Philosophy of Mathematics Learning and Teaching - Essay Example

Running head: MATHEMATICS PORTFOLIO Mathematics teaching portfolio Name Course Instructor Date MATHEMATICS PORTFOLIO Philosophy of Mathematics Learning and Teaching Over time, I have developed a philosophy on Mathematics learning and teaching, it encompasses my beliefs regarding mathematics and why it is important as well as my believes on what the best approach to teaching the subject is. The different aspects of this philosophy have been supported by different mathematics education scholars. However, the philosophy identified is not borrowed from any particular scholar. It is my own, developed over time, and validated by my own experience with students. As a maths teacher I acknowledge the importance of technology in simplifying tasks and enhancing efficiency. I strongly believe that children should be taught how to use various technological platforms while they are still young. In order to realise this, it is important to incorporate technology in different learning activities. Children should be provided with equal opportunities and ample time to incorporate technologies such as computers in learning different concepts in mathematics (Betts & MacNaughton, 2003). My beliefs regarding teaching of mathematics include the fact that: Mathematics is one of the most crucial areas of study that plays an integral role in society It is an essential area of study that provides tools for understanding other subjects and provides a firm foundation for students’ ability to handle issues in daily life (Bruss, 2012). Maths also helps students reason logically and connect ideas. As Plato said, arithmetic and geometry invite thought and lead the mind to reflect and hence enabling the mind to reach the truth. It is therefore essential for students to begin learning mathematics early in MATHEMATICS PORTFOLIO life so that they can acquire basic knowledge and skills that will help them solve real life problems (Hardy, 1941). When it comes to learning and teaching mathematics, student-centred and interactive approaches should be used in order to effectively impart knowledge and skills. (Howard & Perry, 2005). It is important to make mathematics be seen by my students for what I believe it to be, i.e. a pattern of Ideas. Learning which should be easy and enjoyable (Hardy, 1941). Values are an important part of the educational process in macro, meso and micro levels. They help one establish a sense of personal identity as a teacher. They also guide in helping students develop their own identities. Values are also a way of demonstrating a teacher’s pedagogical identity . (Chin, Leu and Lin, 2001). My values are as follows: Building strong personalities and cultivating positive attitudes in students (Howard & Perry, 2005). I endeavour to prepare students to become resourceful and useful members of society by transmitting useful knowledge, new ideas, concepts, positive values and ethics (Hardy, 1941). Through various class activities grounded on mathematical concepts, my priority is to reinforce positive behaviours such team work, collaboration, knowledge sharing and positive social interactions (Pirie & Kieren, 1994). I endeavour to provide students with learning experiences that will challenge them to tap into and capitalise on their creativity (Mason & Johnson, 2004). MATHEMATICS PORTFOLIO Intents My goal key as a mathematics teacher is to develop and nurture the problem solving and application skills of students. In essence, I want students to become critical thinkers (Hardy, 1941). To employ a variety of interactive learning approaches so as to enhance the awareness of students towards mathematical concepts and their application in real life situations (Bruss, 2012). To lay emphasis on the practical application of mathematical knowledge rather than memorization (Ah Sam & Ackland, 2006). To incorporate various physical and play based activities ; play-based activities will help enhance the learning of students since they capture the attention of students and help them construct meaning from their experiences (Marzano et al, 2001). To encourage participation of students in planning and implementation of curriculum prerequisites. This notion is based on the Reggio Emilia approach to early childhood learning which suggests that students should have a certain level of control over the content and direction of their learning (Caldwell, 1997). Diversity The worldviews and perspectives of indigenous learners are different from western worldviews and perspectives. In most cases, Western worldviews and perspective tend to dominate key aspects of learning. This in turn alienates indigenous learners from realising positive learning outcomes (Coleman-Dimon, 2000). MATHEMATICS PORTFOLIO When indigenous contexts are incorporated into learning mathematics, indigenous learners will be able to easily derive meaning from their knowledge base like other students. (Ellis, 2001). When teaching, teachers should drawn on to indigenous cultural contexts by using concrete examples and