The Impact of DeltaMath: A Study that Examines the Effectiveness of Online Homework Versus Paperbased Homework on Algebra 2 Students’ Test Grades
Amy Lee
Teacher College, Columbia University
Fall 2014
Professor Walker
MSTM 5061
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Table of Contents
Introduction .................................................................................................................................................. 3
Relevant Research ........................................................................................................................................ 4
Research Design ........................................................................................................................................... 7
Hypothesized Findings ............................................................................................................................... 11
Suggested Modifications to Existing Programs ......................................................................................... 14
Implications for Policy and Practice ............................................................. Error! Bookmark not defined.
Bibliography................................................................................................................................................ 20
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Introduction
In the fifth unit of the Algebra 2/Trigonometry curriculum, students will learn essential
topics about quadratic functions and complex numbers. At the end of the unit, students are
expected to know how to determine the roots of a quadratic function by using completing the
square method, quadratic formula, and the graphing calculator (TI-83/84) (see Table 1).
Students should be able to describe the number of solutions and the nature of the roots by using
the discriminant. Moreover, students can create a quadratic formula based on the given sum and
product of the roots. Students will also learn how to write equivalent imaginary numbers and
perform operation of complex numbers.
New York State Standards Addressed
A2.A.24 Know and apply the technique of completing
the square
A2.A.25 Solve quadratic equations, using the quadratic
formula
A2.A.2 Use the discriminant to determine the nature of
the roots of a quadratic equation
A2.N.6 Write square roots of negative numbers in
terms of i
A2.N.7 Simplify powers of i.
A2.N.8 Determine the conjugate of a complex number
A2.N.9 Perform arithmetic operations on complex
numbers and write the answer in the form a + bi
A2.A.20 Determine the sum and product of the roots of
a quadratic equation by examining its coefficients
A2.A.21 Determine the quadratic equation, given the
sum and product of its roots
A2.A.26 Find the solution to polynomial equations of
higher degree that can be solved using factoring and/or
the quadratic formula
A2.A.3 Solve systems of equations involving one
linear equation and one quadratic equation
algebraically
A2.A.4 Solve quadratic inequalities in one and two
variables, algebraically and graphically
Common Core Standards Addressed
N-CN.7 Solve quadratic equations with real
coefficients that have complex solutions.
A-REI.4.b Solve quadratic equations by inspection
(e.g., for x2 = 49), taking square roots, completing the
square, the quadratic formula and factoring, as
appropriate to the initial form of the equation.
Recognize when the quadratic formula gives complex
solutions and write them as a ± bi for real numbers a
and b.
A-REI.7 Solve a simple system consisting of a linear
equation and a quadratic equation in two variables
algebraically and graphically. For example, find the
points of intersection between the line y = -3x and the
circle x2 + y2 = 3.
A-CED.1 Create equations and inequalities in one
variable and use them to solve problems. Include
equations arising from linear and quadratic functions,
and simple rational and exponential functions.
F-IF.9 Compare properties of two functions each
represented in a different way (algebraically,
graphically, numerically in tables, or by verbal
descriptions). For example, given a graph of one
quadratic function and an algebraic expression for
another, say which has the larger maximum.
Table 1 Comparing the New York State Algebra 2/Trigonometry standards with the Common
Core Standards.
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One of the main issues with the Algebra 2/Trigonometry curriculum is that the
curriculum is compacted with a variety of topics that students must master in a short about of
time. There is barely any room for extension of a topic and review. Each day is a new lesson.
Thus, students can easily be lost in this curriculum. In order for students to master the standards,
students must practice and homework is given to allow students to practice the material in class.
Two issues that arrived with the homework assignment is (1) not enough time was provided in
class to cover all the materials and (2) students were not sure if they were on the right track. The
solution to these issues was DeltaMath, which is an online homework assignment that is tailored
to the New York State Algebra 2/Trigonometry curriculum. Thus, the purpose of this paper is to
determine the correlation between the students’ performance data on the DeltaMath assignments
versus their actual test results.
Relevant Research
Figure 1 This is the Common Core Curriculum for the high school level
In Figure 1, EngageNY has created a curriculum map for the Algebra 2/Trigonometry
course of the sequence of topics as well as lesson plans for each module that is aligned to the
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Common Core standards (NYSED, 2014). The goal of the Common Core curriculum was to
provide a national curriculum where all the standards were the same in every states. A national
curriculum provide a common expectation of what students needed to know in the course and a
common focus of the big ideas (Porter, McMaken, Hwang, & Yang, 2011). A national
curriculum also allows teacher to share ideas across the nation and to provide higher order of
thinking that will prepare students to be college and career ready wherever they go (Porter,
McMaken, Hwang, & Yang, 2011). With this new rigorous, common curriculum, students are
given the opportunity to compete with other nations because they will have a stronger English
Language Art and Mathematics background.
