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"We don't want students to get lost in math." says NINCHY's Evidence for Education. NINCHY (National Dissemination Centre for Children with Disabilities) tries to find answers to their important question: What can we do when disability affects a student's ability to learn math skills?
Daily life involves math-from the check-out counter at the school store to the express line in the grocery, from our most routine jobs to the high-paying, high-profile ones in engineering, technology, and science (Lee, Grigg, & Dion, 2007; U.S. Government Accountability Office, 2005)
In order to not to lose students with learning disabilities (LD) in math what kind of strategies should be followed by teachers? According to research studies many adolescents with LD experience math difficulties (Mcleod& Armstrong, 1982) particularly in problem application and generally perform at the fifth grade level in math.(Cawley & Miller, 1989).Therefore this study researches about effective math problem-solving instructions to students with LD in secondary level. Teachers, parents and also students will benefit from this research. Teachers can find more learning strategies than are used by themselves currently and indirectly students will benefit as a result of teachers' new approaches. Parents can be beneficial also with the help of teachers' guidance about how to behave to these students.
National Council of Teachers of Mathematics (NCTM) has become involved in identifying appropriate mathematics curricula and published the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1980, 1989). These standards advocate teaching mathematics through a problem-solving perspective and emphasize finding mathematical applications in everyday situations. In fact, the council stated that problem-solving skills should be a top priority in math instruction and that problem-solving approaches should be used to teach mathematical content in US.
There appear to be a variety of definitions for problem solving. However, most authorities interpret problem solving within the context of word problems (e.g.,Cawley, Miller, &School, 1987; Fleischner , Nuzum, &Marzola, 1987; Kameenui & Simmons,1990). Therefore the scope of this research will emphasize elementary and complex problem-solving as it relates to mathematics word problems.
Historically, word problems in mathematics have been difficult for many students, but students with learning disabilities (LD) tend to find these problems particularly challenging (Lee & Hudson, 1981).
There is another issue that most of the students with LD receive education in regular education classrooms. However many regular education teachers don't know what these students' needs are and how the lessons can be taught to these students.
In response to the given importance on problem solving and failure of students with LD in this issue, this research tries to find an answer to the question is given below:
"How can regular education teachers improve word problem-solving skills of students with LD in Turkey?"
Research questions are given as following for this study:
What are the characteristics of students with LD?
What kind of learning disabilities do these students have and how does it affect their learning about problem solving?
What are the current teaching methods for problem-solving and are they sufficient to improve students' skills in this way? If they are not, what are techniques lacking?
How are regular education teachers' attitudes to students with LD in classrooms?
Learning disability is a very wide topic because there can be several reasons which cause to learning disability. For example mentally, emotionally or physically problems can cause to learning disabilities. This study cannot answer all types of learning disabilities but sample of the study can find a way to help some students with LD in problem-solving lessons. 1st and 2nd questions serve for this purpose. 3rd and 4th questions try to describe the currently methods teachers used and determine what the lacking or insufficient sides of these methods are. Also questions try to find an answer for how teachers can teach problem solving better to students with LD.
C. LITERATURE REVIEW
"What are the characteristics of students with LD? How are they integrated to regular education classrooms? What is special education and how is it applied in Turkey? What are the general and special education teachers' role in teaching process to students with LD?" are the first questions that interest me for my research project. Although my problem is more specific-teaching math word-problem solving to students with LD, I needed the answers of these questions indicated above to reach my goal. Therefore I chose, articles related to history and current status of special education in Turkey, special teacher training in Turkey. Besides this, I didn't choose about characteristics of students with LD because in each article characteristics of students with LD and affects of disabilities to mathematics learning are described well. This literature review also includes articles to indicate the current strategies used in problem-solving, alternative solutions to teach more effectively.
Research indicates that teachers should receive adequate education to understand students' psychological and academic needs. The third article also gave a chance to notice a possible gap that I would like to investigate further. This article is related to special and general education teachers' perceptions about NCTM Standards for students with LD. I thought that I can modify it to MEB Standards and investigate teachers' perspectives to this issue because the last article in this literature review states that general education teachers are unaware of teaching strategies for students with LD.
In an article by Borkowski (1992), three important points were addressed for guiding the study. First, is the importance of self-regulation and motivational beliefs which are two great components of meta-cognitive theory to solve word problems and how does the lack of these skills affect students with LD? Second, is the reciprocal relationship between self-regulated learning and beliefs about the "self" as a learner and "working models" and their roles in classroom teaching? The last one is applications to the teachings of literacy, writing and mathematics.
