Implementing
Math Strategies with Special Education Students
To
Increase Math Skills
Julie
Pate
Walden
University
Carol
Luetzow, Ph.D.
EDUC
6733P1 Action Research For Educators
May
20, 2012
Implementing
Math Strategies to Special Education Students To Increase Math Skills
I am currently working as a paraprofessional in a special
education selfcontained class of 2^{nd} to 5^{th} grade
students. The class consists of six students now with various disabilities. The
disabilities are Down’s syndrome, moderately intellectually disabled with
speech impediments, mildly intellectually disabled with speech impediments, and
Attention Deficit Hyperactivity Disorder. The school is a Title 1 school where
the population starts with black students who are the majority, and Hispanic
and white students who are at about the same percentage in the school (Georgia
Department of Education, 2011). The curriculum used in the special education class
is the Unique Learning System (2011) curriculum designed for special education
students. This company offers predesigned lessons, stories, and a variety of
activities to use from math, language arts, word list, science, social studies,
and different themes each month. They have online pre and post test for each
students that the teacher can set for each student’s academic level. I started
after Thanksgiving break full time in the class; I worked as a substitute last
year and I can see student’s growth in different areas of academic and social
skills since last year. One of my jobs in the class is to administer the online
pre and posttests with each student. I reflected on the problem areas that
each student seems to have in common which is addition and especially
subtraction problems.
In reflecting about the students’ math skills during
different lessons, online tests, and games, I see where the students have more
problems understanding the concept of subtraction using just the numbers in a traditional
algorithm equation. With the Unique (2011) curriculum, the students have math
problems with pictorial representation of the numbers in the problem. This does
help the students sometimes; however, some of them will not count every picture
thus giving them the wrong answer. I reflect every day about what interventions
could possibly help each student do better with the special education teacher
and the other paraprofessional the room. The teacher and I have a great
relationship, so I feel that I can talk to her about anything I see as a
problem with how a student is learning or not learning content on the level
they should be achieving. I work closely with all of the students because we
have a rotation between students because of some of the students have more
severe behavior issues than other students. Therefore, we have three students who
we rotate from day to day where a teacher will deal with one student because of
his/her major issues. I think that building a math foundation and offering
students multiple ways to calculate equations will help them make the
connections between addition and subtraction; thus, students will start to
develop critical thinking skills in problem solving math problems.. I know that
math is used in all types of employment, so being fluent in math numeracy is
essential in life. I have observed students having math difficulties while
working in small groups and one on one work with students in teaching math
concepts to them. I will say that they need different ways to count backwards
where is makes sense. This is one of the interventions that will be put into
place next year to learn to count backwards in an attempt to grasp subtraction
concepts.
The
following literature review has been conducted to find relevant articles for
the topic of increasing students’ achievement in subtraction; the initial
wondering question I have decided to pursue is how I can increase
students’ understanding of subtraction with single and doubledigit math equations.
Therefore, I found research that showed various ways to solve single and double
digit math equations. Kercood,
Grskovic, Lee, & Emmert (2007) actually brought insight in ways to
help students with attention problems and learning difficulties focus while
solving math problems. Next, Calik & Kagin (2010) discussed teaching special education students
the Touch Math techniques to improve their addition and subtraction skills.
Last, Peltenburg & HeuvelPanhuizen (2011) discussed strategies that show direct and indirect
addition and subtraction methods that can help students learn easier and more effective
ways to solve math problems up to 100.
Kercood
at al (2007) focused on the optimal stimulation theory and the behaviors of
students with attention problems and learning difficulties. The purpose of the
study was to test if students using a tactile toy concurrently while solving
math problems would be beneficial in helping students focus their attention
effectively. The study involved four students who were not diagnosed with ADHD,
but their teacher and parents recommended them because of having attention
problems and learning disabilities. The
