Teaching Excellence

by Richard Bandler

  building a number counter

  Now it’s time to learn to manipulate much bigger numbers whilst realising that there are really only 10 digits to deal with in any number, however big it gets. The problem with larger numbers is that learners try to hold too many numbers in their mind. What we want them to learn to do is just hold the last and biggest number in their mind, so it is clear and easy to remember.

  The strategy for doing this is to build a number counter in the learner’s mind that is big, bold and clear. So now take a number such as 24 and see it in your mind as big and colourful and clear so it looks something like this:

  Now add 2 to each of the numbers starting with the number on the right.

  4 + 2 = 6 make a click sound, and see the counter move from 4 to the number 6.

  Now take the number next to it on the left: 2 + 2 = 4, make the click sound and move counter from 2 to 4.

  Now this is the only number to have in your mind. A really big colourful number:

  A useful representation to have here is a digital meter, or digital calendar, where the numbers automatically go back to zero after 9. As they roll around, add a ‘click’ sound to the movement, you know it has moved on.

  So now the only number we have in our mind is 46. Add 2 to each number and hear the click each time the number rolls around: 6 + 2 (click) the counter moves to 8. 4 + 2 (click) the counter moves to 6. Now the new number is 68 and this is the only number in your mind, big, bold and clear:

  Move on, adding 3 to the number and then 4 to the number until the machine inside your head is working by itself.

  Move on to adding a number to a 3-digit counter such as 423, so the meter gets bigger and bigger.

  Keep going until the meter has as many numbers as you want it to have. Keep playing the game, (remember the ‘click’ each time) until the counter is encoded in your student’s mind and they are ready to move on to the real addition strategy.

  Remember, we are building a head full of numbers to be manipulated and played with for fun!

  Now there comes a point where the numbers added together are 10 or more. So now it’s time to talk about real addition.

  real addition

  Once the digital meter is working all by itself, the next step is to adjust the numbers every time the number goes around the clock past zero. So start with the big number 99 and add 11 to it:

  The 9 moves up to 0 and the next 9 goes to 0 plus 1 and a new column appears with 1 in it, so the new number is 110 and this is the only number in your mind:

  110

  Move on to even more interesting numbers! Have 626 as big and bold at the top an`d underneath have the number 333, still clear but a little smaller. Like this:

  Now add the top right number (6) and the bottom right number (3) and turn the number on the meter to 9. Next do the same (remember the click) with the top middle number (2) and the bottom middle number (3) and turn the top number to 5. Now do the same with the left hand numbers 6 plus 3 and turn the top number to 9. Now you have a new big number at the top:

  Carry on adding 333 to this number. This time, when the number goes past 0 click the big number up one -so 3 plus 9 is 2, and the 5 clicks to 6. 3 plus 6 is 9. So the left hand numbers are 3 plus 9 so this number clicks to 2 and a new number appears in a new column 1292 and this is the only number in your mind:

  It doesn’t matter how big the numbers get, it’s just as simple as adding one column and moving to the next using just the numbers 1 to 9 plus 0.

  When learning maths, children are instructed to ‘show your working out’. With this process there is nothing to show on the outside - it’s all happening on the inside. Showing how the answer has been achieved is often for the teacher’s benefit so that they can be sure the child has gone through the process of carrying numbers and number placement. This is not the purpose here. We are building machines inside our learners so that the basics of mathematics happen automatically. With these strategies installed inside young minds, multiplication, division, algebra, and geometry become easier processes and maths is fun!

  Subtraction

  Once we have the machine in our heads to add numbers, the same machine can subtract numbers. So all the same games can be played in reverse to build a subtraction machine. Start with the counting game, but this time the counting counts back from 100. Move on to the take away game, which is the same as the addition game, but this time with the question ‘who can get to zero first?’

  Doubling up becomes halving, getting to 10 becomes getting to 0, and the addition game becomes the subtraction game. Starting with two numbers and take off 2 at a time, move on to 3 or 4 first and then make the big number at the top drop down one rather than adding a number on.

  Maths is about finding the simple solutions and the easy way to do things. Adding 4+4+4+4 may seem quite a bit of work, but if your mind sees that it is the same as 2 lots of 8 and double 8 is 16, it’s easy. It doesn’t matter how big the number is, it is always made up of smaller numbers.

  Humans are neurologically wired for the decimal system. We have 10 fingers and 10 toes. Discomfort with numbers is often because we haven’t found the right voice and the right place to have the good feeling; when we find the most comfortable place you can find the solutions and strategies for your students to have inside their minds.

  summary

  In this chapter you have discovered strategies that work on the inside to build mathematical machines. You have learned that maths isn’t just a process of manipulating numbers on a piece of paper, but creating a mind full of numbers and strategies that make manipulating numbers and solving problems easy. You have built the basis of a mathematical mind with counting, addition and subtraction - and next we explore how the fun multiplies.

  references

  1 Reply, according to Dr. Felix T. Smith to a physicist friend who had said “I’m afraid I don’t understand the method of characteristics,” as quoted in The Dancing Wu Li Masters: (1979) by Gary Zukav, Bantam Books,

  2. Skilled for life? Key Findings from the Survey of Adult Skills Ref http://skills.oecd.org/documents/SkillsOutlook_2013_KeyFindings.pdf

  3. Gunderson E, Ramirez G, Levine S. C, Bellock S.L, 2011The Role of Parents and Teachers in the Development of Gender-Related Math, Published online

  activities

  The digital meter in your head Install the digital meter to learn addition and subtraction. Practise adding bigger and bigger numbers to each other as above.

