Experiencing School Mathematics - Literature review Example

Running Head: THE ISSUE AND THE TEACHING OF MATHEMATICS The Issue and the Teaching of Mathematics [The Writer’s Name] [The Name of the Institution] The Issue and the Teaching of Mathematics Introduction There are many psychological studies about children and adolescents' problems with mathematics. For teachers to help their students overcome their problems with mathematics, reviewing such studies and implementing the suggested strategies is of the utmost importance. Teachers must first recognize the common problems and their causes in order to apply the most efficient teaching methods with the goal to get students to correctly use mathematical concepts in the classroom and out. The problems discussed here will involve mathematical representation and vocabulary, parental influence, and ineffective class activities. To help the students with such problems, teachers and parents should provide various representations with meaning and be positive in their approach to discussing mathematical issues. In order to prepare mathematically literate citizens for the 21st century, classrooms need to be restructured so mathematics can be learned with understanding. Teaching for understanding is not a new goal of instruction: School reform efforts since the turn of the 20th century have focused on ways to create learning environments so that students learn with understanding. In earlier reform movements, notions of understanding were often derived from ways that mathematicians understood and taught mathematics. What is different now is the availability of an emerging research base about teaching and learning that can be used to decide what it means to learn with understanding and to teach for understanding. (Wagner, 1989) Literature Review Many students from elementary to high school have the same problems with understanding mathematics. One of students' biggest problems 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 Fennell's 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 question's representations became numerical (Fennell 2001). Many other students' problems lie in their weakness in mathematical vocabulary. Math, 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 (Thompson, 2000). Another problem is that many people "feel it's 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) Emphasis on speed on math classes may also be a problem for a large number of students. Answers are expected too quickly and that puts stress on the students. "We lose many, many students because of the emphasis on speed in math classes. We have children waking up at night, fearing the next day's speed test." (Dedyna 2002) 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). It's clear that concepts must be meaningful for children to retain them. Dedyna, Katherine (2002) points out how so many students cannot do times-tables yet can remember every word of their favourite 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). As far as negative attitudes about mathematics, Dedyna, Katherine (2002) blames parent’s attitudes. "Adults will talk openly about how they always hated math and how they did not do well -- they're 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 person's life"- it is not realistic to have people tested in this fashion (2002). There are strategies in teaching mathematics that may reduce some of the problems. Teachers should use a variety of representations and even some relatively unconventional tools. 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 (Fennell 2001). Parents should also learn their children's 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). 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." Choral responses have also been shown to reinforce meanings of mathematical terms. A teacher can ask, for example, what a logarithm is and the students are to answer, "A logarithm is an exponent" in unison. Another interesting technique was studied by Thompson who found it beneficial for students to invent their own terms for mathematical concepts (Thompson 2000). 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. (Hiebert, 1997) In order to meet the challenge, some mathematics teachers have had to reformat their courses with less traditional techniques. Much of the newer techniques involve the interaction of the students in the room. Some teachers who offered students a choice of what representations to use had students with good representations share their work with the class and then discuss together. This provided students with examples of representations they had perhaps not thought about (Fennell 2001). It also puts more focus on the thinking and less on the answer. It is like "teasing" how to think as opposed to teaching how to think (Dedyna 2002). Discussions like this also allow the teacher to see how that particular class thinks and which representations are used most frequently (Fennell 2001). To provide students with more opportunities to "talk math", some teachers have set up small groups in their classes to do work on problem solving orally. (Fuson, 1997) 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). 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 have adopted 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 correcting them if needed (Thompson 2000). Dedyna, Katherine (2002) suggests that parents use daily life to apply basic mathematical principles such as fractions and exponents, which corresponds with Thompson's study where students used the "think twice" exercise. (Tall, 1977) It involved students identifying from their own lives a situation in which they used or could have used mental math. They then posed a mathematics question and write two different mental-math approaches (or representations) to answer the question (Thompson 2000). The most untraditional method of all was the math journals which were discussed by Thompson (2000). Students wrote about their experiences in class, what they learned, vocabulary terms, and then edited other journals. By reading journals of varying quality, students could see what were good and bad ways of expressing mathematics and could pick up on different perspectives on the terms and problems. Along the same reasoning, some teachers modified their take-home assignments to include explanations for students thinking when they worked out answers (Thompson 2000). Issues and Problems in Teaching Mathematics Teachers can help their students to overcome problems in mathematics by four processes. The first is to know what the most common problems are for the students. A teacher must then understand the causes of that problem. If certain strategies implemented while teaching math can reduce some of the common problems. Getting students to think mathematically