My Philosophy of Education
My philosophy of teaching and learning has evolved over the past ten years since completing my undergraduate degree. As I interact more with students, observing their behaviors, learning styles, and individual differences, I have come to view education in general as much less fixed as previously. I used to have an absolute view of how students learn, and more so, how mathematics as a subject was universal and unchanging. However, I believe there are great social and cultural influences in society, the school, and classroom that greatly impact how I teach and how students learn. I recognize more and more each day how unique each student is in his or her background, history, interests, personality, and learning style. Furthermore, I now see that not only does education itself evolve, but mathematics is subject to social and cultural contexts as well. Because of this shift, my personal epistemological position has grown to incorporate more social aspects than ever before.
I used to believe that mathematics was a body of knowledge of universal truths that could be taught, learned, and passed on to future generations. However, thanks to much reading and research, particularly in the area of math history, I have seen that mathematic practices and procedures differ greatly across countries and cultures. As Skemp (1976) writes, “What constitutes mathematics is not the subject matter, but a particular kind of knowledge about it” (p. 15). Yes, there are specific standards and topics that should be covered as part of mathematics education, but more importantly, students should learn mathematics in order to develop critical thinking and problem solving skills. Mathematics is not a body of knowledge, but a way of thinking, organizing concepts, and investigating our world. In this way, my philosophy of mathematics education is firmly fallibilistic in that it is the result of social processes and is always open to revision (Ernest 2012). “The fallibilist view does not reject the role of logic and structure in mathematics, just the notion that there is a unique, fixed and permanently enduring hierarchical structure” (Ernest, 2012, para. 35).
Due to this fallibilistic view of mathematics, I want students to also view learning mathematics as an interactive process, understanding that there is no one way to go about solving a particular problem. From a teaching standpoint, this requires listening well to all proposed ideas and asking probing questions to further and/or redirect student thinking. Learning mathematics must include the active doing of mathematics. Students should be engaged in reflection on their own cognitive processes, expressing them verbally in group collaboration or class discussion, or in written form.
My philosophy of mathematics education is best supported by the social constructivism theory of learning. While social constructivism can refer to a wide variety of positions, according to Ernest (1994), all social constructivism positions maintain the belief that the social domain significantly influences the individual’s development in a formative way. Furthermore, each individual constantly is “constructing (or appropriating) her meanings in response to experiences in social contexts” (Ernest, 1994, p. 306). Interactions in the classroom, both student to teacher and student to student heavily influence the individual’s cognitive development of mathematics. From this constructivist viewpoint, building new cognitive pathways always requires reflection (von Glasersfeld, 2001), and collaboration provides an excellent means of collective reflection. Ernest (2011) summarizes my personal epistemological beliefs well in that “social constructivism offers the possibility of a philosophy of mathematics which accounts for the objectivity and utility of mathematics, as well as its fallibility and culture-boundedness” (para. 19).
Because mathematics is simultaneously objective and constantly changing, the aim of math education, and all schooling, is not to learn what to think, but rather how to think for oneself (von Glasersfeld 2001). I believe that with appropriate prompting, scaffolding and guidance from the teacher, every student is capable of learning to think critically, problem solve, analyze and create. Students will not be able to attain the same levels of abstraction at the same pace, but all are able to advance their thinking processes. As a teacher, I believe it is my responsibility to provide differentiated opportunities for students to engage in mathematics in a variety of ways. I strive to spark their interest and engage them in order to build upon and challenge prior knowledge in a way that will lead to new schemas for thinking and learning (Pirie & Kieren, 1994).
I believe the optimal learning environment for students is first and foremost, a place that is safe and caring so that the individual feels comfortable asking questions, taking risks in their thinking, and learning through missteps, as safe classroom environments have been shown to impact not only what, but how much students learn (Holley and Steiner, 2005). Students must be given opportunities to participate and engage with the subject matter through student-centered activities. I believe that students possess a variety of learning styles and types of intelligences, so teaching and learning should be varied to accommodate these differences. According to Pirie and Martin (2000), mathematical understanding is a recursive and interactive process, so collaboration and group work is an essential piece of an optimal learning environment because it allows students to reflect together as they work through problem-solving steps.
My philosophy of teaching and learning greatly impacts my teaching strategies. I believe it is my role as a teacher to provide this optimal learning environment for all students so that they can become independent thinkers. “Teaching does not begin with the presentation of sacred truths, but with creating opportunities to trigger the students’ own thinking” (von Glasersfeld, 2001, p. 10). I do not want my students merely to have instrumental understanding of mathematical processes. Instead, I want them to have relational understanding so that they are able to move deeper in their critical thinking and problem solving skills (Skemp 1976). Von Glasersfled (2001) refers to this as conceptual learning because students are actively conceiving and building new cognitive pathways. He continues, “There is no infallible method of teaching conceptual thinking. But one of the most successful consists in presenting students with situations in which their habitual thinking fails” (von Glasersfed, 2001, p. 9). I agree that this is a great way to foster relational understanding: to present students with a situation in which their previous knowledge fails and they are forced to collaboratively access prior knowledge and apply it in new ways. In such a student-centered, collaborative environment, my role as a teacher then becomes one of intervention: leading, shepherding, reinforcing, clue-giving, modeling, and praising in order to move students to greater understanding (Towers, Martin, and Pirie, 2000).
