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Domain 1:
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Professional knowledgeTeaching mathematics is a ‘specialisation’ within the general field of teaching, While the uniqueness of the specialisation is apparent throughout these Standards, it is perhaps most sharply defined in the two components of this knowledge domain that identify the expected knowledge of mathematics and its learning — content and pedagogy. The third area of knowledge that is vital to the work of excellent teachers of mathematics is their knowledge of their students as this is a major component of their understanding of the ‘learning context’ and is fundamental to ensuring that their teaching maximises each student’s learning.
Excellent teachers of mathematics have a strong knowledge base to draw on in all aspects of their professional work, including their decision making, planning and interactions. Their knowledge base includes knowledge of students, how mathematics is learned, what affects students’ opportunities to learn mathematics and how the learning of mathematics can be enhanced. It also includes sound knowledge and appreciation of mathematics appropriate to the grade level and/or mathematics subjects they teach.
1.1 KNOWLEDGE... of studentsThe focus needs to be on knowing and understanding aspects of students’ backgrounds as they relate to their learning of mathematics. This complements the need to ‘know’ students in order to build productive interpersonal relationships as part of general good teaching practice.
Excellent teachers of mathematics have a thorough knowledge of the students they teach. This includes knowledge of students' social and cultural contextsThese aspects of students’ backgrounds shape their response to school and learning. Doing mathematics pervades all cultures and social groups. Its form, and how people view it, varies substantially between cultures and social groups. Hence, excellent teachers of mathematics see it as a priority and take deliberate steps to build a knowledge base and understanding of relevant social and cultural practices within the school community. This can relate to distinct cultural groups, or to overall socio-economic groupings. There are also particular factors for individual students. These relate to general aspects of personal background and can include particular issues such as parents’ attitudes to mathematics, or their aspirations for the student. The emphasis is on the mathematical, as part of knowledge of the ‘whole’ student. , the mathematics they know and useClearly, a thorough knowledge of the mathematics students have done and learnt at school is essential if teachers are to ‘build on where their students are’. When they first come to school, they can already have extensive experience with mathematics. As they progress through school, students do and learn mathematics outside of school as well — in the family and with friends; playing, working and in a wide range of other activities. This may or may not be uses of mathematics learnt at school. Inevitably, this engagement with mathematics (even if students and others do not recognize it as such) is clearly linked to the contexts, interests and needs of the students’ lives outside of school. In order to have a full picture of their students’ mathematics, excellent teachers of mathematics strive to know about their students’ school and ‘other’ mathematics. , their preferred ways of learningStudents can learn mathematics in different ways, and some approaches can be more effective for an individual than others. Some learn best when they work collaboratively, others when they work by themselves. Some look for extensive scaffolding and guidance; others prefer open-ended exploration. There can be preferences for learning through verbal means, with technology, using concrete materials and so on. Excellent teachers of mathematics build their knowledge of students’ preferences to inform their choice of practices. They seek to work with the student’s preferences to enhance learning, and to support them to learn to learn in ways that may be unfamiliar to them, thus expanding their repertoire of means they use to learn mathematics. , and how confident they feelLack of confidence is a major inhibitor to a student’s learning of mathematics. Students’ confidence with mathematics, and their attitudes to mathematics in general, can deteriorate quite quickly over time. Often this can be triggered by a relatively minor happening. Reversing this is most often a much more difficult task, and takes sustained effort and support from the teacher. Excellent teachers of mathematics use strategies to maintain positive attitudes and confidence, and to build confidence in those students who lack it -- monitoring and knowing about their students’ confidence with, and attitudes to, mathematics in an ongoing way enables them to do this most effectively. about learning mathematics.
1.2 KNOWLEDGE... of mathematicsTeachers’ knowledge of mathematics is the basis for their teaching. The reason for having Standards that deal specifically with the teaching of mathematics is the belief that teachers need to have a solid basis in the discipline to give them confidence with the mathematics they teach. They need to know it well, knowing it in a way that enables students to learn it as coherent and connected.
