Running Head: ALGEBRA REASONING CURRICULUM
Algebra Reasoning Curriculum for Adult Learners
John Voight
Post University
Author Note
This paper was prepared for EDU 699.91 Capstone Project, taught by Professor AjtumRoberts.
1
2
ALGEBRA REASONING CURRICULUM
Capstone Algebra Reasoning Curriculum for Adult Learners
The “Mathematical Reasoning for Adult Learners” capstone project was designed to
improve the educational experience of adult learners. The school district that piloted this
program was the Connecticut Department of Correction’s Unified School District #1. This
curriculum was implemented in Adult Basic Education 2 classes, which was the equivalent of
fifth to eighth grade, as well as a specialized GED Mathematics courses, (ABE 3 / High School),
as early as November, 2014. This curriculum was created to improve algebraic thinking and
reasoning skills for incarcerated adult education students due to their relevance and importance
in areas such as decision making, problem solving in daily life, and in the workplace (U.S.
Department of Education, 2011, p. xi).
Rationale
The purpose of this capstone project was to design a curriculum unit using Understanding
by Design and become a model of successful practice for the Connecticut Department of Correction
Unified School District #1. The mission of this school district was to provide the incarcerated
population with educational programs focusing on providing academic knowledge, as well as
vocational, technological and life skills necessary for life-long learning and a successful transition
back to society. This Algebra curriculum unit integrated technology, active learning strategies,
differentiated instruction, collaborative learning as well make real world and career connections
throughout the curriculum unit. The standards listed in this curriculum were taken from the
Common Core State Standards for Mathematics, College and Career Readiness Standards for Adult
Education, and the Unified School District #1 GED and College Readiness Curriculum standards.
3
ALGEBRA REASONING CURRICULUM
The significance of this project was to prepare correctional students for life after their
incarceration as well as success on the 2014 GED test. This population had displayed a current
trend of low mathematics scores especially in Algebra. This section of the new GED test
constituted to 55% of the exam. Algebraic reasoning is essential to adult learners in solving real
world problems by exploring patterns and relationships amongst numbers, data or graphs. This
Algebra Reasoning curriculum not only provides teachers with enduring understandings,
instructional strategies, and lesson plans but also students the opportunity to acquire new
knowledge, to expand on that knowledge, and transfer their learning to real world situations.
This project enabled correctional educators the opportunity to rejuvenate their instructional
strategies, lessons and activities to meet the ever-changing needs of the adult learner to ensure a
successful reintegration back to the workplace and everyday life upon their release. This
curriculum can be used for professional development and as a supplemental resource. This
curriculum is also used by adult educators looking for teaching activities and lessons that
incorporate algebraic reasoning that correlates to common core, career & college readiness, GED
and school district standards
The changing methodology of the GED test has led adult educators
to initiate curriculum reform to better prepare students for the new test as well as the ever changing
society and workplace. This curriculum would fit the district’s current need by providing a
comprehensive algebra reasoning curriculum that coincides with the changing test while adhering to
state standards.
Research
The literature review revealed education for the incarcerated adult learner is vital in many
ways. An education while in prison reduces the recidivism rates among released offenders as
well as provides them with knowledge, skills and confidence to gain employment (Berridge &
Goebel, 2013). Berridge & Goebel stated: “The GED, a job, and a greater sense of self-worth
ALGEBRA REASONING CURRICULUM
were seen as elements of a new life story, a path leading to a better attitude about going and
applying myself. Education was widely seen by these inmates as the path to financial success.
Some saw the GED as a stepping-stone to community college, a trade school, or a specific
degree” (p. 75). The U.S. Dept. of Education (2011) stated: “Adults bring with them
experiences, both positive and negative, that influence their learning. Mathematics instruction
should attend to these experiences and proceed in a practical, coherent, and integrated way, with
careful monitoring of student progress” (p. xi). This department reiterated the importance of
improved mathematical knowledge: “Some researchers have suggested that success in the
workplace requires mathematical knowledge and problem-solving abilities broader than those
fostered by instruction that emphasizes procedures. Across multiple workplace environments,
commonly used skills include algebraic thinking, estimation, judging the necessary degree of
accuracy, logic, and managing and interpreting data” (p. 9).
