Demographic Data: Please answer the following questions. 1. Sex: Male Female 2. Age (years): 3. Semester at university: 4. In which country / state, or in which provinces did you go to school? 5. What type of secondary school did you attend? (Please describe.) 6. Did you attend basic or intensive mathematics courses at high school? 7. Indicate which lectures and/or seminars in the study of mathematics that you have had: Analysis I Linear Algebra Elements of Analysis Didactics of Analysis Didactics of Geometry Computer Programming Analysis II Stochastics Elements of Algebra Didactics of Algebra Didactics of Stochastics History/Philos. of Math Page 1
Please indicate lectures and/or seminars that you have taken, but that are not included in the list above: 8. Why do you want to be a mathematics teacher? Perceptions of Mathematics: Please answer the following questions completely. Give an example, if it helps you to answer the question. 1. What is mathematics? 2. What do you like most about mathematics? (e.g., in your studies, while learning of mathematics) Please explain your answer. 3. What do you dislike most about mathematics? (e.g., in your studies, while learning of mathematics) Please explain your answer. 4. What is your favorite subject (or field) in mathematics? Please explain your answer. Page 2
5. Why do you think should people deal with mathematics? Please explain your answer. 6. Are there any differences between school and university mathematics? If so, identify them and explain your answer. 7. Are there any similarities between school and university mathematics? If so, identify them and explain your answer. 8. Please explain what charactarizes mathematics within the area of analysis. Content Questions: Please answer the following questions completely. 1. When is a mathematical statement true? 2. The Pythagorean theorem states: Let a, b, c be the sides of a triangle with the side c (where the hypotenuse is always opposite the 90 angle, which is formed by a and b), then the square on c is equal to the sum of the squares on a and b. What does the term theorem mean? Explain why the above statement is a theorem. 3. How do conjectures (such as the conjecture about twin primes Goldbach s Page 3
conjecture) differ from theorems? Explain your answer. 4. Is it possible that both of the following mathematical statements in the context of mathematics can be true? Justify your answer. (i) The sum of the interior angles in a triangle is 180. (ii) The sum of the interior angles in a triangle is greater than 180. 5. Explane the following therms and their role in mathematics: - Definition - - Axiom (or Postulate) - - Conjecture - - Theorem - 6. What constitutes a proof in mathematics? Page 4
7. What is the function of proofs in mathematics? Below three arguments for the proposition The sum of the interior angles in a triangle is 180 are given. Argument I: Let ABC be a triangle, and let one side of it, BC, be produced to D. Draw CE parallel to AB. Since AB CE and AC has fallen upon them, the alternate angles BAC and ACE are equal. Also, since AB CE and BD has fallen upon them, the exterior angle ECD is equal to the interior and opposite angle, ABC. It follows that the exterior angle ACD is equal to the sum of two interior and opposite angles (in triangle ABC), BAC and ABC: ACD = CAB + ABC. Add on both sides ACB. On the left we get two right angles; on the right, the sum of the angles in triangle ABC. Fig. a Argument II: Draw any triangle. Tear or cut off each angle (Fig. a). Arrange the angles so that the angles vertices meet at a point (Fig. b). Since the three angles form a straight line, the sum of the measures is 180. Argument III: Given triangle ABC, draw auxiliary line DE through B and parallel to side AC. DBE is a straight angle and measures 180. By angle addition, 1 + 2 + 3 = DBE. Since alternate interior angles have the same measure, 1 = A and 3 = C. Finally, by substitution, A + 2 + C = 180. Page 5
8. Indicate the argument that you think is the best. Justify your choice. 9. Indicate which argument that is the most convincing for you. Justify your choice. 10. Would you describe one or more of the arguments as a mathematical proof? (If so, indicate your choices.) Justify your answer. Page 6
Agree Somewhat Agree Neutral Somewhat Disagree Disagree Mathematics in My View 1. Mathematics is a collection of methods and rules, which precisely determine the solution of a task. 2. Mathematics is a logical, indisputable thought process with clear, precisely defined ideas and unequivocal, provable statements. 3. Almost any mathematical problem can be solved through the direct application of familiar rules, formulas, and methods. 4. Mathematics requires new and sudden ideas. 5. Doing mathematics demands a lot of practice in following and applying calculation routines and schemes. 6. Mathematics particularly requires formal, logical derivation and one s capacity to abstract and formalise. 7. Doing mathematics means: understanding facts, realising relationships and having ideas. 8. Mathematics is the memorising and application of definitions, formulas, mathematical facts, and methods. 9. Central aspects of mathematics are flawless formalism and formal logic. 10. Above all, mathematics requires intuition as well as thinking and arguing, both relating to contents. 11. Doing mathematics requires extensive practice in correctly following rules and laws. 12. Mathematics originates from setting axioms or definitions, then, by deducing theorems according to formal logic. 13. Clarity, exactness, and unambiguity are characteristics of mathematics. 14. Mathematical tasks and problems can be solved in various ways. 15. Any person can invent and re-invent mathematics. 16. Mathematics is of general, fundamental use to society. 17. Only a few things learned from mathematics can be employed later in life. 18. Mathematics helps to solve daily tasks and problems. 19. With regard to application and its capacity to solve problems mathematics is of considerable relevance to society. 20. Mathematics is a game free of purpose. It is occupying oneself with objects without any solid relevance to reality. Page 7