students were tested in an unfamiliar classroom when gathering the data, but in
a room where distractions were limited. They were first tested to get baseline
data. After the first set of data were collected, the students were shown the
tactile toy and how to use it. After using the tactile toy, they were tested
again using the toy along with observers taking anecdotal notes on their
behavior. “Results suggest that two of the four students, Bill and Clem,
performed better with the fine motor activity with tactile stimulation, answering
55% and 45% more problems correctly, respectively, on average than in the
Baseline condition (Kercood, Grskovic, Lee, & Emmert, 2007, p 306).”
Moreover, “All four students engaged in more off task behavior in baseline than
in intervention (Kercood, Grskovic, Lee, & Emmert, 2007, p 307).” The
tactile toy “was presented concurrent with the task. It is unclear why the
intervention reduced off task behavior but did not result in more accuracy in
problem solving (Kercood, Grskovic, Lee, & Emmert, 2007, p 309).” The study was productive in showing that
using a different type of added stimulation helped students start to develop
some selfcontrol of unwanted behaviors. However, the questions that were
brought up for future discussion involved whether this type of fine motor
stimulation could help the students in their regular classroom with diverting
distractions. That future studies could involve tactile stimulation during
lectures or reading groups (Kercood, Grskovic, Lee, & Emmert, 2007).” The
theory in this study will help students focus in learning Touch Math that is
elaborated on in the next article.
Calik
& Kargin (2010) discussed the effectiveness of teaching the Touch Math techniques to mild
intellectual disabled students in the second grade. The students were inclusion
students who had problems correctly solving addition problems like their
nondisabled peers do using memory techniques. Touch Math techniques involve dots,
rings on numbers 1 through 9 where students used their pencil or finger to
count the dots, and rings in adding and subtracting single digits to single
digit numbers. The “dot notation method (touch math) involves
visual, auditory, and tactile learning (Calik & Kargin, 2010)”. Probes taken of the three students established a baseline before the direct
instruction of the Touch Math techniques was
taught. The direct instruction in a one on one format gave the students the
individualized attention they needed to learn the concepts of count all, and
count on strategies using the Touch Math method to improve their addition
skills. The students were tested in maintenance sessions 10 and 20 days after
the final probe sessions to see if the skills taught to the students were
sustained at the 100% level of the final probe sessions. Allowing time for the
students to use the skills learned from Touch Math techniques has showed that
they learned the technique of Touch Math instead of just memorizing the method.
Teachers were mostly in agreement with the questionnaire given after the
research was complete. They agreed that Touch Math is an effective way to teach
addition to students. The researchers noted for future considerations that the
study was limited because of the number of students, the number of teachers and
their years of service, and the math problems were limited to single digit
equations among other issues.
Peltenburg &
HeuvelPanhuizen (2011) discussed different ways for
students to solve addition and subtraction problems using direct and indirect
methods. The participants consist of
special education students between the ages of 812 years old who were at the
2nd grade math level. Early findings showed that some students were using the
indirect subtraction methods without even knowing they were using the method to
solve math problems. The purpose of this study was to challenge an assumption
that Special Education students should be taught only one method of subtraction
to improve their progress level (Peltenburg & Van den HeuvelPanhuizen,
2011). That those “weak learners do not have the necessary insights to choose an
approach that fits to a particular task (Peltenburg & Van den
HeuvelPanhuizen, 2011, p1).” That teaching regular education and special education
students the indirect addition and indirect subtraction methods depends on
various factors such as previous procedures, teaching characteristics,
students’ characteristics, and problem characteristics (Peltenburg & Van
den HeuvelPanhuizen, 2011, p 3). They had two groups of students within the study: one
group was taught how to use the different methods, and the other group was not
taught how to use them. The results showed that the students who received
instruction on how to use the procedures had a higher percentage than those who
did not receive the instruction. In addition, that some problems would
be solved easier by stringing or splitting problems making the calculating
faster. “Indirect
Addition was most frequently applied in smalldifference problems, and Direct Subtraction
