Archived material
Some aspects of the documents in this section no longer apply, however they have been archived for reference.
The Nature of the Learner versus the Nature of Instruction
by Dr G D Hotchkis
From the proceedings of a forum conducted by the Board of Studies NSW on 26 October 1995
Return to the Nature of the Learner Forum Table of Contents
An historical overview of research on effective teaching practices
Background to the Forum
The Board of Studies NSW has a responsibility to develop curricula
and support documents for pupils K-12 in the NSW education system.
In the course of deliberations at Board level over the content and
direction of Board documents, the author commented upon a tendency
for Board documents to emphasise a student-centred/discovery learning/constructivist
philosophy. Across the full range of Mathematics, a common statement
of the nature of mathematics learning is endorsed. As an example,
the Board's documents on 'The nature of mathematics learning' in
the Mathematics Years 9-10 Syllabus and Support Material (Draft)
(Board of Studies, 1995) were cited. In this document, emphasis
is placed upon:
students learn best when motivated ... through interacting ... through
investigating ... through language ... in the context of intellectual,
physical and social growth. (See Attachment
1, below)
In this document, strong emphasis is placed upon concepts such as stages of development, everyday life experiences, student curiosity, cooperative learning, the discovery and creation of patterns, problem solving, language development, individual differences, formally and informally gained knowledge, multiple paths to understanding, and movement from the concrete to the abstract.
The present author objected to this single-minded approach and suggested that a substantial research base existed, which was supported by alternative learning propositions (Catania, 1992; Mazur, 1994; Mercer & Snell, 1977). Some members of the Board indicated that if the author's criticisms were substantiated that several other of the Board's curricula and their supporting documents may be likewise affected. The Board considered this issue to be of sufficient importance to warrant the conduct of this forum with the three stated aims:
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to explore the research on how students learn and effective pedagogy;
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to examine some of the significant issues in the area of effective teaching methodology and classroom best practice within the context of curriculum development; and
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to assist the Board to develop principles and guidelines for student learning based on current research.
It is argued here that the Board must consider the variety of research-based approaches available rather than continue to espouse a single philosophy as at present. The views expressed in the remainder of this paper as to the nature of the learner are:
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All students conform to the same laws of human learning.
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At any point in time, each student has the prerequisites for some new learning to occur.
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Students with more substantive learning bases may progress faster, with less direction and with greater cognitive leaps than will a less well prepared peer
Time does not permit an elaboration of the massive theoretical and educational research literature which impinges on both human learning and school-based education. I would, however, make these observations. The reality of a learning situation must focus upon at least these four components: the stimulus materials, their sequencing, the response and the consequences of making that response. However, the debate in education and learning theory is frequently not concerned with these events but with the way in which these are articulated, the lexicon used, and the concepts that are utilised by the various theorists and educators. Indeed, our problem is that if we could assemble the most eminent learning/educational theorists we would get different accounts of the same pristine student behaviour in a learning situation. This foray into words frequently renders the fundamental learning situation relatively insignificant, while greater focus and debate are brought to bear upon the cognitions of the theorist rather than those of the student.
Often, this leads us in education to the establishment of the 'educational guru' and those who become followers transgress one step further from reality in debate and discourse about what the 'guru' says rather than what the pupil does. This issue, which is still current today, was raised as long ago as 1971 (Wynne, 1971). An example of this is evident today in the concern that many teachers and academics have to identify particular student learning styles. I must admit that I have long had the wish as a teacher, and academic, that research-based data would enable us to clearly categorise students in such a way that each category could be readily provided with its own particular curricula and teaching methods. Unfortunately, research-based data does not support the notion of Aptitude-Treatment Interaction (Hammill & Larsen, 1974). This educational theory tends to re-emerge over time, and so we find Arter and Jenkins (1979) review the literature to that time and caution the 'educational gurus' of the time that there was still no empirical foundation for this position. No doubt, because of its intrinsic appeal, logic and common sense, teachers will continue to be encouraged to embrace the 'learning styles' movement. To the best of my knowledge, the conclusion reached in 1974 by Ysseldyke and Salvia is still valid:
To engage in experimentation is science, to bypass experimentation and to implement unvalidated ability training programs in educational settings is both unscientific and premature. (p 184)
An historical overview of research on effective teaching practices
Since the 1970s, there has been a consistent emphasis in teacher training on the cognitive theories of learning originally proposed by Bruner (1960) and Piaget (Piaget & Inhelder, 1969). Initially, the focus was upon discovery learning and, more recently, the constructivist position has emerged. It is interesting to note that Bruner's major contribution, The Process of Education (1960), which was an attempt to dramatically upgrade the American education system as a consequence of the Russian success in the launching of Sputnik, apparently failed to revitalise American education.
