The study sets out to investigate the way in which numeracy is taught and learned at a Further Education College in Staffordshire. Secondary research has been carried out by the author into the following areas; numeracy, standards and curriculum, assessment, teaching and learning numeracy, formative assessment and feedback, staff and staff training and good practice in numeracy.
“Numeracy is the knowledge and skills required to effectively manage and respond to the mathematical demands of diverse situations. Numerate behaviour is observed when people manage a situation or solve a problem in a real context; it involves responding to information about mathematical ideas that may be represented in a range of ways; it requires the activation of a range of enabling knowledge, factors, and processes” (Gal et al., 2003, p4, online).
Masters and Forster (2000, online) agree that numeracy ability depends on the adult learner’s ability to apply mathematical knowledge and skills in a variety of personal and social contexts. A useful summary of the different contexts of numeracy use can be seen in Appendix a. Appendix b also shows the contexts in which effective numerate behaviour is necessary.
Numeracy skills for everyday life can be found in tasks such as handling money, comparing prices when shopping, time management, making travel and holiday plans, playing games of chance, understanding sports scores, reading maps and using measurements when cooking or doing DIY, according to Gal et al. (2003, online). The contents of Appendix a and Appendix b support this claim.
Unfortunately, both numeracy and mathematics are widely disliked; some people may even be numerophobic and have an irrational and illogical fear of numbers. Pert (2009, online) suggests many people will actually confess to hating number work and do what they can to limit their engagement in this area.
The following sections detail information about some elements of the adult numeracy learning infrastructure (Figure 1.1) in the order of the learning cycle depicted in Figure 2.1, as a result of secondary research carried out by the author.
Figure 2.1: The learning cycle (TUC, 2004, p76)
g. Progression to other education, training or employment
a. Need identified
Initial and diagnostic assessments
Information, advice and guidance
Negotiated, realistic, relevant targets
Meaningful and relevant to reflect ILP
Monitored, recorded progress with feedback
Meaningful and relevant to reflect ILP
Monitored, recorded progress with feedback
-Standards and Curriculum (elements 1 and 2 of Figure 1.1)
The Adult Numeracy Core Curriculum has been based on the National Standards (DfES and BSA, 2001). The curriculum specifies the numeracy skills, knowledge and understanding that are required to meet the nationally agreed benchmarks at each of the five levels (Entry Level 1, Entry Level 2, Entry Level 3, Level 1 and Level 2).
-Assessment (elements 3, 4, 6 and 8 of Figure 1.1 and sections a, b, e and f of Figure 2.1)
Various evidence suggests that good assessment processes are important for effective teaching and achievement of learners, including that of Black and Wiliam (2003) and Clarke, Timperley and Hattie (2003).
In addition to this, The Department for Education and Skills (DfES) (2002a) acknowledge that different assessment processes are needed at the different stages of the learning journey. Beevers & Paterson (2002) report that the purpose of assessment includes informing students of their strengths and weaknesses in order to enable them to improve and become more confident. DfES (2002b) agree that helping learners to understand their learning strengths and weaknesses gives them confidence in their ability to improve their skills.
DfES (2002c) summarised the process of numeracy assessment as shown in Figure 2.2.
Figure 2.2: Summary of Numeracy Assessment (DfES, 2002c)
Screening For possible need
Initial Assessment For level of skills
Diagnostic Assessment For detailed learner profile to inform ILP
Formative For regular review of progress to inform learning programme
Summative For National Test or qualification, completion of ILP
Figure 2.2 shows that, typically, numeracy learners will have a screening, initial assessment and diagnostic assessment to identify their strengths and areas for development, thus enabling teachers to place them on a course at the appropriate level. This procedure is also recommended by Sewell (2004, online), McIntosh (2005, online) and CERI (2008, online). DfEE (2001, online) and Stott and Lillis (2007, online) identify that lack of these assessments is a major factor in contributing to the failure of adults to participate and progress, therefore consolidating the need.
