Truly the mind is such an amazing thing. It is difficult to comprehend how a small amount of grey matter stuck inside our skull can handle and process so much complexity. With a number of unique minds inside the classroom, the teacher is always left to wonder and ask questions such as ‘what did my students learn today?’ ‘What new and deep mathematical knowledge, concepts and strategies were constructed?’ And equally important, ‘how did the process of learning happen?’
Answering the first two questions will require the teacher to have a clear review of the instructions used during the procedure and to conduct an assessment to identify and measure learning outcomes. On the other hand, the last question will require not only the understanding of the continuity in the design of the curriculum but of the abstraction phase as well.
Abstraction is the process of how one builds new knowledge. Understanding its dynamics has been the central aim of research in mathematics education (Dreyfus, 2012). One of the approaches used in investigating abstraction is the Abstraction in Context (AiC) theoretical framework which emphasizes that the materialization of new knowledge to learners involves three stages namely the need, emergence and consolidation of new constructs. On the second stage of the AiC, the RBC model suggests that the emergence of new mathematical construct involves three epistemic actions specifically Recognizing (R), Building-With (B) and Constructing (C).
This is well expressed by Kidron and Monaghan (2009) when dealing with the need thatpushes students to engage in abstraction, a need that emerges from a suitable design andfrom an initial vagueness of the learner’s notions:
The learners’ need for new knowledge is inherent to the task design but this need is animportant stage of the process of abstraction and must precede the constructing process,the vertical reorganization of prior existing constructs. This need for a new constructpermits the link between the past knowledge and the future construction… At the moment when a learner realizes the need for a newconstruct, the learner already has an initial vague form of the future construct as a resultof prior knowledge. Realizing the need for the new construct, the learner enters a secondstage in which s/he is ready to build with her/his prior knowledge in order to develop theinitial form to a consistent and elaborate higher form, the new construct, which providesa scientific explanation of the reality.
AiC illustrates the ability of the students to learn new things even on their own. When faced with situations that will force or give them the opportunity to formulate and construct new strategies using the previous ones they already have, their minds will work in such a marvelous way. For instance, after being taught on how to add two one-digit numbers such as 3 + 1, grade 1 pupils have been asked to find the sum in 2 + 3 + 4. Even though addition involving three addends has not yet been taught, the pupils can manipulate their previous learning to build a technique that will enable them to find the answer for the given question and even extend it to four or more addends.
To be specific, when the students are asked to find the sum in 2 + 3 + 4, they immediately recognize that this question is different from what they were previously taught hence they see the need for another strategy to find the answer to the question. This scenario illustrates the first phase of the AiC which is the need for a new construct.
They will now enter the second phase which is the emergence of a new construct where they will device the strategy required by the question. This phase can be described using the three nested stages suggested by the RBC model. The first stage is Recognizing where students identify the previous constructs relevant to create a new construct. In our case, the students will then recognize that finding the sum of three addends may require the skill in finding the sum of two addends. The Building-with stage, on the other hand, comprises the combination of recognized constructs, in order to achieve a localized goal such as the actualization of a strategy, a justification or thesolution of a problem (Dreyfus, 2012). In our example, this phase will involve discussions among the students on how to apply the rules on finding the sum of two addends with the question at hand. Finally, Constructing consists of assembling and integrating previous constructs by vertical mathematization to produce a new construct (Dreyfus, 2012). The students will then come to realize that to find the sum in 2 + 3 + 4, they need to add 2 and 3 first and add the answer to 4.
The last phase of the AiC is consolidation which is a never-ending process through which students become aware of their constructs, the use of the constructs becomes more immediate and self-evident, the students’ confidence in using the construct increases, and the students demonstrate more and more flexibility in using the construct (Dreyfus &Tsamir, 2004). This is where students apply what they have learned and even extend it to solve the sum of four or more addends.
Understanding how students think and create new knowledge could be one of the greatest assets or trade secrets that any mathematics teacher must have or would have wished to have. Knowledge of how the abstraction process works and the variables or conditions to which it may succeed or fail will pave way to a better appreciation of mathematics and of the teaching-learning process in general.
REFERENCES:
Dreyfus, T. (2012). Constructing abstract mathematical knowledge in context. 12th International Congress on Mathematical Education.
Dreyfus, T., &Tsamir, P. (2004). Ben’s consolidation of knowledge structures about infinite sets. Journal of Mathematical Behavior, 23, 271-300.
Kidron, I., & Monaghan, J. (2009). Commentary on the chapters on the construction ofknowledge. In B. B. Schwarz, T. Dreyfus & R. Hershkowitz (Eds.), Transformation ofknowledge through classroom interaction (pp. 81-90). London, UK: Routledge.
By: Samuel A. Quiroz | Teacher II | Emilio C. Bernabe High School | Bagac, Bataan
