transitioning to mixed-attainment mathematics · 2018-07-15 · – low threshold/ high ceiling –...
TRANSCRIPT
Mixed-attainment maths conference 4 Saturday 16th June 2018
Tom Francome – [email protected] - @TFrancome
Transitioning to
mixed-attainment
mathematics
“‘Obvious’ is the most dangerous word in
mathematics”
Eric Temple Bell
• Teach on PGDipEd at University of Birmingham
• Previously head of maths at KNGS
• Mixed attainment groups 2008
• Research into learner and teacher beliefs and practices.
• Current research around Practising Mathematics
https://www.atm.org.uk/Shop/Practising-Mathematics---Developing-the-Mathematician-as-Well-as-the-Mathematics-Book-and-Download/ACT107pk
• Our approach to teaching MA mathematics
• Notions of ability / background
• Why I wanted to change
• How the change happened
• Ways of working
Our approach to teaching MA mathematics
Becoming a mathematician • Thinking for yourself
• Asking questions
• Making conjectures
• Being organised and systematic
• Writing down what you notice
• Being sure
• Testing and explaining your ideas
More ways to be good at
maths means more people can be good
“Genius is 1% talent and 99% percent hard
work...” - Albert Einstein (maybe)
What’s different? • Work in depth on topics
• Pupils will cover 6 broad topics per year as well as having space and time for some interesting diversions along the way.
• Pupils get to work on their own maths ideas
What’s different? • High expectations
• Problem solving embedded
• Everyone does the same ‘common tasks’
• Lots of support and challenge
• Learning that being stuck is desirable and how to get unstuck
Put these rectangles in order of square-ness
How do warmsnug
double glazing come up
with their pricing? • Problem solving approach
• Everyone does the same ‘common tasks’
• Lots of support and challenge
• Learning that being stuck is desirable and how to get unstuck
Everyone can be a mathematician • Pupils are not grouped by
‘ability’ – KNGS Mathematics Teachers are part of the KCL
Expert Panel on Mixed Attainment teaching
• No one is limited by prior attainment
• Use low threshold/ high ceiling tasks
• Everyone has the opportunity to think at the highest levels
•Francome, T. J. (2016) 'Everyone can be a mathematician', Mathematics Teaching, 250, pp. 14-18.
Results improved every year
Best Practice in Grouping
Notions of ability / background
“Ability” and Types of grouping
• Mixed-ability – where classes contain a range of attainment levels
• Streaming – where pupils are taught in the same ‘ability’ groups for all lessons
• Setting – where pupils are allocated to ‘ability’ groups for each subject.
– (Hodgen, 2010).
‘Ability’ grouping depends on some underlying assumptions;
• Pupils have different ‘abilities’,
• These abilities can be known,
• Pupils will learn most effectively with others of similar ‘ability’.
• Self-concepts of low-attainers will be damaged by working with high-attainers
• Teaching setted groups is easier. – (e.g. Oakes, 1986)
‘Ability’ Grouping
Setting and streaming does not appear to be an effective strategy for raising attainment
… but there is evidence of consistently lower outcomes for middle and low-attaining students
– (e.g. Higgins et al. 2014)
‘Ability’ Grouping
“‘Setting’ is a widespread practice in the UK, despite little evidence of its efficacy and substantial evidence of its detrimental impact on those allocated to the lowest sets.
[…] setting is incompatible with social justice”
– Archer et al., 2018
‘Ability’ Grouping
Privileged students in top sets (white, middle-class)
‘Ability’ Grouping
‘Disadvantaged’ students in lower sets
(working class, Black and FSM)
– Archer et al., 2018
Privileged students see setting as
‘natural’
‘deserved’
‘Ability’ Grouping
‘Disadvantaged’ students see setting as
‘negative’ and
‘unfair’
– Archer et al., 2018
“If three pupils with the same scores on entrance to school were placed in different sets,
one in a top set, one in a middle set and one in a low set,
the performance of the pupil in a top set would be significantly higher and
that of the pupil in the bottom set significantly lower.”
– (Ireson et al., 2002)
‘Ability’ Grouping - sets
• Most pupils are set on the basis of a single test – (Ireson, Clark & Hallam, 2002)
• Behaviour
• Ethnicity
• SES
‘Ability’ Grouping - sets
‘Ability’ Grouping - sets
About half the students are in the ‘wrong’ set
Wiliam, D., 2001, Reliability, validity, and all that jazz. Education 3-13 , 29 (3) pp. 17-21.
