The Understanding Of How Children Learn Mathematics

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Learning theories are a vital part of understanding how children learn Mathematics. They can be used to investigate how children learn and can help to assist teachers in teaching by applying these theories into practice.

This assignment explores the development of scaffolding and its usefulness in other aspects such as Vygotskys Zone of Proximal Development and Woods levels of contingency. It analyses an existing lesson and builds scaffolding into it to improve the lesson an then uses scaffolding to design a sequence of lessons to see how it aids in teaching of the Mathematics concept of straight line graphs, a topic which students find very difficult and struggle with. Overall, the assignment looks at the development and usefulness of scaffolding in the teaching and learning of Mathematics.

Scaffolding emerged from the constructivist theory as a scientific theory from the work of Piaget (1926), Dewey (1933) and was refined by Vygotsky (1962, 1978). Piaget observed conversational practice of children to identify developments in the way they understood the topics and codes of dialogue. He codified these developments into approximate stages and arranged them in a process that moved from ego-centric focus on immediate, non-specific demands into a more specific state in which concepts could be understood and interpreted by one another. Meanwhile, Dewey (1933, pg. 156) developed the idea that 'thought should be trained by special exercises designed for the purpose', and argued that such exercises must be 'conditions that will arouse and guide curiosity... connections in things experienced that will on later occasions promote the flow of suggestions, create problems and purposes that will favour consecutiveness in the succession of ideas' (ibid., pg. 157).

Considering these two ideas conjunctively gives an idea of what constructivism is ultimately about and how the two theoretical areas infuse each other. Learning is a matter of moving through developmental stages, and is best accomplished by intellectual conditions that create a desire to move and connect things learnt in the exploration of those conditions and apply that learning to the next topic. Vygotsky (1962, pg. 12 - 58) critiques what he sees as Piaget's belief in early learning as self-focused, on the basis that even egocentric speech is addressed to a listener for a purpose. Piaget's observations are based upon 'the absence of any sustained social intercourse between the children of less than seven or eight'; Vygotsky argues that the social intercourse that occurs at this age is playful, characterised by a language 'of gestures, movements and mimicry as much as of words' (pg. 27). He claims that 'the processes of inner speech [egocentric statements that are formed in the mind but not spoken] develop and become stabilised approximately at the beginning of school age and that this causes the quick drop in the egocentric speech observed at this stage' (pg. 33). In other words, children develop spoken social interaction that considers others at around the same time as they begin to be taught through that medium.

Vygotsky goes on (1978, pg. 79 - 92) to codify the process of learning through social interactions with the term 'zone of proximal development' (ZPD). He noted that children performed better at new tasks, problems or contexts of learning when discussing the task with an adult and by articulating and refining their thinking in a social context with an adult. The zone of proximal development encloses those tasks which are within the child's capacity but require social interaction with an adult in order for that capacity to be reached. It can be used to 'delineate the child's immediate future...allowing not only for what already has been achieved developmentally but also for what is in the course of maturing' (pg. 87) In other words, to predict what the child is able to learn using what they have already learnt or prerequisite knowledge, provided a teacher is present for the child to interact with in applying their existing knowledge and skills.

This concept of remaining one-step ahead of the learner and providing the context for them to reach this step in their own way, became a defined method when Wood, Bruner and Ross (1976, pg. 89) determined that 'the acquisition of skill in the human child can be fruitfully conceived as a hierarchical program in which component skills are combined into "higher skills" by appropriate orchestration to meet new, more complex task requirements.'

The process of firstly teaching component skills, then helping the learner combine the skills, and finally withdrawing to allow the learner a chance to practice the combination of those skills autonomously, was defined by Bruner and Garton (1978, pg. 254) as 'scaffolding'. Scaffolding operates within the zone of proximal development, providing a framework for social interaction between teacher and learner, refining the learner's knowledge through discussion and therefore their movement through their zone of proximal development.

Roberts (2003, pg. 101) identifies scaffolding in schools as the modelling of tasks by a more skilled person (teacher/instructor), and leading or probing questions which extend or elaborate existing knowledge. She explains that 'as the learner's competence grows the scaffolding is gradually reduced until the learner is able to function autonomously in that task and generalise to similar circumstances'.

