Luke Robbins
Dr. Gareis
Assessment Analysis & Critique
Course: SEM Precalculus (2nd block) at Lafayette High School
Student Teacher: Luke Robbins
Cooperating Teacher: Rachael Smith (SmithR2@wjcc.k12.va.us)
Class size: 23 students
Unit: Sequences, Series, Mathematical Induction, Binomial Theorem (Chapter 11, SOL MA.4-6)
Content (Source, VA Standards of Learning):
MA.4:
The student will expand binomials having positive integral exponents through the use of the
Binomial Theorem, the formula for combinations, and Pascal’s Triangle. The student will use
problem solving, mathematical communication, mathematical reasoning, connections, and
representations to:
 Expand binomials having positive integral exponents.
 Use the Binomial Theorem, the formula for combinations, and Pascal’s Triangle to
expand binomials.
MA.5:
The student will find the sum (sigma notation included) of finite and infinite convergent series.
The student will use problem solving, mathematical communication, mathematical reasoning,
connections, and representations to:
 Use and interpret the notation: Σ, n, nth, and a .



n
Given the formula, find the nth term, an, for an arithmetic or geometric sequence.
Given the formula, find the sum, Sn, if it exists, of an arithmetic or geometric
series.
Model and solve problems, using sequence and series information.
MA.6:
The student will use mathematical induction to prove formulas and mathematical statements.
The student will use problem solving, mathematical communication, mathematical reasoning,
connections, and representations to:

Prove formulas/statements, using mathematical induction.
Content
Expand
binomials
having positive
integral
exponents.
Use the
Binomial
Theorem, the
formula for
combinations,
and Pascal’s
Triangle to
expand
binomials.
Use and
interpret the
notation: Σ, n,
nth, and an.
Given the
formula, find
the nth term,
an, for an
arithmetic or
geometric
sequence.
Given the
formula, find
the sum, Sn, if
it exists, of an
arithmetic or
geometric
series.
Model and
solve
problems,
using sequence
and series
information.
Prove
formulas/
statements,
using math
induction
Table of Contents
Knowledge Comprehension Application Analysis


11, 12a,
11, 12a,
12b, 12c,
12b,
13a, 13b
12c,
13a, 13b
+
+
11, 12a,
11, 12a,
12b, 12c,
12b,
13a, 13b,
12c,
14
13a, 13b

