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Transforming Primary Mathematics

I have spent time reading this informative and thought provoking book by  MikeAskew (@mikeaskew26). Below are some of the key points that resonated with me.

Introduction

Contrary to popular opinion, most children rise to the challenge of ‘hard’ mathematics rather than shy away from it…dependent upon a particular style of classroom ethos, close attention to the mathematical challenges presented and support for children in their efforts.

The nature of teaching is, and always will be, an adaptive challenge, rather than a technical problem…adaptive challenges require solutions that have yet to be found. We need to work with a view of mathematics teaching as an adaptive challenge. That means trying out new ways to teach and in particular allowing pedagogies to emerge rather than imposing them.

Current practices establish norms about different abilities…Society at large also labels people.

To meet the challenges of mathematics teaching, ways of working in classrooms need to emerge through the joint activity of teachers and children. Learning does not only happen in the minds of individual children – classrooms are learning systems. By attending to how the classroom community grows and learns (teacher and children together) it is possible to create classrooms where children: engage with meaningful mathematics; learn that they can learn mathematics; develop socially and emotionally; realise the importance of inter-dependency.

Thinking about learning

“Learning…is more of a reaching out than a taking in. It is participation…The agent’s activity and identify are inseparable from his, her or its knowledge. Knowing is doing is being.”

It is the combination of the following 3 together that makes mathematical teaching powerful:

  • Learning as a collective activity
  • How learning involves becoming as well as acquiring
  • Integrating the maths that ‘emerges’ as children work on rich problems and investigations with pre-planned learning intentions

Being meaningful is not merely about relating contexts to their ‘real’ lives; it is a meaningful context they can ‘mathematise’. (E.g. Place Value in Market Stalls example)

Teachers often say that the children can understand the mathematics but cannot apply it, rather than question the assumption that there is a logical connection in going from the abstract to the application. Starting from realistic contexts and mathematizing these may help the reapplication of this mathematics to other contexts later.

How do we encourage a classroom community that is a co-operative collective rather than a collection of individuals?

Thinking about curriculum

There is little point in teaching something if 3 months later the children show no understanding of it.

Most skills in maths only make sense in relation to other ones, so picking the off in isolation isn’t the most sensible way to address them…Outside of school most learning comes about through engaging in whole activities rather than learning discrete actions or behaviours.

Reflective teaching needs to focus on the activity, the experience, of the learner, not on the actions of the teacher.

“Learning originates in the experiences of the learner, not those of the teacher.” (Bernie Neville)

If we want children to engage in maths then teaching has to be based around collective problem solving, more closely matching the communities of practice that learners are engaged in outside of school.

One of the biggest difficulties in teaching maths is the assumption that what the child brings is not significant.

Thinking about teaching

Teaching and Learning is somewhat mysterious and unpredictable and we need to accept and work with that rather than behave as though it were completely controllable.

Complicated systems: particular actions determine particular, predictable results

Complex systems: effects of particular actions are much harder, if not impossible, to predict. Gardens are typical examples as they involve multiple feedback loops that dynamically change the whole structure. The results of actions depend upon the actions, but they do not uniquely determine them.

Teaching and learning is a complex system: learning is dependent upon teaching but cannot be completely determined by it. Accepting this complexity is liberating. It means teachers accept things as they are and work from that reality, rather than wish things were different.

The role of a teacher is to optimise experiences of learning, and in doing so maximise the likelihood of learning.

Mathematical activity: mindful or fluent?

“Trying to solve a maths problem in a way dictated by the teacher is different from attempting to test one’s own hypothesis. The teacher who tells students to solve a problem in a prescribed manner is limiting their ability to investigate their surroundings and to test novel ideas.” (Langer)

We need to pay more attention to the process of coming to know rather than the end results.

Successful performance depends on engaging in co-constructing emergent mathematical activity.

Is our focus more on finding answers to calculations or more on becoming mindful of the underlying mathematics?

Do the children need to learn their tables? The point of being fluent in addition and multiplication bonds is not as an end in themselves, but how they free up working memory when tackling more interesting and engaging pieces of maths.

As learners become more fluent and confident so they become more engaged and involved in maths lessons.

It is the lack of experience that limits what children can do.

Asian children adopt addition strategies based on partitioning (e.g. 6 + 8 as 6 + 4 + 4) sooner than international peers.

Variation theory

VT provides a framework for thinking about how to maximise the likelihood of learning.

“Exposure to variation is critical for the possibility to learn, and that what is learned reflects the pattern of variation that was present in the learning situation.” (Runesson)

Directing the children to look for and think about possible connections.

Transforming the learner

“The primary aim of every teacher must be to promote the growth of students as competent, caring, loving and loveable people.” (Noddings)

Research shows that attending to relationships in maths lessons helps to raise standards.