I believe that technology can be a powerful teaching strategy if it is used to promote new ways of engaging with mathematics material (Cheung & Slavin, 2013). I strive to utilize technology in the classroom not because it is the latest trend in education, but because it can greatly advance conceptual learning as it allows for experiences and simulations not possible in the traditional classroom setting.
Finally, as my philosophy of teaching and learning, I consider the acquisition of content knowledge as far less important than the process of learning how to learn. Students are in school and in my classroom to learn essential life skills beyond equations and formulas. They are here to learn how to creatively approach a problem, to analyze potential solutions, to engage in critical thinking and reflection, to develop an understanding of how they like to learn, and to learn the value of hard work. It is my desire to foster independence in each student so that they can learn not simply what to think, but how to think for themselves.
I used to believe that mathematics was a body of knowledge of universal truths that could be taught, learned, and passed on to future generations. However, thanks to much reading and research, particularly in the area of math history, I have seen that mathematic practices and procedures differ greatly across countries and cultures. As Skemp (1976) writes, “What constitutes mathematics is not the subject matter, but a particular kind of knowledge about it” (p. 15). Yes, there are specific standards and topics that should be covered as part of mathematics education, but more importantly, students should learn mathematics in order to develop critical thinking and problem solving skills. Mathematics is not a body of knowledge, but a way of thinking, organizing concepts, and investigating our world. In this way, my philosophy of mathematics education is firmly fallibilistic in that it is the result of social processes and is always open to revision (Ernest 2012). “The fallibilist view does not reject the role of logic and structure in mathematics, just the notion that there is a unique, fixed and permanently enduring hierarchical structure” (Ernest, 2012, para. 35).
Due to this fallibilistic view of mathematics, I want students to also view learning mathematics as an interactive process, understanding that there is no one way to go about solving a particular problem. From a teaching standpoint, this requires listening well to all proposed ideas and asking probing questions to further and/or redirect student thinking. Learning mathematics must include the active doing of mathematics. Students should be engaged in reflection on their own cognitive processes, expressing them verbally in group collaboration or class discussion, or in written form.
My philosophy of mathematics education is best supported by the social constructivism theory of learning. While social constructivism can refer to a wide variety of positions, according to Ernest (1994), all social constructivism positions maintain the belief that the social domain significantly influences the individual’s development in a formative way. Furthermore, each individual constantly is “constructing (or appropriating) her meanings in response to experiences in social contexts” (Ernest, 1994, p. 306). Interactions in the classroom, both student to teacher and student to student heavily influence the individual’s cognitive development of mathematics. From this constructivist viewpoint, building new cognitive pathways always requires reflection (von Glasersfeld, 2001), and collaboration provides an excellent means of collective reflection. Ernest (2011) summarizes my personal epistemological beliefs well in that “social constructivism offers the possibility of a philosophy of mathematics which accounts for the objectivity and utility of mathematics, as well as its fallibility and culture-boundedness” (para. 19).
Because mathematics is simultaneously objective and constantly changing, the aim of math education, and all schooling, is not to learn what to think, but rather how to think for oneself (von Glasersfeld 2001). I believe that with appropriate prompting, scaffolding and guidance from the teacher, every student is capable of learning to think critically, problem solve, analyze and create. Students will not be able to attain the same levels of abstraction at the same pace, but all are able to advance their thinking processes. As a teacher, I believe it is my responsibility to provide differentiated opportunities for students to engage in mathematics in a variety of ways. I strive to spark their interest and engage them in order to build upon and challenge prior knowledge in a way that will lead to new schemas for thinking and learning (Pirie & Kieren, 1994).
I believe the optimal learning environment for students is first and foremost, a place that is safe and caring so that the individual feels comfortable asking questions, taking risks in their thinking, and learning through missteps, as safe classroom environments have been shown to impact not only what, but how much students learn (Holley and Steiner, 2005). Students must be given opportunities to participate and engage with the subject matter through student-centered activities. I believe that students possess a variety of learning styles and types of intelligences, so teaching and learning should be varied to accommodate these differences. According to Pirie and Martin (2000), mathematical understanding is a recursive and interactive process, so collaboration and group work is an essential piece of an optimal learning environment because it allows students to reflect together as they work through problem-solving steps.
