The literature seems to define a multiplicity of conceptions and interpretations as to what a skill may consist of and as such these are almost always context dependant and often located in a particular situation. For example, the physical skill of passing a ball, the skill of using a calculator to perform complex
calculations or the skill of playing the piano are all examples of skills and context dependant. Skills range on a continuum from those which are purely task-related such as someone’s skill at making a cake or riding a bicycle to those which are
90 related to some natural ability such as playing a musical instrument. It can be
argued that mathematical skills encompass the entire continuum.
Lowe and Muller (2010) define a skill in terms of what can be done or what can be tested: “a skill is both a modal notion (what somebody is able to do even while not doing it) and has an empirical side (skills can be tested)” (p. 265). Combining Lowe and Muller‘s definition with Dreyfus and Dreyfus (2004) five stages of skill acquisition model Novice, Advanced Beginner, Competent, Proficient and Expert (table 2.7) we are able to explain how skills might be developed.
Novice: Application of context-free rules through
information processing.
Advanced Beginner: Application of rules also based on perceived similarity with prior examples.
Competent: Application of a hierarchical procedure of
decision making (problem solving).
Proficient: Deep involvement, experiencing situations from a
perspective “holistic similarity recognition”; Dreyfus and Dreyfus (2004, p. 28); decisions grounded analytically.
Expert: No need for rules. They normally do not solve
problems or make decisions they do what normally works.
Table 2.7 Dreyfus – Dreyfus (2004) model of the Five Stages of Skill Acquisition
Interestingly whilst the Dreyfus-Dreyfus model was empirically grounded and employed to describe professional skills such as those acquired by nurses, it is strange to note that the model was not applied to mathematical skills especially given that Herbert and Stuart Dreyfus are a mathematician and philosopher respectively. Lowe and Muller (2010) claim that mathematical skills can be viewed in terms of those skills which a professional mathematician might use. They argue for mathematical skills in terms of knowledge “that” (eg that 4+3 = 7) and knowledge “how” (eg how to convert metres into feet). They theorise that
in the process of mathematical research, a lot of skills are involved in a successful research episode: a mathematician tackles a research
question, asks the right people who give her ideas helping on her way to the correct proofs, finally finds the proof, writes it up in a way that she can communicate it to the experts, gives a number of seminar talks on the proof, receives comments from peers in these talks, fixes a number of inaccuracies and uncertainties in the proof, types a journal paper, submits the paper, goes to international conferences reporting on the result,
receives a referee report with revisions, revises the paper, and finally publishes it (Lowe and Muller, 2010, p. 272).
91 Lowe and Muller’s 2010 view of mathematical skills is interesting and entirely
recognisable by a teacher of mathematics, but their view may not entirely relevant to the general population and this can create tensions for the teacher. However, employers often require employees to possess a different set of
mathematical skills from those above such as numeracy, reasoning and problem solving (DfBIS, 2016) which enable individuals to build constructive working relationships. For employers mathematical skills are embedded in a wide range of other softer skills, which might include
1. Communication skills 2. Decision making skills 3. Self-motivation skills 4. Leadership skills 5. Team-working skills 6. Thinking skills
7. Time management skills and ability to work under pressure (Skillsyouneed, 2017) All of the soft skills above have at least one common feature that of being dependent upon an action. Taking thinking skills as an example and exploring more deeply this soft skill encompasses two broad categories; cognitive and metacognition including Bloom's Taxonomy, DeBono's thinking tools and
Lipman's modes of thinking (Moseley et al., 2005), so soft skills have a tendency to be linked with underlying features and attributes.
So where do mathematical skills lie in the above spectrum? The argument from the national curriculum DfE(2013) and Skillsforyou (2017) is that the soft skills are developed alongside, and in conjunction with, those more recognisable mathematical skills. Returning to Lowe and Muller (2010) and their suggestion that mathematics skills be viewed “as professional skills” leads them to conclude that if professional skills can be tested in examinations then they “may well be that the aim of mathematics education is best characterised not as instilling mathematical knowledge, but as teaching mathematical skills” in order to pass examinations (p. 266). So rather than mathematical and soft skills being simultaneously developed the teaching of mathematical skills often takes precedence because of examinations.
92 This led me to think about examinations as assessments of skills and the often
heard phrase “teaching to the test”. However, research does show that merely teaching to the test creates surface or instrumental learning rather than deep, connected relational learning (Skemp, 1976) and therefore “may not adequately capture students’ mastery of the mathematics” (Jennings and Bearak, 2014, p. 387). So a mathematical education reliant on teaching skills as defined by Lowe and Muller (2010) may not be in the best interests of learners.
