Copyright © University of Cambridge. All rights reserved.
Music teachers challenge students to listen and participate.
English and History teachers invite students to journey in other
worlds.
Art and Drama teachers offer students opportunities to explore.
What are we to offer students if they are to function
mathematically?
What do you understand by 'functioning mathematically'?
What characteristic behaviours do your highly achieving
mathematicians exhibit?
Three key characteristics
 working systematically
 generalising
 proving and explaining
Many students 
Highly attaining students 
work randomly 
work systematically 
notice obvious patterns $\quad \quad$ 
conjecture and generalise 
make statements 
offer explanations and proofs 
What do these behaviours look like in practice?
Many numbers can be expressed as the sum of two or more
consecutive integers.
$15 = 7 + 8$
$15 = 4 + 5 + 6$
$15 = 1 + 2 + 3 + 4 + 5$
Look at numbers other than 15 and find out all you can about
writing them as sums of consecutive whole numbers.
Working
randomly:
$5=2+3$
$4+5+6+7=22$
$1+2+3+4+5+6=21$
$11=5+6$
Working
systematically:
$2$ 
$3$ 
$4$ 
$1=0+1$ $\quad \quad$ 


$2=$ 


$3=1+2$ 
$0+1+2$ $\quad \quad$ 

$4=$ 


$5=2+3$ 


$6=$ 
$1+2+3$ 
$0+1+2+3$ 
$7=3+4$ 


$8=$ 


$9=4+5$ 
$2+3+4$ 

$10=$ 

$1+2+3+4$ 
Noticing obvious patterns
 Odd numbers can be written as two consecutive numbers.
 Multiples of $3$ can be written as three consecutive
numbers.
 Even numbers can be written as four consecutive numbers.
Conjecturing and
generalising
 Multiples of $3$ can be written as the sum of three consecutive
numbers.
 Multiples of $5$ can be written as the sum of five consecutive
numbers.
 I wonder if multiples of $7$ can be written as the sum of seven
consecutive numbers'?¦
 I wonder if multiples of $9$ can be written as the sum of nine
consecutive numbers'?¦
 I wonder if multiples of $x$ (where $x$ is odd), can be written
as $x$ consecutive numbers'?¦
Making statements
 Multiples of $3$ can be written as three consecutive
numbers.
 If you give me any multiple of three, I can tell you the three
numbers by dividing by three and that will be the middle
number.
 Even numbers can be written as four consecutive numbers.
Offering explanations and
proofs
 If you give me three consecutive numbers I can always turn them
into a multiple of three.
 $x + (x+1) + (x+2) = (x+1) + (x+1) + (x+1) = 3 (x+1)$
 Same will apply to five, seven, nine and any odd number because
you can pair off numbers on either side of the middle number.
Creating opportunities for students to develop their
mathematical thinking
 Promoting a conjecturing atmosphere
 Careful use of questions and prompts
 Low threshold high ceiling tasks
 Modelling behaviour
 Whole class discussion
 Highlighting behaviour that you want to promote
 HOTS not MOTS
Promoting a conjecturing atmosphere
 Accepting 'messy' work
 Valuing risktaking and halfformed ideas
 Encouraging discussion
 Providing thinking time
Careful use of questions and prompts
 What have you found out so far?
 Do you notice anything?
 Is that always true?
 Can you convince us?
 Can anyone think of a counter example?
 What if...?
 What might you try next?
 Is there a way you could organise your findings?
Low threshold high ceiling tasks
 Using problems that everyone can 'get into' but which also
offer opportunities for working at a higher level
 There is no need to 'label' students in advance
Modelling behaviour
Students should be in the presence of a teacher who is
 curious
 intrigued
 wanting to find out more
 appreciates the value of proof
 expects to understand why things happen
Whole class discussion
 Early on: to share

 Half way through: to share

 possible findings
 useful strategies
 new questions
 Towards the end: to share

 conclusions
 explanations
 proofs
Highlighting behaviour that you want to promote
Drawing attention to and valuing process as well as outcome.
HOTS not MOTS
H igher O rder T hinking S kills
not
M ore o f t he S ame
Each task requiring lower order thinking has a higher order
thinking equivalent/partner. Identify which is which and sort them
into pairs. (see Lower and Higher Order Thinking document)
How can you generate questions that promote these HOTS in a
mathematical context?
 Ask students to work backwards
 Ask "what if...?" questions
 Set questions which have a variety of solutions
 Set questions which can be solved in a variety of ways
Ask students to
work backwards
Instead of:
Find the area and perimeter
of a $3cm \times 8cm$
rectangle
ask:
If the area of a rectangle
is $24$ cm² and
the perimeter is $22$ cm,
what are its dimensions?
Ask "what
if...?" questions
Instead of:
Find the area and perimeter
of a $3cm \times 8cm$
rectangle
ask:
What if the area of a
rectangle (in cm²) is equal to the perimeter (in cm),
what could its dimensions be?
Set questions
which have a variety of solutions
Instead of:
Find the perimeter of a
$3cm \times 8cm$
rectangle
ask:
Find a rectangle which has
unit sides and a perimeter of $100$ .
How many answers are there
and how do you know you've got them all?
Set questions
which can be solved in a variety of ways
Instead of:
Find the area and perimeter
of a $3cm \times 8cm$
rectangle.
ask:
If the area of a
rectangle is $24$cm² and the perimeter is
$22$cm, what are its dimensions?
How did you work this out?
"A teacher of mathematics has a great opportunity. If
he fills his allotted time with drilling his students in routine
operations he kills their interest, hampers their intellectual
development, and misuses his opportunity. But if he challenges the
curiosity of his students by setting them problems proportionate to
their knowledge, and helps them to solve their problems with
stimulating questions, he may give them a taste for, and some means
of, independent thinking."
Polya, G. (1945) How to Solve it
"I don't expect, and I don't want, all children to find
mathematics an engrossing study, or one that they want to devote
themselves to either in school or in their lives. Only a few will
find mathematics seductive enough to sustain a long term
engagement. But I would hope that all children could experience at
a few moments in their careers...the power and excitement of
mathematics...so that at the end of their formal education they at
least know what it is like and whether it is an activity that has a
place in their future."
David Wheeler
This article was originally
presented to the Council of Boards of School Education in India
Conference,"Addressing Core Issues and Concerns in Science and
Mathematics", in Rishikesh, India in April 2007.