1. Mathematics NQT/RQT
Dave Hewitt

2. Overview Support Sharing good ideas Maths meet ups
Within school
Outside school: local support
Outside school: national support
Outside school: support from Loughborough
Sharing good ideas
Maths meet ups
Some influential theoretical perspectives on education

3. Within school
You will have support built-in during your NQT year – make good use of it;
Talk with your department colleagues, share lessons which went well and be prepared to talk about things which did not go well. Seek advice;
Does your department have part of its department meetings talking about teaching different topics or sharing ideas for teaching? If not, ask if this is possible;
Take time to explore the resources which the department has, such as worksheets, ideas for lessons, practical resources, e-resources;
Sometimes you can get support from someone who teaches a different subject, possibly another NQT. What is important is that you find someone who you feel comfortable talking about your lessons and sharing strategies about working with children.

4. Outside school: local support
Across the country there are a number of Maths Hubs which co-ordinate CPD sessions. Do you know your local Maths Hub? Are you receiving automatic s informing you of courses which are available?
Go to:

5. Maths Hubs Cornwall and West Devon Yorkshire Ridings
Bucks, Berks and Oxon
White Rose
Jurassic
Kent and Medway
Yorkshire and the Humber
Boolean
Surrey Plus
GLOW
South Yorkshire
Solent
Central
East Midlands West
Sussex
Salop and Herefordshire
East Midlands East
London North East
East Midlands South
London Central and NW
North Mids and Peaks
Enigma
North West One
London Central and West
Cambridge
North West Two
Matrix Essex and Herts
London South West
North West Three
London Thames
Great North
Norfolk and Suffolk
London South East
Archimedes NE

6. Outside school: branches of professional associations
There are local branches of each of the following associations:
Association of Teachers of Mathematics (ATM)
Mathematical Association (MA)
Institute of Mathematics and its Applications (IMA)

7. Outside school: branches of professional associations
ATM:

8. Outside school: branches of professional associations
MA:

9. Outside school: National support
You can get a lot of ideas and support from many different places.
Join one of the main professional associations so that you receive their journal(s) with lots of ideas and reflections upon teaching mathematics. It will keep you up to date with ideas and events.
ATM:
MA:
Both associations also have YouTube channels

10. Outside school: national support
Both ATM and MA have many publications which are excellent sources of ideas.
Go to their websites and consider purchasing some, or ask your department to buy some.

11. Outside school: national support
The next Association of Teachers of Mathematics conference will be from 15th – 18th April 2019.
Further details will be on their website later on in the year.

12. Outside school: national support
Register on the National Centre for Excellence in the Teaching of Mathematics (NCETM) website. It is free and has lots of professional development support and resources

13. Outside school: national support
There are many websites full of ideas, some better than others! Be selective!
I can recommend the following for ideas which can be different to what is usually found:
NRICH:
RISP (for A level):
MEDIAN:
Underground mathematics (for A level):
Dan Meyer:

14. Outside school: national support
Explore the National STEM Centre. This has an extensive archive of texts and resources from the past. There are many excellent resources there, full of ideas. Remember that many of the ideas around now are just re-workings of what has already been published in the past!
Register (it is free) to get access to the resources:

15. Outside school: national support
There is excellent support for Further Mathematics A level and also A level Mathematics from the Further Mathematics Support Programme (FMSP).
They have got a channel on YouTube where they have videos which go through solutions to a number of examination questions. (Go to YouTube and search for “FMSP Revision Videos”).

16. Outside school: national support
Another organisation is Mathematics in Education and Industry (MEI) who also manage the government funded Further Mathematics Support Programme (FMSP). MEI run a number of courses and also have an annual conference.
They also have a YouTube channel.

17. Outside school: support from Loughborough
NQT/RQT day
contact
Remember that we have our own collection of ideas and resources which are on our shared Google Drive! These are being added to all the time – you can add your good resources as well.

18. Sharing good ideas

19. Local maths meet-up
A group of us meet up once a half-term in a pub and each bring a maths problem for everyone to work on.
We also, of course, gossip about our schools/universities.
By the end of the evening we have also sorted out many of the wrongs currently in education.
What follows are some problems we worked on in our last two meetings.

