Below are the notes that I wrote as I listened to the speakers at @BrewEdMaths 2020. If you want further clarification on a point, it would be best to contact the speakers directly, as they will be able to elaborate much better than I will. The notes are written in a bullet point format, as that was how I took them on the day. Some talks were more visual than others, so there may be fewer notes – that does not mean the talks weren’t as good! Thank you again to all the speakers who took part and everyone who came. We are in discussion about making it an annual event!

Dr Helen – @helenjwc – ‘Pass It On’

- Helen’s talk was mostly about the use of dialogic talk in EYFS and how we should make the maths to do with the children, including building their decision-making, curiosity and predictions.
- She got a few groups to do an activity known as ‘Pass It On’, while the rest of us observed and had to comment on what we saw happening.
- In the game, each person playing had a handful of items. They would roll a die and then give/take the number of items on the die to/from another person.
- The whole activity was focused around talk and how the individuals had to engage with each other – e.g. when one person rolled a 4, but they only had 1 item left, they had to engage in conversation with another person in their group about borrowing from them.
- Helen explained that maths games should focus on the maths and not on the rules.
- Think about having a single rule and children then negotiate around it through the maths they are learning
- She went on to explain that a review body for ELGs had secondary school teachers on it and that she didn’t think this was fair as she wouldn’t put herself forward for a GCSE board as an EYFS specialist.
- She suggested we read up on ‘Re-proposal’ from Italian preschools – which is about turning children’s thinking back on them
- The aim of her game was to increase dialogue between pupils – “Vocabulary isn’t the issue, dialogue is the issue. We need more dialogue in maths”.
- Message her if you would like to see research about the ELGs pilot
- “Mathematics could be like rollerblading, but usually it’s like being told to stop rollerblading and come in and tidy your room”. (Winter, 1992).

Matt Swain – @mattswain36 – ‘25’

• Matt talked about the idea of ‘corridor kids’ – children who towards the end of primary school can spend most of their school day in booster interventions, often in poorer environments like school corridors, away from their peers. He noticed from speaking to these children that they seemed disenfranchised and they felt separated from their class.

• 25% of children (mean taken from the past 4 years) didn’t make the EXS standard at KS2 leaving without life maths – see that as being 25% worse off than their peers.

• ‘Mastery’ – this means all children can understand the same concept, but not necessarily at the same time.

• Pareto principle – 80% of effects come from 20% of causes

• The only currency we have in school is time

• Teach, do, practice, assess, behave. Schools don’t encourage enough of the ‘behave’ part. What Matt means by that is children using what they have learnt as a behaviour – applying it themselves.

• Marking is a dirty word – but it does have value when done in a lesson. Marking after the fact is a waste of time.

• Matt suggests there 5 things to focus on:

1.) Hardwiring facts – there is still place for learning facts – e.g. 7×8 came up 3 times in the SATs last year

2.) Worked examples – the 25% need these. Problem pairs – show the class’ worked example alongside an example for the class to solve

3.) Error tracking – when looking at whiteboards get each row/group/table to go down one row/group/table at a time. Everyone must put them up at the same time, so they don’t copy each other. Give them space for their answer to be put down very clearly on worksheets – e.g. a clearly separate box. During work, don’t talk to kids as you go around, instead just note down misconceptions and then deal with it all afterwards so you can get to every child during the activity.

4.) Exit ticket – single multiple choice Q with 4 possible answers on the thing you really wanted them to learn. Return them to original learning. Make the 3 wrong answers misconceptions you have seen during the activity/ones you know children traditionally have.

5.) Same day keep up – any child who needs an extra 20 mins stays with you after lessons, like during assembly that day – giving the 25% that time they need. This can prevent them from being separated from the rest of their class during lessons in the afternoons. This way every day the 25% get to feel success.

David O’Connell – @doctalkteach – ‘Lesson study and the 5 domains’

• Focus: challenge for all

• Follow @ban_har – the ‘yoda of maths’ – wrote Singapore Maths textbooks

• Lesson study has replaced performance management at David’s school

• You need a knowledgeable other in order to undertake lesson study – this can be a person, a book to refer to etc – something to refer and compare to

• Study, plan, observe, reflect, revision, reteach – 3 day cycle

• His school uses Singapore maths

• 5 part lesson – the anchor task, journalling, reading and reflection, guided practice, independent practice

• A focus on dialogue – talking together, helping each other – teacher enriches by going around and supporting

• There are research questions that guide the lesson study – how do we ensure all children understand the lesson whilst also challenging the advanced learners? What do generalisations look like and how are these addressed so misconceptions are not embedded? How are rules formed?

