mastery approaches to maths ma2m+ · 2018. 3. 20. · greater depth. rapid graspers. high prior...

Mastery Approaches to Maths MA2M+

Monday 20th November 2017: Teaching for Mastery Maths: PrinciplesMonday 04th December 2017: Teaching for Mastery Maths: In your classroomMonday 05th February 2018: Teaching for Mastery Maths: Lessons from ShanghaiMonday 19th March 2018: Teaching for Mastery Maths: Greater depthMonday 23rd April 2018: Teaching for Mastery Maths: Next steps

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Presenter

Presentation Notes

This is the session 4 twilight on mastery approaches to maths… Carlisle NCETM funded NNW Maths Hub and LED Research Centre at the University of Cumbria

Greater depthRapid graspers

High prior attainersGifted and talented

More ableMathematicians

Masters

Presenter

Presentation Notes

Rapid graspers refers to children who pick up the concept being taught, more quickly than their peers. This is not a rewording of high attainers or gifted and talented because with a teaching for mastery approach, no attempt is made to identify which particular children will be rapid graspers on any particular day or topic, recognising that children’s aptitudes vary between topics and over time.

What is GD/ What is not GD• using bigger numbers• more of the same• Maths problems presented in a wide range of different,

complex ways• reasoning mathematically • solving problems • fluency and conceptual understanding • moving onto next year groups objectives

Discuss with a partner …

All children should be …reasoning mathematically solving problems developing fluency and conceptual understanding

• Greater depth is• Maths problems presented

in a wide range of different, complex ways.

Greater depth is not• using bigger numbers• more of the same• Moving onto next year

groups objectives.

What does the National curriculum say?

Presenter

Presentation Notes

�So whilst children working at greater depth will be fluent and will solve problems and reason mathematically, we can't use those indicators to define 'Greater Depth' in maths. The National Curriculum document does give us another clue, however:e might define children who work at greater depth as still working within the expected standard but at a deeper level; this is how theInterim Teacher Assessment Framework (ITAF) classifies them. These children will most likely be children who 'grasp concepts rapidly' - let's assume the two are synonymous. For these children, the ones working at greater depth, we should provide 'rich and sophisticated problems' and we shouldn't just be getting them to move on to the next year group's work - this is made clear in the NC document and the language of the ITAF: working within the expected standard. So, as an indicator, those working at greater depth should be able to access 'rich and sophisticated problems'.��But what about 'mastery'? A word mentioned only twice in the National Curriculum document (in relation only to English and Art) but one which has been bandied about a lot since its publication. If a child demonstrates mastery, could they be considered to be working at greater depth? In a word: no. The NCETM have this to say: "Mastery of mathematics is something that we want pupils - all pupils - to acquire, or rather to continue acquiring throughout their school lives, and beyond." Again we see that word 'all'. The NCETM say that "at any one point in a pupil’s journey through school, achieving mastery is taken to mean acquiring a solid enough understanding of the maths that’s been taught to enable him/her move on to more advanced material" - mastery is something which allows children to move on to be taught new content (c.f. to the NC) whereas working at greater depth pertains to working on current content, but at a deeper level. Notice those words 'solid enough' - a child working at greater depth won't just have 'solid enough' understanding - they'll have something more than that

• The NCETM have this to say: "Mastery of mathematics is something that we want pupils - all pupils - to acquire, or rather to continue acquiring throughout their school lives, and beyond."

• Again we see that word 'all'. The NCETM say that "at any one point in a pupil’s journey through school, achieving mastery is taken to mean acquiring a solid enough understanding of the maths that’s been taught to enable him/her move on to more advanced material"

• - mastery is something which allows children to move on to be taught new content (c.f. to the NC) whereas working at greater depth pertains to working on current content, but at a deeper level. Notice those words 'solid enough' - a child working at greater depth won't just have 'solid enough' understanding - they'll have something more than that.

https://www.ncetm.org.uk/resources/47230

https://www.ncetm.org.uk/resources/49450

A pupil really understands a mathematical concept, idea or technique if he or she can:

•describe it in his or her own words;•represent it in a variety of ways (e.g. using concrete materials, pictures and symbols –the CPA approach)•explain it to someone else;•make up his or her own examples (and non-examples) of it;•see connections between it and other facts or ideas;•recognise it in new situations and contexts;•make use of it in various ways, including in new situations.

