When it comes to knowledge construction, the theory of constructivism perfectly resonates to me. I share the view that knowledge is not a thing that can be simply given by the teacher at the front of the classroom to students in their desks. Rather, knowledge is constructed by learners through an active, mental process of development; learners are the builders and creators of meaning and knowledge. In a math classroom, in the most general sense, knowledge is constructed when students are encouraged to use active techniques, that is experiments, real-world situation, and problems solving, to create more knowledge and then to reflect on and talk about what they are doing and how their understanding is changing. David Tall (2004) introduced the notion of the three different worlds of mathematics, that is (1) conceptual \u2013 embodied world, (2) proceptual \u2013 symbolic world, and (3) axiomatic \u2013 formal world. Tall (2004) came to this conclusion after he found that there are three fundamentally different ways of operation in mathematics, one through physical and mental embodiment, including action and the use of visual and other senses, a second through the use of mathematical symbols that operates as process and concept in arithmetic, algebra and symbolic calculus, and a third using formal language in increasingly sophisticated formal mathematics in advanced mathematical thinking. This offers a useful categorization for different kinds of mathematical context encounter by the students in mathematics. Each category has its own individual style of cognitive process and together they cover a wide range of mathematical activity. Also, in each category we usually have abstract and concrete topics. The concept of abstract and concrete usually depends on the cognitive level of the students. The knowledge is constructed when the student\u2019s understanding transitions from abstract to concrete objects. Sfard (1991) defines interiorization, condensation, and reification as the necessary phases to transition from abstract to concrete knowledge.<\/p>\n
Scheiner (2016) argues that a concrete object is an object for which an individual has established rich representations and several ways of interacting with, as well as connections between it and other objects. He suggests two interesting ways through which the students can construct mathematical knowledge, (1) abstraction-from-action: that is the students first learn processes and procedures for solving problems in a particular domain and later extract domain-specific concepts through reflection on actions on known objects; (2) abstraction-from-object: that is the students are first faced with specific object that fall under a particular concept and acquire the meaningful components of the concept through studying the underlying mathematical structure of the objects. These ways of knowledge can be apply to Tall\u2019s (2004) three different worlds of mathematics (conceptual, symbolic, and formal world).<\/p>\n
In general, the students would construct knowledge in math when they actively engage in learning, practicing and reflecting on their understanding. They generate knowledge when they individually reflect upon their experiences and their ideas. The social context is also important for knowledge construction in math. The students by sharing their experiences and their ideas with others, modify their knowledge, altering concepts that need to be altered and rejecting concepts that are not correct. I agree with Scheiner (2016) when he says that the learner must make sense of the mathematical concept through restructuring knowledge structures that are built on previously constructed knowledge pieces. In fact, because the students usually make sense of the mathematics they learn by connecting them with the mathematics the already know. The mathematics curriculum is vertically aligned in such a way that knowledge is gradually added onto concepts as the students are progressing in their learning.<\/p>\n
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References<\/p>\n
Scheiner, T. (2016). New light on old horizon: Constructing mathematical concepts, underlying abstraction processes, and sense making strategies.\u00a0Educational Studies in Mathematics<\/em>,\u00a091<\/em>(2), 165-183.<\/p>\n Bryant, P., & Nunes, T. (Eds.). (2016).\u00a0Learning and teaching mathematics: An international perspective<\/em>. Psychology Press.<\/p>\n Tall, D. (2008). The transition to formal thinking in mathematics.\u00a0Mathematics Education Research Journal<\/em>,\u00a020<\/em>(2), 5-24.<\/p>\n","protected":false},"excerpt":{"rendered":" When it comes to knowledge construction, the theory of constructivism perfectly resonates to me. I share the view that knowledge is not a thing that can be simply given by the teacher at the front of the classroom to students in their desks. Rather, knowledge is constructed by learners through an active, mental process of […]<\/p>\n","protected":false},"author":53876,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1669399],"tags":[],"class_list":["post-5796","post","type-post","status-publish","format-standard","hentry","category-b-knowledge-diffusion"],"_links":{"self":[{"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/posts\/5796","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/users\/53876"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/comments?post=5796"}],"version-history":[{"count":1,"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/posts\/5796\/revisions"}],"predecessor-version":[{"id":5797,"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/posts\/5796\/revisions\/5797"}],"wp:attachment":[{"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/media?parent=5796"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/categories?post=5796"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ubc.ca\/stem2018\/wp-json\/wp\/v2\/tags?post=5796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}