Assessment for Learning in
the Mathematics classroom
Enrichment workshop
Sessions 2 + 3
"The most important single factor influencing
learning is what the learner already knows.
Ascertain this and teach him accordingly.”
Ausubel (1968)
Educational Psychology: A Cognitive View
The challenge:
To use evidence about learning to adapt teaching
and learning to meet student needs.
Five Key Considerations…
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Assessing learners’ prior knowledge
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Adapting lessons in response to what learners do
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Looking for evidence of learning
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Offering constructive feedback
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Questioning for depth of understanding
1. Assessing learners’ prior knowledge,
identifying misconceptions
Teacher’s role:
watching, listening, questioning
Factors and Multiples Puzzle
Completing Quadrilaterals
I started drawing some quadrilaterals - can you
complete them?
Isosceles Triangles
Draw some isosceles triangles with
an area of 9 cm2 and a vertex at
(20, 20).
If all the vertices have whole
number coordinates, how many is it
possible to draw?
2. Starting with big ideas, sharing
learning intentions
Teacher’s role:
adapting the lesson in response to what learners do
Low Threshold – High Ceiling tasks
Offer accessible starting points and opportunities
for interesting mathematical journeys.
Careful use of questions and prompts
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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?
Tilted Squares
Can you find a quick and easy method to
work out the areas of tilted squares?
Odds and Evens
To play the game, mix up the
balls and randomly pick two
balls out together.
If the total is even, you win.
If the total is odd, you lose.
Is this a fair game?
How many odd and even balls do you need for a fair game?
Cuboid Challenge
Cut a square from each
corner and fold up the
flaps to make a box
without a lid.
What is the maximum
possible volume of the
box?
What’s Possible?
Give me a whole number…
And another…
And another…
What questions does this prompt?
What’s Possible?
Can you find a way to write each number from 1 to 30 as
the difference of two squares?
Can you write any of them in more than one way?
If I give you a number, can you tell me all the possible
ways to write it as the difference of two squares?
3. Choosing activities that promote
discussion and collaboration
Teacher’s role:
looking for evidence of learning, offering
opportunities to refine thinking
Building a community of mathematicians
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Creating a safe environment for learners to take risks
Promoting a conjecturing atmosphere
Encouraging discussion and collaboration
Valuing half-formed ideas and learning from mistakes
Accepting 'messy' work
Valuing a variety of approaches
Providing thinking time
Dialogic Teaching
“Children, we now know, need to talk, and to
experience a rich diet of spoken language, in
order to think and learn. Reading, writing and
number may be acknowledged curriculum
basics, but talk is arguably the true foundation
of learning.”
Robin Alexander, 2004
“…language provides us with a means for
thinking together, for jointly creating
knowledge and understanding.”
Neil Mercer, Words and Minds (2000)
M, M and M
Can you find five positive whole numbers
that satisfy the following properties:
Mean = 4
Mode = 3
Median = 3
Can you find all the different sets of five positive whole
numbers that satisfy these conditions?
Dicey operations
Find a partner and a 1-6 dice,
or preferably a 0-9 dice.
Each of you draw an addition grid.
Take turns to throw the dice and
decide which of your cells to fill either fill in each cell as you throw the
dice or collect all your numbers and
then decide where to place them.
Throw the dice nine times each until all the cells are full.
Whoever has the sum closest to 1000 wins.
Dicey operations
Roll a dice 5 times.
Fill in the grid after
each roll.
Who can get closest
to 10 000?
What will your strategy be?
Kite in a Square
What fraction of the
total area is shaded?
Printable cards:
Coordinates
Similar figures
Pythagoras
Reflection
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Choose one activity that we have discussed this morning.
Explain what makes it an Assessment for Learning activity.
Describe how you will apply/adapt it for your own practice.
Be prepared to share.
Lunch!
4. Providing feedback that moves
learners forward
Teacher’s role:
offering constructive feedback
Encouraging a growth mindset
Carol Dweck has shown that learners and
teachers who believe that intelligence is
flexible, and the goal is to learn as much as
they can, are more successful than those who
believe in finishing tasks and passing tests.
An expert learner?
WD-40 is the trademark of a widely
used penetrating oil spray, developed
in 1953 by Norm Larsen.
WD-40 stands for “Water
Displacement, 40th attempt",
a name which came from Larsen's
laboratory notebook – he was
attempting to concoct a formula to
prevent corrosion by displacing water,
and arrived at the formula on his 40th
attempt!
Square It
Players take it in turns to
click on a dot on the grid.
The winner is the first to have
four dots that can be joined
by straight lines to form a
square.
Got It - a game for two players
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Decide who will go first.
Imagine there are 18 biscuits on a plate.
The first player can take 1, 2, 3 or 4 biscuits.
The second player can then take 1, 2, 3 or 4 biscuits.
Keep taking turns until all the biscuits are gone.
The person to take the last biscuit wins.
Play the game several times.
Can you find a winning strategy?
Got It continued…
What will you do:
– if there are a different
number of biscuits?
– if you can take a
different number of
biscuits?
Dozens
Do you know a quick way to check if a number is
a multiple of two?
How about three, four or six?
Route to Infinity
Can you list the coordinates
of the points in the order in
which they're visited?
Can you develop a strategy
for doing this?
5. Focusing on solutions, not answers
Teacher’s role:
question for depth of understanding
If you want to build higher, dig deeper
“You can know the name of a bird in all the languages
of the world, but when you're finished, you'll know
absolutely nothing whatever about the bird...
So let's look at the bird and see what it's doing –
that's what counts.”
Richard P. Feynman
Valuing higher order mathematical thinking
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Thinking strategically
Looking for connections
Exploring and conjecturing
Reasoning and generalising
Justifying and proving
Valuing mathematical Habits of Mind
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Natural curiosity
Thinking mathematically
Working collaboratively
Being determined
“A focus on learning can enhance performance.
A focus on performance can depress performance.”
Chris Watkins
Wipeout
Take the numbers
1, 2, 3, 4, 5, 6
and choose one to wipe out…
Pair products
Choose four consecutive whole numbers.
Multiply the first and last numbers
together.
Multiply the middle pair together.
What do you notice?...
Warmsnug
Double
Glazing
Which window has
been given the
incorrect price?
Mind Reader?
Choose any two digit number, add together both
digits and then subtract the total from your
original number…
Followed by Always a Multiple?
Five strands of
mathematical
proficiency
NRC (2001) Adding it up:
Helping children learn mathematics
Reflection
Identify three ways that assessment for learning could
enhance learning in your classroom.
What changes will you make to your classroom practice as a
result of these sessions?
Further resources from NRICH Maths
Enriching the Secondary Curriculum:
http://nrich.maths.org/enriching
What we think and why we think it:
http://nrich.maths.org/whatwethink
Support materials and resources
All Cambridge endorsed,
and suggested support
materials can be found
in the Resource Centre
on the public website
www.cie.org.uk/i-want-to/resource-centre/
We’ve produced a
series of resources to
support the teaching
and learning in your
school. They explore
different aspects of
educational practice,
from designing a
curriculum to improving
the quality of classroom
activity.
http://cie.org.uk/teaching-and-learning/
Core documents
Question papers, mark schemes
and principal examiner reports
PDF copies of these can be
found on:
• Teacher Support
http://teachers.cie.org.uk
• www.cie.org.uk
• Syllabus Support CD-ROM
Also available in hard copy via
publications list.
Break time!