Indeed, Bruner in 1971 published a document in which he attempts to outline the reasons why this initial venture failed (1971). This document is provided as an appendix to this paper (see Attachment 2). I have significant reservations as to whether Bruner has, in fact, identified the real causes of student failure in his attempt to implement The Process of Education into the American school system. Furthermore, research (Bryant, 1974; Mpiangu & Gentile, 1975) has, in the past, cast significant doubt upon Piaget's theories. My point in making reference to these early studies (and there are many others in similar vein in more recent times) is that the academic community has been permitted to frequently engage in a biased, selective process whereby any research base, irrespective of its quality, is avidly taken as endorsement and contrary research is conveniently ignored. This is how 'educational gurus' are made! The point is, therefore, that the Board must be wary of appearing to sponsor views of the learner that are equivocal and must therefore broaden its scope to take into consideration the many other views of learning that exist and for which there are substantial research bases and strong academic advocacy.
The second aim of this forum is to explore the research on teaching methodology and classroom best practice. By way of introduction, let me refer you to the now somewhat dated Handbook of Research on Teaching (Wittrock, 1986), a volume of 1000 pages that contains only one paragraph devoted to discovery learning. At best it is argued that discovery learning procedures may be effective for high ability learners (p 620). In this volume, several chapters have been devoted to alternative approaches to instruction, including the following general model of effective instruction, which still has currency in some teacher training programs (Rosenshine & Stevens, 1986, p 379) -- see Attachment 3 which provides more detail of the following general points.
A general model of effective instruction:
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Review, check previous day's work (and reteach, if necessary).
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Present new content/skills.
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Guided student practice (and check for understanding).
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Feedback and correctives (and reteach, if necessary).
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Independent student practice.
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Weekly and monthly reviews
This general model of effective instruction has been produced by educational researchers who have their basic foundations in the 'Classroom Interaction' approach to the articulation of critical teacher behaviours in classroom settings.
The original focus was on the verbal interactions in teaching sessions with particular emphasis on the role of various types of teacher questioning on pupil responding and subsequent retention of learning. This work was largely based upon an analysis of transcripts of audiotaped lessons (Dunkin & Biddle, 1974). Focus has shifted in recent times. Rosenshine and Stevens have conducted a detailed analysis of a large body of educational studies across a broad spectrum of theoretical and research frameworks and across a variety of content/curriculum areas (Rosenshine & Stevens, 1986). From this analysis, they conclude that the above principles, when present in the teaching learning process, consistently and reliably promote superior levels of student performance. Since this 1986 report, the particular model has been refined and more precisely articulated (Algozzine & Ysseldyke, 1992; Christensen, Ysseldyke, & Thurlow, 1989; Pressley & McCormick, 1995; Westwood, 1995). The above findings focus upon the manner in which teachers should present their lessons.
Two further areas of educational research and thinking must also be taken into consideration. The first of these is focused upon 'instructional design' and looks in detail at the way in which the curriculum, the syllabus and individual lessons should be planned and prepared (Kameenui & Simmons, 1990; Smith & Ragan, 1993). A considerable tradition of educational thinking has emerged in the last twenty years in this regard including the work of Romiszowski, Gagne and Briggs, and Smith and Ragan (Gagne, Briggs & Wager, 1988; Romiszowski, 1981; Romiszowski, 1984; Smith & Ragan, 1993). Work in this area has focused upon many areas of the teaching-learning situation, for instance as can be seen in Attachment 4, Smith and Ragan devote chapters to the analysis of the learners, the learning task, the context of learning and instructional strategies. They also focus upon the assessment of learner performance and the evaluation of instructional design. These authors make a substantial claim that different instructional design is required for different situations, some of which may in fact occur within a single lesson. They argue that different strategies are needed by both the teacher and the student for successful learning in each of the following categories of lesson content: declarative knowledge lessons, concept lessons, rule lessons, problem-solving lessons, psychomotor skill lessons. They also focus upon strategies for attitude change, motivation and interest.
