MATHEMATICS IN THE MIND
Derek J. SMITH, CEng, CITP
University of Wales Institute, Cardiff
smithsrisca@btinternet.com
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
REMEMBERING
NUMBERS ETC.
FOR SHORT
PERIODS
DECIDING
WHAT TO
DO NEXT
MOVING
OUR
EYES
DECIDING
WHAT TO
SAY
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 .....
Say cheese .....
So let's see if we've got the hang of this.
TEST THE VOLUNTEERS ONE BY ONE
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 NUMERACY
WHAT 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 NUMERACY
WHAT 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
etc -3
IMAGINARY
-2
-1
0
1
REAL
PLUS INFINITY
2
3 etc
IMAGINARY
BRAIN DAMAGE AND NUMERACY
WHAT 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 NUMERACY
WHAT 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 NUMERACY
WHAT 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 NUMERACY
WHAT 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 NUMERACY
ACQUIRED 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 NUMERACY
TESTS 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 NUMERACY
TESTS 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 NUMERACY
TESTS 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 NUMERACY
TESTS 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 NUMERACY
ACQUIRED 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 NUMERACY
ACQUIRED 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
approximately
(to check up
on him)
I do sums
exactly
BRAIN DAMAGE AND NUMERACY
ACQUIRED 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
.....
MECHANIC
NURSE
PLUMBER
BUILDER
MATHEMATICS
FARMER
CARPENTER
PARAMEDIC
DOCTOR
TEACHER
DRIVER
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.
Thomas, what's three
times five?
er .....
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 .....
I have to learn all this. That's why it takes me
so long.
REMEMBERING
NUMBERS ETC.
FOR SHORT
PERIODS
DECIDING
WHAT TO
DO NEXT
MOVING
OUR
EYES
DECIDING
WHAT TO
SAY
BEING NICE TO
PEOPLE
DOING
SUMS
• 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.
91
90
89
88
87
+1
*(n=(n+1);
(n=(n+1))
?break)
SAY
HEAR
From
Until
Count
eighty
Out
told
loud
on
seven
to by
stop
1
// nIn-tee-wun
/nIn-tee
ay-tee-nIn
ay-tee-ate
/ / //
SEE
• And here's the same thing at a more lifelike
speed .....
91
90
89
88
87
+1
*(n=(n+1);
(n=(n+1))
?break)
SAY
HEAR
From
Until
Count
eighty
Out
told
loud
on
seven
to by
stop
1
// nIn-tee-wun
/nIn-tee
ay-tee-nIn
ay-tee-ate
/ / //
SEE
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 elearning resource, and subject only to acknowledging Derek J. Smith's rights
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