2. Capítulo II: El juicio estético
2.1.1 El problema del gusto
The way that numeracy is taught for teachers as part of their training course is important in terms of a ‘cyclical transmission’ of attitude and perception of their own students towards the subject. Teacher educator approaches to delivering the numeracy elements required for initial teacher training are important in terms of what information, values and pre-conceptions they
65
transmit and how. If ‘absolutist’ methods prevail numeracy will be difficult to digest and will become a hoop to jump.Through teacher training a prevailing paradigm can be seen to be perpetually transmitted, however, there are indications that these paradigms are slowly altering to shift the focus of mathematics and numeracy teaching. This shift in focus which includes new teaching methods demonstrating a possible change in classroom pedagogy has been apparent in the forward movements of generic teaching skills.
Vorhaus (2006) examined teaching and learning in numeracy classrooms, through direct observations, finding that the quality of teaching was not always directly correlated with the standards of learning, in some instances, ‘poor’ teaching did not correlate with learners
progress being poor and conversely noting within classroom observations that the opposite was also the case, where teaching was good this did not correlate with high gains for learners in every instance. Vorhaus concluded that the characteristics of the learners themselves may have a bigger part to play in their success than expected and ‘went on to recommend features which could be included within programmes of teacher training, especially useful for integration of subject specific knowledge, and the pedagogy of the teacher.
These recommendations were presented with the aim of promoting the involvement of learner characteristics within the classroom, and included: Improving teachers’ confidence and skills in classroom management (balance of whole group, small group and individual work) supporting teachers to become more skilled and flexible with different teaching approaches and involving learners more in the learning experience (Vorhaus 2006).
Hands on experiences in mathematics can aid the development of skills, by forming genuine experiential memory for learners. Locke (1632 – 1704) argued that ‘experience’ was the strongest form of knowledge and ultimately, learning through experience created the most enduring type of knowledge. Cole (2002 P.3) expounded the same principle, referring to Aristotle for support;
“Aristotle emphasised experience filtered through logic as the way to gain this abstract knowledge”.
66
Swan and Swain (2007) provided a more up to date rationale for hands on teaching, resulting in experiential learning of significance in mathematics as the approaches were presented for mathematics teachers in particular by the Department for Education. Swan and Swain advocated a more experiential type of learning for both teachers and students that did not detract from the essential knowledge base, ensuring that this was key at all times.The Cockcroft report(1982) proposed an ideal for mathematics teaching in terms of the structure of sessions, and the different methods which should be employed to aid learning for students. These did include the more traditionalist methods such as teacher exposition but also included practical work, problem solving and discussion.
Within the different methods described, there exists the opportunity to differentiate tasks to cater for different learning preferences, ‘discussion’ for instance aids the auditory learner, whilst ‘practical work’ will enhance the learning experience for the kinaesthetic learner in particular (if examining learning preferences using a VAK continuum). Ollerton (2003) disputes the idea that people can simply be classified in terms of their learning styles clearly, indicating that other factors affect the learning experience for the individual including context and stimuli, the only way to account for all of these elements is to use vastly different methods and resources in the classroom. Emphasis has often been placed on the ‘three part lesson’, but the
recommendations for the amount of methods would indicate a lesson in an infinite number of parts, rather than limited to three distinct elements of beginning, middle, and end.
Development of teachers is a valuable tool for improving the experience of learners. Often, teacher development is based on administrative details and target driven collections of information. If the methods advised for learners in the classroom were applied to the trainee teachers learning, the results may prove to be intrinsically valuable, not only for the teachers but for their students.
The report of the Advisory Committee on Mathematics Education (ACME) in 2002, examined the CPD of mathematics and numeracy teachers. Several lengthy recommendations were made by ACME, including that CPD programmes aimed at teachers of the subject should take into account opportunities which would allow teachers to relate theory and practice to each other within the classroom environment, supported by Coben et al, (2003).