This article brings three research together. The aim of the author was to draw the three articles into a more consistent and integrated perspective based in large part on recent advances in metacognitive theory.
At the beginning of the applications to the teaching of literacy, writing and mathematics part of the article a characterization of students with LD and tips for teaching effectively in math and literature are given. Likewise in an article by Babbitt, Beatrice, Miller and Peterson (1996), deficits of students with learning disabilities in problem-solving is discussed and the finding shows that language difficulties are very common among students with LD.
This article discussed strategies which are used for problem-solving. According to the article there are three important strategies for problem solving. First one is cognitive and metacognitive ones. Second one is using computers to teach math to students with LD. Third one is applying hypermedia to problem-solving instruction. Hypermedia is multimedia information linked accessed by a computer. (Lynch, 1991) In other words hypermedia includes graphics, sound, animation and video. Hypermedia can be used in a wide range in mathematics from simple diagrams of mathematical shapes to visualization of complex topics such as trigonometry. (Babbitt and Miller, 1996)
Similar conclusion with first article is given at the end of the article. Both of articles make offers for teachers to build their own 'working models'.
One of the main goals of this study is to find helpful tips for regular education teachers to teach effectively word problem solving to students with LD. In a research article by Maccini and Gagnon, (2002), teachers' perceptions about applications and limitations of National Council of Teachers of Mathematics (NCTM) Standards with students LD (Learning disabled) and ED Emotionally disturbed). 129 secondary general and special education teachers participated in a mail survey in USA. At the end of this research most of the special education teachers tell that they didn't know about NCTM standards, participants also reported that instead of teaching NCTM standards to students with LD and ED, they teach basic math skills to these students. The reason for not being able to apply activities successfully depends on NCTM standards is inadequate materials. Moreover teaching basic math skills versus teaching higher level math such as algebra and geometry is discussed by teachers who participated in a survey. Basic math skills are very important; on the other hand teaching higher level math to students with LD can be possible by using appropriate tools.These results are related to USA so it is one of the limitations for this study.(Maccini and Gagnon, 2002) However this research gives an idea to think about perceptions of special and general education teachers in Turkey. Is there a possible way to teach MEB standards to students with LD? The last two research replies this question.
In an article by Cavkaytar (2006), special education programs in Turkey and services for special education is provided by MEB are presented. There are ten undergraduate special teacher education programs on Turkey for different special education categories. Also special education programs include courses such as "Teaching Mathematics to the Mentally Handicapped", "Preparing Integration Programs", "Developing Individualized Education Programs and Evaluation". In addition to that there are three services in Turkey. First one is that special education is given as a formal education in private schools. Another possible service is that children with special education needs are educated in mainstream schools and special classrooms. Besides these there are also supporting services as physical therapy, speech therapy, occupational therapy in rehabilitation centers. Second service which is available in Turkey is related to this study. Nowadays mainstreaming is known as Individualized Education Program in Turkey. Students with special education needs have a chance to take courses in general classrooms. Therefore teachers should know these students' interests and needs. However an important conflict is stated in the article. Research indicates that special education teachers take courses to teach math to students with LD. By contrast regular education math teachers don't take any courses special to these students. As a result of this, teachers don't know how to approach to these students and integrate them to lessons well.
The last article supports the views in the previous article. In a research article by Melekoglu, Cakiroglu, Orhan, Malmgren, Kimber W. (2009), progression of special education and problems are faced with in Turkey in this area are addressed. Since 1983 Ministry of Turkish National Education has supplied a policy of integrating students with special needs into the general education classrooms by providing students with support services. After this policy, special education developed very fast. Yet there are still some conflicts in this area. Research indicates that number of special education teachers and suitable schools are not enough for special education teachers. The next conflict is general education teachers in classrooms include students with LD (These classrooms are called as "inclusive classrooms") don't receive adequate training in special education. Furthermore another important conflict is newly graduated special education teachers choose to work in private schools rather than public schools, because their salaries and conditions are better. The last one is education of all students in regular education classrooms is not common as it is in other countries. After problems are stated, article offers solution ways. Article suggests that regular education teachers should get sufficient experiences in inclusive teaching. Schools should be supported financially to provide better services for students with disabilities. Also teachers' economical situations should be improved.
