Mental Mathematics: Teaching Integers - Essay Example

Running Head: TEACHING INTEGERS Teaching Integers of the of the Teaching Integers Introduction A lot of research has been performed in past 30 years to find ways of improving numeracy in students. (Chin, 2004, 16) Modern ways of teaching Mathematics have been introduced to benefit both the teachers and the students. This is observed that in most of the cases even highly qualified individuals face difficulty in understanding Mathematics and its laws. Its main reason is lack of skilled educationists, psychological blocks, and idealistic misconceptions regarding character of Mathematics. There are strategies in teaching mathematics that may reduce some of the problems. All authors believe that teachers should use a variety of representations and even some relatively unconventional tools. Following is the explanation of various approaches towards teaching mathematics in schools. Mental Mathematics Mental mathematics, linking cubes, drawings, mental images, concrete materials, equations, base-ten blocks, computer programs can all be used with the goal of students to find the representations they personally can manipulate (physically and or mentally) with most ease and meaning. By choosing the representation that represents how they personally think, students can attach meaning to the situation and it becomes more accessible. Parents should also learn their childrens learning style even if it differs from their own. Parents are also encouraged to "pretend" they enjoy mathematics and can do them "as well and anybody" (Dedyna, 2002, Press). Concept building For Thompson, one of the simplest ways of teaching vocabulary is to explain the concepts first, and then attach the term to it. This strategy can be used in conjunction with etymologies (word origins) because when students know these roots, they can make connections between common English words with which they are familiar and mathematics terms. Constructivist Perception After the space race focus, researchers world wide have in the last twenty years taught us a lot about childrens understanding of numeracy and how they come to develop these ideas. Young-Love ridge (2002) drew attention to the fact that many of the understandings children had on entering school were not well matched to the curriculum and what they were taught. Baroody et al (2005) mentions the work of ten different researchers or research groups who assisted the understanding of childrens mathematical thinking in the last twenty years. Young-Love ridge (2005) acknowledged that teachers in Australia and United States that were given a framework, were better able facilitate their students learning. She valued the work of Fuson and Renick in the United States and has used their framework in her work. Also she uses Baroodys increasingly abstract models for building concepts of double-digit numbers. During this period, the change from behaviourist thinking to a constructivist perception of learning has brought about fundamental changes in instruction practices. The belief that knowledge is actively created or invented and not passively received has changed the approach of Mathematics facilitation. Characteristics of Constructivism Students are encouraged to discuss and share their ideas and construct their own ways to solve increasingly complex Mathematics tasks. (Heibert and Carpenter, 1992, 72) The statement of the MSEB and National Council of Research in 2005 that in reality no one can teach Mathematics, Effective teachers are those who can stimulate students to learn Mathematics, has given rise to major changes in understanding of how research facilitates teaching Mathematics. Researchers have used interviews and other qualitative research methods to find out how students think mathematically and this has revolutionized understanding of how children think mathematically. This research evolves frameworks of how children gain competency over integers. The last decade has seen considerably more development in understanding of numeracy and the development of number frameworks. (Tytler, 2004, 12) The work of Askew et al (2001) found that highly effective teachers had a particular set of beliefs underpinning a series of classroom practices and this included what it meant to be numerate and which presentation and intervention strategies were effective. Mathematical Thinking Development One of the biggest challenges for teachers is to get their students to think mathematically inside the classroom so they can think mathematically outside the classroom. In order to meet the challenge, some mathematics teachers reformat their courses with less traditional techniques. Much of the newer techniques involve the interaction of the students in the room. Some teachers who offer students a choice of what representations to use make students make good representations, share their work with the class and then discuss together. This provides students with examples of representations they had perhaps not thought about (Fennell, 2001, 295). It also puts more focus on the thinking and less on the answer. Liedtke calls this "teasing" how to think as opposed to teaching how to think (Dedyna 2002, Press). Discussions like this also allow the teacher to see how that particular class thinks and which representations are used most frequently (Fennell, 2001, 297). To provide students with more opportunities to "talk math", some teachers set up small groups in their classes to do work on problem solving orally. This makes the students feel less intimidated and the teacher can walk around and discreetly correct students in order to reinforce proper use of terminology (Thompson 2000). Silent Teacher Method Usiskin (1996, p. 236) noted, "If a student does not know how to read mathematics out loud, it is difficult to register the mathematics" so some teachers adopte the "silent teacher" method to strengthen students skills in reading mathematics. Here the teachers puts up overheads of word problems involving equations and symbols and asks students to read them out loud and correct them if needed (Thompson 2000). Two recent and continuing publications to help teachers in Mathematics have been published. Learning Media Ltd