We help our students build conceptual understanding by actively engaging them on a daily basis in the process of having to reason from concepts (and prior knowledge) to answer questions and solve problems, both within applications and within purely mathematical contexts. How?
Questioning that forces students to actively engage in reasoning from concepts:
Open, unfamiliar, unguided questions.
Questions that require students to articulate and explain their reasoning.
Students work on questions (individually or in small groups) prior to establishing consensus as a class - both when developing an initial understanding of the concept and when expanding/applying that understanding in synthesis.
Learning Environment that supports and promotes the development of reasoning from concepts:
Collaboration: In pairs, small groups, together as a class, and together with the instructor
Expectations and Trust: Students know what is expected of them, they know nobody (including the instructor) is going to just give them the answer orprovide example solutions, and they trust that the class will eventually come together with the Instructor to clarify points of confusion.
In a typical, traditional mathematics classroom, the instructor begins by presenting definitions and explanations of new concepts to the class, perhaps also providing some level of rationalization or context by linking the new concepts to past and future classwork. The instructor then works some sample problems for the students, illustrating ways in which this new concept can be applied to answer questions or solve problems. During this process, students may be active in some way, but generally in this traditional classroom, it is the instructor alone that actively engages in the challenging task of reasoning from his/her own understanding of a concept to the solution of a problem or the development of a new concept. The students, on the other hand, are passively watching most of this process. They may understand the teacher's explanations and reasoning. They may even agree that the steps the teacher took build logically from the concept. They may even by actively working through portions of the problem (e.g., completing algebraic work, computations, etc.), but they never have to reason themselves from their own understanding of the concept to arrive at an approach and an answer to a given question. When these students are then faced with a question or problem to solve themselves, they are necessarily approaching it by considering the sample problems that they've seen, selecting one that is most like this new question/problem and using (perhaps even adapting slightly) "the method" of that sample problem to find the answer. In doing this, they are essentially mimicing their teacher's reasoning which requires only minimal understanding of the concepts involved. Students come away from this type of learning able to solve a lot of useful problems, but their abilities are limited to the types of problems that they've already seen. Another concern, especially in a discipline that is widely accepted as building strong problem solving and critical thinking skills, is whether this type of learning truly requires critical thinking on the part of the students.
In our classroom, we strive to push our students beyond this traditional approach to learning mathematics so as to broaden their problem solving abilities and to strengthen their critical thinking skills. Specifically, we want them to be able to reason independently from a deep understanding of concepts (or prior knowledge) to answer questions and solve novel problems. Providing plenty of opportunities (some guided and some unguided) for students to engage in this type of reasoning in novel situations (where there is no example to work from) slowly builds the confidence and ability to reason in this way independently. It is this very experience, in turn, that also helps build further conceptual understanding.
From the very beginning of the course, students have to reason from concepts (and prior knowledge) to answer questions. Questions that we pose prompt students to discover the concepts of the course as opposed to being "presented" with the concepts through reading the text or listening to an instructor's explanations. Following these initial discovery questions, we pose additional "synthesis" questions that provide students with the opportunity to adapt, extend and integrate their understanding of the concepts (in new settings) as this is an integral part of developing their ability to reason using those concepts. These synthesis questions may be embedded in an applied modeling investigation or they may be used to discover new concepts. In mathematics, as in any subject, a student is truly learning when they are asked to carefully examine their current understanding and then use it to develop new understanding.
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