A conceptual conversation is one that has a diminished emphasis on technique and procedure, and an increased emphasis on images, ideas, reasons, goals, and relationships. People conversing conceptually speak in ways that make their meanings clear to others in the conversation. They speak about ideas and ways of thinking.

People conversing conceptually avoid speaking in ways that hide their meaning. Conceptual conversations are meaningful conversations.

This means that teachers must expect students to take significant responsibility for explaining their reasoning, raising questions about tasks, and making decisions about appropriate assumptions, decisions, and conventions. Unfortunately, students (and sometimes teachers) often explain themselves only by way of saying what arithmetic they had done or would do, or by what procedures they would follow. When this occurs, other participants in the converssation have no way to judge the appropriateness of the speakers' implicitly-made decisions, and hence those judgments cannot be objects of reflection by others or by themselves. The technique of holding conceptual conversations about "what is going on" tends to orient students away from rash judgments about procedures and techniques.

There are concrete actions teachers can take to help turn a superficial conversation into a conceptual conversation. They are to routinely ask questions like these:

Conceptual conversations can be initiated by the the simple act of asking students to say what a description is describing or what a question is asking. But merely asking this question is insufficient. Students must come to understand that you expect them to answer in terms of objects, their properties, and relationships among them. You should also help them clarify potential ambiguities (multiple ways in which a statement can be interpreted) and hidden assumptions (e.g., "they are assuming that you move at a constant speed") that could be consequential for the choices and judgments they make.

It often happens during instruction that students are at a loss as to what to do. A teacher can help students clarify situations for themselves by asking what various numbers stand for. Also, when they write an expression using a variable, students often do not have clearly in mind that a variable represents a number and that that number is a number of *something.* In constructing explanations of what each number stands for, it is common for students to become aware of meanings and relationships that had eluded them.

Teachers often ask the question, "What are you trying to find?", at the outset of solving a problem, having in mind that whatever you are trying to find will be the problem's "answer." Another way to think of this question is to ask it __ every time you or a student considers an operation.__ "What about this situation will you find by multiplying those two numbers?"

Teachers must have patience in laying the foundation for long-term development of meaningful reasoning. Having students engage in the cycle of analysis-implementation-reflection may initially appear to be time consuming, but the long-term benefit will be more than worth the initial investment.

The question "What did this calculation give you?", or put another way, "What did you find by doing this calculation?" might seem to be redundant with the question "What are you trying to find?" discussed in the previous paragraph. However, it is not uncommon for students to have an idea of what they are trying to find by a particular calculation, yet be unable to say what they found after calculating.

Often students write expressions or equations as just one step in an answer-getting ritual. It is useful to routinely ask them, and expect them to ask themselves, what they have represented when they write an expression. If they cannot answer this question, then that is a signal that they must return to the situation and make better sense of it.

(This example requires *Quicktime*. Click here to get *Quicktime for Windows*. Click here to get *Quicktime for Mac*.)

$6070 is 5/7 the price of a car. What is the price of the car? |