  The BIG counter

  Using a smart board, create a BIG counter on the screen and play the counting games with the whole class

  Elicit excellent strategies

  Elicit strategies from 2 different people for subtracting large numbers. Find out what they do inside their minds to make it easy. Try them on and see which one works best for you.

  The meter method

  Teach someone who is convinced they are no good at maths to add numbers using the meter method outlined in this chapter. Simplify your language so concepts are really easy to connect to the learner’s previous experience and create states of certainty and comfort in the learner.

  Extension activity

  How could you utilise this method for teaching decimals and negative numbers?

  This eBook is licensed to Dominic Luzi, dluzi@managementalchemy.com on 10/18/2018

  chapter 7

  Mathematical Magic

  TAP THIS TO SEE THE VIDEOS

  ‘Doing mathematics should always mean

  finding patterns and crafting beautiful and

  meaningful explanations.’ (1)

  Paul Lockart

  In this chapter

  Mastering Multiplication

  Fast and effective Division

  Geometry made easy

  Engineering successful strategies

  Mathematics is often taught as a conceptual subject with no real application to the world; only those students who enjoy the logical process love maths, while others can see no real purpose to
learning multiplication and division. However, maths is part and parcel of everyday life. The chef knows how to multiply and divide ingredients, the traveller needs to estimate arrival times and calculate money exchange rates and distances, and artists and photographers need to calculate angles, depth and perspective. It’s a great stress saver; can you imagine the headache of trying to solve a problem without the short cuts and easy tricks that Mathematics teaches us?

  When many of us think about learning multiplication tables we are transported back into the classroom, sitting at a small table and monotonously repeating our tables over and over. Then, at the end of the week or on Monday morning there was the ‘test’, which was often frightening. This traditional method attempts to install times tables auditorily, and creates unpleasant feelings of dread and fear of failure. There is rarely a visual element to the strategy. However, people who manage to learn their tables easily somehow work out that making a picture of the numbers is a good way to remember them. It’s unlikely that anyone showed them how to do this; the phrase ‘learn your tables’ does not specify to the student what to do inside his/her mind to find the easiest way to remember.

  Multiplication

  Adding and subtracting are processes that are carried out consciously in the mind. Multiplication is just a fast way to remember repeated additions of the same number – a quick way to add a series of the same number to each other. Multiplication happens unconsciously because we memorise the sequences rather than add each number to the next and the next. So ask a person who has learned a multiplication table really well, ‘what is 5 times 5?’ and the answer just pops into their head. This memorising process is the one that we as teachers need to teach by specifying precisely what the learner needs to do on the inside of their mind to achieve this. There is a huge difference between the instruction to ‘learn these tables’ , and an instruction to ‘picture these tables and make the numbers jump out and grow really big!’ Just notice for yourself the difference in the quality and detail of your mental processes when you say these two instructions to yourself. Once the process has been correctly installed, multiplying becomes unconscious and natural.

  The good news is that when a learner has learned to count up in sequence (see Skip Counting in Chapter 6) they already have the basis of the multiplication tables and have encoded the numbers so that multiplication and division are easy and already familiar.

  The trick is to enable learners to visualise the times tables as a big chart. The instructions to your students are to see the big square, which has the (by now) familiar sequences of numbers completing the grid, as one complete image.

  The human brain likes patterns and learns them easily, so before you begin with the mechanics of multiplication introduce the big picture first. This also means that the student knows where they are going, so they begin to learn to track forwards and to track backwards on the grid.

  Start by exploring the grid and looking for familiar sequences and patterns. Look at the unit tables and across the top row:

  Now the learner can see that when there is nothing to times a number by there is nothing to do!

  Now look at the next row:

  Well fancy that – it’s just counting! Now look at the next row:

  How familiar are these numbers now? Because in the last chapter we played at racing our way up to 100 in 2s, we can go way beyond this chart. Each row has the same sequence of numbers that you learned when skip counting to 100, and each column has the same set of numbers too. Also notice that the pattern is symmetrical along the diagonal, so you only need to remember half the numbers because the answer is the same for, say, 8x9 as it is for 9x8.

  So now the task is not to multiply the numbers - the task is to track to the number that is in the left hand column horizontally, and track to the number from the top row vertically until the two numbers meet in the square and make the number jump out in your imagination. This trains the mind to pop the right number up automatically.

  The image above stops at the 12 times table, but of course your mind is not limited in size, so the chart inside a learner’s head can be as big as they want it to be and the numbers can be coloured in any way that makes the learner happy and comfortable.

