Transcript Slide 1

Which of these are true?
Every even number greater than 3 is
the sum of two prime numbers?
Every number is the sum of no more
than 4 square numbers?
The sum of 7 and 3 is always 10?
If there are over 30 people in a room it is
more likely than not that two of them
have the same birthday?
Christian Goldbach (1690-1764)
He was born in Konigsberg, Prussia on March 18, 1690. His
father was a Protestant minister in Konigsberg. He attended
the Royal Albertus University.
He became a mathematics
professor and historian
at St. Petersburg Academy
of Sciences in 1725.
He tutored Czar Peter II in
1728.
Goldbach's conjecture
remains unproven to this
day.
If you analyse HMI and OFSTED reports over the last 90
years the concerns noted most often are (in order):
Investigational mathematics
Technical aspects of writing
Speaking/Listening
Reading for meaning
Writing for meaning
Standards of behaviour
The concern over investigational maths is directed at the opportunities
provided for the use of higher order thinking skills.
The difference between:
knowing about maths
and then using knowledge to work like a
mathematician.
Why is
26
special?
Three key tasks face us if we are to challenge
children to think in the way mathematicians
do:
To enable them to look at data
and hypothesise /suggest
reasonable explanations
To enable
them to ask
questions as
well as
answer them
To enable them to
make connections
between things
they already know
Combining Bloom’s Taxonomy and the Key Thinking Skills
REMEMBERING
Recognise, list, describe, identify, retrieve, name ….
Information-processing skills: locate, collect and recall relevant information
UNDERSTANDING
Interpret, give examples, summarise, infer, paraphrase …..
Information-processing skills: interpret information to show they understand relevant concepts
and ideas analyse information eg sort, classify, sequence, compare and contrast understand
relationships
APPLYING
Implement, carry out, use …
Enquiry skills: ask relevant questions pose and define problems
research predict outcomes, test conclusions and improve ideas
plan what to do and how to
ANALYSING
Compare, give attributes, organise, break down into different parts…
Reasoning skills: give reasons for opinions draw inferences and make deductions use
precise language to explain what they think,
EVALUATING
Check, make a critical judgement, hypothesise ...
Evaluation skills: evaluate information they are given
judge the value of what they read,
hear and do develop criteria for judging the value of their own and others' work or ideas
CREATING
Design, construct, plan, produce, use the new knowledge in a different situation ...
Creative thinking skills: generate and extend ideas suggest possible hypotheses be
imaginative in their thinking, look for alternative innovative outcomes
*Bloom’s Taxonomy in brown
*Key Thinking Skills in blue
To what extent are there opportunities
for children to:
Make connections between
different elements
Look for alternative and/or
innovative outcomes
Suggest possible hypotheses/
explanations
Generate and extend ideas
Pierre de Fermat
1601 - 1665
Pierre de Fermat was a French lawyer and government
official most remembered for his work in number theory;
in particular for Fermat's Last Theorem.
It was his theorem, that deemed him fitting of the title,
"The greatest French mathematician of the seventeenth
century’’, and yet he was only a part-time mathematician.
Commemorative stamp to mark
the re-proof of the theorem by
Andrew Wiles
Fermat’s Last Theorem
x² + y² = z²
Thus 3² + 4² = 5²
(9 + 16 = 25)
B U T ..................
x³ + z³ never = z³
Sir Michael Atiyah
Sir Michael Atiyah is an honorary professor at University of
Edinburgh and has been awarded maths’ equivalent of the
Nobel Prize
“The creative side of maths is all to do
with dealing with ideas and trying to
understand how things are related in
various ways. At school, often you’re
simply given the questions: the creative
part of maths is searching for exactly
what question to ask, what unknown
you’re going to explain.’’
The Park
Opening Times
March 2nd
Number
of people
in
The Park
April 1st to September 30th
Monday to Friday:
Gates open at 8.30am
and close at 9 pm
36
Weekends: Gates open at
8.00 am and close at 9 pm
24
October 1st to March 31st
Monday to Friday: Gates
open at 8.30 am and close
at 5 pm
16
Weekends: Gates open at
8.00 am and close at 6 pm
4
32
28
20
12
8
Time of
day
Weather for Wednesday March 2nd
8 - 8.30
am
8.30 - 9
am
9 - 9.30 9.30 am
10 am
10 10.30
am
10.30 11 am
Create a mathematical representation of a
supermarket visit of twenty minutes
(from time entered store and leaving after
having checked out).
