yearly assessment report – mathematics 2016-17 · pdf file ·...

Yearly Assessment Report – Mathematics 2016-17 The following prompt guides discussion regarding the history of the outcome and learning being assessed, evidence of improved learning, evidence of improved teaching/pedagogy, and evidence of improved assessment. Please address each heading separately with relevant examples. I. Student Learning Assessed Identify the student learning outcome(s) being assessed along with a description or definition of that learning outcome suitable for an outside audience.

1. Students will be able to construct and evaluate mathematical proofs.

Description: Students will be able to write mathematical proofs and determine if mathematical proofs are correct. A mathematical proof is a deductive argument consisting of a formal series of statements showing that if one thing is true then something else necessarily follows from it.

2. Students will be able to solve complex, real-world problems using appropriate technologies. Description: A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics.

3. Students will demonstrate an understanding of undergraduate Mathematics including Calculus, Linear and Modern Algebra. Description: Calculus is a branch of mathematics that deals mostly with rates of change and with finding lengths, areas, and volumes. Linear and Modern Algebra are branches of mathematics that are concerned with mathematical structures closed under the operations of addition and scalar multiplication.

II. Changes to the Outcome(s), Program, and Assessment Identify changes made to the outcome(s), program, and/or the assessment of the outcome(s) since the last time it was assessed. Please note any changes made based on the results of the previous assessment. One of the artifacts used to assess programmatic outcome #1 was changed from “Final exam problems from Math 210” to the “Basic Proof Skills Test from Math 210.”

III. Artifacts Used in Assessment Describe the artifacts and methods of assessing the learning and how they illustrate that knowledge set or skill. If multiple artifacts or methods are used, please describe each. Attach any assessment tools, such as rubrics and assignment sheets, used in the process. Provide the number of artifacts collected since the last time this outcome(s) was assessed, a breakdown of the amount used in the assessment process, and why those were used (e.g. 20 artifacts collected per year with seven randomly selected artifacts from each year used in the study).

OUTCOME

ARTIFACT FOR ASSESSMENT

NUMBER OF ARTIFACTS COLLECTED AND USED

#1 Basic Proof Skills Test from Math 210 Math Skills Inventory from Math 400

22 11

#2 Calculus Mastery Exam from Math 307 Math Skills Inventory from Math 400

0 11

#3 Calculus Mastery Exam from Math 307 Math Skills Inventory from 400

0 11

The Calculus Mastery Exam is a test consisting of 15 calculus problems, 9 basic and 6 advanced. It is administered to all math majors taking MATH 307, which is required to be taken by all math majors. Due to an extremely unfortunate oversight, in Fall of 2016, the Calculus Mastery Exam was not administered to the students in MATH 307. The Math Skills Inventory is a test consisting of 8 problems in a variety of areas of undergraduate mathematics. It is administered to all math majors taking MATH 400, which is required to be taken by all math majors. The Basic Proof Skills Test evaluates students’ abilities to work with algebraic expressions, logic, and evaluate and write mathematical proofs. This test is administered to all math majors taking MATH 210, which is required to be taken by all math majors.

In each case, learning is assessed and mastery of each knowledge set or skill is measured by how well the students performed on these tests and problems. Accompanying this document are the Basic Proof Skills Test, the Calculus Mastery Exam, and the Math Skills Inventory.

IV. Results Provide the results of the assessment. Due to an extremely unfortunate oversight, in Fall of 2016, the Calculus Mastery

Exam was not administered to the students in MATH 307 so we have no results from this test for this year. The average score on the Math Skills Inventory in 2016-17 was 42.55%. The average score on the Basic Proof Skills Test from Fall 2016 was 60.87%.

V. Conclusions Provide conclusions regarding student learning based on those results as well as how those artifacts led to those conclusions. Also, provide conclusions about the assessment process, particularly when using multiple artifacts and/or methods.

Students’ abilities to write proofs and solve advanced mathematical problems are slightly below satisfactory this year. Performance on the Math Skills Inventory and the Basic Proof Skills Test led to this conclusion since our goal was to have students get 60% of these problems correct. In previous years, the scores on the Math Skills Inventory have always been at or above 60%. This year’s score of 42.55% is probably just an anomaly.

Again, due to an extremely unfortunate oversight, in Fall of 2016, the Calculus Mastery Exam was not administered to the students in MATH 307 so we have no results from this test for this year. We believe our assessment instruments do an adequate job of measuring what they were designed to measure.

VI. Assessment Use Provide the steps the program intends to take in regard to 1) the program and/or outcome itself, 2) improved student learning, and 3) the assessment process based on those conclusions.

No changes are planned at this time regarding the program and/or outcome itself or regarding student learning.

Regarding the assessment process, we plan on taking rigorous measures to be sure that the Calculus Mastery Exam is administered to majors in MATH 307 in the future, as it is supposed to be.