was most frequently applied in largedifference problems,” as noted by Peltenburg
& Van den HeuvelPanhuizen (2011, p 6). Furthermore, when the students had the
opportunity to use the empty number line they choose not to use it. Peltenburg
& Van den HeuvelPanhuizen (2011) concluded
that, “this study has shown that solely focusing on strategies (splitting, stringing,
and varying) or solely on procedures (Direct Subtraction and Indirect Addition)
is a too restricted way of investigating students’ ability to solve number
problems. Both should be taken into account as
our study showed that the best predictor of a correct answer is the combination
of Indirect Addition and stringing (Peltenburg & Van den
HeuvelPanhuizen, 2011, p 8). ” That the
special education students “are able to use
Indirect Addition spontaneously, without
being asked to do so, are able to use Indirect Addition spontaneously, without
being asked to do so, and are quite successful when solving subtraction
problems by Indirect Addition (Peltenburg & Van den
HeuvelPanhuizen, 2011, p 8).” Future research
would include students in more schools, and student’s inventory of prior
instruction that could help in seeing if prior instruction had any influence of
applying indirect addition. This study showed how special education students
could use more than one procedure in solving math problems involving addition
and subtraction. That teacher instruction seems to be a large factor in helping
students to solve more problems.
The wondering question I started
with is how can I increase students’ understanding of subtraction
with single and doubledigit math equations? First, I learned that the optimal
stimulation theory (Kercood et al. 2007)
could be a useful tool in helping students with disabilities focus better to
deter off task behaviors in the classroom. This theory can help us find selfstimulations
the students need to be able to focus better that they have in the past. I learned that Touch Math ((Calik
& Kargin, 2010) is very fundamental in teaching
students the four main operations in math. Students learn that using Touch Math
is easy by counting the dots and rings on the numbers as a way to stay focused
on the task because the students do lack focus at times and as they easily
distracted by other activities in the room.
Peltenburg
& HeuvelPanhuizen (2007) discussed
teaching indirect addition and indirect subtraction strategies that will be
useful in showing the students that by making the problems easier that they can
solve math equations in multiple ways.
I discussed the wondering question for
the action research paper with the special education teacher who works in the
same room. We talked about if subtraction was enough of a topic. She looked
over the students’ IEP goals for next year, two students have addition and
subtraction solving goals, and the other students have addition goals. So with
this in mind, I have revised my wondering question to be the following: How can
I effectively implement Touch Math techniques in conjunction with the number line to increase students’ addition
and subtraction skills? Other questions I have are what other math strategies
will effectively help special education students comprehend math concepts? Will
learning just Touch Math and the number line be enough?
The plan is to teach more Touch Math
techniques to the upper grade students in order for them to get eventually to
step three of counting without the dots and rings. I will give a probe to
gather baseline information, take anecdotal notes during the data collection we
gather on a weekly basis. I plan to teach the students how to use the number
line and how to make a problem easier to some of the students who could grasp
this higher order thinking skill. I will do a posttest to see how effective the
students have learned to solve math equations with touch Math and the other
math strategies.
Collaboration
in the school is not a problem where I work. The special education teacher
expects to collaborate daily on student’s work and behavior issues with me and
the other paraprofessional. The principal, the instructional coach, and the
special education supervisor are very accessible when I need to collaborate
about any topics or issues for my class work, or anything with the students at
school. The special education teacher shares the passion for my wondering
question. I collaborated with her about the data collection tools I plan to
implement and why I felt the students needed to improve in their math skills.
We discussed different ways to use the organizer (Jimenez, 2011, p 26) I plan
to use with the students among other strategies to implement next year to
increase math skills. “Through collaboration with others, teacherinquirers
find a crucial source of energy and support that keeps them going and sustains
their work ((Dana & YendolHoppey, 2009, pg 60).” We both bring energy and
creative ideas to the classroom in order to increase student learning. I plan
to collaborate with the principal and the instructional coach about my progress
during the data collection process. I plan to collaborate with the special