Test reliability 0.9, predictive validity 0.7, 100 students, 50% in ‘correct’ set
Why is setting a bad idea? High sets
• High pressure environment.
• Too fast
• Teachers expect students to follow procedures, without detailed help or thinking time
• Teach as if ‘one ability’
Low sets
• Low expectations
• Low level work – boring, repetitive
• Less thinking, less discussion.
• Less-experienced/ non-specialist teachers
• Can’t move up
Boaler, J. Wiliam, D. & Brown, M. (2000),
Students' Experiences of Ability Grouping -
Disaffection, Polarisation and the Construction
of Failure, British Educational Research
Journal, 26(5), 631-648.
Ireson, J. Hallam, S. Hack, S. Clark, H. &
Plewis, I. (2002), Ability Grouping in English
Secondary Schools: Effects on Attainment in
English, Mathematics and Science,
Educational Research and Evaluation: An
International Journal on Theory and Practice,
8(3), 299-318.
See Boaler et al., (2000); Ireson et al., (2002)
• It puts pupils (and teachers) into a fixed-mindset which stops them from taking on challenging tasks and making mistakes they can learn from.
• It tells pupils:
• You’re good at maths…
so you don’t have to try.
• You’re not good at maths
so there’s no point in trying.
Why is setting a bad idea?
Francome, T. J. (2016) 'Everyone can be a mathematician', Mathematics Teaching, 250, pp. 14-18.
So why don’t we all teach in mixed groups?
Vicious circle of avoidance of mixed attainment grouping
Taylor et al., 2016
Why did I ‘buck the trend’?
‘How some schools buck the trend and implement mixed attainment teaching’ Hodgen, BSRLM, 2017 Creating a need to change (Hodgen & Johnson, 2004; Holland et al, 1998) Imagining a different culture (Wenger, 1998)
What kept me up at night…
What kept me up at night…
6 groups
4 schemes of work
What kept me up at night…
Teachers looked forward to some groups…
What kept me up at night…
… and dreaded others
What kept me up at night…
Teachers felt they must ‘teach to the test’ in order to improve results
What kept me up at night…
Pupils thought maths is about remembering the right method
What kept me up at night…
Pupils think maths is about avoiding mistakes
What kept me up at night…
Too many pupils didn’t enjoy mathematics
What kept me up at night…
People think some people are good at mathematics and some people are not
Dweck, C.S. (1999) Self-Theories: Their Role in Motivation, Personality and Development. Philadelphia: Taylor and Francis/Psychology Press.
What kept me up at night…
Self-fulfilling prophecy?
How the change happened…
What I wanted… • A scheme of work with space for pupils to
explore and work on their own mathematics whilst developing a deeper understanding of content.
• A five year curriculum where no pupils were limited by prior attainment.
• More pupils to enjoy mathematics, feel it was useful and feel they could succeed in the subject.
• (Results not to go down)
Baby steps or Giant leaps?
• ‘Gradual’ move to mixed groups
• “so wide they may as well all be mixed”
• Does the scheme of work matter?
• Changing teaching habits is hard
– “You can’t think your way into a new way of acting, you have to act your way into a new way of thinking” – Dylan William
Ingredients
• Talk about teaching
• Nurture ‘first followers’ – TED Derek Sivers
• Make the thing you want the easiest thing
• Collaborative planning of units
– Key ideas together, detail separately
– i.e not about what font on the PowerPoint!
Department meetings
• Teaching not admin
• Information giving by email
• Department bulletin
• See people
Department meetings
• Share…
• Resources
• Questions
• Challenges
• Assessment opportunities
• Interesting things to pursue or look out for
Not ‘one way’ to teach mixed groups…
Ways of working…
• Rich Starting Points – Low threshold/ high ceiling
– Students make conjectures and ask/answer their own questions
• Open-middled tasks – “Focus of the learning in the classroom is on the process, not the answer”
– Number talks etc.
• Using mistakes and misconceptions – Big Blue Box’ - Standards Unit - Improving Learning in Mathematics
– Classifying mathematical objects, Interpreting multiple representations, Evaluating mathematical statements, Creating problems, Analysing reasoning and solutions.
• ‘Economic’ whole class teaching – Working with awarenesses, Sufficient complexity, Carefully sequenced and broken down,
Practice through progress (Subordinate practice) – See Hewitt, 1996
– Assisting memory and Educating awareness
Circular Geoboards • Join three points on the circumference.
