Showing posts with label Maths. Show all posts
Showing posts with label Maths. Show all posts

Monday, 9 September 2019

From @Matr_org: Understanding Maths Anxiety: A Parents’ Guide On How To Overcome This Primary School Problem


"I remember finding ways to get out of maths lessons as a youngster.

My favourite ruse was to offer to tidy up the teacher’s cupboard – I even clearly remember stacking the maths textbooks neatly on the shelves, feeling inwardly smug that I did not have to open them and attempt the questions inside.

I recall my dad spending what seemed like hours with me trying to help me to understand negative numbers and how to calculate them – unfortunately, his pictures of eggs and egg cups didn’t help at all although I appreciated his efforts!"

https://matr.org/blog/understanding-maths-anxiety-parents-guide/

Friday, 8 February 2019

Times Tables Fluency and the KS2 SATs

How important is times tables fluency for the KS2 SATs? I'd say quite important.

When we are fluent in speaking a language, we can speak it without thinking much about it. That kind of fluency will be useful for year 6 children to have when it comes to the tests in May.

I looked through the 2018 SATs papers to see just how many questions required some times tables knowledge.

Here's what I discovered:

  • In Paper 1 (Arithmetic) there are 19 out of 36 questions which definitely require children to have fluent times tables knowledge.
  • In Papers 2 and 3 (Reasoning) there are 18 out of 44 questions which also require children to have fluent times tables knowledge.
But why might fluency be important? Can't children just work out the times tables without knowing them by heart?

Well, yes, they could, but it would cost them time.

For Paper 1 children are given 30 minutes, meaning that they have less than a minute per question. For Papers 2 and 3 there are 40 minutes per question - this means children have about 2 minutes per question.


In a question where 8 different times tables facts must be recalled (see above), it is obvious that this needs to be done quickly so that children can focus on the procedure of answering the question. In the same questions accuracy is essential too: if children are fluent with the times tables facts they are less likely to make mistakes.

If children are spending too much time working out times tables facts they risk going over that l or 2 minute per question; in turn they risk not having time to finish the test.

But, looking at the 2018 tests revealed something else: most of the times tables facts that children needed to use to answer the questions were fairly easy: the sort of times tables that are learned in years 1, 2 and 3. The times tables grid here shows exactly which times tables facts are required. The hardest times tables facts (such as 7 x 8, 9 x 8, 11 x 11) weren't required. The most common facts needed were below 6 x 6 with majority of the additional facts coming from the 2, 3 and 4 times tables.

But children need to be able to do more than recall them quickly; they need to be fluent enough to use and apply them. It's not just about remembering the facts but being able to recognise relationships between numbers. For example, the questions below require children to spot related facts:



I've put together a PowerPoint presentation which contains all the questions that require some times tables knowledge. I've animated the working out and answers for each question too so that these can be used flexibly. The PowerPoint (which the images used above are taken from) can be downloaded here: https://www.tes.com/teaching-resource/powerpoint-to-demonstrate-the-need-for-times-tables-fluency-in-sats-12065667. I used it with year 5 parents who found it useful to know how important times tables were going to be for SATs.

Sunday, 2 December 2018

Making Geometric Stars: An Investigative Christmas Maths Teaching Sequence

So Christmas is upon us and, as a conscientious teacher, you don't want the learning to stop (or the craziness to begin), but you also want to make the most of the Christmas context/the kids are pestering you for a 'fun' lesson. You've got something for writing (please don't let it be letters to Santa - so much wrong with that - go with writing a list of presents they'd like to give others, preferably not bought ones either) and RE is sorted - so's reading as there are so many Christmas-themed texts out there.

