Steve Chinn MATHS LEARNING DIFFICULTIES, DYSLEXIA AND DYSCALCULIA SECOND EDITION k h it it ■ Maths Learnin Difficulties, Dyslexia and Dyscalculia Second Edition Steve Chinn Jessica Kingsley Publishers London and Philadelphia Contents Introduction.......................... 7 1. Dyslexia, Dyscalculia and Maths Learning Difficulties ... 9 2. Why Children May Not Learn Maths............. 13 3. Maths Anxiety......................... 18 4. Cognition and Meta-Cognition in Maths.......... 21 5. Key Numbers.......................... 28 6. Early Number Experiences.................. 31 7. Two-Digit Numbers......................' 36 8. Moving from One-Digit to Two-Digit Numbers, from Two-Digit to Three-Digit Numbers...and Back . ... 39 9. Basic Facts for Addition and Subtraction.......... 45 10. Addition and Subtraction................... 56 11. Basic Facts for Multiplication and Division......... 66 12. Multiplying and Dividing by 10,100 and 1000 (Part 1). . . 77 13. Multiplication and Division.................. 87 14. The Development of Multiplication: The Area Model ... 97 15. Fractions............................ 102 16. Decimals............................ 110 17. Multiplying and Dividing by 10,100 and 1000 (Part 2). . . 117 18. Percentages.......................... 120 19. Word Problems........................ 123 20. Measurement......................... 125 21. Time.............................. 127 22. Estimation: An Essential Maths Skill?............ 129 References and Further Reading............... 137 Publications by Steve Chinn................. 139 Index.............................. 140 Chapter 6 Early Number Experiences 'and operation) concepts are unequivocally fundamental to : : r'ess. Get the foundations secure, not just in long-term memory as feds, but as interlinking concepts. I sometimes visit bookshops to browse the books about numbers that are targeted at young children. Often, in fact, usually, they are -z-e about design than content. Sadly, this observation applies to some books that target older learners, too, where the inclusion :r crawings of cute animals in clustered groups blur the concept of -_~:e" that they are supposed to illustrate. I realise that meerkats •ray not naturally cluster themselves in manageable and identifiable :-a::erns in real life, but then the illustrations in maths books are rarely S'awn to look like real life anyway. I believe that two factors are influential here. One is the dominant' X)wer of the first learning experience (Buswell and Judd 1925) and the other is the negative influence of apparent inconsistencies in many of :~e aspects of learning. However, we must remember that consistency -=> take different manifestations for different children. For example, two ways could be used to represent numbers visually: A linear arrangement: □ m lid mm i i i i i i 1 2 3 4 5 6 7 8 31 32 Maths Learning Difficulties, Dyslexia and Dyscalculia Which could include a differentiation to show the contribution of 5: □ m cm iiiii i i i i i i 12.3 4 5 I I II ! U i I i 1 ..LlEJ INI ITTH 6 (as 5 + 1) 7 (as 5 + 2) 8 (as 5 + 3) The second representation is as a pattern (see also Chapter 4): □ □ □ □ □ □ 1 2 3 4 5 □ □ □ □ 6 (as 5 + 1) 7 (as 5 + 2) □ □ □ □ □ □ □ 8 (as 5 + 3) 9 (as 10 - 1) The linear presentation may suit the children who like to count, often one by one. The pattern presentation is a way of presenting numbers in clusters, using a form of subitising, a way of recognising a quantity, for example, 5 from its pattern rather than counting each square. The clusters are based on the key numbers of 1, 2, 5 and 10. Early Number Experiences The squares may not be as attractive as the cartoons of worms or "*eerkats or ladybirds, but they can be used to show, in an uncluttered h , the relationships between numbers. For example, the patterns fwt-1 I—1-1 [-f _1 1 I I_ and i—i r—i can be used to show that 3 is more than 2, that 3 is bigger than 2, that 3 is one more than 2, that 3 is 2 + 1, that 2 is one less than 3, that 2 s 3 take away 1 (and the 'take away' is a literal interpretation if the square is actually taken away). Even for such basic relationships the :;abulary can be confusingly varied. I could have said, '1 is subtracted ^om 3' or (3 subtract 1' (thereby changing the order for the digits). ^he two patterns ■ ■ ... can be used to show that 4 is 'double 2', 'two times two', 'two plus two' .vhich is repeated addition and thus linked to multiplication, 2 x 2) or mat 2 is half of 4 (showing division as the inverse of multiplication). There are learning/teaching implications for this approach. The oattems, both the visual patterns and the symbol patterns, can be used at a range of levels. They are developmental. They take a basic concept and can be used to develop that into other concepts. This could be seen as sowing the seeds of maths development, setting down the roots for future growth, there to be referred back to time and again as the maths develops. For example, the question, (!s it bigger or smaller?' encourages children to appraise their answers in a reasonably low-stress Maths Learning Difficulties, Dyslexia and Dyscalculia way, using an aspect of estimation. The relationships between 2 and 4 can set the foundations for repeated multiplication (x4 as x2x2 and x8 as x2x2x2 or x23), for division and for the doubles facts. For early maths it helps to know where the maths is going; for later maths it helps to know where it has come from. For these early number experiences the visual representations are used to develop a sense of number and how numbers and operations inter-relate. This is using single-digit numbers, but that will progress to become two-digit numbers (and beyond). Materials and manipulatives As a teacher of physics, in the early days of my teaching career, it would have