ࡱ> 7 XbjbjUU .j7|7|yTl 8  ,2DL:-//////$  SSh--P:,-8 BlY  -~0R  - Graduate Diploma in Secondary Education Mathematics Method HEG1606 Mathematics Review Student ID: 3002472 Student Name: Graham Taylor Date: 3rd August 2003 Constructivism in the Mathematics Classroom Theory of Constructivism The principle idea of constructivism is that the learners actively construct their own knowledge, rather than passively receive it. Constructivists argue that the term knowledge is problematic because it evokes a static, rather than dynamic image of learning, and prefer to talk about learning or knowing, interpreting and making sense of experiences (Malone J, Taylor P. preface). A popular conception of constructivism claims that learners can only construct meaningful understanding in relation to their prior knowledge. Johnson-Laird said: Human beingsdo not apprehend the world directly; they possess only internal representation of it, because perception is the construction of a model of the world. They are unable to compare this perceptual representation directly with the world it is their world (cited Orton A, Wain G. p41). This mapping that we have of how the world works is used in our construction of further knowledge. Constructivists value enquiry and discovery instead of theorems and proof, consequently teaching and learning shifts focus from teacher delivery of mathematical knowns to learner investigation of mathematical unknowns (Malone J, Taylor P. preface). Constructivist Pedagogy Von Glasersfeld claims (cited Mathematics of Primary Importance. p329) that what we come to know is created from within ourselves: we do not exchange ideas or knowledge. Most people would argue against this by saying that much knowledge has been transmitted to me from other people or books. The point that von Glasersfeld makes is that we may in fact hear or read the representation of an idea, however, the construction of knowledge comes from our own interpretation of the idea. A transmission view of teaching carries particular pedagogical messages and inadequacies. A classroom in which the teacher passes the necessary knowledge and skills to the students is a classroom that promotes dependency. This in turn leads to inactivity or constrained activity, with the students doing only what the teacher wants in the manner dictated. Students in this kind of classroom are not necessarily expected to have views on what they are learning. Such learners often have an image of mathematics as set of mystical algorithms and rules that have no relation to themselves or the world they live in (Malone J, Taylor P. p9-10). When a real world problem does arise students will struggle to find the correct algorithm to apply to the problem. They find mathematics uninteresting and may only take pleasure in their ability to please the teacher by reproducing meaningless (to them) algorithms correctly. Evolving teacher practices incorporate real world experiences, immersions in activity, language based pedagogies, use of concrete materials, active child centred learning, and co-operative group work to encourage students to express and develop understandings. Although these practices generally fit within the general ethos of discovery learning, the sole inclusion of these practices does not automatically give rise to constructivism. Students do not always see the connection between an apparatus and the mathematics. Real world experiences are not always relevant or interesting to every child in the classroom. To facilitate constructivism the teacher has the responsibility to create problems that can challenge mathematical interpretations and capture the students imagination. Over time the sophistication of the students methods of problem solving increases as they use them (Mousley J cited Mathematics of Primary Importance. p328). Teachers become mediators of students encounters with their social and physical worlds and foster intellectual, social and moral growth. Constructivist pedagogy requires the receptivity and responsiveness characteristics of an ethic of care (Davis R, Maher C. p49). Constructivism and the Mathematical Curriculum Implications on the curriculum assuming a constructivist approach to learning demand a move away from lists of content and lessons defined by particular assessments towards a view of learning as an interaction between existing knowledge, beliefs and skills, experiences, challenges and opportunities for resolution. The resolution that is a new state of knowledge, beliefs and skills is the result of learning. Tony Knight argues that problem-based skills be merged into the curriculum. A strategy can then be adopted to involve the students in the solving of these problems. In the classroom setting, students will learn to solve important social and personal problems. This is developed through daily practice with responsible and effective decision-making, and the deliberate construction of knowledge. Here, education is understood as the construction of meaning, rather than merely the transmission of knowledge. Proposed is that the curriculum is organized to encourage all students to become effective problem solvers (Knight T. p1). Inclusion of problem solving in the mathematical curriculum can be recognised as a partial inclusion of constructivism. However, care must be taken to ensure the delivery and resolution of problems is done