What’s the concept?

No one would object to the idea a student needs to understand concepts to achieve mastery of subject matter. Otherwise, the student would be no better than a calculator, waiting for inputs and indifferently spitting out answers. However, what type of concept is relevant? How big; how small can the concept be?

In building its Intelligent Tutor, xThink breaks down the concepts required by students to solve a problem as far as possible. For example, to master multiplication a student needs to have a sense of the sequence of numbers and to know that 4 times 5 is a shortcut form of addition, the equivalent of 4 + 4 + 4 + 4 + 4. We call these conceptual components “metarules.” And metarules can be pretty minute.

How minutely should the Intelligent Tutor’s metarules disaggregate math concepts? The answer is; the smaller, the better. That way, the tutor can meet each student at his or her level of comprehension, then raise the bar step by step through activities, examples, and exercises. Whenever a student shows mastery of one or more concepts, it is time to move on to new conceptual challenges.

Could the Intelligent Tutor explain things so minutely that the student loses interest? Yes, that’s a theoretical possibility. However, the tutor’s job is to adjust dynamically to the performance of a student. Each student is like a snowflake, unique. The amount of time a student needs to spend on exercises or with any minute concepts varies, and tutoring software can adjust accordingly.

Warning. It is risky to declare one student to be behind or ahead of others. Mastery of any one particular concept might be the key that permits a “slow” student to leap forward in learning. Learning need not be — might never be — a linear, incremental process. Learning is cumulative, yes (especially math), but not necessarily linear. Furthermore, fastness is not universally good, nor slowness universally bad. Consider Einstein’s self-assessment: “It’s not that I’m so smart, it’s just that I stay with problems longer.”

Are some students a lost cause? Never! Intelligent tutoring software has all the time in the world to generate customized, dynamic instruction and encouragement. Research shows that students have deeply rooted curiosity and joy of mastery, which an Intelligent Tutor can nurture.

Cognition and learning can be deceptive. A student can remain on one plateau of understanding and then suddenly display the confidence and insight to climb to new heights of mastery. Each child assimilates knowledge and builds skill differently. Intelligent tutoring software observes student history and performance and asks, “Were the concepts broken down sufficiently?” and “Was sufficient practice time allowed for confidence building and skill building?”

The preceding ideas and examples provide a glimpse of the issues of cognition and learning at the heart of building an Intelligent Tutor. And this glimpse helps frame the discussion of concepts. Perhaps all can agree that tutoring (and teaching) must convey concepts, so that students achieve deep understanding. However, some people might doubt the efficacy of breaking down concepts to a very minute level and encouraging phases of repetitious practice and drill. xThink Intelligent Tutor will provide proof.

In sum, conceptual understanding is a generally accepted goal of learning. Less familiar is the idea that learning is based on sets of minute concepts that each student can coalesce in good time, with proper instruction.