‘Making Sense of Mathematics’ is a chapter featured in the first edition of ‘Enhancing Primary Mathematics by Malcom Swan. Swan concentrates on discussing and analysing the psychological challenges children have while learning mathematics. In this chapter, Swan explains the processing of mathematics with children and the misconception of the teachers roll in ‘fixing’ the children’s mistakes. He expands on the point of that mistakes are not a bad thing and actually help build a child’s cognitive development and their further understanding of maths, as well as mistakes helping the teacher understand how the child is incorrectly interpreting the problem with their “alternative conception” of the mathematics. As It is also a constant theme of how natural mistakes are, as they can come from different reasoning of lack of concentration to more complex reasoning, and how learning is a collaborative effort between child, parent and the teacher which is strongly highlighted and encouraged within the Curriculum for Excellence. Swan also introduces the idea of the child needing to have something within their schema to link concepts to before understanding different parts of mathematics and expanding the difficulty of it whether this is understanding counting money, measurements or sharing. Thus, in this essay I will explore these concepts highlighted throughout this chapter with links to theories and relevant research.
Currently if a child does not understand something, the teacher repeats the same maths to them but slower to try and make the information concrete. Swan took this idea and tried to explain why children have misconceptions of mathematics, or alternative conceptions, and how trying to drill information into children is not a successful route. When learning new concepts, whether in maths or life in general, children need to be able to make direct links to previous conceptions to be able to try and understand new information. He uses examples such as decimals, appearing in measurements and money but how this can be confusing with the concept of time and remainders or even understanding sharing as it can be linked to division. The concept of making connections to previous things within a child’s knowledge to allow smoother learning is the idea of a schema – a database of information expanded over time from a young age to consolidate and expand ideas, for example learning what a bug is, to then later learning about characteristics of caterpillars and ladybirds. This is linked to a theory of cognitive development, as he believed there was 6 steps within a child’s sensorimotor stage (0-2 years) which involves developing awareness and becoming more understanding to their environment such as connecting visual movements to sound, to help aid them in the world (Piaget, 1952). This is shown throughout a child’s development, from knowing the shape and noise of a rattle to what happens when you strike a drum. When the child is older the 5 senses are introduced to add to this, where you learn that a kettle or curling iron is hot, and touching it causes pain. Piaget believed that learning was also caused by facing interactions our schema cannot process, so we learn from it but that development still happened in stages. Whereas Vygotsky believed that although we learn from new experiences to the schema, this interaction with the environment around us is how we develop, not so much the idea of stages (Vygotsky, 1978). Throughout school, in my experience, it was never about learning to understand, it was about memorising facts until you can recite them whether it was multiplication or capital cities. This is an example of teaching for transmission; putting information from the teachers’ brain to the students (Miller & Selller, 1990). It is a heavily teacher-centred approach to teaching, and more outdated with the idea of the teacher just dispensing information to a child and assessing how well they have remembered and can recite this information. This style of teaching is appropriate for learning to pass assessments, rather than learning for understanding and knowledge which can be viewed as heavily disadvantageous. On the other hand, teaching for transaction is based around the concept of students being able to interact with the knowledge being provided and being able to build their own ideas, understanding and interpretations of the information (Miller & Selller, 1990). This is a more interactive way of learning and allows children to link up the new concepts to their schema. In my opinion, this is a better take on absorbing information as each child learns differently, and has different experiences so it is unfair to believe all children can learn from information being drilled into them. In conclusion, Swan emphasises misconceptions are made by children due to the unfitting environment in which they are taught, and the way the teacher allows the child to comprehend the overall understanding from the class.