education supervisor and the special education teacher weekly about the data
collection progress and any insights they want to share.
The
first data collection will be a probe to establish a baseline along with a
probe at the end to evaluate the effectiveness of the lessons. One of the data
collection tools I will use is anecdotal notes; they will be collected during
the lesson being taught, during the teacher and students interactive work sessions,
and during student independent work sessions. The anecdotal notes will be vital
in detailing their behaviors, student’s responses, teacher’s questions, and
observing areas of strength and weakness each student exhibits. As stated by
Dana and YendolHoppey (2009), “ Field notes are not interpretations but rather
focus on capturing what is occurring without commenting as to why the action
might be occurring or how one judges a particular act (pg 74).” With this in
mind, I plan to develop a key or coding of the notes concerning the students’
behaviors and work in progress.
The
next data collection tool I will use is taking digital pictures during teacher
questioning, small group sessions, and one on one work sessions with students.
Using media can provide visual representation of the students work, this will be
essential to use from the beginning of the data collection process to the end,
and will show how each student’s work has either progressed or not during the
instruction and class work. Digital
pictures can be used as a way for students to reflect on their work in class
from working in groups or to evaluate the progress of an assignment (Dana &
YendolHoppey, 2009, pg 86) Digital pictures are a type of data that can be
used alongside text in assessing student learning and progress.
Another
data collection tool I will use is students’ work samples. This data will show
how each student is learning or is not learning the concepts taught in the
small group. Using a variety of worksheets and games made with manipulatives
will provide different views of students’ understanding of the concepts along
with transferring the concepts to other content. Work samples provide different
perspectives of students’ beginning critical thinking and increasing critical
thinking skills.
Next,
I need to design the plan I will use to collect and analyze the data. After
looking at the Calendar for Action Research excerpt (Dana & YendolHoppey,
2008), I have decided to start my data collection in August when school starts
back; in addition, I will be using a monthbymonth format for the
timeline.
August
·
Look over students files
·
Administer probe for baseline data
·
Teach lessons on Touch Math
incorporating the number line.
·
Collect student work samples, take
pictures, and anecdotal notes
September
·
Look for patterns and discrepancies
(Dana &YendolHoppey, 2009)
·
Small group work with Touch Math using
games and the number line laminated mat
·
Take pictures during groups and whole
group instruction
·
Evaluate student performance, and make necessary changes if needed
October
·
Small group work with students doing
independent work
·
Student led games using the
manipulatives
·
Administer the final probe
November
·
Analyze data, write paper, and start
writeups
·
Prepare action research presentation for
school meeting
·
Publish on blog and send to journals for
publication.
The
data I collected for the action research project are student’s work samples,
informal test scores, and field notes. The tools I used are worksheets, digital
pictures of working with write boards with dry erase markers, and field notes.
The reason I choose these data collation tools is that the students are very
concrete learners and need visual representation when they are learning and
working on concepts. The worksheets allowed them to work somewhat
independently; the white boards allowed them to work while I modeled a new
concept and then I practiced with them and then they practiced independently.
The field notes do have a code used by the special education department for the
different prompt levels used with the students. The process I used in
collecting the data started with a preassessment to start a baseline. Next, I
reviewed and introduced the concept of Touch Math skills in adding the dots and
rings to the numbers; next, how to add and eventually how to subtract with the
Touch Math technique. I had students place the dots and rings in each number.
From this activity, the students started working on addition worksheets using
the Touch Math technique. Some worksheets also included manipulatives on the
sheet to help the students count. The second data collection tool involved
using white boards and dry erase markers. I used this so the students could
follow along gradually as I modeled how to calculate an addition problem using
the Touch Math technique. I would then have students work on problems using the
white boards as practice before completing more worksheets. The field notes
consisted of prompt levels I had to use with each student. The prompt levels
