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1. Relational Understanding versus Instrumental Understanding (Module 1 - March 2016)

1.1. Relational Understanding - potentially a longer teaching/learning process BUT allows learners to understand the purpose and relationship of skills to then provide stronger foundations to continue to refine, interpret and apply to future problems

1.2. Instrumental Understanding - teaching and learning process of applying formulae to mathematics tasks as separate processes; potentially a quicker short term teaching/learning process; can allow learners to feel skill is mastered quickly but has potential to not provide sufficient understanding of the skill to refine, interpret or apply to future problems

1.3. Philosophy Statement #1: For children to learn maths effectively, teachers must ensure that they are providing a wide variety of mathematical experiences and interactions. These should include the opportunity to develop Relational Understanding versus only Instrumental Understanding. Students should have the opportunity to make connections with their building mathematics knowledge and skills and learning should be scaffolded to support this.   Students benefit from the incorporation of a mixture of teaching methods, including the use of concrete and visual resources, as well as the opportunity to engage in meaningful dialogue about the learning they are discovering.

2. Reflective Discourses (Module 1 - March 2016)

2.1. Reflective discourse is necessary for each learner to engage in to make connections between new learning and existing knowledge. Just listening to others discuss, however, doesn't appear enough. The benefits appear to be gained by learners are participants rather than bystanders

2.2. Philosophy Statement #2: Add: Teachers must ensure that ample opportunities are provided for learners to engage in reflective discourse, but must be skilled to manage these interactions so that learning is enhanced rather than confused. Teachers need to be congisant of who is engaged and who is not as bystanders' benefits are significantly reduced.

3. Open-ended Questions and Problem Solving (Module 2 - April 2016)

3.1. Open-ended Questions: are tasks where there is more than one method of finding a correct answer, and sometimes there may be more than one correct answer. Open tasks also allow for children of varying abilities to participate in at least part of the same task, with those needing support able to get started and those capable of extension work having an ability to investigate the task further. Good questions require:  1. more than recall 2. educative component 3. open

3.2. Problem Solving: the question is missing a piece of information that is needed for it to become a traditional ‘maths’ question; often presented as a word problem, although it don’t have to be.

3.3. Advantages: Encourages Mathematical Thinking; focusses on the processes used to consider, plan, trial possible solutions; allows learners to develop flexibility in their thinking (a key atribute to becoming a successful maths learner)

3.4. Philosophy Statement #3 Add:  Through the use of challenging yet achievable open-ended questions and problem solving, learners are provided with opportunities to develop: 1. Conceptual Understanding 2. Procedural Flluency 3. Strategic Competence 4. Adaptive Reasoning 5. Productive Disposition. Learners can see connections between their learning; can talk deeply about these connections; defend and explain their processes and answers with justification and proof; critique others; reflect on their own thinking and learning.

4. Inquiry-based Learning (Module 3 - April 2016)

4.1. Inquiry-based learning works around student-centred investigations. "When the content of learning is both relevant and requiring active engagement, student learning is maximised (Murdoch. n.d., n.p.). Teachers act as facilitators or co-learners alongside students, learning is student-driven adding to ownership and relevance. The class environment is supportive and inclusive where students can work independently and collaboratively to investigate, test, hypothesise, reflect and share processes, strategies, mistakes and success.

4.2. Philosophy Statement #5 Add: The mathematics classroom should provide inquiry-based learning opportunities whereby investigations are student-centred. Educators must remember that when we "connect with students’ lives we activate prior knowledge and motivate the desire to think beyond the known” (Murdoch, 2010, p. 3).

5. Effective Assessment (Module 4 - May 2016)

5.1. Assessment in the mathematics classroom should take on many forms. It is crucial that teachers are aware of WHOM they are assessing for, WHAT they are assessing, and WHY they are assessing. It is also equally important that teachers assess and evaluate their own performance as a teacher and reflect upon their pedagogical practice in order to improve as well.