in a constructivist manner. Lessons should not be conducted using a fixed set of examples from a particular chapter in the textbook but should evolve with the students understanding. The teacher is required to actively engage the students through discourse and pursue the students questions and enquiries (Malone J, Taylor P. p10). Students are encouraged to work collaborative, to listen, to reflect, to challenge, to negotiate and re-negotiate meaning. It can be argued that this type of teaching will increase the amount of time it takes to get through the curriculum. Why can we not just teach the algorithm !. It can be shown that students understanding and application of formal paper-and-pencil methods are often poor. Studying grade 3 children with multi-digit computation, Carraher, Carraher and Schliemann found that children were very successful (80-90% correct) when they solved problems mentally in real contexts but this fell to only 20-30% when they used school taught algorithms (cited Exploring All Angles. p269). In the long run time is saved from the necessity to re-cover supposably known concepts. Students will perform better if they are able to create their own algorithms for solving problems that they have given thoughtful interpretation to. Other concerns are raised about constructivism in the mathematics classroom and in general. The curriculum bounds the learning of students to outcomes and assessment. To allow constructivism to occur freely it is necessary to give students the opportunities to explore outside of these boundaries. Where then do the boundaries lay in discovery learning ? Should students be given free rein to explore in different directions that may be outside the topic area ? How is this type of learning viewed outside of the school. Will parents complain that the teaching never tells my child anything ? Are teachers hamstrung by bureaucracy ? These concerns need to be addressed in the curriculum to allow teachers the freedom to implement constructivist pedagogy without restriction. Possibly we should consider having middle year schooling structured closer to what we find in primary schools. In this environment classes can be free of time constraints. Class activities can be based more on the discussion and solving of problems related to real experiences. Resolution of problems can integrate various key learning areas. This will give students motivation and purpose to their schoolwork and prepare them for interaction within society. Affects of Constructivism on Assessment Current assessment of required outcomes generated by transmission does not sit easily alongside knowledge learnt using constructivism. Most final assessment is carried out using tests that do not allow for constructivist thinking. Questions on these tests will be discrete, the answer is right or wrong, and require the student to reproduce taught algorithms. Students who cannot regurgitate this information are judged to have low mathematical ability. To assess mathematical thinking there must be changes to the way we test students knowledge. Questions should be constructed to allow students to display their understanding and communication of mathematical concepts, to draw conclusions and to find convincing arguments to problems. Questions may not always have a simple right or wrong answer. Assessment of mathematical knowledge should be extended to problem solving, concepts and history. Individual projects and investigation work can be used as other types of assessment. Assessment should also be performed continuously in the classroom to support constructivism. Teachers must be able to discovery where a students knowledge is at to devise strategies to improve the students knowledge. However, continuous assessment is nothing new in the classroom and is not the sole domain of constructivism. Formal assessment in VCE years is used as a ranking system to allow universities to select students to progress into further education. Formal assessment is rarely used in the selection process for students leaving school to enter the workforce. What, then, is the purpose of formal assessment prior to VCE ? We could argue that the school has the responsibility to provide parents and students with a discrete formal assessment of the students achievements. We could also argue that discrete formal assessment may also give students motivation to set goals for higher achievement. The opposite may also be true. With this type of assessment children already begin to get categorized into low and high achieving groups. This can be very discouraging for a young person. Why do we need to put this burden on these children when they are at such an early and impressionable stage of life ? I would argue that broadening the assessment criteria will allow students to fully display their mathematical knowledge. The removal of discrete formal assessment will release the constraints imposed on teachers allowing teachers to implement constructivist pedagogy. Constructivism a text book review This is a critical review of a mathematical textbook in relation to a constructivist pedagogy. I have chosen the textbook Profile Mathematics, second edition for CSF II by Merrigan, Haralambous, Hyland, McLynskey. The year 7 textbook states that it is designed to cover the Curriculum and Standards Framework (CSF) and to identify and individual students