This project would focus on the Mathematical Reasoning section of the 2014 GED exam
and specifically the Algebra Module due to the many changes with methodology that were
implemented from the previous test produced in 2002. The 2002 exam was based on Bloom’s
Taxonomy. According to the GED Assessment Guide for Educators (2013), Bloom’s Taxonomy
focused on the learner’s activity such as analyzing or creating while Webb’s DOK models
“emphasis is on the complexity of the cognitive processes that each of those activities requires
on the part of the learner” (p. 23). This new exam is based on Webb’s Depth of Knowledge
(DOK) that incorporates recall, basic application of skills, and concepts as well as strategic
thinking (p. 2.4). The new GED exam utilizes the first three levels culminating with the strategic
thinking level which: “Requires reasoning, developing a plan or sequence of steps; have some
complexity and more than one possible answer” (p. 2.4). This project came to light as a result of
discussions between adult educational professionals who came to the realization after viewing
4
5
ALGEBRA REASONING CURRICULUM
poor test data, that many incarcerated students lacked knowledge of mathematical reasoning and
problem solving skills, including Algebra, due to either a discontinuance of high school or other
extenuating circumstances. U.S. Department of Education (2011) stated the importance of
mathematics for adult learners: “Adult students should be able to demonstrate all aspects of
mathematical proficiency: conceptual understanding, procedural fluency, strategic competency,
and adaptive reasoning. Through their learning experiences, they also should be developing a
productive disposition toward learning and using mathematics” (p. 24).
Implementation
This curriculum consisted of two units based upon particular conceptual skills, each with
four module applications referenced from the Unified School District #1 GED and College
Readiness Curriculum standards. Each module application was scheduled to take two weeks to
complete depending on academic level as well as student understanding and each unit was
planned to be completed in eight weeks. Each unit had enduring understandings, essential
questions, detailed descriptions of what students will know and what they will be able to do with
this knowledge. Each unit featured performance tasks, assessments evidence and learning
activities. For example, unit one focused on the conceptual skill: Linear Equations and
Inequalities. This module application included: Evaluating linear expressions, solving linear
equations, solving linear inequalities and using Algebra to solve real world problems. See table
1.
ALGEBRA REASONING CURRICULUM
Table 1.
The enduring understandings of the unit include:
Real world situations can be represented symbolically and graphically.
Real world problems can be translated into algebraic expressions, equations, and
inequalities.
Algebraic expressions and equations generalize relationships from specific cases and can
solve real world problems.
Using algebra is essential for passing the GED exam and preparing for college and work
readiness.
Essential questions for this unit include:
How can expressions, equations and inequalities help to describe patterns in our world?
How can we use expressions, equations, and inequalities to model and solve real-world
problems?
How can a system of equations and inequalities support you in solving real-world
problems?
What makes a strategy both effective and efficient?
How do mathematical models/representations shape our understanding of mathematics?
This Algebra Reasoning curriculum also included what students would know and learn
from each module. As evidenced from the Unified School District #1 GED and College
Readiness Curriculum, students would know from Module 1, that a linear expression is a type of
algebraic expression in which no term can have two or more variables, nor can it have exponents
or roots and numbers that stand alone (without a variable) are constants. They would also know
that numbers that are multiplied by a variable are called coefficients and that performing
operations with expressions may include: multiplying expressions using the Distributive
Property, simplifying expressions by combining like terms, or evaluating expressions by
substituting a value for the variable.
Students would also be able to display what they could do such as write algebraic
expressions from phrases, identify and label the variable, the coefficient, and the constant in a
linear expression. They are able to demonstrate the use of the Distributive property in
multiplying expressions and simplify algebraic expressions by combining like terms as well as
6
ALGEBRA REASONING CURRICULUM
evaluate linear expressions by substituting values for the variable and add and subtract linear
expressions. Students can also identify examples of when evaluating linear expressions is used
on the GED exam and in college readiness and write and/or interpret word problems requiring
the use of linear expressions.
The performance tasks and key criteria for the assessment section of this curriculum were
referenced from McGraw Hill Education’s Common Core Achieve Instructor Resource Guide.
One performance tasks of Module 1 would be to have students use algebraic expressions to
represent the cost of buying shares of stock in two companies and evaluate the expressions to
decide how much they can spend. The learning activity had students decide which companies
they would like to invest in and how much those investments will cost. Students were given a
table with the names of seven companies, their current stock prices per share, and the change in
their stock prices from yesterday. Students were asked to write an expression for each company
representing the amount invested in terms of the price per share and the variables representing
the number of shares they plan to buy. Students must decide how many shares of each stock
they want to buy and complete the table showing how much it will cost. The total invested can
be no more than $1000. The closing activity, or exit ticket, would be to ask students what are
dome reasons most stock prices change every day and how do these changes affect the
expressions used to determine the value of x shares?
