Teaching Math to Dyslexic Students - Dr Steve Chinn

Audio may be better at this link: http://bit.ly/157Oz9U
Dyslexic Advantage webinar with teaching strategies to help students with dyslexia learn math.

Dr Chinn uses animations, visuals, and simple patterns to address issues such as math facts, arithmetic, multiplication and division, and more complex topics such as algebra. If you have trouble with audio try this site: http://bit.ly/157Oz9U

This is the best short program we have ever seen for teaching math to students with dyslexia and dyscalculia. Thanks, Steve!

Join the movement at: http://dyslexicadvantage.com and like this video to spread the word. Thank you!

What an incredible delight today to have

Steve Chinn with us, he is one of the

worlds authorities on teaching Math to

Dyslexic students. He has a number of books

that are available to Amazon and other

places that I think many of you will find

very useful. Steve has his doctorate in

Applied Physics, but has had many years as

teaching headmaster of Mathematics and

Science, it's a real pleasure to welcome

him today, and we look forward to hearing

everything he has to tell us about teaching

Math.

Thank you for that, let me see if I can do

this...A quick overview: I am not, in 45

minutes be able to deal with everything,

but I want to give you an overview, give

you some practical examples and

illustrations of what I think are the

principals of teaching Math to dyslexic

students. It's no good trying to use the

traditional methods, whatever they are.

In my experience, there are no quick fixes,

as there aren't any for reading dyslexia.

In many cases, it's about starting the math

in the place where the problems become

apparent, and tracking back quite a long

way before you get to a place where they

all disappear. Or often in children that

miss some of the few concepts, I'll talk

about that today. Of course we have to

remember that all students are individuals.

I've looked at the factors that are

involved, I've build them into the talk

rather than just list them at the start,

things that might create barriers to

learning & I'll break at various points so

that you can ask questions. I'll do my best

to answer. If I think it's a long answer, I

might put you off until the end of the

session. So, let's get going, and see if I

can start at the beginning with some

guiding principles and quotes. Let me

start with a lovely saying of Margaret

Rawson, a pioneer in the studying of

Dyslexia, from the U.S.A., a lady who

lived to be 104.

"Teach the subject as it is to the child as

she (or he) is."

Just such wise words, in a sense,

summarizes what I am trying to do, today.

Something I once said:

"It's quite complicated making things simple."

Quite often, people don't realize what

they think is simple, can be complicated to

make it simple, to do.

So once that's said,

"It's usually best to start at the beginning

and to be aware of where this stuff

is heading." ..."Stuff" is a technical

term I'll use for mathematics.

I want to look at the

development trajectory in maths, which

sometimes means you start looking back,

and sometimes means you go forward.

You have do both of those things.

A little bit of background here, relavent

to our populations. The NRC, National

Research Council for the U.S.A., a book

that was published in 2000, called"How

People Learn", a wonderful book, even more

wonderful in that it has just 3 key findings:

kind of succinct! The first of those key

findings is this one: Students don't come

into classrooms with an empty head, they

have preconceptions about how the world

works. It happens to be the case, how

Mathematics works. You've got to deal with

that right at the start, or they won't

pick up the new concepts that you're trying

to explain to them, and the information

which you are giving! Or, which often

happens, they may learn them just for the

purposes of the test, maybe just read them,

but only for 24 hours, and then go back to

their original learnings and

preconceptions, that they brought to the

classrooms. This, in fact was known , in

a really good research paper in a

monograph from Chicago way back in 1925:

When we learn something that is new to us,

what we learn is a ~dominant entry~ to the

brain. Quite often when we are teaching

maths, we've got to do some unteaching.

It's one of the very good reasons for using

manipulatives and visual images, to

try and create new images for teaching. So,

that's the "so what?" of this. We're not

just starting from scratch, we often have

to work with things that are, maybe, not

right. Then maths has an emotive, an

emotional side. There aren't many subjects

where people write books about anxiety. You

get books on maths anxiety. I guess,

less books on geography anxiety, art

anxiety. A lot of the times this is down to

children giving an illusion of learning.

Children with good memories often start off

doing quite well in maths. But then as that

load on memory gets greater and greater,

they tend not to fair as well, because the

memory often stops them from understanding,

as the understanding then helps the memory.

This is one of the strengths for

dyslexics. When I lecture to teachers, as

Fernette said, I do this all around the

world, I often ask a vague question, that

is, "At what age do enough children give up

on maths in your classes for you to notice?

The most frequent, popular answer is 7

years old. That is a big worry, and I'm not

just talking about Dyslexics in this area,

an International problem for many children.

I think we have to look in maths at the

role of failure and getting things wrong

because maths is incredibly judgemental. My

experience in working with children, and

students whatever age, is that failure

rarely motivates. People maneuver to avoid

failure in a number of ways. One of the

good ways of avoiding failure is giving up.

not trying.

But, people will make mistakes, and we have

to look at that, and we have to deal with

that, within the lessons.

So what we are trying to do is to build

security. One of the things, in general,

when we learn when we are right--it helps

us feel secure, is the role of consistency.

When things are consistent, means we can

often see patterns--it helps us to be secure

and when we ar secure, we tend to learn

better. We often think

that maths is a pretty consistent subject,

but like spelling of the English language,

which has some rules, but huge numbers of

exceptions. Let's look at that right now:

That's my first break, so if you have any

questions...If you can't hear me, or

whatever, lets have a little time now

before I launch into the next little bit

of the talk, now. I'm going to get going.

