MATHEMATICS IN THE MIND Derek J. SMITH, CEng, CITP University of Wales Institute, Cardiff [email protected] [email protected].

MATHEMATICS IN THE MIND

Derek J. SMITH, CEng, CITPUniversity of Wales Institute, Cardiff

[email protected] http://www.smithsrisca.co.uk

Specially written to support

Monday 30th March 2009

PART 1

LOOKING AT OUR BRAINS

This is what our brains would look like if we could take off the top of our heads and look inside.

This is what our brains look like from the left side.

"The Biggest Brain in Wales" allows us to "go inside the mind" to see what's really going on.

"The Biggest Brain in Wales" was previously exhibited 14th March 2006 as part of National Science Week 2006 (Smith and Livesey, 2006).

• Our brains do all our thinking for us. But they are very complicated because different parts do different things. One part does SEEING, another part does HEARING, another part does COUNTING, and so on .....

• So the first thing we need is a map of some of the different things the brain does, and where it does them .....

• DISTRIBUTE

SEEING

BALANCING

KNOWIN

G

ALL

ABOUT

THE

WORLD

HEARING

MOVING OUR EYES

DECIDING WHAT TO

SAY

REMEMBERING

WHERE W

E'VE

BEEN AND W

HAT

WE'V

E DONE

DECIDING WHAT TO DO NEXT

REMEMBERING NUMBERS ETC.

FOR SHORT PERIODS

FEELING PEOPLE TOUCH U

S

MOVING O

UR BODIES

BEING NICE TO PEOPLE

DOING SUMS

OK, now we've got our map let's take a trip around the Big Brain.

Let's start with the bit which does HEARING .....

Here it is. Now we need some helpers .....

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

What about SEEING?

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

What about UNDERSTANDING?

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

What about DECIDING WHAT TO DO NEXT?

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

What about SPEAKING?

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

What about MOVING YOUR ARM?

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

What about BALANCING ON ONE LEG?

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

What about MOVING YOUR EYES?

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

So let's see if we've got the hang of this.

TEST THE VOLUNTEERS ONE BY ONE

Say cheese .....

OK, let's try something more difficult. Look at these objects for a moment, and then we'll try to remember what's where .....

HOW MANY THINGS WERE THERE?

WHAT WERE THEY, AND WHERE ?

What about REMEMBERING THINGS FOR SHORT PERIODS?

This needs someone to come and stand here.

But only if you don't mind having your photograph taken for publicity purposes .....

Let's try something even harder.

I'm going to say "move your arm".

Which of our volunteers do we think is going to have to do all the work?

Let's see if we were right.

• The problem is that the words we HEAR .....

• ..... have to move to the part which does the UNDERSTANDING .....

• ..... then to the part which DECIDES WHAT TO DO NEXT .....

• and finally across to the part which actually MOVES YOUR ARM.

So that's how clever our brains are.

Sadly, nobody really knows what keeps all the parts working properly together.

However, we can learn a lot by watching what goes wrong when parts of our brain

stop working .....

• Brain damage might stop us HEARING .....

• ..... or UNDERSTANDING .....

• ..... or DECIDING WHAT TO DO NEXT .....

• ..... or MOVING YOUR ARM .....

• ..... or combinations of these impairments.

"Move your arm please"

The science of learning about MIND by studying BRAIN is known as "cognitive

neuropsychology".

The doctors who help mend injured brains are called "neurologists"

PART 2

THE COGNITIVE NEUROPSYCHOLOGY

OF NUMERACY

BRAIN DAMAGE AND NUMERACYWHAT IS A NUMBER?

• Shout out the answer to this question as fast as you can .....

• How many dwarfs were there in this movie?

BRAIN DAMAGE AND NUMERACYWHAT IS A NUMBER?

• But what does it mean to think "1", or "2", or "7", or "10,000"? Do we think of that number of counters on a table top, or what?

• Many of us rely on an infinitely long mental "number line" of some sort .....

MINUS INFINITY ZERO PLUS INFINITY

IMAGINARY REAL IMAGINARY

etc -3 -2 -1 0 1 2 3 etc

BRAIN DAMAGE AND NUMERACYWHAT IS A NUMBER?

• So did you have to count the dwarfs? Or did you just know how many there were when you looked at the picture? [SHOW OF HANDS]

BRAIN DAMAGE AND NUMERACYWHAT IS A NUMBER?

• The point where we stop "just knowing" and have to count things instead is called the "span of apprehension", and can be accurately measured .....

BRAIN DAMAGE AND NUMERACYWHAT IS A NUMBER?

• What about fractions. If you had to remove 12/28ths of the dwarfs, how many would that be?

Sleepy, Dopey, and Bashful, YOU'RE FIRED!!

Not you, Grumpy, there's something about you I like.

BRAIN DAMAGE AND NUMERACYWHAT IS A NUMBER?

• And what about decimals? What does 0.3 really mean?

• 3/10ths, perhaps. And what does that really mean? Do we still think back to the activities by which we were first taught fractions at school?

BRAIN DAMAGE AND NUMERACYACQUIRED DYSCALCULIA - EARLY CASES

• Walther Poppelreuter was a WW1 military surgeon who published his observations of head injured soldiers. His book contains 52 detailed neurological case reports, each one including the results of a calculation test. There were repeated instances of calculation impairments with damage to the rear left brain [that's the area marked DOING SUMS on your diagram]. This type of disorder is called dyscalculia (literally disordered calculation).

BRAIN DAMAGE AND NUMERACYTESTS FOR FINGER AGNOSIA

• Neuropsychologists like fancy words. The word gnosis means “knowing”; hence an “agnosia” is a state of not knowing what something is.