been bizarre not to use apparatus to set up experiments to demonstrate concepts, even at Advanced level. So, I find the use of materials to illustrate maths concepts quite a natural thing to do. I believe that every maths classroom should have a maths kit readily available, either in a cupboard or in a toolbox of the type you can buy in Do It Yourself stores, so that the contents are ready to use when explaining some maths process or concept that a pupil or pupils find confusing. Match the material to the concept and the vocabulary and, when possible, to the learner(s), using the materials and language alongside the maths symbols, so that the link is made between the 'bricks' and the 'numbers'. There are many examples of materials being used throughout this book. Before we move to the next development, two-digit numbers, I want to introduce a structure that will be used with each topic in the Early Number Experiences : : " z chapters. The key learning factors will be considered for nc topic. These are: • vocabulary and language • images, symbols and concepts • the relevance of the topic/concept to developing maths skills • meta-cognition (thinking about how and what you are thinking) • things to do and practise. Chapter 7 Two-Digit Numbers Two-digit numbers start at 10 and end at 99. Vocabulary and language The English language does not offer consistency for the two-digit numbers, particularly from 10 to 19. This must be very confusing for young children when they meet them for the first time. The vocabulary does not support the pattern of the symbols. The words for the first two-digit numbers after 10 are exceptions: eleven and twelve. The numbers that then take us towards 20, the teen numbers, suggest an order that is the reverse to what we write with digits, for example, we say 'fourteen' which hints at 4 and 10, but we write 14: 10 and 4. As well as the vocabulary, there is a very sophisticated concept here, the concept of place value. Images, symbols and concepts The symbols for fourteen are 14, a 1 and a 4 in a specific order. Change the order and 14 becomes 41, forty-one. There are two places where we could place the digits: __which can be _1__A_ or _4__1_ If we write 1 in the first place and 4 in the second place then that 14 is fourteen, 1 ten and 4 ones. 36 Two-Digit Numbers If we write 4 in the first place and 1 in the second place then that 41 is forty-one, 4 tens and 1 one. if we use coins as our kinaesthetic/visual images, then fourteen, 14, i one lOp coin and four lp coins and forty-one, 41, is four lOp coins a"d one lp coin: So, the place where we write the digits (relative to each other) in the '-mber determines the value they represent. In this example, 1 can 'epresent 1 ten or it can represent 1 one. 4 can represent 4 ones or 4 tens. This is the logic of place value. One key reason for this being a big problem is that this first experience that children have of place value as a concept, a very vital and pervasive concept, is not supported by the vocabulary. Things improve in the twenties, thirties, forties and on. For example, we say forty-five and write 45. Sadly, we don't say 4 tens and 5. We learn to understand that the '-ty' is a distortion often'. Another problem for some children is discriminating (aurally) the sounds of'thirteen' and 'thirty', 'fourteen' and 'forty', and even 'twenty' and 'twelve5, but if teachers and parents are aware of the potential for confusion then they can guard against it, possibly by using visuals or materials (such as coins or base 10 materials/Dienes blocks). Bead strings and coins are good materials for supporting the concept of place value as are base 10 blocks. The effective strategy is to show the materials alongside the numbers and to connect these and to talk the learner through the relevance of the illustration. Maths Learning Difficulties, Dyslexia and Dyscalculia The relevance of the topic/concept to developing maths skills The concept of place value underpins much of the arithmetic part of mathematics. For example, addition and subtraction, multiplying and dividing by tens, hundreds, thousands and so on, and decimals. As well as developing a flexible understanding of numbers, for example seeing' that 46 is 30 + 16 and that 9 is one less than 10, learners need to understand the role of zero in place value. Roman numerals do not include a zero. Zero was introduced when the Hindu-Arabic system of number entered the UK some seven hundred years ago. Two of the major goals for teachers are to develop a clear understanding of place value in their pupils and to wean them off counting in ones. Chapter 8 Moving from One-Digit to Two-Digit Numbers, from Two-Digit to Three-Digit Numbers...and Back The process of'crossing' the tens, for example, from 9 to 10 (and the hundreds, for example, from 99 to 100, and so on for the thousands, etc.), is one of the fundamental concepts of maths. It is a very important part of place value. Vocabulary and language 'Crossing' the tens means crossing from 10 ones to 1 ten. This is sometimes known as 'trading' or as 'carrying' or as 're-naming' when doing addition problems. When the 'crossing' is going back, it is crossing from 1 ten to 10 ones. This can be called 're-grouping' or 'decomposing' or 'borrowing' or 'trading'. (My preferred word for both crossings is 'trading'.) This is used in some subtraction problems. Images, symbols and concepts The images used below are coins. Note that the symbols, the digits, are written next to the images. You must link the visual to the symbol. 