Secondly, considerable attention has been paid to the complex task of teaching concepts; see Englemann and Carnine (Engelmann & Carnine, 1982), Tennyson (Klausmeier, 1990; Tennyson, 1986) and Howard (Howard, 1987). These authors each present a different paradigm for the teaching of concepts. Englemann and Carnine focus on the incorporation of both examples and non-examples into a concept teaching sequence, whilst Tennyson has developed a 'best example' theory and Howard attempts to translate the Piagetian model into a concept learning framework.
It is argued here, therefore, that any attempt at the establishment of guiding principles for school-based learning must acknowledge and employ the knowledge base that now exists in at least the three areas discussed above, ie effective classroom teaching practices, competent instructional design and forthright teaching of concepts. The factor that emerges constantly throughout these educational positions is that the role of the teacher is critical in the appropriate selection and sequencing (Wang, Haertel & Walberg, 1994), and mode of presentation of content, together with precise feedback to pupils (Brangert Drowns, Kulik, Kulik & Morgan, 1991). Professional teacher judgement, which must be data-based, is crucial in the monitoring of ongoing instructional programs (Guskey & Gates, 1986). This is obviously at odds with the prevailing view which emanates from Board documents that learning is a student-centred process in which the teacher need only fill the role of guide and mentor.
The argument presented in this paper so far is that, firstly, the Board of Studies at present endorses a student-centred approach which does not appear to appropriately recognise the variety of sound alternative views supported within the educational community. It would therefore appear to be imperative that, if the Board is to enter this arena, it has an obligation to acknowledge all of those alternatives and their various implications for teaching so that teachers can be made aware of the variety and encouraged to judiciously select stratagems appropriate to the specific instructional needs of their students. If the Board is not prepared to do this, it would be far better if it withdrew from its attempt to articulate advice for teachers. This alternative would raise a significant problem for the Board in the preparation of syllabus support documents, as it seems self-evident that these cannot be created without considerable recourse to teaching/learning models and instructional design principles. As an aside to the main theme of this paper, it is suggested that the above arguments have equal currency in the area of teacher training. Indeed, it should be required of both pre-service and in-service teacher education programs that the range of effective instructional practices/design, as validated in sound research, should receive detailed coverage, together with a study of theories of human learning.
The above conclusions obviously endorse an emphasis upon establishing and maintaining teacher awareness of the ongoing development of theories and practice in education. It is argued that this approach is necessary for three main reasons. The first of these is that there is, and probably always will be, a significant number of alternative philosophies of education attempting to influence teacher behaviour. For instance, there are educators who hold to either a cognitivist point of view or a behavioural point of view with a fervour usually associated with religious matters (Kamil, 1995). It must be clearly understood that these philosophical differences will probably never be resolved through research. One reason for this is that an educator who holds firmly to one particular belief system will conveniently discard (if they ever read) material generated in what is seen as the opposing 'faith'. What must be done, however, is to hold academics and teacher trainers accountable for the even-handed presentation of current educational thinking to teachers in training. If this is not adhered to, then universities and teacher training institutions will continue to be 'seminaries for the induction of new true believers into a particular faith'.
Secondly, little solace for any particular educational faith can be found in the educational research literature. Over the last twenty years, educational research has tended to be published in a growing plethora of journals, each with its own particular affiliation to some faith or other. In addition, there have been strong moves to vary the nature of educational research from the predominance of a statistically analysed logical positivist research paradigm through to what is now known as post-modernism. Post-modernism tends to focus upon the nature of the intricate interrelationships that are established in the teaching learning situation in naturalistic settings.
Finally, the nature of educational research tends to be determined by a series of limiting factors (Mosenthal & Kamil, 1991). Firstly, research within a particular educational domain will reflect the varied beliefs of the various researchers as each adheres to the tenets of their 'chosen faith' in an attempt to produce confirmation that is deemed satisfactory by other members of the 'faith'. Furthermore, there is a recognised need for academics to 'publish or perish' and to do this in what always appears to be a hostile environment in respect of research time, funding and resources. Because of these limitations, organisations such as the Board of Studies and educational service delivery systems rarely have access to large-scale research programs that might provide directions for appropriate decisions in such large-scale systems. This inevitably means that organisations such as the Board of Studies have to make calculated, far-reaching and, indeed, expensive decisions that are frequently based upon articles of faith supported by the convenient extrapolation from the results of smaller scale research. It therefore is incumbent upon the Board of Studies and the systems of education to ensure that the particular biases of the decision-makers are not given free rein and that, as a matter of their accountability, deliberate opportunity remains for the acknowledgement and/or implementation of the various educational paradigms as they emerge.
