EMAT 8990 First-Year Doctoral Seminar Brad Findell, Fall 2006 Mathematical Knowledge for Teaching Many teachers have quick answers to and explanations for the questions below. If, as part of an oral exam or a job interview, you were asked these questions, how would you respond? For authenticity, answer off the top of your head, and speak your answers aloud. Be sure to provide an explanation. 1. Is 0 even, odd, neither, or both? 2. Why is a negative times a negative positive? 3. Is 1 prime, composite, neither, or both? 4. Is a line parallel to itself? 5. Are parallel lines everywhere equidistant? 6. Is a parallelogram a trapezoid? 7. Is a square a rectangle? 8. Is a rhombus a square? 9. If you triple all the lengths in a pyramid, what happens to its volume? What happens to its surface area? 10. What is the slope of a vertical line? 11. What is the sum of the angles in a triangle? 12. What is the area of a circle of radius r? 13. What is a0? 14. What is a-n? 15. What is am/n? 16. What is ax, if x is irrational? 17. If f is a function, what does 2fmean? 18. Is xxcos1cos1=!? 19. Is xxf1)( = continuous? 20. How can you construct an equilateral triangle, given one side? 21. How do you construct a tangent to a circle from a point outside the circle? 22. Is 24 ±=? 23. What is 2/0? 24. What is 0/0? 25. To divide fractions, is it okay to convert to a common denominator and divide numerators? 26. To divide fractions, it is okay to divide the numerators and divide the denominators? 27. Is .99999… = 1? Take a moment to jot down some notes on how well you think you did. Then step back from this activity and consider the following meta-questions:• What kinds of answers are reasonable to expect from high school students? • What kinds of answers are reasonable to expect from high school mathematics teachers? • What kinds of answers are reasonable to expect from high school mathematics teacher educators? Take a moment to jot down some thoughts characterizing the differences among the answers you might expect from these different populations. For most of the above questions, commonly given answers and explanations are wrong, incomplete, or neglect to attend explicitly to choices among various meanings of an idea. In fact, for many of the questions, the best answer is, “It depends,” because different definitions or axioms can lead to different answers. Which of the questions do you think fall into this category? Rather than explore all of these questions, we will use a Geometer’s Sketchpad (GSP) environment for exploring some of these questions in a non-Euclidean geometry. The GSP file is available at http://jwilson.coe.uga.edu/Findell/EllipticDisk9.gsp. The file includes some explanatory pages describing the geometries, the model, and the tools. In preparation for Friday’s seminar, spend some time reading the pages and testing the tools, and then explore the questions that are explorable within this geometry. Be ready to share your findings. Finally, consider the following question: What does this activity have to do with the title of this