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DfEE (2001, online) recommends that a senior member of staff should be defined as having overall responsibility for the initial and diagnostic assessments and for the production of the Individual Learning Plan (ILP). Pert (2009, online) substantiates this stating that organisational good practice includes having a core team with responsibility for diagnosing numeracy concerns, a robust system of screening, initial assessment and diagnostic assessment to determine learners’ skills levels in numeracy and a named tutor who is responsible for reviewing and monitoring a learner’s progress.
– Teaching and Learning Numeracy (elements 5 and 7 of Figure 1.1 and section d of Figure 2.1)
Pratt (1998) identified five main perspectives on teaching, summarised in Table 2.1.
Table2.1: The Five Main Perspectives on Teaching (Pratt, 1998)
Characteristics of teachers
Focus on content and determine what learners should learn and how they should learn it. Feedback is directed at learner mistakes
Value learners’ prior knowledge and direct learning to the development of reasoning and problem-solving skills
Provide learners with authentic tasks in real-life settings
Focus on the interpersonal elements of learning and listening and respond to learners’ emotional and intellectual needs
Relate ideas explicitly to the lives of the learners
Benseman, Lander and Sutton (2005, online) found that the majority of teachers include only one or perhaps two of the five perspectives during a teaching session but will incorporate all of them in their teaching over a period of time. In contrast, Derrick and Ecclestone (2006, online) suggest it is commonly (although wrongly) thought that mathematics is about “truths” and can only be taught through a “transmission” approach where learners are treated as “passive” recipients of information.
Masters and Forster (2000, p3, online) confirm the view that
“students are more likely to become successful, independent learners when they are encouraged to appreciate learning as a lifelong process of individual growth through the development of new skills, deeper understandings, and more positive attitudes and values.”
Ciancone (1988, p8, online) states that
“The numeracy tutor must establish an open relationship with the learner in order to be aware of the individual’s needs and at the same time must be familiar with the learning of mathematical concepts and the structure of the hierarchy of skills in order to determine an appropriate agenda of instruction.”
van Groenestun (2003, p233, online) agrees that
“The art of teaching is to create and facilitate learning environments in which learning is possible and to guide learners in their learning activities.”
Acknowledging this, Ginsburg, Manly and Schmitt (2006, online) state that numeracy tutors need to be familiar with the mathematics needed to manage the demands of family, workplace, community and further education.
Skemp (1971) distinguished between two different approaches to teaching and learning mathematics known as instrumental understanding and relational understanding. The former involves memorising and routinely applying procedures and the latter involves helping the learner to develop their own understanding by teaching for meaning. Van Groenestun (2003, online) also believes that it is not enough to merely consider which numeracy skills need to be taught but that it is vitally important to take into account the way in which they are taught and the way in which they are learned by adults.
If a particular level of numeracy is not completely understood, the learning of any other concepts could be hindered. Ciancone (1988, online) refers to a study carried out by Skemp which compared schematic (conceptual structure) and rote learning. In terms of percentage recall, more than double the number of candidates who were taught by schematic methods remembered what they had learned than those taught by rote. In addition, after four weeks only 15% of those taught by a schematic approach had forgotten their new knowledge compared with 75% of those taught by rote. It can therefore be seen that the way numeracy is delivered affects the way a skill is remembered which in turn affects the learning of other concepts based on that skill.
Several researchers cited in Coben et al. (2005, online) state that the capability to do mathematics is localised within the brain and that many of the difficulties that adults face when learning actually stem from the primate brain architecture. Other evidence also suggests that gains in numeracy may be generally more dependent on characteristics of learners and classes rather than on those of teachers and teaching styles used (Coben et al., 2006, online).
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Van Groenestun (2003, online) suggests the way adults learn in out-of-school situations differs from the way children learn in school, regardless of the teachers. Adults tend to process new information by “learning by doing” and thus need to be able to read, watch or listen to information, identify key points, communicate and discuss with others and reflect on possible implications of their new knowledge. From this it can be seen that the literacy skills of an adult can affect their acquisition of numeracy skills.
