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1. A number of commentators (e.g. Bellert, 2009; Graham & Pegg, 2010; Hollingsworth, Lokan, & McCrae, 2003) have observed that during the 1990s and into the present decade mathematics curriculum documents in many countries, including Australia, have advocated mainly a student-centred, problem-based and investigative approach based on constructivist principles of learning. The constructivist view stems from an assumption that teachers cannot directly transmit conceptual understanding to learners because such understanding can only develop through an individual’s own first hand experiences and cognitive activity (Pelech & 6 P. Westwood Pieper, 2010; Piaget, 1965) PAGE 6 and 7

2. The question to be addressed was why these feelings of confusion arose in the Žfirst place. Primary school teachers teach mathematical concepts they would have encountered when they were young children. A reasonable assumption would be that primary school teachers who have problems with the early stages of mathematics acquired these difŽ culties during their own primary years. Teaching mathematics during this phase involves the use of real life, concrete, visual and human resources, each of which involves spoken language to establish understanding. This suggests that inappropriate or imprecise use of spoken language could play a part in the formation of imperfect knowledge and even misconceptions in mathematics. PAGE 46

3. How will we ensure progression within and through levels? As children and young people develop concepts within mathematics, these will need continual reinforcement and revisiting in order to maintain progression. Teachers can plan this development and progression through providing children and young people with more challenging contexts in which to use their skills. When the experience or outcome spans two levels within a line of development, this will be all the more important. PAGE 2

4. Principles of curriculum design There are seven broad principles that practitioners should take into consideration when planning children’s learning: Challenge and enjoyment Breadth Progression Depth Personalisation and choice Coherence Relevance.

5. Neuromyths

5.1. ...that children should be identified as either ‘left-brained’ or ‘right-brained’ learners, because individuals ‘prefer’ one type of processing3 . Teachers are told that the left brain dominates in the processing of language, logic, mathematical formulae, number, sequence, linearity, analysis and unrelated factual information. Meanwhile, the right brain is said to dominate in the processing of forms and patterns, spatial manipulation, rhythm, images and pictures, daydreaming, and relationships in learning. PAGE 2

5.2. Other courses for teachers advise that children’s learning styles should be identified as either visual, auditory or kinaesthetic,

6. Goswami, U. (2006) Neuroscience and education. from research to practice?, Nature Reviews Neuroscience, vol 7, no 5, pp. 406-413

6.1. Neuroscience in the classroom

6.1.1. Neuroscience in the classroom These neuromyths need to be eliminated. The dominance of these myths obscures the important strides being made by cognitive neuroscience in many areas relevant to education. For example, our understanding of the neural bases of the ‘3 Rs’ — reading, writing and arithmetic — is growing rapidly. So is our understanding of how to optimize the brain’s ability to benefit from teaching. Good instructional practice can be undermined by brain-based factors such as learning anxiety, attention deficits and poor recognition of social cues. All of these factors disrupt an individual’s capacity to learn, and also have an effect on other learners in the same classroom. PAGE 3

6.2. Children with developmental dyslexia also show significantly less activation in the usual left hemisphere sites. If targeted remediation is provided, usually through intensive tuition in phonological skills and in letter-sound conversion, activity in the left temporal and parietal areas appears to normalize. PAGE 2

6.3. Attention, emotion and social cognition. The short attention spans of some children pose continual problems for their teachers. Children with attention deficit/hyperactivity disorder (ADHD) are particularly challenging to educate, as they are inattentive and impulsive, cruising the classroom instead of focusing on their work. Of course, all young children experience some difficulties in sustaining attention and inhibiting impulses. Perhaps attentional training might benefit all preschoolers47, leading to educational advantages? A recent brain imaging study claimed that 5 days of attention training significantly improved performance on tests of intelligence in 4- and 6-year-old children48. The children were given training exercises to improve stimulus discrimination, anticipation and conflict resolution. PAGE 4

7. Week 2 - Active Learning Taking active learning into the primary school: a matter of new practices? Christine Stephena *, Jennifer Ellisb and Joan Martlew International Journal of Early Years Education Vol. 18, No. 4, December 2010, 315329

7.1. The guidance goes on to list spontaneous play, planned and purposeful play, investigating and exploring, events and real-life experiences and focused learning and teaching as all contributing to this process. Furthermore, it calls for a wide range of resources to be available, space for children to learn through ‘social, sensory, creative, constructive and dramatic activities’ and for learning opportunities to be appropriately paced and challenging in order to meet the needs of individuals. Teachers are asked to plan responsively and intervene with sensitivity PAGE 317

7.2. https://laurawattblogblog.files.wordpress.com/2018/09/stephen-ellis-and-martlew-2010-active-learning.pdf

7.3. Teachers are asked to plan responsively and intervene with sensitivity. However, there remain expectations of and targets for the acquisition of specific knowledge and skills (e.g. for reading and numeracy) within school and local authority accountability systems PAGE 317

7.4. Changing pedagogy involves not only changing practices but also thinking differently about the process of learning and the role of the learner and teacher. PAGE 328

8. Mindsets and Math/Science Achievement Carol S. Dweck, Stanford University 2008

8.1. You have a certain amount of intelligence, and you can’t really do much to change it (fixed mindset item). No matter who you are, you can significantly change your intelligence level (growth mindset item). PAGE 1