From ‘Assessing Mastery’ –Introduction. Adapted from a list in ‘How Children Fail’, John Holt, 1964.

What can I use to help ?• White Rose Materials • National Curriculum Assessment Materials• NRich• http://www.londonsouthwestmathshub.co.uk/curren

t-projects/project-resources/greater-depth-2016-2017

• Numerous books• https://nicolaballantine.wordpress.com/2018/02/13/

dive-deeper-with-the-high-attainers/

http://www.londonsouthwestmathshub.co.uk/current-projects/project-resources/greater-depth-2016-2017

https://nicolaballantine.wordpress.com/2018/02/13/dive-deeper-with-the-high-attainers/

Presenter

Presentation Notes

Some resources give specific examples related to areas of Maths like the NC assessment materials

Could this be used for decimals?

Year 2- Fractions

Mastery

Mastery with Greater Depth

Year 3 – Multiplication and Division

Year 4 – Addition and Subtraction

Mastery with Greater Depth

• Write three calculations where you would use mental calculation strategies and three where you apply a column method.

• Explain the decision you made for each calculation.

Year 5- Geometry

Which of these statements are correct? • A square is a rectangle. • A rectangle is a square. • A rectangle is a parallelogram. • A rhombus is a parallelogram. Explain your reasoning.

Year 6 -

Mastery of mathematics: a view from Ofsted Jane Jones HMI, National lead for mathematics 20 June 2016

Would this cater for all children?Which elements would challenge greater depth?

General Strategies

• ‘Working on’• Another and Another• How many ways

Presenter

Presentation Notes

Three general strategies for greater depth… lets consider each in turn…

‘Working on’ versus ‘working through’

When working on any task or problem make sure you ask yourself:● Can I explain the idea to others? Orally, in writing, using diagrams, etc.

● Can I make up my own question and can I do the question that I have set myself?

● Can I write (in my journal) a description of what you need to do to be able to answer these sorts of questions well?

Presenter

Presentation Notes

Maybe have this on your classroom wall…children get used to the idea that if they feel they are close to finishing a task they can use these three working on steps… it becomes a routine

When learners are asked to make up an example of something they have to think themselves about what this particular object is, what its particular properties are.The invitation to make up “another and then another and then another …” can result in deeper thinking.

•A triangle•A pair of numbers that sum to 2•A shape with an area of 7 square units

Another and another

Another and another

Presenter

Presentation Notes

Other resources are more general and can be applied to different areas of Maths seamlessly.

How many ways?In how many different ways can you …?• Halve a rectangle• Calculate 25 ×29• Find the area of this shape

How could you use these examples in your classroom?

• ‘Working on’• Another and Another

• How many ways


So what’s the answer? It cannot be complex or require yet more work on the part of the teacher – we’ve simply reached capacity. In fact, if

the answer were to somehow minimise work, that would be a very welcome solution indeed

Presenter

Presentation Notes

For example, if the class was engaged in a fluency activity, such as one which explored the concept of multiplying unit fractions, a student who completed the task early could then write a Maths story which could cement a connection between the abstract algorithm and the real world, i.e. three friends each had ⅓ of the ½ of a cake left over from a party, ⅓ x ½ = ⅙. Next, if the reasoning task asked for the student to explain why the product of two fractions is smaller than the factors, they could extend their learning by drawing a pictorial representation, proving why this is true. Lastly, after completing a problem solving task requiring them to apply their understanding in a new context, they could then deepen their learning by demonstrating what they deem to be a common mistake, such as adding the denominators rather than multiplying them, and they could explain why this error sometimes occurs. By engaging in these deepening tasks, students are able to sit with their learning just a little bit longer, devoting more thinking time to understanding the structure of the concept as well as unpicking any of their own misconceptions.


Children to write an explanation for an absent pupil or children in

another class on how to complete skill?

This is very challenging and needs a high level of understanding.

It will often involve generalising, a higher order problem solving skill.

Mark MoodySecondary Teaching for Mastery Lead

Variation andIntelligent Practice

Presenter

Presentation Notes

This session focuses on variation theory… in mastery approaches we are using theory to shape our practice…

Mark MoodySecondary Teaching for Mastery Lead

Avoiding Knitting

“Some still justify children spending a lot of time working through examples by saying children like it and I can understand why.”