But what've you got for maths? Some Christmas-themed word problems? If I eat 24 mince pies over the Christmas season and mince pies come in boxes of 6, how many boxes of mince pies have I eaten? Bit tenuous. Let not mighty dread seize your troubled mind. Try this investigative teaching sequence which culminates in making some lovely geometric stars:

Step 1: Investigate the size of internal angles in regular polygons

Begin with triangles, discussing what they already know about the sum of the internal angles in any triangle. A tangent here could be to check a number of different triangles, not just equilateral ones, to see if indeed all the angles add to 180 degrees - doing this will also provide important protractor use practice for later on. It might be worth pointing out that the half-circle protractor also measures 180 degrees.

Move onto squares, rectangles and other quadrilaterals. Again, call on prior knowledge: all of a square's internal angles are right angles, right angles are all 90 degrees and that multiplying 90 by 4 is 360 so a square, and a rectangle, has interior angles which add up to 360 degrees. You could mention at this point that circles also contain 360 degrees. You could also get them to check a range of different quadrilaterals, reinforcing their names and other important shape
vocabulary.

Although probably too early, you could ask children to make conjectures about what the internal angles of a 5-sided shape - a pentagon - might total. Some may point out that from the triangle to the square the number doubled so may predict that a pentagon's internal angles might add to 720 degrees. Others may point out that from the 3-sided shape to the 4-sided shape the number of degrees increased by 180 and therefore predict that the pentagon might have internal angles totalling 540 degrees. If both of these conjectures are brought up you can discuss how it is too early to be sure of any pattern and that it is important to continue testing the hypotheses.

Provide a printed sheet containing at least a pentagon - it's also worth including a hexagon, a heptagon and an octogon (you could even include a nonagon and a decagon). Allow the children to further investigate the sum of the internal angles in these shapes. At the same time ask them to create a table to record their findings.

Shape
Number of sides
Each angle (in a regular shape)
Sum of internal angles




Triangle
3
60 o
180o
Quadrilateral
4
90 o
360o
Pentagon
5
108 o
540o
Hexagon
6
120 o
720o
Heptagon
7
128.57… o
900o
Octagon
8
135 o
1020o

After measuring and totalling (this could be by multiplication or repeated addition - discuss efficient methods) the internal angles of each shape they should make further conjectures about what the next one will total - before moving onto the hexagon they should be almost certain that each time a side is added to a shape another 180 degrees are added to the sum of its internal angles.

It should be noted that some of their measuring almost certainly won't be accurate and that some mathematical reasoning will have to be applied, for example:

"If I measured one angle as 107, another at 109 and another at 108, which is most likely to be?"
"I've predicted the sum of the internal angles to be 720 degrees but it is coming out as 723 degrees - which is wrong, my prediction or my angle measuring?"

Once findings are recorded in a table it becomes easier for children to begin to find a way of expressing a rule for finding the sum of the internal angles of a shape with any number of sides. I have worked with year 6 children who have managed to generate a formula for this. Even if they cannot yet write it down, some will be able to verbalise the rule:

"Number of sides subtract 2, then multiply that by 180"

To get to this point it helps to talk about the triangle being the first shape, the quadrilateral being the second shape, the pentagon being the third shape, and so on. With this as a starting point they can generate something like this:

Shape
Shape Number
Number of sides
Sum of internal angles




Triangle
1
3
180o
Quadrilateral
2
4
360o
Pentagon
3
5
540o
Hexagon
4
6
720o
Heptagon
5
7
900o
Octagon
6
8
1020o

Now that they have the shape number next to the number of sides in the shape they will much more easily be able to see that the difference is two therefore subtracting two from the number of sides results in the number that 180 must be divided by to find the sum of the internal angles.

