, they had to begin by asking their own questions, forming their own theories, and then working within those theories¬–or, when the need arose, rebuilding those theories from the bottom.
The difference between the students in my research and students in a traditional classroom setting should now be obvious. In a traditional classroom setting, the students learn according to the textbook’s structure and language. Textbooks typically present the mathematical concepts and formulae that students simply must accept. The students are then asked to solve certain problems that fall within the realm of the formulae, based on template examples to be imitated. Speiser’s class worked the other way around: He asked questions first, and these questions invited students to formulate their own mathematical theories. Thoughtful students in a traditional classroom who desire to look further are often caught in a maze of textbook formulae, not knowing how to get out. Because they are taught according to the formulae, they often do not know how to work outside of them.
In contrast to a traditional classroom, in MathEd 306, students felt comfortable with being outright skeptics, and therefore demanded rigorous explanations from each other. Instead of feeling insecure or personally threatened by such requests, these students eventually rose to the challenge and found methods to convince, and in effect prove, their position. Because of the rigorous investigation needed to establish a given theory in this classroom, once a theory was accepted, the students had gained enough experience to work comfortably within that theory to answer questions that were significantly different from the ones that motivated the construction of the theory.
In the students’ search for justification, they often turned to previous tasks and understanding for guidance. Clearly, the models they had constructed during prior tasks played integral roles in the understanding that they transferred to the later tasks.
In my research, I also saw how the role of teacher can be played not only by professionals but by students as well. Students who have explored a concept in their own way may have reached a level of understanding from which they can effectively demonstrate their understanding to others. In addition to the possibility of helping the listeners, I think the student playing the role of teacher can benefit as well. Any theories that are forming in that person’s mind can be further solidified (or perhaps even reformulated) through explaining them to others. Also, a student who is in the course of working out a theory can become more familiar with that theory and surer of it as she discusses it with others.
It is the mathematics teacher’s job to help students understand mathematics. As we have seen, this does not necessarily mean that teachers should take upon themselves the responsibility of not allowing their students to consider incorrect solutions. Instead, I believe their responsibility should be to encourage students to explore options carefully and critically, and to offer them sufficient tools to explore and test those options. Even if a student seems to understand a certain concept, this student should be able to understand it well enough to convince others. Students should not simply be told results that they could very well discover on their own, as many textbooks do. Instead students should discover such results, and gain experience, as far as possible, as builders (and also critics) of their own ideas and understanding.