1, 2, 3, 4, 5, 6,
7, 9
Synthesis
Evaluation

10a, 10b
+
1, 2, 3, 4,
5, 6, 7, 9
1, 3, 4, 9
5, 6, 7, 9

1, 2, 3, 4,
5, 6, 7, 9

1, 2, 3,
4, 5, 6,
7, 9

8a, 8b
+
8a, 8b

8a, 8b
Out of a class of 23 students, 22 students have taken the test so far [one poor soul has
been sick for over a week, which will be a threat to the reliability of the assessment for him when
he does get around to taking it]. The test was graded out of 50 points. The scores were:
46, 30, 31, 40, 28, 36, 48, 50, 47, 44, 48, 41, 47, 42, 46, 46, 41, 49, 47, 46, 43, 38
Aggregate Data Analysis
Mean = 42.45 = 84.9%
s.d. = 6.3 = 12.6%
Median = 45 = 90%
Mode = 46 = 92%
Min = 28, Q1 = 40, Med = 45, Q3 = 47, Max = 50
The mean was an 84.9%, and the standard deviation was 12.6%. So, if we assume that
the data is a normal curve (which it is not, but let us pretend), then we can infer that 68% of the
students scored between a 72.3% (C-) and a 97.5% (A+), and 95% of students scored better than
a 59.7% (F+). The median was 90%, and the lower quartile was an 80%. This means that 75%
of the students scored an 80% or better on the test (B- or better). Overall, I believe that the
intended learning outcomes were met for this class. The test was written using the intended
learning outcomes and table of contents, which allowed me to create a valid and reliable
assessment. The test was administered by me on Wednesday, April 17, 2013 during 2nd block
(8:50am – 10:15am) at Lafayette High School. I gave the students the full block (85 minutes) to
the complete the test. The test was administered in a fair and reliable setting, as I separated the
desks before the students arrived. I also monitored the students constantly, keeping my eyes
peeled for cell phones or wandering eyes. I believe that my method of administering the test
prevented cheating entirely.
This is the score breakdown by letter grade, followed by a graph representing the data:
A (90-100): 11
B (80-89): 6
C (70-79): 2
D (60-69): 2
F (50-59): 1
12
10
8
Number
6
of students
4
2
0
A
B
C
D
F
Letter Grade
Only one student in the class of 22 failed, with a score of 28/50 or 56%, which is close to a
passing grade.
Aggregate Analysis with regard to the ILOs:
For the purposes of aggregate analysis, I will divide the ILOs into three distinct sections and
classify each problem as such. These are the categories:
1. Sequences & Series
2. Math Induction
3. Binomial Theorem
This is how the class scored on each section:
1. Sequences & Series: total of 8 questions
a. Data points: 4, 5, 6, 6, 3, 7, 8, 7, 7, 8, 5, 7, 5, 8, 7, 5, 7, 7, 7, 6, 6, 5
b. Mean = 6.18/8 = 77.3%
2. Math Induction: total of 2 questions
a. Data points: 0, 0, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2
b. Mean = 1.59/2 = 79.5%
3. Binomial Theorem: total of 7 questions
a. Data points: 7, 5, 3, 6, 7, 7, 5, 5, 5, 5, 6, 5, 5, 5, 6, 7, 7, 4, 1, 5, 3, 1
b. Mean = 5/7 = 71.4%
*Note: in determining correct/incorrect for individual items, the author has ‘rounded’ to integer
values for each student. This means that if a student answers a question “mostly correct” then it
is counted as correct in the data analysis. Similarly, if a student answers a question “mostly
incorrect” then it is counted as incorrect in the data analysis.
Looking at the student scores per section in the aggregate we can begin to draw
inferences about the nature and degree of student learning. The section breakdown shows us that
the students, as an aggregate, performed slightly worse on the binomial theorem section (71.4%)
than on the sequences & series and math induction (77.3% and 79.5%, respectively). This means
that student mastery of the binomial theorem was less than that of sequences & series and math
induction. I will infer that this trend is correlated with the fact that binomial theorem was the last
section taught before the test, and it was taught soon before the test. In the future I could allow
ample time between teaching the last section of a unit and administering the unit test.
We will now analyze the scores of two particular students, Student Z, the lowest-scoring
student, and Student X, a randomly-chosen student:
Student Z, Lowest-scoring student:
Aggregate Score: 28/50 = 56%
Items missed: #1, 7, 8b, 11, 12a, 12b, 12c, 13a, 13b, 14
Score by category:
1. Sequences & Series: 6/8 = 75%
2. Math Induction: 1/2 = 50%
3. Binomial Theorem: 1/7 = 14.3%
By analyzing Student Z’s results for each particular section, we can see that Student Z’s
biggest issue on the test was binomial theorem, scoring a devastating 14.3% correct on this
section. This indicates that for Student Z, some of the intended learning outcomes were met and
some were not. My near-term instructional decision, with respect to this particular student, is to
remediate this topic for him/her. I could perhaps offer an extra-credit assignment to earn back
points on this test, which will require the student to revisit that particular section of the unit.
This assignment could involve reading and outlining the section of the textbook or watching a
video on Youtube about the material. At any rate, Student Z, needs to try another method of
learning the content in the binomial section. Student Z also could use remediation or review of
math induction. Student Z performed satisfactorily on the sequences and series section. My
long-term instructional decision based on Student Z’s performance is to re-teach or spend time
reviewing binomial theorem before the final exam. Also, I plan to teach binomial theorem
differently next year, as that appears to be the main overall issue with regard to this unit.
Student X, a randomly-chosen student:
Aggregate Score: 41/50 = 82%
Items missed: #4, 5, 6, 8a, 13a, 13b, 14
Score by category:
1. Sequences & Series: 5/8 = 62.5%
2. Math Induction: 2/2 = 100%
3. Binomial Theorem: 5/7 = 71.4%
Student X scored worse on the sequences & series section than the other two sections.
This indicates to me that Student X could use review of the sequences and series material before
the final. However, the variance between the three sections was far less for Student X than it
was for Student Z. The specific items that Student X missed map back onto the table of contents
as application or analysis questions.
I believe that my instructional strategies were aligned with the expected cognitive level
that I had of the students. I think that this indicates the nature of student learning was in line the
expected content and cognitive levels, but the degree could have been higher. However the class
as an aggregate performed very well.
I believe that this I constructed and administered a valid and reliable test. I would edit
this test by changing the numbers, so that students next year cannot get a copy of the test from
this year’s students (which would greatly threaten the reliability). I would also teach binomial
theorem earlier on so that it is not so close to the test date. I would also do summative
assessment of binomial theorem in case I need to quickly reteach it.
To answer the ultimate question, yes, the students did learn from my instruction, and I
know because their performance individually and collectively demonstrated their understanding
(and in many cases mastery) of the content. I am more confident in my ability to teach an entire
unit after administering this test. I also am very thankful that I had such talented and hardworking students in my class! They deserve as much of the credit as I do!