The information to be shown is
time passing, distance travelled
and number of items bought in
each minute.
BEWARE – you will be
asked to describe/explain
the information contained.
The answer is 35
What is the
(mathematical)
question?
2004 KS2 SATS Question
Use these clues to guess the number.
It’s
It’s
It’s
It’s
more than 20
less than 40
a multiple of 5
a multiple of 3
Start with the sequence of non-zero digits
1 2 3 4 5 6 7 8 9
The problem is to place plus or minus signs
between them so that the result of thus
described arithmetic operation will be 100
One answer:
12 + 3 - 4 + 5 + 67 + 8 + 9 = 100
Can we find another?
Dr Alison Steadley
Alison Steadley was Professor of Philosophy and
Creative Thinking at Philadelphia University
‘’If we want to be creative, we must
be able to explore connections
between different areas. This means
we should have a good knowledge
base as the ingredients for creativity
depend on the store of ideas and
knowledge that are available to us.’’
This number
9
goes, according to a rule in
my head, with one of these.
Which one, and why?
6 27 25 15 12 99 11
The challenge is to get 4 people across this bridge at
night with a torch in no more than 15 minutes.
WARNING
The bridge will collapse WARNING
If anyone is on the bridge
if more than two people stand on it. the torch must be on the bridge also.
TO MAKE IT DIFFICULT
If anyone
steps on the bridge they must then go
all the way across.
BE CAREFUL The torch must be on the
bridge. If they cross in pairs they can
only travel at the speed of the slowest.
Person 1: can only travel slowly
and needs 8 minutes to cross
Person 2: can travel a little
quicker, but still takes 5 minutes
to cross
Person 3: is quick and can cross
in only 2 minutes
Person 4: is really quick and can
get across in only one minute
Key Thinking Skills Questions Template
An ‘easy-to-view’ selection of maths questions that might be useful starting points
to initiate specific types of thinking.
Information-processing
Why did ……happen?
What does…… mean? List the (numbers, shapes etc
that….) How many…… are/were there?
What is a ……?
Who has the widest/ shortest/ biggest/
smallest……? When did ……… happen?
What happened after/before……? Which of these ……..
happened first/latest etc.)?
Which way would I go to……? How much would……cost?
Reasoning
Looking at the chart/ timeline/graph etc. can you tell me……?
Can you sort these ……into
groups? What is the pattern? What caused …… to happen? Can you draw diagram that shows………?
Can you use the diagram to find out……? Can you explain how ……?
In what order did ………….
happen?
Can you give an example of……? What kind of a ……is this?
Which one doesn't belong in
this group?
What is the purpose of…? What is the relationship between …and…?
Evaluation Explain why you…….… Explain how………
What would you have done/do if…………?
Why? Which is the best solution to the problem of …… ? Why do you think so?
How could ……be
improved? What has changed from the beginning of …. and the end?
What does it mean that ……
happened? Is there another way to find a solution to……? What generalization can you make from this
information?
Enquiry
How is …like…? How are……and ……different? How could we find out if……?
What
questions could we ask in order to find out…? What do we know that..…? What do we need to know in
order to be able to……? How could we prove that……..?
Is it true that….? How do you know?
If we
changed/altered ….. what would happen to ……?
Creative
Can you create a pattern using……?
What will/would happen if ……? If we changed/altered
… what would happen to …? Can you create a way to ……? Can you find a solution to the problem of…?
How are these things the same/different…? Is there another way to show that…. ?
How could...be used in
another way?
How might we link/use these things together..? How many ways can you find to explain
why …….might have happened?
How many ways can you find to……..?
1/4
1/2
1/16 3/16
What might this pie chart be telling us?
1
2
3
4
5
6
7
Why is
26
special?
‘’Can you tell me this:
Why does a ball bounce?’’
Pascal's Triangle
This pattern is named after the French mathematician Blaise Pascal
who brought the triangle to the attention of Western mathematicians
(it was known as early as 1300 in China, where it was known
as the "Chinese Triangle").
The triangle itself is made by arranging numbers.
Each number in the triangle is the sum of the pair of numbers
directly above it (to the above left and above right).