20. Local maths meet-up Three of a digit to make 6 Use three 0s…
Use three 9s

21. Local maths meet-up Where am I?
A square has corners labelled anti-clockwise A, B, C and D
I am inside the square
I am 12m from A
I am 60m from B
I am 84m from C
How far am I from D?

22. Local maths meet-up
Factor links 5 3 2 4 7 6 8 10 9

23. Local maths meet-up A set of five whole numbers:
x is the mean of the five numbers
y is the mode of the five numbers
z is the median of the five numbers
How many solutions can you find for x, y and z? 10 3 x y z

24. Local maths meet-up A set of six whole numbers:
x is the mean of the six numbers
y is the mode of the six numbers
z is the median of the six numbers
r is the range of the six numbers
How many solutions can you find for x, y, z and r? 1 6 x y z r

25. A brief history of some influential theoretical perspectives on education

26. Skinner: Behaviourism Piaget: Cognitive Development
These are some of the names/theories which have been influential in shaping practice in schools:
Skinner: Behaviourism
Piaget: Cognitive Development
Bruner: Constructivism
Vygotsky: Social Constructivism
Gardner: Multiple Intelligences
Barbe, Milone, Swassing: Visual/Audio/Kinaesthetic Learning Styles
Skemp: Instrumental/Relational Understanding
Gattegno: Subordination of teaching to learning

27. Burrhus Skinner ( )
Skinner’s box – apparatus in which the behaviour of rats was modified by giving rewards
Operant conditioning – behaviour is determined by its consequences:
Positive reinforcement – something desirable received, leading to repetition of behaviour being more likely
Negative reinforcement – something undesirable removed, leading to repetition of behaviour being more likely
Positive Punishment – something undesirable received, leading to repetition of behaviour being less likely
Negative Punishment – something desirable removed, leading to repetition of behaviour being less likely
All behaviour is externally controlled and is a function of genetic and environmental conditions.

28. Education is what survives when what has been learnt has been forgotten
"New methods and new aims in teaching", in New Scientist, 22(392) (21 May 1964), pp
The consequences of an act affect the probability of its occurring again Accessed 19/02/2018
It is a mistake to suppose that the whole issue is how to free man [sic]. The issue is to improve the way in which he [sic] is controlled
Accessed 19/02/2018
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29. Skinner Influence and Issues
Basis of many of the reward/punishment models used in schools
Some misunderstandings of what is meant by “behaviour” – it is not just about managing pupils’ conduct in the classroom, but also has implications for intrinsic/extrinsic motivation for learning
The notion that behaviour can be explained without the need to consider internal mental states or consciousness has been challenged (e.g. Chomsky, 1971).

30. Jean Piaget (1896-1980) Schemas – ways of organising knowledge
Stages of cognitive development – children think in different ways to adults:
Sensorimotor (0-2);
Preoperational (2-7);
Concrete Operational (7-11);
Formal Operational (11- )
Adaptation Processes – enable transition between phases to produce equilibrium:
Assimilation – incorporate new information within existing schemas
Accommodation – change pre-existing schemas.

31. The principle goal of
education is to create men [sic] who are capable of doing new things, not simply of repeating what other generations have done – men [sic] who are creative, inventive and discoverers. The second goal of education is to form minds which can be critical, can verify, and not accept everything they are offered
1964 November, The Arithmetic Teacher, Volume 11, Number 7, Piaget rediscovered by Eleanor Duckworth, Start Page 496, Quote Page 499, Published by National Council of Teachers of Mathematics
Each time one prematurely teaches a child something he [sic] could have discovered himself, that child is kept from inventing it and consequently from understanding it completely
Piaget, J.(1970). Piaget's theory. In P. Mussen (Ed.),Carmichael's manual of child psychology (Vol. 1, pp. 703–772). New York: John Wiley & Sons. Page 715
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32. Piaget Influence and Issues
Focus on individual learners who construct their own understanding
Encouraged experience-based learning opportunities
Research methodology has been challenged – small sample of children of well-educated parents
Stages of Development may not be as clear cut as he suggested
Notion of “readiness” has also been challenged (see Bruner)
Ignores cultural background/environment as an influence on Stages of Development.

