Eratosthenes and Us,
It Just Keeps Going and Going and Going…

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Eratosthenes

Eratosthenes (276-197 BC) was born in Cyrene (now Shahhat, Libya). After studying in Alexandria and Athens he became the director of the Library in Alexandria. (The Library at Alexandria, while functioning as what we think of as a library today, also had elements of what we think of as a research laboratory.) He is best known for his accurate calculation of the Earth's circumference, within 15% of the modern day value, and this work is often referenced in introductory astronomy textbooks. One of the great beauties of his observation and measurement is the combination of clever insight, or the "Aha moment" and careful measurement. The "Aha moment" is in his realization that since the rays of the Sun striking the Earth can be taken as parallel, that on any given day the solar zenith angle at solar noon of a point, A, on the Earth is the same size as one with its vertex at the center of the Earth with one ray pointing to point A and the other pointing directly to the Sun (i.e. the point on the Earth where the Sun is at the zenith, the overhead point O on diagram). The careful measurement part is in measuring this angle and the distance between O and A. Eratosthenes knew that at the summer solstice the noon Sun was reflected in a well dug at Syene (modern Aswan). Thus, this was the overhead point. Eratosthenes then determined via shadow measurements that the Sun as observed at Alexandria was south of the vertical by about 1/50^th of a full circle. The Earth's circumference would thus be 50 times the north-south distance between the sites.

What is Constructivism?

Constructivism is a theory about acquisition of knowledge and learning process, not a teaching technique. "Learning from this perspective is understood as a self-regulated process of resolving inner cognitive conflicts that often become apparent through concrete experience, collaborative discourse, and reflection." (Brooks and Brooks, The Case For Constructivist Classroom. 1993). Students come to class with an understanding of how the world works. When they encounter new experiences, the students synthesize and assimilate them based on their prior knowledge. The students adjust their prior understanding with their current one by formulating new rules. Students engage with each other and the task at hand as independent and active learners. They experience and evolve the concept through the activity and their own questions. As they discuss and experience their discoveries, they share their thinking and listen to each other. Via oral or written expression, the students are always reflecting on what they know and what they need to know. This metacognitive process, done as individuals and in group, is a continuous checking point of students' self assessment.

Thus, students are active learners who consciously or unconsciously build their own new understanding based on their prior knowledge. In science, students come to class with a preexisting perception of natural phenomena, which they use to make sense of new situations. Taking advantage of students' prior information, they and their teachers can develop new concepts and foment solid insights and learning as well as correcting misconceptions.

In a constructivist class students are actively engaged whether they are interacting with each other or self-reflecting. Students are encouraged to work in groups, listen to each other and be heard, as they reflect on their ideas and that of their friends, they become flexible thinkers. Students are exposed to different, methods, ways of thinking, approaches and formulating questions. They are constantly encountering new disequilibrium wherefore they revise their schemata and reformulate new concepts.

The activities and/or deliberations are different at each group and self-directed making the classroom abuzz with discussions, movement and hard work. The activities and focus of the discussions are designed by the students according to what they want to explore at that moment creating a high level of interest, engagement and motivation. If the teacher is the designer of the group activity it is usually based on the students' needs maintaining still the high level of student interest and commitment. This group model encourages critical thinking skills, deeper and authentic learning, and participation on the part of the students.

Even though the students are self-directed, they are bound by a curriculum that gives focus to the constructivist class. One misconception about constructivism is that children do what they want and therefore dictate the curriculum to be taught. This could not be further from reality. The curriculum is ever present in the teacher's mind, leading her/his planning and actions throughout the lessons. This facade of chaos is apparent to outsiders but not to the classroom teacher. The teacher observes how the students are approaching a problem, the kind of questions they are asking and how they answer them, what concepts are solid and which ones are still evolving. The teacher assesses what the students needs are and how to best help them in order for them to evolve their concept. In this respect the students influence the planning of the lessons and oftentimes shape the curriculum; but they do not dictate it. The role of the constructivist teacher is significant in that she/he has to be aware of the students at all times in order to set up the projects and plan the next lessons based on what students were working on the prior the class. The educator has clear what fundamental concepts she/he wants the students to learn, as established by the classroom curriculum, and steers the students to remember the questions they set out to answer. The educator refocuses the students, suggests a new direction, or encourages the students' selected path. The teacher is more of a coach, a designer and a facilitator than a presenter of "knowledge" and information.

The constructivist teacher organizes the curriculum around conceptual clusters from which the students develop their understanding. In more traditional settings, the teacher may "give" the students what the students "need" to learn and often these are a series of facts isolated from each other and the learner. The student is passive in that he/she takes what it's been given as a series of must-be-memorized essentials. In a constructivist class the activities the students do are relevant to them and the unit they are learning, i.e. they have to use the knowledge acquired as a tool to solve the larger question they are trying to answer. Geometry, science, history, language arts are tools that students use when they need them in meaningful ways.

In more traditional classrooms, students ask questions with the expectation of being given the answer. Their question is more of curiosity for a fact than investment. In a constructivist class the students' questions are very different. Oftentimes, they will ask a question of which they expect no answer from the teacher but are an indication of their own reflection and future plans. The teacher too asks questions not always expecting an immediate answer but to motivate student critical thinking and reflection. Students and teacher have an even intellectual exchange in which both are learning and discovering new elements together.

The bond between student and teacher in a constructivist class is strong. What links the classroom community together is having a common goal; we are learning and exploring together. The teacher is as much a learner as the students are teachers. There is basic respect for each other's opinions and knowledge. The students feel acknowledged and their sense of importance equals their contributions to the class. The teacher feels free to express her/his limitations and appreciate when the students teach her/him something she did not know, thus making a contribution in the teacher's growth. The classroom experience becomes a two way street in which all are responsible for each other and oneself.

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