Monday, March 4, 2013

Problem-Based Research Project












Implementing Math Strategies with Special Education Students
To Increase Math Skills
Julie Pate
Walden University









Carol Luetzow, Ph.D.
EDUC 6733P-1 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 self-contained class of 2nd to 5th 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 double-digit 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 & Heuvel-Panhuizen (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 self-control 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 non-disabled 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 & Heuvel-Panhuizen (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 8-12 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 Heuvel-Panhuizen, 2011). That those “weak learners do not have the necessary insights to choose an approach that fits to a particular task (Peltenburg & Van den Heuvel-Panhuizen, 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 Heuvel-Panhuizen, 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 small-difference problems, and Direct Subtraction was most frequently applied in large-difference problems,” as noted by Peltenburg & Van den Heuvel-Panhuizen (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 Heuvel-Panhuizen (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 Heuvel-Panhuizen, 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 Heuvel-Panhuizen, 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 double-digit 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 self-stimulations 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 & Heuvel-Panhuizen (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, teacher-inquirers find a crucial source of energy and support that keeps them going and sustains their work ((Dana & Yendol-Hoppey, 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 Yendol-Hoppey (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 & Yendol-Hoppey, 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 & Yendol-Hoppey, 2008), I have decided to start my data collection in August when school starts back; in addition, I will be using a month-by-month 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 &Yendol-Hoppey, 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 write-ups
·         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 pre-assessment 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: 3-modeled by the teacher; 4-verbal cues; 5-independent 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 & Yendol-Hoppey (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 & Yendol-Hoppey (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 self-contained class is 3rd through 5th 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 & Heuvel-Panhuizen ,2007) about making a problem easier helped D in adding and subtracting double-digit problems. I would draw a dotted line in-between 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 write-ups will follow the four steps stated by Dana and Yendol-Hoppey (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/daily-happenings.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 double-digit 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), 195-204.
Dana, N. F., & Yendol-Hoppey, 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., & Yendol-Hoppey, 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, S., Grskovic, J. A., Lee, D. L., & Emmert, S. (2007). The Effects of Fine 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=36f37d93-2694-4f9a-bfbb-7e7341a60587%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
Meijer, P. C., Verloop, N., & Beijaard, D. (2002). Multi-method 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 Heuvel-Panhuizen, 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_Peltenburg-etal.pdf
Unique Learning System. (2011). Retrieved from: http://unique.n2y.com/





Appendix A
Touch
Math
Addition
Pre test



2nd step


Student
8-8
8-22
9-4
9-6
9-12
9-19
10-3
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
10-16
11-5
11-12
11-16
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
8-8
8-28
10-13
10-25
10-29
10-31
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
11-1
11-6
11-7
11-16
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