An example of the key criteria for this performance task included probing questions to
assess knowledge such as: Suppose they share prices of the two companies you bought dropped
by 15% ? What expression could you write to show how much money you would have lost? On
the other hand, suppose the share prices went up by 15%. Now what expression could you use to
show how much your investment is worth? Other evidence of assessment would include teacher
observation of students working on tasks, assess understanding through class conversation/
7
ALGEBRA REASONING CURRICULUM
questioning by having students answer essential question and provide their interpretations or any
misunderstandings. Students would also be asked to solve real world Math problems, workplace
skill question and skill practice questions.
An example of student self-assessment for this unit would include exit slips, index cards,
one- minute essay and/or student reflection. This self-assessment section was cited from the
Alexandria City Public Schools Algebra I Curriculum. See table 2.
Table 2
Exit Slips:
In the closing moments of class, students are asked to respond to questions geared towards
assessing their own self-assessment such as: What do you understand better now regarding
reasoning with equations and inequalities that you did not understand before this module?
Index cards:
Side 1- Based upon the module topic, summarize a big idea that you now understand such as
enduring understandings.
Side 2 – Identify either as a statement or question something about this module that you do not
fully understand regarding solving systems of equations.
One minute essay:
Students are asked to complete a one-minute essay summarizing what they now understand
about linear equations.
Self-reflection:
Students are asked to reflect on how he or she contributed to their own learning process and
evaluate how they could have done better or if they require additional support on a particular
area by completing a short write exercise.
This report included two lesson plans that were designed using Understanding by Design
and referenced from McGraw Hill Education’s Common Core Achieve Instructor Resource
Guide. One of these lesson plans included in Module 1/ Unit 1 and the other lesson plan would
be included in Module 1/ Unit 2. These lesson plans include desired results, assessment
evidence, learning plans, materials and activities. Any educator interested in obtaining these
lesson plans can retrieve them from the author’s blog entitled:
educationbehindbars.wordpress.com. Also available on this blog is the Unit 1 and Unit 2
8
9
ALGEBRA REASONING CURRICULUM
Algebraic Reasoning Curricula as well as this report and additional resources relevant to
educating incarcerated adult learners.
Evaluation
This project was evaluated by two correctional educators that have over thirty-six years
of correctional education between them. One teaches ABE 3 & GED classes and specializes in
English Language Arts. She is also on the English Language Arts curriculum board for the
district. The other teacher is a culinary instructor who specializes in career skills and is on the
affective domain board for the district. Affective domain includes student’s attitudes, values,
and providing them with a positive learning environment and engaging lessons to motivate them
to learn. They both stated that their evaluation of this curriculum provided them with an
opportunity to reflect on their own teaching practices and resources as well as motivated them to
update their current curriculum and instruction. One of my colleagues particularly commented
on the vital piece of using essential questions to focus on the “big understandings” and why it is
important to their students to understand Algebraic Reasoning and how it is relevant to their
lives. She believed that by focusing on the real life applications of the lesson, students become
more engaged in their own learning and see how vital higher order thinking and problem solving
skills are to their lives. She also stated that this curriculum was thorough, comprehensive and
very useful for the incarcerated population. The culinary teacher stated the rubric that I adapted
and referenced from Kinnelon Public Schools curriculum writing rubric, was specific to the
incarcerated population and represented Understanding by Design as well as student learning,
curriculum implementation and student assessment/evaluation. They both believe by
incorporating the latest andragogical practices and curriculum, this will continue to enhance their
teaching instruction as well as student learning and understanding. A copy of one of my
10
ALGEBRA REASONING CURRICULUM
colleague’s evaluation and the rubric used to compile this project evaluation can be seen in
Appendix A.
Professional Growth
This project was a compilation of learned processes which included the development and
organization of lessons and units of instruction which would build on students’ prior knowledge,
skills and interests. This curriculum incorporated higher level learning of content skills and
concepts to actively engage students to think critically, creatively and problem solve. This
project also displayed knowledge of curriculum design using Understanding by Design standards
and criteria “with a focus on developing and deepening students’ understanding of important
ideas” (McTighe & Wiggins, 2004, p. 3). This curriculum reflected the relevant stages of
Backward Design which included identifying desired results, determining acceptable evidence
(assessment) and planning learning experiences and instruction (p.8).