So, how do we write numbers? Because,

ultimately, children have got to learn to

write numbers. My grandson, when he was

age 2 and 3, was able to say numbers. There

is a big difference between saying numbers

and writing them. The symbols we use to

represent numbers are going to be very

important. Particularly for children who

have difficulty with the information, as

presented...something I guess that dyslexics

have in common. I want just to introduce

you to the numbers and the symbols to 9.

These are symbols for single digit numbers,

just one digit to represent 9. I'm going

to start in this section via illustrating

one of the things that's very important for

me, in the philosophy/principles behind

the way I teach--that's the question I ask

myself: What else am I teaching? What else

are my student's learning? Lots of times

the actual maths might be stuff these

learners have done many years ago. What

I am looking for is the way I can teach

that in a way that makes it acceptable and

understandable to older learners, who might

have failed at learning this stuff many

times before. So what else am I teaching

what else are people learning,

when I'm dealing with any topic

in maths? I also want to try to make people

be more comfortable with numbers. One of

the things I'm going to show you in this

next section 'what else are you teaching',

is showing people how to 'see numbers' in

numbers. I'm going to explain that in a

moment. So here is a question, you don't

have to answer, it's rhetorical...but it is

a very key question, in terms of the number

system, we use. And that is, what is

nothing? What is zero?...Here we go!

There's a slide--nothing more. What's very

important in a number system is that we

actually have a symbol for nothing. There

it is..Zero. That's a key symbol, and I'll

explain why it is key, and why it is the

root of many misconceptions in Mathematics.

If I want to generate problems in arithmetic

calculations, I put zeroes in the numbers.

It really makes things more challenging. So

let's look at the next symbols and do a

little bit about what else we are teaching.

That's the symbol '1' for one thing, the symbol

'2' for two things, the symbol '3' for three

things. I'm using the red color for the

odd numbers, and blue for the even numbers.

I'm not going to deal with that right now,

but it would be a 'what else are you

teaching' topic. So, I've got to four. It

would depend on the learners--I might say,

"Now look at this four. It's two plus two.

It's two times two. That's two operations

I've talked about; addition and

multiplication. It's a square. If it's an

older learner, I might say that we represent

four as two, with a small two up here,

meaning 2 squared. Wouldn't do that with a

six year old. This is how you can take

basic information, support all principles,

early learning principles and make them

more age appropriate for children. Let's

get 5 up there. You'll notice, I've

used a pattern, and it's one that's probably

familiar because it's a pattern on dice,

playing cards, dominoes. I might also just

use this as a random pattern to just check

out other issues with recognizing quantity.

Right now my agenda is to show a couple of

specific points. The pattern I'm using,

here is the difference. This is not the

pattern you see on cards, dominoes, and

dice. This "What else you are teaching here"

is I'm showing that 6, the number within

numbers, is made up of 5 and 1. I was

also showing that 3 is made up of 2 and 1.

A lot of the time learners who don't have

good number sense, I'm going to build that

number sense by using key numbers.

One, two, and five. Then, I'm going to look

for the key numbers in other numbers. So

that was my 6: it's 5 and 1. My 7 is 5

and 2. Eight, is 5 and 3. If I move this

counter (I would use real counters...get

some tactile experience)...If I move this

one to ~here~, I would have 2 and 2 and 2

and 2. So that 2 plus 2 plus 2 plus 2, we

have 8. (Eight up there...) I would have 4

plus 4. I would have 2 cubed, if I was

working with older learners. Four times two

2 x 2 x 2 x 2. So again, it is using numbers

where hopefully the numbers are not a

barrier to illustrate some of the

mathematical operations, some of the

principals. Eight...now we want to get 9.

There's 9. I'm still in single digits. I

haven't got to 10 yet. What I've done, I've

introduced the learner to all 10 symbols we

use for making any number. I've done that by

by counting forwards and adding one on

every time. I'll move through this next bit

a little quicker. Quite often we focus on

counting forward, and therefore develop

additions skills, which tends to be the

default operation. For learners who are

struggling with maths.But shouldn't we also

introduce counting backwards because when

we count backwards, we learn (let's keep

that gentleman for a moment)...we learn

about taking away. This flexibility and

seeing numbers in numbers, in interrelated

addition and subtraction...that's a skill that

some people have almost intuitively. They

have an intuitive number sense. In some

research I did when I worked in the states

back in the 80's, we, my colleagues

Dwight Knox and John Bath,

called those people grasshoppers. People

who take numbers very literally-- they see

9 as 9, they see 4 as 4--they don't see it

as 1 less than 5...we call those inchworms.

I just popped this little slide in here,

this grasshopper is scratching his head,

just to give you a little image, we are trying to get

some intuitive thinking, some good number

sense, some grasshopper skills. So to come

back, we take away one every time. We are

introducing subtraction. The counting goal

was introducing addition, adding one

every time, we want to get better at

adding one. Subtraction we are introducing

as taking a whole number away every time,

so in this particular example the quantity

is getting smaller, and we are returning to

zero. Now with some examples of what else

we are teaching. What I have done, is count

to 9 and come back again.