published the connected series, which in a similar format to school journals focuses on technology, science and Mathematics. The stories and articles are to show technology, science and Mathematics in the context of the childrens everyday lives. Also the Figure It Out series which replaces the School Mathematics series has been published and distributed by Learning Media Ltd too. These colourful booklets provide curriculum support in practical everyday mathematical content at level 3 and 4. Further booklets in the series at level 4 are planned for distribution in 2002-2003. Think Twice Exercise Harrison (2004) suggested that outside the classroom, parents may use daily life to apply basic mathematical principles such as fractions and exponents, which corresponds with Thompsons study where students used the "think twice" exercise. They then posed a mathematics question and write two different mental-math approaches (or representations) to answer the question (Dowker, 2005, Press). The most untraditional method of all is the math journals which are discussed by Thompson (2002). Students write about their experiences in class, what they learn, vocabulary terms, and then edit other journals. By reading journals of varying quality, students can learn what are good and bad ways of expressing mathematics and can pick up on different perspectives on the terms and problems. Thinking Representation One of the biggest problems faced by the student is the inability to represent their thinking. Representations can be oral, numeric, drawn, concrete, on a computer, etc. A student may understand a problem in its oral form, for example, but the written version of the same problem may stump the student because they incorrectly make the transition from the words and symbols on the paper to their mind when they attempt to reason out the answer. This was the case in Fennells experiment where an elementary student gave the correct response to an oral math problem in the form of a story but could not give the right answer when the same questions representations became numerical (Fennell, 2001, 289). Many other students problems lie in their weakness in mathematical vocabulary. Understanding of the language Mathematics, of course, is taught through the medium of language, so Thompson believes that "students need to master this [mathematical] language if they are to read, understand, and discuss mathematical ideas." Students often cannot remember the correct vocabulary terms and their meanings, and other times, misuse terms resulting in further confusion. Weakness in mathematical vocabulary can make understanding and explaining a mathematical concept extremely difficult (2000). Another problem, according to Werner Liedtke, an education professor in math, is that many people "feel its alright to muddle your way through what some people call some basic notions of arithmetic" and this attitude is easy to adopt if a person has had earlier problems with elementary mathematics. (Dedyna 2002, Press). Speed of solving a Mathematics Problem Emphasis on speed in mathematics classes may also be a problem for a large number of students. Answers are expected too quickly and that puts stress on the students. Liedtke expresses, "We lose many, many students because of the emphasis on speed in math classes. We have children waking up at night, fearing the next days speed test." (Dedyna 2002, Press). Negative Attitude towards Mathematics As far as negative attitudes about mathematics, Liedtke blames parent’s attitudes. "Adults will talk openly about how they always hated math and how they did not do well -- theyre proud of it" and "parents are the most important factor as far as the future achievement and attitude toward math." Neither does not help that so much focus is put on speed. Really, "twenty seconds is not along time in a persons life"- it is not realistic to have people tested in this fashion (2002). Teacher’s Understanding of the Subject The professional development on numeracy in recent years has increased teachers pedagogical and content knowledge. The framework has given teachers the skills and knowledge to identify where the children are at in their understanding of numeracy. Teachers in the Numeracy Project talk about their increased confidence and improved facilitation with number strand of the Mathematics curriculum. Teachers now have a clearer understanding of how children learn numeracy. The increased enthusiasm for Mathematics, this work has encouraged, is of considerable benefit. Many primary teachers are not confident of the mathematical understanding. In the Education Gazette 5 March 2001 one of the key elements of the numeracy project 2001 was stated as; the national professional development programme in which the framework is the core component for all primary school teachers during the next three to five years. Hyland the Ministry of Education curriculum facilitator said in the same issue that the project is to improve the performance of all students; particularly Maori and Pacific Island students. The article also goes on to talk about continued support for schools that participated in the Count Me in Too project so gains are locked in. Many teachers lack the mathematical knowledge to engender confidence and enthusiasm towards Mathematics for their students. Professional development has taught us that the teachers’ understanding of a subject is one of the most crucial factors in the successful implementation of it. Teaching in a foreign accent Bobis (2000) cited in Thomas et al (2001) said, the professional development of teachers is accepted almost universally as critical to the advancement of educational effectiveness. Askew et al (2001) valued the importance a positive attitude towards Mathematics and a strong understanding of the connectedness of numeracy. Studies have revealed a number of effective ways of delivering a thought provoking Mathematics lecture. A very effective technique is to teach in a foreign accent. In this regards, useful accents include accents of Chinese, Hindi or German etc. In Ministry of Education has commenced website development and support for teachers. On the Mathematics community page teachers can find news, support and materials. The site