• Time – allow ample processing time for exploration, look at a problem informally and at low stakes, and eavesdrop on pupil conversations

• “The hardest thing to do as teacher is to shut up and eavesdrop”

• 5 domains of challenge is the differentiation

1.) Cognitive – the actual maths

2.) Metacognitive – pupils being able to recognise mistakes, recognising and selecting the most effective methods

3.) Social Collaboration – contributing to the learning of others

4.) Affective – showing perseverance, restraint and enjoyment within the lessons

5.) Mindsets – having a productive set of beliefs

- Metacognition is built through children writing journals, of which there are 4 types:

- Descriptive – students explain different ways to attempt a question
- Evaluative – students make a judgment of efficiency of select methods
- Investigative – students may be asked to explore if a certain method will always work
- Creative – students created their own problem for a friend to solve

- Ask chn to break down their method into 5 steps. Often children will just say, “I don’t have a method, I just know it”.

• Take focus off marking and change it to dialogue in class

• “Learning floats on a sea of talk”- Britton (1970)

• David recommends you read Gaunt and Stott’s book –*Transform teaching and learning through talk*

• Dialogic teaching – it should be collective, reciprocal, supportive, cumulative, purposeful (Alexander, 2010)

• Try to get children to ask more questions. The proportion of questions is often heavily swayed in favour of the teacher. Kids do ask questions a lot in maths, but are they questions that will further their learning?

Andrew Jeffrey – @AJMagicMessage – ‘Magic, Maths and the Rosetta Stone’

(The notes on this talk aren’t as detailed as I spent most of the time laughing!)

• John Holt said, “Maths is not about what you know, but how you behave when you don’t know” – Andrew believes maths to involve a lot of magic

• We should visualise our mathematics

• Numicon pieces weigh the same – one 8 piece weighs the same as a 3 piece and a 5 piece (mindblown!)

• It is all about the language we present and how we code it (saying 21 = 3x rather than 3 x 7 = 21)

• Algebraic reasoning can be done very young, especially when using the language as shown above

• “You can’t do arithmetic unless you learn algebra first” – Caleb Gattegno

• Comparing the Rosetta stone with algebra – the Rosetta stone had hieroglyphics that they couldn’t translate, but they used two of the languages on it (ancient Greek and Demotic) to translate it – we can do that with algebra using Numicon – If a 2 piece and two envelopes (with Numicon inside them) weighs the same as one envelope and an 8 piece then 2 + 2e = 8 + 1e, which can be changed to 2 + 1e = 8 and so on

• “It’s important to get stuck at maths, even if you’re quite good at it!”

• Andrew made his case for being against the National Lottery and how slim your chances are of winning, although his friend Grant won about 10 prizes in the raffle!

• He has a new book out called ‘Greater depth in Primary Mathematics’ – you can buy it from his website here – https://www.amazon.co.uk/Greater-Depth-Primary-Mathematics-Strategies/dp/170169123X

Peter mattock – @MrMattock – ‘Let It Make Sense!’

It is very hard to recreate this talk as it used a lot of visuals, but these are the points I did take from it that are worth sharing. Peter made a great case for Cuisenaire rods and Algebra tiles and how versatile they are in explaining abstract concepts such as -1 x -1 = 1 and 2/5 ÷ 3/4.

- Mathematics makes sense – it is consistent within itself. Focus on this. It must always make sense.
- Peter recommends that we read up about ‘Arbitrary and Necessary knowledge’ (Hewitt, 1999)
- Arbitrary – all students need to be informed of the arbitrary by someone else e.g. the x axis is horizontal
- Necessary – some students can become aware of what is necessary without being informed
- Arbitrary – cannot be worked out, might be so
- Necessary – can be worked out, must be so (mattock’s emphasis)
- Patterns in maths are there to be explained
- 3 x 1 and -3 x 1 as reflection of each other or rotating 180 around the point of 0
- -1 x -1 = 1 and 2/5 ÷ 3/4 use visuals to teach these problems not mnemonics like KFC (keep it, flip it, change it)
- Just telling a child 2/5 ÷ 3/4 = 2/5 × 4/3 is not good enough
- What was clear from his talk was that the more increasingly abstract the concept/teaching becomes, the more important visual representation is (and it’s always important) – especially for fractions and algebra which we all know to be difficult topics for all ages
- Peter has a book out called Visible Maths which you can buy here – https://peter-mattock.visiblemaths.co.uk/search/label/Visible%20Maths%20Book

Jo Morgan – @mathsjem – ‘Bursting Our Method Bubbles’

- People are defensive about their methods – we need to burst our method bubbles – just because it is the one we learnt as a child, doesn’t make it the best one or better than others

• There are loads of interesting methods and people just don’t know about them

• Are you aware of how an individual concept is taught in India? Scotland? North of England? The classroom next door? It is worth researching this to broaden your knowledge.

• Speaking to a colleague might get you to change the methods that you teach

• 84 – 27 written method can be done as decomposition as 70 + 14

• Or equal addition like this 84 – 27 can be done as 94 – 37 (which was the dominant method before decomposition)

• She recommends reading ‘Relative merits of the methods of teaching subtraction’

• 9 different methods of subtracting in her book! And she recently learnt a 10^{th}! How many subtraction methods can you think of?