Indeed, for organisations such as the Board of Studies, an additional impressed force of political policy and direction can, and frequently must, take precedence in guiding the Board's deliberations. For instance, there has been recent political direction towards the creation of curricula that are divided into 'prescribed stages' and are articulated in terms of student 'outcomes' within these stages. Much debate has occurred within (and outside) the Board over 'outcomes' resulting in a need to focus upon student competencies in curriculum development - see, for instance, McGaw, 1995:
Overall, recent Australian experience suggests that the best standard setting occurs when curriculum and assessment considerations are married ... A curriculum framework specified in terms of desired student outcomes provides a structure to which assessments can be keyed so that the results provide a curriculum-linked evaluation of students and teaching. (p 4)
This implies a clear shift from 'means' to 'ends' in curriculum statements and may provide for the future establishment of an appropriate foundation for teacher accountability to their students, the community and employers.
Both the Board of Studies and the educational delivery systems in New South Wales should encourage experimentation and diversity in approaches to instruction but only where full accountability, in terms of student outcomes across the full range of learners, is provided. Successful programs should be promoted by the systems to encourage the spread of accountable and effective instructional practices across schools.
Under the Education Reform Act, 1990, the Board should move to the registration and re-registration of schools only where there is sound evidence of the presence of proscribed student competencies at all stages of curricula.
The Board should revise its prescriptions so that schools are expected to ensure that basic literacy and numeracy skills are a prime focus and, where necessary, penetrate the more traditionally accredited courses across the KLAs. This may require a revision to present regulations in respect of the minimum required course hours to accommodate a mix of emphasis between literacy/numeracy skills building and other specific KLA content focus.
The arguments presented in this paper also have some strong implications for the professional development of teachers. If we accept that there is a diversity of educational philosophies presently available and the fact that, on average, teachers in NSW are more than twenty years removed from their initial teacher training, then a systematic and comprehensive teacher training program and in-service training scheme must be developed to ensure the provision of the best possible service provision to the school students of New South Wales. Indeed, the professional accreditation of teachers should be brought into line with other professions (eg lawyers, accountants, medical practitioners) whereby continued registration as a teacher requires evidence of regular completion of accredited courses, together with evidence of accountability of service delivery to the pupils of NSW.
If this is not accomplished, we may well see the day in the near future when teacher competence is challenged at law, as has happened recently with members of the professions cited above.
This paper has argued that educational emphasis must shift to the provision of appropriate instruction from the present focus on a detailed analysis of learner characteristics. Therefore greater emphasis should be placed on instructional methodologies that can lead to improved educational outcomes for students. To achieve improved outcomes, the responsibilty for designing the most effective learning program lies with the teacher, not the learner.
References
Algozzine, B, & Ysseldyke, J E, 1992, Strategies for Effective
Instruction, Sopris West, Longmont.
Arter, J A, & Jenkins, J R, 1979, 'Differential diagnosis -- Prescriptive
teaching: A critical appraisal', in Review of Educational Research,
49, pp 517-555.
Board of Studies, NSW, 1994, Mathematics 9-10 Syllabus and Support
Material (Draft), Board of Studies, NSW, Sydney.
Brangert-Drowns, R L, Kulik, C C, Kulik, J A, & Morgan, M, 1991,
'The instructional effect of feedback in test-like events', in Review
of Educational Research, 61, pp 213-238.
Bruner, J S, 1960, The Process of Education, Harvard University
Press, Cambridge.
Bruner, J S, 1971, 'The process of education revisited', in Phi
Delta Kappan, September, pp 18-21.
Bryant, P, 1974, Perception and understanding of young children,
Methuen, London.
Catania, A C, 1992, Learning, Prentice-Hall, Englewood Cliffs.
Christensen, S L, Ysseldyke, J E, & Thurlow, M L, 1989, 'Critical instructional factors for students with mild handicaps: An integrative review', in
Remedial and Special Education, 10, pp 21-31.
Dunkin, M J, & Biddle, B J, 1974, The Study of Teaching, Holt,
Rinehart & Winston, New York.
Engelmann, S, & Carnine, D, 1982, Theory of Instruction: Principles
and Applications, Irvington Publishers, Inc, New York.
Gagne, R M, Briggs, L J, & Wager, W W, 1988, Principles of Instructional
Design, Holt Rinehart and Winston, New York.