and out loud has shown to improve their understanding of concepts significantly. The main problems for math students discussed in the present paper were 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 favourable attitudes towards mathematics in everyday life. (Skemp, 1971) Students need to be encouraged to keep trying to work past their boundaries, to see them as goals and not as obstacles. When we start labelling students they begin to feel different, as if they cannot do the same things that their peers can and we end up encouraging an outcast feeling. All students need to learn a variety of problem solving skills because we all have different learning styles, why then must we influence children with learning disabilities to think that they have an excuse to just give up? They do not; they simply have to discover a new skill to break past that block in the road to success. The dependence of Mathematics on culture would be a subject that most people would tend to agree on. The expected answer, of course, when someone is asked whether or not Mathematics is influenced by culture, is a "no". (Secada, 1997) This is much like the Human and Natural Sciences; our first guess would be that they are not dependent on culture, but after some thought and through considering a few examples, it becomes apparent to us that other points of view are valid. The same applies for Mathematics. We must keep in mind that Mathematics is completely man-made; it is not a creation of God or something that the world's population inherited with the Earth. It is a solely man-made creation. This compels us to consider Mathematics an outcome of human culture. The rather "unnatural" creation of Mathematics should prove to be enough evidence that it is dependent on culture. However, it is possible to escape this through a minor technicality: we are discussing the dependence of knowledge with Mathematics on culture, not the creation of Mathematics. It is fair to say that since the knowledge related to Mathematics today can all be traced back to the original creation of Mathematics, and hence it is dependent on culture, but for the purposes of this discussion, we will address the knowledge of Mathematics as an independent entity, separate from the subject's creation.  The knowledge of Mathematics is something that we can consider independent of one's culture. This is because no matter where the subject is being taught, the mechanical and theoretical aspects will remain the same; the country or language in which it is being taught has no effect on the content to be learned. An example comes from my own experience. After moving to Indonesia from the United States after 9th grade, I began 10th grade at a local international school. The following summer, I moved to Canada. In the years prior to living in the United States, I had also lived in India and Spain. Throughout my relocations, spanning five countries over three continents, the cultures of the countries had no effect on the mathematics taught; the only variable was the difficulty of the course and the teacher. (Askew, 1995) Mathematics is a pure subject, and the knowledge within Mathematics is not dependent on culture.  Someone may argue against it, citing language as an aspect of culture and claiming that Mathematics taught in French is different from its English or Swahili counterpart. While the language of the Mathematics certainly is different, it does not have any effect on the ability of a student to do a Mathematics problem in English. Mathematics is a universal language; and the knowledge of Mathematics is therefore not dependent on culture. (Boaler, 1997) History is an Area of Knowledge that could fall into either category regarding its dependence on culture, due to a well-supported argument and one controversial example. Firstly, it may be assumed that History is not dependent on culture simply because it is based on fact. Although it is a valid reason, there are many Problems of Knowledge that arise from this assumption. How can we know that these facts are true? After all, the facts we learn our History from are all results of record keeping - how can we just blindly trust the record keepers? Are we not to question any of these records? Even though it is based on only one argument, and Problems of Knowledge do result, History's position as a fact-based Area of Knowledge is a strong argument, and makes it a legitimate position to take.         On the other hand, there is an example that shows how History is affected by culture. The education system in Tennessee has long been criticized for teaching its students that the Confederate Army won the US Civil War, despite the successful Emancipation Proclamation that declared all slaves free (and allowed the Union to reach its goal). Even though the 49 other states believe otherwise, there is nothing to prove the Tennessee State Board of Education wrong. It is simply a matter of their culture - their pride of the South, their respect for soldiers of the past - that results in this educational discrepancy between students who live in different states. As a result, History gives rise to problems regarding its dependence on culture, and it is because of this reason that History can legitimately be considered to be both dependent and independent of culture. In this way, the Areas of Knowledge vary widely both in the extent of their dependence on culture and the way in which they depend on culture. Some, such as the Arts, are heavily culture-based; while knowledge in Mathematics is quite independent of culture. The similarities and differences between the Areas of Knowledge and their degree of dependence on culture are an intriguing topic, to say the least. Children begin to construct mathematical relations long before coming to school, and these early forms of knowledge can be used as a base to further expand their understanding of mathematics. Formal mathematical concepts, operations, and symbols, which form the basis of the school mathematics curriculum, can be given meaning by relating them to these earlier intuitions and ideas. For example, children as young as kindergarten and first grade intuitively solve a variety of problems involving joining, separating, or comparing quantities by acting out the problems with collections of objects. Extensions of these early forms of problem-solving strategies can be used as a basis to develop the mathematical concepts of addition, subtraction, multiplication, and division. Conclusions Students need to be encouraged to keep trying to work past their boundaries, to see them as goals and not as obstacles. When we start labelling students they begin to feel different, as if they cannot do the same things that their peers can and we end up encouraging an outcast feeling. All students need to learn