If children don't have prior experience of writing this as a formula they can be shown how to record this:

Sum of interior angles = (n-2) × 180° (where n = number of sides)

And that each angle (of a regular polygon) = (n-2) × 180° / n

Step 2: Investigate drawing stars within circles

Nrich has a couple of great activities for this:

Path to the Stars: https://nrich.maths.org/1097 (their printable resources page has circles with pre-printed dots on it for this ativity: https://nrich.maths.org/8506)

Stars: https://nrich.maths.org/2669 (this is an interactive resource)

Round and Round the Circle: https://nrich.maths.org/86

In these activities it is worth drawing out rules such as:
  • if you draw a line straight to the next dot you get a regular polygon
  • with an odd number of dots, if you a draw a line which skips a dot you get a star shape
  • with an odd number of dots which isn't a mulitple of 3, if you draw a line which skips two dots you get a star shape with longer points than when you just skip one dot (doesn't work for 5 dots as there aren't enough dots - skipping two is the same as skipping one in the opposite direction)
  • if you skip just one dot when there are an even number of dots you get a regular polygon with have the number of sides and vertices as the the original number of dots
There are many more rules such as these to notice and discuss.

You could also experiment with Nrich's Mystic Rose activity (another great interactive resource: https://nrich.maths.org/6703) which does more than just create star shapes.

Step 3: Practise drawing regular shapes using a protractor


Model to children how to draw the shapes from step 1 using the angles they discovered and by deciding on a particular side length. Impress upon them the importance of accuracy in measurements - perhaps demonstrate how even being a few degrees/millimetres off once or twice will result in an irregular shape.

Children should mark a starting point and draw a line of the side length they have decided. Then they should measure the internal angle according to their findings in step 1 and draw a second side of the same length to the shape. This should be repeated until they reach their starting point again.

It will be best to do this exercise on scrap (and/or large) paper as often the children will find that their chosen side lengths lead to their shape becoming too big for the paper! In this case they will have to readjust and start again.

Step 4: Make stars!

Once children have mastered the drawing of regular shapes in step 3 they can move onto making their stars. Give children coloured/decorated card in festive hues/patterns to draw their regular shapes out onto.

From protractor to tree!

Once they have drawn out their shapes they can use what they experimented with in step 2 to join the vertices of their shape in different patterns to form stars.

If you carried out the Mystic Rose activity in step 2 you will need to ensure the children can identify a regular pattern where they will cut (there will be many options). It is a good idea to give them a pen to go over the lines that they want to cut before they take their scissors to their carefully drawn out shape. The mystic rose patterns that they have drawn will provide interesting decoration to their finished stars.

Alternatives:

If you don't have time to follow the whole sequence, or teach children too young to be able to do all aspects of it, there are alternatives to the above sequence which avoid the lengthier steps 1 and 3:
  • Just do one of the activities from step 2 then make large print outs of the Nrich templates on coloured card so that children can make their stars.
  • Teach children to use a pair of compasses to draw their own circles then teach them to use a protractor to divide the circle into equal sections by dividing 360 by the number of points they want ( 5 points = 72°; 6 points = 60°; 7 points = 51.42...°; 8 points = 45°; 9 points = 40°; 10 points = 36°). They can then use these to create star or Mystic Rose patterns on coloured card to cut out.
A colourful hexagonal mystic rose pattern - the green part will have been cut off to form a star.

A well-drawn nonagon with mystic rose pattern and a heptagon with mystic rose pattern that has already been transformed into a star.

If completed on plain card children can decorate their stars - perhaps in more festive colours than this one!

If children can draw a perfect hexagon then they can also make and fold shapes which can be cut into snowflake shapes which will have a realistic six lines of symmetry.

    Wednesday, 14 November 2018

    From the @TES Blog: Times Tables Check: What Do I Need To Know?


    We’ve known about the proposed key stage 2 multiplication check for a while now, but have so far been waiting for more information about exactly what the check will entail. With the publishing of the 2018 'Key stage 2 multiplication tables check assessment framework' this month, we now have a greater insight into what we can expect of the tests.

    Click here to read the rest on the TES site: https://www.tes.com/news/times-tables-check-what-do-i-need-know

    Tuesday, 6 November 2018

    Explicitly Teach Metacognition to Boost Maths Skills


    Anyone who has ever taught primary maths will, most likely at many points, have asked themselves, ‘Why on earth did they do it like that?’.