• What questions might a mathematician ask?
www.nRich.maths.org/2883
Circular Geoboards • Join three points on the circumference.
• What questions might a mathematician ask?
www.nRich.maths.org/2883
/8506
Circular Geoboards • Join three points on the circumference.
• What questions might a mathematician ask? – What are the angles?
– How many different triangles are there? - What are their angles?
– Can you sort them by type (isosceles, right-angled etc.)?
– What do the right triangles have in common?
– What happens if you allow the centre?
– Can you make any conjectures about the sums?
– ...What happens for more dots?
– ...What happens for more sides?
Empty Protractor
Francome, T. J. (2016). Empty Protractor. Mathematics Teaching, 253, 32-33.
Circular Geoboards
Circular Geoboards
Circular Geoboards
Circular Geoboards
Circular Geoboards
Circular Geoboards
Circular Geoboards
Circular Geoboards
Geoff Faux
https://www.atm.org.uk/shop/Primary-Education---View-All/Exploring-Geometry-with-a-9-Pin-Geoboard-Book-and-Download-
Trapeziums
• Find as many different ways of calculating the area of a trapezium as possible.
• Explain them clearly using diagrams or other representations.
• Everyone in the group should be able to explain the methods when asked.
Trapeziums
• Find as many different ways of calculating the area of a trapezium as possible.
• Explain them clearly using diagrams or other representations.
• Everyone in the group should be able to explain the methods when asked.
Always, Sometimes, Never
• Statements are often made by students (and teachers) in mathematics classrooms.
• Often, a good question to ask is “is that Always true, sometimes true or never true?”
Always, Sometimes, Never
• Numbers in the 5 times table end in a 5
• Two negatives make a positive
• a² + b² = c²
• sin2x = 2sinx
Task: • in pairs produce a poster which shows each statement classified
according to whether it is
always, sometimes, or never true and:
• if it is sometimes true, then write examples around the statement to show when it is true and when it is not true;
• if it is always true, then to give a variety of examples demonstrating that it is true, try to use things you know about adding, subtracting, multiplying or dividing fractions to show the two expressions are equivalent;
• if it is never true, then try to say how we can be sure of this.
• Take turns to place a card in one of the columns and justify your answer to your partner. Challenge them if the explanation has not been clear enough.
Algebraic Expressions
Task: • in pairs produce a poster which shows each statement classified
according to whether it is
always, sometimes, or never true and:
• if it is sometimes true, then write examples around the statement to show when it is true and when it is not true;
• if it is always true, then to give a variety of examples demonstrating that it is true, try to use things you know about adding, subtracting, multiplying or dividing fractions to show the two expressions are equivalent;
• if it is never true, then try to say how we can be sure of this.
• Take turns to place a card in one of the columns and justify your answer to your partner. Challenge them if the explanation has not been clear enough.
Grid Algebra
'We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development'. (Bruner, 1960, p. 33) https://www.mixedattainmentmaths.com/uploads/2/3/7/7/23776169/grid_algebra_t_francome_developing_fluency_with_formal_notation_mixedattainmentconference_170617.pdf
Bruner, J. S. (1960). The Process of education.
Cambridge, Mass.: Harvard University Press.
Benefits of MA teaching…
• Shared workload
• Collaborative planning
• Collegiate atmosphere
• Hard at the start but then tweaking
• Quality assurance
• Happier teachers
• Get to teach in line with their beliefs – (Francome and Hewitt, in press)
• Happier pupils
• Happier parents
Mixed-attainment as Trojan horse?
• Mixed attainment teaching gets teachers talking to each other
• “mixed-ability groupings may be a catalyst for improving pupils’ experiences of learning mathematics”
http://etheses.bham.ac.uk/5601/
Think about
• To what extent do other teachers in your school know the evidence or implications?
• Do teachers and pupils recognise that a measure of current attainment is not a measure of potential?
• How are you monitoring the impact of your grouping approaches on your pupils’ attitudes and experiences
• How do you ensure your grouping approaches work for all pupils including the lowest attainers?
• Could you transition to wider bands?
https://educationendowmentfoundation.org.uk/evidence-summaries/teaching-learning-toolkit/setting-or-streaming/
Mixedattainmentmaths.com
#mixedattainmentmaths
http://www.ucl.ac.uk/ioe/news-events/events-pub/jun-2018/what-if-kids-love-maths
Everyone can be a mathematician
• By doing the things that mathematicians do.