BASIC PROOF SKILLS TEST 1

NAME:

1. Find the set of all values of x for which

|2− x2| < 2

(a) (0,√2) (b) (0, 2) (c) (−2, 0) (d) (−

√2, 0)

(e) (−√2, 0) ∪ (0,

√2) (f) (−2, 0) ∪ (0, 2) (g) (−2, 2) (h) (−

√2,√2)

2. Let f(x) = x2+3 and let g(x) = 2x−7. Select all expressions that are equivalent to (f ◦g)(x).

(a) 4x2 + 52 (b) (2x− 7)2 + 3 (c) 2(x2 + 3)− 7 (d) 4x2 − 28x+ 52

3. If, in a class of 30 students, 8 own a cat, 15 a dog, and 5 own both, how many own only a cat?

(a) 8 (b) 3 (c) 12 (d) 0

4. In a school of 300 students, 80 students are in the band, 200 students are on sports teams, and50 students participate in both activities. How many students are involved in either band orsports?

(a) 30 (b) 70 (c) 150 (d) 230


5. Do the following two statements mean the same thing?

“If I am healthy, then I will come to class.”“If I come to class, then I am healthy.”

(a) Yes

(b) No

6. Do the following two statements mean the same thing?

“If it is Wednesday, then I lift weights.”“If I do not lift weights, then it is not Wednesday.”

(a) Yes

(b) No

7. Consider the statement:

“If I do not walk my dog today, I will walk my dog tomorrow.”

If the previous statement is true, determine if the following statement is true or false.

“I walk my dog on both days.”

(a) True

(b) False


For each of the statements below, indicate whether each statement is

(a) always true: true for any choice of the variables

(b) sometimes true: true for some variable choices, but not for all choices, or

(c) never true: not true for any choices of the variables.

8. For real numbers u and v:If u2 < v2, then u < v.

(a) always true

(b) sometimes true

(c) never true

9. For real numbers a and b,(a+ b)2 = a2 + b2

(a) always true

(b) sometimes true

(c) never true

10. For real numbers x and y, √x2 + y2 < x

(a) always true

(b) sometimes true

(c) never true


11. Below is a statement and three proofs. Select the proof of the statement that is correct andcomplete.

“For any positive numbers a and b, a+b2≥√ab.”

Proof A: Assuming that a+b2≥√ab,

Multiply both sides by 2 a+ b ≥ 2√ab

Squaring (a+ b)2 ≥ 4ab

a2 + b2 + 2ab ≥ 4ab

a2 + b2 − 2ab ≥ 0

(a− b)2 ≥ 0

Which is true for positive numbers. So the assumption was true.

Proof B: For all positive numbers

(√a−√b)2 ≥ 0

a− 2√a√b+ b ≥ 0

a+ b ≥ 2√a√b

a+ b

2≥√ab

So the result is true.

Proof C: Look at the diagram. The area of the large square is equal to (a+b)2. The unshadedarea is equal to 4ab. Since the area of the whole square is larger than the unshaded area, wehave

(a+ b)2 > 4ab

a+ b > 2√ab

a+ b

2>√ab

So the result is true.

a b

a

b


12. Below is a statement and three proofs. Select the proof of the statement that is correct andcomplete.

“If n2 is an odd number, then n is an odd number.”

Proof A: If n was an even number, then n = 2m for some m. Then n2 = 4m2 = 2(2m2) willalso be an even number. So, as n2 is an odd number, then n must be an odd number.

Proof B: If n was an odd number, then since odd × odd = odd, n2 is also odd. Then n2 + nwould be the sums of two odd integers and would therefore be even. Since n2+n = n(n+1) itis also the product of two consecutive numbers and so it certainly is even. Therefore n is odd.

Proof C: Since n2 is odd, we have n2 = 4m2+4m+1 for some m. This means n2 = (2m+1)2

for some m. So n = 2m+ 1, which is odd.


13. True or False: For every real number a, there exists a real number b such that a− b = 4.

(a) True

(b) False

14. True or False: There exists an integer x such that for every integer y, x+ y = 3.

(a) True

(b) False

15. Consider the statement“At least two of my friends are blond.”

Which statement is the correct negation of the statement above?

(a) “None of my friends are blond.”

(b) “At most one of my friends is blond.”

(c) “More than two of my friends are blond.”

(d) “Less than two or more than two of my friends are blond.”

16. Consider the statement“The number q is odd or it is less than three.”

Which statement is the correct negation of the statement above?

(a) “The number q is even and it is greater than or equal to three.”

(b) “The number q is even or it is greater than or equal to three.”

(c) “The number q is even and greater than three.”

(d) “The number q is even or greater than three.”

CALCULUS MASTERY EXAM

1. What is f ′(x) if f(x) = x5 + sin(x) + ex + 7?

A. 16x

6 − cos(x) + ex + 7x

B. 5x4 + cos(x) + ex

C. 5x4 + cos(x) + ex + 7

D. 0

E. 5x4 + cos(x) + xex−1

2. Find the equation of the line tangent to the graph y = x2 at x = 3.

A. x = 3

B. y = 6x

C. y = 6x + 3

D. y = 6x− 9

E. Not enough information.

3. Determine limx→∞x2+7x+6

4x2+11x−15 .

A. −25B. ∞C. 0

D. 14

E. 1

4. Determine∫ 2

0(3x2 + 6x + 2)dx.

