Showing posts with label national curriculum. Show all posts
Showing posts with label national curriculum. Show all posts

Thursday, 12 December 2019

Beware The Reverse-Engineered Curriculum (or The Potential Pitfalls Of Going For Retrieval Practice Pell-Mell)

Aren't Knowledge Organisers brilliant? Isn't Retrieval Practice just the bees knees? As for Powerful Knowledge... sigh - the stuff dreams are made of.

Over the last couple of years, many of us have taken the outcomes of relatively recent research and applied it to the way we teach and we are pleased with how it's going - children are actually learning, retaining and retrieving information, something which, if we're honest, didn't always happen before.

But what seems to me to have happened is that we have found something that actually, reliably works, and we have made our curricula work for it. We've realised that retrieval practice does make a difference and we've begun to design a curriculum which focuses on what can be learned by that method.

Sure, there are other arguments for teaching Powerful Knowledge - it's supposed to ‘enable students to acquire knowledge that takes them beyond their own experiences’ (Young et al., 2014, p. 7) and for this reason it is often held up as essential for children from disadvantaged backgrounds who we see as needing to be upwardly-mobile socially. Powerful Knowledge is seen to be the answer to closing the gap between the rich and the poor.

Perhaps it isn't even Powerful Knowledge that we're even talking about here, but actually what is known as Declarative Knowledge - facts and information stored in the memory. I'm not sure Powerful Knowledge could be defined so narrowly. Powerful Knowledge is supposed to be more than that.

Whatever terminology we use, the focus on knowledge is potentially narrowing what is taught and how it is taught. And this narrowing is perhaps now coming as a result of the most vaunted method of teaching the knowledge from a knowledge-rich curriculum: retrieval practice. And more specifically, the activity that seems to have become synoymous with retrieval practice: quizzing.

Because retrieval practice works so well, we seem to have searched out the things that best suit the method: the sort of information that can be retrieved and recited e.g. history dates, word definitions, geographic processes, scientific theories and outlines of philosophical concepts, for example.

But is it right to have allowed a certain understanding of the concept of Powerful Knowledge to reduce our curriculum to the identification of exactly which pieces of information we are going to teach, and to the teaching of that information using retrieval practice techniques?

Indeed, possessing Powerful Knowledge, and being able to retrieve it, is supposed to be so much more than rote learning; more than memorising and regurgitating facts. Done right, it should, lead to better understanding and it should improve complex thinking and application skills.

But is an approach to teaching and learning which is satisfied by children who can simply recall information really what's best? The recall of information is not the end of the story. A high score on a (supposedly low stakes) quiz should not mean job done.

Actually, I'd argue, once the facts are memorised, that's when the real teaching and learning begins. Once children know the facts, that's when they can start to use and apply them in various ways. Powerful Knowledge is supposed to allow us to generalise and use what we've learned to think beyond the immediate context.

And don't forget procedural knowledge - how to do things. You can't teach writing, artistic techniques or how to use a tool just by teaching facts. Any time we model something (and this is something we should do a lot in teaching) we are teaching procedural knowledge. This procedural knowledge is separate from 'common sense' knowledge which we gain from everyday life, and therefore lands it more in the realms of Powerful Knowledge. However, the focus on bolstering the curriculum with declarative knowledge has the potential to leave procedural knowledge behind. A balanced approach is needed.

Does your recently-re-written curriculum (all 'i's dotted - intent, implementation and impact) allow the learning to go beyond the retrieval of facts? Or has a child who has learned all the dates on the knowledge organiser and filled in the gaps in the booklet succeeded in all you set out to do? And did that child get the opportunity to do anything else that half term, or did they spend all of their time ensuring they knew all the facts?

If we reverse-engineer the curriculum based on one teaching method which allows one aspect of what might be taught to be taught, then we might not end up with the curriculum we really need.

Those of us involved in curriculum design or review should think carefully about the problems we are trying to solve in order to decide on the criteria we need to set for our curriculum development. We should definitely identify the declarative knowledge we want children to learn - the facts, the information - but we should also be thinking about other kinds of knowledge too.

Things to think about

  • Don't narrow your curriculum down to just the things that can be learned through quizzing. 
  • Think more broadly about how a greater range of retrieval practice techniques can be used to help children learn things from a wider knowledge base than just simple facts and figures. 
  • Ensure that your curriculum really meets the needs of the children in your school and let them be your starting point.

Friday, 29 November 2019

History Key Questions To Ask When Learning About A Person, Event or Period in KS1

Last year I provided a list of Key Questions linked to the KS2 National Curriculum. Since then , a few people have asked for a KS1 set of questions. When it came up as a potential need in my own school, I decided to act.

Here are a set of questions, split into three categories (People, Events and Periods of Time) that teachers can use to structure their planning and teaching. Some of the questions may be suitable for children to ask themselves, others might be better used as guidance for teachers as they plan content. Many of the questions across the three categories are very similar although there are one or two more category-specific questions.

to download these questions as a Word document, go to TES.com: https://www.tes.com/teaching-resource/key-questions-to-ask-and-answer-during-ks1-history-units-12217633

People

Questions to ask about historical figures who are studied in years 1 and 2:

Characteristics:

What are the most important facts about this person?
What do these important facts tell me about this person? (focus on understanding, rather than knowing facts)

Where (linked to KS2 Elsewhere):

Where did this person come from?

Evidence:

How do we know about this person?

Significance:

What did this person achieve or help to achieve?
Did this person’s actions change anything for the future? How did they make a difference?

Timeline:

When in history did this person live? (birth dates and death dates)
Did this person live before or after [another person/event they have studied] lived/happened?
How many years before or after [another person/event they have studied] lived/happened did this person live?
What period of time did this person live in?
Did this person live within or beyond living memory? (living memory: can be remembered by people who are still alive now, not children’s own living memory)

Events

Questions to ask about historical events which are studied in years 1 and 2:

Characteristics:

What are the most important parts of (key facts about) this event?
What do these key facts tell me about this event? (focus on understanding, rather than knowing facts)

Where (linked to KS2 Elsewhere):

Where did this event take place?

Evidence:

How do we know that this event happened?

Significance:

Did this event change anything for the future? How did it make a difference?

