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TEACHING MATH.   Term Paper ID:26245 Click to get this paper

Essay Subject: Examines problems, evolving theories & techniques, examples, problem-centered instruction, curriculum, measuring performance, teaching standards, proposes field trip to encourage problem solving.... More...

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Paper Abstract: Examines problems, evolving theories & techniques, examples, problem-centered instruction, curriculum, measuring performance, teaching standards, proposes field trip to encourage problem solving. Paper Introduction: In response to unacceptable levels of mathematics achievement in the United States many groups with an interest in the teaching of mathematics have called for reform (Niemi, 1996). The calls for reform are based on recent advances in cognitive theory that call for a more constructivist view of learning (Wood & Sellers, 1996). Traditionally, mathematics instruction in the primary grades has consisted of teaching computational skills, drills, and simple word problems. Concepts now included in mathematics curricula are being changed to stress problem-solving skills, but the resources and teaching methods available to primary grade teachers are not keeping current with the new standards. In 1989, the National Council of Teachers of Mathematics (NCTM) began emphasizing the teaching of mathematical concepts and problem solving at all grade levels. In the years Text of the Paper: The entire text of the paper is shown below. However, the text is somewhat scrambled. We want to give you as much information as we possibly can about our papers and essays, but we cannot give them away for free. In the text below you will find that while disordered, many of the phrases are essentially intact. From this text you will be able to get a solid sense of the writing style, the concepts addressed, and the sources used in the research paper. The teacher and the textbook determinewhat the problems are and how they are to be solved. G. Assessing conceptual understanding inmathematics: Representations, problem solutions, justifications, andexplanations. Promoting a problem-posing classroom.Teaching Children Mathematics, 4(3), 172-18 . The teacher is a coach, or facilitator to the student(Bay, B. Deepening the Analysis: Longitudinalassessment of a problem-centered mathematics program. Last the foundsolution should be checked against the original question to see if it isreasonable and accurate. These changes seem to be most difficult to achievein the primary grades where students are least familiar with mathematicalconcepts and skills. There are many ways to create a classroom environment which is richwith problem-solving opportunities. A teacher using a standards basedcurriculum needs time to observe her students. M. (1998). The atmosphere of the classroom mustbe open to multiple ideas of how to solve a problem. Were the all of the goals and objectives of theproject met? Portfolios and journal serve the purpose of documenting a child'sprogress and at the same time they give the opportunity to fulfill anothergoal of the California framework and of the NCTM standards--that childrenbe given the opportunity to explain their thinking to other students and tothe teacher. At the weekly meeting, these journalentries will be shared and discussed to increase each teachers' resource ofproblem-solving situations, to gain ideas, and to support each other. Rose, T. [Infotrack: GeneralReference Center]. The Clearing House, 7 ,137-141. There is usually morethan one method which could be useful in solving a problem. (1997). The standardalgorithms are taught and used. The purpose of this project are twofold (a) todetermine what factors and conditions contribute to, and are necessary forthe development of a rich mathematics problem-solving environment in theprimary grade classroom, (b) to develop a curriculum project which willinclude all elements necessary for primary students to develop a richrepertory of problem-solving skills. As opportunities present themselves the teachershould promote the use of mathematical problem solving across thedisciplines. After ayear, teachers shift their beliefs, in how children learn about mathematicsand how to teach mathematics, in the direction proposed by the standards(Woods & Sellers, 1997). The primary focus of mathematics instruction is shiftingfrom simple computational skills to problem-solving applications in allgrade levels. Inadequacies have been found in students' abilities to solve and posemathematical problems (Christou & Philippou, 1998; English, 1997, 1998).Problems posed in a linear fashion, when the mathematical equation can bemapped directly from the word structure of the problem, are fairly easy formost children to solve. 