• Gerstmann (1924, 1930) noted that one subtype of agnosia - known as "finger agnosia" - was regularly associated with defects in literacy and numeracy. It shows itself as confusion and inaccuracy in keeping track of where your fingers are, and what they are up to.

• WE SHALL NOW PAUSE A MOMENT TO TEST WHETHER YOU RECOGNISE YOUR OWN FINGERS!

BRAIN DAMAGE AND NUMERACYTESTS FOR FINGER AGNOSIA

• Doctors have devised a series of tests of finger agnosia.

• Finger Naming: Ask the patient to point to fingers on his/her own hand named by the examiner [e.g., left index finger, right thumb, etc.].

BRAIN DAMAGE AND NUMERACYTESTS FOR FINGER AGNOSIA

• Touch-Point: The patient places their hand (palm up) behind a screen. The examiner uses a pencil to touch fingers one at a time in random order. Each time, the patient must reach out with their other hand and touch the finger which has just been stimulated.

• Double Touch-Point: Same, but touching any two fingers at a time.

BRAIN DAMAGE AND NUMERACYTESTS FOR FINGER AGNOSIA

• In-Between: With the patient's eyes closed, touch two fingers on the patient's hand and ask how many fingers are in between the two touched.

BRAIN DAMAGE AND NUMERACYACQUIRED DYSCALCULIA - RECENT CASES

• Neurologists continue to be fascinated by the search for numerical skills in the brain. Warrington (1982) reported on patient DRC, another person with rear left brain injury.

• DRC could read and write numbers, and could give reasonable rough estimates of magnitude for variables like height. When asked to multiply 3 by 4 he replied "13, roughly".

• However, for simple problems such as 5 + 7 his performance was slow and inaccurate. He commented that he often knew the rough answer to a problem but could not come up with the exact answer.

BRAIN DAMAGE AND NUMERACYACQUIRED DYSCALCULIA - RECENT CASES

• Warrington (1982) saw this as indicating that there are brain processes for rough calculation separate from the brain processes for precise calculation. One of these had been damaged, but the other one was still OK

I do sums exactly

I do sums approximately(to check up

on him)

BRAIN DAMAGE AND NUMERACYACQUIRED DYSCALCULIA - RECENT CASES

• Moreover, even though DRC couldn't do simple additions, he remained able to do "counting onwards": thus 7 onwards from 5 would go 6 - 7 - 8 - 9 - 10 - 11 - 12!

• INTERPRETATION: The ideas of adding and equalling, as well as the number concepts of 5, 7, and 12, were all individually intact, but the mental process which in the rest of us returns a precise sum from two given numbers was somehow faulty.

PART 3

WHY MATHEMATICS IS DIFFICULT TO LEARN

You need science and technology in most jobs .....

PLUMBER

BUILDERFARMER

MECHANIC

DRIVER

PARAMEDIC

NURSE

MATHEMATICSCARPENTER

DOCTOR

TEACHER

This is why good maths teachers are so important

When babies are born, their brains don't have very much inside. They can't talk, and they can't do sums, either.

er .....

Thomas, what's three times five?

Our mums and dads, brothers and sisters, friends and teachers have to teach us everything we know. We store all these lessons away in our brains, and it gradually makes us clever.

Three times five makes fifteen.

What sort of things do babies have to learn?

The answer's on your brain map .....

SEEING

BALANCING

KNOWIN

G

ALL

ABOUT

THE

WORLD

HEARING

MOVING OUR EYES

DECIDING WHAT TO

SAY

REMEMBERING

WHERE W

E'VE

BEEN AND W

HAT

WE'V

E DONE

DECIDING WHAT TO DO NEXT

REMEMBERING NUMBERS ETC.

FOR SHORT PERIODS

FEELING PEOPLE TOUCH U

S

MOVING O

UR BODIES

BEING NICE TO PEOPLE

DOING SUMS

I have to learn all this. That's why it takes me so long.

• It's because our brains have all this to learn, that our teachers have to give us so much practice.

• The secret is that teachers start by teaching each part of the brain on its own, before trying to get all the skills working together smoothly.

In the end we can think so fast we don't know how we do it.

• So how does the brain learn to do counting, then?

• Here's what happens when someone is asked to count upwards from a given number.

From eighty sevenUntil told to stopOut loudCount on by 1

SEEHEAR

87+1

(n=(n+1))*(n=(n+1);?break)

88SAY

899091

/ ay-tee-ate / / ay-tee-nIn / / nIn-tee / / nIn-tee-wun /

• And here's the same thing at a more lifelike speed .....

From eighty sevenUntil told to stopOut loudCount on by 1

SEEHEAR

87+1

(n=(n+1))*(n=(n+1);?break)

88SAY

899091

/ ay-tee-ate / / ay-tee-nIn / / nIn-tee / / nIn-tee-wun /

DIWEDD

Diolch yn Fawr i Bawb

REFERENCES

Smith, D.J. and Livesey, K.M. (2006). A head for sums. Interactive presentation to Maes yr Haul Primary School, Bridgend, 14th March 2006, as part of National Science Week 2006.

Copyright Notice: This material was written and published in Wales by Derek J. Smith (Chartered Engineer), Senior Lecturer in Cognitive Science and Informatics at University of Wales Institute, Cardiff. It forms part of a multifile e-learning resource, and subject only to acknowledging Derek J. Smith's rights under international copyright law to be identified as author may be freely downloaded and printed off in single complete copies solely for the purposes of private study and/or review. Commercial exploitation rights are reserved. The remote hyperlinks have been selected for the academic appropriacy of their contents; they were free of offensive and litigious content when selected, and will be periodically checked to have remained so. Copyright © 2009, High Tower Consultants Limited.

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