The Changing Landscape of Elementary Mathematics Teaching and Learning

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A person wanted to go from his home to a distant well-known natural landmark. He consulted a computer-based program and received written directions telling him step by step what roads to take, how far to go on each road, and where to turn. He left home and began following the directions. Along the way he somehow got off track. After wandering aimlessly for hours he had to stop and ask directions to the landmark. He had good directions to begin with, but when he got separated from the original route his original directions were useless. Eventually he made it to his destination, but he took a lot longer than he needed to, and he experienced great frustration along the way.

On the other hand, this man's sister consulted a map in order to plan her visit to this landmark. As she studied the map she gained an understanding of where the landmark was in relation to her home. She noted the important roads, towns, lakes, and rivers that were in the general vicinity. She noted that in order to reach the landmark she would need to travel northwest about 120 miles from her home. She noted that there were no direct routes but there were a number of north-south roads and a number of east-west roads. She measured and found that the total northerly distance was 75 miles and the total westerly distance was about 94 miles. She reasoned that as long as she kept traveling north and west and kept an eye on the number of miles traveled, she would eventually get to her destination.

Is it obvious which of these travelers had the more powerful approach to reaching the destination? Both had adequate methods, but one was weak while the other was strong. The first method would have been fine if the man had not gotten off track. Its weakness would not have been revealed. If we apply this metaphor to mathematics teaching, we see that the first traveler is like the traditionally educated mathematics student. The student was given step-by-step procedures to follow and learn. With enough practice the student could reduce the likelihood of ever going off track. Understanding of the lay of the land was unimportant and might even be considered to be a distraction. Simply memorize the steps so well that you can follow them flawlessly, and you will reach your destination every time.

In contrast, the second traveler's method was much more powerful. There wasn't one right way to go, there were any number of different "right" ways. Memorization was less helpful than understanding the big picture. The time spent studying the map and gaining insight into the variety of routes virtually ensured that the traveler would not get lost. As long as the system was followed (keep traveling north and west, keeping track of the distances) there was no way to go wrong. If the traffic on one road was bad, the traveler could take the next turn and still know where she was going. If she were talking on her cell phone and missed a turn, there was no problem; she could take the next turn and still know how to get to her destination. A further benefit to this method is that if she had to make the same trip several weeks later she would probably remember the big picture much more effectively than her brother would remember the detailed directions that he was trying to use.

The teaching of mathematics has moved in significant ways in recent years toward the model represented by the second traveler. In the best mathematics classrooms today, students are learning to understand mathematics rather than to just follow steps of procedures. Mathematics is being taught today in rich contexts that make it clear to students "why we need to know this." Mathematics is being taught as a landscape of deeply interconnected ideas and relationships. Students are learning together as they explain their thinking to one another and create ways of sharing ideas with one another. Through all of this, students are learning to think mathematically. It is truly an exciting time to be a student of mathematics!