Table 2 Degree of Congruence of State Standards as Compared to the Common Core State
Standards for Mathematics (Schmidt & Houang, 2012)
Schmidt and Houang determined that alignment of the individual state’s mathematics
curriculum was barely over 30% aligned to the Common Core curriculum (Schmidt & Houang,
2012). In general, New York State mathematics curriculum was similar to the Common Core
curriculum (Table 2). They also compared the cognitive demands of the Common Core and the
State Standards for memorize, perform procedures, demonstrate understanding, conjecture, and
solve non-routine problems. Schmidt and Houang determined that the demands for both
standards were fairly similar.
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Table 3 Contrasting Common Core (CC) and State Standards on Cognitive Demand
(Percentages)
Slavin, Lake and Groff investigated the effective programs in Middle and High School
Mathematics that provided mastery learning (Slavin, Lake, & Groff, 2009). Mastery learning
was defined as an approach to instruction in which students will meet the requirement of a
particular standard. Students must first obtain mastery of the prerequisite skills before they
obtain mastery of the new standards. Formative and summative assessments are given to
measure the students’ mastery level. Thus, the results from the DeltaMath assignment will be
the formative assessment and the results from the test will be the summative assessment.
Cox and Singer did a research on the effectiveness of Web-based homework versus
paper-based homework on calculus students’ test grades (Cox & Singer, 2011). The study
consist of 87 students and the students had Web-based homework from WebAssign and paperbased homework. The result was that the online homework allowed students to work
independently on the calculus problems because there was explanations and resources provided.
Student made a significantly improvement in the course because they had a better understanding
of the materials. This also comes from the fact that the WebAssign homework provided
immediate feedback. Students, however, liked the paper-based homework in addition to the
online homework. The hard-copy homework allowed students to use as reference when they
studied for an exam.
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In this study, the evaluand type was the high school students, who range from 8th graders
to 11th graders. The main question that I will be investigating is whether or not the DeltaMath
assignments improve the students’ test scores. Ultimately, the goal is for every students to make
growth at the end of the year and to pass the New York State Algebra 2/Trigonometry Regents.
This was also a personnel evaluand because I was able to evaluate my instructional method and
modify the lesson based on the areas that students struggled. Thus, questions that I asked were:
o What areas did the students struggle?
o What mistakes are the students making?
o How can I addressed these mistakes?
Research Design
Peter McIntosh, a high school math teacher at Oakland Unity High School, used Khan
Academy as a math engagement strategy (McIntosh, 2014). The students at Oakland Unity High
School had poor math study habits and had low homework completion. McIntosh incorporate
Khan Academy to reengage his students, increased homework completion and increased test
results. The contribution to the success was that the resources were provided for individual
student if they were stuck on a problem. There was immediate feedback for the student and the
teacher. The student was able to assess what areas they need to improve on and how to improve
on it. The teacher was able to use the data to make modification in the lesson plan by going
concepts that the majority of the students did not understand.
My research design is fairly similar to McIntosh. In the Quadratic Functions and
Complex Numbers unit, the students are given approximately four weeks to learn the material.
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The DeltaMath assignments are assigned with two questions from each topic within the standard.
Students must answer two questions correctly, otherwise the score within the standard will go
back to zero. The goal was to allow the students to obtain mastery within the topic. The
deadline was the day before the test. The test consist of 20 questions where there are 16 multiple
choice and 4 short response. Students were given 43 minutes to complete the exam (Figure 2)
Figure 2 Exam #5 – Quadratic Function and Complex Roots
Part I: Multiple Choice
Direction: Determine the best solution for each question. Make sure to write the solution on the line provided and
bubble the correct solution on the scantron. [2 points each]
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
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13.
14.
15.
16.
Part II: Short Response
Direction: Please show all work and write very neatly! [4 points]
17. A rectangular patio measuring 6 meters by 8 meters is to be increased in size to an area measuring 150
square meters. If both the width and the length are to be increased by the same amount, what is the number
of meters, to the nearest tenth, that the dimensions will be increased?
18.
Perform the indicated operations and give the answer in standard complex number form:
19.