39 Maths Learning Difficulties, Dyslexia and Dyscalculia Remember coins are not proportional in size to the values they represent. THE TASK When we count up to, and then past 10, we move from a one-digit number to a two-digit number where one digit represents tens and the other-represents ones according to the place they hold in the number. Let's start at 8 and count up to 12 and let's look at how it works by using lp and lOp coins: 8 10 10 When we reach ten, we can exchange to ten lp coins for one lOp coin. So, as we cross the tens there is an exchange (or trade). Sometimes in maths this is called re-naming or re-grouping. Now as we count onwards, 11, 12, 13, the lOp coin represents, of course, the ten, and the lp coins represent the ones. O # o I o I • Moving from One-Digit to Two-Digit Numbers...and Back exchange happens as we cross the tens and count from 19 :d20: 19 2 0 ~ne zero, 0, in 20 tells us that there are zero ones. It also keeps the tens-c'git in the correct place, where the tens-digits should be. If the zero, D, was not there then we would have just 2. The next exchange, the next time we cross the tens, is from 29 to 30: 9 • • • %^ o 30, shown as 2 tens and 30 as 3 tens 10 ones 20 + 10 When we count backwards the exchange is reversed; it is the opposite procedure. We change one lOp coin for ten lp coins. For example, it nelps to 'see' or visualise 30 as three lOp coins and then as two lOp coins plus ten lp coins, that is as 20 + 10: Maths Learning Difficulties, Dyslexia and Dyscalculia 30-3 x 10 30 = 2 x 10 + 10 x1 o o ^ o \ o • • • 30 shown as three 30 shown as two lOp coins lOp coins + ten lp coins Then we can count back in ones and take away one lp coin each time. The same concept applies when we move from two-digit to three-digit numbers. So, for the sequence, 98, 99, 100, 101 we have: cuo o o v ( ) «rii it o o o o 9 8 . o o o v • • o o o o * 9 Now we get a double cross (!). The next one we add takes the 9 ones to 10 ones, which crosses the tens and we trade the 10 ones (lp coins) for 1 ten (a lOp coin). That makes ten lOp coins so we trade again, but this time we trade ten lOp coins for one £l coin (lOOp). Moving from One-Digit to Two-Digit Numbers...and Back . - 0 O O V V 0„0 CuO rooo'V ©°©#© O as 9 tens and 100 as 10 tens 10 ones The 'trade' " ■ - - ■ mages should be demonstrated first with the coins and -.: ..: .vith discussion and diagnostic questioning to ensure that --: ::-:ept is understood. Hie relevance of the topic/concept to developing maths skills -"ting on builds the foundation for addition and counting back :: :"-; roundation for subtraction. ~~e concepts here are place value and 'trading'. Place value -~: = 'D!ns our number system. Trading will be a key concept when ■:. z onto addition and subtraction. It will be used again in other topics. All the demonstrations above can and should be demonstrated with : 1C blocks (which can be bought online from a range of suppliers). Meta-cognition (thinking about how and what you are thinking) The learner should be encouraged to articulate their perceptions of irese demonstrations. Diagnostic questioning may weli help this, for = ■ =~ Die, 'Can you explain what is happening with the coins when you sfcow me the sequence 38, 39, 40, 41?' Can they now demonstrate 998, 999, 1000 (a £10 note)? Maths Learning Difficulties, Dyslexia and Dyscalculia This is also reinforcing an understanding of coins and notes, which, unlike the base 10 blocks are not proportional in size to the values they represent. (In Australia, the $2 coin is smaller than the 1$ coin! In the USA the dime, 10c, is smaller than the nickel, 5c!) If these concepts were demonstrated with base 10 blocks as well as with coins, then that 'base 10' part of the concept may be demonstrated more directly since the sizes of these materials are in proportion to the values they represent. Things to do and practise Practise crossing the tens, both ways, for a range of examples, such as 49 to 50 and 50 to 49, and 79 to 80 and 80 to 79. Do this with coins and symbols, base 10 blocks and symbols and then only symbols. Practise counting on in tens, starting from numbers such as 17 (27, 37, 47...), 14 (24, 34, 44, 54...), 35 (45, 55, 65, 75...), or 48 (58, 68, 78...). Do a similar practice exercise counting backwards. Do similar exercises for 197 to 202, 499 to 503 and 789 to 801. Can the learner extend the concept to 999 to 1001? Ask the learner to point to the tens-digit in numbers such as 2961, 641 and 1007 and then do similar exercises to point out hundreds and thousands digits. Encourage the learner to articulate their thinking as they carry out the tasks (meta-cognition again). Chapter 9 Basic Facts for Addition and Subtraction These are the 'facts' for the addition of any one-digit number or 10 to another one-digit number or 10. That is, from 0 + 0 to 10 + 10. They are probably known as 'basic' facts because, if students know them, they can work out ail other whole number additions. There is an equivalent collection of basic facts for subtraction, from 20 - 10 to 0 - 0. One of the most useful things about these basic facts is that they can be used to work out a fact that may have been forgotten, or to check it, if the learner is not 100 per cent sure of the answer (and preferable to using finger counting). These basic facts interlink. For example, if you know that 10 + 7 is 17, and you understand that 9 + 7 will give you an answer that is smaller and smaller by 1, so then, from 10 + 7 - 17 you can get 9 + 7 = 16, without counting 7 onto 9, or indeed retrieving the fact from long-term memory. From 10 + 7 you can extrapolate to 100 + 70, 1000 + 700 and so on (and thus re-visit place value). So, if pupils can use number skills to work out more facts and answers from the ones that they do remember, then it is worth considering which facts are the most useful to memorise. 