With the process of long division, the trick is to find the biggest number on your multiplication chart that is below the number you want to divide. So you know that all your brain has to do is scan the chart. Imagine you are dividing 124 by 7. (124 ÷ 7). Your brain finds the biggest number that’s below 124. The question is how many 7s are there in that number so you can tell what is left over? It’s really another multiplication task. So you simply track across the line of multiples of 7 until the number is bigger than 124 and then go back one square and the number pops out. So there it is again, 119. 7 divides into 119, 17 times. (119 ÷ 7 = 17). Subtract 119 from 124. (124 – 119 = 5) which gives you the number left over, known as the remainder. 119 ÷ 7 = 17 r5. The more you train the brain to do this, the more automatically it finds the biggest number and then of course what’s left over is simple subtraction.
It’s important not to underestimate the power of these simple methods and how quick and effective they can be. One day, sitting in the back of the car, 8-year-old Maisy announced, ‘I’m rubbish at maths ’. After 20 minutes of playing number races, she was doing long division easily and enjoying the process.
To divide 377 by 6, we asked her to take the first two numbers in the sum and asked, ‘how many times can you get 6 into 37?’
‘I know it fits into 30, 5 times’ she said.
‘Can you squeeze another one into the 7 that is left?’
‘Oh yes one more – that’s 6 times ’, she said with delight on her face.
So that leaves us with 1, which is in the 10s, and so is 10 and the next number to it is 7. Add these together and we have 17 more to fit 6 into. ‘How many times can you squeeze 6 into 17?’ we asked.
She said, ‘3 times 6 would be 18, but it is too big so it has to be 12, which is 2 times 6’
Yes, so add the 2 to the 60 and the answer is 62 with 5 left over’
A very big sum for a little girl who used to think she was rubbish at maths!
Teaching these processes to your students is now very much easier. At one time all we had as visual aids were posters and grids in books. Now we have computer programmes and phone apps which can create and animate the multiplication charts on the screen. Although some teachers lament the use of mobile phones in maths classes, we don’t agree. When learners can see the process of numbers popping out of a grid on a screen with their real eyes, it’s much easier to create the representation on the inside of their mind with their inner vision.
A computer programme or animation can pop up the number and blow it up so it fills the screen, so that when we ask what is 5 times 7, the number 35 pops up, fills the screen and then shrinks back down again. This way the brain gets used to taking the numbers, going to the intersection of the grid where the two points meet and the picture is BIG. If the question is 5 times 9, it goes to the 9s and to the 5s and then tracks to the intersection of the two points and the number 45 gets BIG. If the question is 45 divided by 9 it goes to number 9 and tracks down to 45 and then across to 5. The brain is always looking for the point of intersection. This way it’s not about taking five lots of nine and adding them all together very laboriously; it’s about geometry, because if you learn maths this way, you are already set up to do geometry.
Geometry
Shapes and form surround our world. Just look out of your window and notice the line of the rooftops, or walk down a city centre street and notice the angles of entrance lobbies. How much geometry can be seen in buildings such as The Shard or The Gherkin in London? The natural world is also full of beautiful geometric shapes - a snowflake, a nautilus shell, a sunflower. Geometry is in our DNA. Just close your eyes and imagine some of these fabulous shapes in our world.
The ability to see and rotate a visual image as if it’s a movie allows learners to extend their visualisation skills to geometry. Once you have the images of the geometric shapes, equations have a context and make sense.
Geometry obviously has to have pictures because it’s about shapes. So when you draw a triangle and add the angles in the triangle, you can start teaching the children to rotate, stretch and shorten the angles and watch the numbers change, so that it happens by itself. Again, years ago you couldn’t do this so easily because we didn’t have computers to create the images, whereas now teachers have built websites to do just this. This way, when you pick two angles, whatever the third one is automatically adjusts. Instead of having to add all the angles up and subtract from 180, the numbers can be there by themselves.
It’s not a question of whether you can name a trapezium (trapezoid), it’s a question of seeing the pictures of shapes and writing the angles in the corners and the lengths of the sides. As the lengths of the sides extend or shrink in the same proportion, it’s still a trapezium. As one line gets bigger, the other one does too. And if the line on one side gets bigger, the other one has to get even bigger too! You can watch the numbers go up and down so that the numbers shrink and expand together.
This process works for understanding shapes and changing shapes from one to another. Take a picture of a hexagon and write hexagon on it, adding the lengths of the lines and the size of the angles. Take a line out and have the picture change shape and as the shape changes to a pentagon so the name changes too. Then you put the angles in the pentagon.