• No one should be limited by prior attainment
• Use low threshold/ high ceiling tasks
• Everyone deserves the opportunity to think at the highest levels
Further reading • Archer, L., Francis, B., Miller, S., Taylor, B., Tereschenko, A., Mazenod, A., ... & Travers, M. C. (2018). The symbolic violence of setting: A
Bourdieusian analysis of mixed methods data on secondary students’ views about setting. British Educational Research Journal. • Boaler, J., Wiliam, D. and Brown, M. (2000) 'Students' Experiences of Ability Grouping - disaffection, polarisation and the construction of failure',
British Educational Research Journal, 26(5), pp. 631-648. • Brown, L., Coles, A., Hewitt, D., Benson, I., Francome, L., Francome, T. J., Haworth, A., Mason, J., Messum, P., Orr, L. and Stansfield, J. (2016)
Mathematical Imagery. Derby: Association of Teachers of Mathematics • https://educationendowmentfoundation.org.uk/evidence-summaries/teaching-learning-toolkit/setting-or-streaming/ • Francis, B., Archer, L., Hodgen, J., Pepper, D., Taylor, B. and Travers, M.-C. (2015) 'Exploring the relative lack of impact of research on ‘ability
grouping’in England: a discourse analytic account', Cambridge Journal of Education, pp. 1-17. • Francis, B., Connolly, P., Archer, L., Hodgen, J., Mazenod, A., Pepper, D., Sloan, S., Taylor, B., Tereshchenko, A. and Travers, M.-C. (2017) 'Attainment
Grouping as self-fulfilling prophesy? A mixed methods exploration of self confidence and set level among Year 7 students', International Journal of Educational Research, 86(Supplement C), pp. 96-108.
• Francome, T. J. (2015) Experiences of teaching and learning mathematics in setted and mixed settings. Thesis, University of Birmingham. • Francome, T. J. (2016) 'Empty Protractor', Mathematics Teaching, 253, pp. 32-33. • Francome, T. J. (2016) 'Everyone can be a mathematician', Mathematics Teaching, 250, pp. 14-18. • Francome, T. J. (2017) 'Random Thoughts', Mathematics Teaching, 255, pp. 20-22. • Francome, T. J. and Hewitt, D. (2017) Practising mathematics: developing the mathematician as well as the mathematics. Derby: Association of
Teachers of Mathematics. • Higgins, S., Katsipataki, M., Villanueva-Aguilera, A., Coleman, R., Henderson, P., Major, L., Coe, R. and Mason, D. 2016. The Sutton Trust-Education
Endowment Foundation Teaching and Learning Toolkit. Durham: Education Endowment Foundation. • Hodgen, J. (2010) 'Setting, streaming and mixed-ability teaching', in Dillon, J. & Maguire, M. (eds.) Becoming a teacher 4th edition: Open University
Press, pp. 210-221. • Hodgen, J. (2017) '‘How some schools buck the trend and implement mixed attainment teaching’ ', British Society for Research into Learning
Mathematics, Birkbeck College (University of London), 4th March 2017.
• http://www.ucl.ac.uk/ioe/news-events/events-pub/jun-2018/what-if-kids-love-maths • Ireson, J., Clark, H. and Hallam, S. (2002) Constructing Ability Groups in the Secondary School: Issues in Practice, School Leadership and
Management: Formerly School Organisation, 22(2): 163-176 • Ireson, J., Hallam, S., Hack, S., Clark, H. and Plewis, I. (2002) 'Ability grouping in English secondary schools: effects on attainment in English,
mathematics and science', Educational Research and Evaluation, 8(3), pp. 299-318. • www.mixedattainmentmaths.com • Oakes, J. (1986) 'Keeping track, Part 1: The policy and practice of curriculum inequality', Phi Delta Kappan, 68(1), pp. 12-17. • Taylor, B., Francome, T. J. and Hodgen, J. (2017) 'Best practice in mixed attainment grouping', Mathematics Teaching, (258), pp. 35-39. • Taylor, B., Francis, B., Archer, L., Hodgen, J., Pepper, D., Tereshchenko, A. and Travers, M.-C. (2016) 'Factors deterring schools from mixed
attainment teaching practice', Pedagogy, Culture & Society, 25(3), pp. 1-16. • https://www.tes.com/news/school-news/breaking-views/listen-what-every-teacher-needs-know-about-setting-professor-becky • http://www.ucl.ac.uk/ioe/departments-centres/centres/groupingstudents • Wiliam, D., (2001) Reliability, validity, and all that jazz. Education 3-13 , 29 (3) pp. 17-21.