45 Maths Learning Difficulties, Dyslexia and Dyscalculia Vocabulary and language There are several words that are used for the addition symbol + and the subtraction symbol -. Learners need to be familiar with the words used for these symbols. So, for example, 7 + 6 can be said as; 7 plus 6 7 add 6 7 and 6 7 and 6 more 7 more than 6 the total for 6 and 7 is and 13-6 can be said as: 13 minus 6 13 take away 6 What is the difference between 13 and 6? 13 subtract 6 What is 6 less than 13? Images, symbols and concepts The images and/or materials that could be used to illustrate these facts include counters, coins, Cuisenaire rods, base 10 blocks and number lines. Creative minds could find many other resources, such as sweets or rulers. The concepts here are that: addition and subtraction are reverse operations' so, the two operations are linked and thus, the number facts involved interlink, e.g. 7 + 6 = 13 13-7 = 6. Basic Facts for Addition and Subtraction There can be an interaction between the vocabulary used and the manipulatives that are used. For example, subtraction can be implied with the words: 'What is the difference between 7 and 4?' or 'From 7 take away 4'. The first can be modelled with Stern blocks, which can be compared to show the 'difference'. 7 The second can be modelled with counters. THE TASK So, which are the most useful facts to learn? Part of the way we find the answer to that question is by asking a second question, 'Which facts are most useful for working out other facts and doing other maths?' Three of the most useful sets of facts are: the doubles, e.g. 7 + 7 the number bonds for 10, e.g. 6 + 4 2 + 8 and the 10 plus a single digit, e.g. 10 + 3 Chapter 10 Addition and Subtraction As the topics and strands of maths develop, learners should move further and further away from counting in ones. The 'basic facts' or 'number bonds' are part of that process. The previous chapter looked at how these facts could be retrieved by methods that supported memory and sense of number. In this chapter the processes of addition and subtraction are extended to all numbers. Whilst using finger counting to work out 6 + 8 can be effective, it can be inefficient and prone to error, but using a similar strategy is not going to be feasible for examples such as 364 + 877 or 2009 - 743. Even if the counting strategy is used for the steps in a procedure, it will slow the procedure down, thus putting more demand on working memory and leading to errors. The strategies that were explained in the last chapter use chunks, for example 7 is chunked as 5 + 2 rather than seven ones. As with all the themes in this book, this idea will be developed and used as the maths develops. The principle is that using familiar, automatic chunks reduces the load on working memory. Vocabulary and language The vocabulary and the language used when addition and subtraction are put into word problems would require a chapter of its own (Chinn 2017a). In this chapter addition and subtraction are only presented as number problems. Later, in Chapter 20, we will look at word problems built around all four operations. 56 Addition and Subtraction The vocabulary used in this chapter will centre on the key concepts, that is, place value and crossing the places (another example of an alternate phrase, used here, is 'bridging the tens'). A variety of words nave been used at different times in the history of maths in schools for the process of'trading' 10 ones for 1 ten and vice-versa, as explained in the previous chapter. For example, changing 1 ten to 10 ones has oeen called 'decomposition' and 're-grouping' and 'renaming'. There was also a variation on this which was called 'borrow and pay-back'. Teachers should make clear which vocabulary they are going to use and model it with materials and/or visual images so that the learner understands. So, if the teacher says, 'I am going to decompose', it is best if the learner knows that this refers to a specific procedure in maths. It is worth bearing in mind that a previous teacher may have preferred, for example, 'rename'. The pupil needs to know that the teacher is referring to the same procedure, but is using a different word. Addition is what is known as 'commutative'. This means that it does not matter in which order you add numbers, the total will be the same, for example, 721 + 52 = 52 + 721 = 773. This is not true for subtraction, for example 600 - 10 does not give the same answer as 10 - 600. Different word order can lead to confusion. For example, 70 take away 50' presents the 70 and 50 in the order for the symbols used in the calculation, 70 - 50. But 'Take 50 away from 70' does not. Images, symbols and concepts The key prerequisite concepts are place value and trading. As ever, maths is developmental. The role of zero may also require specific attention and instruction. As with any topic in maths, if the prerequisite concepts are not understood, then the grasp on the new topic may well be insecure. Maths Learning Difficulties, Dyslexia and Dyscalculia Since trading and place value are key concepts it is likely that base 10 blocks and coins will create effective kinaesthetic and visual images. It is useful to remember that addition and subtraction are complementary operations. Adding is about putting numbers together. Subtraction is about splitting up numbers. 