Taken together students with LD whatever their special needs have problems in word problem solving topic very commonly. Word problem solving skills is closely related to language skills and strategy based learning. To develop these skills teachers should teach self-regulation and make problems more concrete. There is an Individualized Education Program in Turkey for students with special needs to support these students in regular education classrooms. And previous research indicates that word problem solving in math is one of the common problem among students with LD. These students continue to regular education classrooms however regular education teachers' knowledge is inadequate in special education area and they don't have appropriate materials. Thus most of them don't know how to involve them to lessons. It can be assumed from these five literature reviews that lack of teachers' equipment is the biggest obstacle to get involved these students into not only math but also other courses. Therefore research for how regular education teachers have sufficient equipment should be done. Also a specific research about problem solving strategy for students with special needs should be done too. This would be helpful both regular education and special education teachers particularly in problem-solving lessons.
Ethnography is a scientific research method often used in the field of social sciences particularly in anthropology. It generally studies people; ethnic groups and their cultures. However ethnographic research has broad implications for many fields, including education and it is a useful tool for understanding how other people see their experiences. (Ethnography. (2009) . In Encyclopaedia Wikipedia online. )
Students with LD are supported by teachers, parents, counselors and special education teachers which gives this research topic has a broad view. Because of this reason to give a comprehensive response to the research questions, looking from different perspectives would be useful. Ethnography implies "learning from people". Therefore ethnography gives an opportunity to reach my goal in this project.
My informants will be secondary school teachers and a counselor. Jale AtaÅŸalar who is a counselor and also is an instructor in Bilkent University has experienced helping students with LD. At her school there is an "Individualized Education Program" for these students to enhance their understanding and learning. This program also can help teachers to understand these students well and develop different strategies on problem-solving.My last informants are regular education math teachers who have experiences with students with LD. One of the regular education teacher's name is Zerrin Toker. She is working at Ä°hsan DoÄŸramacÄ± VakfÄ± Ã-zel Bilkent Ä°lköÄŸretim Okulu (Ã-BÄ°O) in Ankara and she has been teaching for 7 years. The other teacher's name is Senem AtÄ±lgan. She is working at Çubuk Türk Telekom Ä°lköÄŸretim Okulu and she has been teaching for two years. Her school is in rural area in Ankara. Zerrin Toker is working in a private school, but Senem AtÄ±lgan is working in a public school. Both of the schools have Individualized Education Program (IEP) for students with special needs.
Actually this research is focused on teaching strategies about problem-solving in math for students with LD. Therefore teachers' opinions and their diagnosis about this issue will help directly to make the situation clear.
One of the most common methods for data collection in ethnography is interviewing and to learn different people's thoughts, to explain their stories, interview is an appropriate tool. In addition to that an interview guide approach will be more useful to gather information. Topics and issues are specified in advance, in outline form in this type of interview and I can change sequence and wording of the questions during the interview.
I will make interviews with middle school teachers and a counsellor. As I mentioned above, students with LD work with a wide team, thus I want to learn the team's experiences, feelings about problem-solving strategies. Therefore questions are generally constructed as open-ended. I have to ask different questions to them because they have different perspectives, different backgrounds. However different responses from different perspectives reduce the comparability of responses.
There are also limitations of using interview as a tool. Representing each informant's stories, thoughts without imposing my own opinions will be difficult.
2 regular education teachers replied the same questions in their interviews. However counselor replied different questions.
The responses of two regular education teachers can be grouped into four different categories: percentages of students with LD in their classrooms, types of these students' needs, programs used in schools for students with LD and teaching styles of teachers for these students. There are only 1 or at most 2 students with LD in Zerrin Toker's classes. On the other hand there are 3 students with LD in Senem AtÄ±lgan's class. In addition to that there are 20 students at Ã-BÄ°O in classes but there are 30 students in classes at Türk Telekom Ä°lköÄŸretim Okulu. The amount of the number of the students in class affects the lesson in positive or negative manner. Besides this, students' needs affect the teachers' methods they used. Senem AtÄ±lgan has two students who have mentally developmental delay and one student who has behavioral disorder. However Zerrin Toker has students with learning disorder and attention deficit disorder. Therefore their techniques are different from each other. Nevertheless there are some similarities in their techniques to involve them to problem-solving lessons. Both of the two teachers emphasize that knowing the students' needs and interests make easier to involve them to lessons.
As mentioned above counselor replied different questions in her interview. For this reason I evaluated her responses on her own. Jale AtaÅŸalar's responses can be categorized into four titles: Types of students' needs, role of guidance centre during teaching process, suggestions for enhancing students' problem-solving skills and teaching assistant issue.
