Author Jason Martin
Journal Educational Studies in Mathematics
Title Differences between expert and student conceptual images of the mathematical structure of Taylor series convergence
Affiliation University of Central Arkansas
Address Department of Mathematics, MCS 234, 201 Donaghey Ave., Conway, AR 72035
E-mail ; Office (501) 450-5653;Fax (501) 450-5662
Appendix
Selected questionnaire tasks with selected subtasksreferenced in the paper. Original section and task numbering has been retained.
Appendix Table 1 Common language of structural image tasks and results
Image / Common Languagea / Student Group / TotalCalculus / Analysis
Particular x / You decide the convergence of a Taylor series to a function by plugging in numbers for x and noticing that the series converges to the function for each number x. / 39
(38%) / 10
(36%) / 49
(37%)
Sequence of Partial Sums / You decide the convergence of a Taylor series to a function by considering the sequence of Taylor polynomials {T0(x),T1(x),T2(x),…,Tn(x),…} and noticing that this converges to the function. / 35
(34%) / 10
(36%) / 45
(34%)
Dynamic Partial Sum / You decide the convergence of a Taylor series to a function by considering an nth degree Taylor polynomial and adding more and more terms and noticing that the Taylor polynomial converges to the function as n goes to infinity. / 43
(42%) / 17
(61%) / 60
(46%)
Remainder / You decide the convergence of a Taylor series to a function by considering the difference between the function and an nth degree Taylor polynomial and noticing that this difference goes to zero as n goes to infinity. / 41
(40%) / 14
(50%) / 55
(42%)
Termwise / A series converges whenever . / 60
(58%) / 17
(61%) / 77
(59%)
aSee Tasks 19 and 28 for exact language.
Note. Percents within each student participant subgroup are given in parenthesis within each column. Percents of all novices are given in parenthesis in the “Total” column. N=131, and the sample sizes for the calculus and analysis students were 103 and 28, respectively.
SECTION 3: SHORT ANSWER
Legibly write down your answersto the following questions. Please show all you work. Feel free to write down anything that comes to your mind while attempting each task.
5)Using the graph of below, on the same axes sketch two different Taylor polynomials for sine.
6)Using the graph of below, on the same axes sketch the Taylor series for sine.
SECTION 4: MULTIPLE CHOICE
Please completely read each question and ALL responses before circling your answers. Then circle ALL responses that apply to each question. Feel free to provide any other responses that may have came to your mind. The real number system applies to all problems.
19)A student correctly writes the following:
for all numbers x.
The student also correctly writes the following:
We can define by,
,
,
,
etc.
Circle ALL responses that you can correctly conclude based off of what the student has written.
a)You decide the convergence of the series to by plugging in numbers for x and noticing that converges to for each number x.
b)You decide the convergence of the series to by considering and noticing that this converges to .
c)You decide the convergence of the series to by considering and adding more and more terms and noticing that converges to as n goes to infinity.
d)You decide the convergence of the series to by considering the difference between and and noticing that this difference goes to zero as n goes to infinity.
[There were additional parts to Task 19 not related to this article]
21)A student correctly writes the following:
Taylor’s Inequality states that if there exists a number M such that for all x in an interval, then the remainder function of the Taylor series for at satisfies the inequality for all x in the interval.
The student also correctly notes the following:
A Taylor series for is given byand for all numbers x.
The student now relates Taylor’s Inequality to the function and correctly writes:
Since for all x in the interval , the remainder functionof the Taylor series for at satisfies the inequality for all x in .
The student then correctly notes the following:
The for all numbers x.
Please read ALL responses before circling.
Circle ALL responses that you can correctly conclude based off of what the student has written.
a)Since for all numbers x, for all numbers x.
[There were additional parts to Task 21 not related to this article]
SECTION 5: TRUE / FALSE
Circle T for True or F for False. If you circle either T or F but you are not confident please also circle G for Guess. If unable to make an educated guess please only circle DK for Don’t Know.
28)A series converges whenever...... T / F / G / DK