In unsophisticated use of quantity, you can

use tallies. If you see learners using lots

and lots of tallies, that's worrying,

because it means they are not understanding

how to group numbers, they are only seeing

numbers in ones, and that is very limiting.

Their mathematics is not going to progress.

So you see something like this, just

a string of tallies, very rarely are they

grouped...just strings of tallies. The less

you'll remember, you won't remember how

many are there just by looking. Back to

Forty three just happened to be

how many I could squeeze in on this

particular side. Now I'm going to show you

32 tallies. What I've done is grouped in 5

and 5 and 5 and 5 and 5 and 5, I'm going to

group them again, in 10's. ~Now~ I can see

10-20-30 and 2. So number systems will

eventually group ones into quantities. More

often than not those quantities are tens,

then we have a two units left over,

and that's 32.

Why do we use 10 as this grouping quantity?

Remember, the answer is here: we have 10

fingers on two hands. Five and five. So

hands go '1', for one digit, '2', for two

hands, '5' for the five fingers, and the 10

-the core key facts on which we conclude

other numbers. To be a little mischievous,

let me just ask you a question, that I'm

not going to answer. Some of you do

research, and when you do, where did 12

come from? It's rather used for time!

Some of you do research! So this

idea of clustering numbers into groups to

make counting and manipulation of numbers

easier-- well, 3000 years ago the Egyptians

used this symbol [image] to represent

10 things. They used this symbol [image]

to represent 100 things. A couple of

thousand years ago, the Romans used an 'x'

to represent 10, and a 'C' to represent 100,

and we still see 'C' in century and cent.

We still see that today.

In the number system we use, the Hindu-

Arabic system, you're familiar with seeing

one-zero (10), and one-zero-zero (100).

Where these symbols are quite distinct, we

are only using 2 of those symbols are

introduced while we are on this session,

one followed by zero, and a one followed by

two zeroes. What seems to be common, is

that the tally is I, and the one unit is

[inaudible]

So, I've introduced the idea of using one

tally. We have also introduced the idea of

9 single digit numbers, symbols of 1,2,3,4,

5,6,7,8, 9. The key moment when we are

teaching mathematics, is this change from

single digit, 9- to double digits, 10. Here is

the 10, two 5's....here the digits go from

9 to 10. Beneath the key moment is when we

erase that process, and we go back

from 10 to 9. That's going to help children

do more complicated addition problems,

more complicated subtraction problems, and

thus, we lead to multiplication. That's a

key moment, and it's a very conceptual

moment, because we are using one symbol,

then we are using two. That's kind of

interesting. What makes the Hindu-Arabic

system clever is that the digits, the

symbols hold places. Here is 9 units, we

write 9. When we get to 10, we can't write

one, because we've used one in the units.

We've got to come up with something that

distinguishes one unit from one ten. We use

1-0, we use 2 symbols. This means we have

one 10, it means we have zero units. The

implication there is that we have to get there

using the right place. If we put a zero

here, and a one here, that would mean

zero 10's and one unit. It's the order, the

sequencing. Children who have problems

with sequencing, this can be an issue.

Maths isn't always friendly if the learners

have issues like sequencing issues. Now

you can guess that eleven is going to be

one 10, one unit. The symbol allows us to

bring the numbers. Unfortunately, learning

numbers in English is not a good thing.

There are some real vocabulary problems.

If I was learning these early numbers in

Cantonese, I wouldn't be saying eleven, I'd

be saying one ten, one unit. Twelve--

one ten, two units. We don't, we say

eleven and twelve. The language, vocabulary

does not support symbols, does not

support the concept. Children learn these

things because they practice them, their

parents practice them with children, and

they give this illusion of one, but they

are not learning this key issue of place

value. This one, is different than the place

value of this one. In terms of what it

conveys, mathematically...that's bad

enough, but then we get to the teen numbers

and the vocabulary is telling us 3, not

very well, and 10, not in the language,

but it is suggesting three

and ten, but children have to write 1 and 3.

Four and ten, children have to have to write

1 and 4. When children come to practice

two digit numbers for the first time, the

vocabulary does not support the symbols

that they want. A lot of children will have

difficulty grasping the concept because

language, our medium of communication,

is not supporting the concept. This needs to

be modeled with materials and visual

images, which are not.[inaudible]

Let me get to twenty--well, twenty is not

too bad it, it suggests 2 in the tens, but

then we do say, twenty-one, twenty-two,

twenty-three. At least the vocabulary is

supporting the concept. I sometimes think

maybe we should never teach children these

until we teach them 21, 22, 23.

That's a little different.

So, if you are scared of big numbers, look

away now. I want to show you how this

system .........[inaudible] just clever,

and how, for older learners, the same

pattern. It goes on and on...so if you

didn't get it in the first learning, you

are not going to get the

extension of the maths.

So, the unit is the base. Ten is 10 units,

the next place value. One hundred is 100

units. Now, we've moved up to third place

value...we have a 3 digit number, a 2

digit number, and a 1. As we go up in value,

increasing the number of digits. Again,

I've modeled this in base 10 which is more

likely, these blocks very good material.