    You know, when a child completes a full written calculation just for adding 10 to another number or attempts to divide a huge number by reverting to drawing hundreds of little dots.

    And it’s an absolute certainty that every primary teacher will have asked the following question, but this time with a little more frustration: ‘Why didn’t they check their answer?’.

    Click here to read the rest on the Teach Primary website: https://www.teachwire.net/news/we-need-to-explicitly-teach-metacognition-to-boost-maths-skills

    Friday, 13 April 2018

    From Teach Primary Magazine: KS2 World Cup Maths Lesson


    I wrote a lesson plan for Teach Primary Magazine to go along with their feature on lessons inspired by the World Cup.

    This lesson was one I taught during the last World Cup - an event which also coincided with an Ofter inspection at my the school where I was working at the time. The inspectors commented that they hadn't seen much use of ICT so of course being the computing lead I was asked to tweak a lesson for the next day. Whether or not I'd agree with this sort of thing these days is another matter but suffice to say I met the request and this lesson is what I came up with.

    If I remember correctly (I do but I'm trying to be modest) the school's maths lead and one of the inspectors couldn't really find any 'next steps' for me when they gave feedback and only had positive things to say. That's not to say that this is a failsafe Ofsted outstanding lesson - there's no such thing, and it's mostly in the delivery - but that hopefully it will provide a good starting point for a lesson for other teachers.

    The whole lesson plan/article is available online so you don't have to squint at the photo above.

    https://www.teachwire.net/teaching-resources/ks2-lesson-plan-make-predictions-using-real-time-statistics-from-the-2018-football-world-cup

    Friday, 16 March 2018

    From The @thirdspacetweets Blog: What Every KS2 Teacher And Maths Lead Needs To Know About NEW KS1 Maths Assessment Frameworks

    From The @thirdspacetweets Blog: What Every KS2 Teacher And Maths Lead Needs To Know About NEW KS1 Maths Assessment Frameworks
    Valentine’s Day 14th February 2018 brought KS1 teachers not one but two lovely treats: the teacher assessment frameworks for the 2017/18 academic year and the same document for the 2018/19 academic year.

    While there are no changes for the current cohort of Year 2, the current Year 1s will be teacher-assessed on a new and amended framework.

    Of course, the biggest question on everyone’s lips is…are the changes to the KS1 assessment framework for Maths an improvement?

    To find out more, read on here: https://thirdspacelearning.com/blog/new-ks1-assessment-frameworks-maths-insights-ks2/

    Saturday, 24 February 2018

    From The @thirdspacetweets Blog: EEF Report Summary: Putting Evidence To Work


    My work with Bradford Research School has really turned me on to the work of the EEF. So when they release another a guidance report I'm always keen to read it first to find out what its implications are. The latests one applies to all subjects and all schools but here, in this blog post for Third Space, I outline how I should have used it had it been published in time, and how I will use it in the future to introduce any new changes.

    Winds of change blew in the world of primary Maths when the 2014 National Curriculum was introduced. We now had to teach some things sooner, other things later, some things not at all and there were additions too (hello, Roman numerals!). The ‘new’ holy trinity of Maths teaching and learning were introduced: fluency, problem solving and reasoning.

    Then the SATs gradually changed. The calculation paper had already been done away with; next to go was the mental Maths test, replaced by the arithmetic test. And the reasoning tests appeared to begin to assess how pupils were doing on the 2014 curriculum ahead of schedule. The two new reasoning papers were perceived by many to be more difficult than before.

    And so, up and down the land, Maths leaders and teachers have been making changes to the way the subject is taught in their schools...