are the following: 3modeled by the teacher; 4verbal cues; 5independent work
by the students. The assessments were informal and summative because the
worksheets informal work as they needed many verbal cues in learning to stay
focused. The posttest was summative as it tested both addition and subtraction
skills they had learned. I did collaborate with the Special education teacher I
work with and she did give me information that I might try with them; however,
the strategies she suggested I was already trying with the small group. I did
find other strategies that I incorporated with the students to help them
understand how to subtract better. I found that when using the white boards
they could easily remove dots when doing subtraction problems. In addition, I
make a backwards counter ruler from a Touch Math worksheet where they could
count from twenty to one starting from the left side. I had the students use a
gummy bear as the counter to move or jump when we did subtraction problems (see
Appendix B). This did help them start to grasp the concept a little better that
the Touch Math subtraction technique. The timeline was very useful as I could
look to see when I needed to start a new step such as the second step in the
Touch Math addition technique. It was also helpful in making decisions of when
I needed to test the students on the skills they had been learning, and to keep
track of how long a concept was taught to the students and how they responded
in learning the skill
The
analysis process I will start with is similar to the jigsaw metaphor (p 119)
emphasized by Dana & YendolHoppey (2009). I think that spreading out the
data and start asking questions about what groups I could sort or categorize
the work into will be the first step. I will then use the four steps outlined
by Dana & YendolHoppey (2009,) “which are description, sense making,
interpretation, and implication drawing (p 120).” These four steps will help me
ask insightful questions during my analyzing of the data in determining its
validity. I will also be looking for patterns in the data for convergence or for
dissonance (Farmer,
Robinson, Elliott & Eyles, 2006, p 378.) within the data as
part of multi method triangulation (Meijer, Verloop & Beijaard, 2002, p 146). I feel
that using coding for the anecdotal notes will help save time by using symbols or
numbers to abbreviate behaviors or actions students’ exhibit. This will allow
me to graph the data collected from the coding to show increases or decreases
in student performance levels. Then I will proceed to finding my claims and
doing the write up of the inquiry process. How I will anticipate discrepancies
is by looking over the data at least four to five times and using probing
questions that come to mind. I feel that some of the discrepancies might be the
intellectual level of each student, the age group because the state standards
will be a little different since this selfcontained class is 3^{rd}
through 5^{th} grade.
In
analyzing the data, I first made a list of the dates of student’s work samples.
I found that when trying to make a graph or chart of the data, I had trouble
showing the difference between the addition and subtraction worksheets together
for each student. It was easier to separate the addition from the subtraction
worksheets and make different charts (see Appendix A) for each because there
were more addition worksheets compared to the subtraction. I then had to think
about how I would value the prompt levels in determining their growth as they
learned the concepts. After first trying to make a graph with the additional
assessment data, I found it was hard to accurately give a percentage for the
prompt level and correct problems for each assessment. I decided to use a chart
instead to organize the data and look for patterns of prompt levels and
percentages of problems correct. I wanted to keep the same number of questions
on each assessment; I felt this help would give consistency to show the
progress of each student. I then used
the pictures as a visual to show the different strategies I was using to help
them grasp the subtraction process and look for patterns of what strategies
worked or not. The charts helped to show the students’ comprehension in the
Touch Math technique of addition and subtraction. The data showed the students’
slow but ever increasing comprehension of adding single digits using the Touch
Math technique even making progress in learning the second step of Touch Math
addition. However, the Touch Math subtraction technique was very hard for the
students to grasp counting backwards or taking off the dots or circles without
using manipulatives. The worksheets had objects in addition to the Touch Math
digits. They could then cross out the number to subtract or if they objects
were already crossed out they just had to count what was not crossed out to
find the answer. The data showed that using the Touch Math technique with MoID
students who are very concrete learners did answer my wondering question. That
using the Touch Math subtraction technique as it is supposed to be used is hard
