Use what you have learned from your assigned readings as well as your article reviews and our class discussions to write a paper addressing the questions below. You may either answer the questions separately or weave you answers into a coherent paper. In either case, be sure to refer to the readings you have done since the beginning of the class.
What constitutes proof in mathematics?
Why is proof considered important in high school mathematics? What should be proved?
Someone in class said that too much emphasis is put on proof in geometry over proof in other areas of high school mathematics. Do you believe this? Why do you think that proof has been concentrated in geometry?
What relationship does proof have to logic?
What relationship does proof have to the physical world?
What is the difference between deductive and inductive reasoning?
What is the relationship between proof, reasoning, conjecture, inquiry, and discovery?
What does the van Hiele model have to say about the teaching of formal deductive proofs in geometry?
Read the case discussion and outline answers to the following questions. (This part does not have to be polished, nor handed in -- you will use it in a class discussion.)
Looking at their worksheets, what do you believe that Justin and Chris understood about proof? What did Justin mean when he said "[The lines are parallel] because Nicole and the computer made them parallel."? How would you respond to Justin?
What do you believe that Nicole understood about the relationship between the tangents? What would you ask her?
Do you think the students in Ms. Wilson's class had an understanding of proof? What evidence leads you to believe that?
What would you say to Officer Jenkins when she asserted that "proofs are a crucial part of learning how to reason?"
What tacit message is Milburne High giving to Nicole, her mother, and other class members by emphasizing proofs in upper level courses but not in average level courses?
Ms. Wilson gives the students a worksheet that directs them to use the computer to "draw a circle with a diameter and two lines tangent to the diameter" and to use the drawing to state a relationship between the tangents. How could she have been clearer in this direction?
In asking why the lines are parallel, Ms. Wilson tells Justin and Nicole "... You need to write the geometric laws we have studied that are making the lines parallel." Do you think she wants them to write a formal proof? Is that necessary?
Is the computer drawing program an effective aid to discovery in this exercise?
Would the construction of the tangents using a compass and straightedge offer any advantages over a computer drawing program? What advantages, if any, are there in using the computer over pencil and paper construction?