25. The teacher'sattitude affects the students' acceptability of multiple approaches andsolutions. [Infotrack: General ReferenceCenter]. Children's problem posing within formal andinformal contexts. The purpose of teaching problem solving in the primary grades is toenable the student to understand the relationships between an event and themathematical model of that event. Journal for Research inMathematics Education, 29, 436-443. An example event: There are threechildren outside. Strategies for teachers to use to promote a variety of problem-solving techniques to students include: (a) having students act the problemout, (b) making a diagram, (c) looking for a pattern, (d) constructing atable (e) systematic listing, (f) guess and check, (g) working backwards,(h) sorting relevant and irrelevant information, (i) solving a similar butsimpler problem, (j) changing your point of view, (k) choosing anoperation, (l) using logic, (m) using a model, (n) using a formula (LaidlawEducational Publishing, 1996, pp. In the above problem, children will be predisposed tounderstand the word "more" to mean addition (English, 1998). The NCTMstandards also state that children should develop strategies to solve avariety of problems, check their solutions against the original problem,and become confident in their abilities (NCTM, 1991). A better overall picture of the child's developmentand learning is constructed. This differs from a class which is using astandards based curriculum (a curriculum based on the standards of theNCTM). Teachers often informally meet and discuss problems and classprojects over lunch. The shift from computation skills to applying mathematical conceptsto problem-solving situations requires changes in teacher philosophy,teaching methods, the skills taught (the scope and sequence), andassessment strategies. The CaliforniaMathematics Framework also includes problem-solving instruction beginningin kindergarten. With training in posing problems, the children'sability to pose problems from informal contexts increased. Niemi, D. The teachers will be meeting andreporting to the principal on the groups progress and activities.Evaluation The teachers will provide the principal or his representative with awritten weekly report on the project's status and the classrooms'activities. An initialapproval will be solicited from the principal at this time. Likewise, teachers who taught a problem-centeredmathematics class, for at least one year, also changed their views onmathematics instruction. (1996). Journal of Educational Research, 89, 351-364. & Reys, R. And any otherquestions or further thoughts, comments or modifications which might betaken in future projects. The Clearing House, 72, 51-55. Attendance and lunch countsare opportunities for real-life problem solving. It will also state ifthe project was viable. (1998). Students may learn from each other aswell as the teacher. References Bay, J. (1997). Traditionally, mathematicsinstruction in the primary grades has consisted of teaching computationalskills, drills, and simple word problems. In reality themathematical operation called for is subtraction. The teacherand the textbook are the authority. In contrast, problems which do not have clearmapping, between the problem situation and the mathematical operationneeded, are found to be difficult; these problems include comparison typeproblems. Teachers should be open to seeing and solvingproblems mathematically when they are found regardless of the subjectmatter. This project report will outline theyear's problem solving and fund-raising activities. (1997). Atthe end of the project, after the field trip is taken, each teacher willsubmit to the group leader a formal evaluation on whether the field triplearning activity has met its goal and objectives. The primary consideration is theteacher's attitude and willingness to take time to see and solve problemsas they arise in daily classroom activities. Were the students able to successfully raise enough money tofund their field trip or were school funds needed to supplement the classraised moneys? The mathematical model: 3 children = 3 glasses of lemonade are needed 3 glasses - 2 glasses = 1 more glass of lemonade is neededMathematics is studied in school so that we can resolve real-life problems(Krech, 1999). Throughout the year,journals can be kept by the students to record weekly progress toward theirgoal. Reys, 1999; Wood & Sellers, 1996). The students who received instruction in classrooms which usedproblem-centered instruction methods gained a greater fluency withmathematical concepts, their representation, and changed their ideas aboutwhat mathematics. The standard for grades kindergarten through fourthgrade states that; The study of mathematics should emphasize problem solving so that students can--use problem-solving approaches to investigate and understand mathematical content; formulate problems from everyday and mathematical situations; develop and apply strategies to solve a wide variety of problems; verify and interpret results with respect to the original problem; acquire confidence in using mathematics meaningfully. Allowing for different approaches to solving a problem, theteacher must allow each group of students or individual the opportunity toexplain why they used their method to solve the problem. The value of collaboration between teachers when theyare striving to implement and improve the problem-solving environment intheir classrooms is proven (Bay, B. [Infotrack:General Reference Center]. School budgets being nonexistent,administrative support should be obtainable.Project Goals and Objectives The project goals are to raise money for the field trip and to enrichthe children's problem-solving environment. [Infotrack: General Reference Center]. The