1 Group 3 Report Sequences & Series Marc, Joceline, Scott, Annie, John An important property of series is whether or not they converge.

2 Our problem for your consideration: Test  (1/n 2 ) for convergence. We conjectured that students might say this series converges because 1/n 2  0. This is a wrong reason for a correct answer.

3 Of course, some students could give a correct reason for the correct answer:  (1/n 2 ) converges by the Integral Test. We would not know which unless we ask them: Why is this true? That is, unless we ask them for a warrant.

4 Eventually students should ask themselves the question “Why is this true?” AND go on to check whether their warrant is OK by looking for a counterexample. For example, a student might know that  1/n) diverges, but note that 1/n  0.

5 We also conjectured that students who give the wrong reason might have the following misconception: They are confusing  a n converges implies a n  0 with its converse a n  0 implies  a n converges.

6 One FIX for this kind of behavior would be to encourage students to ask themselves “Why is that true?”, that is, to justify their answers, especially in the case of relatively new knowledge.

7 One might also want to address the possible misconception of assuming the converse is true by: 1.Giving an everyday example of a conditional where the converse does not hold. 2.Giving a (different, but easier) mathematical example where the converse does not hold.

8 We noted that in France a series is an ordered pair ( {a n }, {a 1 +a 2 +…+a n } ), which is not how series are presented in U.S. calculus texts.

9 Students need to know when to take examples to test whether they have gotten an answer right.

10 For example, in the Students-and- Professors problem: Write an equation for the statement: There are 6 times as many students as professors at this university. Use S for the # of students and P for the # of professors. Many people say: 6S = P. But those who test their equation with numbers will say 6P = S.

11 Getting students to ask of themselves, “Why is this true?” takes time, perhaps as much as 2 months of the instructor asking this question of students. We see this as an example of repeated reasoning, as well as practice. We noted that this (asking why something is true) is not specific to the calculus curriculum.

12 We also discussed the question: Find at least one solution to the equation 4x 3 – x 4 = 30 or explain why no such solution exists. together with the result that some students had used the Rational Roots Test. We conjectured that had the students asked themselves “Why is this true?” they might have been able to regroup and answer the problem correctly

13 We also discussed one more question from the Seldens’ nonroutine test: ax, x<= 1 Let f(x) = { bx 2 + x + 1, x > 1. Find a and b so that f is differentiable at students set ax = bx 2 + x + 1; 8 of those substituted x = 1 to get a = b+2 and stopped.

14 They could do the associated routine problems, but not this nonroutine problem. Their knowledge appears not to have been sufficiently connected.