Then we get to 1000, and 1000 is kind of

like a watershed. Up to now, we've done 10

units, 100 units, 1000 units. Now, for the

~bigger~ numbers, I'm going to work in

thousands. So I am going to do 10 thousand,

a 100 thousand, and a thousand

thousand. So I am using this ten-hundred-

thousand sequence--I've got to 1000 again,

we have to have a new word, that's a

million. Now I will carry on using

10 million, 100 million, 1000 million,

which, as you know, is a billion. You can

work out what a trillion will be. There is

a logic in this form. Just to show you how,

working with all the learners, I want to

use very basic maths, illustrating complex

symbols and complex representations. What

maths tries to do is represent complicated

ideas presciently, and sometimes called

very elegant. A hundred is 10 times 10,

ten lots of ten. So it is 2 tens multiplied

together. Mathmeticians symbolize this with

this 10, with this little 2 tucked up here in the

top right hand corner, 10 squared. This is

very difficult for children with dyspraxia,

with untidy writing, sequencing issues,

it's making all those demands. This too,

has to be small, otherwise it will be 102.

It's critical. These are points of danger

for learners, and points that teachers have

to pre-empt by demonstrating them and

making them very clear. A thousand is

10 x 10 x10, three times only, three tens

all together, so we write it with a 3 up here.

There's a logic, there's a pattern. Don't

just teach by rote. Ten Thousand is 10 to

the 4th, skip a hundred thousand, go on to

a million, which is

10 x 10 x 10 x 10x 10 x 10. Instead of

writing all that, mathematicians write 10

to the 6th, and the 6 tells you, there are

six 10's multiplied together. Those of you

that are looking for patterns, what do you

think a mathematician might write,

that doesn't [inaudible] might write for 10?

Think of the sequence. Four, three, two,

the answer is, they write 10 to the

power of 1. If you really want to be

confused, but there is a logic, for a 1, we

write 10 to the power of 0. That's the

older learners, the ones doing the [inaudible].

Here's another natural break time, I'm

ready for some questions if you have them,

but equally, I'm ready to go on if you don't.

Okay, I'm going on...

I want to just look for a moment at number

facts, what are sometimes called

[inaudible] facts. A great idea that's

that's come from some key researchers in

the U.S.A., is not to call these facts, but

to call them number combinations. When I

started working with Dyslexics, thirty or

more years ago, I discovered them because

my learners taught me. What they didn't

remember, they often worked out.

If they knew 5 + 5 was 10; 5 + 6 was 11.

That's what this number combination means.

Children and adults use key parts, to work

out all the facts. Back to my 1,2,5 and 10.

This is a great skill because for some

facts, there's no reason, unless you

understand numbers very well why the

answer repeats...what it is.

Let me explain a little bit about numbers.

The research on maths learning, Mathematics

learning, what we call the numbers calculating,

this is stuff the kid's lack confidence in.

It seems to me, from the research, you can

do endless practice on this, in some ways,

and the entry to the brain, is pretty short

lived. The thing I hear time and time again,

from parents is:

"He learned it last night, but he's

forgotten it this morning. " That is very

common. If you have problems with

remembering facts, and I'm going to do

Algebra problems, coming to that shortly,

don't turn off, all those of you going to

do Algebra, [inaudible].

What you really want to learn are the number

combinations that add up to ten. The reason

they are useless to do lots with

them, I'll try to be straight in a little bit,

And the doubles, the numbers like 5 + 5,

and 6 + 6, and facts, ......a bit of algebra

in here. This is a number, any number, this

is the same number. This is a way of

generalizing, this could be 5 + 5, this could be

7 + 7, so in a sense that's a very good

example of algebra we use to generalize,

to show patterns, things are the same.

This is some manipulatives,

again, to try and help people see what

numbers, what combination of numbers,

make 10. I'm going to start with 10 on this

side, and none on this side, so zero--this

is a combination that makes 10. If I move

one over, I've still got 10, I conserved

the 10, that's an important concept,

this side now, I have only 9, I

have 1 on this side. If I move 2 over, I

now have 8 and 2; 8 on the left side, and 2

on this side. I can keep moving one over,

7 and 3, 6 and 4, until I reach double key

number. If the number combination for 10

is key, this particular 5 + 5, is super key,

mega-key, whatever you want to call it.

It's the 5 fingers on one hand, 5 fingers

on the other.

It's a key number. If I move over again,

moving one over, I've now got 4 and 6.

Instead of 6 and 4, which we had,

we've now got 4 and 6: still 10. An

important rule for you here...if two

numbers, 4 and 6 add up to make 10, then

those numbers in the other rule, 6 and 4,

will also make 10. Sometimes people want to

know why 7 and 6 makes 13, but they also

do know, because I've pointed it out to them,

that 6 and 7 makes 13. This of course,

goes on with 3 and 7, and 2 and 8, until

you end up with 0 and 10. So those are the

number bonds group. Why are they grouped?