illustrations that indigenous learners can easily relate to. Teachers should also use interactive, active or practical approaches to teaching mathematics since most indigenous people develop their knowledge through experiencing and doing (Warren, Cooper & Baturo, 2004). The values and intents mentioned above are instrumental in dealing with different aspect of diversity in mathematics learning. To teach numeracy to indigenous students, active learning is a particularly effective approach, it caters for the students by giving them with an opportunity to learn through hands on experiences which convey the social meanings of mathematical ideas. It converts mathematical concept from being merely symbols on paper to useful instruments in real life situations (Bruss, 2012). The classroom will offer the students an opportunity to co-operate and coexist in their learning. I seek to ensure that the classroom fosters affection and respect and also encourage s personal decision making. Digital technology will be used to represent data to the students in a non linguistic form which is one of the strategies known to improve student performance (Marzano et al, 2001). Digital technology is also important because it takes advantage of visual manipulatives and applets to offer different ways in which the indigenous students develop their understanding of mathematics (Suh, Johnston & Douds, 2008). Strong relationship between teachers and students reduce direct questioning while promoting practical experience and group cooperation. Rephrasing of questions when necessary to ensure that indigenous students do not have a problem understanding the questions should they have problems with language. The model below shows elements of active learning and how they MATHEMATICS PORTFOLIO work together to help bring out positive outcomes in mathematics learning and teaching (Caldwell, 1997). Two week Mental Math’s Plan & Overview The key aim of this two-week mental mathematics plan is to develop student’s number sense and interest in mathematics. The learning activities incorporated in this plan aim at enabling students to learn how they can work with numbers mentally without using computers, calculators or paper and pen on numbers 1-100. This plan mainly targets Year 2 students and seeks to develop their basic mental math skills in addition, subtraction, multiplication and division. Furthermore, this plan aims as developing student’s problem solving and reasoning skills. In order for this plan to be effective, students should possess prior knowledge of how the number system works and how to carryout basic addition, subtraction, multiplication and division mathematical calculations. MATHEMATICS PORTFOLIO Week One Focus Objective Prior knowledge Activity Assessment Resources Grouping Diversity Monday Addition of single and multiple denominations involving ‘5’ Students should be able to carry out basic additions of single and multiple denomination involving ‘5’ Students should have an understanding on how to add single and multiple denominations Students will be divided into three groups, (Blue, Red and Green) The groups will compete against each other. A set of numbers will be dictated to the students. Students will be asked to add the numbers mentally. The first group to add the number correctly will be awarded 5 points. The accuracy of answers provided by each group will be assessed. Moreover, the participation of each student in their groups will be assessed. White board and ink pen for recording points of each group Students will be divided into three groups , each group will comprise of a balanced number of children with both low and high abilities. Both simplified and complex addition calculations will be dictated to the three groups . Students with low abilities will be given priority when it comes to answering simplified addition calculations. The fact that the activity will be undertaken in groups will give indigenous students a sense of cooperation which will enable them learn effectively. Dictation of the problems will also reduce the language barrier Tuesday Subtraction of numbers with multi Denominations ons involving Students should be able to carryout basic subtraction of multiple denomination involving ‘10’ Students should have an understanding on how to carryout basic subtraction of multiple denomination involving ’10’ Students will be divided into groups of two. Each group will be provided with a set of different math sums involving subtraction of multiple denomination involving ’10” One student will dictate the provided set of sum while the other student will carryout the sum mentally. The student dictating the sum will then performance of the other student. Students will subsequently interchange their roles. Students will be keenly observed while participating in their respective groups. The accuracy of answers provided by students in each group will be assessed in order to see how well the learners have understood. Paper and Pen Low and high ability students will be paired Subtraction sums for students with low abilities will be