Wood (2001, pg. 282 - 283) chronicles the evolution of scaffolding in more detail, introducing the idea of 'contingency'. Contingency defines specific qualities of the ZPD by defining sub-zones that can be developed within a particular lesson. He defines the concepts of 'domain contingency', which are skills or knowledge that naturally lead and refer to one another (interconnect) e.g. hundreds, tens and units have to be understood before rounding can be understood so these ideas have domain contingency, and 'temporal contingency', which is the state in which learners are able to identify when their solutions to a problem are failing and attempt alternative solutions. A learner who is able to do this without teacher intervention has progressed through their ZPD and no longer requires scaffolding; they have learned the skill or knowledge that the problem was structured to teach them. If teacher interventions are required, they must achieve 'instructional contingency' with the learner's efforts to continue and guide the learner through a process of development rather than enforcing a stop and restart. This is the process by which scaffolding works. The teacher does not stop the learner and say "that's wrong, let me tell you the right answer" but instead says "you know how to do this, you know how to do that, so try and put them together" and gives hints and prompts to keep the learner going to discover the solution for themselves.

Mathematics is often suggested as an application for scaffolding. De Corte (1995) suggested it as a way to modernise Mathematics practice, citing Vygotsky in constructing his view of learning and suggesting scaffolding as a design principle for effective learning environments which achieve 'balance between discovery learning and personal exploration on the one hand and systematic instruction and guidance on the other' (pg. 41). Shayer and Adhami (2007) convey a critique of instructional technique which draws similar conclusions, suggesting that conventional teaching of Mathematics comprises of systematic instruction in techniques.

Van der Stuyf (2002) suggests a ground-level example of scaffolding in Mathematics; introducing the 'hundreds, tens and units' representation of numbers and using this in rounding numbers up and down. With this as a suggestion in mind, I propose to explore a simple four-stage lesson plan on rounding (BBC, 2009), looking for ways to implement the five conditions of successful scaffolding, as identified by Applebee (1986):

1. The task must allow students to make their own contribution.

2. The task should build upon existing knowledge and skills, but be difficult enough for new learning to occur.

3. The environment must provide a natural sequence of thought that approaches the task.

4. The teacher must collaborate with the learning, and not evaluate it.

5. As the learner becomes more confident, they should take greater responsibility in controlling the task.

The lesson which I am analysing (BBC, 2009) is about rounding off. It starts with a follow-me chase game about multiplication (starter), a worksheet about rounding with suggested sentences for teacher intervention (main), a discussion of significant figures (development) and a return to the chase game using roundings-off (plenary).

In theory, the plan fulfils three conditions in its starter and plenary activities. By beginning with a game and asking learners what they had to do in order to succeed in the game allows learners to identify the topic of the lesson themselves (self-discovery), rather than being told (condition 1). This creates a natural sequence of thought between an activity and learning how to improve at that activity (condition 3). Returning to the same game at the end, with learners creating their own challenges, fulfils conditions 3 and 5, concluding the sequence of thought with an application of learning. However, the game included in the plan impairs the process of scaffolding to an extent: although it is a plenary, and the teacher does not have to stop the lesson to explain it, it is about mental multiplication, not rounding, so consequently fails to instigate a natural sequence of thought. To scaffold the process of learning appropriately, the starter needs to be on the same topic as the lesson, highlighting the existing knowledge and skills that are to be built upon, and so a card chase game on decimal points would be more appropriate for this lesson.

The main activity involves presenting a worksheet to the learners without attempting to contextualise the task by modelling the skill involved in a less abstract context. The context in which the skill of rounding off can be used by learners does not appear until the extension; therefore those who are struggling will never reach that point. It might be preferable to begin with an application and demonstration of the skill: for instance, the teacher could ask students for the time, and collect answers of varying precision that can be used to demonstrate how people round off the time to varying degrees of significance, e.g. 14:36:03 becomes 14:35 or "about half past two". Then, the concept of significant figures can be introduced by encouraging learners to find their own examples beyond the telling of time, with teachers using prompt questions like "how many children do you think there are in the school?" and providing contexts for different levels of accuracy, i.e. "who needs to know, and what for?" (the school register needs to be accurate, the local newspaper can afford to round off). This fulfils conditions 1 and 2, introducing the idea of expressing the same numerical value more or less precisely in a familiar context and developing it through discussion (conditions 3 and 4). As the process is modelled using examples, learners who are comfortable with the process can collect their own worksheets and begin to apply the learning (conditions 3 and 5), before creating a card for the chase game plenary.

These changes adapt the lesson plan to build in scaffolding into the lesson - using the familiar concepts of decimals and time to scaffold significant figures, and then using those to scaffold rounding off, by explaining it using examples of it in application. By the end of the lesson, the scaffolds of significance and context are removed, and the learners are rounding off arbitrary figures of their own choice. Therefore, instead of being an instruction of technique, the lesson has presented the learners with questions to answer and contexts to find, ensuring that they are engaged and take an active role in moving through their zones of development. By the end of the lesson they are creating their own applications for the technique and solving them without assistance, indicating that the technique of rounding off has been 'learned', in the terms of Vygotsky.