This might be something I would do with an

older learner, not a 5,6,7 year old. I want

to show you again how there are links,

patterns. If you know these facts, you'll

know your combinations for 100. For an older

child whose doing decimals, they'll know--

they'll be able to work out their number

combinations for decimals. So, 1 and 9

make 10, 10 and 90 make 100. Practicing

place value and your two digit plus two

digit makes three digit in this particular

example. Here I'm practicing decimals, but

look: the 1, the 2, the 3, the 4...the 1,

the 2, the 3, the 4...the 1, the 2, the 3,

the 4, combined with the 9,8,7,6...

the 9,8,7,6...the 9,8,7,6, are there in

in all these examples, just are different

place values. Again, there are lovely key

points and I might go here, and here

and for reason, and

5 and 5 makes 10, and 50 and 50 makes a

hundred. This isn't hard at all. So again, you

as the teacher or parent look at the child,

You look at that, and it's working [inaudible]

and that's kind of important. Here I've

used symbols, again, I've used

manipulatives. [inaudible] Okay...

Algebra...let's just play around a litle

bit here before we get to Algebra, let's

be patient...in moments...

In this particular slide I'm starting off by

showing you there is a relationship

between addition and subtraction. Those

inchworm learners, those original learners

that I've talked about, don't see that

connection unless it is pointed out. I do

want to point it out, so here's a little

problem: 100 minus 0 is what? What would

I put in that little space there, what

would be my answer? It's 100. Let me

now make that an addition problem. What

would I add on to 0, what would I put in

this little box here to make 100? Again, I

hope you are seeing, that's 100. So I am

relating subtraction and addition, using

addition facts to make subtraction facts.

Let's do-what would I put in the box, for

this subtraction? I would put 80, same as

I would put here in this box. The question

is, what happens when I subtract 20

from 100? I take 20 away from 100, again

the calvery comes in, I get 80. What do I

add on to 20 to get 100? I add 80.

Fifty, the same deal. Seventy, same deal.

Relating addition to subtraction.

You need to understand that

is a law, because if you don't you're not

going to get the Algebra. I've subtly

introduced Algebra, because I've asked you,

what number goes in here, to make this

side equal that side. Let's put that square

away for a moment, and let's replace it,

with an 'x'. ~All~ I've done is taken the

square away, and replaced it, with an 'x'.

Now my question is, what number would I

write instead of 'x' to make this equation

the same on both sides. So what number

do I add to 20, to make 80 ?

In order to get 'x', you have to tell me,

what number 'x' represents. In maths we

often use the instruction, find 'x'. I've

seen the classic cartoon where the child

looked all around and said,

says,"Here it is!". That's not what we meant.

Let's do it again, with the next set. Just

get rid of the box, let's put in the

letter, this time use 'y'. What is the

value of 'y' that makes this side 100,

make it equal to that side.

If you don't understand the numbers,

you don't understand the Algebra. If I was

teaching Algebra, I might want to go back

to the early numbers, that is important.

Sorry, Coleen, fractions, we're not going

to have time for that tonite, but I will

mention them, not very constructively.

With the four operations we use: addition,

subtraction, multiplication, division...

they are all linked. There is a hugely

confusing language around addition and

subtraction. Sometimes with subtraction,

we say 'take away', sometimes 'what's the

difference', sometimes we say 'subtract'.

Just as a quick illustration here, I want

to use these number combinations of 10

again. What is the difference between a

thousand, and 438? I've drawn a number

line, so here's my 1000, here's my 438--

this is the difference. What is this bit

here? That's what I want to know.

I'm going to do that, by adding. I'm

going to add, first of all, 2 to the 8, to

make it 40, that little tiny bit there,

headed towards a thousand. What do I add

on to 440, to make the hundred? Sixty.

These are my number bonds for 10.

I've got to 500. What do I add on to 500

to make 1000? Five hundred. Now I've gone

from there to there, I've found this

difference by addition. That's the way,

since we can't do money much today, that's

the way when I was a boy, we used to make

change. We could make it by adding on from

the purchase price to the money that the

customer gave us. So, that's the answer.

I've done it in using number bonds of 10,

a 100, and a 1000, and I"ve done it in

stages to help with that. If you want to do

the take away, 1000 minus 438, is so

difficult, when you do it as a take away.

Can I suggest you get on google or whatever

search engine you use, and search for

'Tom Lehrer and New Math'. Tom Lehrer was

a math professor at Harvard and a great

song writer, a tremendously, often black,

sense of humor, and see how he explains

how you do this type of calculation. It's

wonderful, much more entertaining than I

can be. This "what else are you teaching?"

this principal, and then I'll move on to a

new topic in a moment..."9". Grasshoppers,

these intuitive thinkers, love 9, because

it's close to 10, they don't see 9,

they see "9" as unique. Inchworms see 9

as lots of fingers, they DON'T see it next

to 10, they see it very, very literally.

They add 9 by adding 1,2,3,...til they get

to 9. And they add 99 in the same way,

they see it singly.

Shops, yet again, a bit about money. Shops

use 99 cents or 99 pence at the end.

Something is priced at 14 dollars and 99

cents [$14.99] because they know

psychologically a lot of people see the 14

dollars, they don't see the ".99", they

don't see it as close to 15 dollars. I'd

want to teach that you have 10, you round

up as a way of avoiding the psychological

price. Inchworms, again, you might convince

them that adding 10 can happen, but then

they have trouble, knowing they have to

subtract. Why? Because it's addition, and

they don't interrelate operations and

numbers. These are skills we are teaching

to make people more comfortable

with number in everyday life, often,

with money.