Timeline:

When in history did this event happen? (day/month/year(s))
What period of time did this event happen in?
Did this event happen before or after [another person/event they have studied] lived/happened?
How many years before or after [another person/event they have studied] lived/happened did this event happen?
Did this event occur within or beyond living memory? (living memory: can be remembered by people who are still alive now, not children’s own living memory)

Periods of Time

Questions to ask about historical periods of time which are studied in years 1 and 2:

Characteristics:

What is similar about the way people lived in this time period and [another time period they have studied]?
What is different about the way people lived in this time period and [another time period they have studied]?
What are the most important things (key facts) to know about this period of time?
What do these key facts tell me about life in this period of time? (focus on understanding, rather than knowing facts)
What important events happened in this time?
Which important people lived in this time?

Where (linked to KS2 Elsewhere):

Did the things that happened in this time period happen in a particular place?
Were things the same everywhere in the world during this time period?

Evidence:

How do we know about this period of time?

Significance:

How did life change during this period of time?
Did this time period change anything for the future? How did it make a difference?

Timeline:

When did this period of time begin and end? (specific years and approximate number of years duration)
Was this period of time before or after [another person/event /time period they have studied] lived/happened?
How many years before or after [another person/event/time period they have studied] lived/happened was this period of time?
Did this period of time occur within or beyond living memory? (living memory: can be remembered by people who are still alive now, not children’s own living memory)


Also available:

Geography Key Questions for KS1 and 2: http://www.thatboycanteach.co.uk/2019/06/geography-key-questions-place-national-curriculum.html

Friday, 14 June 2019

Geography Key Questions To Ask When Learning About A Place

When reviewing and revising our curriculum to ensure better humanities coverage I began to think about how there were potentially missed opportunities for children to be revisiting and re-enforcing geography knowledge and skills.

For example, when children learn about the Ancient Romans, they might learn about Italy, or indeed any of the other places in the Roman Empire. As well as ensuring that children can place the historic period on a timeline and so on, I think it is also worth them knowing about the places where events took place.

I began to think that a common approach to learning about these places might help teachers to provide sufficient and consistent information about them. I decided a set of questions that could be asked and answered whenever a new place was 'discovered' might be a good way to structure this common approach.

The hope is that, with these questions, children will begin to build up a) a good knowledge of the world, and b) a good bank of questions that they might begin to ask more autonomously whenever they come across the mention of place that they do not know much about.

I used the National Curriculum as guidance for the following questions with the intention that NC objectives would be covered multiple times during a child's time in school.

As well as sets of questions, I've also proposed some actions that might be undertaken each time a new place is learned about - one of the main aims of these actions is that children know where in the world each place that is studied is located.

A downloadable version of the below is available for free on TES: https://www.tes.com/teaching-resource/geography-key-questions-linked-to-ks1-2-national-curriculum-12133248

Thanks must go to Geography Meg on Twitter for inspiring me with her CLOCC acronym.

KS1

When learning about a new place (for example, during non-geography-based units, such as history-based units) always ask and answer these questions:

COWWS:

  • CONTINENT – Which continent is it in?
  • OCEANS AND SEAS – Which oceans or seas are nearby?
  • WEATHER – What is the weather like there? Is it hot or cold there? Is it near the equator or the poles?
  • WHO AND WHAT – Who (people) and what (animals and plants) live there?
  • SEE – What would we see there? What is natural? What has been made by humans?

A pre-populated COWWS grid - a blank
version can be found in the TES download.
When learning about a new place (for example, during non-geography-based units, such as history-based units) always carry out these actions:

• 1st: Locate it on a map of the county/region it is in (and show and discuss, using simple compass directions and locational language, where it is in relation to: other places previously studied; our country; our location)
• 2nd: Locate it on a map of the country it is in (and show and discuss, using simple compass directions and locational language, where it is in relation to: other places previously studied; our country; our location)3rd: Locate it on a map of the world (and show where it is in relation to: other places previously studied; our country; our location)
• Locate it on a globe (and show and discuss, using simple compass directions and locational language, where it is in relation to: other places previously studied; our country; our location)
• Locate it on a plan perspective or on aerial photographs
• Show images of the place (avoid only showing stereotypical images, especially when studying a whole continent or country)

KS2

When learning about a new place (for example, during non-geography-based units, such as history-based units) always ask and answer these questions:

General questions to ask about location:

HOTCLUB:

  • HEMISPHERE - Which hemisphere(s) is it in?
  • OTHER PLACES - Where is it in relation to other places we have studied or know about, including countries and continents (using 8 points of a compass)?
  • TIMEZONE - Which timezone(s) is it in?
  • CLIMATE - Which climate zone(s) is it in? (Tropical/Dry/Temperate/Continental/Polar)
  • LATITUDE - Where is it in relationship to the main lines of latitude (using 8 points of a compass)? (Arctic Circle/Tropic of Cancer/Equator/Tropic of Capricorn/Antarctic Circle) What is its latitude and longitude?
  • US - Where is it in relation to our village/town/city/county/country?
  • BODIES OF WATER - Which bodies of water are nearby?


A pre-populated HOTCLUB grid - a blank
version can be found in the TES download.
Questions to ask about the location…

…Of a continent:

• Which countries are in this continent?

…Of a country:

• What is the capital city?
• Which major cities are in this country?
• Which other countries are nearby?

…Of a city/town/village:

• Which country is it in?
• Which continent is it in?
• Which other cities/towns/villages are nearby?
• Which county/region is it located in?
• What is its grid reference?
• What are its origins?

General questions to ask about any continent/country/city etc:

Human Geography

• Who lives there?
• Which major landmarks are found here?
• What human-made features are found here?
• How was the land used here now and in the past?
• What types of settlement are found here?
• What kinds of economic activity happen here?
• Which natural resources can be found here?
• What is its population?
• (If studying a country) What do they export and where do they export it to?
• (If studying a country) What do they import and where do they import it from?

Physical Geography

• Which (terrestrial) biomes are found here? (Rain Forest/Deciduous Forest/Desert/Temperate Grassland/Tropical Grassland/ Taiga/Tundra)
• What lives there?
• What is the elevation like?
• Which major rivers and valleys are found here?
• Which major mountains are found here?
• Which natural disasters are known to happen here?