Guskey, T R, & Gates, S L, 1986, 'Synthesis of research on the
effects of mastery learning in elementary and secondary classrooms',
in Educational Leadership, 43, pp 73-80.
Hammill, D D, & Larsen, S C, 1974, 'The effectiveness of psycholinguistic
training' in Exceptional Children, 41, pp 5-14.
Howard, R W, 1987, Concepts and Schemata, Cassell Educational,
London.
Kameenui, E J, & Simmons, D C, 1990, Designing Instructional Strategies:
The Prevention of Academic Learning Problems, Merrill Publishing
Company, Columbus.
Kamil, M L, 1995, 'Some alternatives to paradigm wars in literacy
research', in Journal of Reading Behavior, 27, pp 243-261.
Klausmeier, H J, 1990, 'Conceptualizing', in Dimensions of Thinking
and Cognitive Instruction, B F Jones & L Idol (eds), Lawrence Erlbaum,
Hillsdale, pp 93-138.
Mazur, J E, 1994, Learning and Behavior, 3rd edn, Prentice-Hall,
Englewood Cliffs.
McGaw, B, 1995, 'Standards for Curriculum and Assessment', Newsletter
Supplement, ACER.
Mercer, C D, & Snell, M E, 1977, Learning Theory Research in Mental
Retardation: Implications for Teaching, Charles E. Merrill, Columbus.
Mosenthal, P B, & Kamil, M L, 1991, 'Epilogue: Understanding progress
in reading research', in Handbook of Reading Research: Volume II,
R Barr, M L Kamil, P B Mosenthal, & P D Pearson (eds), Longman,
White Plains, pp 1013-1046.
Mpiangu, B D, & Gentile, J R, 1975, 'Is conservation of number
a necessary condition for mathematical understanding?', in Journal
for Research in Mathematics Education, May, pp 179-192.
Piaget, J, & Inhelder, B, 1969, The Psychology of the Child, Routledge
and Kegan Paul, London.
Pressley, M, & McCormick, C B, 1995, Advanced Educational Psychology,
Harper Collins, New York.
Romiszowski, A J, 1981, Designing Instructional Systems, Kogan
Page Ltd, London.
Romiszowski, A J, 1984, Producing Instructional Systems, Kogan
Page Ltd, London.
Rosenshine, B, & Stevens, R, 1986, 'Teaching functions', in Handbook
of Research on Teaching, M C Wittrock (ed), Macmillan, New York,
pp 376-391.
Smith, P L, & Ragan, T J, 1993, Instructional Design, Merrill,
New York.
Tennyson, R D, 1986, 'An empirically based instructional design
theory for teaching concepts', in Review of Educational Research,
56, pp 40-71.
Wang, M C, Haertel, G D, & Walberg, H J, 1994, 'What helps students
learn?', in Educational Leadership, 51, pp 74-9.
Westwood, P, 1995, Effective teaching, paper presented at the Priorities,
Partnerships (and Plum Puddings): North West Region Inaugural Special
Education Conference, Armidale.
Wittrock, M C, 1986, Handbook of Research on Teaching, 3rd edn,
Macmillan, New York.
Wynne, L, 1971, 'Behavior analysis and behavior synthesis', in
The Psychological Record, 21, pp 171-9.
Ysseldyke, J E, & Salvia, J, 1974, 'Diagnostic-Prescriptive Teaching: Two models', in Exceptional Children, 41, pp 181-5.
Attachment 1
Extract from Mathematics 9-10 Syllabus and Support Material
(Draft 1994), © Board of Studies NSW, 1994.
The nature of mathematics learning
Students learn best when motivated
Mathematics learning is more effective when it is interesting,
enjoyable and challenging.
Implications
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Learning activities should provoke curiosity, should be appropriate to students' stages of development and should be related to everyday life experiences.
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The actual experiences of students should be used as the source of many learning activities.
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Students should be encouraged to become aware of the relevance of mathematics to their lives.
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Students should often experience success in mathematical activities. A positive attitude towards mathematics and towards oneself will be promoted by emphasising the students' achievements.
Students learn mathematics through interacting
Mathematics learning should involve interaction with the physical and social environment, leading to the abstraction of particular mathematics ideas encountered.
Implications
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The understanding of mathematical ideas is promoted by interaction with people and manipulation of materials in a wide variety of learning situations.