The UK mathematics national curriculum identifies the following skills that should be developed in learners:-
Problem solving skills Calculation skills
Algebraic manipulation skills Graphing skills
(DfE, 2013)
These mathematical skills do appear to be fairly self-contained or perhaps related to specific mathematical topics. Even this list is not universally accepted and the UK national curriculum documents do not give a definitive definition of what constitutes a mathematical skill. In the 1970s there was a general acceptance that basic mathematical skills (appendices 40 and 41), fall into a number of categories which are fundamental to pupils’ mathematical development these being:-
1. Problem Solving,
2. Applying Mathematics to Everyday Situations, 3. Alertness to the Reasonableness of Results, 4. Estimation and Approximation,
5. Appropriate Computational Skills, 6. Geometry,
7. Measurement,
8. Reading,
9. Interpreting and Constructing Tables, Charts, and Graphs and Using Mathematics to Predict,
10. Computer Literacy.
(NCSM , 1977, pp. 4-6) Some recent attempts to formulate a definition of mathematical skills in terms of features and attributes has been attempted, for example, the influential expert report “Maths Counts” by Smith (2004) has over 200 references to the phrase “mathematical skills” but no single helpful clear unambiguous definition. Other
93 expert reports have tried to come to a conclusion about mathematical skills by
trying to define skills in terms of the needs of society
At the one extreme are those young people for whom mathematics is about the most basic of life skills, like telling the time and counting the money in their pockets. At the other end of the spectrum there are those who will go on to create and work with some of the most sophisticated ideas known to mankind (Vorderman et al., 2011, p. 17).
Earlier, Cockcroft (1982) had offered a number of definitions of mathematical skills purely in terms of the needs of employers, for example, in the
manufacturing industry mathematical skills might include being able to add, subtract, multiply and sometimes divide whole numbers, perhaps with the help of a calculator (p. 35). As can be witnessed from this definition the context is all important and the implication is that the definition of a skill then becomes context dependant. But given society is developing at an ever increasing rate this type of skills definition is likely to become outdated and even obsolete. As Stigler and Hiebert (1999) report, defining skills in this manner leads to mathematics teaching consisting of large periods of time spend acquiring isolated
mathematical skills through the repeated practice of similar banks of questions (appendices 40 and 41).
Defining mathematical skills in terms of learning intentions (AfL, 2017) which include what pupils should know (the mathematical knowledge) and understand (the mathematical understanding) might be a better way forward. Thinking of a mathematical skill in terms of learning intentions might also correlate much more directly with the design and planning of lessons. As a means of exemplification this might be best explained through a standard problem in the secondary mathematics curriculum, that of knowing how to construct linear graph (for example y = 3x + 2).
For Lehtinen et al. (2017) mathematical skills are intimately related to the
development of conceptual and procedural knowledge (see section 2.2). The two types of knowledge are bound to the construction of deep learning (conceptual) and a sequence of steps or actions (procedural). Procedural knowledge is often developed and enacted by the repetition of routines and drills to be practised (Hiebert and LeFevre, 1986; Baroody, 2003). Lehtinen et al. (2017) argue that “mathematical skills in educational contexts can be characterized as drill-and-
94 practice that helps automatize basic skills, but often leads to inert routine skills
instead of adaptive and flexible” (p. 1). Many studies equate the acquisition and facility of mathematical skills with drill-and-practice type questions (Tournaki, 2003, Fuchs et al., 2010) and that these can be developed by learners through interacting with computer programs. However, according to Ericson (2016), the practising of questions has to be deliberate, planned and supported by
knowledgeable people such as teachers.
A mathematical skill could be presented to the learner in the form of a worksheet or a set of practice questions designed by the teacher which might be taken from a textbook. It is a closed piece of learning which is fairly tightly defined and controlled by the teacher and requires an exposition or explanation that might be followed by a teacher intervention. A skill has its basis in behavioural learning theory and described in teaching plans in terms of what pupils might be expected to do by the end of the learning episode.
A mathematical skill viewed in terms of a learning intention as advocated by AfL (2017) might, for example, focus on how to construct a graph. The mathematical skill in terms of a learning intention would then be a combination of knowledge (graphs) and understanding (how to construct). This wider view of a
mathematical skill in terms of a learning intention might enhance the
transferability of mathematics skills both within, and across, different employment sectors and academic disciplines (Britton, 2002). However, it is not a view that most teachers would recognise and so it was not adopted for this study.
95 Summarising the literature on skills:
The review of literature has therefore pointed me towards a definition for a skill as being
A Skill: is an aspect of pedagogy designed by a teacher where a new piece of
mathematical knowledge is presented and the learner is required to complete a number of prescribed questions to practise and perfect the new learning, an example would be the division of two fractions.