How to learn the way around a new city? In a more traditional approach, the teacher, in the classroom, explains some basic turns and landmarks and even models how to get to a specific location, then hands the student the keys to find someplace specific for homework. In this approach, the teacher is pulling on past experience with the city, and the explanations and modeling on the board are a short-hand for that experience. The student needing extra help (intervention) has no or at best a mixed experience to make accurate sense of the short-hand.

In a second approach, the teacher and student get in the car together but the teacher drives, pointing out the turns and landmarks he or she knows and again asks the student to find a specific location for homework. Better, but it is a second-hand experience for the student.

A third approach is to have the student drive with the teacher in the passenger seat not telling where or when to turn. The teacher first orients the student to a map of the city to be experienced. With good questions, the student notices landmarks on the map and is able then to begin making connections between the map and what is experienced while driving, “Oh this turn coming up is right here on the map.” “Good,” says the teacher, and that’s it, no lecture on the history and use of gazetteers. The teacher keeps the student in the driver’s seat.

If the student makes a wrong turn, the teacher simply asks the student to show on the map what they did and where they wanted to go. The student retraces the steps and notices that a left turn was needed not a right turn. “Good,” says the teacher and the student turns left. The city is slowly internalized as it is experienced, guided by the teacher and the map.

A good graphic representation of the math, not explained but oriented to and guided through, can be the map of the conceptual and procedural processes students have to navigate through.