Mathematics Education and Constructivism - Assignment Example

Mathematics Education 136: Assessment 1 Marking Report Student name (to be completed by the student): Student ID (to be completed by the student): Submission meets specified submission requirements No Yes (All required aspects are included – purpose, overview, justification, examples, summary/conclusion and reference list; page/font/video/audio specifications are adhered to. If the assessment does not meet the requirements adequately, it will be returned unmarked for revisions and resubmission.) Presentation, spelling, grammar, APA referencing, and weak satisfactory excellent writing style _______________________________ (This component does not contribute to the mark on the assessment submission. However, if this component is weak the assessment will be returned for revisions and resubmission.) Score Criteria: Purpose of the report 1 Lacks focus or clarity of articulation and/or lacks relevance to the specified task. 2 Relevant and stated clearly and succinctly. Score (x 2) Criteria: Overview of key principles and ideas that guide your approach to mathematics curriculum and related teaching practices 1 Statements are vague and/or not clearly articulated with regard to ‘best practice’ in mathematics teaching and learning; Statements lack scope/breadth in that they are limited in number. 2 Statements are clearly and concisely articulated with regard to ‘best practice’ in mathematics teaching and learning; Statements lack scope/breadth in that they are limited in number. 3 Statements are clearly and concisely articulated with regard to ‘best practice’ in mathematics teaching and learning; Statements have scope/breadth with regard to curriculum and/or teaching/learning practices. 4 Statements are clearly and concisely articulated with regard to ‘best practice’ in mathematics teaching and learning; Statements have scope/breadth with regard to curriculum and/or teaching/learning practices; Statements are cohesive in that they present a holistic, consistent approach. 5 Statements are clearly and concisely articulated with regard to ‘best practice’ in mathematics teaching and learning; Statements have scope/breadth with regard to curriculum and/or teaching and learning practices; Statements are cohesive in that they present a holistic, consistent approach that also demonstrates insight into what constitutes ‘best practice’. (insight: demonstration of perceiving and understanding the true nature of something) Total Score (x 2) Criteria: Justifications for your approach and related practices for mathematics teaching 1 Justifications/explanations are limited and/or not clearly articulated; Use of references is included but needs further development with regard to appropriateness and/or number. 2 Justifications/explanations are clearly articulated but provide a partial argument only; Use of references is included but needs further development with regard to appropriateness and/or number. 3 Justifications/explanations are clearly articulated and are sufficient in number and nature to make a relevant case for the approach; Use of references is included in a comprehensive way. 4 Justifications/explanations present a comprehensive, cohesive/integrated and convincing case; Use of references is included in a comprehensive and well integrated way. 5 Justifications/explanations present a comprehensive, cohesive/integrated and convincing case; Use of references is included in a comprehensive and well integrated way; the overall justifications/case presented also demonstrate insight into what constitutes ‘best practice’. (insight: demonstration of perceiving and understanding the true nature of something) Total Score (x 2) Criteria: Specific examples of how you put your teaching approach into practice in mathematics lessons 1 Examples do not include space/geometry and measurement. 2 Examples are appropriate and include space/geometry and measurement, but are limited in scope with regard to what they demonstrate concerning ‘best practice’ in mathematics teaching and learning. 3 Examples are appropriate and include space/geometry and measurement, and do so in a comprehensive way but are not always clearly linked to key principles/ideas and/or justifications. 4 Examples are appropriate and include space/geometry and measurement, and do so in a comprehensive way clearly links them to key principles/ideas and/or justifications. 5 Examples are appropriate and include space/geometry and measurement, and do so in a comprehensive way clearly links them to key principles/ideas and/or justifications; the choice of examples demonstrates insight into what constitutes ‘best practice’. (insight: demonstration of perceiving and understanding the true nature of something) Total Score (x 2) Criteria: Summary/conclusion 1 Lacks focus or clarity of articulation and/or lacks relevance to the specified task. 2 Relevant and stated clearly and succinctly. 3 Relevant and stated clearly and succinctly; Provides a convincing statement for the target audience of parents and teachers. Score (x 2) Criteria: Overall demonstration of analysis, synthesis and evaluation of aspects of ‘best practice’ in mathematics teaching and learning 1 The report presentation is limited with regard to the degree to which it is valid, comprehensive and/or well-articulate; It is also possibly limited in inter-connectedness or the degree to which it intellectually engages the audience. 2 The report presentation overall is valid, comprehensive and well-articulated, but is limited in inter-connectedness or the degree to which it intellectually engages the audience. 