Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. 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/

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, 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, 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, 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

    Friday, 6 January 2017

    Key Stage 2 SATs Results 2016 Explained: 15 Insights That Will Change How You Teach Year 6 Maths in 2017

    Given that I'm maths leader at my school you'd expect that my blog would contain more than just one post about maths, but it doesn't. Until now, that is. And even this one's not a full and proper post, only a link to a piece of work I've done for Third Space Learning.

    I spent some time with the Question Level Analysis document produced by RAISE online, working out what the most difficult aspects of the KS2 tests were in 2016 so that hopefully we can all prepare our children well enough for the 2017 tests.

    Click here to read the full in depth analysis: Key Stage 2 SATs Results 2016 Explained: 15 Insights That Will Change How You Teach Year 6 Maths in 2017

    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.

    Photo Credit: WoofBC via Compfight cc