Swan also highlights a theme of fluency versus meaning. This derives from the National Numeracy Strategy (NNS) saying that they want children to be ‘fluent’ in mathematics as if it was a foreign language. To become fluent in a language involves practicing it until it becomes natural and little psychological effort is needed. Swan backs this view with “Framework for Teaching Mathematics from Reception to Year 6”, who expand on how to become ‘fluent’ in mathematics, as they go to describe how a maths lesson should take placed. This method consists of ‘remembering facts’ and being able to recite the facts and answers to calculations quickly, especially when being vocally tested. This is more about being able to know that the answer is correct rather than knowing why the answer is correct. Although being correct is advantageous to an extent, Swan goes on to say that there is more to the necessary elements of knowing mathematics than being ‘fluent’ in it, as maths is also about developing meaning. This involves having their own understanding of an idea rather than just reciting what they have been told which is beneficial to the child as they have more of a chance of remembering the concept and will be able to apply it in different situations as it is hopefully part of their schema. Furthermore, most children are fluent in some part of mathematics, but if this is not combined with meaning then the child may not be able to explain and apply their knowledge further on especially if rushed, according to Swan. To help consolidate the information, it is important for children to discuss their ideas with the teacher and other students to prevent misconceptions and encourage alternative conceptions, where the teacher can tell where exactly the child did not pick up the idea that is being taught as this will lead to less lesson time being spent on repeating the basic concepts in the long term. In a classroom situation, it is beneficial to all learners, including the teacher, to find out how everyone else solved problems and why they did it a certain way as it allows everyone to learn from their mistakes and learn how other people interpret problems despite being taught the same way. During placement, I saw this was the case as no matter how many times the teacher repeated herself the child did not understand the mathematics, and therefore could not advance to the harder problems with their friends. It was only from an explanation and discussion between the child, the group at the table and myself that she was able to comprehend a way of understanding the basic problems. Thus, in conclusion, I agree with Swans statement of needing a combination of being fluent and having ‘meaning’ in mathematics, except I believe meaning is vastly more important as it encourages discussion and wider learning to prevent misconceptions.
During this chapter, Swan explains the 4-step plan for how he perceives successful teaching for meaning; Uncovering the problem, creating an awareness that new learning is needed, resolving the conflict and consolidating learning. These were the principles that were discovered in research in Newcastle. The first step is not teaching the children, but facing them with a task that allows them to try and recall information they have learnt previously and asking them to be able to justify their working and answers. This allows the child to recall information ready for the lesson, test themselves as well as allowing the teacher to assess the progress of the class as well as also seeing how well they explained a certain topic. After this, the inconsistencies of answers between the children is opened to discussion, without the teacher commenting on whether they are correct or not. Swan explains a few methods in which this can be done but he makes it clear that the teachers involvement to the resulting answer is kept to a minimum. This is to encourage discussion and interest from the children at home as well as in the classroom. After this stage, Swan suggests holding a class discussion about the ‘issue’ to try and resolve it. After the discussion, the teacher would explain how they would have done it to get the correct answer, while also explaining how there are several ways to solve the same problem correctly and get the same answer. Finally comes the last stage, where the children apply their knowledge by creating their own versions of sums related to the topic and solving them, or other ways of interactive learning to further the classes mathematics as well as personal development. Personally, think Swans 4-step plan to a successful way to teach for meaning is promising as it allows communication between pupils, their parents and the teacher in a way that they can further their learning or have more clarity on the concept. However, I think this would only theoretically work on smaller classes, as class sizes of 30+ have a large variation of abilities and concentration is a lot harder to grasp so some children would slip and fall behind.
in conclusion, Malcom Swans chapter ‘Making Sense of Mathematics’ analyses the psychology of the children in different teaching situations, and how the idea of misconceptions is incorrect, it’s simply alternative conceptions which is necessary to a child’s learning experience. Swan explained how important cooperative styles of teaching are as it gives children of different learning styles a chance to understand and allow everyone to be part of the same discussion. He also heavily concentrates on the idea of that children are not ‘blank slates’, but instead need time to be able to absorb the information and to link it to previous ideas they have within their schema. This was further linked to the different styles of teaching as well as the idea of having an understanding versus being fluent in mathematics. Swan addresses in the conclusion his faults, such as that his suggesting methods would not work for all age groups and acknowledging that he did not discuss all angles of mathematics relevant to his points. Overall, the chapter highlighted different views of teaching – from Swan, to Piaget, to Vygotsky – which helped create a well-rounded opinion on my behalf, and opened for discussion the different methods and their effect with ranging capabilities and how to allow most of them to understand a concept and its meaning instead of it being ‘fluent’.
Miller, J. & Selller, W., 1990. Curriculum: Perspectives and Practice. Torronto: Copp Clark Pitman.
Piaget, J., 1952. The Origins of Inteligence in Children, s.l.: New York: International University Press.
Vygotsky, L., 1978. Interaction Between Learning and Development, s.l.: s.n.