Two builders were building a house extension and were asked if they would use Pythagoras to ensure the building was square to the original wall. ‘No’ , they said, ‘we are using a 3 - 4 - 5 triangle!’ They were using geometry the easy way. Learners often believe that Pythagoras’s theorem is a long complex equation that has to be computed, when the truth is that it’s a way of solving a puzzle. If we know the length of two sides of a right-angled triangle, we can find out how long the third side is. It is quite complicated to figure out the area of a triangle, but quite easy to find out the area of a square, so you put two triangles together, work out the area of the square and just chop it in half! When learners can see the process of the squares forming on the edges of the triangles and merging together, the brain knows what it is aiming for with the calculation. So the learner knows where they are going and why.
As soon as you think of a circle as a whole bunch of Πs (Pi) then it’s just a whole bunch of triangles. So all you need to do is work out one triangle and multiply it to figure out how many triangles are in the circle. Suddenly it’s easy. You can use the Πr² formula, but really all you do is figure out the area of one triangle and multiply the triangles. So if you look at the circle and know how many triangles are in it you know that the reason you are taking the radius and computing it is just because of the curve on the circumference. This way learners understand that a formula is just a shortcut, and as long as you follow the formula it gives you more information about the object.
Of course, plane geometry is about flat shapes. When you take the object and make it 3-dimensional it becomes a number of shapes added to the sides of the original shape. So a square becomes a cube by adding more squares to it. When you want to think about the surface of an object as opposed to the bulk or mass of the object, you want learners to look at the object and rotate it in their minds. Learning to observe objects in motion enables a learner to begin to make the connections between objects in space. If you are shown a still picture of a triangle and then shown another still picture of a triangle, it’s hard to make the connection to a pyramid. But when you are shown how to morph a triangle into a pyramid, it’s easy to realise that the surface of a true pyramid is just 4 of the original triangles. This way it’s all connected and every machine you build in your mind is a machine that works.
Engineering successful strategies
We can build the machine that visualises moving shapes in the minds of learners. To do so we need to make the steps very explicit, and teachers don’t always know how to do this. Sometimes we need to reverse engineer the problem to find out what the learner needs to be able to do, that they are not doing currently.
Paul was a really bright 11-year-old b
oy with good literacy and numeracy skills. He had a problem with non-verbal reasoning questions. His teacher’s strategy was to give him more past papers to practise, which actually meant he just got very good at not being able to do non-verbal reasoning questions.
The questions showed simple geometric objects on a page and asked the learner to identify which object was the same as another drawn from a different angle, or how many pyramids would fit into a box. Paul didn’t like drawing and actually said he wasn’t any good at it. Often, children who say they are no good at drawing are judging themselves by comparing what something looks like with what they have on the page. If it’s not accurate enough by their reckoning, they will decide they are no good at drawing!
Now think about it. What does he need to be able to do to work out these problems? Firstly, he needs to look at a 2-dimensional drawing of some lines on a page and turn the lines into 3-dimensional objects in his mind. Then he needs to do the same thing with each of the other drawings so they become solid objects.
Look at the drawing of the cubes above. Represented in this way it is simply 9 lines on a page, which isn’t a cube! Paul also said he had a horrible feeling in his tummy when he thought about tackling the problems. This wasn’t surprising, as every time he tried to do the questions he got them wrong and didn’t know what else to do.
So to make sure that the horrid feeling changed at the outset, all the activities we gave to Paul were designed to be fun. Some good suggestions were given about how the teacher may have just forgotten to teach some pieces and when Paul knew what they were, he could surprise her!
The first activity was to exercise his ‘imagination muscle’. He spent a fun evening painting his thoughts and feelings whilst listening to Vivaldi’s Four Seasons violin concerti and imagining the seasons. He said it was fun and not like homework.
The next task was to take digital photographs of cubes, pyramids and balls. Using lots of coloured pens, he drew lines around the edges of the shapes and traced them onto plain paper. This activity reversed the process so that he could see a 3-dimensional shape and conceptualise it as a 2-dimensional line drawing, making the link between how a cube looks in real life and what it looks like when we represent it on paper. You see, he hadn’t got that bit; previously, when he was asked which of the drawings on the page was the cube, his answer was ‘none of them, they are just squares with extra lines!’
Now the representations he had in his mind of the shapes looked more like this:
Next he did some drawings directly from shapes to see how many ways he could represent each shape. What did it look like from the bottom, the top, the side? And finally what does it look like when it’s behind my back? Once he could go from a 3-dimensional object to a 2-dimensional representation of the object, he could do this in reverse and go from a 2D representation to a 3D object. He could rotate the shapes in his mind and put them back on the paper as a 2D representation, compare it to the ones on the paper and tick the right box!
Algebra
Some mathematicians are said to be ‘intuitive mathematicians’ - they know the answer and work backwards to check the answer is correct. Others do it by working out step by step. This is inductive and deductive reasoning. But when we say intuit, this simply means that the number just jumps off the chart. Then they go back and do it the slow way to verify the answer.
This is good because you should always have two ways of doing everything to know you have arrived at the right place. That’s why you solve an equation one way and then go back the other way to make sure you are right.
