4/27/2016
Active Learning in
Mathematics
Camilla Rudin
• Academic & Professional Qualifications
Camilla Rudin
South Africa
April 2016
– PGCE in Mathematics
– BSc Joint Honours Mathematics & Psychology,
University of Birmingham, UK
• Professional Experience
– Head of Faculty & Teacher of Mathematics in British
International Schools in Central America, Middle East
and Europe
– International examining and marking.
Mathematics is not a spectator sport
Exploring
→ Noticing patterns
→ Conjecturing
→ Generalising
→ Explaining
→ Justifying
→ Proving
Encouraging curiosity
Active learners need to be
Curious
Thoughtful
Collaborative
Determined
The most exciting phrase to hear in science,
the one that heralds new discoveries,
is not Eureka!, but rather,
“hmmm… that’s funny…”
Isaac Asimov
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Some questions that might emerge:
Is this the only rectangle
with an area of 24 cm²?
Is it possible to have an odd number perimeter and an
area of 24 cm²?
6cm
What is the smallest possible perimeter? And the largest?
4cm
More generally, is it possible to have a fractional perimeter
but a whole number area? And vice versa?
If I give you the area and perimeter of a rectangle, will you
always be able to work out its dimensions?...
Another starting point…
Work out the mean, median, mode and
range of:
2, 5, 5, 6, 7
What do you notice?
Are there other similar sets?
May lead to:
Is it possible to find five positive whole numbers where:
Mode < Median < Mean
Mode < Mean < Median
Mean < Mode < Median
Mean < Median < Mode
Median < Mode < Mean
Median < Mean < Mode
Summing consecutive numbers…
Some numbers can be expressed as the sum of
two or more consecutive whole numbers…
Encouraging mathematical thinking
10 = 1 + 2 + 3 + 4
11 = 5 + 6
12 = 3 + 4 + 5
…
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If I ran a school, I’d give all the average grades
to the ones who gave me all the right answers,
for being good parrots.
I’d give the top grades to those who made lots of
mistakes and told me about them and then told
me what they had learned from them.
Buckminster Fuller, Inventor
The Factors and Multiple Challenge
You will need a 100 square grid.
Choose which numbers you want to cross out, always
choosing a number that is a factor or multiple of the
previous number that has just been crossed out.
Each number can only be crossed out once.
e.g. 40, 10, 60, 3, 33, 66, 22, 11…
Try to find the longest sequence of numbers that can
be crossed out.
Cryptarithms
Encouraging collaborative work
Rules for effective group work
All students must contribute:
no one member says too much or too little
Olympic measures
The cards contain some interesting measurements
and records from events at the Olympic Games.
Every contribution treated with respect:
listen thoughtfully
Unfortunately they have been muddled up.
Can you regroup them correctly?
Group must achieve consensus:
work at resolving differences
Every suggestion/assertion has to be justified:
arguments must include reasons
Neil Mercer
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Reflecting Squarely
In how many
ways can you fit
all three pieces
together to
make shapes
with line
symmetry?
Encouraging a growth mind-set
What’s it worth?
… our studies show that teaching people
to have a “growth mind-set”, which
encourages a focus on effort rather than
intelligence or talent, helps make them
into high achievers in school and life.
Carol Dweck
What can you
deduce from
this image?
Each symbol has a
numerical value.
The total for the symbols
is written at the end of
each row and column.
Can you find the missing
total that should go
where the question mark
has been put?
What can you
deduce from
this image?
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What can you
deduce from
this image?
What can you
deduce from
this image?
Product Sudoku
Product Sudoku
Like a conventional
Sudoku, but this time
the numbers in the cells
are the product of the
digits in the cells
horizontally and
vertically adjacent to
the cell…
The numbers in the
cells suggest the
order in which they
could be filled; it is
just one possible
route through the
problem…
Five strands of
mathematical
proficiency
Conceptual understanding -
comprehension of mathematical concepts, operations, and relations
Procedural fluency -
skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
Strategic competence -
ability to formulate, represent, and solve mathematical problems
NRC (2001) Adding it up:
Helping children learn mathematics
Adaptive reasoning -
capacity for logical thought, reflection, explanation, and justification
Productive disposition -
habitual inclination to see mathematics as sensible, useful, and worthwhile,
coupled with a belief in diligence and one’s own efficacy.
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… 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.
George Pólya, How to Solve it
What next?
What teachers can do
Aim to be mathematical with and in front of learners
Aim to do for learners only what they cannot yet do
for themselves
Focus on provoking learners to
• use and develop their (mathematical) powers
• make mathematically significant choices
John Mason
Thank you
Cambridge resources:
– https://teachers.cie.org.uk/
– http://www.cie.org.uk/teaching-and-learning/
• NRICH Maths materials:
– http://nrich.maths.org/enriching
– http://nrich.maths.org/whatwethink
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