first standard calls for mathematics as problemsolving (NCTM, 1991). & Stupiansky, N. A problem-centered curriculum in mathematics also influences thestudents' belief system about mathematics (Wood & Sellers, 1997). (1998. Make time for tune-ups. By allowing the child to place work into the portfolio orrecord entries into a journal, the child can see his own progress inmathematics, which will help him to develop confidence in his own abilitiesto solve problems mathematically. Students who spent two years in problem-centered mathematics instruction tended to be task oriented. Children'sinterpretations of standard formal number sentences appears to be limited.The diversity of types of word problems generated from an informal contextwas greater than with a formal number sentence where children tried tomatch the word problem's syntax to the operation in the number sentence.Even with training in posing problems, the types of word problems given forany number sentence was limited, although the children could change thecontext of the problem. Reys & R. The developmental nature ofability to solve one-step word problems. Krech, B. Wood and Sellers have demonstratedthat children placed in a problem-centered curriculum have significantlyhigher achievement in mathematics than those in traditional textbook basedinstruction. As the date of the trip approaches in spring, the distance, timeneeded for travel, and return time can be calculated. How many students areeating hot lunch or cold lunch each day. The calls for reform are based onrecent advances in cognitive theory that call for a more constructivistview of learning (Wood & Sellers, 1996). Journal for Research in Mathematics Education, 29, 83-97. When given a formal mathematical sentence and asked to create aproblem to go with it, students will likely create a problem with the samestructure as the mathematical sentence (English, 1998). Theimportance of collaboration was second only to administrative support whenteachers ranked the elements necessary to successfully implement astandards based curriculum. Top 1 elements thatmust be in place to implement standards-based mathematics curricula[database]. The ability of childrento correctly translate a problem into mathematical terms depends on theirlevel of mathematical development and their fluency with mathematicalterminology (Niemi, 1996). This last step fulfills the goal of havingstudents reflect on their learning. Amonthly report will be given to the principal on student activities and themoney raised to date.Timeline Before school releases in June, second and or third grade teacherswill be approached to determine interest in a joint project. p. The CaliforniaMathematics Framework lists five strategies to help students developattitudes and learning strategies useful to problem solving. Teachers will meet weekly to discuss problem-solving activities related to trip planning and funding activities. The shifting emphasis to problem solving in the mathematicscurriculum means that strategies for assessing the work of an individualstudent must be altered. If a recyclingproject is chosen, objects can be sorted, weighed, and value calculated.An initial budget for the trip can be made. These types ofproblems are useful for teaching primary students numeration,classification, and seriation which are all taught in the primary grades(Micklo, 1997). Reys, 1999). Other opportunities for real-life problems come fromsocial studies--map reading, calculating distance, travel time,populations, and contrived economic situations (Richstad, 1998).Literature can be a source of material for formulating problems (Burns,1997; English, 1998). Also, students experience a disconnectionbetween what they perceive as school mathematics and the mathematics theyuse in informal, non-school situations. Students tendto misinterpret key words in the problem to mean the opposite mathematicaloperation. At the end of the projectchildren will be able to (a) add and subtract amounts of money, (b)complete a bar graph, (c) weigh and measure, (d) classify and compareobjects, (e) calculate distance, (f) add and subtract units of time, (g)explain how they have solved a problem using mathematical terminology. The number of multi-step, operationallycomplex problems was greater than structurally complex problems where thechild must identify and relate part-to-part,and part-to-whole (English,1998). Traditionalmathematics classes do not utilize varied types of word problems in theirtextbooks (English, 1998). After school begins in September, students will be informed of theneed to raise funds for a school field trip. How manymore glasses of lemonade does Mother need to bring outside if all childrenwill have one glass of lemonade? [Infotrack: General ReferenceCenter]. Reys, & R. I propose a problem-solving venture toallow the students to raise money to go on a field trip (a real-lifeproblem). A student who is capable of solving problems in one subjectcan generalize the knowledge to other subjects. The logical steps to solving problems do not vary across thedifferent subjects learned in school. English, L. & Philippou, G. (practicing mathematicsproblem-solving skills). Traditional textbook instruction is teacher centered. 