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2014 11-13
1. What do students at UW get out of Calculus I?
Work for the Assessment Committee
Dr. Michelle Chamberlin, Dr. Nathan Clements
November, 13th, 2014
2. Agenda
1 Methods and context for the study
2 Students overall performance on the
3. nal exam
3 Results for four exam questions
4 Conclusions and next steps
5 Questions and discussion
Michelle
4. Motivation for the study
Each year we are required to assess the learning of our
undergraduate students in mathematics
Needed assessment activities for the 2013-2014 academic year
Decided to draw upon student work that Nathan had
collected in Calculus I
Michelle
5. What is assessment of student learning?
A process that all programs at UW are required to undertake.
Involves an iterative process of:
Posing a question about how well are students learning?
Gathering evidence about students learning
Interpreting and analyzing such evidence
Using the results to enhance teaching and learning within the
program
Repeat
We are asked to engage in such
cycles yearly
Michelle
6. 2013-2014 Assessment Question
Question: What understandings of the basic concepts and skills
from our calculus sequence do students exhibit on the
comprehensive
7. nal exams?
Assesses Objective 1 under Goal 1 of our Undergraduate
Learning Goals and Objectives
Goal 1: Students shall demonstrate a solid understanding
appropriate for the 1000- and 2000-level mathematics required
in their majors.
Objective 1: Show pro
8. ciency in basic skills and concepts
embedded in their courses
Michelle
9. Data Collected: Student Work on Final Exam
To target our eorts, we elected to focus on Calculus I
Students take a comprehensive and common
10. nal
Decided to analyze students work on selected questions from
the
11. nal exam
Provide a picture of student learning across dierent sections
and instructors
In Fall 2013, scanned student work on the
12. nal exam and
entered students overall scores and scores on each question
into a spreadsheet
To select speci
13. c exam items, we looked for questions with a
spread of scores
We selected 4 questions from the
15. Analysis of Student Work on Exam Questions
For each question:
1 Randomly sampled work from approximately 35 students
2 Articulated the procedures, skills, and understandings that
were associated with each question
3 Read through students work, coding common approaches,
dierent ways of thinking, and errors
4 Generated frequency information about the dierent
approaches, ways of thinking, and errors
5 Prepared an overall conclusion about students learning
Michelle
16. Overall Performance on Final Exam
Mean 70
St Dev 17.91
Q1 59.5
Median 72.5
Q3 84
Q1 59.5 3 8 1 6 4 0 2 Median 72.5 5 9 3 8 8 4 5 Q3 84 6 9 4 8 8 5 5 Max 98 7 9 7 8 8 5 5 All Scores (all sections)
Score
Frequency
0 10 20 30 40
5 10 20 30 40 50 60 70 80 90 100
Problem: 2 (all sections)
150 200
50 60
Nathan
18. First Question Analyzed
Question 3 Express the integral below as a limit of Riemann sums.
Do not evaluate. Z
0
sin(5x)dx
Solution lim
n!1
Xn
i=1
sin
5i
n
n
Michelle
20. nite integral represents the area under the curve
between two given xvalues (a and b).
We often use a sum of the areas of rectangles to approximate
the integral.
This approximation is more accurate as we use rectangles with
smaller widths
The value of the integral is found by computing the limit as
these widths go to zero, i.e., as the given segment along the
xaxis is broken into more and more (an in
21. nite number) of
subintervals.
The width of each rectangle is given by (b a)=n where n is
the number of partitions and is represented by x.
The height of each rectangle is found by evaluating the
function at any point within each subinterval (x
i )
Z b
Thus,
a
f (x)dx = lim
n!1
Xn
i=1
f (x
i )x
Michelle
22. 1st Q: Understandings and Procedures (cont.) {
R
0 sin(5x)dx
In addition, students need to understand:
How the limit relates to the variable n and how this interfaces
with the variable x
How the summation occurs and relates to the variable i
Limit and summation notation (and function notation)
Michelle
23. 1st Q: Results {
R
0 sin(5x)dx
Of the 264 students completing the exam, 54 earned full credit,
approximately 20%
Of the 35 analyzed:
All but one student understood that an integral represents the
area under the curve, that we use a sum of the areas of
rectangles to approximate this area, and that this is more
accurate when we use more rectangles
34% were able to determine the width and height for each
rectangle in the limit
89% struggled with the concept and formality of expressing
this idea as a limit
This struggle seemed compounded by students diculties with
understanding limit and summation concepts as well as limit,
summation, and function notation
Michelle
24. 1st Q: Common Errors {
R
0 sin(5x)dx
Taking the limit as x goes to in
25. nity, as x goes to n, or as x
goes to
Forgetting to write the limit before the summation notation
Forgetting to multiply by the width of each rectangle
Evaluating the sum from n = 0 to n = or from i = n to
i =
Michelle
26. 1st Question: Conclusion {
R
0 sin(5x)dx
Students appear to be leaving Calculus I with an understanding that:
The integral represents the area under the curve
It is estimated by summing the areas of rectangles, becoming
more accurate with more rectangles
However, they struggle with expressing this idea as a formal
limit
Likely compounded by diculties with limit and summation
concepts as well as associated notation
Michelle
27. Second Question Analyzed
Question 9 Evaluate
Z 1
0
x
(x2 + 1)3 dx
Solution: Requires using integration by substitution where
u = x2 + 1, and so du = 2x dx.