Additional, non-essential questions to ask (a non-exhaustive list):

• What is the place famous for?
• What kind of food is eaten there?
• Which religions are followed there?
• Which famous people are from there?
• What are houses and buildings like there?
• What happened there in the past?
• Which sports are played there?
• What is it like to live there?

Geography Units

If carrying out a geography-specific unit use the majority of the questions from STEEP in addition to the above questions in order to ask more in-depth questions about the place:

• Social
• Technological
• Economic
• Environmental
• Political

(Thanks to Geo Josie on Twitter for STEEP)

When learning about a new place (for example, during non-geography-based units, such as history-based units) always carry out these actions:

• 1st: Locate it on a map of the county/region it is in (and show where it is in relation to: other places previously studied; our country; our location; lines of latitude; hemispheres)
• 2nd: Locate it on a map of the country it is in (and show where it is in relation to: other places previously studied; our country; our location; lines of latitude; hemispheres)
• 3rd: Locate it on a map of the world (and show where it is in relation to: other places previously studied; our country; our location; lines of latitude; hemispheres)
• Use computer mapping (e.g. google maps) to zoom in to and out of the place, discussing location in relation to other known places
• Locate it on a political map (and look at nearby countries and borders)
• Locate it on a physical/topographic map (and look at elevation, mountains, rivers, bodies of water)
• Locate it on a climate map (and look at the colours used to show different climatic areas)
• Locate it on a map with a satellite image overlay
• Locate it on a globe (and show where it is in relation to: other places previously studied; our country; our location; lines of latitude; hemispheres)
• Locate it on an Ordnance Survey map (and identify its grid reference and use symbols to locate local features)
• Show images of the place (avoid only showing stereotypical images, especially when studying a whole continent or country)

Useful resources for preparing information to help children answer HOTCLUB questions:

HEMISPHERE - Which hemisphere(s) is it in?

Use Google: type in ‘ + hemisphere’ e.g. ‘Algeria Hemisphere for the following, or similar:


OTHER PLACES - Where is it in relation to other places we have studied or know about, including countries and continents (using 8 points of a compass)?


https://www.climate-zone.com/ - gives information about where any country is located in relation to nearby places


TIMEZONE - Which timezone(s) is it in?


Use Google: type in ‘ + timezone’ e.g. ‘Madagascar timezone’ for the following, or similar:



https://www.timeanddate.com/worldclock/personal.html - use this customisable site and the snipping tool to get a snip of clocks showing GMT and any other city in any other time zone e.g.:



https://www.worldatlas.com/ - click on a continent/country; use the ‘time’ tab to find out about timezones


CLIMATE - Which climate zone(s) is it in? (Tropical/Dry/Temperate/Continental/Polar)


Tropical (A)/Dry (B)/Temperate (C)/Continental (D)/Polar (E) are the main 5 classifications in Köppen’s climate classification system. More information can be found at the following sites:


https://en.wikipedia.org/wiki/K%C3%B6ppen_climate_classification


https://www.mindat.org/climate.php - use the colours on the map to ascertain climate zone


https://www.nationmaster.com/country-info/stats/Geography/Climate - some of the terminology here is slightly different to Köppen’s climate classification system but it is useful nonetheless.


LATITUDE - Where is it in relationship to the main lines of latitude (using 8 points of a compass)? (Arctic Circle/Tropic of Cancer/Equator/Tropic of Capricorn/Antarctic Circle) What is its latitude and longitude?


https://www.climate-zone.com/ - gives the geographic coordinates for any country


https://www.worldatlas.com/ - use a world map from this site (or similar) to find and describe a place’s location in relation to the main lines of latitude. This site also gives latitude and longitude of any country and its main cities.

US - Where is it in relation to our village/town/city/county/country? 


Use a map to locate both places and describe position using compass points with reference to the prime meridian (Longitude 0º, Greenwich Mean Time) as this runs through the UK e.g. ‘India is to the south east of the prime meridian making it south east of the UK’


Use Google: type in ‘ to distance’ e.g. ‘UK to India distance’ for the following, or similar (add ‘km’ to search to change unit of measurement):



BODIES OF WATER - Which bodies of water are nearby?


Use a map to locate nearby major oceans and seas


Use Google: type in ‘rivers longest’ or similar e.g. ‘rivers France longest’ for the following, or similar:



Generic resources, useful for answering most of the questions:


https://www.climate-zone.com/ - gives official name, capital, area, climate, location, geographic coordinates, comparative area, land boundaries, coastline (length), terrain and elevation extremes for any country.


https://www.worldatlas.com/webimage/flags/flags.htm - flags of the world


https://www.worldatlas.com/ - for each country there are various maps (including physical, outline and location maps), basic information (including area, population, population density, currency, largest cities), history, famous natives, weather, general facts, historical timeline and more.


https://www.mapsofindia.com/worldmap/ - a wide selection of world maps can be found here including physical, political, outline, climate, tectonic plates, world time and religion maps (these maps cannot be downloaded but can be viewed on screen)


https://askabiologist.asu.edu/sites/default/files/resources/articles/biomes/world-biomes-map.gif - a useful map of the world showing biomes closely matched to the biomes selected to use (its accompanying article is useful too for defining the different biome types: https://askabiologist.asu.edu/explore/biomes)


https://www.internetgeography.net/wp-content/uploads/2017/12/world-map-biomes-.png - a useful map of the world showing biomes closely matched to the biomes selected to use (its accompanying article is useful too for defining the different biome types: https://www.internetgeography.net/topics/what-is-a-biome/)


https://www.britannica.com – use this site to search for information about a place – best for countries, but also works for cities too – information for towns is limited.


https://www.kids-world-travel-guide.com/


https://www.natgeokids.com/


https://tutorful.co.uk/blog/learning-geography-useful-websites-and-resources-that-will-rock-your-world - a huge list of websites and apps to support geography learning

Wednesday, 10 April 2019

From the @TES Blog: 7 Ways To Make Art Inspiring At Primary

Actually '7 Ways To Choose Artists, Artworks and Artistic Movements For Your Primary Curriculum' would have been a more accurate title...



As well as developing skills within the realms of drawing, painting and sculpture, and producing their own creative works of art, children, according to the National Curriculum, should also ‘know about great artists, craft makers and designers, and understand the historical and cultural development of their art forms’.