My philosophy of teaching and learning greatly impacts my teaching strategies. I believe it is my role as a teacher to provide this optimal learning environment for all students so that they can become independent thinkers. “Teaching does not begin with the presentation of sacred truths, but with creating opportunities to trigger the students’ own thinking” (von Glasersfeld, 2001, p. 10). I do not want my students merely to have instrumental understanding of mathematical processes. Instead, I want them to have relational understanding so that they are able to move deeper in their critical thinking and problem solving skills (Skemp 1976). Von Glasersfled (2001) refers to this as conceptual learning because students are actively conceiving and building new cognitive pathways. He continues, “There is no infallible method of teaching conceptual thinking. But one of the most successful consists in presenting students with situations in which their habitual thinking fails” (von Glasersfed, 2001, p. 9). I agree that this is a great way to foster relational understanding: to present students with a situation in which their previous knowledge fails and they are forced to collaboratively access prior knowledge and apply it in new ways. In such a student-centered, collaborative environment, my role as a teacher then becomes one of intervention: leading, shepherding, reinforcing, clue-giving, modeling, and praising in order to move students to greater understanding (Towers, Martin, and Pirie, 2000).
I believe that technology can be a powerful teaching strategy if it is used to promote new ways of engaging with mathematics material (Cheung & Slavin, 2013). I strive to utilize technology in the classroom not because it is the latest trend in education, but because it can greatly advance conceptual learning as it allows for experiences and simulations not possible in the traditional classroom setting.
Finally, as my philosophy of teaching and learning, I consider the acquisition of content knowledge as far less important than the process of learning how to learn. Students are in school and in my classroom to learn essential life skills beyond equations and formulas. They are here to learn how to creatively approach a problem, to analyze potential solutions, to engage in critical thinking and reflection, to develop an understanding of how they like to learn, and to learn the value of hard work. It is my desire to foster independence in each student so that they can learn not simply what to think, but how to think for themselves.
References
Cheung, A.C.K., & Slavin, R.E. (2013). The effectiveness of educational technology applications for enhancing mathematics achievement in k-12 classrooms: A meta-analysis. Educational Research Review, 9, 88-113.
Ernest, P. (1994). What is social constructivism in the psychology of mathematics education. In Ponte, J.P. da, and Matos, J.F. Eds Proceedings of the 18th Annual Conference of the International Group for the Psychology of Mathematics Education, Lisbon, Portugal: University of Lisbon, 1994, Vol. 2, 304-311.
Ernest, P. (2011). Social constructivisim as a philosophy of mathematics: Radical constructivism rehabilitated? Retrieved August 24, 2013 from http://people.exeter.ac.uk/PErnest/soccon.htm.
Ernest, P. (2012). What is the philosophy of mathematics education? Retrieved August 20, 2013 from http://people.exeter.ac.uk/PErnest/pome18/PhoM_%20for_ICME_04.htm
Holley, L.C., & Steiner, S. (2005). Safe space: Student perspectives on classroom environment. Journal of Social Work Education, 14 (1), 49-64. Retrieved September 7, 2013 from http://dx.doi.org.ezproxy.gsu.edu/10.5175/JSWE.2005.200300343
Pirie, S. & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26 (2/3), 165-190.
Pirie, S. & Martin, L. (2000). The role of collecting in the growth of mathematical understanding. Mathematics Education Research Journal, 12(2), 127-146.
Skemp, R.R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
Towers, J., Martin, L., & Pirie, S. (2000). Growing mathematical understanding: Layered observations. In M.L. Fernandez (Ed.), Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ, 225-230.
Von Glasersfled, E. (2001). Radical constructivism and teaching. Perspectives 31(2), 191-204.
Cheung, A.C.K., & Slavin, R.E. (2013). The effectiveness of educational technology applications for enhancing mathematics achievement in k-12 classrooms: A meta-analysis. Educational Research Review, 9, 88-113.
Ernest, P. (1994). What is social constructivism in the psychology of mathematics education. In Ponte, J.P. da, and Matos, J.F. Eds Proceedings of the 18th Annual Conference of the International Group for the Psychology of Mathematics Education, Lisbon, Portugal: University of Lisbon, 1994, Vol. 2, 304-311.
Ernest, P. (2011). Social constructivisim as a philosophy of mathematics: Radical constructivism rehabilitated? Retrieved August 24, 2013 from http://people.exeter.ac.uk/PErnest/soccon.htm.
Ernest, P. (2012). What is the philosophy of mathematics education? Retrieved August 20, 2013 from http://people.exeter.ac.uk/PErnest/pome18/PhoM_%20for_ICME_04.htm
Holley, L.C., & Steiner, S. (2005). Safe space: Student perspectives on classroom environment. Journal of Social Work Education, 14 (1), 49-64. Retrieved September 7, 2013 from http://dx.doi.org.ezproxy.gsu.edu/10.5175/JSWE.2005.200300343
Pirie, S. & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26 (2/3), 165-190.
Pirie, S. & Martin, L. (2000). The role of collecting in the growth of mathematical understanding. Mathematics Education Research Journal, 12(2), 127-146.
Skemp, R.R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
Towers, J., Martin, L., & Pirie, S. (2000). Growing mathematical understanding: Layered observations. In M.L. Fernandez (Ed.), Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ, 225-230.
Von Glasersfled, E. (2001). Radical constructivism and teaching. Perspectives 31(2), 191-204.