Z 1
0
x
(x2 + 1)3 dx =
3
16
Michelle
28. 2nd Q: Understandings and Procedures {
R 1
0
x
(x2+1)3dx
Procedure Understanding or Skill
Recognize the integrand as
a composed function
Recall substitution as a technique for integrat-
ing a composed function
Identify the associated
compositions of functions:
g(x) = x2 + 1,f (x) = 1=x3,
Being able to decompose a composite function
Selecting and substituting
u = x2 + 1
12
Skillfully selecting a u-substitution such
that its derivative occurs in the integral,
e.g., du = 2xdx, which implies du = xdx
Determine the antideriva-
tive
Application of antidierentiation formulas and
Rproperties of the integral; in particular
un du = 1
n+1un+1 + C; n6= 1
Using appropriate limits of
integration
Understand that with the change of variables,
the coordinate system upon which we are inte-
grating has changed
Apply FTC
R b
a f (x) dx =
F(b)F(a) where F0(x) =
f (x).
Michelle
29. 2nd Q: Results {
R 1
0
x
(x2+1)3dx
Of the 264 students completing the exam, 34% earned full credit.
Of the 35 analyzed:
85% realized the need for u-substitution
However, 89% of these students incorrectly determined the
anti-derivative, often misapplying the anti-dierentiation
formulas
Most students proceeded to evaluate the (often incorrect)
anti-derivative with the correct limits of integration
Michelle
30. 2nd Q: Common Errors {
R 1
0
x
(x2+1)3dx
Failing to realize substitution for du includes the x term
(leaving the x term in the numerator after usubstitution)
Failing to take an antiderivative before evaluating
Challenges with determining the antiderivative:
Misapplication of antidierentiation formulas to numerator and
denominator independently
Perceiving the antiderivative for u3 involves ln
Incorrect use of the Power Rule (subtracting the power by 1
rather than adding one for the antiderivative)
Forgetting to multiply by the coecient of 1
2
Michelle
31. 2nd Q: Conclusion {
R 1
0
x
(x2+1)3dx
Approximately 1/3 of students appear to be leaving with an
understanding of how to use the Substitution Rule to compute
a de
32. nite integral
Most other students are able to recognize when
usubstitution is warranted, are able to decompose the
associated functions, and can make an appropriate
usubstitution
Diculties arise when students try to determine the
antiderivative
Nearly all students appear able to apply the FTC
Michelle
33. Third Question Analyzed
Question 11 Find the value of the limit. If it does not exist, put
DNE.
lim
x!1
ln( 1
x )
Solution: 1
Mean 2.75
St Dev 1.41
Q1 2
Median 3
Q3 3
Score
Frequency
0 20 40 60 80 100 0 1 2 3 4 5
Frequency
Problem: 11 (all sections)
Score
Frequency
0 20 40 60 80 100 120
0 1 2 3 4 5
Frequency
0 20 40 60 80 100
Problem: 14 (all sections)
150
150
Nathan
34. 3rd Q: Understandings and Procedures { lim
x!1
ln( 1
x )
Using limit arithmetic to identify that this limit is of a
determinate, rather than indeterminate, form
Such limit arithmetic in this case includes analyzing the end
behavior of the function as x ! 1
Understanding that a limit may still exist even when unde
35. ned
at a particular point, e.g., for a limit we are interested in the
behavior of the function near a point, not necessarily exactly
at the given point, e.g., just because ln(0) is unde
36. ned does
not necessarily mean this limit does not exist
Understanding that lim
x!1
1
x = 0.
Understanding that the lim
x!0
ln x = 1.
Knowing when to appropriately use L'H^opitals Rule, which is
not called for in this problem.