Excellent teachers of mathematics have a sound, coherent knowledgeIn order to help their students learn mathematics as a richly connected means for exploring and understanding the world, teachers themselves need to have a similar knowledge and understanding of mathematics. The ‘amount’ of mathematics will vary according to the ages of the students and their aspirations (see of the mathematics appropriate to the student level they teachIt is clear that the detail of teachers’ formal knowledge of mathematics will depend on the level(s) being taught. This does not mean knowing only the mathematics of ‘year 5’ or of ‘year 10’. Rather, excellent teachers of mathematics have a knowledge base in mathematics that contributes to their capacity to deal with all the demands that can and do emerge when teaching at a particular level. , and which is situated in their knowledge and understanding of the broader mathematics curriculumAs well as knowing the mathematics relevant to the particular class(es) they teach, teachers need to know, in crude terms, ‘where the maths is heading’ and, equally importantly, ‘where it has come from’. Continuity in students’ learning requires teachers to have this knowledge, and to use it in their planning and practice. Knowing what follows allows teachers to set students up for that further learning, as well a providing the option of extending students for whom it is appropriate. Knowing previous learning enables them to connect with where the students ‘are at’ and, particularly, to understand and work with those students who have gaps in their backgrounds. . They understand how mathematics is represented and communicatedA key outcome of school mathematics is that students can use and work with different representations of mathematics -- diagrams, graphs, symbols and so on -- and to make effective use of mathematics in their communication with others. These elements have emerged as explicit foci for teaching and learning in mathematics in more recent years, and excellent teachers have integrated these into their mathematical knowledge base. , and why mathematics is taught. They are confident and competent users of mathematics who understand connectionsThis is fundamental to the view of mathematics that modern curricula and teaching practices seek to develop in students and is therefore a core area of knowledge that underpins excellent teaching of mathematics. The connections exist at a range of levels... within aspects of mathematics -- e.g. the connections between fraction, decimal and percentage representations -- between aspects of mathematics -- e.g. the quantification and comparison of probabilities using these representations -- within the school curriculum -- e.g. the use of probabilities in genetics (Science) or for analysis of health risks (Health and Physical Education) -- or with the world beyond -- e.g. considering the connection between problem gambling and poverty or trying to initiate action on an environmental issue. within mathematics, between mathematics and other subject areas, and how mathematics is related to societyRelating learning of mathematics to the students’ world can be a powerful means for engaging students with that learning, although this may not always feasible and/or realistic. Teachers need knowledge of how mathematics is/can be used outside of the mathematics classroom in order to be able to incorporate this in their teaching when it is appropriate. Given that mathematics and its uses are continually evolving, excellent teachers strive to maintain and extend their knowledge base in this area .
1.3 KNOWLEDGE... of students' learning of mathematicsTeachers’ knowledge of mathematics is the basis for their teaching. The reason for having Standards that deal specifically with the teaching of mathematics is the belief that teachers need to have a solid basis in the discipline to give them confidence with the mathematics they teach. They need to know it well, knowing it in a way that enables students to learn it as coherent and connected.
Excellent teachers of mathematics have rich knowledge of how students learn mathematics. They have an understanding of current theoriesResearch in mathematics education is, for the most part, designed to explore and understand issues relating to the teaching and learning of mathematics. ‘Theories’ about students’ mathematical learning emerge from this work. These range from the grand (social constructivism, SOLO taxonomy, van Hiele) through to smaller scale suggestions about novel and improved ways of introducing or assessing a particular topic. Through their commitment to their own ongoing professional development, excellent teachers of mathematics maintain a currency with research that is relevant to them and their teaching context. They are able to discuss this with colleagues, and critically appraise new, research-based information for adaption or adoption in their work in the classroom, given the opportunities and constraints of the context.
relevant to the learning of mathematics. They have knowledge of the mathematical development of students including learning sequencesCurricula typically outline sequences for learning. Excellent teachers of mathematics have a detailed knowledge of these, and understanding of why they are arranged as they are. The sequences represent something of a ‘best guess’ in terms of students’ development, and these teachers are alert to individual students following different pathways. Based on their sound knowledge of the key learnings (ie the ‘bigger picture) they are flexible and able to work to ensure that these students still achieve these, albeit through a different route.
, appropriate representationsA key outcome of school mathematics is that students can use and work with different representations of mathematics -- diagrams, graphs, symbols and so on. Excellent teachers of mathematics are aware of the need to continually develop students’ exposure to and use of different representations, and of the ways they can use different representations to enhance students’ learning.
, modelsThrough their work at school, students need to develop a capacity to use their mathematics to investigate and understand their world. Excellent teachers of mathematics pay explicit attention to this aspect of students’ learning of mathematics, based on their knowledge of the mathematical models that can be introduced and developed as an integral part of the students’ learning of mathematics.
and languageCrucial to students learning to communicate effectively with and about mathematics is their development of appropriate knowledge of the language of mathematics. Excellent teachers of mathematics know (and use) mathematical language appropriate to the students, how they can develop the sophistication of students’ use of mathematical language and how relevant, language-rich approaches can enhance students’ learning.
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They are aware of a range of effective strategies and techniques for: teaching and learning mathematics; promoting enjoyment of learning and positive attitudes to mathematics; utilising information and communication technologiesAs in virtually all other areas of human endeavour, information and communication technologies continue to have a profound impact on school mathematics. The increasing use of ICTs to ‘do’ mathematics challenges what is important for students to learn. ICTs also have great potential for enhancing learning of school mathematics. Excellent teachers of mathematics are aware of developments in both these areas, and maintain the currency of their knowledge through ongoing professional development.
; encouraging and enabling parental involvementThe value of parental involvement in students’ learning has long been acknowledged and capitalized on in the early years of schooling. General recognition of the worth of involving parents has gradually extended beyond these years. Similarly, the emphasis on parental involvement in students’ mathematics learning is relatively novel in many schools. Excellent teachers of mathematics are aware of strategies that are appropriate in their teaching context.
; and for being an effective role model for students and the community in the ways they deal with mathematics.
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