    Click here to read on: https://thirdspacelearning.com/blog/eef-putting-evidence-work-report-slt-summary/


    Friday, 22 December 2017

    Teaching Mathematical Problem Solving: What The Research Says


    Recently the EEF published their guidance report for KS2 and KS 3 maths. It gives 8 recommendations for improving the teaching of mathematics:


    In this blog post for Bradford Research School I focus in on problem solving but touch on the use of manipulatives, developing a network of mathematical knowledge and other areas of the guidance. In the article I outline a maths lesson which follows much of the advice given in the guidance (the cube trees at the centre of the lesson):

    https://bradford.researchschool.org.uk/2017/12/20/teaching-mathematical-problem-solving-what-the-research-says/

    Saturday, 2 December 2017

    Mathematical Misconceptions And Teaching Tricks: What The Research Says

    Imagine a factory. Think of the vast machines clanking away. Think of the whirring, the turning, the raw materials becoming a finished product. Beneath those metallic exteriors cogs, cams, belts and levers are working together to effect that change. But all but the most initiated don't really understand how the machines do what they do, they just know that if they put the right parts in at one end, the machine will produce the desired item.

    And this is how many children feel about maths. They know that putting some numbers into a calculation will give the desired answer, but they don't really have a clue what goes on inside the 'machine' of that procedure. This is all well and good until that child has to apply this learning - having no understanding of the mechanics of mathematics makes it very difficult to use procedures in context.

    In my blog post for Third Space Learning entitled 'Maths Tricks or Bad Habits? 5 Bad Habits in Maths We're Still Teaching Our Pupils' I make several suggestions for how to use visual representations to teach good conceptual understanding of some tricky aspects of the maths curriculum, such as the ones below:



    The recent EEF guidance document on improving maths in KS2 and KS3 backs up the importance of modelling good conceptual understanding in maths lessons, rather than relying on tricks that work but don't help children to have an understanding of the 'why' and the 'how':
    Recommendation 4: Enable pupils to develop a rich network of mathematical knowledge 
    "Pupils are able to apply procedures most effectively when they understand how the procedures work and in what circumstances they are useful. Fluent recall of a procedure is important, but teachers should ensure that appropriate time is spent on developing understanding. One reason for encouraging understanding is to enable pupils to reconstruct steps in a procedure that they may have forgotten. The recommendations in this guidance on visual representations, misconceptions, and setting problems in real-world contexts are useful here."
    In order to teach maths well, and in order for children to succeed in maths, teachers need to make sure children understand what is going on when they carry out a mathematical procedure. A great way of developing this understanding is using manipulatives and representations:
    Recommendation 2: Use manipulatives and representations 
    "Manipulatives and representations can be powerful tools for supporting pupils to engage with mathematical ideas. However, manipulatives and representations are just tools: how they are used is important. They need to be used purposefully and appropriately in order to have an impact. Teachers should ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept. The aim is to use manipulatives and representations to reveal mathematical structures and enable pupils to understand and use mathematics independently.
    Teachers should: Enable pupils to understand the links between the manipulatives and the mathematical ideas they represent. This requires teachers to encourage pupils to link the materials (and the actions performed on or with them) to the mathematics of the situation, to appreciate the limitations of concrete materials, and to develop related mathematical images, representations and symbols."
    As I wrote in the guide to Bar Modelling that I produced for Third Space Learning (click to download for free):

    If we don't do this, we run the risk of allowing children to proceed in their mathematical education with misconceptions:
    Recommendation 1: Use assessment to build on pupils’ existing knowledge and understanding 
    "A misconception is an understanding that leads to a ‘systematic pattern of errors’. Often misconceptions are formed when knowledge has been applied outside of the context in which it is useful. For example, the ‘multiplication makes bigger, division makes smaller’ conception applies to positive, whole numbers greater than 1. However, when subsequent mathematical concepts appear (for example, numbers less than or equal to 1), this conception, extended beyond its useful context, becomes a misconception. 
    It is important that misconceptions are uncovered and addressed rather than side-stepped or ignored. Pupils will often defend their misconceptions, especially if they are based on sound, albeit limited, ideas. In this situation, teachers could think about how a misconception might have arisen and explore with pupils the ‘partial truth’ that it is built on and the circumstances where it no longer applies. Counterexamples can be effective in challenging pupils’ belief in a misconception. However, pupils may need time and teacher support to develop richer and more robust conceptions."
    When we do teach children using appropriate models and images so that they understand the mathematical concepts behind the procedures (or the 'tricks'), we provide children with something that they can actually look at and explain. Explaining something that is concrete is easier than explaining an abstract concept.