8.2. The Role of Parents and Educators For the last few decades many parents and educators have been more interested in making students feel good about themselves in math and science than in helping them achieve. Sometimes this may take the form of praising their intelligence or talent and sometimes this may take the form of relieving them of the responsibility of doing well, for example, by telling them they are not a “math person.” Both of these strategies can promote a fixed mindset. PAGE 8

8.3. Our research (Cimpian, Arce, Markman, & Dweck, 2007; Kamins & Dweck, 1999; Mueller & Dweck, 1998) has shown that giving students praise for their intelligence, as opposed to praise for process (such as effort or strategy) makes students think that their abilities are fixed, makes them avoid challenging tasks (so they can keep on looking intelligent), makes them lose confidence and motivation when the task becomes hard, impairs their performance on and after difficult problems, and leads them to lie about their scores afterwards. Process praise (such as praise for effort or strategy), in contrast, leads students to seek and thrive on challenges. In a recent study, Good, Rattan, & Dweck (2007 PAGE 8

8.4. As part of regular classroom teaching or in school-sponsored growth mindset workshops (such as those described in Dweck, 2007; see also Aronson et al., 2002; Blackwell et al., Dweck (2008) Mindsets and Math page 9 2007; Good et al., 2003), students can be taught that the brain is like a muscle that gets stronger and works better the more it is exercised. They can also be taught that every time they stretch themselves, work hard, and learn something new, their brain forms new connections and that, over time, they become smarter. In these workshops, they are also taught how to apply this message to their schoolwork. PAGE *

8.4.1. Thus, students must learn that passion, dedication, and self-improvement – and not simply innate talent – are the road to genius and contribution. As we have shown in our research (Good, Rattan, & Dweck, 2008), this message may be especially helpful for females, who may not believe that they fit the image of the born genius in math and science. PAGE 9

9. Numeracy - Week 4

10. The problem with problems DISCUSSION PAPER The problem with problems: Potential difficulties in implementing problem-based learning as the core method in primary school mathematics Peter Westwood* Education consultant and writer Australian Journal of Learning Difficulties Vol. 16, No. 1, May 2011, 5–18

11. POLICY - CFE Mathematics Principles and practice

11.1. We should regularly encourage children and young people to explore different options: ‘what would happen if...?’ is the fundamental question for teachers and learners to ask as mathematical thinking develops. PAGE 2

11.2. As they tackle problems in unfamiliar contexts, can they confidently identify which skills and concepts are relevant to the problem? Can they then apply their skills accurately and then evaluate their solutions? PAGE 3

11.3. Mathematics is important in our everyday life, allowing us to make sense of the world around us and to manage our lives. Using mathematics enables us to model real-life situations and make connections and informed predictions. It equips us with the skills we need to interpret and analyse information, simplify and solve problems, assess risk and make informed decisions. Numeracy and mathematics https://education.gov.scot/scottish-education-system/policy-for-scottish-education/policy-drivers/cfe-(building-from-the-statement-appendix-incl-btc1-5)/curriculum-areas/Numeracy%20and%20mathematics

11.4. Scotland has a maths problem. Too many of us are happy to label ourselves as “no good with numbers.” This attitude is deep-rooted and is holding our country back educationally and economically.https://beta.gov.scot/binaries/content/documents/govscot/publications/report/2016/09/transforming-scotland-maths-positive-nation-final-report-making-maths-count/documents/00505348-pdf/00505348-pdf/govscot:document/

12. Cambridge Journal of Education, Vol. 32, No. 1, 2002 Spoken Language and Mathematics ANDREA RAIKER De Montfort University

12.1. This document recognises the import- ance of language by emphasising ‘the correct use of mathematical vocabulary’ in the raising of standards. PAGE 45

12.2. However, the language used in a mathematics lesson is not conŽ ned to such words. The discourse of discipline, instruction, explanation, demonstration, questioning, discussion, other subject areas and social interaction is all present. It follows that teachers should be aware of the language they use and be able to alert the children when mathematical vocabulary is being taught. PAGE 45

12.3. The learning objectives of mathematics lessons are concerned with the acqui- sition of mathematical concepts by the children. The teaching and learning experiences organised by teachers, which can take days or even weeks to complete, are concentrated in single words or phrases, for example origin, cuboid, repeated addition, equivalent fractions. These words or phrases have precise meanings in mathematics, even though their meanings in non-math- ematical language may not be so precise. This can be demonstrated by consid- ering the word ‘origin’ as used in the mathematical context of the point where horizontal and vertical axes cross and as used in ‘origin of the species’.

12.4. The question to be addressed was why these feelings of confusion arose in the Žfirrst place. Primary school teachers teach mathematical concepts they would have encountered when they were young children. A reasonable assumption would be that primary school teachers who have problems with the early stages of mathematics acquired these difŽ culties during their own primary years. Teaching mathematics during this phase involves the use of real life, concrete, visual and human resources, each of which involves spoken language to establish understanding. This suggests that inappropriate or imprecise use of spoken language could play a part in the formation of imperfect knowledge and even misconceptions in mathematics.

12.5. There are parallels in other subjects, particularly in science, to suggest this could be the case. Sizmur & Ashby (1997) observed that ‘becoming a scientist was as much about learning and using a spoken language as it was about carrying out experiments’

12.6. It follows that teachers should be aware of the language they use and be able to alert the children when mathematical vocabulary is being taught. The identiŽcation of key vocabulary and focused teaching on meaning should be detailed in lesson plans. Understanding of this key vocabulary should be included in the assessment of achievement of learning objectives. PAGE 45 and 46