teaching mathematics for mastery
TRANSCRIPT
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Teaching Mathematics for Mastery
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•I know how to do it
•It becomes automatic and I don’t need to think about it- for example driving a car
•I’m really good at doing it – painting a room, or
a picture
•I can show someone else how to do it.
What does it mean to master something?
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What is mastery?
If you drive a car, imagine the process you went through…
• The very first drive, lacking the knowledge of what to do to get moving
• The practice, gaining confidence that you are able to drive
• The driving test, fairly competent but maybe not fully confident
• A few years on, it’s automatic, you don’t have to think about how to change gears or use the brake
• Later still, you could teach someone else how to drive
However not all of us know exactly how the car actually works!
Learning to master driving takes time and a lot of practice!
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Mastery of Mathematics is more…..
• Achievable for all
• Deep and sustainable learning
• The ability to build on something that has already been sufficiently mastered
• The ability to reason about a concept and make connections
• Conceptual and procedural fluency
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Teaching for Mastery
• The belief that all pupils can achieve
• Keeping the class working together so that all can access and master mathematics
• Development of deep mathematical understanding
• Development of both factual/procedural and conceptual fluency
• Longer time on key topics, providing time to go deeper and embed learning
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A mastery curriculum
NC 2000 NC 2014
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Achieving mastery
Mastery of the curriculum requires that all pupils:
• use mathematical concepts, facts and procedures appropriately, flexibly and fluently;
• recall key number facts with speed and accuracy and use them to calculate and work out unknown facts;
• have sufficient depth of knowledge and understanding to reason and explain mathematical concepts and procedures and use them to solve a variety of problems.
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National Curriculum
Progress in mathematics learning each year should be assessed according to the extent to which pupils are gaining a deep understanding of the content taught for that year, resulting in sustainable knowledge and skills.
Key measures of this are the abilities to reason mathematically and to solve increasingly complex problems, doing so with fluency, as described in the aims of the National curriculum:
‘The national curriculum for mathematics aims to ensure that all pupils:
• become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
• can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.’ (National curriculum p3)
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9
●Factual & Procedural
Fluency
●Conceptual
Understanding
●INTEGRATION
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Is there evidence
of conceptual
understanding?
Is there procedural
fluency and
efficiency?
ANGHILERI et al FROM INFORMAL STRATEGIES TO STRUCTURED PROCEDURES: MIND THE GAP!
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Sally knows all her tables up to 12 x 12
When asked what is 12 x 13 she looks blank.
Does she have fluency and understanding?
11
12 x 12 = 132
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Solve the following (on your own!)
+ 17 = 15 + 24
99 – = 90 – 59
12
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Procedural without conceptual Conceptual without procedural
Computation without meaning Computation which is slow, effortful and
frustrating
Inability to adapt skills to unfamiliar
contexts
Inability to focus on the bigger picture
when solving problems
Difficulty reconstructing forgotten
knowledge or skillsDifficulty progressing to new or more
complex ideas
Fluency
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Developing conceptual understanding
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• Number Facts
• Table Facts
• Making Connections
• Procedural
• Conceptual
• Making Connections
• Chains of Reasoning
• Making Connections
• Access
• Pattern
• Making Connections
Representation
& Structure
Mathematical Thinking
FluencyVariation
Coherence
Teaching for Mastery
Small connected steps
are easier to take
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Features of Teaching for mastery
• Whole class teaching
• Differentiation (but not as we know it!)
• Carefully structured lessons
-Step by Step approach
• Going deeper
• Application of variation
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Meeting the needs of all pupils The Road to Differentiation
Inclusion is important, but maybe
we need to think about it in a
different way
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What about ‘differentiation’?
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Coherence
• A comprehensive, detailed conceptual journey through the
mathematics.
• Small steps are easier to take
• Focusing on one key point each lesson allows for deep and
sustainable learning
• Certain images techniques are pre cursors to later ideas - getting the
sequencing of these right is an important skill in planning and
teaching for mastery
• When something has been deeply understood and mastered, it can
and should be used in the next steps of learning
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Teacher Pupils Teacher…
Ping Pong
Approach
• Provides a clear and
coherent journey
through the mathematics
• Provides detail
• Provides scaffolding for
all to achieve
• Provides the small steps
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Pupil Support
One of the most important tasks of the teacher is to
help his students…
If he is left alone with his problem without any help
or insufficient help, he may make no progress at
all…
If the teacher helps too much, nothing is left to the
student
(Polya 1957)
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Let them go, but then reign them back in
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Let Go
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●Providing Textbook Supports for Teaching Math Akihiko Takahashi
https://prezi.com/s1nvam1gllv9/providing-textbook-supports-for-teaching-math/
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Reining back in
25
●Providing Textbook Supports for Teaching Math Akihiko Takahashi
https://prezi.com/s1nvam1gllv9/providing-textbook-supports-for-teaching-math/
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Let go
26
●Providing Textbook Supports for Teaching Math Akihiko Takahashi
https://prezi.com/s1nvam1gllv9/providing-textbook-supports-for-teaching-math/
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Reining back in
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●Providing Textbook Supports for Teaching Math Akihiko Takahashi
https://prezi.com/s1nvam1gllv9/providing-textbook-supports-for-teaching-math/
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Going deeper
• The role of memorization in deep learning
• Learning Tables
• Using STEM sentences
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• Expect children to use correct mathematical terminology and to express their reasoning in complete sentences
• The quality of children’s mathematical reasoning and conceptual understanding is significantly enhanced if they are consistently expected to use correct mathematical terminology and to explain their mathematical thinking in complete sentences.