progress through the CSF levels, even in the context of mixed-ability groups (preface). The textbook contains chapters that are related to CSF II indicators and learning outcomes. Each chapter is divided into five sections called core materials, chapter tests, projects and problem solving, review, and extension. The textbook claims that taught concepts are introduced using clear and concise explanations and a range of teaching strategies. Immediately I was concerned with the use of the word explanations when describing methods to teach new mathematical concepts. The problem solving section uses the concepts covered in the chapter to apply their knowledge in a variety of ways (preface). I found that topics (chapters) where introduced without an opportunity for the student to investigate or develop a sense of meaning for the material to be covered. The following introductions where given for the chapters on fractions and percentages. Fractions are used everywhere, so you need to know how to use fractions properly (p133), and, Per cent means something out of one hundred. The words per cent come from the Latin words per centum(p243). I found the introductions to topics to be inadequate. The introduction should be the springboard from where the students interests are generated to allow the student to undertake the work with some meaning and enthusiasm. The introduction may include some history, an anecdote, an interesting application of the topic, and real world experiences. The textbook could provide links to reference materials that provide the teacher with resources to introduce a topic. The students may spend a lesson discovering what the topic is about by performing research, solving problems, worksheets, tasks, discussion or reading. Introduction of new concepts where all given orally. A percentage can be written as a fraction. This fraction should be written in simplest form. Since per cent means per hundred, divide each percentage by 100 to make a fraction. 11%=11/100 (11 per hundred). This is transmission of algorithms without discovery learning that will not allow students to construct their own meaning. In this particular example an alternative to introducing fractions could be to give children a problem relating to the determination of fractions. An example would be to ask students how much percent they would have been given for a test if they had 70 questions right out of 100. What percent would they have been given if they had 7 questions right out of 10. What percent would they have been given if they had 14 questions right out of 20. Can they see a pattern ? Can they create an algorithm to calculate percent ? This type of problem is real to the students because it relates to something that they are faced with in school all the time. It will also allow them to construct their own understanding of percentages and devise an algorithm to compute percentages. The problem solving sections in the chapters where quite good as it allowed for students to work on real world problems. The chapter on percentage had a problem based on the calculation of percentages for the AFL ladder (p254). The algebra chapter devised a problem which required the students to use algebra to calculate posts required for a farmers fence (p118). These sections of the textbook where quite good as it generally related to real world problems where students could apply their mathematical knowledge to find a solution. The other sections of the textbook gave students the opportunity to practice the skills they acquired. The level of difficulty in the questions gradually got harder further into a topic which meant that high achieving students could remained challenged. I found the textbook needed to incorporate more history and meaning into their introductions of topics, incorporate more problem solving type activities and replace transmission type explanations of concepts with discovery type learning tasks. Constructivism a teachers resource Using humour in the classroom is a tool for learning that is often overlooked by teachers. Selected humour is healthy, friendly and attention grabbing. It also works to improve your relationship with the students. The curriculum urges teachers to introduce instances of mathematical ideas studied for interest or aesthetic reasons rather then for immediate utilitarian reasons (cited Mathematics Shaping the Future. p218). You may not always get a laugh, however humour is an ideal vehicle for constructivist thinking. Mathematicians trained in logic can fully appreciate the humour of jokes that rely upon illogic (Lannen B. cited Exploring All Angles. p252). Many jokes have a punch line. Students dont always see the punch line and this is where the joke begins to challenge their mathematical understandings. They will investigate the relationship between the joke and their concepts to find where the anomalies lie. They reconstruct their knowledge of mathematics and are rewarded by finally recognising the punch line. I personally find cartoons an excellent resource for constructivist learning. Cartoons provide a springboard for posing investigation and discussion of misconceptions. Cartoons are simply stimulating and fun (White A. cited Mathematics Shaping the Future. p218). Allan White creates a Dry Rot cartoon series which creates some excellent material for discussion. His cartoons can be found in the web site maintained by the Cambridge University in the UK (HREF1), and in the Square One journal produced by the Mathematics Association of NSW.  