is: www.tki.org.nz/e/maths/ also on the Mathematics website teaching resources for statistics, measurement and algebra are available. This site is: www.nzmaths.co.nz these resources are to support classroom teachers and assist them present mathematical concepts in a meaningful context. In spite of these publications the Educational Review Office (ERO) in 2000 criticised the lack of resources and guidance to Mathematics teachers. Why do students suffer from these problems? The representation problem, first off, may be because the students are not given the choice of how to represent their thinking. The representation is imposed on them whether they are strong with that form or not. Students have their own methods of learning best using the representations they find most logical but if they are put in a situation where they cannot use those methods, the math problems become of little meaning to them. Fennell believed that the mistake made by his student was because "the number and symbols that she used were not representations of her thinking"- translation: what the student read on the paper did not have the same meaning as it did orally because it did not look the same as what she "saw" it in her mind when it was in story form. Fennell added, "as soon as symbols become replacements for thinking, we [teachers] have missed our learning target" (Fennell, 2001, 290). It is clear that concepts must be meaningful for children to retain them. Liedtke points out how so many students cannot do times-tables yet can remember every word of their favorite songs. Their songs have meaning to them. This also applies to the remembering of vocabulary words. When students cannot remember the correct meaning of a word, they will not use it to their own detriment and even when they do understand the correct significance, it is often forgotten if not used frequently enough (Thompson 2000). Conclusion In conclusion, the main problems for math students are the inability to represent, or exemplify, their thinking. Teachers can solve this problem by providing students with a variety of representations and letting them discover their own methods that are meaningful. Allowing students to use the representations that they have the most facility with also improves their ability to retain mathematical vocabulary because of the meaningfulness that they can attach to the terminology. To further their improvement in mathematics vocabulary, teachers can create activities in class that encourage mathematical language during discussions and journals about experiences with math can be assigned and then peer-edited. Parents, who act as math teachers at times, should also follow the guidelines all while expressing favorable attitudes towards mathematics in everyday life. To help the students who face problems, teachers and parents should provide various representations with meaning and be positive in their approach to discussing mathematical issues. It is solely a teacher’s responsibility to get the students involved in learning Mathematics. An extremely effective tip for teachers is to avoid giving well-organized lectures. Sometimes, when a lesson is extremely simple, students start to take it for grated and they do not show much interest to it. So the best way is to, make them prepare to come out with the new material in their own way. By doing so, a teacher will develop confidence in them to use their new learning in a way they like to use. An unclear, confusing and disorganized lecture will leave them with lots of questions in their minds. And to fulfill their curiosity, they will make an effort to develop their understanding of the subject on their own. Sample Lesson Plan References Askew, M., Brown, M., Rhodes, V., Johnson, D., & Wiliam, D. (2001). Effective Teachers of Numeracy. London: Kings College, Baroody, A. J. & Ginsburg, H. P. (2005). Childrens mathematical learning: A cognitive view. Journal for Research in Mathematical Education, pg 51=64 Biddulph, F. Gehrke, J. Taylor, M.& Carr, K. (2002). International maths results – Should we be concerned about primary school Mathematics in New Zealand? New Zealand Principal. 17-20 Chin, C (2004) ‘Questioning students in ways that encourage thinking’ Teaching Science, 50(4) 10-20 Clements, D. H. & Battista, M . T. (1990). Constructivist learning and teaching. Arithmetic Teacher, 38 (1). Dedyna, Katherine (2002, October) Hated Math? Dont let the children know. Southam Newspapers, Press Dowker, A.(2005) Individual Differences in Arithmetic. Psychology Press, Hove Sussex Fennell, Frenacis (2001) Representation: An important process for teaching and learning mathematics. Teaching Children Mathematics, Reston; Jan 2001; Vol. 7, Iss. 5; pg. 288-300 Harrison (2004) ‘Teaching and learning Science with analogies’ Allen and Unwin, 160-180 Heibert, J and Carpenter, T.P (1992) Learning and Teaching with understanding’, New York, Macmillan, 60-90 Hughes, M. (2000). Can preschool children add and subtract? Educational Psychology 1 (3) 207-219 Kemp, M. & Hogan, J. (2000). Planning for an emphasis on numeracy in the curriculum, Contextual paper commissioned by AAMT. Canberra, ACT: Commonwealth of Australia p 4-5. Ministry of Education (2002b). The Diagnostic Interview. Wellington: Ministry of Education. (51) Paulos, J. A. (2001). Innumeracy: Mathematical Illiteracy and its consequences. New York: Hill and Wang. Resnick, L. B. (2005) Developing mathematical knowledge. American Psychologist.44 (2) 162-169. Thomas, J. & Ward, J. (2001). An Evaluation of the Count Me in Too Pilot Project. Ministry of Education Tytler. R (2004) ‘Constructivism views of teaching and learning’ Allen and Unwin, 10-35 Velde, M. (2001). Every Child Counts. Education Gazette Tukutuku Korero 80 (3), 5-8. Wright, V. (2000). The development of numeracy frameworks. New Zealand Principal. 4-7 Young-Love ridge, J. M (2002). The Development of Childrens Number Concepts: the First Year of School. New Zealand Journal of Education Studies. 24 (1) Young-Love ridge, J. M. (2005). The acquisition of numeracy. Set: Research Information for Teachers, No. 1. 12 (8) Young-Love ridge, J. M. (2006). Helping children move beyond counting to part-whole strategies. Teachers & Curriculum 5. 72-78 Read More