Tasks and learning
mathematics
Anne Watson
University of Oxford
DfE 2010
Nature of evidence
• Published research, usually comparative studies
• Review of research on pedagogy to Smith report
• Key Understandings in Mathematics Learning
(Nuffield)
• 20 years of teaching and curriculum development
with teachers, teacher educators worldwide
• Research into the practice of exceptional teachers
working in the English context
• International knowledge about task design
Being careful with research
findings
• Do evaluation strategies match curriculum
aims?
• Innovation v. roll out
• Cultural context
• Research shows ‘what is good/possible’,
not ‘what is best’ or ‘what will always work’
International Congress of
Mathematics Education (ICME) task
workshop
• Task types known to develop knowledge,
understanding and application:
– Draw on students’ perceptions and past experience,
and offer extensions
– Afford conjectures and feedback on effects of actions
– Constrain choices to fit models, images & alternative
representations of conventional mathematics
– Exploit students’ search for familiarity,
similarity/difference, and effect
Example
• £2500 is invested at 2% per annum. What
is it worth after 2 years, 10 years, 7 years,
27 years?
Find the number mid-way
between
28 and 34
280 and 340
2.8 and 3.4
.00028 and .00034
1028 and 1034
38 and 44
-38 and -44
40 and 46
Example
Example
• You have a sheet of black card 150 cm. x
150 cm. You have to make a Hallowe’en
witch hat. What possible heights can you
make?
Concepts
• Puzzles, problems, situations which can be
understood, but not resolved, using current
knowledge
• Classifying mathematical objects - new
classifications
• Interpreting multiple representations - new
notations and new methods
• Evaluating mathematical statements - truth,
usefulness, domain of applicability
Advantages
• Flexible, adaptable, knowledge
• Can construct and reconstruct meaning
• Misconceptions may arise for usual
reasons, but are resolved through
microtasks
• Mathematics with meaning
• Students do better in test questions that
require adaptation than …
Shortcomings
•
•
•
•
Takes time (coverage)
Takes time to establish appropriate habits
Memory maybe, and fluency
Teacher knowledge is challenged
(secondary)
Applying concepts
• problem to read and understand
• decide whether to use statistical, algebraic, logical or
ad hoc methods
• identify and select variables
• coordinate mental, graphical, numerical,
representations
• select facts, operations and functions to apply
• … and how to apply them
• apply appropriate knowledge of situations and
operations to interpret
Advantages
• Realistic application of known mathematics
• Creates a need for new mathematical methods
and knowledge
• Draws on everyday reasoning in mathematical
contexts, so more students are included
• New opportunities with every new task - less
chance for students to ‘lose track’
• Habits of identifying variables and deciding
methods
• Good preparation for workplace
• Students do better with unfamiliar test items and
multistage problems than …
Shortcomings
• need to understand both context and maths
• need experience of making good and bad
choices
• ad hoc, numerical or visual approaches
dominate
• purpose can be confused: to understand the
situation better, or the maths, or to learn new
mathematical ideas?
Procedures
• for fluency: repetition and not much change
• for understanding: careful change of variables;
reflect on outcomes
• for retention: layout, visual, representations,
pattern and rhythm
• to challenge usual misconceptions/errors:
special cases, comparing similar cases, focus on
meaning, correction
Advantages
• Can anticipate answers and difficulties
• Can generalise and extend methods to more
complex situations
• Can develop algorithmic understanding
• Automatisation of key procedures
• A page of ticks boosts confidence
• Can be automated (online worksheets with good
quality feedback and adaptation)
• Understood by society (parents, outsiders etc.)
Shortcomings
• Misapplication of methods; repetition of errors; hard to
adapt to unfamiliar situations; difficulty of multistage
problems
• Hard to recognise when to apply methods
• Textbooks need research-based principles (China)
• Associated with dislike of subject and boredom
• It is what machines can do
• Need to reflect on answers in order to fully understand
method
• Confusion between purpose: fluency or conceptual
understanding or retained knowledge
• Does not prepare students for higher study
• Traditional use does not develop the potential
advantages
Learning from effective teachers
(www.cmtp.co.uk)
Microtasks more important than task-types:
• Exemplifying and specialising
• Completing, deleting, correcting
• Comparing, sorting, organising
• Varying, reversing
• Conjecturing, generalising
• Explaining, justifying, convincing, refuting,
proving
• Representing
Stop press GCSE results …
Examples of microtasks
•
•
•
•
•
•
•
•
•
Is it always, sometimes, never
true?
What do you get if you change …
to …?
Make up three examples like this
Make a connected chain from …
What is the same and what is
different about…?
Provide the missing steps in …
Can you swap the property and
the definition and define the same
objects?
Of what is this a special case?
Explain the role of … in …
•
•
•
•
•
•
•
•
What is wrong with …?
Verify that … means the same as
…
What cases does/doesn’t this
work with?
Provide missing steps in ‘if … then
…because …’ arguments
When is … a good notation for …?
When is … a good method for …?
What has to be included in … to
make …?
Find a relationship between …
and …
Example
38
7
3a  8
7
3a  8
7b
2 5
3
2x  5
3
2x  5
3y
2
4
3
2a
4
3
2a
4
3b