## Sunday, December 16, 2007

### The Monk Problem

The Monk Problem poses this question:  A monk hiked to a mountaintop at starting daybreak and arrived at the mountain top at just at sunset.  After two day’s meditation, the monk starts his downward journey at daybreak and arrives at the bottom of the mountain again at sunset.  Was the monk ever at the same point on the trail at exactly the same time of day?  The problem can be mentally modeled by imaging the two trips superimposed on each other.  As ascending monk  starts from the bottom, the future self start on the trip down.  At some point they will pass each other –they will have been at the same place at the same time of day.  An alternate model is to graph distance from the bottom against time.  The first day’s graph is increasing while the return journey’s graph is decreasing.  Where the two graphs intersect represents the same distance from the starting point and the same time – the graphic case of the monk meeting himself on the trail.

When I first ran across the problem I shared it with my wife – also an experienced teacher – and she argued that the answer was not so clear to her.  She thought is was quite possible to never be at the same point at the same time.  After perhaps 30 minutes of explanation, acting out,  and resorting to hand waving and raised voices ( a sure fired way to increase comprehension) we  found the root our  miscommunication.  My wife saw the trip up the mountain as discreet – the point the person occupied was the footprint .  The monk had obviously not trodden the entire trail; the trip was a series of points of contact.   My mental image of the trip was a continuous motion of an idealized point.   The incident made me wonder about teaching and how we think with language.  Even though we have similar backgrounds and training my wife and I made different tacit assumptions about the meaning of point and this resulted in two different interpretations of the Monk Problem.

If my wife and I have this much trouble sharing mental models, we face an even greater challenge sharing our mental models with our students with their varied backgrounds and experiences..  If my wife had taken my word as authority she would have rejected her meaningful and accurate construction to accept a senseless one. This is why telling isn’t teaching. She certainly could have repeated my explanation and that is why saying isn’t knowing.

## Friday, December 14, 2007

### Computing Power and 6th Grade Math

As I was watching our middle school kids trying to unravel the intricacies of dividing decimals (move the decimal, move the other decimal and bring it up) I started wondering just how much computing power was sitting in their backpacks. Each student has been issued a MacBook with a 1.2 gig RAM and a 2 GHz processor. In comparison, the Apollo Guidance Computer that guided the spacecraft to the moon had 2k RAM operating at about 2 MHz. That means our 6th graders have 600,000 times the memory operating 1000 times faster than the computer that did the rocket science for the Apollo mission. In fact, the computing machine in their hands could do all the paper and pencil calculations these kids will do in the rest of their public school career before the kid could get to the pencil sharpener and back.

This has to change how we think about teaching math. If an algorithm exists, a computer can do it. I don’t think we have a good map of this new ground before us, but I do feel we need to throw the old map away. Teaching computational skills at the expense of conceptual understanding and technology-based problem solving is just no longer appropriate.