20.
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All of the questions were taken from previous Algebra 2/Trigonometry regent exam.
There is an example of the rubric provided in Figure 3 for question #17 on the exam. I will be
analyzing students’ response for question #17.
Figure 3 Rubric for Question #17 on Exam #5
At the end of the exam, students were asked to fill out an online survey that was posted on my
Web Site at lee01m539.weebly.com.
Figure 4 Online survey
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Hypothesized Findings
I teach three classes of Algebra 2/Trigonometry and 75 students took the exam. Based on
Figure 5, at least 75% of the students passed the test. Group A and Group B had at least 50% of
the scores above 80%, but the data from Group B was more spread out. The range was
approximately 80%. Group C did not do so well as Group B and Group C. In addition, there
seems to be an outlier present in Group B.
Group C
Group B
Group A
0%
20%
40%
60%
80%
100%
Test Result (Percentage)
Figure 5 Student Test Result in comparing to the classes.
In Figure 6, at least 75% of students from all ethnicity passed the test, with Multi-cultural
students outperforming Hispanic, Asian, Caucasian, and African American students. The Asian
students had the most spread of data, and an outlier seems to be present.
Multi-cultural
Hispanic
Asian
Caucasian
African American
0%
20%
40%
60%
80%
100%
Test Result (Percentage)
Figure 6 Student Test Result in terms of Ethnicity
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There are 39 girls and 24 boys. Based on Figure 7, 100% of the girls scored higher than
64%. The data for the boys were more spread out and distributed normally.
Boys
Girls
0%
20%
40%
60%
80%
100%
Test Result (Percentage)
Figure 7 Student Test Result for Girls v. Boys
Based on Figure 8 and Figure 9, there seems to be no correlation between the result of the
DeltaMath assignments and the result of the test. Thus, the result of the DeltaMath assignment
could not be used to predict students’ test score. However, students who completed the
DeltaMath assignment two weeks prior to the test had above 95% on the test.
.
Group A
Group C
Linear (Group A)
Linear (Group B)
Linear (Group C)
y = -0.0342x + 0.8022
R² = 0.0059
100%
Quadratic Function Test Result
(Percentage)
Group B
80%
60%
y = 0.9995x - 0.0292
R² = 0.2755
40%
y = 0.167x + 0.6147
R² = 0.1394
20%
0%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
DeltaMath Assignment Results
(Percentage)
Figure 8 Student Exam Result v. Student DeltaMath Assignment Result
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Quadratic Function Exam Result
(Percentage)
Overall Exam v. DeltaMath Assignment Result
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
y = 0.5816x + 0.2705
R² = 0.0801
25%
35%
45%
55%
65%
75%
85%
95%
DeltaMath Assignment Results
(Percentage)
Figure 9 Overall Student Exam Result v. Student DeltaMath Assignment Result
After the exam, students had to go online to complete the survey. The students were asked
whether they refer the DeltaMath assignment or the Paper-based assignment or both. Majority of
the students chose DeltaMath because they liked the fact that it provided immediate feedback
and that DeltaMath provided an example to model how to do the problem.
50%
40%
30%
20%
10%
0%
DeltaMath
Paper-Based
Both
Figure 10 Results of Students’ Response to the Online Survey
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Suggested Modifications to Existing Programs
One suggested modification is to use a “flipped classroom approach.” Moore and Gillett did
a study using the “flipped classroom approach” where they had the students watch the lesson at
home and the students did their homework in class (Moore, Gillett, & Steele, 2014).
Table 4 Features of the Flipped Classroom Implementations
Gillett, Moore, and Steele’s research showed that the flipped classroom made a significant
impact on students’ test scores. They discovered that this approach allowed students to be more
independent. This allowed students to increase the amount of time spent on practicing on the
math problems in class and allowed teachers to make better use of planning time and technology.
In addition, Web 2.0 tools that were provided create a variety of learning experiences to enable
the development of students’ habits of practice (McCoy, 2014). A Web 2.0 tool enables the
student to enter data and create multimedia products using text, graphics, audio, and video. One
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Web 2.0 tool that is popular amongst students and teachers is glogster. A glogster is an online
learning sit that provides users with digital educational content. Users can create posters as
shown in Figure 11.
Figure 11 This glog demonstrated a group understanding and application of the distance and
midpoint formula
In addition, students can investigate the properties of quadratic functions and making the
connection to calculus by using the common difference between two points as shown in Figure
12.
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Figure 12 Curve Stitching Worksheet
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