Analysis of Regular Education Teachers' Responses
Percentages of students with LD in their classrooms: 10% of students have LD in Senem AtÄ±lgan's class but 5% or at most 10% of students have LD in Zerrin Toker's classes. Also Senem AtÄ±lgan's size of class is 30 but Zerrin Toker's is 20. Senem AtÄ±lgan emphasizes that she can't help to students with LD enough because of the numbers of students in the class while Zerrin Toker doesn't have similar problem.
Senem AtÄ±lgan: The class is crowded. In addition to that 3 students who have learning disability make my job more difficult. Even I cannot take care of other students, to be honest I cannot take care of students with LD enough. They are doing other works durin the lesson they cannot catch their peers.
Individualized Education Program offers one or two students with special needs in classes. Each of them has special needs and teachers have to prepare appropriate tasks for these students. Also teachers have to monitor each student in the class. Senem AtÄ±lgan's words show that to be helpful to students with LD, the sizes of classes should be reduced.
Types of students' needs: Senem AtÄ±lgan has two students who have mentally developmental delay and one student who has behavioural disorder. She describes her students' needs with following words:
"Students with mentally developmental delay just do four basic arithmetical operations. They are coming back from their peers. They have also difficulty in understanding what the text says. When I teach problems in class, I give them paper -which I prepared before and it includes arithmetic operations only - they deal with this paper. I cannot involve them to lesson because I don't know how I will do it."
"A student with behavioural disorder cannot stay in his desk during the lesson and he cannot focus on the lesson. Instead of this he walks around during the lesson, takes care of other things such as looking posters on the walls, sharpening his pencils. He can only focus on the things which he is interested in. Therefore I try to link the topic with his interests to involve him to the lesson."
Zerrin Toker has students with learning disorder. They have learning difficulties in reading and mathematics. They have difficulty in understanding what problem text saying. Besides she indicates that most of students have learning disorder also have attention deficit disorder.
As can be seen from teachers' explanation, attention deficit disorder and difficulty in reading and understanding are common disorders among students of Zerrin Toker and Senem AtÄ±lgan.
Programs used in schools for students with LD: Both of two schools apply Individualized Education Program prepared by Ministry of Turkish National Education to students with special needs. However in terms of application, there are some differences between two schools. Except from inclusive classrooms there is a special education class in Türk Telekom Ä°lköÄŸretim Okulu whereas there are only inclusive classrooms at Ã-BÄ°O. There are only two students in special education class at TTÄ°Ã-. They are separated from the other students because their needs are more than other students with LD. Senem AtÄ±lgan told that she doesn't know what these students' needs are but they have a special education teacher. In fact she is not a special education teacher. She is working as a substitute teacher at TTÄ°Ã- and there isn't any special education teacher in the school. For this reason she is working as a special education teacher. Another difference is the support given to these students outside of the classroom.
Zerrin Toker: Students with learning disorder take an extra hour support lesson per week from the teachers of the lessons which they have difficulty with. This extra hour is used for making practices about the topic or explaining the issues which student doesn't understand in the lesson. This support helps students to catch their peers. Moreover their exams are prepared differently from other students. Curriculum remains the same but questions are chosen according to students' capabilities.
I made interview with Zerrin Toker first. After I learnt that they give support outside the classrooms I asked to Senem AtÄ±lgan whether her school gives any support outside the classroom but I learnt that they don't. However they prepare different exam for students with LD.
It is clear that students with LD in private schools and students in public schools don't have equal opportunities. These students need more time than their peers to understand the topic. However in public schools classes are crowded and teachers don't help them enough even in the lessons. Apart from that they don't take any support from the teachers outside of the school. On the contrary students in private schools have more chance to catch their peers and attend the lessons effectively.
Teaching styles of teachers for students with LD: Although teachers don't have equal materials and technological equipments, techniques they used similar in problem solving lessons. Teachers explained the techniques they used for all class. Then they added special techniques for students with LD if they have.
Zerrin Toker: I introduced students to George PolyaÂ´ s1 problem solving process. I expect from them to use it during problem solving lessons.
Students also can find different solution methods during the lessons. They can draw pictures about what they understand from the problem or they can simplify the language of the problem. Moreover I am using group activities. They work in groups of two or three. Also I am using problem constructive activities. Students write their own problems in groups of two or three too. Students with LD try to understand the problem by using mind maps or drawing tables like understanding the reading text in Turkish lessons. They prefer reading the problem out and express what they understand. In particular they like real life problems rather than abstract ones.