5+2=7 7-2=5 Algebra often expresses ideas succinctly, and it generalises and shows patterns, which makes it such a shame that people shy away from it. In this example: Addition is: A + B = C e.g. 25 + 63 = 88 Subtraction is: C - B = A e.g. 88 - 63 = 25 The relevance of the topic/concept to developing maths skills Adding and subtracting are important skills for life, particularly for money (though maybe less so these days with touch cards). They are also prerequisite skills for multiplication and division. Another maths example is calculating a mean or average which also requires accurate addition skills. THE TASK Written procedures for adding and subtracting usually follow the sequence of place value, that is, ones, tens, hundreds, thousands and onwards. The process moves from right to left on the page, small place values to large place values. This is opposite to our norma! writing and reading direction. Early maths so often appears to be inconsistent. Addition and Subtraction The first examples shown below do not require any bridging (or crossing) of place values. The procedures can be modelled with base 10 blocks, but only symbols are used here. 7432 +2516 Ones: 2 + 6 = 8 Tens: 3 + 1 = 4 Hundreds: 4 + 5 = 9 Thousands: 7 + 2 = 9 9948 (8) (40) (900) (9000) 9948 -7432 Ones: 8-2 = 6 Tens: 4 - 3 = 1 Hundreds: 9 - 4 = 5 Thousands: 9-7 = 2 2516 (6) (10) (500) (2000) For the second group of examples, there is 'trading' from ones to tens for addition and from tens to ones for subtraction. The examples are modelled with images of base 10 blocks. Apart from the 'trading' these examples are the same as the first examples. The ones are added, then the tens are added. 57 57 +26 +26 First add the ones: 7 + 6 = 13 7 + 6= 13 50 + 20 = Then add the tens: 50 + 20 = 70 83 Now add the two sub-additions: 70 + 13 = 83 Maths Learning Difficulties, Dyslexia and Dyscalculia With base 10 blocks: First add the ones: 7 + 6 = 13 Trade ten one cubes for a ten block Now add the tens, that is 5 tens + 2 tens plus the 'traded' ten, 5 + 2 + 1 = 8 tens. The final answer is 83. Now look at subtraction. It is about reversing the process. (Note that some learners find reversing a process quite difficult, so the steps need to be presented comprehensively and clearly.) The 'trading* is illustrated with base 10 blocks. Ones □ □ □ 83 -26 The blocks show 8 tens and 3 ones. Start with the ones, 3-6 There are not enough ones for the subtraction, so, 1 ten is traded for 10 ones, making 13 units. The subtraction in the ones column is now ready 13 - 6 = 7 □ □ □ □ □ ' □ □ □ □ □ The exchange or trade of 1 ten for 10 ones leaves 7 tens in the tens column. The subtraction is then 7 - 2 = 5 (70 - 50) Addition and Subtraction The answer is 57, which, of course, tallies with the numbers used in the corresponding addition problem. Subtracting when zero is involved ~"e subtraction problems that often cause most difficulties are those that involve a zero, for example 304 - 67. However, the logic -emains the same...there will have to be some trading to create more ones for the subtraction in the ones column. The confusion arises tsecause there are no tens in the tens column for use for trading; trading will need to start at the first available place, in this case, the hundreds column. if this is understood as the objective, to get some ones into the ones column, then the logic is to use the knowledge of the place value system to do this. The first practice could be confined to trading to get more ones: For 201, trading makes it 1 hundred plus 9 tens, plus 11 ones = 100 + 90+ 11 Hundreds Tens 11 [—i ■ i Ones 201 is now 1 hundred and 10 tens and one Now a trade of 1 ten for 10 ones creates more ones □ □ □ □ □ □ □ □ □ a Maths Learning Difficulties, Dyslexia and Dyscalculia The 2 hundreds and 1 one has become 1 hundred, 9 tens and 11 ones and the subtraction can begin: Hundreds Tens Ones ■ LI □ □ □ a □ □ □ □ □ □ 10Q □ 90 +11 201 100 90 11 This procedure might be written without the 'scaffolding'/support of place value columns as: 9 11 1 \ 0 1 Or, perhaps more clearly by using two extra lines for the two trading steps: H 1 T I O 2 0 1 1 10 1 (trade 1 hundred for 10 tens) 1 9 11 (trade 1 ten for 10 ones) The skill is to take a number and change the way it is made up, breaking it into parts without changing its value. This skill can be used for trading in additions and subtractions, but remember it was also used for basic facts such as 7 + 7, which can be changed to 5 + 2 + 5 + 2 = 5 + 5 + 2 + 2 = 10+ 4 = 14. Addition and Subtraction Maths can progress by taking an idea or skill and extending it to new areas and problems. This is much better for memory and understanding than seeing each extension of an idea as something completely new, instead of a development of previous understanding. Meta-cognition (thinking about how and what you are thinking) As number sense develops, students can start to estimate answers before starting to compute and appraise answers after computation. The level of sophistication used for this can be refined as confidence grows. For example, for 57 + 26, the first attempt could be 50 + 20 = 70. The next attempt could be to consider the ones, which will add to make more than 10 (since both are more than 5), taking the estimate to 80. Each problem should be overviewed to see if there are alternative and better ways to tackle it. As is stated above, it can be very helpful to understand how ideas develop and inter-relate: to know why you can use a method or a formula to solve a problem. An alternative method for 57 + 26 could be 55 + 25 + 2 + 1 = 80 + 3 = 83. Place value is very much a part of subtraction and addition, in particular knowing how to exchange/trade/regroup/rename numbers, as in seeing 673 as 500 + 160 + 13 or seeing 507 as 490 + 17. Students need to be able to see' numbers in different forms and combinations. A pragmatic example of meta-cognition is the strategy of casting out tens'. One of the goals I have for a learner is that they become less of an impulsive problem solver and more of a reflective problem solver. I want them to overview and reflect on a problem before they attempt the solution. I want them to think about how they will think about the problem. 