Inequality of attainment in the primary years may be more a result of children’s different experiences than their ‘innate’ mathematical ability.

You find yourself in the flow with an optimum level of challenge that stretches your capabilities.

Build a classroom culture that emphasises listening to each other, working together and trying things out rather than waiting for the teacher to provide help.

Any worthwhile mathematical experience is going to lead at time to some difficult emotions: frustration, confusion and irritation. Confusion is a necessary part of learning mathematics and can never be removed from the process.

Building mathematical community

“A sense of belonging, of continuity, of being connected to others and to ideas and values that make our lives meaningful and significant – these needs are shared by all of us.” (Sergiovanni)

Mathematical communities needs to promote: trust, friendliness, inclusion as well as resilience, perseverance and curiosity, and be inviting, engaging and welcoming.

Work towards building and creating shared goals and values, rather than imposing rules and regulations that create an orderly class but not a community.

Learning needs periods of incubation – over more time.

‘Sharing’ needs the vital component of ideas and solutions being built upon by other learners.

Tasks, Tools, Talk

Maths not based on procedural fluency but involves understanding means learners are active constructors of knowledge, not passive recipients of it.

Setting up tasks with a certain amount of uncertainty is a way to make learners engage mindfully and bring their sense making to the activity.

Introducing models (10 frame, numberline, arrays…) takes time. Learners will only appreciate them through repeated exposure, and it takes them different time to take them on as tools for thinking…they need to be part of the pedagogical furniture of the classroom.

Talk is central to maths lessons…it mean mathematical vocab becomes part of classroom discourse.

Making sense of problems by explaining them to someone else, putting them in your own words and comparing your answer with others all help meaning to emerge.

Crucial Primary Maths Knowledge

So, in the news this week Nick Gibb has confirmed that from 2019, Year 6 pupils will undertake a Times Tables test alongside their other SATs tests.

Nick Gibb Times Tables test announcement

“Multiplication was a “very important” part of a child’s mathematics knowledge, Mr Gibb said…It is my view that there should be a multiplication check.”

To be fair I don’t disagree at all with Nick Gibb’s view that an accurate and quick recall of times tables facts (and the linked division, fraction, decimal and percentage facts) are very important. Not to pass a test (or check) but to allow pupils to focus on application of these facts when undertaking complex or lengthy calculations and problem solving. In my opinion having a secure and accurate recall and understanding of some basic mathematical knowledge is crucial in order for pupils to think and work as mathematicians.

After all if someone was learning a musical instrument they would need to know key information, such as how to play certain notes and how read music before we could expect them to play whole pieces of music fluently and expertly.

I’m not sure I even have a problem with there being a test in Year 6. By then all pupils should know these facts. But what about other facts…

Since the introduction of the Phonics screening check in Y1 and Y2, schools have invested a great deal more time on teaching phonics. Again personally I think this has had benefits, but it has also potentially minimised time and focus on other aspects of reading, and other strategies required to become a fluent and confident reader.

So what are the other aspects of Crucial Primary Maths Knowledge? And will some of these be sidelined to some extent in the drive to show high achievement in a national test linked to school accountability?

At our school we have a series of “Maths Learn Its” that go home each term (these can be viewed at Maths Learn Its or on the Numeracy Shed, thank you @grahamandre). Within school we have ‘Regular Drip’ time, which is when what we think are key reading, writing and maths knowledge is practised during registration times, and in those 5 minute slots that sometime appear before lunch or going home time.

These are then balanced with the pupils being engaged in more contextual practice and application in more open-ended problem solving lessons.

For me Crucial Primary Maths Knowledge would include:

  • Counting on and back (in different amounts: 1, 2, 5, 10, 100, 1/2…)
  • Finding 1 more or less (moving onto 10, 100, 1000, 0.1…)
  • Number bonds to 10 (and all single digit numbers) (moving onto to 20, 100, 1000, 1…)
  • Place Value knowledge and understanding, initially Tens and Ones (moving onto Hundreds, Thousands… and Tenths, Hundredths)
  • Time tables to 10 x 10 (and learning how these link to division facts, fractions, decimals and percentages)
  • Doubling and halving
  • Multiplying and dividing by 10, 100 and 1000

 

In his excellent book “Transforming Primary Mathematics” Mike Askew (@mikeaskew26) explains his view on ‘Elements of fluency’ he states that:

“In moving up through the years of primary mathematics children are hampered if they are not fluent in

Elements of fluency

  • adding or subtracting a single digit to any number
  • adding a multiple of 10 or 100 to any number
  • counting on or back in ones from any starting number
  • counting on or back in twos, tens or fives from any given number
  • recalling rapidly the multiplication facts up to 10 x 10
  • multiplying any number by two or ten”

He then goes on to share a second set of skills which he calls ‘Procedural fluency’. He states that these would include:

“Procedural fluency

  • knowing what to add to a number to make it a multiple of 10 or 100
  • halving any number
  • multiplying any number by five ( by multiplying by ten and then halving)
  • knowing the division facts associated with multiplication facts”

 

I do wonder what the views of other primary colleagues are. If you had to pick a Top 5 aspects / sets of maths knowledge / skills for your pupils to be absolutely secure, fluent and confident with by the end of Year 6 what would they be. I’d be very interested to hear.

