## Second Year

Our team was composed of a NASA scientist (Paul), sixth grade teachers in Glenarden, Maryland, two fifth grade teachers in Amherst, Massachusetts (Liliana and Tamara) and a Massachusetts high school teacher of home schooled students at Pathfinder Learning Center. In addition to Paul, Karl Martini, a professor at Amherst College was another scientist the Amherst teachers worked with. As a team, and as individuals, we all felt comfortable accepting our own strengths and limitations. Paul would often ask for the teachers' help on how to introduce a concept to the students. The teachers shared with Paul the students' scientific and mathematical problems, as well as their own, to have his guidance and knowledge to better help the students. The exchange among all members of the team was rich in content and context.

The first order of business was to draw up a lesson plan. For this we drew in part upon lesson plans in the Project Astro book: The Universe at Your Fingertips, produced by the Astronomical Society of the Pacific. While The Universe at Your Fingertips does not have a specific activity that duplicates Eratosthenes work, it does have activities that are useful (B6 and C2 cover understanding shadow length and movement, C3 covers Eratosthenes' reasoning, and M1 relates Eratosthenes to Columbus). Liliana helped us to focus the plan along constructivist lines and presenting the material to the students as a problem to be solved by them. We emphasize that our lesson plan was a working plan for us. We were not afraid to modify it according to the level of the students, for instance, the high school students attacked the problem at their level, use of trigonometric functions, planetary motion, etc. or to modify it as we went along.

One key part of the project is that the students need to know the Euclidean geometry fact that opposite interior angles formed by a line intersecting two parallel lines are equivalent. However, we did not tell the students this fact. Instead, we guided the students in a discovery process via construction i.e. drawing parallel lines and then drawing an intersecting line and measuring the angles. They also drew non-parallel lines and an intersecting line to discover the converse, that if the angles are not the same then the two lines are not parallel.

The teachers never told the students how Eratosthenes used opposite interiors angles to measure the circumference of the earth, thus they had to discover, or rediscover, how he used it as a tool. This made learning geometry meaningful and relevant to students rather than being taught in isolation as it is generally done in schools. In Amherst, the students were measuring the length of shadows at solar noon without yet being aware of how Eratosthenes had used this information. Tamara asked the students what they were doing and they explained they were measuring the shadows. The teacher then asked the two students, "Do you have any idea how Eratosthenes might have used this information?" The students explained that they thought that the tomato stake was the line crossing the parallel lines and that the sun's rays were the parallel lines in the Euclidean geometry lesson they had done the week prior. Tamara had never told the students what they "needed" to know, the students evolved their understanding on their own. The teacher, delighted, told them to continue thinking in this way, that they were "Totally on the right track." The students were actively involved, piecing the parts together to understand how Eratosthenes used shadows, angles and parallel lines to determine the circumference of the earth.

Evolving one's knowledge in this way is a powerful way to learn. The material presenting presented has helped it stay withhas stayed with at least one student, Will, to this day. Will's mother recently related the following story to us. Will, an eighth grader now, was in math class when his teacher announced that he was going to talk about Euclid and how to measure opposite angles. Will told him that he knew how to do that and he even knew how to use them in "real life too." The teacher showed disbelief but played along with him. He asked him, "When did you learn this?" Will answered that it was in fifth grade. The teacher told him to show him what he knew. Will proceeded to show him not just about geometry but how to use it to measure the circumference of the earth. The teacher was incredulous that a fifth grade student had learned this in class, understood it and recalled it three years later.

Because we encouraged the students to question what they were doing and why they were doing it we engaged in many discussions with them and among ourselves. In Amherst, the big question for the students was, "Are the Sun's rays really parallel?" This originated from a student who was quite adamant that the sun's rays were not parallel. This challenged the teachers, especially when, via consultation with Paul, they learned that the student was correct, the Sun's rays are not exactly parallel. The teachers then shared what they learned with the students and all gained a deeper understanding of Eratosthenes' process through this student initiated exchange. Interestingly, while the students eventually understood that since the Sun is so very far from the Earth we could assume its rays to be parallel for the project, they did not question how could Eratosthenes know this. (He knew because of previous work by Aristarchus of Samos on the Earth-Moon distance and the Earth-Sun distance.)

For the Glenarden students the major challenge was with making the measurement. We have found through the years that while the students can grasp the theory behind the measurement, it can be hard for them to make the required careful measurements of shadow length. We also had a data anomaly in that the final data set of solar zenith angle was bimodal. Both modes were a normal distribution, but while one was centered on the correct value the other was centered 3 degrees away. This opened the doors to a "teachable" moment for the students about when and how do you average data?? (The average of the all of the data was, of course, in between the modes where there was no reported value! A sure tip off that something is wrong.) This in turn lead to a "teachable" moment for Paul as he realized that to him, a scientist, average means reducing the random error in a measurement, to elementary students average means average performance, as in a teacher averaging their test scores to get there their grade for the semester.

With the IDEAS funds we gave a teacher's workshop and developed materials that we have used and have been used by others. A video, produced by Tamara, was made, demonstrating our collaboration and how the project was put in practice in the classroom. This video has been broadcast on the Amherst PBS several times, shown at national conferences, and used at the University of Massachusetts in the Constructivist Educator Training Program (CETP) to train teachers on how to teach science and math in a constructivist class. A lesson plan/guide was developed and has been shared at the above conferences and workshops and others. The set of lesson plans includes student work to give samples of what the project looks like in the classroom to encourage teachers to reproduce the project.

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