a variety of problem solving skills because we all have different learning styles, why then must we influence children with learning disabilities to think that they have an excuse to just give up? They do not; they simply have to discover a new skill to break past that block in the road to success. Mathematics instruction is changing in the western industrialized countries, and there is little need to repeat time and again that mathematics counts, to admonish youth to choose exact sciences, and to call for "agendas for action. Mathematics lessons frequently are planned and described in terms of the tasks students engage in. Tasks can range from simple drill-and-practice exercises to complex problem-solving tasks set in rich contexts. Almost any task can promote understanding. It is not the tasks themselves that determine whether students learn with understanding: The most challenging tasks can be taught so that students simply follow routines, and the most basic computational skills can be taught to foster understanding of fundamental mathematical concepts. For understanding to develop on a widespread basis, tasks must be engaged in for the purpose of fostering understanding, not simply for the purpose of completing the task. Standard mathematical representations and procedures involve symbols and operations on those symbols that have been adopted over centuries and have been constructed for the purposes of efficiency and accuracy. The connections between symbols and symbolic procedures and the underlying mathematical concepts that they represent are not always apparent. As a consequence, practicing formal procedures involving abstract symbols does little to help students connect the symbols or procedures to anything that would give them meaning. Although the selection of appropriate tasks and tools can facilitate the development of understanding, the normative practices of a class determine whether they will be used for that purpose. In classrooms that promote understanding, the norms indicate that tasks are viewed as problems to be solved, not exercises to be completed using specific procedures. Learning is viewed as problem solving rather than drill and practice. Students apply existing knowledge to generate new knowledge rather than assimilate facts and procedures. Tools are not used in a specified way to get answers: They are perceived as a means to solve problems with understanding and as a way to communicate problem-solving strategies. The classrooms are discourse communities in which all students discuss alternative strategies or different ways of viewing important mathematical ideas such as what is a triangle. Students expect that the teacher and their peers will want explanations as to why their conjectures and conclusions make sense and why a procedure they have used is valid for the given problem. In this way, mathematics becomes a language for thought rather than merely a collection of ways to get answers. The mathematics to be taught and the tasks and tools to be used might be specified by an instructional program, but without requisite understanding of mathematics and students, teachers will be relegated to the routine presentation of (someone else's) ideas neither written nor adapted explicitly for their own students. In short, their teaching will be dominated by curriculum scripts, and they will not be able to establish the classroom norms necessary for learning with understanding to occur. They will not be able to engage students in productive discussion of alternative strategies because they will not understand the students' responses; neither will they be able readily to recognize student understanding when it occurs. Understanding mathematics for instruction involves more than understanding mathematics taught in university mathematics content courses. It entails understanding how mathematics is reflected in the goals of instruction and in different instructional practices. Knowledge of mathematics must also be linked to knowledge of students' thinking, so that teachers have conceptions of typical trajectories of student learning and can use this knowledge to recognize landmarks of understanding in individuals. Teachers need to reflect on their practices and on ways to structure their classroom environment so that it supports students' learning with understanding. They need to recognize that their own knowledge of mathematics and of students' thinking, as well as any student's understanding, is not static. Teachers must also take responsibility for their own continuing learning about mathematics and students. Class norms and instructional practices should be designed to further not only students' learning with understanding, but also teachers' knowledge of mathematics and of students' thinking. Tasks and tools should be selected to provide a window on students' thinking, not just so that the teacher can provide more appropriate instruction for specific students, but also so the teacher can construct better models for understanding students' thinking in general. References Askew, M. and Wiliam, D. (1995) Recent Research in Mathematics Education 5-16, London: HMSO. Boaler, J. (1997) Experiencing School Mathematics: Teaching Styles, Sex and Setting, Buckingham: Open University Press. Dedyna, Katherine (2002, October) Hated Math? Don't let the children know. Southam Newspapers Fennell, Frenacis (2001) Representation: An important process for teaching and learning mathematics. Teaching Children Mathematics, Reston; Jan 2001; Vol. 7, Iss. 5; pg. 288 Fuson K. C., De La Y. Cruz, Lo A. M. Cicero, Smith S. S., Hudson K., & Ron P. (1997) towards mathematics equity pedagogy: Creating ladders to understanding and skill for children and teachers. Manuscript submitted for publication. Hiebert .J., Carpenter T. P., Fennema E., Fuson K. C., Wearne D., Murray H., Olivier A., Portsmouth, NH: Heinemann. & Human P. (1997). Making sense: Teaching and learning mathematics with understanding. Secada W. G. & Adajian L. B. (1997). "Mathematics teachers' change in the context of their professional communities". In E. Fennema & B. S. Nelson (Eds.), Mathematics teachers in transition (pp. 193 -219). Mahwah, NJ: Lawrence Erlbaum Associates. Skemp, R. R. The Psychology of Learning Mathematics, Harmondsworth, Penguin, 1971, p. 14 Tall, D. O. ‘Conflicts and catastrophes in the learning of mathematics’, Mathematical Education for Teaching, 2, 4, 81, 1977, pp. 2-18 Thompson Denisse R; Rheta N Rubenstein (2000) Learning mathematics vocabulary: Potential pitfalls and instructional strategies. The Mathematics Teacher, Oct 2000, Vol. 93, Iss. 7; pg. 568 Wagner, S. and Kieran, C. (1989) Research Issues in the Learning & Teaching of Algebra, Research Agenda for Mathematics Education, vol. 4, Reston: Lawrence Erlbaum & NCTM. Read More