Nathan
37. 3rd Q: Results { lim
x!1
ln( 1
x )
Of the 264 students completing the exam, approximately 20% com-
pleted this item
Of the 30 analyzed:
50% indicated a clear misunderstanding of limit forms
(determinate and indeterminate).
Of those 50%, half concluded that the limit does not exist
since the determinate form did not exist as a number.
The other half of the 50% misinterpreted the form as an
indeterminate form and applied L'H^opital's rule.
Five of the thirty got the right answer but provided no
justi
38. cation for their reasoning.
Many misapplied logarithm laws.
Nathan
39. 3rd Q: Conclusion { lim
x!1
ln( 1
x )
Understanding limit behavior in a symbolic way seems dicult
for many students.
Recognizing that an expression, though unde
40. ned
algebraically, produces a meaningful answer for large x is a
skill that is frequently not gained.
This work is evidence that students are not gaining a big
picture conceptual understanding of limits.
Nathan
41. Fourth Question Analyzed
Question 12 Find the value of the limit. If it does not exist, put
DNE.
lim
x!1
x2ex
Solution:
lim
x!1
x2
ex = lim
x!1
2x
ex = lim
x!1
2
ex = 0
Mean 1.83
St Dev 1.9
Q1 0
Median 1
Q3 3
Score
Frequency
0 20 40 60 80 100 120
0 1 2 3 4 5
Score
Frequency
0 20 40 60 80
0 1 2 3 4 5
Frequency
0 50 100 150
0 Problem: 11 (all sections)
Score
Frequency
0 20 40 60 80 100 120
0 1 2 3 4 5
Problem: 12 (all sections)
Score
Frequency
0 20 40 60 80 100
0 1 2 3 4 5
Frequency
0 20 40 60 80 100 120 140
0 Problem: 14 (all sections)
Problem: 15 (all sections)
Nathan
42. 4th Q: Understandings and Procedures { lim
x!1
x2ex
Recognizing it is an indeterminate form, but that L'H^opital's
Rule cannot be applied in its current form
Skillfully writing the product as a quotient such that
application of L'H^opitals Rule leads to a form for which the
limit can be determined
Recognizing the need to and applying lHospitals Rule twice
Nathan
43. 4th Q: Results { lim
x!1
x2ex
Of the 264 students completing the exam, 14% earned full credit.
Of the 30 analyzed:
43% either indicated that the limit was of an indeterminate
form or applied l'Hospital's Rule.
However, students then struggled with the indeterminate
form; most often failing to express the function as an eective
quotient.
Other common diculties included
incorrectly using limit arithmetic,
thinking the limit does not exist,
and stating that the limit is zero without providing justi
44. cation
or work.
Three students did determine the limit correctly using a
numerical approach, and one student wrote about the
dominant nature of ex over x2 .
Most students proceeded to evaluate the (often incorrect)
anti-derivative with the correct limits of integration
Nathan
45. 4th Q: Conclusion { lim
x!1
x2ex
Identifying and then evaluating limits of indeterminate forms
appear challenging for students.
While some students recognized the indeterminate form, they
still presented diculties with applying l'Hospital's Rule,
especially in this more challenging case that calls for applying
l'Hospital's Rule twice.
A few students appeared successful however by approaching
this problem through numerical means and by thinking about
which function dominates as x ! 1.
Students' work on this problem peaked our interest because it
allowed use to aske the question: Just how do students think
about indeterminate forms?
Nathan
46. Overall Conclusions
Recall we intentionally selected four questions where
performance was mixed.
These four items may appear to indicate that students are not
meeting Objective 1 of our Learning Goals: show pro
47. ciency
in basic skills and concepts embedded in the course.
However, students did do well on a signi
48. cant number of
other items on the exam, meaning Calculus I is meeting
Objective 1 in many ways.
From these other items, it appears that many students are
leaving Calculus I understanding
how to dierentiate functions;
how to graphically interpret functions, derivatives, and
anti-derivatives;
how to use rates of change to solve real-world problems;
how to use approximating rectangles to estimate the area
under a curve;
and how to evaluate integrals.
Nathan
49. Possible Future Plans
Develop some study resources including possible exercises,
class activities, and/or problems that could be used in future
semesters of Calculus I for preparing for the
50. nal exam or in
Calculus II as a review at the beginning of the semester.
Use what we learned about students' understandings in these
areas to write more conceptually-based exam questions for
future
51. nal exams across all calculus courses.
Consider using small-case interviews to further investigate
students' thinking on topics that emerged in this analysis;
e.g., how do students think about indeterminate forms when
evaluating a limit?
Nathan