3 The report presentation overall is valid, comprehensive and well-articulated, and is presented in an integrated way that presents a holistic approach and engages the audience. 4 The report presentation overall is valid, comprehensive and well-articulated, and is presented in an integrated way that presents a holistic approach; The report shows a sense of ownership of ideas (i.e., rather than borrowed ideas) by demonstration of innovation in the use of examples and/or the presentation approach (e.g., the presentation approach actively engages the audience intellectually). 5 The report presentation shows a sense of ownership of ideas (i.e., rather than borrowed ideas) by demonstration of innovation in the use of examples and/or the presentation approach (e.g., the presentation approach actively engages the audience intellectually); intellectual initiative is displayed through analysis, synthesis and evaluation of ideas that also shows potential leadership as a mathematics educator. Total your work was not well written. You needed to clearly use headings such as those on your assessment rubric. Your paragraph structure was poor and needs to be addressed. There are many key principles that needed to be developed and supported in this report and you do not address these. Your end of text referencing was poor and you lacked in text reference support. Mark 14 /50 Circle if needed: Re-submit / Fail Communication Session Report: Mathematics Year 4 Purpose of report The approaches used focus on what has been described by Booker et al (2009) as a constructivism. By this, the authors meant that a more practical approach has even greater impact towards learning of mathematics. It engages the students in real-life scenarios of solving problems. Solving math problems using such real life experiences has proved to have a greater impact on the student. Additionally, it is more interesting and engaging than the traditional approach. It is less cumbersome and more entertaining, thus capturing a child’s imagination. This way, the child tends to understand more and also learns to apply various approaches to real-life situations. This report seeks to explain the learning theory (constructivism) and how it improves children’s capabilities not only towards learning maths but also towards learning in general using a number of activities in which the children were involved. Overview of key principles and ideas that guide various approaches to mathematics Mathematics over the years has been described as a discipline that is difficult (Bell, Bellis & Bond, 2002). This has in more than one way interfered with the teaching of the subject. There is need to change this view of math. Considering that we are living in very different times, globalisation has influenced every aspect of our society today, from the way we trade to the way we communicate with each other (Booker et al, 2009). Mathematics is an exact science, with facts and no arguments over them. It involves great explorations; and new designs and inventions are successfully realised through mathematics. Hence, it forms the basis of numerous sciences. In spite of the importance of the subject, not everyone has an aptitude for mathematics. Some can only work through the more general principles of mathematics while others have keen interest and can develop theories in advanced mathematics. At school, children meet the challenges of mathematics at a very tender age, and they often progress according to their aptitudes towards the subject (Wadhwa, 2004). Regardless of whether one is at the advanced or general level, mathematics has an elemental importance. It develops logical thought and helps people to reason things out. This applies even to everyday activities as some areas of mathematics deal with the known factors, and through the logical processes, the student arrives successfully at the correct solution. Other areas of mathematics make larger demands on logical processes as the factors that are involved are unknown and must be adduced (Wadhwa, 2004). In the past, learners went through mathematics studies at a slower pace, but today with technological innovation, mathematics has greater demands on the students, and since not all students are good in the subject, streaming has become necessary. Streaming helps to give every learner a chance to do well in particular areas he or she is not comfortable with (Zevenbergen, Shelley & Robert, 2004). There is therefore a need to come up with activities that make the learning of mathematics easy and interesting. Specific examples putting teaching approach into practice in mathematics lessons A series of activities are being performed by students in year four to evaluate the application and the eventual impact of these contemporary approaches to teaching. The activities are aimed ensuring learners build on their own experiences by practising counting, geometry, graphing, patterning, measurement, and beginning computation as discussed by Moore (1998). Activity I: The first activity was in geometry. Here the students were placed in several groups at random and facilitated with geoboards and a series of questions that they were supposed to solve. The geoboards were used to construct and identify the specific shapes as dictated in the question and also create a 3 dimensional figure that can help the students discuss and identify with the issues at hand; they will then construct three different shapes. The teacher ensured