So that X equals B+C and then you can solve Y, you should be able to solve back and forth in both directions. If you can multiply a number, you can divide it and go back the other way. In Mathematics, everything has those checks and balances within it.
When teaching algebra the checks and balances can be felt kinaesthetically . Watch a great mathematician working on a really long equation on a blackboard and one thing you will notice is that s/he stands back every now and again and looks at the board and feels whether it is balanced or not.
To begin teaching algebra, it’s good to present the process as a detective story where you are solving the mystery. You are trying to find out who the main character is. Most algebra teaching stops with the quantity, but the nature of the quality you are trying to discover isn’t introduced until higher maths. But to find out that Y is the main character because 2 Xs equal Y means that Y is the more significant character.
Maths, by definition, is a human endeavour. It’s not about the real world, it’s about how the human mind formulates. As is often the case when strategies are elicited from mathematicians, those we have studied didn’t know consciously how they were doing it, but we discovered that they did maths in the way we have explained it. The fact that they don’t see the chart in their mind anymore doesn’t mean it’s not there. They say, ‘the number just pops into my head’ , and the same thing is true with triangles in geometry and many other learnings.
The mathematician who gets to the answer and then goes back to do it the way s/he was taught in school to check it’s right is really doing reverse mathematics. It’s reverse engineering again; if you intuit the answer and don’t double check the answer that popped up, you might be getting the wrong square. So it’s useful to constantly check you have the right answer. Just as if you write a word in your head, you should look at the word on paper to make sure you have the right letters. It’s the same process; just with maths it’s a little more linear than it is with spelling.
When you go to school you still have to go home. If you are going somewhere and only looking at where you are going, you don’t know how to get back to where you were! A friend of ours was visiting Copenhagen and went out to dinner. He got in a taxi at the end of the evening, the driver asked where he was going and he said ‘back to the hotel’ . Which hotel? Oops, he’d forgotten to take notice of the name!
When you park your car at the airport you look at the car and notice where it is so you can find it again. You have to do this with numbers. So if 5 x 5 is 25 then 25 divided by 5 will be 5. The motivation to learn to do this is that down the road, life will be easier. So through the process of proving your work, and the process of learning to multiply, you learn to divide. This is a better way of doing maths. Jump to the answer and then work back. You track your way back until you get to the answer and if you can’t then you track your way forward until you get to the answer.
Have you noticed how the answers to the maths tests are in the teacher’s book? Because whoever wrote the teacher’s book knew that was the best way to do it. The trick is not to GET to the answer. We are taught that the whole rationale is to get the right answer but it’s really to get the right process , so people will always be able to add subtract and multiply.
Teachers can create the desire for the children to go BOOM! I got the answer , check if it’s the right answer, and if it’s not, go back to the process. Instead of having to wait a day to find out if you did it correctly, the student should know which process was wrong and which was right immediately. One of the problems with our current methodology is that the teacher has the certainty (the right answers) and the student has doubt until the teacher gives them certainty! When we reverse this process, the student has the certainty of when they are right or not and the teacher has the doubt, so they constantly check that the students are learning in the right way. Fortunately, with the advent of learning platforms more and more learners are able to know the answers straight away, correct what they are doing and move on, so the motivation is to get better and better and faster and faster.
As part of the New Wave Project(2) to teach adults maths in the workplace with the technology of NLP, we ensured that the answers were not just at the end of the book, but at the end of each section. Interestingly, even some of the learners questioned this, saying ‘well we will just cheat and look at the answers!’ Of course they didn’t do this. What they did was get more and more
enthusiastic about working the answers out and seeing immediately whether they were right. When they were correct, they felt great. If they were not, they went back to find out why and work it out again. Answers at the back of the book are great motivators for learning when it feels good to solve a puzzle and know you are right.
As we mentioned, most great mathematicians are not just working visually; they are working kinaesthetically too, in that they are trying to get to the good feeling. You get that good feeling by bending the numbers. The more you build in the strategy that makes a child hungry for the right answer rather than dreading the wrong answer, the more successful the child will be. Knowing that this one is right, this one is right, this one is not right yet so go back a few steps, installs the strategy that works rather than installing the dread of what doesn’t work. Whether it is maths or any other subject, the more you get it right, the better the good feeling. Education is about Design Human Engineering® for kids. We are teaching children to do things that are conceptual, not natural. The target is to have the mechanism or strategy of learning connected to the learning itself and connected to other learnings. So instead of looking for mistakes they are looking for how the dots fit together. So they become better than inductive or deductive thinkers; they become learning machines. In future generations, knowledge is going to change so fast that what you learn won’t be useful by the time you get out of school. What will be useful will be the rudimentary skills of learning. So what you learn may change, but how you do it will not.
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Teaching Excellence Page 11
