Mathematics

Introduction Through the years that I have been teaching, seldom have parents questioned our math program, or even mentioned much about math. Reading has been the focus of the questions about our curriculum. Do you do phonics' How much do you work on reading' And so on. With our changing world and technological advances, math is every bit as important and powerful as reading. Mastery of math is vital to the development of citizens in the new century. As our society has become more technologically oriented, math and science have become gateways to future careers. Everyone is affected by science and math on a daily basis. Students that do not receive a good mathematical basis or who drop out of the higher math classes, will greatly lesson the job options they have in the future. Mathematical literacy is more important now than ever before. The ability to make sense of information, the development of critical reasoning skills, and problem solving are critical to becoming contributing citizens. It is thought that information technology jobs will be the hot careers of the next century. Along with our changing world, our students are changing. The fast paced electronic world is much different than many of us experienced. They can sit at the computer listening to music, watching television, while surfing the web and chatting with friends through instant message. This multi-tasked student is the same one that will be sitting in the classroom the next day. What appears overload to many is their way. Their motivations have changed. Traditional teaching methods are unlikely to satisfy the changing conditions. A shift to more active learning and stimulating classrooms seems necessary. Research has shown that attitudes towards math are positive in the early years of schooling. By the end of elementary school, children see themselves as less successful. They loose interest in mathematics through high school and by adulthood fear math. Is this because of the decontextualized way that mathematics is taught in our schools' The challenge we have is to help students see the connection between math and their everyday lives. We need to make it relevant and real to them so they see the need for math. We need to teach more than just how to do division, students need to understand when to use division and why. This vision of mathematics is supported by much research and practice. The math classroom should be one where students are actively engaged in work, inquiring and working to make sense of mathematical ideas. There are many aspects to math, and I have decided to focus on these major areas: 1. Integration of math in the curriculum 2. Technology in the curriculum 3. Teaching to all learners 4. Professional Development 5. Mathematics in kindergarten Integration of Math in the Curriculum Brophy (1987) explained the expectation x value theory as the effort children make in learning math, is equal to how competent they feel times the value they see in the math activity. The value a student places on a task is a major factor in determining the effort they expend on the task. This in itself speaks to the habit of tracking. Those students not in the highest group certainly must feel less competent. Those in the lowest group would probably given this explanation not try very hard. In addition, if the task seems useless to them it does not look as though much effort would go into the task. This having been the traditional approach makes it obvious why by high school students, other than the high math students, aren’t much into math. Isolating the subject of mathematics from other parts of the curriculum would certainly cause a student to question the value of a task. Methodology that puts math instruction in real world context will increase the students’ motivation to learn, as well as their understanding. Techniques that put emphasis on problem solving and reasoning, and less emphasis on one right answer will also aid in making many more students feel competent in math. In the kindergarten curriculum there are many opportunities to integrate math. For example when we do our restaurant unit, the children figure out what we will serve to the customers. We talk about what types of foods we will need and how much of each item. We have to figure out how many people will be eating and how mach food we will give each customer. We set the price, and also earn money by helping with the preparation. We can use this money to buy food in the other restaurants. The children are involved in every step of the process from writing the menus to setting up the classroom to preparing the food. The problems that come up that must be solved are real problems and they learn how to figure out solutions. They learn how to work together and listen to ideas of others to come up with a common solution. These types of opportunities could be used in higher grade levels in a more complex way. One of the methods that seems like it could work well is the Project Approach. This is an in depth investigation of a topic the students want to learn more about. Because the students select the topic, it will have more meaning and the students will take more ownership than if it were a topic selected by the teacher. The project is focused on finding answers. In this method, a variety of types of problems must be solved. The children are consistently challenged to solve mathematical problems and scientific questions. They become more aware of numbers and quantities. Often they need to use reason, they do sorting, classifying, make comparisons, and arrive at conclusions. The teacher must take part in the process by encouraging and knowing what type of questions to ask to challenge the students further. When showing results, students often make graphs or diagrams. They must tell how the process went and verbalize what they are doing during the process so they are learning to communicate math concepts. This approach can be used from preschool throughout the years of schooling. It is unfortunate that the early years are the only years that processes like this seem to be the norm. Even the use of manipulatives, which is done so much in preschool and kindergarten, is phased out, and by second grade many feel the use of manipulatives is for babies. The values of the use of the Project Approach would seem to be many, but certainly the value of each child as a participant and the fact that many divergent ways of thinking and solving problems are accepted. All students could feel competent as mathematicians and the value of the task would