The artwork of others is one of the greatest sources of inspiration and information when it comes to children learning, and then applying, skills to their own pieces of work. So it is important that we teach them to appreciate the human creativity and achievement of the world’s most renowned artists.

But it is also important that the artwork, artists and artistic movements we expose our children to isn’t just a line up the usual suspects. A planned, whole school approach to developing an art curriculum is essential. Perhaps due to a lack of knowledge or time, teachers will often wheel out the same old, same olds, often meaning that children see similar things over and over again.

Read more here: https://www.tes.com/news/art-curriculum-inspiring-primary

Thursday, 7 March 2019

From The @TES Blog: 6 Ways To Get D&T Right At Primary


“Design and technology is an inspiring, rigorous and practical subject…”

But is it really, in your school? I mean, it could be. But how do we ensure in primary schools that it is?

Read the article here: https://www.tes.com/news/6-ways-get-dt-right-primary

Friday, 21 December 2018

A Model For Teaching The Wider Curriculum

After two half terms in my new school, and since curriculum planning and delivery is a hot topic, I thought I'd share a document I put together to help us visualise and explain the logistics of how we teach the wider curriculum.

The purpose of organising the delivery of the curriculum this way is to achieve the following, which we consider to be aspects of the school's culture which help us to deliver on our vision and values:
  • Responding to misconceptions through same day intervention
  • Setting children learning challenges (Apprentice Tasks) that are open-ended and encourage decision making (and time management)
  • Setting up inspirational areas of provision within the environment
  • Providing frequent masterclasses which communicate age-appropriate skills in all areas of the curriculum
  • Supporting children to critique their own work and that of others
By delivering the curriculum this way we hope to ensure that all areas of the curriculum are covered and that they aren't being squeezed out by Maths and English. We also hope that as a result of taking this approach children will not produce near-identical pieces of work. As well as this we aim to provide children with less structured time which gives them opportunities to engage in decision making and time management. Because there is less structured time than in a more traditional timetable teachers are also freed up to spend time on same day interventions based on feedback gained during all lessons, including Maths and English.

The green sequence shows what the adults are doing during the sessions; the turquoise sequence shows how children are grouped.
Units of Work

Each unit of work runs for one half term. The length of the half term will dictate the number of Apprentice Tasks set and the number of Masterclasses that take place.

Each unit of work is based on a book. Units of work cover National Curriculum objectives as well as objectives taken from the school’s own skills continuums for painting, drawing, clay work, woodwork etc. Long Term Planning documents ensure coverage of all objectives.

Each unit of work is also centres around a question which should be answered by the end of the unit using information learned during the half term.

Units of work usually cover a range of national curriculum subjects although there is often a predominant subject e.g. Space covers mainly Science but also some History and Geography, Castles covers mainly History but also some Geography. Currently most Science is taught discretely by a cover teacher during teachers’ PPA.

Key Fact Sheets

Knowledge teaching is supported by Key Fact Sheets which contain 10 key facts for the topic and 10 key pieces of vocabulary. This information is learned by heart supported by various retrieval practice activities. A Key Fact Sheet is produced per unit of work prior to the planning of the unit to ensure teachers know what it is they want children to know by the end.

Facts on the Key Fact Sheets should spark intrigue and should be a gateway to further learning. They should provoke children to ask questions and to want to find out more.

Key vocabulary words should be linked to the theme of the unit and should be words that will be used regularly in both spoken and written language during the unit. Child-friendly definitions should be written by teachers.

Diagrams and useful images may be included on the Key Fact Sheet.

The Showcase

The Showcase event provides an audience and purpose to all the apprentice tasks. It might be in the form of an exhibition, gallery, exposition or a screening. Alternative audiences/purposes might be a website, a tea party (e.g if the unit is formed around Alice in Wonderland) or a show. This event is decided upon before planning the Apprentice Tasks to ensure all tasks feed into this final event.

Apprentice Tasks and Masterclasses

Apprentice Tasks are open-ended tasks which allow children to operate with some freedom and creativity. However, each task has a set of objectives that should be demonstrated in the final piece. The expectation is that each child produces unique and original pieces of work.

Each Apprentice Task, or sequence of Masterclasses, is typically controlled by one member of staff: they source or make exemplars, research information further to the core information contained on the Key Fact Sheets, deliver the masterclasses and support children during the independent application stage.

One Apprentice Task might require more than one sequence of Masterclasses running consecutively. For example, an Apprentice Task which requires children to produce a painting might have two sequences of Masterclasses: drawing skills and painting skills.

During a Masterclass focusing on creative skills such as woodwork, painting, drawing or clay work, children will create studies which will help them to practise the skills they will need to complete the Apprentice Task.

Not all Masterclasses focus on skills teaching. There are also regular Masterclasses focusing on knowledge teaching, particularly linked to Science, Geography and History. These Masterclasses expand on the Key Facts from the Key Facts Sheets.

Some Masterclasses may focus on producing a final piece for an Apprentice Task – this would occur when children need more adult input, for example if it is too soon to expect independent application of the skills.

Some Apprentice Tasks may be group tasks, most are individual tasks.

Some Apprentice Tasks may be worked on as part of the English lessons, particularly where writing is a major component e.g. a script for a documentary, a poem, a story, a report. In this case, the Masterclasses become the whole class/half class teaching inputs.

Logistics and Organisation

Although a detailed Medium Term Plan is produced, logistical and organisational planning takes place weekly to ensure best use of time and adults. This might sometimes making decisions to provide whole class inputs rather than repeated group inputs, making decisions about length of time needed to complete a Masterclass carousel and so on. No two weeks look exactly the same where timetabling is concerned.

Most of this work takes place in afternoons once Maths and English has been taught. However, English is sometimes taught in half-class (or smaller) groups whilst some children complete a Masterclass or work on their Apprentice tasks.

Materials needed to complete Apprentice Tasks are readily available either in classrooms or in shared areas. Most of them are displayed in sight and not kept in cupboards – children can access what they need when they need it without needing to ask for it.

The Environment

As well as the Apprentice Tasks and the Masterclasses there are also further activities (linked to prior teaching in all subjects) which children can access (usually independently) during the time set aside for work on the wider curriculum. These will be set up in classrooms in the same way that Early Years classrooms have activities set up in areas of provision.