 

Teaching, Learning and Assessment morning

I am running a free morning of CPD for local primary teachers on Friday 27 January 2017. It is taking place at Cornerstone CE Primary school (PO15 7JH) in Hampshire (Junction 9 off the M27).

I will be sharing our journey so far in developing our Teaching and Learning practice and policy, and linked Assessment procedures (since September 2014). Colleagues attending will hopefully be sharing their ideas, the practice in their classroom and schools, and hopefully we will all go away with more ideas and greater clarity.

I have attached a copy of the presentation below, but undoubtedly the professional dialogue will be the most valuable aspect of the morning.

If you live or work locally and would be interested in joining us, you would be very welcome.

Please contact the school on 01489 660750 or adminoffice@cornerstoneprimary.hants.sch.uk to book a place.

Teaching Learning Assessment 27.1.2017

 

Top 5 posts of 2016

Below is a list of the most popular posts on my blog during 2016.

#teacher5aday #wintercalendar

My December contribution with @vivgrant to @MatrynReah’s important Teacher Wellbeing initiaive.

Assessment Journeys 2016

The principles and processes behind our school’s developing Assessment practices, whihc aim to focus securely on the learners and making it useful and manageable for teachers.

Learning First

My thoughts about the conference I attended in September orgainsed by @AlisonMPeacock and @JuleLilly to focus on Assessment Beyond Levels.

TLT 16

A summary of the thoughts shared by a range of speakers at this year’s event in September at Southampton University.

Big Ideas in Primary Maths

A summary of our staff’s professional learning and development from a day with @mikeaskew26. Thought provoking, insightful and highly helpful.

Maths Learn Its

In seeking a balance between quick recall and more in depth understanding we are developing a range of Maths ‘Learn Its’ posters. The aim is that each week within class one is focused on during a 5-10 minute slot or two during registration. It is also that they are shared with parents each term or half-term as ongoing practical maths Home Learning (that don’t require work coming in to teachers to be marked, but give parents guidance on how they might encourage their children to think and talk about maths in the world around them).

We are completing the ‘Learn Its’ for each term for Y1-Y4 as we go through this academic year, and are very happy for other schools to use, adapt, improve or ignore them. If any colleagues have any suggestions, improvements, or similar sheets / posters we would very much appreciate having a look at them.

yr-autumn-learn-its

yr-spring-learn-its

YR Summer Learn Its

 

y1-autumn-1-learn-its

y1-autumn-2-learn-its

y1-spring-1-learn-its

Y1 Spring 2 Learn Its

Y1 Summer 1 Learn Its

 

y2-autumn-1-learn-its

y2-autumn-2-learn-its

y2-spring-learn-its

Y2 Summer Learn Its

 

y3-autumn-learn-its

y3-spring-learn-its

Y3 Summer Learn Its

 

y4-autumn-learn-its

y4-spring-learn-its

Y4 Summer Learn Its

 

 

 

 

Deepening Learning with SOLO

As part of our school’s development of both our understanding and professional practice of ‘Mastery‘ of the curriculum for all and ‘Deepening‘ or ‘Enriching‘ of learning for our ‘Higher Attainers / Deeper Learners’, we have been creating SOLO Learning Journeys to accompany each domain the Maths National Curriculum.

However to ensure we are sufficiently challenging our ‘Higher Attainers / Deeper Learners’ we are drafting an overview for using SOLO across the curriculum and within individual lessons. The first draft is attached below. It includes: our school’s definitions for ‘Mastery’ and ‘Higher Attainers / Deeper Learners’; how our school’s Learning Values fit within the structure; and the type of Learning Behaviours would should plan to provide for and expect to observe in our learners.

We would be interested for any thoughts or feedback from other colleagues on similar journeys. Many thanks.

solo-deepening-learning

 

 

Learning Journey Prompts

Questions to encourage pupils to think about and reflect on their learning (not what they are or have been doing). The power of meta-cognition: getting them to think about their thinking.

Linked to idea of packing / going on / unpacking at the end of a journey. (They’ve all got a memory of a car journey on holiday!)

learning-journey-prompts-prism

 

 

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