A suggested activity for the students would be to display a cartoon and then ask the students to write down the answer to the following questions; Why do you think this cartoon is funny ? How would you explain the cartoon to a friend or a younger brother or sister ? What mathematics destruction has happened ? (White A. cited Mathematics Shaping the Future. p220). A cartoon may also be used as an introduction to a topic.  A cartoon can be used as a connection to life outside of the classroom.  Other suggested activities using humour may be to ask the students to search for mathematical jokes and cartoons on the Internet. The results can be collated then ranked by the whole class. This activity provides cross curriculum tasks, investigation work and also an opportunity to assess students mathematical understanding. An anecdote will also require the students to challenge their mathematical knowledge and construct new meaning. Here is an example of an anecdote of the fictional discovery of a mathematical theory by Liethagoras (cited Mathematics Shaping the Future. P224). Liethagoras' Theorem (say Lie-thag-or-rus) Get it? Liethagoras was a long distant cousin of Pythagoras. He was very jealous of Pythagoras, because his theorem had given number triples that gave right triangles such as 3,4,5 or 5,12,13 or 7, 24, 25. So he thought, 'If only I could come up with a theorem that was simpler than Pythagoras, then I too would be famous'. So he thought and he thought until he had a wonderful thought and he rushed off to tell the mathematicians. His theorem was: In any right triangle the square of the smallest side equals the sum of the other sides. For example. In the 3, 4, 5 Triangle 3 x 3 = 4 + 5 In the 5, 12, 13 triangle 5 x 5 = 12 + 13 Does it work for the 7, 24, 25 triangle? Instead of becoming famous, the mathematicians just laughed at him. Why did they laugh at him? Let's investigate: 1. Can you find a number triple that obeys Liethagoras' theorem but doesn't obey Pythagoras' theorem and so doesn't give a right triangle? 2. Can you find a number triple that obeys Pythagoras' theorem but doesn't obey Liethagoras' theorem? 3. Liethagoras' theorem only works on Pythagorean triples that .........? The use of humour in the classroom is a resource well worth implementing to promote constructivist thinking. Students are often threatened when their mathematical misunderstandings are challenged by traditional methods. The difficulty seems to be that students invest some of their ego when solving problems. When the teacher explicitly exposes a students error, there is the opportunity for a negative effect upon the students confidence and attitude towards mathematics (White A. cited Mathematics Shaping the Future. p220). Students are less intimidated when they are required to be critical of work that is not their own or other students. Humour provides a vehicle for students to challenge their mathematical misunderstandings without the threat of damaging their confidence. References Malone J, Taylor P.C.S. (1993). Constructivist Interpretations of Teaching and Learning Mathematics. Perth, Curtin University of Technology. Davis R.B, Maher C.A. (1993). Schools Mathematics and the World of Reality. Massachusetts, Allyn and Bacon. Orton A, Wain G. (1994). Issues in Teaching Mathematics. London, Cassell. Gates P (2001). Issues in Mathematics Teaching. New York, RoutledgeFalmer. Mathematics Association of Victoria. (2000). Mathematics Shaping the Future. Melbourne, The Mathematical Association of Victoria. Mathematics Association of Victoria. (1998). Exploring all Angles. Melbourne, The Mathematical Association of Victoria. Mathematics Association of Victoria. (1996). Mathematics Making the Connections. Melbourne, The Mathematical Association of Victoria. Mathematics Association of Victoria. (1993). Mathematics of Primary Importance. Melbourne, The Mathematical Association of Victoria. Merrigan R, Haralambous H, Hyland M, McLynskey O. (2000). Profile Mathematics Second Edition for CSF II year 7. Melbourne, MacMillan Education Australia Pty Ltd. Backhouse J, Haggarty L, Pirie S, Stratton J. (1992). Improving the Learning of Mathematics. London, Cassell. Bobis J, Perry B, Mitchelmore M. (2001). Numeracy and Beyond. Sydney, Michael Mitchelmore. Knight T. (paper in progress 2003). No Student is an Island: Middle-School Curriculum. Melbourne, Victorian University. HREF1. Cambridge University Press. NRICH Archive.  HYPERLINK "http://www.nrich.maths.org.uk/mathsf/journalf/rb_archive.htm" http://www.nrich.maths.org.uk/mathsf/journalf/rb_archive.htm. Viewed 29-7-03. HREF2. What is Constructivism.  HYPERLINK "http://hagar.up.ac.za/catts/learner/lindavr/lindapg1.htm" http://hagar.up.ac.za/catts/learner/lindavr/lindapg1.htm. Viewed 24-7-03. HREF3. OTRNet. Mathematics texts, online maths investigations and resources for teaching and learning mathematics.  HYPERLINK "http://www.otrnet.com.au/" http://www.otrnet.com.au/. Viewed 24-7-03. HREF4. Drexel University. Math Forum Constructivism in Mathematics Education.  HYPERLINK "http://mathforum.org/mathed/constructivism.html" http://mathforum.org/mathed/constructivism.html. Viewed 24-7-03. 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