Equipment for all subjects is available to the children at all times enabling them to continue to practise skills learnt in Masterclasses.

The following are some images of the studio area we have developed outside of the classroom as an additional learning environment. The classrooms in year 5 are set up as fairly traditional classrooms with a bank of 5 computers each - the size of the rooms and the size of the children meant that to provide the aforementioned items in our environment we had to use some other space.







Monday, 25 September 2017

What Does 'Greater Depth' Look Like In Primary Maths?


What do we mean by 'Greater Depth' in maths? What would a child working at greater depth be doing? How can we support children to work at greater depth? With a little detective work we can piece together a good idea of what we might be talking about.

At first, we might think that to be working at greater depth in maths children should be fluent in their mathematical ability, and that they should be able to solve problems and reason well. But that can't be it as the National Curriculum states that those are the aims for ALL pupils:


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:

We might define children who work at greater depth as still working within the expected standard but at a deeper level; this is how the Interim 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 Key Stage 2 ITAF does not contain any information about what a children working at greater depth should look like by the end of year 6 so we have to look to the Key Stage 1 ITAF for more clues. Thankfully Rachel Rayner, a Mathematics Adviser at Herts for Learning, has done a great piece of work on this already. Her article 'Greater Depth at KS1 is Elementary My Dear Teacher' identifies three key differences between the statements and exemplification material for working at the expected standard and working at greater depth within the expected standard: she says that for pupils to be working at greater depth they should confidently and independently be able to deal with increases in complexity, deduction and reasoning. Please do read her article for more information about, and examples of, these three areas.

Complexity

Complexity is not about giving children bigger numbers, nor is it necessarily giving them more numbers (for example, giving children more numbers to add together, or order). Complexity needs to be something more as, based on curriculum objectives, giving bigger numbers is just a case of moving children onto the content of a following year group.

So, how do we provide more complex work which will challenge those children identified as working at greater depth? One consultant advises that "in order to provide greater challenge we should keep the concept intact while changing the context." And, anyone who has witnessed a year 6 class doing their SATs will know that if there's one thing that throws them more than anything, it's the context of the questions. The test writers come up with endless ways of presenting maths problems but children working at greater depth are very rarely phased by these, whereas children working at the expected standard will come up against a few that they cannot answer.

The best bet for increasing the complexity of the maths but continuing to work within the expectations for the year group is to present the problems differently, and in as many ways as is possible. The more children are exposed to problems presented in new ways, the more confidently they will approach maths problems in generally - gradually, nothing will phase them and they will have the determination to apply their maths skills to anything they come across.

The NCETM Teaching for Mastery documents, although designed for assessment purposes, contain a wide range of complex problems under the heading 'Mastery with Greater Depth'. Organised under the curriculum objectives, these provide a great starting point for teachers to begin thinking outside the box with their maths questioning. Here's an example from the Year 1 document:


A working group from the London South West Maths Hub have also begun putting together some similar documents, focusing initially on number, place value, addition and subtraction and again categorised under NC objectives - those documents can be downloaded here. Here's an example of one of those, taken from the year 3 documents:


It's also worth looking at the KS1 and KS2 tests to get an idea of the question variety. The mark schemes will help you to decide which year group's content is covered in each question. When picking a question from the tests, decide whether or not it could be considered as an example of greater depth, rather than just mastery. Here's an example (from last year's year 6 test) of how different the questions can look:


Reasoning

Reasoning is defined in the NC document as "following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language." 

As already discussed, reasoning is a skill that we want every child to have. But the greater depth exemplification makes more of reasoning than the expected standard exemplification so we need to be able to differentiate between those who are reasoning at the expected standard and those who are reasoning at greater depth. When it comes to assessing children on their level of depth in reasoning, NRICH have a very useful progression of reasoning:


I would suggest that those working at greater depth would be able to work at at least step 4: justifying. The NRICH article gives excellent examples and analysis of children's reasoning work so it is a must read to become more familiar with recognising reasoning at these five different levels.

For further discussion of reasoning skills, please read this article, also on NRICH, which discusses when we need to reason and what we do when we reason.

Deduction (and asking mathematical questions)

Making deductions, a key part of reasoning, is similar to making inferences when reading and is all about looking for clues, patterns and relationships in maths. Once they have found clues they need to make conclusions based on them, and to then test them out. To be able to make conjectures, generalisations and to follow a line of enquiry, children need to ask their own questions. They need to look a sequence of numbers and ask themselves, 'Does the difference between each number in the sequence is the same?' - this is all about wonder: 'I wonder if...'.

In order for children to ask questions about maths, so that they can begin to deduce things such as patterns and rules they need to be provided with activities that encourage them to do this. But even more importantly, initially they need to have these questioning skills modelled to them by an adult. They need to be taught and shown that maths can be questioned because many children think that every maths problem just has one set answer to be found.

NRICH is the go-to place for such activities, but don't just give children a problem and expect them to be able to get on with it on their own - they need to have had much practice in questioning mathematically. Only when children are asking questions about maths, testing out their hypotheses and following lines of enquiry that they themselves have set, will they be able to reason at those higher levels set out by NRICH.

Confidence and Independence

In order for children to be working at greater depth we would expect to see a certain confidence not seen in all children. We would also want to see that they were working independently on the three areas outlined above. As already mentioned, children may need plenty of modelling before they become confident and independent - especially those children who are currently working at the expected standard who could work at greater depth with some extra help. A key indicator of whether or not children are working at greater depth will be their levels of confidence and independence (especially the latter, as some children are of a more nervous disposition yet are still highly capable).

In Summary

To answer our original questions we would hope to see that children who are working at greater depth would confidently and independently:

  • access maths problems presented in a wide range of different, complex ways;
  • be able to justify and prove their conjectures when reasoning;
  • ask their own mathematical questions and follow their own lines of enquiry when exploring an open-ended maths problem.
In order to make provision for children working at greater depth we must:
  • model higher-level reasoning skills (justification and proving) and encourage children to use them;
  • model mathematical questioning during open-ended maths problems and encourage children to ask them;
  • provide complex maths problems (open and closed) with a variety of contexts and support children initially to access these, until they can do them independently;
  • motivate children to be confident and resilient enough to do the above.