33. Jerome Bruner (1915 – 2016)
Learning is an active process in which learners construct new ideas based on their current and past knowledge and experiences
Development is a continuous process rather than a series of stages
Modes of representation
Enactive
Iconic
Symbolic
Scaffolding – reduce the degrees of freedom in some task, so the child can concentrate on a particular element (this was influenced by the work of Lev Vygotsky)
Spiral Curriculum – topics revisited with increasing levels of complexity.

34. http://www.azquotes.com/quote/600207 Accessed 19/02/2018
We begin with the hypothesis that any subject can be taught effectively in some intellectually honest from to any child at any stage of development
Bruner, J. S. (1960). The Process of education. Cambridge, Mass.: Harvard University Press. Page 33
Good teaching is forever being on the cutting edge of a child’s competence
Accessed 19/02/2018
Ideally, interest in the material to be learned is the best stimulus to learning, rather than such external goals as grades or later competitive advantage
Bruner, J. S. (1960). The Process of education. Cambridge, Mass.: Harvard University Press. Page 14
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35. Bruner Influence and Issues
Discovery Learning
students are more likely to remember concepts developed for themselves, as opposed to a transmission model
Promotion of autonomy of the learner.

36. Lev Vygotsky ( )
Social environments influence the learning process
Social learning precedes development
Learning takes place through interactions with peers, teachers and other experts
Zone of Proximal Development (ZPD) – the distance between the actual developmental level as determined by independent problem solving, and the potential development as determined through problem-solving under adult guidance, or in collaboration with more capable peers.

37. the only "good learning" is that which is in advance of development.
A teacher must adopt the role of facilitator not content provider Accessed 19/02/2018
the only "good learning" is that
which is in advance of development.
Vygotsky, L. S. (1978). Mind in Society. The Development of Higher Psychological
Processes. Cambridge, Massachusetts: Harvard University Press.
Human learning presupposes a specific social nature and a process by which children grow into the intellectual life of those around them
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds.). Cambridge, Massachusetts: Harvard University Press. p. 88
What children can do with the assistance of others might be in some sense even more indicative of their mental development than what they can do alone
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds.). Cambridge, Massachusetts: Harvard University Press. p. 85
Every function in the child’s cultural development
appears twice: first, on the social level, and later,
on the individual level; first, between people
(interpsychological) and then inside the child
(intrapsychological)
Vygotsky, L. (1978). Interaction between learning and development.
Readings on the development of children, 23(3),
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38. Vygotsky Influence and Issues
Provides theoretical underpinning for pupils learning collaboratively
Teachers seen as facilitators of learning
Knowledge construction within social context involving student-student and expert-student collaboration on real world problems or tasks
Focus on language as main tool to promote thinking

39. Howard Gardner ( )
Different and independent ways of processing information (multiple intelligences or MI)
Verbal-Linguistic
Logical-Mathematical
Musical-rhythmic
Visual-Spatial
Bodily-Kinaesthetic
Interpersonal
Intrapersonal
Naturalistic
Later proposed two additional intelligences
Existential
Moral

40. The biggest mistake of past centuries in teaching has been to treat all students as if they were variants of the same individual and thus to feel justified in teaching them all the same subjects the same way
Howard Gardner (in Siegel & Shaughnessy, 1994), quoted in: Cara F. Shores (2011), The Best of Corwin: Response to Intervention, p. 51
All human beings have all of the intelligences. But we differ for both genetic and experiential reasons, in our profile of intelligences at any moment Accessed 18/02/2018
Multiple Intelligences is not a statement about learning styles Accessed on 18/02/2018
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41. Gardner Influence and Issues
Encouraged moves away from a general concept of “intelligence”, by recognising pupils’ different areas of expertise
Criticised for lack of experimental evidence
Sometimes misguidedly used as support for some “Learning Styles” approaches

42. Walter Barbe (1926 -), Michael Milone (?-), Raymond Swassing (?-2016)
Individuals have different learning styles (VAK)
Visual
Auditory
Kinaesthetic
Teachers can identify an individual’s learning style and provide appropriate activities
Matching instructional style to learning styles will produce better learning
Later adaptation (Neil Fleming) with the addition of “Read/write learning” to form VARK.