Mathematical Vocabulary
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The role of repetitionI say, you say, you say, you say, we all say
This technique enables the teacher to provide a sentence stem for children to
communicate their ideas with mathematical precision and clarity. These sentence
structures often express key conceptual ideas or generalities and provide a
framework to embed conceptual knowledge and build understanding.
For example:
If the whole is divided into three equal parts, one part is one
third of the whole.
Having modelled the sentence, the teacher then asks individual children to repeat
this, before asking the whole class to chorus chant the sentence. This provides
children with a valuable sentence for talking about fractions. Repeated use helps to
embed key conceptual knowledge.
https://www.ncetm.org.uk/resources/48070
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Year 1
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Year 1
Repetition of key sentences supports memorisation.
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Securing learning
Reapplication of a key idea in different contexts deepens and secures learning
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Forms of Stem sentences
• Sometimes they are generalisations that are reached at the conclusion of a lesson:
For repeated addition we can calculate by using multiplication
• Or used to emphasise the correct language
Addend plus addend equals sum
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Repetition and Chorusing
• Set structures – fill in the blanks to apply to different contexts
• Example
There are ______ boats altogether, and _____
children in each boat.
Identifies the multiplier (the number of groups) and minuend (the size of the group)
Later the nouns might also be replaced (boats and children) – supporting conceptual
variation – lets look at the same mathematics in a different context
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Year 3
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There are 9
of 9 is equal to ___ 13
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There are 12 .
of 12 is equal to ___ 14
Write three more sentences about the 12
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What might be the purpose of repetition and memorisation in learning?
• Maintaining children’s focus• Enabling them to recognise what’s important and
what needs to be remembered for later learning
• Reducing cognitive load to enable learning to happen
• Returning to and enabling ideas to be connected
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15
15
2 paper tapes were broken, can you guess which original paper tape is longer?
Why? How do you get your answer?
The answer is only the beginning...
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•The central idea of teaching with variation is to
highlight the essential features of the concepts
through varying the non-essential features.
•Gu, Huang & Marton, 2004
Teaching with Variation
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Variation Theory in Practice
Consider how variation can both narrow and broaden the focus
Taken from Mike Askew, Transforming Primary Mathematics, Chapter 6
Compare the two sets of calculations
What’s the same, what’s different?
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Intelligent Practice
•In designing [these] exercises, the
teacher is advised to avoid mechanical
repetition and to create an appropriate
path for practising the thinking process
with increasing creativity.
•Gu, 1991
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Intelligent Practice
Noticing things that stay
the same, things that
change, providing the
opportunities to reason
make connections
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• Variety
• ‘Pick and mix’• Most practice exercises contain variety
• Variation
• Careful choice of WHAT to vary
• Careful choice of what the variation will draw attention to
Variation versus Variety
Mike Askew 2015
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Procedural VariationFocusing on relationships
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Making Connections
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Another and another
• Take a number ending in 7 and add 6
• Repeat for another and another……….• What do you notice?
50
Now answer the following:
467 + 6 =
1,487 + 6 =
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Procedural VariationAnd Calculation Strategies
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Procedural Variation
Provides the opportunity
• To focus on relationships, not just the procedure
• To make connections between problems
• To use one problem to work out the next
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Conceptual Variation
• An important teaching method through which students can definitely master concepts. It intends to illustrate the essential features by demonstrating different forms of visual materials and instances or highlight the essence of a concept by varying the non essential features. (Gu 1999)
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Conceptual Variation
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What’s the Same What’s different?
same shape same size same amount
An example of conceptual variation
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Shanghai Textbook Grade 3
Providing Challenge
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Carefully chosen examples
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True or False
Conceptual and Non Conceptual Variation
12
13
12
14× ×√ ×
Why, explain?
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Variation and Problem Solving
1 Varying or extending a problem
2. Multiple methods of solving a problem (eg the area problem)
3. Multiple application of a method, by applying the same method to multiple types of problems
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Using the same strategy to solve multiple problems
• Peter has 4 books
• Harry has five times as many books as
• Peter. How many books has Harry?
64
4
4 4 4 4 4
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Which problems can you solve using the same model
Sally and Tom share stickers in the ratio 1:5. If Sally has 6 stickers, how many
stickers does Tom have?
Sally and Tom share stickers in the ratio of 1 to 5. If together they have 60
stickers, How many does Sally have?
Sally has twice as many stickers as Tom. If together they have 60 stickers, how
many does Tom have?
Harry had £3.00 pocket Money. He saved 1/6, how much did he save?
Write your own problem that could be solved using the above model
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• CONCEPTUAL VARIATION to provide pupils with multiple perspectives
and experiences of mathematical concepts.
• PROCEDURAL VARIATION to provide a process for formation of
concepts stage by stage, in which pupils' experience in solving problems
is manifested by the richness of varying problems and the variety of
transferring.
• INTELLIGENT PRACTICE: when designing exercises, the teacher is
advised to avoid mechanical repetition and to create an appropriate
path for practising the thinking process with increasing creativity.
• Gu, 1991
Teaching for Mastery