Constructivism in Mathematics - Literature review Example

CRITIQUE OF FOUR ARTICLES RELATING TO CONSTRUCTIVISM IN MATHEMATICS Abstract The current essay is a critique of views expressed by Dr. Max Stephens (Identification and Evaluation of Teaching Practices that Enhance Numeracy Achievement), Joanne Lobato, David Clarke, Amy Burns Ellis (Initiating and Eliciting in Teaching: A Reformulation of Telling), Harkness, Ambrosio and Morrone (Preservice Elementary Teacher’s Voices Describe How their Teachers Motivated them to do Mathematics), and Tracey Muir ( Setting a Good Example: Teacher’s Choice of Examples and their Contribution to Effective Teaching of Numeracy) on how effectively and constructively Mathematics can be taught in classrooms. The 1990s saw significant progress in the field of children’s literacy and numeracy programmes all over the world. The impetus was on children between 3 to 8 years,which made sure that children participated in rewarding literacy and numeracy-based activities on a routine basis. Extensive research programmes have been undertaken to merge cultural studies with early childhood education. Early childhood literacy and numeracy learning revolves around the teacher who plays a pivotal role in educating and moulding the children.Individualised learning got replaced with community learning or learning as a group. Apart from meeting individual needs, this type of learning focused much on imbibing different cultures which ultimately help in shaping the individual identity of children In his article, Dr. Stephens discusses policies and initiatives undertaken by the Australian government, regarding school numeracy tutoring in the milieu of the National Literacy and Numeracy Plan agreed by Commonwealth, State and Territory Education Ministers. He provides a detailed study of particular areas in school numeracy education, including: * information on relevant international and Australian research findings, data, thinking and directions; * information and findings from relevant work in other areas, including literacy and general teaching methods; * information on and understandings of current programs and policies in Australia; * identification and discussion of relevant issues. He argues that a national policy for numeracy must address student’s needs at different stages of schooling. The focus should be on practical applications of mathematics which matches the increasing maturity and experience of the students. Models of tutoring which depend on classroom lectures using chalk, provide little hope for students who are weak in school mathematics. Australia has had less big scale set of courses and instruction projects. A current progress is the financial support by Victoria Ministry of Instruction of an Early Numeracy Research Project. The main objective of the venture is to assess the effect on learner numeracy outcomes of prominent design rudiments of the nation’s early numeracy curriculum and related professional improvement actions. He opines that it is vital that lessons of successful professional improvement and classroom performance resultant from literacy programs be applied to a State policy for numeracy. This is necessary because the same teachers and schools will be involved. Nevertheless, while having a clear idea of what has worked for literacy programs, there are quite a few significant differences. Several primary instructors, inclusive of those who are teaching in the early years, convey great lack of self-confidence in their own understanding and knowledge of mathematics. There is also a need to improve insight of children concerning mathematics. Dr. Stephens observes that a lot of children perceive mathematics only as ‘sums’ and after one year some have begun to tell maths as uninteresting. There is no common understanding among instructors of the ‘great ideas’ of numeracy in the primary period of schooling combined with a lack of evaluative instruments and methods which echo the rich conceptions of school mathematics(Dr. Stephens 18). The instruction outline put forward by the National Numeracy Project and now advocated in the report The Implementation of the National Numeracy Strategy (1998) is that all primary schools should offer a daily mathematics session of between 45 and 60 minutes for all children with excellent teaching. In particular the report comments that a majority of these lessons should center on building numeracy skills and quite a lot time than was former the case in mathematics should be used training the whole class together. He argues that these recommendations are definitely aimed against a manner of teaching where mathematics is freely and at times disjointedly incorporated into the learning of other subjects. This is not to underestimate the significance of incorporating mathematics in real contexts. It is simply to say that it requires very knowledgeable teachers to make these links clear to students and to develop consistent and coherent threads of mathematical understanding. Dr. Stephens opines that the function of calculators in the primary and succeeding years of school is a controversial issue globally, and in Australia there is an apparent need for recommendation regarding their suitable use. If intellectual computation , a growing number sense and an increasing understanding of the number system are among the most imperative goals of these years, as seen in the core curriculum documents of all States and Territories, then there is a powerful case for some limited access to calculators and for clear direction to instructors about what is to be used. The paper also analyses the ways in which school-home links can be supported. The Family Math Program devised a number of models for linking parents in school mathematics. Homework is one part where there is great deal of opportunity for schools and parents to work together. Such programmes encourage parents to engage their children in games with a numeracy focal point such as snakes and ladders and counting games. The back up of main parent Organizations such as the Australian Council of State School Organizations(ACSSO) and the Australian Parent’s Council(APC), will be crucial in reaching parents. Above all there has to be a change in the perception that it is alright to say ‘I’m no good at Maths.’ It is that rearrangement of outlook which is at the core of Australia’s state literacy and numeracy aims, which Dr. Stephens has introduced in his paper. Constructivism is a recent philosophical viewpoint in the field of mathematics teaching and learning. Following the trails of Piaget and other prominent educational psychologists, constructivism holds the view that individuals construct their own knowledge through experience. The learner is an active participant here. Constructivist teachers promote active and meaningful learning by taking into consideration students’ perspectives and background. Lobato, Clarke and Ellis discuss how academic action of telling has been methodically downplayed in the mathematics tutoring literature. Telling has conventionally meant lecturing information or demonstrating events (Smith,1996). A perceived inconsistency between telling and constructivism, an understanding of the potential negative consequences of relying too heavily on telling, and a focus on the development of a range of “non-telling” actions have all contributed to a de-emphasis of telling. Lobato, Clarke and Ellis rearrange telling in three ways: 1) relating to function (which requires concentration to the teacher’s purpose, the nature of the teaching endeavor, and the students’ comprehension of the action) rather than the type of educator’s communicative acts,2) in relation to theoretical rather than practical content of the innovative information, and 3) in relation to connection with other actions rather than as a remote act. This rearrangement resolves some of the concerns with training as telling and helps to set up the authenticity of offering new information within a constructivist viewpoint on education(para 1). They enumerate the demerits of traditional telling actions: The ‘teaching as telling’ practice is undesirable when it minimizes the opportunity to learn about students’ ideas, interpretations, images and mathematical strategies; b) focuses only on the procedural aspects of mathematics; c) emphasizes the teacher’s authority as the ultimate arbiter mathematical truth rather than developing the students’ responsibility for judgments of mathematical correctness and coherence; d) minimizes the possibility of cognitive engagement on the part of students; e) communicates to students that there is only one solution path; and f) represents premature closure of mathematical; exploration.( : www.beloit.edu/~newb/spring2006/initiating%20and%20eliciting.pdf para 9) In their study, Harkness, Ambrosia and Morrone analyze data in the form of (preservice teacher) student voices taken from mathematical autobiographies, written at the beginning of the semester, and end-of-semester reflections, were analyzed in order to examine why preservice elementary school teachers were highly motivated in a social constructivist mathematics course in which the teacher emphasized mastery goals(para 1). Here, the authors connect two very different theories – motivation goal theory and social constructivist learning theory – and examine the ways in which the students describe Problem Solving and their learning. Constructivist teachers encourage and accept student autonomy and initiative, use raw data and primary sources along with manipulative, interactive, and physical materials, allow student responses to drive lessons, shift instructional strategies, and alter content, inquire about students' understandings of concepts before sharing their own understandings of those concepts, encourage students to engage in dialogue both with the teacher and with one another, engage students in experiences that might engender contradictions to their initial hypotheses and then encourage discussion, allow a waiting time after posing questions, nurture students' natural curiosity through frequent use of the learning cycle model. Constructivist classroom structures make mastery goals salient and elicit mastery-oriented patterns of motivation because students are allowed the freedom to share ideas, ask questions, and make mistakes. Students learn about working hard to achieve understanding; the emphasis is on effort. In Problem Solving, the students learned that the teacher expected them to work hard to understand the mathematics embedded within the tasks. They were open to share their thoughts and strategies, to make mistakes, and to ask questions of not only the teacher but also of their contemporaries. Because both small group and large group construction of meaning were important aspects of the context of the course, students were not compared with others and again, the emphasis was not on performance, but on learning. The authors assert that if we want future teachers to have a positive disposition toward mathematics and to be motivated by it, then we must consider the goal orientations of our students and how we make different goals salient in our classrooms. Tracey Muir’s paper reports on teachers’ selection of examples and the part they play in students’ construction of knowledge. Choosing a suitable example is a tough task for trainers, with both the teacher’s content and pedagogical content knowledge being a determining factor in the selection process. According to Askew (2005): effective teaching of numeracy involves helping students acquire knowledge of and facility with numbers, number relations, and number operations and assisting them with building an integrated