for the students to comprehend. Thus, I had to use other methods from the
literature findings to help them better understand how to subtract single and
double digits. I found that the strategy from (Peltenburg & HeuvelPanhuizen ,2007)
about making a problem easier helped D in adding and subtracting doubledigit
problems. I would draw a dotted line inbetween the one sand tens column and
have her calculate the problem. This worked well for her after she learned to
place her answer under the correct side. Another strategy I used was creating a
backwards counting ruler that started from the left at twenty and counted to
zero all the way to the right side. I used different objects for the students
to place on the number and then jump the number they were subtracting. Two of
the three students did well after working with this strategy for several weeks.
The other student had a hard time counting on the first move, and this caused
her not to get the right answer at first and I had to help her with the first
jump then she would stay on track and get the right answers most of the time.
In addition, it helped when I added another strategy of writing “start” next
the number they started at, “jump” next to the number they were to move their
counter, and “end” next to the number where their counter would stop at on the
ruler. The anecdotal notes gave me insights as to what strategies were working,
when the students were ready to move to the next step of Touch Math addition,
and to progress monitor their comprehension of the skills taught. Therefore, I
found that for teacher practice using anecdotal notes give me insights into how
well each student comprehended the skills being taught, that not overwhelming
the students with too many problems help them from becoming frustrated with
completing the assignments. I feel the Touch Math addition technique worked
well with the student; however, I did have to use other methods in helping them
grasp the concepts of subtraction using research findings from the literature
review.
In
sharing my action research story,
I
have decided to make a power point presentation to share with my Walden
colleagues. Part of the collected data will be students’ work sample, pictures,
and teacher anecdotal notes that will be useful in visually seeing the
strategies applied with each student and whether the strategies made a
difference or not. My writeups will follow the four steps stated by Dana and
YendolHoppey (2009) of “providing background information, sharing the design
of the inquiry (procedures, data collection, and data analysis), stating the
learning and supporting the statement with data, and providing concluding
thoughts (p 136).” This guideline will help me stay focused on presenting the
findings in an organized format. The risk I am willing to take with sharing my
research study is to present the action research study during a faculty meeting
at the school I work, and submit it to an action research journal for
publication, and I will publish it on my blog, http://juliescollaborations.blogspot.com/2012/04/dailyhappenings.html.
Presenting the action research findings to my colleagues will show that I can
utilize data to make informed decisions during the research and the data
collection process. The impact I hope my action research will have is to help
other teachers who have students with intellectual challenges implement the
techniques I used in the action research study in their classrooms. I hope
primary grade teachers will see how effective Touch Math techniques can be for
regular education students who are first learning addition and subtraction
algorithms. I hope my wondering will foster questions for other educators in
their practice where they will want to start an action research study to
improve their teaching and student learning. (Pate, 2012)
I also wanted to add
that after reading one of my colleague’s discussion post (Ward, 2012), I am
planning to look at the website she is planning to use called SlideShare.net as
another vehicle in sharing my action research story.
Finding
a wondering question, researching the topic, collaborating with colleagues is
just the start of an action research project. Action research is what it
implies, action from an inquisitive educator who wants to find answers to a
question about his or her teaching practice, students learning levels, or
school policies that are causing the individual to stop and ask whether changes
need to be made. Finding answers is a
combination of using one of the triangulation methods stated by Meijer et al.
(2002) along with research findings, and collaborations with colleagues with a
structured timeline for collecting the data. I feel that the data collection I
am planning to use will bring insights for making changes that will help
students improve their mathematic skills, and will help to improve my teaching
practice as an assistant teacher and paraprofessional.
The
changes I will make will start with how I would teach subtraction concepts to
students with disabilities. I found that using a visual ruler helped them