certainly be obvious to the student. Research compiled by the George Lucas Educational Foundation has supported the use of project-based learning in schools. Among this research the results have shown lower absenteeism, more cooperative learning skills, and improved test scores among the schools that use project-based learning. Also research dating back to Brownell (1945, 1947) has shown that teaching for meaning and understanding has positive effects on learning, which includes better initial learning, greater retention of the learning, and the chances that the students will use this knowledge in new situations is greater. The students who learn in meaningful situations develop a better conceptual understanding, which will help them for procedural knowledge later. Also being a way to make them feel like a more competent learner. Reaching all learners Among the educational concerns in mathematics is the evidence that minorities, women, and other disadvantaged groups have not excelled in the subject, as have white males. The discourse that white males are superior in math skills has perpetuated through the years. Women and minorities have actually been discouraged from taking more challenging math classes. Some teachers have accepted the belief that it is better to prepare students for the jobs that will be available to them, therefore discouraging females, and minorities to continue in advanced math classes. Studies have shown that female’s math achievement in elementary grades is equal to males, but decreases in the middle schools. Studies have also shown that gender differences in math do not surface until age 10 (Callahan and Clements, 1984). This might suggest that it is the result of a pattern of socialization rather than a gender difference in actual ability. There are cognitive differences in the way students’ process information and approach problem solving. Culture differences in students certainly would explain different perceptions. The context in which you grow up has much to do with the way you see things. Different ethnic groups look at things through a “ different lens”. This way of looking at things is different from that which educators have identified as “normal”. Our curriculum has been developed using this “normal” way of learning as the basis. Obviously, those that do not fit the norm will be at a disadvantage. Special needs students have also been counseled to take less challenging courses instead of being encouraged to work harder to master a more difficult course. These attitudes need to go out with the traditional math curriculum. Staff should believe that all students have the ability to learn and need to be committed to removing educational inequities relating to sex, race, ethnic background, or disability. Teachers need to use diverse approaches to teaching in order to reach all learners. Information needs to be taught in familiar, meaningful contexts that the students can relate to. Rather than focus on competition, correct answers, and who can get the answer first, an environment of cooperation and collaboration should prevail. Students should see the diverse ways that you can confront a problem and learn that there are many different approaches to solving problems. Again, I think of the project approach, or a project-based learning situation as one way to diversify instruction. In this way all students can get involved, feel productive, and feel like they are competent. Also, they would be seeing the problems in a meaningful context, so the effort they put out would most likely increase. Technology in the kindergarten classroom Digital literacy is one of the most important tools a student can develop. Most jobs in the future will require some type of technology. Not only knowing how to use computers effectively is important, but also knowing how to make sense of the multitude of information one can receive from the Internet is important. Computers have not revolutionized the world overnight, and will never take over the job of a teacher. They do, however, give us another way of reaching students. Computers also give us access to information not yet in textbooks or the library. Because of the expense of becoming up to date with technology, and given the rate of improvement in systems, schools have often set up “ computer labs”. In this lab there may be 15-20 computers and each classroom is to use the lab at a designated time during the day or week. This may work well in instructing students on the use of a computer, or practicing a particular skill, but it is only a small part of the impact a computer can have on learning. Computers can enhance our lives in so many ways but they need to be available in the classroom for the incidental learning opportunities. The practice of “computer labs” prevents this technology from being “integrated” in the curriculum and becoming a natural part of learning. Integrating learning is one of the main thrusts of the new math curriculum. Without true integration, the benefits of technology to aid in learning cannot be achieved. Papert (1993) feels that by removing computers from the classroom and putting them in a lab, schools are minimizing the impact that computers can have on learning. Shade and Watson ( 1990) have stated that children gain the most valuable skill, that of using computers as natural tools for learning, only when they are in the classroom and integrated into the curriculum. In this way, they become another way of learning like manipulatives and books. Often in kindergarten, the computer is used only for practice of skills. For example, Bailey’s Book House is an excellent program for reading readiness as well as Millie’s Math House for mathematical skills. While these are good programs and have a place in the classroom, the computer can be used in more ways in the kindergarten room. For example, doing a unit on fish, the teacher might use an informational CD-Rom encyclopedia where the children can gather information about fish. Then she may choose to use a Kid Pix program where the child can dictate a story and add drawings. It is important to note here that we need to use computers with kindergarten children within the context of what we know are good teaching. The teacher will need to instruct the students on the use until they become comfortable and familiar with technology. As the students gain experience, the teachers’ role will become one more as a facilitator. To provide guidance or assistance, the teacher may choose to have an assistant or parent volunteer facilitate the use of the