Mike Askew - Transforming Primary Mathematics (2011)

Variation versus Variety

“I loved doing pages of 'sums'. Setting everything out neatly … working through and being reasonably confident of getting everything right, then the reward of a full set of red ticks and 10/10 and, joy of joy, a sticky star, made everything in the world seem right and in order.Of course, I was one of the lucky ones.”



“But I wasn't really engaged in mathematical activity. Pages of sums like this - routinized and easy to do if you have mastered the technique -are closer to knitting than mathematics.”



Presenter

Presentation Notes

Refer to Variation section as part of Big 5

Variety• ‘Pick and mix’• Most practice exercises contain variety

Variation versus Variety - WHAT

Variation• Careful choice of WHAT to vary• Careful choice of what the variation will

draw attention to

“By encouraging students to use only one method (algorithmic) to solve problems, they lose some of their capacity for flexible and creative thought.

They become less willing to attempt problems in alternative ways, and they become afraid to take risks.

Furthermore, there is a high probability that the students will lose conceptual knowledge in the process of gaining procedural knowledge”

Narode, Board & Davenport (1993)

Variation versus Variety - HOW

Presenter

Presentation Notes

Narode, Board & Davenport (1993) Algorithms supplant understanding: case studies of primary students’ strategies for double digit addition and subtraction.

Intelligent practice is a term used to describe practice exercises where the teacher avoids mechanical repetition and creates an appropriate path for practising the thinking process with increasing creativity.When practice is “intelligent”, attention is drawn to the mathematical structures and relationships to assist in the deepening of conceptual understanding, while at the same time developing procedural fluency.

Intelligent Practice

Helen Drury - How to Teach Mathematics for Mastery

The central idea of teaching with variation is to highlight the essential features of the concepts through varying the non-essential features.

When constructing a set of activities / questions it is important to consider what connects the examples; what mathematical structures are being highlighted

Teaching with Variation

5

13

5

13

13 5

Conceptual VariationPositive variation

Standard variation

Non-standard variationConceptual Variation

Negative variation

Standard variation Non-standard variation

Conceptual VariationPositive variation

Standard variation

Non-standard variationConceptual Variation

Negative variation

What represents

½and what does not

True or False?

49

510=

35

310= 3

7614=

368 16=

Conceptual Variation

Procedural Variation

Varying a problem by changing conditions

Multiple methods to solve a problem (“find three ways …”)

Procedural Variation

Apply a method to several related problems

“It is better to solve one problem five ways than five problems one way.”

George Pólya

Presenter

Presentation Notes

Eg 40/5 = 8 then 8x3=24 Or 3/5=60% 40x 0.6=24 Or 40x3=120 then 120/5 = 24 etc

120 - 90

Variation - Subtraction

235 - 180

502 - 367

122 - 92119 - 89237 - 182

Set A


120 - 90

Variation - Subtraction

235 - 180

502 - 367

122 - 92119 - 89237 - 182

Set A Set B

120 - 90

235 - 180

502 - 367

122 - 92119 - 89237 - 182


www.cimt.org.uk/projects/mep

http://www.cimt.org.uk/projects/mep

MEP Year 3

MEP Year 6

MEP Year 7

median donsteward.blogspot.co.uk

Adapting resources

• Remove part of the question• Give the answer, find the question• Questions with the same/related answer• Find the smallest / largest / closest to ...• Find another way ... and another ...• Give another example ... and another ...• True / False• Always / Sometimes / Never• Find a rule / generalise

Presenter

Presentation Notes

10ticks website – Mastery section – free samples

Presenter

Presentation Notes

12 questions – always, sometimes, never; emphasis on all 12 on a worksheet, colour in the RAG answer ...

explain why

explain why not

• give examples• explain when

it does and when it doesn’t

• explain why / why not

• find another rule

• and another• what about

subtracting?• ...

Presenter

Presentation Notes

Each question could be the opportunity to dig deeper

5cm

2cm

3cm4cm

8cm

6cm

Divide this shape into two rectangles and use the area of the rectangles to help you find the area of the shape.