simplified to involve the use of two denomination numbers (e.g. 30-10) Subtraction sums for students with high abilities will be expanded to involve the use of three denomination numbers (e.g 212-10). Group work will enable students work in an environment familiar to the aboriginal students Wednesday Subtraction and addition of numbers with multiple denominations Students should be able to carryout basic additions of single and multiple denomination Students should have an understanding on how to carryout basic subtraction and addition of multiple denomination A power point presentation on an interactive white board will be showcased. The presentation will comprise of 100 picture icons. The accuracy of answers provided by each student will be assessed by the teacher who will mark the student’s books. computer Whiteboard Students will work independently . Students with low abilities will be given priority when it comes to answering simple subtraction and addition calculations. MATHEMATICS PORTFOLIO Week Two Monday Multiplication of single digit numbers Students should be able to memorise and reflect on multiplication of single numbers Students should have knowledge in multiplying sums involving single digits An image projector will be used to project a set of small boxes arranged in different number of rows and columns Students will be required to determine the number of boxes present in an entire set by only counting one row and column The accuracy of answers provided by each group will be recorded and assessed. Computer Interactive display device or a projector Students will work independently Simple box images will be projected for students with low abilities whereas more complex images will be projected for better students The projector will help indigenous students associate maths with real life situations Tuesday Division of numbers with multiple denominations with ‘5’ and “10” Students should be able to carryout basic subtraction of multiple denomination involving ‘10’ Students should have an understanding on how to carryout division using multiple digit number Students will be keenly observed while participating in their respective groups. The accuracy of answers provided by students in each group will be assessed. Paper and Pen Tray of marbles Students with Low and high abilities will be grouped separately Students with low abilities will work with less marbles while those with high abilities will work with a lot of marbles. The fact that they work in groups will be good for indigenous students Wednesday Multiplication and division of numbers with multiple denominations with 0”s at the end Students should be able easily to carryout sums involving multiplication and division of multiple digits with 0”s at the end Students should understand how the numbers systems works They should also have be able to carryout sums involving multiplication and division A brief interview (5minutes) will be conducted with each students. During the interview the teacher will key in 100 in a calculator. During the interview the teacher will add or remove a particular number of 0’s and subsequently ask the student to determine which number has been multiplied or divided from the number keyed in the calculator. The accuracy of answers provided by each student during the interview will be assessed Calculator Students will work independently Less 0’s will be keyed in during interviews with students with low abilities and vice versa Assessment Item: Diagnostic Assessment in Mathematics Diagnostic assessment plays an integral role in decision making when it comes to teaching mathematics. This is mainly because it provides information about student's existing skills, knowledge, interests, attitudes and learning styles (Warren, Cooper & Baturo, 2004). Using information obtained from a diagnostic assessment, teachers are able to effectively establish students’ level of knowledge. (Howard & Perry, 2005). Diagnostic assessment acts as a very important tool for enhancing student’s learning mental mathematics skills. For instance, prior to teaching or assessing mental math skills, teachers may want to establish student’s prior knowledge, in such a case, teachers may use diagnostic assessments methods such as tests and interviews to determine students’ prior knowledge. After which teachers can effectively design suitable mental mathematics tasks for the students (Marzano et al 2001). The use of technology in diagnostic assessment can help to enhance accuracy, speed and the effectiveness of the assessment process. Advanced technologies may enable teachers to MATHEMATICS PORTFOLIO administer embedded assessments in the course of the learning process, extract multi-faceted and ongoing information from learners and respond immediately in helpful ways. The technologies that may be used include automated scoring and machine-based reasoning techniques (Shute & Underwood, 2006). Cultural diversity, student’s learning style and level of ability are some of the key aspects of diversity that must be taken into account when using diagnostic assessment. Given