It is believed that learners who have a good awareness of how they learn are more effective at setting their own goals, developing a variety of learning strategies and evaluating their own progress (Centre for Educational Research and Innovation (CERI), 2008, online). Kirby and Sellers (2006, online) recommend that tutors should engage learners in “metacognitive awareness” so that they can investigate how they learn best as an individual. The development of numeracy ability involves the acquisition of knowledge and skills as well as their application in real situations (Dingwall, 2000, online). Thus, the more independent the learner, the more likely they are to be able to apply their knowledge and display numerate behaviour.
Brookes et al. (2001) concluded that evidence about the impact of general adult numeracy tuition was sparse and unreliable. Benseman, Lander and Sutton (2005, online) also found that there are very few empirical studies of actual numeracy practice. Interestingly, Benseman, Sutton and Lander (2005, online) were unable to identify any research during their review that discussed factors associated with progress in numeracy or assessment and its effect on learning outcomes. .
According to research carried out by Benseman, Lander and Sutton (2005, online) most teachers tend to teach the way that they themselves were taught because they were successful in those formal learning environments and because they do not have enough knowledge of adult learning theory and alternative models of delivery. They also found that numeracy diagnostics had taken place and numeracy teaching was clearly linked to the diagnostic results and that numeracy teachers frequently focus their teaching on specific errors being made by learners. Ironically, in the same year, Bhattarai and Newman found that adult numeracy programmes which actually respond to learners at their existing level of mathematics were extremely rare which links with the findings of DfEE (2001), four years previously, that only 15% of providers carry out an initial assessment to determine numeracy needs, showing that little progress has been made in this area.
Coben et al. (2006, online) discovered that the most common methods of class organisation were whole class teaching or individuals working on their own. Very little group work was found. Very few teachers used concrete objects, games, computers or calculators. Worksheets were used extensively but very few teachers used text books. The majority of teachers used a range of activities although alarmingly, only approximately 50% differentiated work and made connections to other areas of maths. Benseman, Lander and Sutton (2005, online), found that teachers talked for up to 60% of the observed session and there were few opportunities for learners to discuss their new skills. Many questions were asked by the teachers but these tended to be “closed” and were not used as scaffolds for further learning. Most teachers used a relatively small range of teaching methods.
In contrast to Benseman, Lander and Sutton (2005, online), Coben et al. (2006, online), discovered that numeracy teaching activities most commonly used by tutors include using everyday materials, problem-solving, worksheets, estimating activities, using concrete materials, co-operative problem solving, using calculators, demonstrations, critical numeracy activities, computers, small group work, puzzles and games and vocabulary building activities.
-Formative Assessment and Feedback (element 5 and 7 of Figure 1.1 and section e of Figure 2.1)
Black and Wiliam (1998) define assessment generally as activities which are undertaken by both teachers and learners in order to assess themselves and provide information which can then be used to modify teaching and learning. They suggest that assessment only becomes “formative” when the information gathered is actually used to change the teaching in order to meet the needs of the learner and take their learning forward.
Pert (2009, online) points out that even when groups have been set up according to their numeracy level, learners will still have a range of individual needs. It is therefore good practice to include small, regular assessments to ensure learners have understood a taught topic before progressing onto the next topic. If necessary, revision of learning goals documented on the ILP can then take place. Defined by CERI (2008, online), formative assessment actually refers to frequent, interactive assessments of student progress and understanding which are used to determine future learning needs and differentiated teaching.
Formative assessment is “assessment for learning” and is a key component in good teaching and learning practice. Information gained from formative assessment should be used to adapt teaching and learning activities and can be used to set targets for future learning. Many researchers agree that formative assessment should emphasise progress and achievement and increase learners’ motivation (Ciancone, 1988, online, ALI, 2002, Beevers and Peterson, 2002 and Stott and Lillis, 2007, online). DfES (2002b) confirm that assessment should inform the development and review of ILP’s and that assessing is an essential part of the planning process.