The goal of this capstone project and Algebra curriculum reform was to enhance
students’ understanding and achievement while providing the incarcerated adult learner with
essential work and life skills necessary for a successful transition into society. The Algebraic
reasoning test was changed to reflect changes in society, technology, career and college
readiness, and common core standards. By using backward design, students saw the overarching theme or big picture. Units of instruction and lessons vary to meet the individual needs
of all students by offering differentiated instruction, relevant performance tasks, designated
content goals and objectives, use formative assessments to show evidence of learning, and
develop real world and career connections to engage and motivate learning.
11
ALGEBRA REASONING CURRICULUM
Key M. Ed. Outcomes
The “Algebraic Mathematical Reasoning for Adult Learners” capstone project incorporated
the following core program outcomes:

Students will develop a vision for the future of education, a personal critical and creative
perspective on issues and changes in education, and several means to adapt to future realities.

Students will develop expertise in designing and delivering instruction to support the
achievement of a diverse population of learners of all ages in a variety of settings.
This project reflected a vision for the future since it identified a potential issue and
enacted curriculum reform that reflected the methodology and technology of the new 2014 GED
exam. This new exam featured increased usage in technology and content based material that
needed to be addressed to prepare students properly for the exam. Adult education professionals
have to adapt their instruction and learning objectives to keep pace with the changing landscape
of adult learning as well as today’s workplace. They also have to design and deliver instruction
to a diverse population including incarcerated men from varying backgrounds, experiences, and
academic performance levels. This capstone project of curriculum reform reflected on numerous
aspects and techniques of designing and delivering of instruction to meet the needs of all
students.
Teaching and Learning Outcomes
This project also incorporated the following outcomes in the teaching and learning
concentration:

Students will be able to design and develop curriculum using models of curriculum design
such as Understanding by Design (UbD)
12
ALGEBRA REASONING CURRICULUM

Students will be able to use selected advanced instructional strategies appropriate for specific
learning outcomes.

Students will be able to apply a variety of formative and summative assessment strategies to
evaluate and improve teaching and learning.
This capstone project incorporated many facets of this master’s program teaching and
learning concentration. By developing this new curriculum, students will be able to learn and
understand new knowledge concerning Algebraic reasoning due to the changing methodologies
of the 2014 GED test including Webb’s Depth of Knowledge, Common Core, and college &
career readiness standards. Adult education professionals could also use this curriculum for
professional development to incorporate in their classrooms as well as engage students with
updated activities and lessons to improve learning and understanding. Ongoing formative
assessments will be used to guide teaching and learning to improve student learning and
performance (McTighe & Wiggins, 2004, p. 3). These assessments will also be aligned with the
learning goals and be used to modify instructional decisions if necessary.
Technology
Another aspect of the capstone experience that was learned and incorporated was the use
of technology. With the incorporation of the new 2014 GED Exam, students will be required to
become proficient with technology such as navigating and keyboarding since the test is computer
based. Due to security concerns, a correctional classroom is unable to integrate web-based
technology but computer skills software can be used as well as smart-board technology to
enhance student learning. If a correctional classroom was capable of web-based technology,
students could become accustomed to the new test format by participating in practice GED tests
13
ALGEBRA REASONING CURRICULUM
online to become comfortable taking the whole test on a computer. The only type of hand held
technology that is allowed in a prison classroom is the calculator. Lessons were incorporated
into this curriculum to get students accustomed to using this technologically advanced calculator.
Students need to become comfortable in using this Texas Instrument Scientific TI-30XS
calculator since it is imbedded into the mathematical reasoning section of the new computerized
GED test.
This author has had the pleasure of creating and designing a blog entitled, Education
Behind Bars, and has learned how to “tweet.” Technology will continue to play a major role in
education not only in the classroom but outside the classrooms as well by enabling teachers to
communicate with one another to share instructional ideas, lesson plans and resources through
the creation of personal learning communities. This Algebra Reasoning Curriculum and
Capstone Report are available on the author’s blog for teachers from all educational
environments to use and share. This capstone project was a culmination of learned concepts and
processes to not only improve educator’s pedagogical practices but also improve their students’
learning, engagement and understanding.