that the materials used were as unique as possible to make the students innovative. According to Hech (2008), teacher effectiveness is one of the means through which students can improve their performance in mathematics. Thus after identifying the shapes, the students were as well requested to come up with formulas for reaching several conclusions for example the circumference and the area per measurement square. Activity II: The second activity involved a different approach but still retained the aspects of the first activity. The students were also furnished with geoboards, but this time they were not in groups and worked as individuals. Students were required come up with more shapes other than the ones they created when in groups. The similarity in shapes did not matter. In this activity the students were asked to calculate once more the circumference where applicable, the area per measurement square and the total surface area. Here the teacher went around and assisted any student to a specific limit to solve any of the issues as need arose. Activities yet to be done Activity III: In activity three, an element of technology will be introduced. Students in groups of two will be furnished with a computer. The computer has been programmed with several mathematical soft wares that will engage the students in solving several geometric problems. The students using the software will also be made to construct several shapes in both two and three dimensions. After coming up with the shapes they will be asked to combine the shapes to make other complex geometrical shapes as their capabilities can let them. The shapes can take any form that mathematical principles can allow. The students will then make a presentation and in it they will be asked to calculate several aspects like the area in three dimensional and in two dimensional and the surface area. Activity IV: In the next two activities, the students will interact both at the individual level and the group level. Activity four will engage the students in groups. They will also be furnished with several toys of different lengths and sizes. The students will therefore be made to measure and tabulate the lengths of the different toys and in the end come up with a comparison of which toy or item is larger and which item is smaller compared to other items in the same group of items. The various groups will subsequently present their findings briefly to the rest of the class. Activity V: Each student will work alone. They will be once more furnished with a computer. This time they will participate in a temperature measuring game, where they will be given several facts about temperature. For example, they will measure their body temperatures in degrees Celsius and convert them to Fahrenheit. Finally, they will be asked to come up with a conclusion of how many degrees Celsius make up a Fahrenheit and so on, but only if they are able to complete the main exercise. The teacher will also move around to assist any student who needs help. Observations Several observations were made during the whole exercise. It should be noted that the exercise though similar in various aspects, involved the students participating with each other and also alone. This was meant to analyse the level of understanding within the two situations and secondly gauge the interaction of the students within a group. The purpose of education is to mould a student into a whole; both in academics and in social communication (Clark, 2005). In the first activity, the students successfully constructed several mathematical shapes in the two possible dimensions. Each in groups was also seen to participate in the exercise as it was needed. When it came to coming up with the formulas, the level of engagement increased as each keenly followed the proceedings and raised up issues when one student made a mistake, for example forgetting to include a measurement or made a wrong calculation. Also notable is that the students created several shapes and then in consensus came up with the three they felt they were capable to handle. This was noted in each of the groups and highlights the significance of working in groups as noted by Ioannou-Georgiou and Pavlou (2003). The authors note that children usually enjoy working in groups, and evaluating them as a group feels much safer than individual exposure. To evaluate this, the teacher timed them to evaluate their working speed individually and in groups. In the second exercise the students were left to work alone but in essence they were to use the same principles used in the first exercise. They therefore came up with different shapes and also reached several conclusions. The principles used were quite the same. The students reached conclusions successfully though there was a few who needed assistance but most insisted in doing the exercise alone. Of note is the pace that each used to conduct the exercise. The students were quite fast and each completed the assigned work. The answer varied as the shapes did but most of the answers submitted were correct. In the third activity, the students will work in groups of two that will be totally random. They will construct various shapes and figures. They will also use a computer to do simple calculations and arrange figures using simulations by a computer programme. The teacher will encourage the students to interact in order to develop and share computer skills. Many research works on child development suggest that this is one way ensuring that children develop the skills required for