Estimation is what grasshoppers do

naturally. They are good at that, they have

great number sense. They may not be great

at other things, but that's one of their

great skills. Inchworms, not natural

estimators. With this 9, I am teaching

estimation, inter-relating numbers and

operation, rounding up to the nearest 10

and so on. I am also teaching people to

appraise their answer. After it all, I want

them to know, is your estimate bigger or

smaller than the real answer. Estimation

is taking away that pressure of decision,

and introducing a new idea upon them.

I need to get moving, I've been talking

for a long time. Any questions again at

this break?...[pause] I'll move on to

conceptualizing. Money is quite good

for introducing conceptualization, it

combines two things there. In terms of

place value, the money in the U.S.A, you

have this coin called a quarter, which

is not as good for maths, for place values,

kind of good for fractions, not as good

for regular mathematics. In our currency,

and in lots of other countries we have a

1 cent, 2 cent, 5 cent, 10 cent, 20 cent,

50 cent...there is a good pattern for

breaking down numbers. But you can use

trading with money, so this crossing from

units to tens. If you get 10 single cents, you

can trade them for a dime. You can trade

10 dimes for a dollar. In the U.K., it

would be 10 pennies for a ten pence coin,

and 10 ten pence coins for a pound. You

can use coins to relate when transforming

and conceptualizing.

Let me motor on because I am talking too

much which I always do. Again, BIG research

Children/Adults don't know their

multiplication facts, their times table

facts. This, again, when I ask teachers,

BIG problem, 50 to 70% of children age 10

in the U.K. don't master this task. Again,

what you need to learn, what you've got to

learn are 0's, 1's, 2's, 5's and 10's. So

what I'll do, is to try to show you how

these can be done...got to keep working.

Here we go...what else are you teaching?

Rote learning is not good for a lot of

dyslexic students. I nearly destroyed my

relationship with my first bunch of

dyslexic students by trying to enforce rote

learning, the times table facts upon them.

This can be an advantage because if you

don't rote learn them, you have to learn

them in other ways, so I go back to the

motto, "Teach the subject as it is, to the

child as he is." If rote learning doesn't

work, let's go for something deeper. Let's

make a demotivating experience a positive

experience. We need to know what

multiplication is: let's just remind

ourselves that addition is about adding

together numbers, multiplication is about

adding "lots of" the same number. Now,

lots of sometimes means ~many~ in everyday

life. Lots of in maths can be 2 lots of,

or 3 lots of, it doesn't have to be mega

numbers. Here is 3 "lots of " 4, added

together. I like that it is 3 'x' 4,

three times four [inaudible]

Again, symbols are communitive property[??]

I wouldn't expect a child to add

all of these together, that would be very

time consuming, as someone mentioned,

working memory, it certainly swamped

the working memory. This is a tough fact,

it's one of the facts that is very hard to

learn. I don't actually know why, but it

is incredibly hard for children. Let me

give you a quick lesson on how to help

with this, and teach something else

at the same time. So here is me,

representing 4 and 4 and 4 visually. I

could get an early estimate by popping this

square in here, that square in here...that

would give me two fives, and give me 12.

That's not what I want to do right now.

What I want to do right now is take this

image of 4, and change it into a new image,

all in a line. Two reasons: Here is

3 x 4 I'm doing visually. I could put them

all together and have 3 x 4, or I can

seperate them, and have 2 lots of 4, and

have 1 lot of 4. Two easy facts, rather

than one hard fact. Similarly, for 7 x 6, I

can line up 7 separate 6's, or I can make

it an area, I've got 7 lots of 6, or I could

break it up into 2 parts, five 6's and

two 6's, trying to introduce this idea of

a multiple that you might know, and a

multiple that you might know, and putting

them together. That's part of the deal.

In symbols, I'm showing you here how to

use a partial products to work out the

facts they don't know. Sometimes, you

might want to make a few notes to support

the working memory, but what we are

teaching here is the basics of long

multiplication. That's good, we aren't just

wasting time, restraining [???] the child

with rote learning that doesn't work.

Once you know this principal, looking for

easy numbers, 2 and 1 are 3, 5 and 2 are 7.

Twelve breaks down into 10 and 2. I used

to teach the 12x's tables, but we don't

use them very often anymore. But I know my

10's and I know my 2 facts, and I combine

them. Fifteen---I don't teach that, but

I know my 10, and I know 5's, I know

that 5 is half of 10, so I can use this

strategy to go by. When we go up to the

cheat-sheet, yeah, I think sometimes "just

give them the cheat-sheet" so we can

concentrate on the concept and not lose

out all the time if I'm not getting these

facts. I kind of like the idea of spending

some time on teaching this method because

it is a conceptual lesson, and it's not

just teaching how to access the fact--it's

teaching the principals of multiplication,

and indeed, division. What I'm saying here

is adding each separate number is a many

step procedure. This, the cheat-sheet

method or rote learning is the one step

method. I'm kind of offering an alternative

in between, the two-step method. You can't

do this in one, can you do it in two?

It's not as much as the many, it's the two.

I try to make the two easy, and these are

automatic, low load on working memory.

I want to get people off this total

reliance on rote learning, because it is

so hard. That's why an alternative to the

cheat-sheets in some situations--These are

the core facts, you combine them to make

other facts. So I combine 'this' and 'this'

one 8 and five 8 are six 8. Conception,

one can understand that combination. Five

8's and two 8's make seven 8's. Two 8's and

ten 8's to make twelve 8's. What I'm

trying to do is give strategies that are

consistent. Combining facts that you know,

using facts that you know to extend the

concepts, to extend the facts you do.