    In the bar modelling guide (click to download for free) I pointed out that:


    By developing children's skills to represent and explain their understanding using a model, we develop their independence and motivation:
    Recommendation 5: Develop pupils’ independence and motivation
    "Teachers can provide regular opportunities for pupils to develop independent metacognition through:
    • encouraging self-explanation—pupils explaining to themselves how they planned, monitored, and evaluated their completion of a task; and
    • encouraging pupils to explain their metacognitive thinking to the teacher and other pupils."
    Next time you plan a maths lesson question how you will ensure that children have a good conceptual understanding of the content you teach. Often, concrete or pictorial representations will be the best way to show children the inner-workings of the concepts you cover. Following Psychologist Jerome Bruner's research-based CPA (Concrete - Pictorial - Abstract) approach means that children (and adults) are more likely to understand what is going on inside the maths machine as calculations and processes take place.

    Further Reading and Resources:

    Wednesday, 29 November 2017

    On The Third Space Learning Blog: Maths Tricks or Bad Habits? 5 Mathematical Misconceptions We Still Teach Pupils (And How To AvoidThem)


    Whilst I'm sure I've been guilty of all of these 'tricks' during my time as a teacher, undertaking my role as maths lead and learning more about best practice has prompted me to become rather passionate about avoiding trick-based teaching in maths

    It is also certain that the root of my desire to eradicate this kind of teaching which does little to support conceptual understanding can be found in my own school experience. I remember asking one question constantly in maths: "Yes, but why?". Teachers expected me to rote learn and regurgitate maths procedures but I struggled to remember them because I didn't understand them.

    Whilst the list of tricks I've outlined in my latest blog post for Third Space Learning is by no means comprehensive, it will hopefully serve to provoke thought on this matter and will be a starting point for some who are not yet teaching so that children truly understand the maths:

    https://thirdspacelearning.com/blog/maths-tricks-bad-habits-we-teach-pupils/

    Thursday, 16 November 2017

    My Guide To Bar Modelling


    My guide to bar modelling, written for Third Space Learning, is now available to download. The guide includes information on different types of bar model, how to use them across the primary phase and in different areas of the maths curriculum.

    The download also includes a PDF of PowerPoint slides which can be used for staff training purposes.

    https://www.thirdspacelearning.com/resource-ultimate-guide-bar-modelling/

    Monday, 13 November 2017

    On The Third Space Learning Blog: 2017 Maths SATs QLA Analysis

    It's often helpful to use data to inform teaching but finding time to sit down and go through it with a fine enough tooth comb isn't easy.

    The Question Level Analysis for the ks2 tests now provided on ASP (the RAISE online replacement) contains national and school data which can be useful to key stage 2 teachers to inform their future teaching. It's also useful for year 7 teachers, but they don't often get access to this information.

    So, for the benefit of many teachers and children, here's a breakdown of the parts of the 2017 maths tests that children scores the country struggled with the most:

    Thursday, 19 October 2017

    Poster: Maths Written Feedback Comments

    marking feedback policy maths thatboycanteach
    Many teachers will still be operating in schools where feedback policies require a certain amount of written feedback. Some schools have begun to adopt no-marking policies but these are in the minority; most teachers, in order to follow policy, have to provide written feedback: marking.

    For primary teachers, this is fairly simple in English but is a little trickier in maths. My team and I sat down and analysed a selection of marking comments which we found in maths books and reduced them to question/statement stems. We tried hard to make them as succinct as possible in order to make the task of perhaps having to mark 30 books a little less onerous.