2-3). Concepts now included inmathematics curricula are being changed to stress problem-solving skills,but the resources and teaching methods available to primary grade teachersare not keeping current with the new standards. The ability of students to give multiple meanings to formalmathematics symbols increases their understanding of concepts and theirability to pose different types of word problems (English, 1998; Niemi,1996). Micklo, S. What problems were encountered? Parent volunteers will be recruited to assist the children. National Council of Teachers of Mathematics, Curriculum andEvaluation Standards for Mathematics. E. The value of the new standards, calling for the inclusion of problemsolving in mathematics curricula, written by the NCTM is shown by researchstudies (Wood & Sellers, 1996, 1997). Project design Encouraging mathematical problem solving in the classroom willincrease the learning of mathematical concepts by students and increasetheir test scores. 25). Laidlaw Mathematics, Series 2 (pp.2-5).Riverforest Ill.: Thatcher and Madison. What benefit did theteachers gain from collaborating? Instructor (199 ), 1 8, 54. The teacher's job in the classroom is to facilitate the learningprocess in mathematics. In 1989, the NationalCouncil of Teachers of Mathematics (NCTM) began emphasizing the teaching ofmathematical concepts and problem solving at all grade levels. D. Instructor (199 ), 1 8, 11-13. Part of the difficulty children have in posing a variety of typesof word problems situations may lie in the lack of exposure to manydifferent types of problems in the classroom environment. Christou, C. Laidlaw Educational Publishing. Wood, T., & Sellers, P. From these finalevaluations from the teachers, the group leader will prepare a groupproject report for the principal. In theyears following, the NCTM have strengthened their stance on the inclusionof problem solving within mathematics curricula. The standards based curriculum places more responsibility on thestudent to learn. Wood, T., & Sellers, P. Assessments which place more importance onthe how and why of a child's solution are available. Studentsin problem-centered classrooms were motivated by a belief in the importanceof finding their own solutions to problems and believed that to besuccessful in mathematics they must be able to explain their logic andreasoning to other people (Wood & Sellers, 1997, 171). In October, the children will begin to raise funds. H. The plan should carried out and a solution found. Childhood Education, 73, 194-2 . She needs time to learn tolearn to use scoring guides to grade projects, to interview her students,to help students develop a portfolio of their work, and to grademathematics journals (Bay, J. Teachers will coordinate a fund-raising activity for the children andsuccessfully raise money to fund a field trip, estimated cost total forthree classes, 3 dollars. Instructor (199 ),1 7, 72. For example: Sue has seven cookies. Students with higher levels ofrepresentational knowledge are able to produce better justifications,explanations and were able to use mathematical representations moreeffectively in problem-solving tasks (Niemi, 1996). & Schuncke, G. Phi Delta Kappan, 8 , 5 3. Waite-Stupiansky, S. Funds needed oralready acquired can be added, graphed, and tabulated. Students withless that two years experience in problem-centered mathematics instructionbelieve that mathematics is about solving problems the teacher's way.Textbook instruction in mathematics leads students to believe that there isonly one method used to solve problems. What do students need to becomeeffective problem solvers? Assessment of a problem-centeredmathematics program: Third grade. Mother brings out two glasses of lemonade. Review of the Related Literature In 1991, the NCTM gave the community of mathematics teachers a newset of standards. Data can be graphed, percentagescalculated, estimates made for each day, which menus are most favored, andleast favored (Waite-Stupiansky & Stupiansky, 1998). E Reys, 1999; Micklo, 1994).These types of assessments allow teachers to focus on the child'sindividual development of mathematical concepts instead of justcomputational ability. Burns, M. Journal for Research in MathematicsEducation, 27, 337-353. A child's ability to pose or solve a problem mathematically islinked directly to the child's ability to form a representation of theproblem with mathematic symbols. Richstad, B. (2) Create a classroom atmosphere in which all students feel comfortable trying out ideas, (3) Invite students to explain their thinking at all stages of problem solving. D. First the problem must beidentified and understood. Reston Va.: NCTM. She has five more cookies than Jan. E. Journal of Researchin Mathematics Education, 28, 163-186.----------------------- 1 (1999). M., Reys, B. Do theteachers wish to repeat the project in future years? (p. (1999). The teachershould: (1) Model problem solving behavior whenever possible, exploring and experimenting along with students. Solutions to weekly problems can be recorded with the children'sreflections on their activities. These same studentsdid not believe that they would succeed in mathematics by being quiet orcompleting their work neatly. M. How many more cookies does Jan have?In this type of problem there is an unknown reference set. Educated candies: a classroom businessplanned and managed by students [database]. J. Finding multiple ways to solve a problem.