'Casting out tens' can be used an early experience of this strategy. < 64 Maths Learning Difficulties, Dyslexia and Dyscalculia Here is an example. Add the numbers: 25 47 81 16 33 54 82 79 68 The addition could be done sequentially, adding the ones digits in the order they are given, then doing the same for the tens digits. (This could also be considered as an inchworm strategy.) This requires several addition steps, each of which may be a challenge (and a potential source of error) if the pupil is not secure in the retrieval of number facts. If the numbers are overviewed, then it is usually possible to find number combinations (number bonds) for 10, for example: 25 47 81 16 33 54 82 79 68 10 10 10 10 Only the 5 (from 25) has not been paired. The total for the ones is 40 + 5 = 45. Now look at the tens digits (with the 45 from the ones total included) and find the number combinations for 100. 25 47 81 16 33 54 82 79 68 + 45 100 100 100 100 Only the 80 (from 82) has not been combined. The total for the tens is 400 + 80 = 480, making the total 485. The cognitive strategy is to look at the numbers before starting to add them, to see if there is another way to add, using facts in an Addition and Subtraction encient and user-friendiy way. In a sense this strategy is a variation on r**e chunking strategy. A very basic estimation of the total can be obtained with an overview of the numbers. There are 9 two-digit numbers with a spread of values {not all near 100, not all near 10), so take an average number value of SO and multiply it by 9 giving an estimate of 450. This can prevent 'big' errors such as the place value error, 45 + 44 = 89 (instead of 45 + 440). Things to do and practise 3'actise trading, crossing tens and hundreds with materials such as base 10 blocks or coins, but also with symbols (the digits and numbers). This practising need not involve actual adding or subtracting; just r-arne numbers, for example 72 as 60 + 12 (as the French language does anyway!). Chapter 11 Basic Facts for Multiplication and Division The times tabie facts are probably a child's first experience of persistent failure. Luckily this is not true for every child, but I suspect that it is a very significant percentage, particularly among those who are dyslexic. It should be noted that even when children do commit this body of facts to memory, it does not guarantee that they will understand the principles that underlie them, nor that they will remember them when they are older, when there is less practising and less recall. It is also likely to be a child's first experience of the consequences of a maths belief, implicitly inferred or explicitly stated. That belief is that children, all children, can and should learn the times tables. A secondary belief is that maths will be very difficult if these facts are not retrievable, quickly, from memory. These somewhat simplistic beliefs are not correct for every child and neither are they reasonable. The basis for my counter-beliefs is that there is an alternative and it is a mathematical alternative with cognitive benefits. And they are attainable. But the belief in the efficacy of'learning by heart' is a very entrenched belief, often coupled with a sense of superiority. For many people, the problematic facts lie in the bottom right-hand corner of the multiplication square. It seems like a small problem... unless you can't do it. 66 Basic Facts for Multiplication and Division X 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Vocabulary and language Once again, the vocabulary is not helpful in communicating the concept and once again there is no consistency. Let's start with, 'What is multiplication?' and the standard reply, 'Repeated addition'. Adding a column of different numbers could be interpreted as 'repeated addition'. In this specific case, multiplication, it means, 'Repeated addition of the same number' An example would be 6 + 6 + 6 + 6 + 6 + 6 + 6 = 7x6. Then there is a range of vocabulary to infer multiplication, including, 'times' lots of, 'product' and nothing at all, as in 'What are six sevens?' That is a question where you need to know the maths (lack of) language code. Maths Learning Difficulties, Dyslexia and Dyscalculia If the definition of multiplication involves 'repeated addition' then it would follow that the inverse operation, division, should be 'repeated subtraction of the same number'. The two definitions emphasise the link between the two operations. There is a smaller range of vocabulary for division, including 'share' and 'How many...in...?', 'What is 25 over 5?', 'per' as in 'percentage... divided by 100' and nothing as in 3/5 or J. 5 But there is, as with subtraction, an issue with the order of words. For example, 460 divided by 5 gives the numbers and the operation word in the correct order for keying into a calculator or for writing in symbols, 460 + 5. The order of the wording for 'How many tens in 500?' does not work for a calculator, but it might help for the 'bus stop' presentation: 10 \soo Images, symbols and concepts The symbols used to infer multiplication and subtraction are more varied than the solitary + and - used for addition and subtraction. Again, consistency is not there to reassure the learner. For multiplication we can use: • x as in 5 x 7 • indices as in 82which means 8 x 8 • brackets as in 5 (4 + 3) which means 5x4 + 5x3 • nothing at all, as in algebra, where ab means a x b. Basic Facts for Multiplication and Division For division we can use: • + as in 10 + 2 • the 'bus stop' layout for the division procedure 2 10 • a negative index as in 25"2 • a stroke as in 5/10 • a line with one number above another number as in JL 7 A reminder: all four operations are interlinked, a concept that can be used constructively to provide alternative ways to solve problems. The relevance of the topic/concept to developing maths skills If the definition of multiplication is taken as repeated addition of the same number then aspects of algebra follow, for example: 