Reflection
This program has been instrumental in my development as not only an educator but as a
life- long learner as well. The capstone project was a culmination of the relevant and useful
learning that I have received at Post University. This included being able to design a curriculum
using “Understanding by Design,” developing lessons that are engaging and incorporating the
latest teaching strategies available. This program helped me to be a better educator by providing
us with pertinent content necessary in our field such as cognitive processes of student learning,
integrating diversity in our lessons, using assessment data collection as well as infusing today’s
14
ALGEBRA REASONING CURRICULUM
technology into our classrooms. I also feel as though this program has made me a better lifelong learner and student by improving on my researching and writing skills including web
searches, writing literature reviews and adhering to particular writing styles such as APA.
The
communications between student and faculty provided numerous learning opportunities,
differing perspectives as well as experiences using new technologies which included discussion
boards, blogs and “twitter”. The usage of technology has opened up various opportunities to
communicate with faculty, students and other educators through the formation of personal
learning communities. Through these PLC’s, educators can discuss, share and revise lessons,
curriculums and teaching strategies with the common goal of improving student learning and
understanding.
This algebra reasoning curriculum has become a valuable resource for many Adult Basic
Education classes in my educational setting. My colleagues are also using this curriculum for
their Algebra units since it is relevant for today’s ever-changing educational environment while
adhering to common core as well as the GED, school district, and college & career readiness
standards. My colleagues feel as though this curriculum provides today’s incarcerated adult
learner with knowledge and strategies necessary to succeed on the new GED test as well as
prepare them for the life after prison. By developing and improving student analytical thinking
and problem solving skills, we hope they will be able to make better decisions and adjust to life
after their release.
I recommend that we as a faculty continue to develop new curricula for all content areas
that we teach so we can be proactive in our instruction in preparing our students for what lies
ahead with not only the new GED test but growing demands of today’s workforce as well. I plan
to continue to use the knowledge, perspectives, and skills that I have gained in this program to
make a positive impact on my students to help them lead successful and productive lives upon
ALGEBRA REASONING CURRICULUM
their release from prison. Thank you Post University for providing me with a tremendous
learning experience by offering relevant content courses featuring today’s latest technology and
providing me with opportunities to become a better life-long learner, educator and person.
15
16
ALGEBRA REASONING CURRICULUM
References
Alexandria City Public Schools (n.d.). Learning to live. Loving to Learn: Algebra I Curriculum.
Retrieved from: http://www.acps.k12.va.us/curriculum/design/sample-algebra-course.pdf
Berridge, G., & Goebel, V. (2013). An In-depth Look at a GED Program in a U.S. County Jail.
Journal of Research and Practice for Adult Literacy: Secondary and Basic Education.
Vol. 2 Number 2. Retrieved from EBSCO.
Common Core State Standards Initiative. (2010). Common Core State Standards for
Mathematics. District of Columbia: Author.
GED Testing Service. (2013). Assessment guide for educators. Syracuse, New York: New
Readers Press.
Kinnelon Public Schools: http://www.kinnelonpublicschools.orgboe/Curric/writing/UBD Review
Rubric 0607.doc [30 September 2010]. Retrieved from
http://conference2011.flatclassroomproject.org/file/view/UnderstandingbyDesignUnitPla
nnerRubric.pdf
McGraw Hill Education. (2015). Common Core Achieve: Mastering essential test readiness
skills for high school equivalency exams. Instructor Resource Guide. Columbus, Ohio:
Author.
McTighe, J., & Wiggins, G., (2004). Understanding by design: Professional development
workbook. Alexandria, VA: ASCD.
Pimentel, S. (2013). College and Career Readiness Standards for Adult Education. Berkeley,
CA: MPR Associates.
ALGEBRA REASONING CURRICULUM
Unified School District #1 (2014). GED and College Readiness Curriculum. Wethersfield, CT:
Author.
U.S. Department of Education. (2011). Office of Vocational and Adult Education. Building on
Foundations for Success: Guidelines for Improving Adult Mathematics Instruction.
Washington, D.C.
Wiggins, G., & McTighe, J. (2006). Understanding by design. Alexandria, VA: ASCD
17
18
ALGEBRA REASONING CURRICULUM
Appendix A
ALGEBRA REASONING CURRICULUM
19