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In the last couple of years, we experienced rapid advancements and changes in the practice of teaching and learning mathematics, globally. This has provided us with a variety of techniques to support accessibility to the subject and to improve our understanding of its needs (e.g. choosing the type of assessment that aligns with the module’s scope and intended learning outcomes). Informed decisions on ‘when, where and how’ to use and implement those innovations are crucial for achieving better results.
In this article, I share insights from past and recent readings in mathematics education. Inspired by the contributions of Gattegno, Wheeler, Tall, Hewitt and Claxton, this work is designed with the purpose of enabling practice exchange and discussions between practitioners. I have presented this content during my guest contribution at the Technology Enhanced Mathematical Sciences Education TEMSE seminar of the Maxwell Institute for Mathematical Sciences (Heriot-Watt and Edinburgh Universities). I aim to share it with the wider audience through this blog post.
This article aims to draw on the theoretical perspective of making sense of mathematical expressions to outline how the Principle of Articulation (Chin et al., 2022) and other theories can be used to reinforce a deeper learning in the study of the subject. This piece is divided into three main concepts: Educating Awareness; Mathematisation; Learning Theory.
The famous statement by Gattegno (1987, p.220), ‘only awareness is educable,’ suggests that self-awareness is key to learning, and it is the responsibility of the individual to educate their own awareness, which is only then when someone becomes aware of their own awareness and understanding. Hewitt (2001) then distinguishes between two types of awareness, ‘unconscious awareness’, which is the awareness through senses, and ‘conscious awareness’ through cognitive interactions.
The author tells the story of a child who was tasked to find the total number of counters inside two boxes (each of four counters). The child concluded the total to be eight (using senses). Later, a counter was moved from one of the boxes to the other, and the same child was asked the same question, but in this case, conscious awareness was used (manual counting) to conclude that the total was still eight. Further changes were applied by moving a second counter and then a third to the other box. However, the child was now aware that such a process would not change the total number of counters. This event is an illustration of ‘awareness being educated’. Hewitt adds:
‘Memorising only takes us somewhere, it is by educating awareness that we can have the means to take things further on our own accord and not be limited to reproducing only those things which we have been told.’ (Hewitt, 2001, p.38).
Claxton (1984, p.45) comments: ‘I can not learn if I am not attentive to, or aware of the success or failure of my actions at some, not necessarily conscious, level.’
In response to the statement: ‘becoming aware that my awareness is being educated’ (Hewitt, 2001, p.39), I present the following activity:
Starting by a contribution from Wheeler (1982):
‘It is more useful to know how to mathematize than to know a lot of mathematics. Teachers, in particular, would benefit by looking at their task in terms of teaching their students to mathematize rather than teaching them some mathematics.’ (p.45)
The author states the difficulty of identifying mathematisation and provides some indications of its presence, such as through structuration (finding patterns); dependence (chaining between old and new learning); and infinity (generalisation and universality). Later, a couple of additions were also introduced, like extrapolating and iterating and generating equivalence through transformation (Wheeler, 1982).
A second assignment:
In 1962, Skemp stated the need for a schematic learning theory referring to the theory called ‘Schema’ developed by Piaget in 1950. The theory considers ‘the systematic development of an organised body of knowledge, which not only integrates what has been learnt but is a major factor in new learning.’ (Skemp, 1962, p.133).
The author then continues: ‘The incorporation of new knowledge into an existing schema is called assimilation; and the enlargement of a schema, which may be necessary if it is not adequate for the above purpose in its existing form, is called accommodation’ (Skemp, 1962, p.133). A follow up activity would task the reader to:
Later, in 2013, David Tall developed this learning theory into a new version evolving through: Conceptual embodiment; Operational symbolism; and Formalism.
Mathematical thinking requires the uncovering of different layers of understanding. The first of which is the stage where we try to find a meaning to the mathematical concept through our perceptions and interpretations of the real world, and this could be in the form of finding real life applications or trying to link this new concept to an existing one – mathematically speaking, this sometimes takes the form of studying special cases before generalising it to a more sophisticated level. This level of understanding can be described by ‘embodiment,’ after which comes the stage when we learn about the structure of the given problem (operational symbolism). Here is when we start questioning the different aspects of the statement, e.g. the number of variables, their nature and type, environment (conditions and assumptions), as well as connections between components.
Then follows the need for appropriate mathematical techniques to address the problem, such as definitions, principles, axioms, logic, theorems and proofs. In the same book, the author argues that in order to be able to move from embodiment to operational symbolism, there is a need for ‘articulation,’ which allows this transition to take place and to settle mathematically. This gives rise to the ‘Principle of Articulation’ which was developed by Tall (2019):
’the meaning of a sequence of operations can be expressed by the manner in which the sequence is articulated.’ (Tall, 2019, p. 7).
The principle offers:
‘a meaningful foundation for a long-term learning theory that enables teachers as mentors to encourage learners to take control of their own learning.’ (Chin et al., 2022, p. 658).
Here, I list some of the pedagogical approaches which I have developed illustrating an alignment with the theories (mentioned above).
Those approaches have common objectives aiming to encourage the articulation of mathematics; the development of conceptual embodiment and the sense of association in the practice of mathematics; as well as accessibility to the subject through other means.
The author thanks Professor L. Boulton and the TEMSE organizing committee and its community for their invite and kind hospitality.