A little bit of Algebra coming up here

slowly. So I'm doing six lots, six lots

of 6, transferring that into five lots and

one lot. Six lots of 7, six lots of 8, six

lots of 9. Same split every time into five

lots of 6, one lot of 6;

five lots of 7 and one lot of 7,

five lots of 8, one of 8; five lots of 9,

one of 9. This is not a quick fix, but

you are wasting less learning time, because

you are teaching concepts. This is

conceptual, this is about teaching

mathematics, what is facts.

Let's put this now into symbols, and what

I've now done, is left those visual images

behind, and focused on the symbols.

6 lots of 6, 5 lots and 1; 6 lots of 7,

5 lots and 1; 6 lots of 8, 5 lots and 1;

6 lots of 9, 5 lots and 1. It's going to

work for any number: 6 '21's", 5 "21's"

and 1 "21's". Oh, there's a hand in there,

and I don't know why...This is meant to be

5 lots of any number you want to write...

6 lots of any number you want to write,

5 lots of any number and 1 lot of any

number. Instead of this hand, I can use

an 'n'. I can use an 'n', I can use a hand.

6 lots of any number: 5 lots of any

number plus 1 of any number.

And, as you know, in Algebra, we don't use

times. This question, Christine, about

dyslexic student's that taught the methods

to themselves. Yep, that's where I learnt

it, I learnt it from my dyslexic students.

Just kind of streamlined it a bit, little

bit of order [??] A generation of dyslexic

students taught me all this stuff.

Okay, what is it about division, the tail

end of my session. I'm only doing this as

an illustration, again of the complexity

of maths, and how we can help, to some

extent, so this optimal [inaudible]

can do. I'll just give you a little bit of

data from the U.K., this is some data from

the standardized tests. In the U.K., at ten

years old, about 3/4 of the students

across the U.K. completed this. This

division, less than 1/4 of the ten year

olds can do. At thirteen, it's still there

about 3/4 to just over 40%. At 15, it's

gone up to 83%, still only 44% . Division

is tough, learning it is complicated.

Nearly finished, for those of you that are

listening. Division and fractions,

again, when I ask teachers around the

world what are the subjects that kids

find hard, these are the top two.

Fractions, of course, incorporate division.

It's a culture, the culture is that these

two topics are hard, so it helps people

pre-empt their feelings of failure,

because we know it's tough, and that's

the beginning. Why is it hard? Because

our language is. We say "sixteen

divided by 3". We also say, "How many

threes in sixteen?" That reversal of the

numbers involved will confuse some children

before they even begin. Again, I want this

to be conceptual. I want it to be

understandable. Link things together.

Addition is about adding together numbers,

subtraction is about taking away. They are

inverse, they are opposites.

Multiplication is about adding 'lots of'

the ~same~ number. We can guess, that

division is about subtracting/taking away

lots of the same number. About 5 slides left.

Sixteen divided by three-- how many lots

of 3 in 16? So, I'm using multiplication

language to relate the concept.

So let's just take away one 3...we've got 13

Another 3, we've got 10. Another 3, we've

got 7, another 3, we've got 4...so what

I've done is taken away 5 lots of 3, and

I've got 1 left over, 1 remainder. Let me

just rearrange that for you, and I'm going

to do the same again, "How many 3's in 16?"

Take away 3, and another 3, and another 3,

the fourth 3, and the fifth 3. I've taken

away five 3's, and I've got 1 left over.

That is the way we present short division

in the U.K. How many 3's in 16? Five, and

1 left over. This also relates to a

rectangles, again, what else am I teaching.

I am trying to

teach, this is a rectangle. You see

it's 3 on one side of the rectangle, and

you see it's 5, linking multiplication

and division. Again, I am

doing this very quickly, but I'm trying to

illustrate these with the visuals, and the

icon. Einstein said, among

many other things, "Everything should be

made as simple as possible, but not

simpler." I'm just delighted and very

honored [inaudible] that you are here, thank

you. Let's just quickly look at what children

are often taught, and why this is so hard.

"How many 23's in 16?" There aren't any.

"How many 23's in 168? " How the heck do I

know? It's so difficult, so that's the

barrier; already, some kids are gone.

What we do is, let's say, we work out a

answer of 7. We write the 7 here. How do I

know that? I don't know, so much the child

has to take in faith. We take away that, we

get 7, we bring down the 3, whatever that

means. How many 23's in 73? What is it?

Take away. We bring down the 6. SO MUCH

to remember in that sequence. Make 1

mistake, and your out! There's another way,

where I'll use a different example,

which relates to multiplication, relates to

that times tables. So again, only doing the

same thing again and again and again. What

you might do is link the two. Instead of

having to find out how many 17's in

something, what we are going to do here is

use repeated subtraction. Let's find

something to take away. Our initial

jottings might be, well, whats a hundred

times 17? That would be 1700, that's too

big, so we do some reappraising. I go to my

next easy multiple, and halve that, 50

lots, 850. At least I'm started, I've got the

layer started. I've taken away the 50

times 17, and got 357 left. Then we take

away twenty 17's, using that multiplication

again...and then what? You've still got to

subtract, you've still got to organize your

work in space, but I've made it more

conceptually manageable. And I'm using

key multiples in the subtraction, something

that's I've learned from [inaudible],

many years ago. Total numbers are there.