(mathematics teaching) [database]. Math portfolios in the primary grades[database]. Create a climatefor problem solving. Teachers who volunteer to teach problem-centeredmathematics classes have a more constructivist view of learning than otherteachers before they are exposed to problem-centered instruction. My project goal is toincrease problem-solving opportunities in the primary grade classrooms bycreating in these classrooms a rich problem-solving and problem-posingenvironment. These gains in concept understanding are seen after two yearsof instruction with problem-centered mathematics instruction; computationskills were equivalent to children in the traditional textbook curriculum.Further, these gains in mathematics concepts, computation, and motivationto learn, continue to be evident for at least one year after the studenthas returned to traditional textbook instruction (Wood & Sellers 1997). (1996). J., Reys, & R. (teaching mathematical problems to children)[database]. There is a definite relationship betweena student's representational fluency and success in problem solving,justification and explanation. These skills are of secondaryimportance in a mathematics curriculum rich with problems solving ideas andmethods where the main goal is to teach mathematical logic and reasoning.The how and why of a child's solution is more important than the arithmeticused to compute the solution. (1997). For my project I propose that three primary gradeteachers who share the same lunch period meet weekly to discuss andcoordinate classroom activities on problem solving. Copies of this and the teachers formalevaluations shall be given to the principal. The teacher isresponsible for being open to alternative approaches and encouraging them.She should require all students treat each other with respect and encourageopen mindedness during discussion. Traditional paper and pencil mathematics teststest primarily for computational skills. Their drawback isthat they take more time to utilize. The types ofproblems did not change significantly, but the level of complexity withinthe created problem did increase. Should the project be repeated? If the activity is deemedsuccessful and repeatable, the group leader will disseminate a copy of thefinal report to interested teachers within the school and to other schoolsas requested. [Infotrack: General Reference Center]. (4) Allow for the fact that more than one strategy may be needed to solve a given problem and that problems may require original approaches. (5) Present problem situations that closely resemble real situations in their richness and complexity so that the experience that students gain in the classroom will be transferable.The California Mathematics framework echoes the NCTM standards. In response to unacceptable levels of mathematics achievement in theUnited States many groups with an interest in the teaching of mathematicshave called for reform (Niemi, 1996). Problem solving: The linkbetween social studies and mathematics [database]. J. She can do this most effectively by designing aclassroom rich with opportunities in problem solving and by encouraging herstudents to use a variety of techniques to solve problems. The problems teachers design for their students shouldfollow these parameters: (a) the problem should be mathematicallysignificant, (b) the problem should occur in real-life, include realobjects, or be an obvious simulation of a real occurrence, (c) thesituation should capture the interest of the child, (d) the students shouldbe able to solve the problem and know when he has the solution. Changes in how a childthinks and refines his concepts of what mathematics is and how it is bestused are documented in the child's own hand and with real examples of thestudent's best and significant work (Micklo, 1999). Each teacher will keep a journal of activities and problemssolved with their classes' solutions. English, L. Students all are motivated by field trips so creativity andmotivation should be high with students. A joint classroom project will be designed to promote problem-solving skills for the full year. The use of a mathematics portfolio andjournal are especially useful because they give evidence of the child'smathematical development over a larger time frame. Rose and Schuncke (1997) show howsimilar processes are used to solve problems in social studies andmathematics. E. The amount of money necessaryfor the proposed trip will be discussed. Possible solutions includerecycling projects, bake sales, car washes, candy sales, and other creativemoney-makers. D. (1998). J. During Augustteacher in-service an initial meeting will be held with interested teachersand the principal. Both sets of objectives state the value of cooperative learning inlearning methods of problem solving. Teachers who use the teaching concepts outlinedby the standards come to the conclusion that these methods used to teachmathematics are better and more effective than the methods used in thetraditional textbook approach. Next a plan must be devised to solve theproblem. If this paper is not what you are looking for, you can search again: Search for: or Click here to request an essay written just for you.