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Formative Assessment and Teaching Mathematics: A Didactical Approach? Jeremy Hodgen King’s College London SW Consultants Day: 26 September 2008

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How can we teach maths for understanding and engagement in real classrooms? Know (a lot) about – how children learn – teaching in experimental / “unusual” classrooms – “generic” teaching approaches Know very little about how to enable ordinary teachers to teaching for understanding & engagement in mathematics

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Increasing Confidence and Competence in Algebra and Multiplicative Structures (ICCAMS) 2008 - 2012 Margaret Brown, Dietmar Kuchemann & Robert Coe Survey comparison with 30 years ago Collaboration with teachers – Learning in real classrooms – How to respond to children’s learning needs – Formative and diagnostic assessment – Design (and test) an approach to spreading the ideas more widely

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Participation … Increases in post-16 mathematics but SLOWLY Not enough numerate graduates Numeracy for critical / informed citizenship Enjoyment and appreciation Particular problem with algebra and multiplicative reasoning – Algebra “gap” (Wiliam, et al, 1999)

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Enjoyment? “because it SUCKS and I wouldn't want to spend any more of my time looking at algebra and other crap” “I hate mathematics and I would rather die” (Brown, Brown & Bibby, 2008)

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A difficult subject? “It’s just too damn hard” “Everyone who I have spoken to who is on the course says it is way too hard and is not worth it” (Brown, Brown & Bibby, 2008)

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Relevance? “What’s the use of maths? … when you graduate or when you get a job, nobody’s gonna come into your office and tell you: ‘can [you] solve x square minus you know?’ … It really doesn’t make sense to me. I mean it’s good we’re doing it. It helps you to like crack your brain, think more and you know, and all those things. But like, nobody comes [to] see you and say ‘can [you] solve this?’” (Mendick, 2006)

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Ability and the “clever core”? “All the 3Bs go to the side of us, the 3Cs go in the middle and the 2As, they go to the end” “Table 1, that’s clever, really really clever, table 2 is very clever, table 3 is very clever, number 4 is just clever. I’m on 1.” Y4 children from Hodgen & Marks (2009) “I’ll be a nothing.” Y6 child from Reay & Wiliam (1999)

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Images of Mathematics (Picker &Berry, 2000)

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Mathematicians … have no friends (except other mathematicians) are not married or seeing anyone are usually fat are very unstylish have wrinkles in forehead from thinking too hard have no social life whatsoever are 30 years old have a very short temper. (Picker &Berry, 2000)

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Understanding? Y7 dip: – Absolute dip of 2-4 months end-Y6 to end-Y7 (Brown et al, 2003) KS3 Plateau: – Almost no progress Y7-Y9 (Williams et al., 2003) Exam Comparisons – C “2006” = D “1996”? (Coe, 2008)

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So what can we do didactically … Assessment for learning / formative assessment?

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Assessment for Learning is … … any assessment for which the first priority in its design and practice is to serve the purpose of promoting learning. (Black & Wiliam, 1998)

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Assessment for Learning is … Keep one eye on the mathematical horizon and the other on students’ current understandings, concerns and interests. (Ball, 1993)

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The evidence … Black & Wiliam: Inside the Black Box (1998) etc Numerous other reviews worldwide – Natriello (1987); Crooks (1988); Kluger & DeNisi (1996); Nyquist (2003) All find consistent & substantial effects on … – Attainment and engagement – BUT poorly described in practice.

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More evidence … Hattie’s (2007) meta-analysis: – Feedback is the most effective intervention in education (effect size: 1.14) Wiliam (2007): – Assessment for learning probably the most cost effective way of improving teaching – Better and more achievable than reducing class size or enhancing teachers’ subject knowledge

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Extraordinarily “successful” … Inside the Black Box: > 50,000 copies sold (UK) Working Inside the Black Box: > 40,000 Mathematics Inside the Black Box: > 7,000 Embraced by DCSF / DfES, National Strategies Taken up by schools Hard to find a teacher who hasn’t heard of it

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BUT …

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Be careful what you wish for. (Paul Black)

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Formative assessment, or the ways in which teachers identify and record progress made by the learners; this may be a very creative process with suggestions of video recordings, learner diaries, exhibitions as well as more traditional assignments and tests. (LSC, RARPA)

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It is hard to understand why work which I and many others have carried out over almost ten years, which we have written up with very careful attention to evidence, and which has been widely welcomed by teachers, is either mis-understood or ignored - apart that is from the use of the label. (Paul Black)

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So what is assessment for Learning? … Keep one eye on the mathematical horizon and the other on students’ current understandings, concerns and interests. Feedback Listening & Questioning Talk

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Feedback & Marking Butler (1988)

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Listening to students Evaluative – listening for the correct answer – ‘Almost’ / ‘Nearly’ Interpretive – why do pupils respond in the way they do (Davis, 1997)

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Listening to students 2 Evaluative v Interpretative? When Miss used to ask a question, she used to be interested in the right answer. Now she’s interested in what we think. [Y8 pupil]

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Questions & Wait Time Typical: <0.9s Increase 3 s – More & longer contributions from more students – Increased attainment

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What is rich discussion? Percentages discussion Y7 1 st lesson in a sequence on percentages Task: talk about what you know Excerpt of a 35 minute discussion: 5% > 10%?