that, western construction of knowledge is different from indigenous knowledge construction and students have different learning styles and abilities, all students cannot be assessed in the same way; each student should be assessed using diagnostic assessment methods and approaches that are most suitable for them. Approaches, methods or procedures to be used during the assessment process should be based on students’ cognitive style, cultural knowledge and prior knowledge. It is therefore crucial for teachers to foremost, understand the needs, knowledge constructions, cognitive abilities and cultures of students (Warren, Cooper & Baturo, 2004). Trial Item: Diagnostic Assessment The key aim of this trial exercise was to illustrate how diagnostic assessment can be carried out to determine students’ knowledge in addition and their mental math abilities. Through the use of different diagnostic assessment methods this exercise sought to establish student’s strengths, weaknesses, skills and knowledge in addition and mental math strategies Establishing these aspects will subsequently enable the teacher to make adjustments that will ensure that the individual needs of each student are met (Willis, 2000). MATHEMATICS PORTFOLIO This exercise involved two students in Year 2. They should understand how the number system work and how to carry out basic addition sums. It involved one 40 minutes session with each student and took place within the school settings of the two children. In order to assess the student’s knowledge in addition and mental math abilities, this exercise employed three diagnostic assessment methods namely; math tests, observation and interviews. Procedure/ Activity Students were given six mathematical sums to solve without using pen and paper or a calculator. Sample of sums: 9+ 8= 10+ 4= 20 + 15= 15 + 5= 12 +12= 7 + 8= Both of them got all the answers right with student A taking 8 minutes and student B taking 11 minutes, both said that they found the problems easy. References Ah Sam, M. & Ackland, C. (2006). The curriculum: a doorway to learning. In Phillips, J., & Lampert, J. [Eds.]. Introductory. Indigenous studies in education. Frenchs Forest: Pearson Education Australia Betts, P., & McNaughton, K. (2003). Adding an aesthetic image to mathematics education. International Journal for Mathematics Teaching and Learning. Retrieved December 27, 2012, from http://www.ex.ac.uk/cimt/ijmtl/ijmenu.htm. Bruss, F.T (2012). Image, Influence and Importance of Mathematics as Directives for Public Awareness. Raising Public Awareness of Mathematics. Behrends. E, Crato, N & Rodrigues, F. Eds. pp 297-311 ISBN: 978-3-642-25709-4 Caldwell, L. (1997). Bringing Reggio Emilia Home: An innovative approach to early childhood learning. New York Teachers College Press. Coleman-Dimon, H. (2000). “Relationships with the School: Listening to the Voices of a Remote Aboriginal Community”. The Australian Journal of Indigenous Education, 28(1), 34-43. Chin, C., Leu, Y.-C., & Lin, F.-L. (2001). Pedagogical values, mathematics teaching, and teacher education: Case studies of two experienced teachers. In F.-L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 247-269).Dordrecht, The Netherlands: Kluwer Academic Publishers.  Ellis, A. (2001). Teaching, Learning and Assessment Together: The Reflective Classroom. New York: Eye on Education. Hardy, G.H. (1941) A Mathematician’s Apology. Cambridge: Cambridge University Press. 2005 Edition available as pdf at www.math.ualberta.ca/~mss/books/A Mathematician’s Apology.pdf (accessed June 2005). Howard, P., & Perry, B. (2005). Learning mathematics: Perspectives of Australian Aboriginal children and their teachers. Proceedings from the 29th Annual Conference of the Mathematics Education Research Group of Australasia (Vol 3.pp 153-160) Melbourne, Vic: MERGA 29. Marzano, R.J, Jennifer, S.N, Diane E, Pickering, J & Gaddy, B (2001) A handbook for classroom instruction that works. Alexandria, VA: Association for supervision and Curriculum Development. Mason, J. and Johnston-Wilder, S. (2004) Fundamental Constructs in Mathematics Education. London: Routledge Falmer. Pirie, S and Kieren,T. (1994) ‘Growth in mathematical understanding: how can we Characterise it and how can we represent it?’, Educational Studies in Mathematics, 26 (2–3):165–90. Shute, V. J. & Underwood, J. S. (2006). Diagnostic assessment in mathematics problem solving. Technology, Instruction, Cognition, & Learning, 3(1), 151-166. Suh. M.J, Johnson, C.J & Douds, J (2008) Enhancing mathematics learning in a Technology – Rich environment. Teaching children mathematics. National Council of Teachers of mathematics. Warren, Elizabeth and Cooper, Thomas J. and Baturo, Annette R. (2004) Indigenous Students and Mathematics: Teachers' Perceptions of the role of Teacher Aides. The Australian Journal of Indigenous Education 33:pp. 37-46. Willis, S, (2000), ‘Strengthening numeracy: Reducing Risk’, In Improving numeracy learning Research Conference 2000: Proceedings, Melbourne: Australian Council for Educational Research, 2000.       Read More