2+2+2+2=4x2 5+5+5+5=4x5 8+8+8+8=4x8 a + a + a + a = 4xa = 4a If these repeated additions are collected into chunks, then the process of 'long' multiplication follows, a process that depends on partial products, for example: 7 + 7 +7 + 7 +7 + 7 + 7 + 7 + 7 +7 + 7+ 7 = 10 x7 + 2x 7 = (10 + 2) x 7 The links between multiplication and division are relevant in many areas of maths (see also the next chapter). For example, it helps to be able to relate these three equations and to understand their relationship: Maths Learning Difficulties, Dyslexia and Dyscalculia 6x3 = 18 6 = 18-3 or JL8 3 3 = 18-6 or la 6 This is relevant to many equations/formulas such as: Distance = Speed x Time D = S x T or Speed = Distance Time or Time = Distance Speed Such three component equations are sometimes presented in a triangle as an aide-memoire for the inter-relationships. It's good to be able to understand why. (There is also a clue in the units used for speed 'miles per hour' or 'kilometres per hour'. Miles and kilometres are for distance and hour is for time, so speed is equal to distance per time, and 'per' means divide.) Meta-cognition (thinking about how and what you are thinking) Remembering the basic multiplication and division facts is beneficial, but knowing how they inter-relate - how facts can be combined to Basic Facts for Multiplication and Division find new facts (often via partial products) - means that the benefits will be far greater. In the equation above, S = D x T, it may help to relate it to a basic fact which is known comfortably and accurately, such as 2 x 5 = 10. The two other forms then follow and verify themselves: 5 = 10 and 2 = 10 2 5 Children need to be taught how to understand the concepts of multiplication and division, how to think around the ways they can be used in other applications, such as percentages, area, algebra and fractions. They need to realise that in an equation like xy = constant, then as x gets bigger, y gets smaller. This can lead to understanding sequences such as: 12-12= 1 12-r 6= 2 12-r 3= 4 12+ 2= 6 12+1 = 12 12 + Vi = 24 and thus, to the realisation that, in certain situations, division can make the answer bigger (than the divided number). Although the culture and beliefs around maths seem to be that basic facts are always retrieved in one step, there is a strong case for arguing that two steps are a very viable and mathematically alternative way of thinking about some of these 'facts'. Meta-cognition comes from knowing how partial products work for multiplication and for division. This knowledge leads us to alternative ways to multiply and divide. Maths Learning Difficulties, Dyslexia and Dyscalcutia THE TASK The task is to be able to access all the basic times and division facts, by recall or by strategy, as swiftly as possible (without raising anxiety). One possibility is to use mnemonics. I confess that I am not a great fan of the extensive use of mnemonics. The odd one here and there, like 'Richard Of York Gave Battle In Vain' for the colours of the rainbow 'Red, Orange, Yellow, Green, Indigo, Violet' is great, but to create a whole book of them dedicated to mnemonics for times table facts is making War and Peace out of a nursery rhyme. However, if it works, it works. But remember that recall on its own isn't cognition. A powerful rote-learning strategy is the self-voice echo strategy of Dr Colin Lane, where the facts or information to be learned are recorded, in the learner's own voice on a PC, with a matching visual on the screen. That fact is then repeated back time and again (well, it is rote learning), preferably through headphones. My own research into this (Lane and Chinn 1986) showed that when it works, it works dramatically and with long-lasting retention. But it doesn't work for everyone, which is a valuable lesson for anyone who thinks they may have found 'the' cure for learning difficulties. A method based on meta-cognition relates back to some early maths concepts. First: let's consider the definition and understanding of what this collection of facts is: It is a collection of repeated additions, relating to the vocabulary, 'lots of. For example 6x8 is 8 + 8 + 8 + 8 + 8 + 8 6 lots of 8 and 7x7 is 7 + 7 + 7 + 7 + 7 + 7 + 7 7 lots of 7 Second: you can cluster these additions into chunks rather than add on one number at a time. The chunks' that help are likely to be Basic Facts for Multiplication and Division 73 ::~e ones using the 'easy' numbers: 1, 2, 5 and 10. In terms of cognitive ft. e, this will appeal more to grasshopper than inchworms. The chunks, for example, for 6 x 8 are 5 x 8 and 1x8 which are 40 and 8. These are 'partial products'. (A product is the outcome of a multiplication. Partial products can be combined to make the :omplete product.) So, then the 40 and the 8 can be added to make ~S\ the product for 6x8. -cr 7 x 7 the process can also be done as two partial products: Dartial products can be combined by adding, or extended by multiplying, *or example, 4x7 can be accessed via 2 lots of 2 x 7: 4x7 = 2x7x2 = 2x14 = 28 This strategy re-deflnes what makes a basic fact 'basic'. The key basic -"acts are the minimum number of facts that you need to work out other facts efficiently. They are the lx, 2x, 5x and lOx facts for a "-.mber. For example, for the 8 times table, the core basic facts are: 1x8= 8 2x8 = 16 5 x 8 = 40 10x8 = 80 + 8 5x7 = 35 2 x 7 = 14 35 + 14 = 49 These four partial products can be combined to access all the other 8x table facts, and some extra facts, too. Maths Learning Difficulties, Dyslexia and Dyscalculia 1x8 = 8 2x8 = 16 3x8 -24 3x8 = 2x8 + 1x8 = 16+ 8 = 24 4x8 = 32 4x8 = 2x8 + 2x8 OR 2x(2x8) = 16 + 16 5x8 = 40 6x8 = 48 6x8 = 5x8+1x8 = 40+ 8 = 48 .7x8 = 56 7x8 = 5x8 + 2x8 = 40+16 = 56 8x8 = 64 8x8 = 2x2x2x8 = 2(2 x 16} = 2 x 32 = 64 