Friday, 15 September 2017

9 Important Changes to the Primary Maths Curriculum and Assessment

In response to the DfE's latest documents, I wrote this for Third Space Learning. It's a summary of the key changes in the way primary maths will be assessed over the next few years:

On 14th September, just as we were all getting settled into the new school year, the DfE published not one, but two documents of considerable importance: ‘Primary assessment in England: Government consultation response’ and the 2017/2018 ‘Teacher assessment frameworks at the end of KS2’. Both documents reveal changes that will no doubt affect our approach as teachers and leaders.

Whilst the most imminent and significant changes involve writing and reading, there are also some interesting developments in Maths.

Saturday, 28 May 2016

I Thought I'd Lose My Job.

A few years ago I really thought my career had come to an end. It was definitely an overreaction but for a few days I had that sick feeling in the pit of my stomach 24/7. In my more rational moments I was sure that at least my being trusted to work in year 6 was over.

It was July and my first set of SATs results had come through. I'd been teaching a really high-achieving and compliant (if not a little boring) year group in what might be considered a leafy-lane school. They'd worked well and had aced practice tests. But the results arrived and calculations were made and there were disappointments. Enough disappointments for it to be a problem.

I went into overdrive: worrying, gathering evidence, mentally phrasing and re-rephrasing my defence. I met with the senior leaders and with my partner teacher and the School Improvement Officer was drafted in for a special meeting. Nothing else occupied my mind; I sat glued to my computer compiling page after page of reports based on the year's data (which thankfully I'd kept a good record of). I only remember one moment of peace: I'd cycled home and, in an attempt to clear my head, I lay in my garden listening to a favourite album from my youth: Kula Shaker's 'K'. 'Hey Dude' still reminds me of that time.

"Smell the rose, the sweet, sweet rose,
Catch the sun, find where it grows,
Smell the rose, the sweet, sweet rose,
That grows on castle walls in heaven."

Things were not smelling of roses and my experience did not seem like anything approaching heaven.

In short, the finger was pointing squarely at me. Well-meaning leaders tried to attribute the perceived failure to some difficult family circumstances I'd had that year. The problem was that they had all occurred after the SATs - they were clutching at straws, perhaps because they didn't want to say I was a bad teacher. They couldn't say that anyway as they had no evidence from their own monitoring that would suggest it were true. My carefully collected (and subsequently curated) practice papers and assessment grids were proof that there were no real issues in the achievement and progress throughout the year. I'd been successfully observed, my books had been scrutinised and there had been no issues with my data; pupil progress meetings had gone well and I always followed through with any interventions or changes that were suggested.

All this made it worse because it was so hard to put a finger on what had gone wrong. I doubted myself but at the same time was the only one being proactive about explaining the differences in the data. My confidence was shot yet I had to repeatedly defend myself, having to appear confident in what I had been doing for the year. 

In the end we put it down to an increase in challenge in the tests - these were the 2013 tests, the first year of the SPAG tests and the first time we began to see Tory ideals creeping in (inclusion of an excerpt from classic literature). Many perceived the tests to have already begun moving towards assessing principles from the incoming National Curriculum.

The agony I felt was prolonged until I'd been told which year group I'd be teaching the following year. I knew there was deliberation. I wanted out because I didn't want that pressure again - and my confidence had taken a severe blow. I wanted in because being ousted would have been proof (in my mind) that they thought I was incapable. In the end I was asked to teach year 6 again - that was probably the best outcome. And I've never taught another year group since.

The following year we had a visit from Ofsted. The previous year's data (which I'd had sleepless nights over- not to mention the terrifying days) did not stop the school from getting 'Good' overall (with two areas of 'Outstanding'). I was observed twice - SLT directed the inspectors back to me on the second day so they could see my cross-curricular use of ICT. An SLT member and an inspector told me there were no points for improvement in my lesson. It was noted in the inspection report that provision for reading (the test in which we'd suffered most) was 'Outstanding' - I'd led on reading for a year and a half. I'd already secured my current job by that point - assistant head at another school. The School Improvement Officer conducted a book scrutiny and affirmed that from what she'd seen in my books I'd make a good Maths leader in my next school. Those awful few days from the year before were long forgotten. We had a successful set of SATs results through that July. All was well. 

And I've learned something from all that; something I'd like my readers to learn too. There's probably a cleverly-worded, pithy quote somewhere which will better express this next point, but here it is in my own words: the things we worry about rarely have any lasting impact. A month, term, year down the line they are all but forgotten. Now, whenever I'm worrying about something work-related, the memory of this event reminds me that it probably won't have any lasting consequences. I do all I can to make things right and then let it go - it's a very freeing way to be but if it wasn't for the described event I wouldn't have learnt that lesson. 

Although at the time I was certain I'd lose my position as year 6 teacher, or even my job entirely, I didn't. Even though I worried that it'd harm my chances of procuring a leadership role, it didn't. All that you are most afraid of may never came to fruition - don't worry unnecessarily. Don't allow your fears to limit your potential. That thing you're living in fear of? It'll probably never happen. 

At least, that's how I see it.

"Smell the rose, the sweet, sweet rose,
Catch the sun, find where it grows,
Smell the rose, the sweet, sweet rose,
That grows on castle walls in heaven."

PostScript: It must be said that throughout this whole experience my wife constantly reminded me of what I ended up learning for myself. She reminded me too of the comparative insignificance of the event and of the principle laid out in Romans 8:28: "And we know that in all things God works for the good of those who love him". Her support was, and is, invaluable to me.

Sunday, 21 February 2016

All Aboard!

In 'All In The Same Boat' I touched very briefly on today's subject matter and after a couple of conversations after yesterday's post it became clear that more needs to be said. Previously I wrote "Make sure your leadership are taking responsibility too - don't let them allow you to be alone in the boat" and I'd like to say a little more.

I am going to address this post to year 2 and year 6 teachers, but if you are a senior leader reading this, it is your responsibility to make sure that everything I suggest they do actually happens.

Most leaders will naturally want to be on board - it's their school and their data. Most leaders won't be leaving year 2 and year 6 teachers to hoist the mainsail themselves. Many leaders will now be adopting an 'all hands on deck' approach, but even the best captain needs to know from his crew what is happening in each area of the ship's life. He'll need the quartermaster to inform him when the ship is low on supplies, and he'll need the boatswain to tell him if such-and-such a part is in need of repair. Head teachers, and other members of SLT, will need feedback from teachers in order to understand what the needs and priorities are. And that's where this blog post comes in.