future assignments. For instance, Barbeau and Taylor (2008) argue that technology can enhance the mathematical development of young children by offering them attractive challenging environments. The use of the computer will therefore be meant to children’s logical thinking. Activity four will be a different concept altogether. In groups, the students will measure and compare the various toys and items they have, compare their lengths and come up with a variety of conclusions. Measurement of materials is mentioned by Burke (2010) as one of the methods that enhance children’s metacognitive development. The students will fully participate and compare their various solutions to problems and eventually agree on the right ones. Activity five will also involve measurements but the students will work as individuals. They will measure temperature in degrees Celsius and in Fahrenheit. They will be given several facts to derive various conclusions as needed. Of note in this case each student will be asked to tell how many degrees Celsius make up a Fahrenheit, and vice versa. During the session, the students will seek answers to any questions they have. . Achievements The exercise is among other things meant to improve the student’s interaction and communication skills. In this case understanding the concepts is also very important. One achievement is that the level of understanding is expected to be generally high because of the mix of activities. The application of individual and group work is expected to help the students improve their metacognitive capacity and hence their performance in mathematics. The mix of activities will benefit the students in a multiplicity of ways. For instance, they will have better experiences in counting, geometry, graphing, patterning, measurement, and beginning computation (More, 1998). The students will also be able to tackle very complex issues and understand them using interaction with each other and the use of real-life equipment and gadgets that they have at home. Critical Reflection A notable point will the use of the computers when the students are given computer-related exercises. Here, a significant drop in concentration is expected as compared to the use of other gadgets like geoboards. This can be attributed to the fact that the students don’t know how to properly make use of the computers to learn. To them, computers are used for playing games or simply accessing irrelevant information online. This can be attributed to their computer habits at home. Rarely do they use these opportunities to learn as expected and this could set precedence of poor use of the online resource in the future. Nevertheless, the use of computers will be applied progressively in recognition of the fact that modern mathematics involves the use of calculators and computers. Another issue that has been given attention is interaction. Positive interaction was noted in activity one and two and is also expected in the rest of the activities. Unfortunately this was observed among students who are used to each other. A level of conflict could be noted if the students were grouped with unfamiliar students as each tried to outdo the other in explaining the concepts. This is a matter that should be tackled during the remaining activities. Another issue is the nature of some of the problems to be solved. The problems involved are of varying complexity and this is expected to give the students different challenges as they slowly progress to tackle more difficult problems. Conclusion: In summary, the constructivist approach proves to be a viable way to teach mathematics and at the same time improve the student’s social skills. The activities are custom-made to ensure that the students enjoy learning maths through interaction with several elements in their environment including their friends and fellow students. The approach is therefore fundamental in ensuring that the student is able to use mathematical concepts in everyday life and that is probably why most parents have noticed the students playing more with their toys at home and interacting with their fellow students. This approach is a way of ensuring that the level of participation is improved both in and outside the classroom. Maths is a continuous process and therefore the approach ensures that it is a part of the student’s life. References Barbeau, Taylor, P. J. (2008). Challenging Mathematics In and Beyond the Classroom. New York: Springer. Bell, J. C., Bellis, T. J. & Bond, J. (2002). When the Brain Can't Hear: Unravelling the Mystery of Auditory Processing Disorder. New York: Simon and Schuster. Booker, G. (1997). Teaching Primary Mathematics (2nd Edition). NSW: Pearson Education Australia. Bottle, G. (2005). Teaching Mathematics in the Primary School. London: Continuum International Publishing Group. Clark, M.M. (2005).Understanding Research in Early Education: The Relevance for the Future of Lessons from the Past (2nd edition). New York: Taylor & Francis. Heck, R. H. (2008). Teacher effectiveness and student achievement: investigating a multilevel cross-classified model. Journal of Educational Administration. 47(2): 227-249. Ioannou-Georgiou, S & Pavlou, P. (2003). Assessing Young Learners. Oxford: Oxford University Press. Moore, J. E. (1998). Real Math for Young Learners. London: Evan-Moor Educational Publishing. Wadhwa, S. (2004). Modern Methods of Teaching Mathematics. London: Sarup & Sons. Zevenbergen, R., Shelley, D. & Robert, W. (2004). Teaching Mathematics in Primary Schools. NSW: Allen & Unwin. Read More