43. Modality strength is not a fixed characteristic
The most frequent modality strengths are visual or mixed: each accounts for about 30% of the population … about 25% of the population are auditory, and the remaining 15% kinaesthetic
Barbe Walter B and Milone, Michael N. “What we know about modality strengths” in Educational Leadership, February Page 378
The relationship between modality strength and achievement is still unclear, as is the effect of the school and home environment on development of modality strengths
Barbe Walter B and Milone, Michael N. “What we know about modality strengths” in Educational Leadership, February Page 380
Modality strength is not a fixed characteristic
Barbe, Walter B and Milone, Michael N. “What we know about modality strengths” in Educational Leadership, February Page 378
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44. Barbe, Milone, Swassing Influence and Issues
Encourages teachers to consider different approaches to teaching
Adopted by many businesses, schools and teachers as a basis for planning
Not validated through independent research
Studies which identified students’ learning styles then randomly assigned them to groups which were taught using different approaches reported no differences between matched/mismatched pupils
Association for Psychological Science (APS) in United States in 2009, concluded “there is no adequate evidence base to justify incorporating learning styles assessments into general educational practice … limited educational resources would better be devoted to other educational practices which have a strong evidence base”

45. Richard Skemp (1919 – 1995)
Distinguishes between two types of understanding
Relational understanding – knowing both what to do and why
Instrumental understanding – sometimes described as “rules without reasons”
Advantages of instrumental mathematics (Devil’s advocate)
Within its own context, easier to understand
Rewards are more immediate and more apparent
Less knowledge involved, so right answers can be obtained more quickly
Advantages of relational mathematics
It is more adaptable to new tasks
It is easier to remember
Relational knowledge can be effective as a goal in itself
Relational schemas are organic in quality

46. Learning relational mathematics consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point
Skemp, Richard R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching,77, p25
There are two effectively different subject being taught under the same name 'Mathematics‘
Skemp, Richard R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching,77, p22
[In relational mathematics] there is more to learn – the connections as well as the separate rules – but the result, once learnt, is more lasting. So there is less re-learning to do, and long-term the time taken may well be less altogether.
Skemp,Richard R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching,77, p24
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47. Skemp Influence and Issues
Has encouraged teachers to change their focus away from “remembering rules” to developing an appreciation of the reasons behind them, and the rich tapestry of interwoven mathematical concepts

48. Caleb Gattegno ( )
First translator into English of Piaget’s work
Set up an Experimental School in the Bronx, New York
Created the Association for Teaching Aids in Mathematics (ATAM), which later become the Association of Teachers of Mathematics (ATM)
Teaching should adapt to the way in which students have learned throughout their lives by accessing ‘powers’ which children used as very young children
Learning focused on the education of awareness
Seeing mathematics as an attribute of what it means to be human; everyone operates mathematically every day
As well as focusing on mathematics, he developed new ways to teach language (‘The Silent Way’) and reading
Legendary teacher, being prepared to teach any class in any school

49. Stressing knowing rather than knowledge Only awareness is educable
Gattegno, C. (1982). Thirty years on. Mathematics Teaching 100, pp
Only awareness is educable
Gattegno, C. (1974). The common sense of teaching mathematics. New York: Educational Solutions. P. vii
Out should be all the static treatments, the atomic presentations of unconnected facts; in should be the dynamic and vast wholes, now easily available with film, video and computer graphics
Gattegno, C. (1984). Notes on adolescence. Mathematics Teaching 108, pp
No statement gains its mathematical
attributes unless we perceive in it the
presence of infinity
Gattegno, C. (1984). Infinity. Mathematics Teaching 107, pp
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50. Gattegno Influence and Issues
Brought the use of practical materials into common practice within mathematics classrooms, particularly the use of Cuisenaire rods
Recognition of the impressive learning all young children achieve before schooling and ways in which those powerful ways can be accessed within educational settings
Promoted use of film and movement within mathematics teaching
Some feel his methods of teaching require teachers to have particular sensitivity and awarenesses, and so a Gattegno approach is difficult to become universally adopted

51. Further Reading/Watching
Skinner
Chomsky on Skinner The Case against B F Skinner available at
Piaget
Bruner
Interview with 99 year old Bruner
Vygotsky
Gardner
VAK
Critique of VAK
Skemp
Gattegno
Video of Gattegno teaching and about the Bronx school

52. Keeping in touch…