    When I put these maths marking comment stems on Twitter some people pointed out that these comments were things that we should be planning into our daily lessons, and they are right. Many of the ideas are to do with reasoning and problem solving - something we should be giving all children the opportunities to engage with on a very regular basis. So, these comment stems come with a multiple purpose: plan maths activities using them, and if required, use them to provoke thought in children who have finished the work you planned for them.

    Of course, I would always advocate that much of this kind of 'feedback' is provided in lesson, so these comment stems aren't just for writing - they're for giving verbal feedback too. I have found that sometimes verbal feedback is forgotten - making a quick note (just a few words) in a book might just be enough to jog a child's memory, meaning they won't have to wait for the teacher to come round again for another explanation of what they need to do. Written feedback during lesson time can be useful for this purpose - the fact that these comments are only a few words long makes this more manageable.

    Another point some have made is that it isn't necessary to write one of these in every book - it isn't (although some policies may require it). If all children need the same comment (unlikely), then these comments can be provided whole-class, perhaps by way of writing it on the board.

    A final note on the comments themselves: there are quite a variety - some pertain to mistakes made, others are intended to challenge further; all are supposed to make children think and to help them to improve their understanding in maths.

    For the record: my own school's feedback policy does not require teachers to provide written comments (although they are allowed) but we recognise instances where they are useful and productive. Our maths policy also states that problem solving and reasoning activities should be part of daily lessons.

    Click here to download the poster and the editable Word document of the 30 statements

    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.

    Monday, 11 September 2017

    KS2 Maths SATs On Reflection: Why We Teach For Mastery In Maths

    Here's one I wrote for Third Space Learning: https://www.thirdspacelearning.com/blog/2017/ks2-maths-sats-on-reflection-teaching-for-mastery

    ‘Without reflection, we go blindly on our way, creating more unintended consequences, and failing to achieve anything useful.’ - Margaret J. Wheatley

    Perhaps that’s a little over the top, but there’s something in it. As a teacher it’s always worth reflecting on a year just gone, looking back at what went well and what might need changing for the next year. I spent the year as Maths and UKS2 lead whilst teaching in Year 6.

    As such I have the privilege of being up to date with the changes taking place in primary education, especially with regards to the expected standards in assessment. Now that I’ve got a few weeks of holiday under my belt, my mind is a little fresher. It's on natural then, that I begin to look back upon KS2 Maths SATs 2017. Read on for my reflections on the end of July and the ever-present changes to how Maths is assessed in UK primary schools...

    https://www.thirdspacelearning.com/blog/2017/ks2-maths-sats-on-reflection-teaching-for-mastery

    Tuesday, 4 July 2017

    KS2 Tests 2017 Maths SATs Round-Up


    https://www.thirdspacelearning.com/blog/2017/ks2-sats-results-2017-what-they-mean-what-they-ll-never-tell-you-what-to-do-next

    I produced a quick response to the KS2 maths SATs results for the Third Space Learning blog.

    In it I cover what to do once results are opened: support staff, conduct a marking review, report sensitively to parents and children, learn from the results and look for the positives in the results.

    Monday, 20 March 2017

    On the TES Blog: Why Every Primary Should Be Using Bar Modelling – And Six Steps To Make It A Success

    As a primary maths coordinator, it's been difficult to escape the lure of bar modelling: it's in every new publication, on all the maths blogs and at every coordinator's meeting. And so, when the time was right for my school, I succumbed.

    Bar modelling, for the uninitiated, is not a method of calculation. Instead, it is a way of representing problems pictorially: from simple addition, through to finding percentages of amounts, all the way to complex multi-step problems involving ratio and proportion. Bar models can be used to pictorially represent arithmetic problems, as well as reasoning problems written with a context.

    For a worked example of bar modelling and 6 steps to ensure introducing bar modelling is successful, read on at the TES blog:

    https://www.tes.com/news/school-news/breaking-views/why-every-primary-should-be-using-bar-modelling