At the earliest possible opportunity, call a meeting with phase leaders (UKS2 and KS1), class teachers (Y2 and Y6), the head and any other SLT members. At the meeting discuss the new assessment arrangements (if you have not done so already) and its implications. If you have new thoughts and feelings after last week's revelations then it will be worth having another meeting anyway. It might be a good idea to take some assessment information with you so that you can identify the areas of greatest need. It'll also be good to approach it with some ideas already - if you go with only problems and no solutions the meeting will take longer, plus leaders always like to see a bit of initiative. Arm yourself with a list of questions you'd like to ask too. The meeting then needs to become a practical planning meeting with decisions made on what your school approach will be to this year's assessments. It's also worth considering as a team how you are going to keep a balanced curriculum instead of just doing maths and English (read this excellent blog post on the matter).

Even if you don't get to have a proper meeting, it'd be wise to ensure that the leadership of your school knows the course you are deciding to take with your year 2 or year 6 class. I would also involve them in any changes you're planning to make. Even when you begin to feel like you're pestering them, keep on asking for advice and informing them of your decisions.

The point of all this?
  • So that you're not alone in the boat at your school. 
  • So that you are supported. 
  • So that collective wisdom, and the wisdom that comes from experience, influences decisions.
  • So that you have the chance to suggest that more manpower might be needed. 
  • So that when the data eventually comes in, it is data that represents a team effort. 
  • And so that no leader can make accusations of you, blaming poor results on you alone. This should not be about taking one for the team, but taking one AS a team
It's a sad state of affairs that I'm even suggesting safeguarding yourself against these eventualities but I know it goes on - there are plenty of disheartening stories out there of teachers stuck in schools with leaders who absolve themselves of these responsibilities and then point the finger at the ones who have slaved all year to make as much progress as possible with each child.

In short; make sure everyone is on board with everything that will end in assessment this year. Do everything you can do get the support that you need - even the best leaders need proactivity from their team.

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Saturday, 20 February 2016

All In The Same Boat


Now that we've all experienced the cocktail of initial relief, mild anger and nervous hilarity that the DfE's announcements yesterday generated, it's time to think soberly and wisely about it. My 'DfE Tells Teachers They're All Very Naughty' was a crude response, yet it did seem to voice the opinion and feeling of many teachers upon hearing what Mr. Gibb and Ms. Morgan had to say (@theprimaryhead's post was much better). What we teachers, however, really need to focus on now is making sure that, for our students, the next few months are worthwhile.

The positives that came out of yesterday's communication from the department are that it appears schools will not be judged too harshly based on the outcomes of this year's assessment. Nick Gibb wrote this in his letter to the NAHT:
"I have also written to Her Majesty’s Chief Inspector asking that his inspectors take into account national performance and the contextual factors you have outlined when considering a school's performance on writing at Key Stage 2. All organisations holding schools to account should be aware of the changes being introduced in 2016 and will consider the impact of this in making any decisions about performance or intervention on the basis of 2016 data alone. This should give schools the confidence to engage fully with the vision of the new curriculum and to rise to the new standards."
He's actually right. We should have the confidence to go on teaching, even if he had never said this. Even if we know our RAISE online could go blue next year. Even if we know Ofsted will not look kindly on us. Because what we have to focus on is the children - we have a responsibility to them first and foremost. And in a sense it has always been this way in year 6: teachers have always had the role of gatekeeper, protecting the children from the pressure. Sadly, some teachers have never managed this, instead subjecting children to weeks and weeks of practise papers under exam conditions and taking every opportunity to brow-beat them with "It's only x amount of weeks until your SATs, you know?" Some teachers pass on the pressure they feel - and, at all costs, we must not do that.

"As this is the first year of new accountability measures and new assessments, we will wait until tests have been taken to set minimum expectations for a school’s progress scores."
We must also remember this: the scores will be calculated based on the tests taken nationally and if everyone does poorly then the minimum expectation will be set lower. We're all in this together and if the DfE stick to what they've said this week then perhaps we shouldn't worry as much. 

Although the DfE have tried to shy away from admitting that this has all been a bit shambolic,lines like this give it away: "Significant reforms such as these take time to get right and for the system to catch up." We're all in the same boat; teachers, school leaders, inspectors and government officials should all be chalking this year up to experience.

At the end of my first year teaching in year 6 I thought my world was going to come crashing down around me. Some children hadn't achieved as highly as I had hoped, despite acing many a past paper. There were certain issues when comparing the data to the previous year's data. I thought I was for the chop. I spent hours writing documents to defend each and every poor test score, compiling evidence to prove that my teacher judgments weren't way off the mark. I sat in front of the school leaders and even our school improvement partner to defend myself. It all lasted about a week, and then life went on. They asked me to teach in year 6 again and my current school employed me to teach in year 6 too: last year saw a ten percent rise in children achieving floor standards or above. 

That year when it all went wrong is now long gone and forgotten, in fact it was all gone and forgotten after a few weeks. The school went on to get a 'Good' Ofsted inspection (with two ares being 'Outstanding') despite the data which I thought would end the world (in fact, there was barely any mention of the data). 

And when 2017 rolls around 2016 will be gone and forgotten too. Whatever happens this year will not be career-defining for you. Realising that every year 2 and 6 teacher in the land is in the same position is key to having a more positive outlook on this matter. Knowing that when schools are compared, aside from the usual variations, there will be a national trend. That trend will not necessarily be a trend of 'underachievement' because I know that every year 2 and 6 teacher in the country will be working their socks off to ensure excellent progress and high achievement for all their children. Possibly the tests, and even the teacher assessment based on the interim objectives, will show that we are 'underachieving', but we will all be there together - maybe then the DfE will admit that their handling of the changes was clumsy. Maybe they won't. Whatever happens, it will all blow over and we will all move on.

In the meantime, steel yourself for Monday, plan and teach some exciting lessons and make sure the kids are learning and making progress. Don't foist the stress on to them; they're just kids. Make sure your leadership are taking responsibility too - don't let them allow you to be alone in the boat. Do the wise thing and make sure you teach them according to the curriculum and the interim objectives - do what you can with the short time we've got, but just remember that there are only so many hours in a day, week, half term. Maximise that time so that you can be confident your children will 'perform' to the best of their ability. And that is all you can ask for.