• Parents thinking methods are wrong because it’s not the method they learnt – being defensive about methods, we mustn’t let them influence the children we teach to think the same way

• Jo has a new book out (A Compendium of Mathematical Methods), which you can find here – https://www.amazon.co.uk/dp/1912906600Kieran @Kieran_M_Ed and Lloyd @lwilliamsjones – Purposeful Storytelling - The world we see is a representation applied to what we already know

• Language is a structure that is capable of creating infinite representations

• Late language model – evolutionary leaps for language came around the same time as retrieval ability came for homoerectus/homosapiens – an ape’s memory is purely episodic, ours is not.

• Willingham talks about the 4 Cs – narrative comes from these 4 Cs

• Character – how it influences meaning

• The meaning derived from the story depends in part to the character children relate to/identify with

• Consider a teacher’s role as a ‘real actor’

• The embodiment principle – we learn more when there is gesturing, movement, eye contact etc

• They recommend we read Greg Ashman’s blog

• Pupils should relate to characters

• Remembering is similar to person perception (perception of characters)

• Causality – output depends on current and past input

• In understanding, readers try to explain why characters act like they do

• Predicting doesn’t meant we are going to necessarily understand, but that we will be more likely to remember it

• Conflict – central character faces obstacles and they prevent the goal from being met (purposefully getting a question wrong by missing out a step or incorrectly performing a step could perhaps be an example of this)

• Stories with obstacles were found to be more memorable than those without

• Powerful narratives and their effect on memory and recall, not understanding

• Conflict provides a point of view through which you can remember the story

• Narratives are sticky in the memory

Bernie Westacott – @BernieWestacott – ‘’Genesis: The Birth of Number in Young Children (and rats and chickens)

- Clever Hans the horse from 1904 – he could tap out using his hoof any addition like 3 + 5. It could also do 2/5 + 1/2 and the divisors of 28 (it had been trained to react to its trainer’s eye movements)

• Poupette the dog could tap a paw to add digits and lick a hand to show it was finished

• Rats could press a lever a certain amount of times to release food – press too early and it would reset or the light would go out

• If it needed to press the lever 4 times, it would press between 3 and 7 times

• The bigger the number required, the bigger the range of their error 12: 16-27 – but they still demonstrated a sense of keeping track of the number of presses

• A chimp could compare two numerical amounts – possibly using an approximate comparison

• A chimp had pile of 4 and pile of 3 together and a pile of 5 and a pile of 1 together – after thinking for a while, it went for the piles of 3 and 4 – this suggests that the chimp carried out some form of arithmetic

• Distance effect (the distance between the numbers in two quantities) and magnitude effect (how large each quantity is) affects our ability to compare two quantities using our Approximate Number System (ANS) -ANS is a gross quantity system that does not result in an exact number

• To read more about the chimp experiment, search for Woodruff and Penmark

• Babies looking at slides of 2 dots/items – when they looked away, a new slide was shown (also 2) – the time they looked at each slide got less until a slide of 3 was shown. They spent longer looking at this 3 slide, indicating that they were aware that this was different to 2. This was odne with 6 -7 months old children and later replicated with babies a few days after birth - Clements and Sarama 2019 – check out their website ‘Learning Trajectories’ – including the subitizing song
- Subitizing is direct perceptual apprehension and identification of the numerosity of a small group of items without the need to count them
- Perceptual Subtizing – without counting, to know exactly how many there are when dealing with small quantities. Children can subitize 1, 2, 3 and even 4 objects. Well know patterns, such as 5 dots on a die, are easier to subitize than if the 5 dots were spread out randomly
- Conceptual Subitizing – 8 in a 10-frame is subitized in two groups – a full row of 5 and a group of 3 below that – which are then combined; or as 10 with 2 missing
- Manipulating numbers spatially through use of dice, dominoes and ten-frames plays a significant role in building number sense and understanding of arithmetic
- This can be used to develop addition strategies beyond counting in 1’s. For example, using two frames to model 9 and 5 which becomes 10 + 4 (moving 1 over from the 5 to the 9)
- Some children have a poor start from age 0-3 because parents don’t include a focus on number in their interactions with their children – thus, they may not develop Spontaneous Focus on Number (SFON) tendencies. This may also be the case with children in other environments, such as day care centres
- Research shows that intervening with children as young as 3 to increase their SFON can result in signficant improvement in their SFON and cardinality recognition. The more they notice number all around them, the more practice they get with number, which results in further mathematical development

Follow @BrewEdMaths for all things maths and the BrewEdMaths team – @MorgsEd, @MissSDoherty, @mattswain36, @lwilliamsjones, @MrAlmond_Ed, @Kieran_M_Ed, @RebeccaTurvill