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The teacher … – Knew about learning – Let the children talk – Questioned correct answers HARD – Interested in what they said – Took time

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This isn’t easy “They won’t give us our marks” (Smith & Gorard, 2005) Wait time too long (Hodgen, 2007) Implementing generic strategies mathematically is tricky (Watson, 2006) Talk limited in mathematics classrooms

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… Keep one eye on the mathematical horizon and the other on students’ current understandings, concerns and interests. Didactics Sharing learning outcomes Rich tasks

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Sharing learning outcomes? How can you understand what you don’t know? Most ideas in maths are projects of 10 years or so Children may be interested in something else entirely

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I want you to regard this as a challenge, Molesworth.

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The only way with a maths master is to hav a very worred xpression. Stare at the book intently with a deep frown as if furious that you canot see the answer. at the same time scratch the head with the end of the pen. After 5 minits it is not safe to do nothing any longer. Brush away all objects which hav fallen out of hair and put up hand.

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“Sir?” (whisper) “Please sir?” (louder) “Yes, molesworth?” sa maths master. “Sir i don’t quite see this.” nb it is essential to sa you don’t quite ‘see’ sum as this means you are only temporarily bafled by unruly equation and not that you don’t kno fanetest about any of it.

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“What do you not see molesworth?” sa maths master (Thinks: a worthy dolt who is making an honest effort) “number six sir i can’t make it out sir.” “What can you not make out molesworth?” “number six sir.”

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“it is all very simple molesworth if you had been paing attention to what i was saing at the beginning of the lesson. Go back to your desk and think.” This gets a boy nowhere but it show he is KEEN which is important with maths masters.

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As part of a whole class session, the teacher is working on halving numbers. Each child has an individual white board and marker pen with which to display answers. Teacher: Half of 36? Meg starts to lift her board up to show the teacher. She has written ‘15’, but before she shows it she notices that others around her have ‘18’. She quickly changes it; the teacher does not notice and says, ‘Well done, Meg.’

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Teacher: Half of 72 Meg puts on an act. She takes the top off her pen, pushes it back again and looks puzzled. She appears to be counting - her lips are moving but it is not clear what she is saying. She turns round and sees what George has written then turns back again and wrinkles her face (as if to say, ‘I'm concentrating hard’). Then she looks around at several boards and see what answer others have got. Next she closes her eyes and screws up her face. After a time her face lights up as if she’s just made a big discovery and she writes down ‘36’.

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Sharing learning objectives … … takes time.

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Knotty problems … Two negatives make a positive? Not always! … -1 + -1 = -2 BUT … why is -1 × -1 = 1

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So … … why is -1 × -1 = +1?

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Strategies … What else could it be? Patterns Stories Images Understanding multiplication

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Stories A car is travelling backwards at 1 m per sec. Where was it 1 second ago? Distance [1m] = Speed [-1 mps] × Time [-1 s]

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Images Dominant images of negative numbers – Temperature – Scores in games – Above / below sea- or ground-level & lifts – Bank balances What about directed numbers?

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0 × −1 =0

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(1 + -1) ×−1 = 0

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0 × −1 =0 (1 + -1) ×−1 = 0 1 × −1 + −1 × −1 = 0

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0 × −1 =0 (1 + -1) ×−1 = 0 1 × −1 + −1 × −1 = 0 −1 + −1 × −1 = 0

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Understanding multiplication Identity: + 0 … × 1 … × 0 Inverses: 1 + -1 = 0 Distributive Law [Axiom]: (2 + 3) × 4 = 2×4 + 3×4

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Tackling difficult issues … “I don’t know, but we can find out.” Chinese Teacher (Ma, 1999)

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Making connections Thinking about that it was something no-one really made clear to me at school. You know that something like quadratic equations have a spatial meaning. No-one made the connections between the spatial and the number system. Numeracy Consultant (Hodgen & Johnson, 2003)

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Experiencing pattern … The “easy” can be hard. Counting can be hard.

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Count in twenty-fourths …

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BUT simplify…

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Patterns.. Structure …

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Another way of looking …

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Exploring structure Japanese teachers and lesson study – Groups of teachers work on 2-3 lessons over a year – Present to others – Highly mathematical

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How can we teach maths for understanding and engagement in real classrooms? Assessment for learning BUT in the context of mathematics / mathematics learning

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How can we we work with teachers to enable them to teach maths for understanding and engagement in real classrooms? Feedback Time to think Rich dialogue Sharing learning outcomes Being mathematical

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