Black and Wiliam (1998) and Her Majesty’s Stationery Office (HMSO) (2005) suggest that formative assessment is an essential part of and indivisible from effective teaching and instruction at all levels. Similarly, teaching which includes formative assessment helps students to acquire “learning to learn” skills which should assist them with future learning throughout their lives, also helps to raise levels of student achievement and helps teachers to meet the diverse needs of learners (CERI, 2008, online). It emphasises the process of teaching and learning and involves learners in that process.
Derrick and Ecclestone (2006, online) state that formative assessment should monitor learner performance against set targets, give feedback on the next steps necessary for improvement, measure learner progress, enable learners to take charge of their own learning, encourage independence and promote self-reflection. Bimrose et al. (2007, online) suggest it should be carried out at the beginning of or during a learning programme in order to improve the quality of learning and the results used to review and modify a programme of learning.
According to Black (1999) and Briggs and Ellis (2008), formative assessment is the analysis of students’ learning to discover what they know, understand and can do and the appropriate response during teaching and learning to ensure it informs future planning and teaching. They also all state that formative assessment is the analysis of students’ learning and the appropriate response after the teaching.
In comparison, Beevers and Paterson (2002) view formative assessment as an assessment which is only undertaken during a course or module and not after the teaching has taken place. There is no doubt, however, that formative assessment helps the learner and teacher to review progress and that it is central to the learning process (HMSO, 2005, Briggs and Ellis, 2008 and CERI, 2008, online).
Key features of formative assessment include establishing a classroom culture which encourages interaction, monitoring student progress towards individual learning goals, use of a variety of teaching and learning methods in order to meet the diverse needs of learners, constructive and regular feedback on learners’ performance and the active involvement of students in the entire learning process (CERI, 2008, online).
Furthermore, Black and Wiliam (1998, p19) claim that
“â€¦there is a firm body of evidence that formative assessment is an essential feature of classroom work and that development of it can raise standards.”
Unfortunately, in a study of 15 Skills for Life tutors, Benseman, Lander and Sutton (2005, online), found very few of them used strategies or activities associated with high quality formative assessment. Questions used were closed rather than open and only required recall rather than higher thinking skills.
Extensive research carried out by Black and Wiliam (1998) showed that if formative assessment was improved, significant gains and improvements in learning were also achieved. They argue that the overall quality of teaching and learning can be improved by enhancing teachers’ ability to use formative assessment effectively. It is interesting to note that much of the research also showed that improving formative assessment helps lower attainers more than the higher attaining learners. This implies that effective formative assessment of numeracy should have a greater impact on the success of Skills for Life learners who tend to be lower attainers.
The most common forms of assessment used by numeracy teachers are formative and include teacher observation, portfolios and self-assessment (Benseman, Lander and Sutton, 2005, online).
CERI (2008, online) said feedback can be used to discover the extent of learner understanding and help teachers to pitch their teaching at the correct level so that learners can continue to improve their skills. By providing feedback, teachers are able to focus on what learners do and do not understand and are thus better able to adjust their teaching strategies to meet individual needs. Adapting the teaching and learning process from the results of formative assessment draws upon a teacher’s pedagogical and subject knowledge and also requires a great deal of flexibility and creativity on their part.
Feedback should focus on the issue, be specific and constructive and offer ideas of how the learners could improve. It should not be too lengthy and should never end negatively, according to Derrick and Ecclestone (2006, online). It is essential that feedback includes suggestions about ways to improve future learning performance (CERI, 2008, online).
Staff and Staff Training
Dingwall (2000, online) and Schmitt (2003, online) raise concerns about the maths skills and understanding of teachers delivering numeracy, let alone their numeracy teaching skills. A tutor’s experience of teaching numeracy has been found to positively affect learners’ progress in and attitude towards numeracy (Cara and de Coulon, 2008, online). Therefore, the Government began developing mandatory teaching qualifications for new teachers from 2002 (Cara and de Coulon, 2008, online and Simpson, 2008). The Further Education National Training Organisation (FENTO) developed a new range (Level 2 to Level 4) of teaching qualifications for numeracy tutors.