9x8 = 72 9x8 = 10x8-1x8 = 80- 8 = 72 10x8 = 80 Extra facts: 11x8= 88 11x8 = 10x8 + 1x8 = 80+ 8 = 88 12x8= 96 12x8 = 10x8 + 2x8 = 80+16 = 96 15x8 = 120 15x8 = 10x8 + 5x8 = 80 + 40 = 120 19x8 = 152 19x8 = 20x8-1x8 = 2x10x8-1x8 =160-8 = 152 The principle that underpins this strategy is the principle that underpins 'long' multiplication. Division facts are the complementary or reverse facts to the multiplication facts, for example, 6 x 8 = 48 is a multiplication fact. Two division facts are complementary, for example, 48 + 6 = 8 and 48 + 8 = 6. These two division facts could be written as 6 x ? = 48 and 8 x ? = 48, thus interlinking division and multiplication. If the multiplication facts are known, then the ability to visualise them in this form will give the division facts. The answers to division questions and facts can be accessed from the key basic facts. Where multiplication is repeated addition of the same numbers, division is repeated subtraction of the same numbers. For example, to access 72 + 8, subtract partial products: Basic Facts for Multiplication and Division The two partial products that will be used in this example are: 2x8 = 16 and 5x8 = 40 Start by subtracting 40 (5x8) 72- 40 = 32 then 16 (2x8) 32- 16 = 16 and 16 (2x8) 16- 16 = 0 9 lots of 8 have been subtracted so 72 + 8 = 9 Partial products can be demonstrated with Cuisenaire rods. Take the examples, 9 x 8 = 72 and 72 + 8 = 9. First, for 72 + 8 use 9 'eight rods'. (The area this creates represents the 72.) 9 (rods) 5 (rods) 2 (rods) 2 (rods) 8 This area can be split up to illustrate the partial products being subtracted This image relates to the written presentation for division: 9 8 72 lo demonstrate 9x8, the rods can be put together as three partial □roducts, 5x8 and 2x8 and 2x8. '5 lots of 8, plus 2 lots of 8 plus 2 lots of 8.' 76 Maths Learning Difficulties, Dyslexia and Dyscalculia Things to do and practise Practise and learn the key basic facts described above. Practise combining them to make other facts. Write the key facts on cards and play memory games with them, for example, after spreading a number of (8x or another number) cards face down on a table ask questions such as, 'Find two cards that can •make 56.' (It will be a 5 x 8 card and a 2 x 8 card.) Use the rods to demonstrate how areas are constructed and split up (de-constructed) using partial products. Chapter 13 Multiplication and Division Once the problem of the basic facts has been addressed, the next step : be able to compute longer/harder problems for multiplication and : s:on. The data collected to create the norm-referenced maths test r my book on diagnosing maths difficulties (Chinn 2017b) revealed : ■ evels of performance in these tasks. For example, the percentages from this large (1783) sample of UK school children who achieved the czrrect answers for the questions were very low: For 9 927 10 years, 14.5% 13 years, 31.4% 15 years, 46.8% For 541 x203 10 years, 14.1% 13 years, 15.2% 15 years, 38.2% ::: - abilities with such questions obviously stretch beyond the special -eeds population. This could suggest that methods that are introduced ~cr pupils with specific learning difficulties will help many more children ss well. An example of the developmental nature of maths is that the ?:'ategies used to extend the key basic facts to other basic facts were rased on partial products and these are now used in long multiplication. This concept will be developed further in this chapter. 87 Maths Learning Difficulties, Dyslexia and Dyscalculia The four operations are interlinked, a concept that can be used constructively to provide alternative ways to solve problems. For example, it is possible to obtain an answer to a subtraction problem by adding on. For example, to compute 100 - 63, start with 63 and add 7 to make 70, then add 30 to make 100. 30 + 7 have been added, so the answer is 37. It is possible to divide by thinking of multiplication. For example, 42 + 7 = ? could be written in a different order as ? x 7 = 42, becoming, 'What do I multiply 7 by to get 42?', but to do this there must be an understanding of that link between multiplication and division. Another example of this link is that 'long' multiplication is carried out by adding together partial products and 'long division is carried out by subtracting partial products. Vocabulary and language Multiplication, like addition, is commutative. That means that the order in which you multiply numbers does not change the final answer, for example: 6x5 = 5x6 or 7x8x9 = 9x8x7 This is not true for division, where the sequence of numbers and symbols affects the answer, for example: 30 + 6 = 5 but 30 + 5 = 6 and 30 + 6 = 5 30 + 6 can be stated in words as 'thirty divided by six' where the order of the words matches the symbols. It could also be stated as, 'How many sixes in thirty?' where the order of the numbers is now reversed. There may also be an interpretation issue. 'How many sixes in thirty?' gives no clue as to the process that might lead to an answer. The learner needs to know the mathematical meaning of the vocabulary. Multiplication and Division 2~e of the fascinating errors for division I have encountered is: 0^ 5 35 02 4 28 x guess as to the method used is, 'How many fives in 3? There aren't any, so 0. How many fives in 5? There's 1' And 'How many fours in 2? There aren't any, so 0. How many fours in 8? 2! Many of the errors encountered in the standardising data were probably down to part -r~embered procedures or mis-applications. ages, symbols and concepts ~-ere are some concepts and skills that are prerequisites for these tasks and this might be one reason for their apparent difficulty. The ^requisites have been forgotten. When a maths procedure becomes :-e complex and demands more steps, it obviously becomes more -