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Monday, 4 January 2016

Times Tables: What is Knowing?

@tombennett71: There should be nothing controversial about a mainstream expectation for children to know times tables and we'll look daft if we dispute it.

And I agree. Apart, perhaps, from the part about 'know'. What does 'know' mean? Merriam-Webster defines 'know' thus:

  • to have (information of some kind) in your mind
  • to understand (something) 
  • to have a clear and complete idea of (something)
  • to have learned (something, as a skill or a language)
If a child, when they are tested on their tables in 2017, can choose their own version of 'know' then I definitely agree. When you've wiped away your tears of laughter after watching Nicky Morgan avoid answering 11x12, read what she said: "We are introducing a new check to ensure all pupils know their times tables by age 11." She says 'know'. The 'by heart', 'by rote', 'by memory' rhetoric has been added by the papers who gleefully reported the news, glad at the chance to stick another knife in the back of the profession. So, theoretically children don't have to know their tables by heart.

The reason why this issue resonates with me, and with many others, is that as a child, despite my dad's best efforts, I found it impossible to learn my tables by heart. And I still don't know them all today. What I can do is work out multiplication problems very speedily using Merriam-Webster's second, third and fourth definitions. I understand what happens when you multiply one number by another so I can solve a problem. I have a clear and complete idea of how timetables link to other areas of maths. And I have learned lots of methods (you might say skills) to help me to work times tables questions out before anyone realises I haven't memorised them.

The beginning of my journey out of times-tables-embarrassment-land was when I realised that my dad, at random moments during the day, in an attempt to keep the practice up, would only ever ask me what 6x6 was. So I learnt 6x6 (it's 36 - see, told you I'd learnt it). I soon realised that if I knew that then I could work out 6x7 really quickly.

The next step of my journey was my realisation, in secondary school, that if a teacher tried to get me to learn a method without explaining how and why it was working, then I wouldn't be able to do it. I had to understand the mechanics of the mathematical process in order to be able to solve problems. As my teacher took the time to model processes in a way that I understood them, I began to improve in maths. I started to enjoy it too. In fact, I started to think mathematically, could problem solve, reason and I sure was getting fluent. Recognise those three terms? Yes, the aims of the National Curriculum. If I had only learnt by heart the formula for finding the area of a triangle without understanding why it worked then I'd have been far less fluent and would not have been able to problem solve or reason. So why are so many teachers hellbent on getting kids to memorise stuff like times tables?

OK, if a child can memorise them then great, but teachers beware, I truly believe there are kids out there in year 5 right now who will be better supported this year if you teach them some tricks and tips so that instead of rapid recall, they can do rapid work out of tables. Take it from someone who knows.

Here are a few tips and tricks for how you can help those children once you've identified who they are (probably by giving them one of the hundreds of times table check practise tests that will appear online by the time the month is through):
  • Find out which tables they have learnt by heart - the majority of children will definitely have 2s, 5s and 10s.
  • Assuming children know 1s, 2s, 5s and 10s they already have good reference points for other tables. 3s and 4s could be taught using manipulatives such as Numicon shapes or cubes (or you can get really creative - Ikea's dogs' bums coat hooks are fun for 3s) to reinforce what is happening when multiplying 3 and 4.
  • When learning 4s and 8s make links back to 2 times tables. Lots of simple investigation opportunities here too i.e. Which times tables does the number 16 appear in? If the kids can make these connections themselves they will be more likely to learn skills that they can apply in a test situation.
  • Similarly link 3s and 6s together. Later they can be linked to 9s and 12s.
  • Teach 9s using the finger trick. Make sure children have identified the pattern in the answers: the digits in the answers add to 9 - do investigation so that they find this out for themselves.
  • Teach 11s by looking at the pattern in the answers. 10x11, 11x11 and 12x11 might be a bit more difficult so these might need to be learnt by heart - reducing the number of answers that need to be learned by heart is still helpful.
  • This might sound totally ridiculous... OK, it absolutely will, but Weetabix taught me how to work out my 12 times tables quickly. I know the pack sizes.  Each tube inside a box contains 12 Weetabix. You can get boxes of 12, 24, 48, 72. Help the kids tap into outlandish methods like this - maths in real life will be a saviour to many. So many kids need to know why maths is important and relevant to them SO THAT they can begin to understand it.
  • Your children probably are capable of retaining a few facts. I could always remember 6x6 which inspired me to learn my square numbers. Mathematically square numbers are interesting and are more likely to stick in the head (nice links to actual squares in geometry too as a model). Once you've learned square numbers the world is your oyster, especially if you know the square of 6, 7, 8, 9, 11 and 12. You can use those as a reference point and quickly add or take from them.
  • Many children will be able to get a feel for which numbers sound like correct answers and which don't. Some work on what a prime number is might help - children will learn to avoid 17 as an answer in a tables test because it doesn't sound right. They will begin to know that 56 and 42 do appear somewhere - this gives them a reference point to check their answers by. 
  • I use the idea of The Hard Tables. This reduces the number of times tables that children have to really worry about in year 5 and 6. The Hard Tables (even the name shows a child who struggles to memorise them that you understand their plight) are basically any problem (beginning at 6x6) above the square roots of square numbers i.e. 6x6, 7x6, 8x6, 9x6 and 12x6 (most children will know 10x6 and 11x6)
  • Give the children tests where you model a thought process e.g. "The question is 8x7. So think of your square numbers: 7x7 is 49 so you need another lot of 7. 49 add 7 is... is your answer one of those numbers that we know is an answer in the times tables? Does it sound right?"
Put me in front of the nation and ask me a times table question and I'll answer it right away. Not because I know everything up to 12x12 by heart but because I will THINK MATHEMATICALLY about the answer. I will demonstrate fluency as I link areas of my mathematical understanding together. I will demonstrate, invisibly, my ability to problem solve and reason. I will demonstrate that I 'know' my times tables without actually ever having memorised them all. I will be using one or two of the above strategies to get to answer. It will take me a fraction of time longer than someone who has memorised the answer, but out of the two of us, I'll be the one demonstrating better mathematical thinking.

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