Newly qualified teachers are expected to have a generic teaching qualification; for example, a Certificate in Education (CertEd) or a Postgraduate Certificate in Education (PGCE), as well as a subject specialist qualification (Level 4). Those already teaching Skills for Life have been encouraged to gain these qualifications as well so that by 2010, all post-16 teachers will be fully qualified (McIntosh, 2005, online).
In 2004, the TUC recommended that only fully trained staff should be employed and it is pointed out by Benseman, Sutton and Lander (2005, online) that those staff should undertake regular Continuous Professional Development (CPD) to update their skills. Interestingly, in 2005/2006, only 29% of numeracy teachers were fully qualified and 18% of them did not have any teaching qualifications at all (Cara and de Coulon, 2008, online). In terms of experience, however, Coben et al. (2006, online) found that in a study of 34 teachers, they had, on average, been teaching maths or numeracy for 13 years and teaching adults for 8 years. In comparison with Cara and de Coulon (2008, online), Coben et al. (2006, online) identified that 79% had a qualification in maths and 88% had a teaching qualification but that only 18% had gained the new level 4 qualification in adult numeracy teaching.
There is much evidence to suggest that highly qualified teachers lead to higher achievement of learners but there is also evidence to counter this from research that suggested that over-qualified teachers are sometimes less effective at delivering numeracy to adults (Cara and de Coulon, 2008, online and Cara et al., 2008, online).
It is interesting to note that numeracy specific CPD requested by respondents at a discussion group about the state of numeracy teaching included practical, hands-on workshops focussed on effective diagnostic assessment (Wedgbury, 2005). Mackay et al. (2006, online) found that other priority areas for professional development included dealing with the needs of learners with several disadvantages, developing skills in the use of computers when delivering numeracy and understanding the backgrounds and needs of particular groups of learners. These researchers also found numeracy staff requested that professional development be provided by experts and appealed for the opportunity to share good teaching practice with peers, feeling that this would be an effective way to address any gaps in skills and knowledge.
Currently there is an over-reliance on voluntary or part-time teachers and this presents a barrier to the development of effective practice (CERI, 2008, online). However, having volunteers who have been selected carefully and well trained does enable learning to be further tailored to individual’s needs (McIntosh, 2005, online and CERI, 2008, online). McIntosh (2005, online) recommends that learning is delivered by full-time staff and corroborating research suggests that teaching is less effective overall when delivered by mostly part-time staff because this can lead to lack of consistency in teaching approaches and less participation in CPD (McIntosh, 2005, online and Benseman, Sutton and Lander, 2005, online).
So what is “Good Practice” in Numeracy?
“Most learners on adult numeracy courses have studied the subject of numeracy or mathematics in primary and secondary school. Many have also attended key skills and a Return to Study course, and helped their own children. They have had several different teachers and experienced various teaching/learning approaches. So why haven’t any of these done the trick?” (Kirby and Sellers, 2006 p4, online).
Appendix c provides a summary of best practice in teaching and learning numeracy compiled after extensive research by the author. Perhaps the importance of each practice is emphasised by the number of researchers quoting it as best practice. Many of the practices mentioned in Appendix c are discussed in the following text.
Adult numeracy programmes are thought to be effective if they are designed and delivered in accordance with the “best practices” of adult education, including linking learning to goals, building on previous knowledge and experience, making the learning relevant, focussing on learners and their situations and maximising flexibility (Dingwall, 2000, online). The TUC (2004) expand on this, summarising good teaching as shown in Appendix d.
In terms of length of study, research suggests that a minimum of 100 hours per year are necessary in order for learners to show some achievements (Benseman, Sutton & Lander, 2005, online). McIntosh (2005, online) found intensive courses over a long period of time have proved most successful for students up to Entry Level or Level 1. McIntosh (2005, online) and Benseman, Sutton and Lander (2005, online) agree that learners below Entry Level should have access to 330-450 hours of learning. Those already at Entry Level require 210-329 hours and those at Level 1 need 120-209 hours.