What this method does, and what it doesn't

do is, it's logical, it's conceptual,and

all the stuff we need. Nothing starts

with [inaudible], nothing starts with

[inaudible]. It's the

inverse of the method that I teach for

multiplication. There's less reliance on

memory for procedures, because I'm using

the procedure, that I use for

multiplication, just the inversion,

that's all. It uses core key fact multiples

so you don't need a cheat sheet, which you

don't have, by the way, for 17. It reduces

the chance for errors , and that feeling of

helplessness--"Can't do it! Not even going

to start." You have to be able to subtract,

do your spatial organization, and you do

need a knowledge of place value.

The devil is in the detail. I haven't done

anything that's not mathematical. What

I have done is focused on the details. So

often the details will cause the children

to fail.[inaudible] It's the details in

the language you use, the images you use,

the sanctions we often use. It's about

knowing the math from a learners

perspective, and that's why I said, my

dyslexic ones taught me this. It's about

preventing failure, getting to those

emotional ideas before they become a

block to learning. And....it's complicated!

Thank you all for all your great comments,

I really have enjoyed talking to you in a

virtual way, and I hope the feeling is mutual.

Thanks, bye-bye. Any questions, does

anyone want to keep going? [Dr. Fernette

Eide] Absolutely awesome, thank you so

much! ...Fantastic!...Questions?... I know

you didn't get a chance to use fractions.

Anything, generally? Do you use

manipulatives? Do you do the pie thing?

[Dr. Chinn] Pies are pretty good...pizza's

are cruel because you are just going to put

children on pizza's! What I do use is

square paper, squares of paper, and I

get kids to fold them.

The reason I use squares, I can fold in two

directions. What I also do,

I premark the paper so the kids

aren't messing around trying to fold into

3 pieces or 5 pieces. I make it easier for

them to follow it. With the folding, I can

show that if you fold it 5 times you get

5 parts. If you fold it 3 times, you get

3 parts. So the third is bigger than the

fifth. You look to what are big conceptual

barriers. A big conceptual barrier is that

1 over 3 is bigger than 1 over 5. Language

again is a problem: a half, a third, a

quarter, don't tell you...a fifth, a sixth,

a seventh at least tell you something. It's

challenging, all the stuff that kids have

learned about numbers before you. When you

see 25, you know that the 2 is 2 tens, the

5 is 5 units. When you see 2 over 5, the

2 is 2 units, the 5 is 5 units, but that

sneaky line in the middle is telling you

the 2 is divided by 5. Twenty-five, you

know that the 2 is multiplied by 10.

There's hidden information in fractions,

it's challenging fact consistency. Now

that I mention it, it's a big topic to deal

with, but you've got to do it with visuals

aides and manipulation. Otherwise the

symbols are just gone, not getting constancy.

[discussion about tech details...]

[Fernette Eide] Do you have lesson plans

for teachers? [Dr. Chinn] Ha, it sounds

like my brother asked that question! I'm

working on that right now. In fact, I've

done a great deal of work on that, and I

hope to find a publisher. It's a huge

amount of work, hardly a mixture of

power-points like some of what I've done

today, which could be used as power-points

or guidance, so far as manipulatives, some

topic sheets, talking to teachers about

progress for learning, other resources,

like the squares I

mentioned. There's about 100 worksheets

that are criterion referenced,

and all have been posted with solutions.

so that teachers can use those worksheets

with a guide-master for them. The

worksheets are also set up to highlight

some error patterns, not trying to make

children go wrong, but to allow some other

patterns to happen. We can then deal with

those before they become embedded in

the kids, or the adults mind. I find that

working with adults,like some I mentioned-

just go back to that early stuff and

present it in a way that's acceptable to

adults. These are often the adults who, as

children, gave up at 7 and spent the last

10 years hating maths. It's very sad.

Many levels in mastery, sometimes you can

bypass lower levels. A lot of my students

often found Algebra easier than Arithmetic

not because school had gotten any easier,

but because their number facts are so poor.

There are less number fact demands in

Algebra--it's conceptual, and often my

students are very good conceptually, they

are just prone to error, maths is not

forgiving in that. The question about

mainstreaming, my experience has been what

works for dyslexics, works for anybody.

There is a group just north of Boston who

[inaudible] technology that right now is

talking about [inaudiable] people

like dyslexics, and computics. What we

are learning, work with these students,

these adults, can be mainstreamed. What

we are doing, I've said from the start, is

we are teaching the subject as it is, to

the child as it is, not messing with the

mathematics, implied to keep the

mathematics the same, but I'm trying to

make it more conceptual rather than rote.

[Fernette] We'll close now...Thanks, Steve,

this was incredibly helpful and awesome, I

wish I'd been taught math this way. I think

a lot of people are thinking about this

now. I really appreciate all the time

you've given us. You really have a genius

for all this kind of instruction...[We]

really appreciate it, thank you so much!

[clapping sounds]

[Dr. Chinn] Thank you, everyone, for being

so kind and for such good questions.

[ Dr.Fernette Eide] You are a rock star!

[laughter]