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 – embodied world, (2) proceptual – symbolic world, and (3) axiomatic – 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’s 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.
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’s (2004) three different worlds of mathematics (conceptual, symbolic, and formal world).
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.
References
Scheiner, T. (2016). New light on old horizon: Constructing mathematical concepts, underlying abstraction processes, and sense making strategies. Educational Studies in Mathematics, 91(2), 165-183.
Bryant, P., & Nunes, T. (Eds.). (2016). Learning and teaching mathematics: An international perspective. Psychology Press.
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24.
Hi Vivien
I like the fact that you tied the Scheiner and Tall articles.
Do you think that some students do better with abstraction-from-action and others do better with abstraction-from-object?
Christopher
Hi Christopher,
I think multiple intelligence enables the students to perform in various ways. However, in my experience, the students have usually performed better with abstraction-from-object. Their understanding is meaningful from the outset of a lesson based on abstraction-from-object. The risk with abstraction-from-action is that the processes, and the procedures learned are subject of possible lost. If over a certain period of time, there is no opportunity to extract domain-specific concept that will meaningfully strengthen, and anchor the understanding. I wonder if abstraction-form-action is applied because students’ mental maturity to extract domain-specific concepts is not yet attained.
Vivien
I was very intrigued by your description of the importance of concrete objects as part of the constructivist environment for learning mathematics. Your post made me think of the importance of manipulatives for learning math. The use of manipulatives in teaching mathematics has a long tradition and solid research history. Manipulatives not only allow students to construct their own cognitive models for abstract mathematical ideas and processes, they also provide a common language with which to communicate these models to the teacher and other students. In addition to the ability of manipulatives to aid directly in the cognitive process, manipulatives have the additional advantage of engaging students and increasing both interest in and enjoyment of mathematics. Students who are presented with the opportunity to use manipulatives report that they are more interested in mathematics. (Sutton & Krueger, 2002).
Do you think that digital manipulatives are as valuable as physical ones?
Trish
Hi Trish,
As the saying goes, better late than never. Thank you for your insight on the importance of manipulatives for learning mathematics. I agree, manipulatives are good tools to embody learning in mathematics. The digital manipulatives give students the same opportunity to make meaning and see relationships as the result of theirs actions, just like a physical manipulative. However, as cognitive tools, digital manipulatives have unique characteristics that go beyond the capabilities of physical manipulatives. Many digital manipulatives provide support for sense-making, and make mathematics ideas more explicit as the student interacts with the tool. Physical manipulatives don’t provide specific and directed feedback and interaction, while digital tools react to the student’s actions, provide prompts, and guidance that help the students focus on the mathematics in the task. This advantage that digital manipulatives have over physical manipulative counterparts should not lead us to only use digital manipulatives. I think both type of manipulatives are important. In fact physical manipulatives appear to be more effective for building conceptual understanding through embodied learning, and digital manipulatives at hand reinforce those concepts.
Once conceptual understanding is brought about using physical manipulatives, the subsequent use of digital manipulatives seems to facilitate bridging to the abstract. What do you think?
Vivien