5.2. Diagnostic assessment is useful for teachers to determine a student's current level of understanding and to ensure that lessons and/or units of work are designed to meet all levels of ability. Diagnostic assessment is particularly useful for planning for differentiated activities.

5.3. Formative Assessment is a valid and potentially less intimidating means by which a teacher can provide feedback and observe the learning processes and language taking place whilst the learning is still occurring. Feedback, praise, suggestions, prompts, and/or challenges allow teachers to potentially enhance the learning experience whilst it is still occurring.

5.4. Summative assessment offers a summary of a student's skill, knowledge, and/or abilities at a particular point in time. Summative assessment is often used upon the completion of a unit of work and provides a graded outcome of a student's performance against a specific performance measure (e.g. rubric, correct/incorrect answers). Summative assessments can take the form of tests, exams, assignments, etc.

5.5. Philosophy Statement #6 Add: Assessment is an important component of any good mathematics classroom, however, care must be taken with how it is carried out. A diverse range of assessment strategies should be in place in order to accurately determine a student's learning progress. These will include diagnostic, formative, and summative assessment strategies, and can be complemented by peer and/or self assessments as well. Assessment should provide opportunities students to reflect on their strengths and weaknesses in order to further develop strategies and/or learning programs to continue to progress. Teachers must also be prepared to assess and evaluate themselves, their teaching practice, and their ability to meet their students' mathematical learning needs.

6. Initial Thoughts about Mathematics teaching, learning and assessment (February/March 2016)

6.1. Mathematics learning requires students to have good fundamental understanding of numeracy, mathematical equations (+-x/), quick recall of number facts (times tables) in order to apply these skills to questions and problems.

6.2. Students must be supported and encouraged by their teacher through the use of challenging yet suitable activities, along with the use of aids to help support their learning (manipulatives, visual, ICT, stories, role play)

6.3. An effective mathematics program is one that shows student improvement across these content areas. Teachers will use teaching time to engage in meaningful discussion and observation of student activities. Assessment will be done through series of formal and informal tests (e.g. weekly quiz, 'beat your own personal best' tests)

7. Mathematical Thinking (Module 2 - April 2016)

7.1. What is Mathematical Thinking? Mathematical proficiency defined as 1. conceptual understanding; 2. procedural fluency; 3. strategic competence; 4. adaptive reasoning; 5. productive disposition

7.2. How do I develop these skills and attributes? As the teacher, I must support the learning WITHOUT giving away all of the information or answers! The teacher is NOT the keeper of all knowledge! Students must be developing skills and strategies to be able to investigate and find the solutions themselves, AND to develop the perseverance and resilience to continue to try alternate solutions after experiencing setbacks.

7.3. Philosophy Statement #4 Add: Teachers must critically consider what they believe constitutes a mathematically proficient student in order to develop a mathematics program to accomplish this. Being mathematicaly proficient is much more than being able to recall facts quickly or follow a formula. Proficiency requires an ability to think flexibly and positively about non-standard problems and to select from the wide array of mathematics skills and knowledge to apply the most suitable to solve non-standard problems. It also means being able to consider more than one solution

8. Differentiated Instruction (Assessment Task 2 - May 2016)

8.1. Differentiated instruction is the most effective approach for adequately addressing mixed-ability levels within classrooms. It can mean differentiating the curriculum and instruction methods in order to meet student readiness, interest, and learning profiles. Tomlinson is a major advocate for differentiation and provides countless practical strategies for ways in which teachers can incorporate differentiation into their classrooms.

8.2. Alternatives to differentiation are streaming (ability groups), small group instruction, or 'teaching to the middle', all of which do not adequately address the often wide variety of ability levels within the one class.

8.2.1. Philosophy Statement #6 Add: Teachers are bound to face classrooms with a range of ability-levels among their student group. As such, each teacher needs to employ practical strategies in order to ensure they are providing an engaging and enriching mathematical learning experience for all of their students. Teachers can utilise differentiated instruction as a means to tailor learning to suit student readiness, interest, and/or preferences and can do so by differentiating content, instructional methods, and/or ways to demonstrate and present learning.