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Cooperative learning in small groups provides excellent opportunities for interaction.
Mathematics learning is promoted by the appropriate use of a variety of materials, equipment and personnel.
Implications
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Materials and equipment should be used in imaginative ways to explore, discover and develop mathematical ideas.
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The availability of technological equipment, such as calculators and computers, does not reduce the need for mathematical understanding or the need for competence.
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Some concepts and skills will need to receive greater emphasis with the introduction of calculators and computers, eg place value and decimal concepts, skills of approximation and estimation.
Students learn mathematics through investigating
Mathematics learning should involve the investigation of mathematical patterns, relationships, processes and problems.
Implications
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Students should be given opportunities to discover and create patterns, and to describe and record relationships contained in those patterns.
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Opportunities to use mathematical processes and to compose and solve problems should be provided in all strands of mathematics.
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Students learn mathematics through language
Mathematics learning is promoted by the appropriate use of language. Language, including symbols and diagrams, plays an important part in the formulation and expression of mathematical ideas and serves as a bridge between concrete and abstract representation.
Implications
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Mathematical activities should be regarded as opportunities for teachers and students to use and develop appropriate language.
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It is important that teachers be familiar with the language patterns appropriate to the different mathematical processes.
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Students should be encouraged to use oral and written language appropriate to their particular stage of development to gain meaning from their mathematical learning experiences.
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When developing teaching strategies and learning activities in mathematics, teachers should give consideration to the diverse cultural and linguistic background of students.
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Students learn mathematics as individuals but in the context of intellectual, physical and social growth.
Mathematics learning is promoted when individual differences of students are taken into account.
Implications
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Students vary in the way and the rate in which they learn mathematics.
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Learning experiences should be appropriate to the students' stage of development.
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Teachers should take into account the students' knowledge gained formally and informally outside the school, including the home.
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Recognition should be given that the whole of society has mathematical ability.
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Maximum participation and extension of all students, regardless of sex, is appropriate.
Mathematics learning should be appropriate to each student's current stage of development and should build upon previous experiences and achievement.
Implications
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To cater for the variety of developmental levels that may exist among a group of students, teachers should provide a flexible learning situation where there is a variety of opportunities for involvement.
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Whilst the student's readiness to proceed to new work will depend on previous knowledge and understanding, this does not mean that there is an absolute order in which mathematics learning should proceed for all students. There are many paths to understanding.
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Teachers should respond to emergent opportunities to capitalise on the student's interests and needs and vary the intended sequence of mathematical experiences.
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As each new mathematical concept is encountered, learning should proceed, where possible, from the concrete to the abstract. Concepts should be continually developed and consolidated through a wide variety of learning experiences.
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The development of understanding should, as a general principle, precede a requirement for both automatic recall of factual information and speed and accuracy in performing mathematical computations. Skills should be maintained through meaningful practice and enjoyable drill.
Assessment and Evaluation
In implementing any syllabus based on this Statement of Principles, schools must be careful to evaluate the program offered by the school and to assess the progress of individuals within that program.
Aims of mathematics education K-12
The aims of mathematics education will be achieved in different
levels in the K-12 range according to the stages of development
of the students at these levels. These aims are to develop in students
confidence and enjoyment in doing mathematical activities; knowledge,
skills and understandings in certain areas; and awareness of the
place of mathematics in solving problems of everyday life and in
contributing to the development of our society.
Confidence and enjoyment in doing mathematical activities
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confidence in their ability to do mathematics
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a positive attitude to mathematics as an interesting, enjoyable and challenging subject
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an appreciation of mathematics as a creative activity with aesthetic appeal
Knowledge, skills and understandings in certain specified areas
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thinking which is logical, flexible, fluent and original
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skills in computation and problem solving in all areas of mathematics
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appropriate language for the effective communication of mathematical ideas and experiences
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an ability to recognise mathematical patterns and relationships
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a variety of methods for calculations and problem solving
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an awareness of the basic structure of mathematics by an appreciation of the nature and interrelationship of the various strands of mathematics.
Awareness of the place of mathematics in solving problems of
everyday life and in contributing to the development of our society
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an ability to apply mathematical ideas, rules and procedures to particular situations and problems
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an awareness that the learning of mathematics includes the processes of enquiry, discovery and verification
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an awareness of the uses of mathematics both in and beyond the classroom
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an appreciation of mathematics as a relevant and useful activity
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an appreciation of appropriate uses of technology, including calculators and computers.