67
Kirby and Sellers (2006) examined the practical application of learning styles in the numeracy classroom. The most usual application of learning styles is to assess individual learners, and note down the result (paying a type of ‘lip service’ to the process). Kirby and Sellers went on to develop CPD training in the practical application of learning styles information, informing the pedagogical process in the classroom. The main aim being not just to pay ‘lip service’, but to allow learners to develop a ‘metacognitive awareness’ related to their own numerical skills, and through this to further develop the whole process of teaching and learning. It should be noted that although the subject of much study, learning styles is not considered to be scientific and has fallen out of favour with educationalists over time.Rumelhart and Norman (1981) considered all learning to be defined by a set of schemas or schemata or blocks of knowledge. The process of learning includes ‘assimilation’ which has particular relevance to adult learners, who assimilate new learning to prior knowledge,
regardless of whether their prior knowledge is accurate. Context is important to the process of assimilation for adult learners, a true contextual clue can hinge on to prior knowledge, building on foundations that are already present. The value of making mistakes should not be under- estimated. Learners’ mistakes often lead to independent investigation, confusion and frustration, but in turn these can lead to the development of a re-constructed knowledge in a cyclical logical format developed through problem solving. Wadsworth (1996) saw intellectual and
mathematical development as full of errors and making mistakes in mathematical and numerical learning as not only acceptable but desirable.
2.14 Chapter summary
The literature surrounding teachers’ numeracy has provided some context and background to be able to anchor the research clearly. Several areas have been identified as lacking in information in a general sense; very limited information is available relating to the
implementation of minimum core numeracy in the post-compulsory sector. Teachers’ perception of numeracy and mathematics in this area has not been afforded a great deal of research time by the educational community. Some of the main policy landmarks with reference to numeracy
68
learning within the lifelong learning sector and the compulsory schools sector have beenexamined.
Developments in curriculum, economic effects and the effects on the individual have been reviewed as part of the wider context of the research. Following on from this the minimum core of numeracy for the post-compulsory sector was described and the basic process of the professional skills tests for numeracy presented providing an overview of implementation and teacher skills in numeracy. Comparisons have been made between the compulsory schools sector and the post-compulsory sectors of education in terms of trainee teachers learning numeracy.
Over time, the same information relating to mathematical and numerical learning can be seen appearing again and again. From Locke (1632-1704) to Cole (2002) Swan and Swain (2007) experiential learning is seen as a method which is useful for learning in mathematics but not a method that is necessarily strongly advocated for trainee teachers.
The perception of mathematics and numeracy partially underpins the original research questions formulated and has been developed here more theoretically, examining both the individual and wider perceptions, examined in more detail in phase one of the research.
Training teachers forms a major part of the research and has been viewed from several perspectives in this chapter. An outline of the process of initial teacher education was included here with further exploration into the application of the minimum core of numeracy in training teachers in post-compulsory teacher education and generalised comparisons made with the compulsory schools sector. The process of learning in mathematics and numeracy and the subsequent effects on the individual have been briefly explored, examining the fear of failure and the experience of hatred described by many.
The process of research creates a ‘ripple effect’ between the research itself and its implementation (Robertson, 2000) or effect on practice and pedagogy.This ‘ripple’ is a desirable outcome here as numeracy in teacher education does not appear to be very well received. This needs to be addressed practically to have any real impact. The main purpose of research of any kind is always to develop and create knowledge (Coben, 2000). The lack of knowledge surrounding numeracy and numerical skills required by teachers necessitates the
69
research into this area to develop and support implementation to be more effective in practice and therefore more useful.70
71
3.1 Introduction
This chapter describes the methodology employed to answer the research questions posed in the introductory chapter. Initially a generalised overview includes a comparison of qualitative and quantitative methods and examines the use of a research journal to support the work completed at every stage. This overview is followed by a more in depth account of the methods used in the different investigation phases and those methods which completed the research. The different methods employed are presented in a chronological sequence to aid clarity and include the use of a research journal throughout both phases one and two of the research.
The use of data collected from forum postings found via the internet is the first method outlined in the sequence, followed by a description of the methods used to gain data from online prospectuses relating to teacher training provision within higher education.
The next step in the methodology is the completion of a critical review of resources available to support the implementation of numeracy for trainee teachers. The sequence of methods used culminates in a case study of the delivery of a functional skill mathematics staff development program for teachers in a further education college, with data collected through participant observation, focus groups and a staff e-bulletin board. All the data collected and analysed conforms to the same typology, comments, text, discussion and conversation. This allows for all the findings to be presented together from each research method used and allows for the data to be cross analysed as like is being compared with like data.
An examination of relevant ethical principles for this research is presented and a chapter summary is provided.