A. 0

B. 8

C. 24

D. 20

E. 42

5. Integrate

∫10x

(5x2 − 3

)2/3dx.

A.2

3

(5

3x3 − 3x

)−1/3+ c

B.2

3

(5x2 − 3

)−1/3(10x) + c

C.3

5

(5x2 − 3

)5/3 (5x2)

+ c

D.3

5

(5x2 − 3

)5/3+ c

E. 3503 x

43 − 30 + c

Page 1 of 3


6. Determine the convergence of the infinite series∑∞

n=11n .

A. The series converges absolutely.

B. The series diverges to infinity.

C. The series converges conditionally.

D. The series diverges to negative infinity.

E. Not enough information.

7. Create a vector that starts at the point (4,−1, 2) and ends at the point (3, 1,−5).

A. −1i + 2j +−7k

B. 1

C. 7i + 3j +− 252 k

D. 1i− 2j + 7k

E. 7i− 3k

8. Let f(x, y) = (xy − 1)2. Compute ∂f∂y .

A. 2(xy − 1)

B. 2(xy − 1)y

C. 2(xy − 1)x

D. (xy − 1)2x

E. 1

9. Evaluate the integral∫ 1

0

∫ 1

0x2y dx dy.

A. 14

B. 16

C. 2

D. 12

E. 1

10. A total of L feet of fencing is to form three sides of a level rectangular yard. What is the maximumpossible area of the yard, in terms of L?

A. L2

9

B. L2

8

C. L2

4

D. L2

E. 2L2

Page 2 of 3


11. Given f(x) =√x +√x, find f′(1).

A.√

2

B. 34√2

C. 1√2

D. 3√2

E. 14√2

12. Determine the value of∑∞

n=12n+1

3n .

A. 3

B. 2

C. 32

D. 4

E. ∞

13. Find an equation of the plane that passes through the (noncollinear) points P (2,−1, 3), Q(1, 4, 0) andR(0,−1, 5).

A. 5x + 4y + 5z = 20

B. −12x + 3y + 9z = 0

C. −12x− 3y + 9z = −24

D. 10x + 8y + 10z = 42

E. 10x− 8y + 10z = 42

14. Compute the following indefinite integral:∫x ln(x)dx

A. 12x

2 1ln(x) + C

B. 12x

2 ln(x)− 14x

2 + C

C. ln(x) + 1 + C

D. 12x

2 ln(x) + C

E. 12x

2 ln(x)− 16x

2 + C

15. Solve the following differential equation: y dydx = x(y2 + 1).

A. y2 = cex2 − 1

B. 12x

2 = 12y

2 + ln(y) + c

C. ln(x) = 12y

2 + ln(y) + c

D. 12y

2 = 12x

2(y2 + 1)

E. 12y

2 = x(y3

3 + y)

Page 3 of 3

Skills Inventory of Mathematics MajorsMorningside College

Fall 2016

Please answer the following questions to the best of your ability. Show all work, and mentionwhen technology is used. All work and solutions should be written on a separate sheet ofpaper. Turn in this exam, along with your solutions, when you are done.

1. Give an example of a subset of the real numbers that is

(a) countable and infinite.

(b) uncountable.

2. Show that the sets A = N and B = Z have the same cardinality (have the samenumber of elements in them). Do this by exhibiting a one-to-one, onto function fromA to B.

3. Find the coordinates of the point of intersection of the functions f(x) = ex andg(x) = 2 + sinx, accurate to three decimal places.

4. Evaluate375∑n=1

1

n, accurate to five decimal places.

5. Suppose you are running a machine that both fills and drains a liquid from a container.You know that after 1 hour there are 12 L in the container, after 2 hours there are15 L, and after 3 hours there are 16 L. Find a quadratic polynomial that models thisbehavior. That is, find coefficients a0, a1, and a2 such that

a0 + a1(1) + a2(1)2 = 12a0 + a1(2) + a2(2)2 = 15a0 + a1(3) + a2(3)2 = 16.

6. A rectangle has length x and width 20 − x.

(a) Find a formula for the area of this rectangle, as a function of x.

(b) What domain makes sense for this function?

(c) Of all such rectangles having dimensions x and 20 − x, what are the specificdimensions of the one that has the largest area?

7. Prove by induction that 1 + 2 + 3 + · · · + n = 12n(n + 1) for every natural number n.

8. On Day 1, you are given $1. On Day 2, you are given $2. On Day 3, you are given$4. On Day 4, you are given $8. On each day that follows, you are given twice asmuch as on the previous day. The table below summarizes the situation.

Day Amount You Are Given Total Amount You Have

1 $1 $12 $2 $33 $4 $74 $8 $155 $16 $31...

......

30 ? ?...

......

n ? ?

(a) How much money are you given on Day 6? Day 7? Day 8?

(b) What is the total amount of money that you have on Day 6? Day 7? Day 8?

(c) Find a formula for how much money you are given on Day n.

(d) Find a formula for the total amount of money you have on Day n.

(e) How much money are you given on Day 30? What is the total amount of moneyyou have on Day 30?

yearly assessment report – mathematics 2016-17 · pdf file ·...

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