The Myth of Low Ability

Anne WatsonPGCE Oxford 2015

I wanna be a ...

• Doctor

• Solicitor

• Priest

• Maths teacher

• But I only want to work with well people who want to get fitter and stay well

• But I only want to work for honest people with good incomes, write their wills and do their conveyancing

• But I only want to work with believers who do not have any problems and will take care of the church and do fundraising

• ...

“I really love to teach bottom sets, because that’s where all the people go who learn in different ways from the teachers, and it is really exciting working with them.”

“I miss teaching the full range of attainment – that keeps me thinking.”

What does ability mean?

I have no idea!

What is the usual experience of PLAS?

• repetitive simplified work • step-by-step• arithmetic in imaginary ‘everyday’ contexts• unconnected methods have to be memorised• re-do, again and again, forgotten methods

“It’s boring; I can’t do this; I can do it but I don’t want to; I want to go home; she knows we can do this already.”

Denvir, Stolz and Brown (1982)• Seen as low attaining throughout

school• Across the curriculum• Poor reading skills• Poor in all aspects of maths• Poor language skills• Perceptual problems• Poor motor skills• Immature relationships• Shows no interest in maths• Shows no interest in school• Cannot relate to adults

• Preoccupied• Emotional problems• Social problems• Behaviour problems• High anxiety in school• High anxiety in maths• Sensible in one-to-one but not in front of

others• Physical limitations• High absence• Tired• Frequent changes of school or maths teacher

• ‘ … roughly half of the children who had been identified as having a learning problem in mathematics did not show any form of cognitive deficit …’ (Geary 1994 p.157)

Reasons for low prior attainment:• Disrupted schooling• Cultural differences• Social and emotional difficulties• No social skills for positive attention• Low expectations (home, family, particular group, school)• Lack of specialised teaching• Limited teaching methods• Learned helplessness• Reading and writing difficulties• Language differences • Physical and physiological problems: diagnosed or undiagnosed• Sleep deprivation or hunger/malnutrition• Cognitive problems which affect all their learning, such as short-term memory

deficiency; dyslexia• WHO?

Deep progress means that students:• learn more mathematics,• get better at learning mathematics,• feel better about themselves as

mathematics students

“It’s boring; I can’t do this; I can do it but I don’t want to; I want to go home; she knows we can do this already.”

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• What knowledge is needed to solve this?• How/why would you distinguish between

students who have this knowledge and those who do not?

• What has to be available to enable all to work on the problem?

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Explain how you know the red lines are parallel

• What habits would students need to have?– establish mathematical working practices

• What knowledge?– gradient of knowledge

• How would you distinguish between students who have these and those who do not?– different pathways to the same core idea

• What has to be available to enable work on the problem?– essential knowledge and habits

• What abilities are necessary?

Establishing habits (from ‘Deep Progress’)• listen• effort (think hard about it)• memory• talk about maths• concentration (know the cost of not listening)• homework habits• discussion• continuity of work; personal bag/folder/work in

progress/exercise book• what are the valuable kinds of answer• grow a storehouse of facts and methods• have a pen/pencil/ etc.

Imagine you are going to teach a topic:

Exchange problems

• Bod has six goats worth £25 pounds each; Tod has a bolt of 100 metres of silk cloth worth £10 per metre. Bod wants cloth and Tod wants goats. What can they exchange?

Plan to teach “exchange problems” to a class which includes students who:

• Have no past experience of exploring mathematics but do well in tests

• Are enthusiastic problem-solvers• Have little confidence in unknown situations, know their

tables and have calculators• Are not very strong on multiplication and division, but have

graphical calculators (or experience with graphplotters on the IWB)

• Are not very strong on multiplication or division or anything else, but will have a go if someone is working with them

• Are not very happy with maths at all, and cannot be relied on to recall any past experiences to help them

• Cannot understand English very well

• What knowledge is needed to solve this?• How/why would you distinguish between

students who have this knowledge and those who do not?

• What has to be available to enable all to work on the problem?

• What habits would students need to have?– establish mathematical working practices

• What knowledge?– gradient of knowledge

• How would you distinguish between students who have these and those who do not?– different pathways to the same core idea

• What has to be available to enable work on the problem?– essential knowledge and habits

• What abilities are necessary?

What does differentiation mean?

I have no idea!

IAMP – Improving Attainment in Mathematics Project (2001/2)

• Establishing working habits• Generating concentration and participation• Giving time to think and learn• Working on memory• Visualising• Students’ writing (formats)• Students’ awareness of progress• Giving choice• Being explicit about connections and differences• Dealing with mathematical complexity

What methods do teachers use to differentiate - and when?