  Each time you ask ‘what is four times four? ’, track along the line from 4 on the left side to meet the number 4 coming down from the top line and see the number 16 pop up in the correct square.

  Ask ‘what is four times five?’ , track along the line from 4 on the left side to meet the number 5 coming down from the top line and see the number 20 pop out.

  Then move to the 10s and make the table bigger and bigger, with large numbers that are easy to see. When you do this enough times the correct numbers pop up by themselves, because the numbers are in the squares. It is just a question of paying attention to the two starting points and playing the tracking game lots of times, just for fun!

  The brain is capable of memorising an infinite amount of information. An idiot savant may be able to memorise all the numbers in a phone book and tell you the name of the person each belongs to. For a few such individuals, the information is all there, all of the time. However, this isn’t useful - you don’t want the chart to stay there all the time, you just want the chart to determine the number, so the more you look at your chart, the more the number pops up by itself, so that it becomes unconscious. As soon as somebody says ‘what is 12 times 12?’ rather than having to compute it through some long process, the brain will flash up 144. Once you have all the 10s it’s easy to get the 100s, then you can do the 1,000s, which happens just as easily. This is because it is the mechanism of learning how to make it happen automatically that makes the fast mathematician.

  Some multiplications will be easier simply because they are more familiar. Others may seem harder in the beginning, but when we start to look for what we already know even unfamiliar numbers become easy. For example 3 x 18 might seem quite hard until you notice that you already know that 3 x 20 is 60. The bit of the number that is too big is just the number 2 (the difference between 18 and 20) which just has to be multiplied 3 times to find out how by how much 60 is too big a number. Go back to your table and you easily know that 2 x 3 is 6. Subtract 6 from 60 and you have 54. So 3 x 18 = 54. Easy! Looking for the patterns in what you already know and finding the tricks makes maths fun.

  When numbers get bigger in multiplication, remember you are still only dealing with 9 digits plus 0, just like with addition. Getting to the answer for 17 x 17 might seem like a lot of work, but if your brain thinks in simple terms and breaks the problem into simple steps it becomes easy - like this:

  The 10 times table is easy, so just add a zero to find 10 x 17 = 170

  Using the 10 times table in the same way, 10 x 7 =70

  Add these two numbers together to get 240.

  That just leaves the 7x7 bit to find. 7x7 is on your chart and pops out at 49.

  Add this to the number you already have and even a large multiplication like 17 x 17 = 289 is easily discovered.

  The message for our learners is that they want to look for the easy way to get to the answer.

  Debbie had a complete fear of the 9 times table until she was shown a pattern that she hadn’t been aware of before. The pattern was that all the multiples of 9 add up to 9 when the two digits are added together, so it was a really easy way to be certain the number was correct.

  The pattern is 18, 27, 36, 45, 54, 63, 72, 81, 90. So to find 2 x 9, drop down one number from the multiplier - 2 down to 1 and make this number up to 9, which is 8, so 2 x 9 is 18.

  For 7 x 9 drop the multiple 7 down one to 6 and find the number that makes this number up to 9, which is 3, so 7 x 9 is 63.

  As soon as Debbie saw the pattern and practised a few multiplications she had the pattern installed and felt really good about herself.

  The traditional and somewhat tedious way of multiplying already presupposes that the 10 times table works by itself. So if you multiply 12 x 12 traditionally, you multiply
2 x 2 and then 1 x 2 and add a 0.

  But you still have to be able to multiply 2 x 2, so there is no reason why you can’t multiply 12 x 12. It’s easier to increase the numbers on the grid so they pop up by themselves.

  If for some reason a learner doesn’t know the answer they can go back and dissemble the multiplication, but they will still know some part of the sum which will help them. They already know 10 x 10 equals 100, so all they have to worry about is how to get 144 with the bit that’s left over! They also know that 2 x 10 is 20 and 2 x 12 is 24. So you can do it the long way or you can just do it with the chart in your mind!

  Division

  Here is a secret to share with your learners - long division doesn’t really exist! The process of division is simply multiplication and subtraction with a bit of guess-work thrown in. The bigger the numbers you are dividing, the more often these steps are repeated and long division just repeats the sequence more than once.

  Now that you have the strategies for multiplication you have the basic strategies for division, which also happens to use the same multiplication chart. Imagine you are dividing 108 by 9. You are really asking yourself the question, how many times will 9 fit into 108. Your brain tracks across the multiplication chart to 108. The two numbers that correspond to the intersection with the square 108 are 9 and 12. So 9 fits into 108 12 times. 108 ÷ 9 = 12. As the chart has the same numbers for multiplication and for division, it’s easy.

  Now the chart in your head doesn’t have to stop at 12 so now divide 119 by 7. Remember, you are asking yourself how many times will 7 fit into 119. Your brain tracks across the multiplication chart to 119. The two numbers that correspond to the intersection with the square with 119 in it are 7 and 17. Just as easy when you have a big chart in your head.