Questions asked in the video:
How do you accurately find longitude at sea? Can the stars, planets, and moons be used to accurately find the longitude while at sea? How can you make a clock that will be accurate at sea? How do you create a pendulum that is not effected by the motion of the ship or environmental factors? Could a pocket watch keep accurate time? (PBS - NOVA, 1998)
Discussion:
In the "The Art of Problem Posing" it mentions in Chapter 3 that some mathematical questions may go unanswered simply because our views are restricted (Brown and Walter, 2005). Narrow views can limit our thinking and make certain questions and approaches seem absurd and impossible. This proved to be the case in the video longitude video. At the time, clocks were not reliable enough to accurately tell time at sea and thus the astrologists felt that it was absurd to even consider them as an option. It seemed impossible that a clock could withstand the motion of the ship as well as the other environmental factors that occur throughout a voyage. John Harrison on the other hand disagreed with this view and did not let the current limitations of the clock affect his vision. He spent years creating and testing clocks to overcome the limitations and to create the first reliable clock at sea. Brown and Walter (2005) also mention that modifying a question will bring it into line. I try to remind my students to use this technique when they get stuck on a problem. Unlike John Harrison, many of them will give up if they do not find the answer immediately. John Harrison was having trouble creating a clock that would accurately tell time at sea. He tried several different models but something was missing. Rather than quit, or continue to create models that weren't just right, he decided to modify his question. He was no longer concerned with finding a pendulum to withstand being out at sea. This led him to the pocket watch and thus the answer to the longitude problem.
I think as teachers we can sometimes get stuck in one frame of mind as well. We might teach a particular method because we found it easy, that's how we were taught, or that's how it is presented to us in our curriculum guide. We then expect the students to use these same approaches on their assignments and assessments. Teachers may expect their students to see everything as they do and that is not always the case. I require that my students show all of their work on their assessments not only so I can see that they didn't just guess but so that I can follow their thought process. I have seen some very interesting approaches over the years that I wouldn't have ever considered. I do not take off points if they approach the problem differently. I try to make a point to be open minded but I know that not all teachers are like that.
Exploring Quadrilaterals:
After reading through the article by Richardson, Reynolds, and Schwartz, I considered how I would have approached this problem. I would start the problem using dot paper. I would use dot paper because it is more helpful to have a visual representation in front of me. I would chose the dot paper over the geoboards because I could use different colors to represent the different quadrilaterals. Eventually I would get tired of guessing and testing my way through concrete examples (I truly despise geometry) and I would attempt to find a general rule to identify all of the quadrilaterals. From there I would have moved on to finding the area and perimeter of the quadrilaterals. (I love formulas!) Much like the work in Figure 6 (Richardson, Reynolds, and Schwartz, 2012), I would have to use the Pythagorean Theorem to find some of the exact side lengths.
References:
Brown, S. & Walter, M. (2005). The art of problem posing (3rd ed). New Jersey: Lawrence Erlbaum Associates, Inc.
PBS - NOVA. (1998, October 6). Lost at Sea - The Search for Longitude. Retrieved from http://www.pbs.org/we have/nova/transcripts/2511longitude.html
Richardson, K., Reynolds, A., & Schwartz, C. (2012). Exploring quadrilaterals through flexible approaches. Mathematics Teacher, 106(4), 288-294.
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