Ginsburg and Gal (1996, p16, online), support the model shown in Figure 2.3, stating that tutors should provide opportunities for adult learners to
“…comprehend a situation, decide what to do, and choose the right tool(s) from their “mathematical tool chest” that will enable them to reach a reasonable solution”
because this is what they will need to be able to do in their lives. Ginsburg and Gal (1996, online) also firmly believe that a significant proportion of sessions should be focussed on situational questions so that learners have the opportunity to analyse situations and determine which numeracy skills are required.
“Ultimately, instruction should aim to be more obviously useful (keeping students involved and coming) and more cognitively meaningful (so that students will be more likely to leave the classroom with skills that will be retained and applied)” (Ginsburg and Gal, 1996, p17, online).
Figure 2.3: Model for Numeracy Tuition (Ciancone, 1988, p11, online)
High quality resources should be used to support all numeracy work. According to Pert (2009, online) the main limiting factors to this are the institutional budget allocated to purchasing published resources and the time teachers have available to create innovative and inspiring materials.
It is vital that numeracy teachers recognise learners’ personal approaches to solving particular problems. Teachers should make efforts to understand what the learner is actually doing, how their method actually works, why they have chosen this particular method and the success rate of their chosen method.
“Only if the learner’s method is unsuccessful, laborious and has limited use should you consider imposing an alternative algorithm” (Pert, 2009, p19, online).
Ginsburg and Gal (1996, online) also recommend that tutors should ask learners why they did what they did and what alternative method they could have used. If the method is successful, a teacher would have to be extremely confident that a more traditional method of calculation would hold significant benefits for the learners before swapping, according to Pert (2009, online).
Derrick and Ecclestone (2006, online) found that “student-centred learning” featured regularly in adult education research but it was open to diverse interpretation by teachers. Apparently, numeracy teachers interpret this by thinking that examples should be as visual as possible, teaching aids should be used, examples should be related to the “real world” and skills should be consolidated through revision (Benseman, Lander and Sutton, 2005, online).
Using kinaesthetic materials can increase learner motivation, increase interaction and discussion and improve formative assessment due to teachers being able to observe where learning is taking place or diagnose any difficulties (Kirby and Sellers, 2006, online). They conclude that increased activity in numeracy sessions seems to improve learners’ understanding and makes sessions more fun and interactive, but state that using kinaesthetic and tactile approaches requires more time to be spent preparing resources. However, teachers have said that, regardless of the time spent preparing resources, they felt motivated to use kinaesthetic approaches because of the benefits to the learners. Kirby and Sellers (2006, online) found that increased activity in numeracy sessions seemed to improve learners’ understanding and made sessions more fun and interactive. Teachers should be encouraged to develop a repertoire of questioning techniques and share their ideas with colleagues. Double, leading, rhetorical and closed questions are not thought to be particularly useful when teaching numeracy because they discourage learners from reflecting on the problem or admitting that they do not understand the concept (Derrick and Ecclestone, 2006, online). Black et al. (2006) found a direct link between the types of questioning used by teachers to check learning and understanding and improved motivation of learners.
Traditionally, numeracy is taught to the whole class and then learners work through worksheets individually but this approach allows little collaborative learning with peers in order to share experiences. It is better practice to promote group learning so that learners can learn from each other and help each other to develop solutions to the numerical problems set (Pert, 2009, online). Foster and Beddie, 2005, p6, online agree that
“Human interaction is essential for effective teaching and learning.”
Ciancone (1988, p11-12, online) makes the following recommendations to numeracy tutors when teaching adults which concur with the views of authors already mentioned:
Each small step in teaching a skill should be consolidated before moving on to the next step
It is better to reinforce a learner’s method than to introduce a new method
Lessons and learning materials should be autonomous and self-contained due to the irregular attendance of some adult learners
Be aware of the learner’s reading ability and cultural background
Informal learning using games and puzzles should be introduced sensitively if the learner’s past learning experience was very academic
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