• Set different tasks for different groups/individuals• Provide materials at different ‘levels’

(access/content/literacy)• Encourage learners to respond in different ways:

oral/visual/written/role play• Structure pair/group work according to individual

capabilities• Use other adults or learners to support• Vary pace for different learners as needed• Through interaction: simplify for some; make it

harder for others

What do students say about choice?

• task• order of task(s)• within task - what to do• within task - method• resources and tools• how to present work• what to do after set work is finished• what to revise

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• What habits would students need to have?• What knowledge?• How/why would you distinguish between

students who have these and those who do not?

• What has to be available to enable work on the problem?

• What abilities are necessary?

Scenario (true story)• 2.40 p.m. Yr 10

• Thursday in the final week of term

• There has been a fight at lunchtime

• This is a bottom set

• The TA is attached to one student only

• The teacher is being observed by her tutor

• The teacher has decided to teach simultaneous equations

• They know it is not on their syllabus

– established mathematical working practices– gradient of knowledge– different pathways to the same core idea– essential knowledge and habits

• What abilities are necessary to solve simultaneous linear equations?

• n.b. no choice; no groupwork; no TA general support

Deep progress means that students:• learn more mathematics,• get better at learning mathematics,• feel better about themselves as

mathematics students

“It’s boring; I can’t do this; I can do it but I don’t want to; I want to go home; she knows we can do this already.”

What to do?

• Setting• In class grouping• Intervention

Setting• Setting has slight positive effects on the highest

achieving students• Setting has different effects in different studies on

average students, mainly negative• Setting has been shown time and time again to have

negative effects on the mathematical learning of students with difficulties (limits opportunity and takes little account of differences) (Ireson & Hallam; Boaler; Wiliam; Bartholomew)

• Setting is illegal in some countries and unusual in most countries

In-class grouping

• Seems to be effective (in primary) when it is done with:– Assessment– Understanding reasons for different achievement– A range of activities: see, listen, do, join in– Revision and consolidation– Multi-sensory experience

Teaching assistants

• Only effective when they follow the main idea of the lesson, not simplifying the focus for some students; when they understand the subject and what they do complements what the teacher is intending

What I did last year ....

• Diagnostics, year 7 lowest attaining students according to KS2 test scores.

• What do you think the problems were?• How can you teach if you do not know what

the precise problems are?• CSMS http://iccams-maths.org/CSMS/

What gets missed?

• Counting processes• Counting principles• Written symbols• Place value• Word problems• Concrete/verbal/numerical links• Derived facts• Estimation• Fact retrieval (Dowker, 2004)

Intervention• Developing cognitive skills towards formal operations

(Cognitive Acceleration in Mathematics Education CAME: pisa/pisa)

• Developing arithmetic (diagnostic)• Developing metacognitive strategies (unclear effect)• Derived facts (inverse, commutativity, associativity,

distributivity, relations)• Detailed assessment for buggy understandings• 1-to-1 works best as mathematics understanding is

complex and individual

Bibliography• Denvir, B., Stolz, C. and Brown, M. (1982) Low attainers in mathematics. London: Methuen• Dowker, A. 2004 http://dera.ioe.ac.uk/2505/1/ma_difficulties_0008609.pdf • Dowker, A. (2005). Early identification and intervention for students with

mathematics difficulties. JOURNAL OF LEARNING DISABILITIES, 38(4), 324. (available online, google scholar ‘Dowker 2005’)

• Geary, D. C. (1994). Children’s mathematical development: Research and practical applications. Washington, DC: American Psychological Association.

• Hart, S., Dixon, A. et al. (2004) Learning without limits. Buckingham: Open University Press.• Watson, A. (2026) Raising achievement in secondary mathematics. Buckingham: Open

University Press.• https://www.ncetm.org.uk/files/16801567/CTP0213+Fulford+Final+Report.pdf• http://files.eric.ed.gov/fulltext/ED322565.pdf• http://educationendowmentfoundation.org.uk/toolkit/teaching-assistants/• http://www.nationalstemcentre.org.uk/elibrary/resource/8019/deep-progress-in-mathem

atics-the-improving-attainment-in-mathematics-project• http://www.oecd.org/pisa/keyfindings/pisa-2012-results-volume-II.pdfor google ‘deep progress in mathematics’ and ‘pisa excellence through equity’