understand that they were making the starting digit smaller. I would use the
methods of making a problem easier. For example, separating a doubledigit
addition problem, so it is easier for a student to calculate makes learning
effective. I feel using strategies until one works well for students is
essential in meeting students’ needs. I have shared the results with the
special education teacher about the results that came from the research
project, and she was impressed that they were able to complete the second step
of addition using the Touch Math technique. She was equally impressed that the
other methods I used help the students understand subtraction better than using
the Touch Math technique. The students did well with using the worksheets that
had manipulatives on it to cross out the amount to be subtracted. I plan to
post my research project to my blog, and send an email out to the staff at
school asking them to read the post on the blog and give me feedback on the
results. Teacher inquiry needs to be a constant in the field of education as
gathering data to drive instruction is vital for every educator. Data brings
insights that teachers use in decision making of curriculum changes to improve
student learning. I feel that sharing research results and impact it has on
student learning helps other colleagues see how a different strategy can be
effective in their own practice.
References
Calik, N., &
Kargin, T. (2010). Effectiveness of the touch math technique in teaching addition
skills to students with intellectual disabilities. International Journal Of
Special Education, 25(1), 195204.
Dana, N. F., & YendolHoppey, D.
(2009). The reflective educator’s guide to classroom research: Learning to
teach and teaching to learn through practitioner inquiry (2nd ed.).
Thousand Oaks, CA: Corwin Press.
Dana, N. F., & YendolHoppey, D. (2008). The
reflective educator’s guide to professional development. Thousand Oaks, CA:
Corwin Press.Copyright 2008 by Sage Publications. Reprinted by permission of
Sage Publications via the Copyright Clearance Center.
Farmer, T., Robinson, K., Elliott, S. J., & Eyles, J.
(2006). Developing and implementing a triangulation protocol for qualitative
health research. Qualitative Health Research, 16(3), 377–394. Retrieved
from the Walden Library
Georgia Department of
Education. (2011). AYP 2011 Report. Retrieved from: http://archives. gadoe.org/ReportingFW.aspx?PageReq=103&SchoolId=35993&T=1&FY=2011
Kercood,
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Motor Movement and Tactile Stimulation on the Math Problem retrieved from: http://web.ebscohost.com.ezp.waldenulibrary.org/ehost/detail?vid=9&hid=24&sid=36f37d9326944f9abfbb7e7341a60587%40sessionmgr13&bdata=JnNpdGU9ZW
hvc3QtbGl2ZSZzY29wZT1zaXRl#db=eric&AN=EJ785143
Jimenez,
B. A. (2011, March). Teaching grade aligned math and science to students with
significant intellectual disabilities. In Proceedings
of the Delaware Inclusion Conference. Retrieved from: http://www.doe.k12.de.us/infosuites/students_family/specialed/ NEW/files/March2011.DE.Incl.Conf.pd
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Beijaard, D. (2002). Multimethod triangulation in a qualitative study on
teachers' practical knowledge: An attempt to increase internal validity. Quality
& Quantity, 36(2), 145–167. Retrieved from the Walden Library
Pate, J. (2012, May 30).
Re: Group 3 initial post. [Online discussion group]. Retrieved from: https://class.waldenu.edu/webapps/portal/frameset.jsp?tab_tab_group_id=_2_1&url=%2Fwebapps%2Fblackboard%2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_550023_1%26url%3D
Peltenburg, M. &
van den HeuvelPanhuizen, M. (2011). Special
education students’ ability in soling subtraction problems up to 100 by addition.
Freudenthal Institute for
Science and Mathematics Education. Utrecht University. Retrieved from: www.cerme7.univ.rzeszow.pl/WG/2/CERME7_WG2_Peltenburgetal.pdf
Unique Learning
System. (2011). Retrieved from: http://unique.n2y.com/
Ward, J. (2012, May
28). Re; Group 3. [Online discussion group]. Retrieved from: https://class.waldenu.edu/webapps/portal/frameset.jsp?tab_tab_group_id=_2_1&url=%2Fwebapps%2Fblackboard%2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_550023_1%26url%3D
Appendix
A
Touch
Math
Addition

Pre
test

2^{nd}
step


Student

88

822

94

96

912

919

103

D

3/4

100%(4)

100%(4)

100%(4)

100%(3)

100%(4)

80%(5)

T

0/4

75%(3)

100%(4)

88%(3)

75%(3)

100%(4)

absent

C

0/4

75%(3)

100%(4)

75%(3)

75%(3)

100%(4)

100%(4)

Post
test


Student

1016

115

1112

1116

D

60%(5)

100%(5)

75%(5)

100%(5)

T

100%(4)

100%(4)

75%(5)

75%(5)

C

100%(4)

100%(4)

25%(5)

100%(4)

Subtraction
Methods

Pre
test

Introduced
Backwards
ruler


Students

88

828

1013

1025

1029

1031

D

25%

100%(3)

3

25%(5)

absent

75%(5)

T

50%

100%(3)

3

100%(4)

67%(4)

100%(3)

C

25%

100%(3)

3

absent

100%(3)

100%(3)

Post
test


Students

111

116

117

1116

D

100%(5)

100%(4)

75%(5)

75%(5)

T

100%(4)

100%(3)

75%(4)

100%(5)

C

100%(4)

75%(4)

25%(5)

100%(4)
Appendix B backwards ruler and subtraction method