The first four rows are as follows:
Blaise Pascal (1623-62)
Pascal invented the first digital
calculator to help his father with his
work collecting taxes.
He worked on it for three years between
1642 and 1645.
The device, called the Pascaline,
resembled a mechanical calculator of
the 1940s.
If all men knew what each said of the other, there would not be four
friends in the world.
People are usually more convinced by reasons they discovered
themselves than by those found by others.
I have discovered that all human evil comes from this; man's being unable
to sit still in a room.
We know truth, not only by reason, but also by the heart.
Alan can do the
job in 6 hours.
David can do the
same job in 5
hours.
What part of the
job can they do
by working
together for 2
hours?
‘’Do they like
talking to each
other?’’
‘’Do they need tools
that they couldn’t
both use at once?’’
‘’Just tell me what
the job is and then
I’ll try and tell you
the answer.’’
Debbie buys one
pizza for £2.15.
How much will 5
pizzas cost her?
What are the chances
of throwing a six on a
dice?
‘’Quite good if you
blow into your hand
six times before you
roll it.’’
’’Is there an offer
on?’’
‘’ Sometimes you
get a discount if
you buy three or
more’’
If it takes one woman
nine months to have a
baby how long does it
take three women to
have three babies?
The Three Little Pigs
(From an American website)
Show your work.
Pig 3 got a good deal on his on his phone bill. It cost him $2 the
first month, $4 the second month, and $6 the third month. At this
rate, what will his bill be in the 5th month?
After an exciting game of leap hog, Pig 3 had an idea. To help pay
for their homes, they could open a lemonade stand. They could sell
lemonade for 10 cents a glass. If they sold 10 glasses, how much
would they make?
Pig 1 felt something was wrong. "We're being followed!" he
screamed. "Let's run for home!" The pigs ran and ran. They ran 4
miles in 2 minutes. How many miles did they run each minute?
Both pigs went squealing down the road to their brother, who like
all big brothers said, "I told you so!" And they sat down to watch
TV. Their favorite show, Pigmalion, comes on at 8:00 p.m. It was
7:30 p.m. How long did they have to wait for their program?
Anyway, this wolf wasn't stupid. He knew he couldn't blow down the brick house
without popping a lung so he thought...."I'll just get in my 1963 Volkswagen and
run this house down!" If it's 1999, how old was the car?
The Three Little Pigs
How they used their time each day before and after the tragic death of the Big
Bad Wolf.
Before
Number
of Hours
14
12
10
8
6
4
2
After
Number
of Hours
14
12
10
8
6
4
2
Mrs. Belton’s Class (Yr3 children)
Collective height on the morning of September 15 th:
36 metres and 87 centimetres
This is the distance from the right hand side of the
classroom door to the wall with the heater on it, down
that wall to the bottom, along the wall with the
smartboard on it, along the wall with the door into the
playground on it and then back along the wall with the
classroom door on it until you reach the marker we have
placed there.
We have put a second marker further along that
wall to show where we think we will get to after
measuring our heights again at the end of
March.
‘Fermi’ questions
Enrico Fermi was famous for thinking about physics out loud – devising problems and then
going through a process of reasoning and guessing to arrive at very plausible estimates.
The most well-known example is as follows (the thinking process is outlined as well).
HOW MANY PIANO TUNERS ARE IN NEW YORK CITY?
The number of piano tuners in some way depends on the number of pianos. The number of
pianos must connect in some way to the number of people.
1) Approximately how many people are in New York City? 10,000,000
2) Does every individual own a piano? No
3) Would it be reasonable to assert that "individuals don't tend to own pianos; families
do’’? Yes.
4) About how many families are there in a city of 10 million
people? Perhaps there are 2,000,000 families in NYC.
5) Does every family own a piano? No. Perhaps one out of every five does.
That would mean there are about 400,000 pianos in NYC.
6) How many piano tuners are needed for 400,000 pianos?
If we assume that "on the average" every piano gets tuned
once a year, then there are 400,000 "piano tunings" every year.
7) How many piano tunings can one piano tuner do?
Let's assume that the average piano tuner can tune four pianos a day. Also assume
that there are 200 working days per year. That means that every tuner can tune about
800 pianos per year.
How many piano tuners are needed in NYC?
The number of tuners is approximately 400,000/800 or 500 piano tuners.
Why is
26
special?