computer, so someone is available to the students. Computers can also be a great benefit to students with special needs with physical limitations. Or for a child who interacts more comfortably with technology than classmates. Again, technology must be used in ways that are compatible with the instructional goals. It must be embedded in the program rather than used as a fancy add on. This brings up the problem of inequities among schools that can afford to have the latest technological equipment and those that cannot. Rural and urban schools need up to date technology every bit as much as the suburban schools. Professional Development One of the mistakes that educators make in instituting curriculum changes, is that they fail to recognize the need for professional development. To make changes a teacher has to first of all be convinced of the need to change, and secondly feel confident he/she knows how to implement the new ideas. Teachers need opportunities to attend conferences and seminars, access to professional journals, encouragement to visit with colleagues, and time for reflection. Without time for professional development, teachers often continue doing what they have done in the past. In the research about the Maryland Virtual High School Project, much was said about professional development and need for it. Also, in comparing schools in the United States to Japan and wondering why Japan had better scores, one of the areas looked at was the amount of time the teachers in Japan have for professional development and professional interaction. That was one of the largest differences in our programs. They have much more time to develop professionally, both in the teacher training programs and after becoming a teacher. It is in fact the teachers who are interacting with the students each day. It is the interaction between the teachers and students each day that determine how much and how well a student will learn. If we are seeking to improve the quality of education it seems professional development must be one of the most important parts to the equation. Math in kindergarten When children enter kindergarten they do not “begin” their math education. Most children have begun to develop many mathematical concepts in their preschool years. They have put together puzzles, worked with small numbers, sorted, and compared. They have seen differences in shapes, and learned terms such as smallest and largest. This mathematical learning has developed because of their natural curiosity and enthusiasm. They all bring this natural curiosity and enthusiasm with them when they enter school. This is one of the joys of working with children this age. In kindergarten we can allow them to continue seeking information with projects and hands on activities. Throughout the research I read concerning kindergarten and early childhood mathematics the major theme has been that children this age are able to do more than we give them credit for. Much more mathematical learning can take place if we set up the correct learning environment. During these years, children are building beliefs about what math is and about themselves as learners. These beliefs influence their thinking, performance, attitude and decisions. Our challenge is to keep mathematics connected to real life. We need to build on and extend their knowledge well beyond the rote skills of numeral recognition and counting. Much more should be included in the kindergarten curriculum. The children should understand numbers, that 10 is a combination of other amounts. They should be classifying, estimating, sorting, reasoning, and be encouraged to solve problems. These skills have been included in the curriculum and certainly in good textbooks such as Mathland. But I think we can go beyond what we have been doing. The teachers have set up the learning situation and created the math concepts for them to learn. We have made the graphs, and set out the “estimation” jar, and given the children manipulatives to sort. The children have been responding to the learning situation we set up for them. After reading all of the research, I can see where I can do more to challenge the students to think mathematically. When we need to figure something out, I will have the children help decide how we will do it. For graphs, I will have the children make the graph and display the information. I will let them tell how they would or did solve a problem. Then we can discuss the different ways of solving problems. Research has shown that children need adequate time and opportunity to develop, construct, test ideas and reflect on their understanding. In working this way, they are developing conceptual understanding. Research supports the notion that children who develop conceptual understanding early perform best on procedural knowledge later. Students with low levels of conceptual understanding need more practice in order to acquire procedural knowledge. Conclusion When originally beginning my research for this class I had intended to look mainly at mathematics in kindergarten. However, as I began reading and searching, I found many interesting articles and went beyond the early years. I found out that even though the material we are teaching is different, the philosophy of our teaching methods should be much the same. Regardless of whether we are teaching kindergarten math or geometry to high school students, the goals for all should be the same. We should be instructing the students so that they see the connection between math and their life. That they see the value in mathematics, and are challenged to think, reason, problem solve, and construct knowledge. We should be teaching to all of the students and encouraging all to learn, not discouraging students because it may be difficult. We should include diverse approaches so that we reach all learners, and we should be integrating technology so that it becomes a natural part of the learning process. Bibliography Balas, Andrea K. (1997). The Mathematics and Reading Connection. Eric Digest, 2/7/02 Bloom, L., Perlmutter,J., Roger, A., Rose, T., (1997). Who Uses Math' Primary Children’s Perceptions of the Uses of Mathematics. Journal of Research in Childhood Education, 1997, Vol.12, No. 1 Cebulla, K.J. & Grouws, D.A. (2000). Improving Student Achievement in Mathematics. Part I. Research Findings. Eric Digest, Dec. 2000 Checkley, Kathy (1999). Math In The Early Grades, Laying a Foundation For Later Learning. Curriculum Update, summer, 1999 Davis, B.C. & Shade, D.D. 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