Prestage and Perks (2001)Adapting and Extending Secondary Mathematics Activities

5cm

2cm

3cm4cm

8cm

6cm

Find the area of the shape.

3cm4cm

8cm

6cm

Find the area of the shape in three ways.

8cm

6cm

Find the area of the shape.

8cm

6cm


3cm4cm

8cm

6cm


6

15

4

16

2

4

DifferentiationNational CurriculumPupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content.

Rational or Irrational?



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Presenter

Presentation Notes

This is the session 2 twilight on mastery approaches to maths…

John Kotter

Kurt Lewin

Andy Hargreaves & Michael Fullan

Hargreaves, A. & Fullan, M. (2012) Professional Capital: Transforming Teaching in Every school. New York: Teachers’ College Press.

Presenter

Presentation Notes

Significant change in practice requires a careful preparation in forming a leadership team which in a Primary school MUST include the head teacher plus voluntary teachers who have expertise and interest as well as possibly formal roles such as subject leader… Kotter’s model reminds us of the need to prepare for change…setting the stage and creating a sense of urgency that the change is essential… Lewin’s model reminds us that practice can be frozen in place ‘ways of working here’ and you will need to unfreeze practice to change it…then establish the changes by freezing them into practice going into the future… Above all the professional capital concept of Hargreaves and Fullan remind us that change is about people, about relationships, about trust and about maintaining professional autonomy… top down implementation by ‘heroic’ strong leaders may gain some short term profit but ultimately will fail…

First-order or second order change?Second-order change is perceived as:• A break with the past • Lies outside existing paradigms• Conflicts with prevailing values and norms• Requires the acquisition of new knowledge and skills• Requires resources not available to those responsible for implementing

the innovations• May be resisted because only those who have a broad perspective of

the school see the innovation as necessary

Marzano, R.J., Waters,, T. & McNulty, B.A. (2005) School Leadership That Works: From research to results. Aurora, USA: McREL.

Is development of mastery approaches to maths in your school a second order change?

Presenter

Presentation Notes

Marzano, Waters and McNulty present a challenging idea – that the many failed or forgotten initiatives in schools, that soaked up so much energy at the time, are caused because in busy schools significant changes are often treated as incremental, first-order changes, when they should have been identified and managed as big cultural changes and managed as second-order change…

The characteristics of effective professional learning for teachers:• A collective sustained focus on learning• Increasing ownership, collaboration and trust• Classroom inquiry and experimentation• Focus on pedagogy within curriculum subjects• Focus on formative assessment and impact• Critical engagement with external knowledge

Developing Great Teaching (2015) http://tdtrust.org/about/dgt/

Presenter

Presentation Notes

The TDT review of research on effective CPD for teachers is a useful foundation for leading professional development… if it can be combined with the principles and strategies suggested by the change leadership models… professional inquiry, perhaps in the form of lesson study could be useful…and if designed and facilitated effectively could match the TDT characteristics of effective CPD…

http://tdtrust.org/about/dgt/

Lesson Study as Professional Inquiry1. Collective leadership2. Topic3. Elements of mastery approaches4. Relate to current practice5. Teach the lesson6. Evaluate the impact on learning7. Revise the lesson8. Teach the lesson again9. Evaluate the revised lesson10. Disseminate

Could lesson study be useful in developing mastery approaches to maths in your school?

Presenter

Presentation Notes

See the lesson study handout and consider this question… if not lesson study then what form of professional inquiry will you use? Action learning sets are a possibility… action research is another… but given the focus of mastery approaches on classrooms then lesson study seems appropriate… and has been a huge part of the development of maths teaching in Singapore and Shanghai… so we might take inspiration from that and develop our own version in the UK…

Leading Change: session 5

• 10 minute re-cap especially on definitions of mastery approach to teaching maths

• 60 minutes in three break out groups sharing ideas, discussing a (pre-completed) self-assessment on mastery teaching and action planning

• 20 minute plenary back together in the lecture theatre – key ideas and challenges

• Follow-up online survey to be completed…and a concise project evaluation report sent out by the project team

Presenter

Presentation Notes

Proposal for session 5…



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mastery approaches to maths ma2m+ · 2018. 3. 20. · greater depth. rapid graspers. high prior...

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