network of understanding, techniques, strategies and application skills. In assisting students to construct understanding, teachers often select examples to illustrate particular principles, concepts and techniques. The selection of examples can be an indicator of effective teaching for numeracy, with both the teacher’s content and pedagogical content knowledge being a determining factor in the selection process. (http://www.merga.net.au/publications/counter.php?pub=pub_conf&id=418 para 1) The basic tenet of constructivism is that the learner constructs his/her own knowledge; each learner constructs a unique mental representation of the material to be learned and the task to be performed, selects information perceived to be relevant, and interprets that information on the basis of his or her existing knowledge (Shuell, 1996). The process is an active one and according to Shuell (1996) the most important determiner of what is learned. The construction of an idea will therefore vary from individual to individual even with the same teacher and within the same classroom (Van de Walle, 2007). The teacher’s role is to ensure that students engage with the material to be learned and particularly to foster the connections between both different areas of mathematics and previous mathematics learning. The connectionist teachers identified in the Askew et al., (1997a) study were found to hold beliefs that supported this premise, including the need to explicitly recognise and work on misunderstandings (Askew, Brown, Rhodes, Wiliam, & Johnson, 1997b). Muir points out that in order for a teacher to practice within a constructivist paradigm, knowledge of the subject matter being taught, along with knowledge of the pedagogical principles needed to impart this knowledge to students, is required. One instructional strategy that teachers can use to help students construct meaning and one that plays a central role in the learning of mathematics is the use of examples. Examples may include illustrations of concepts and principles, contexts that illustrate or motivate a particular topic in mathematics and particular solutions where several are possible (Watson & Mason, 2002). Because examples are chosen from a range of possibilities (Watson & Mason, 2002), teachers need to recognise that some examples are “better” than others (Huckstep, Rowland, & Thwaites, 2003). A good instructional example is one which is transparent to the learner, helpful in clarifying and resolving mathematical subtleties and generalisable (Bills, Dreyfus, Mason, Tsamir, Watson, & Zavlavsky, 2006). Bills et al., (2006) maintain that the specific representation of an example or set of examples and the respective focus of attention facilitated by the teacher, have bearing on what students notice, and consequently on their mathematical understanding. Inappropriate examples can lead to a construction of understanding that was not the intention of the teacher. Thus the four articles discussed in this paper, deal with constructivism in mathematics teaching, which is a recent theory. They discuss how innovative techniques can be employed while teaching mathematics and thereby make the students interactive. References Askew, M. (2005). “It ain't (just) what you do: effective teachers of numeracy”. In I. Thompson (Ed.), Issues inteaching numeracy in primary schools (pp. 91-102). Berkshire, UK: Open University Press. Askew, M., Brown, M., Rhodes, V., Wiliam, D., & Johnson, D. (1997b). Effective teachers of numeracy in primary schools: Teachers' beliefs, practices and pupils' learning. Paper presented at the British Educational Research Association Annual Conference, University of York Harkness, Ambrosio and Morrone. Preservice Elementary Teacher’s Voices Describe How their Teachers Motivated them to do Mathematics. Retrieved July 28,2009 from the world wide web: http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=EJ757649. Jaworski,Barbara. Constructivism and Teaching-The Socio-Cultural Context. Retrieved July 28, 2009 from the world wide web: http://www.grout.demon.co.uk/Barbara/Chrcods.html Joanne Lobato, Clarke and Ellis.Initiating and Eliciting in Teaching: A Reformulation of Telling. Retrieved July 28,2009 from the world wide web: www.beloit.edu/~newb/spring2006/initiating%20and%20eliciting.pdf Morrone, A.S., Harkness, S.S., D’Ambrosio, B., and Caulfield, R.: 2004, Patterns of instructional discourse that promote the perception of mastery goals in a social constructivist mathematics course, Educational Studies in Mathematics 51, 19–38. Muthukrishna, N. and Borkowski, J.G.: 1996, ‘Constructivism and the motivated transfer of skills’, in M. Carr (ed.), Motivation in Mathematics, Hampton Press, Inc., Cresskill,NJ, pp. 40–63. Saunders, P. (2004). Characteristics of effective teaching. Retrieved July 26, 2009 from world wide web http://www.wmich.edu/teachlearn/new/char_effect_tch.html Shuell, T. J. (1996). “Teaching and learning in a classroom context”. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 726-764). NY: Simon & Schuster Macmillan. Dr. Stephens Max. Identification and Evaluation of Teaching Practices that Enhance Numeracy Achievement. Retrieved July 28,2009 from the world wide web: www.aamt.edu.au/content/download/1253/25272/file/stephens.pdf Tracey Muir. Setting a Good Example: Teacher’s Choice of Examples and their Contribution to Effective Teaching of Numeracy. Retrieved July 28,2009 from the world wide web: http://www.merga.net.au/publications/counter.php?pub=pub_conf&id=418. Van de Walle, J. A. (2007). Elementary and middle school mathematics (4th ed.). Boston, MA: Pearson. Watson, A., & Mason, J. (2002). Student-generated examples in the learning of mathematics. Canadian Journal of Science, Mathematics and Technology Education, 2(2), 237-249. http://mathforum.org/library/ed_topics/constructivism Read More