Senem AtÄ±lgan: Generally I teach to students how to transfer the words in the problem into mathematical language. For example I expect them to use x for the unknown in the problem. However I used this method for students who don't have any special needs. As I mentioned, I can't involve the two students who have mentally developmental delay because I don't know how I can achieve this. For the other student who has behavioural disorder, I rewrite the problem by considering his interests since he tries to solve the problem which is related to his own world. For example he is making shepherd in summers. If the problem is related to meat or sheep, he tries to solve the problem. But he doesn't like mathematical language. He likes dealing with numbers.
Both of two teachers' responses show that real life examples attract students' attention. Also understanding the problem is the main stage to solve the problem.
Analysis of Counselor's Responses:
Types of students' needs: She told that there are 7 students with special needs and 5 of them have mathematical learning disorders. She told that guidance unit makes meetings with these students.
Jale AtaÅŸalar: In these meetings we ask to a student how he is working math and how he feels during the lessons. We have also meetings with teachers about student's development and sometimes we bring together two sides to make the needs of the students clear. It shows that teachers and counselors take care of these students' needs, they allow time for them.
Role of guidance centre during teaching process: She mentioned about the works of guidance centre related to special education. Guidance unit prepare a seminar for all teachers informed about special education at the beginning of each academic year in Ã-BÄ°O. Unit gives information about what is IEP, learning disorder, attention deficit disorder is. They organize meeting with teachers of students with LD. Teachers are expected to prepare a special annual plan for these students.
Suggestions for enhancing students' problem-solving skills: Her suggestions for teachers in problem-solving lessons are not different from what teachers told. Her suggestions are as following:
"Concrete materials can be used to make the problem meaningful. In addition to that situations in students' life can be transferred into the problem."
Teaching assistant issue: I asked to Jale AtaÅŸalar about teaching assistant issue. There are teaching assistants in foreign countries, for example in USA, UK. They are not teachers but they are dealing with students who have special needs in classes and they are working as volunteers. It is very helpful for regular education teachers because it makes their job easier. With the help of assistant and teacher, student can understand easily. My question was can teaching assistant application be possible in Turkey.
Jale AtaÅŸalar: It looks like a good idea but I think Turkish teachers are not ready for teaching the lesson with another person. They don't feel relax during the lesson. They are afraid of being criticized. Perhaps two teachers can teach the lesson together by considering the students' special needs.
It can be understood that passing through the teaching assistant application takes time.
With respect to the findings in data, currently used methods in problem-solving lessons, characteristics of some learning disability types, affect of the percentages of the students with LD in regular education classrooms are discussed. The responses of informants give an idea for the research however they may not be generalizable. Because informants were from only two schools and special education is a broad topic. In addition to that my study was more specific rather than my findings. Instead of finding specific tips for effective problem solving teaching, I found general comments about the research which were similar to the findings in literature review.
Percentages of the students with LD in classes and the size of the classes affect the lesson negatively. Because these students have special needs and teacher has to monitor and look after them during the lesson. While the class is crowded monitoring and also controlling both of the class and these students can be challenging. There are several types of learning disabilities and helping ways for the abilities changes according to the characteristics of learning disability. For that reason making a generalization may not be possible. Teaching techniques used by teachers were transferring word problem into mathematical language, George Polya's problem solving strategy, mind maps, tables, figures, transferring the problem students' life, using real life applications. While currently used methods were enough for students with LD at Ã-BÄ°O, methods used in TTÄ°O were not enough. Because teacher didn't know which methods suitable for the students with LD in her classroom are.
To sum up, teachers should know students' needs well and they should be informed about their students' problems and solutions of these problems. The size of the classrooms should be reduced in public schools; they should be supported economically by the government. Making students with LD successful not only in math but also in all their lessons their studying and learning process and their feelings should be monitored. They should take support outside of the classroom. In particular using concrete materials, visualization methods, real life applications can be used in problem solving lessons.
The data gave some of the answers of the research. Yet there are still some areas that I recommend for further research. I mentioned that Individualized Education Program is performed by a big team which includes special education teachers and parents in the introduction. However I didn't touch on the roles of these people in students' success. A research about special education teachers' aspects for problem-solving lessons and their interactions between regular education teachers and parents should be done.
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Appendix A: Interview questions made by teachers
Appendix B: Interview questions made by counselor