Improving the learning of numeracy through formative assessment

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Improving the learning of numeracy through
formative assessment

• Dylan Wiliam
• National Numeracy Conference
• Edinburgh, March 2009
• www.dylanwiliam.net

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Raising achievement matters

• Which of the following categories of skill is
  disappearing from the work-place most rapidly?
• Routine manual
• Non-routine manual
• Routine cognitive
• Complex communication
• Expert thinking/problem-solving

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but what is learned matters too
Autor, Levy Murnane, 2003
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The only 21st century skill

• So the model that says learn while youre at
  school, while youre young, the skills that you
  will apply during your lifetime is no longer
  tenable. The skills that you can learn when
  youre at school will not be applicable. They
  will be obsolete by the time you get into the
  workplace and need them, except for one skill.
  The one really competitive skill is the skill of
  being able to learn. It is the skill of being
  able not to give the right answer to questions
  about what you were taught in school, but to make
  the right response to situations that are outside
  the scope of what you were taught in school. We
  need to produce people who know how to act when
  theyre faced with situations for which they were
  not specifically prepared.
• (Papert, 1998)

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Formative assessment
Assessment for learning is any assessment for
which the first priority in its design and
practice is to serve the purpose of promoting
pupils learning. It thus differs from assessment
designed primarily to serve the purposes of
accountability, or of ranking, or of certifying
competence. An assessment activity can help
learning if it provides information to be used as
feedback, by teachers, and by their pupils, in
assessing themselves and each other, to modify
the teaching and learning activities in which
they are engaged. Such assessment becomes
formative assessment when the evidence is
actually used to adapt the teaching work to meet
learning needs. (Black et al., 2002)
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Types of formative assessment

• Long-cycle
• Span across units, terms
• Length four weeks to one year
• Medium-cycle
• Span within and between teaching units
• Length one to four weeks
• Short-cycle
• Span within and between lessons
• Length
• day-by-day 24 to 48 hours
• minute-by-minute 5 seconds to 2 hours

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Unpacking formative assessment

• Key processes
• Establishing where the learners are in their
  learning
• Establishing where they are going
• Working out how to get there
• Participants
• Teachers
• Peers
• Learners

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Aspects of formative assessment
Where the learner is going Where the learner is How to get there
Teacher Clarify and share learning intentions Engineering effective discussions, tasks and activities that elicit evidence of learning Providing feedback that moves learners forward
Peer Understand and share learning intentions Activating students as learning resources for one another Activating students as learning resources for one another
Learner Understand learning intentions Activating students as ownersof their own learning Activating students as ownersof their own learning
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Five key strategies

• Clarifying, understanding, and sharing learning
  intentions
• curriculum philosophy (goals and horizons)
• Engineering effective classroom discussions,
  tasks and activities that elicit evidence of
  learning
• classroom discourse, interactive whole-class
  teaching
• Providing feedback that moves learners forward
• feedback
• Activating students as learning resources for one
  another
• collaborative learning, reciprocal teaching,
  peer-assessment
• Activating students as owners of their own
  learning
• metacognition, motivation, interest, attribution,
  self-assessment

(Wiliam Thompson, 2007)
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and one big idea

• Use evidence about learning to adapt teaching and
  learning to meet student needs

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Keeping Learning on Track (KLT)

• A pilot guides a plane or boat toward its
  destination by taking constant readings and
  making careful adjustments in response to wind,
  currents, weather, etc.
• A good teacher does the same
• Plans a carefully chosen route ahead of time (in
  essence building the track)
• Takes readings along the way
• Changes course as conditions dictate

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Eliciting evidence of student achievement
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Kinds of questions Israel
Which fraction is the smallest?
Success rate 88
Which fraction is the largest?
Success rate 46 39 chose (b)
Vinner, PME conference, Lahti, Finland, 1997
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Draw an upside-down triangle
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Misconceptions
3a 24 a b 16
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Questioning in maths discussion

• Look at the following sequence
• 3, 7, 11, 15, 19, .
• Which is the best rule to describe the sequence?
• n 4
• 3 n
• 4n - 1
• 4n 3

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Eliciting evidence

• Key idea questioning should
• cause thinking
• provide data that informs teaching
• Improving teacher questioning
• generating questions with colleagues
• closed v open
• low-order v high-order
• appropriate wait-time
• Getting away from I-R-E
• basketball rather than serial table-tennis
• No hands up (except to ask a question)
• class polls to review current attitudes towards
  an issue
• Hot Seat questioning
• All-student response systems
• ABCD cards, Mini white-boards, Exit passes

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Questioning in maths diagnosis

• In which of these right-angled triangles is a2
  b2 c2 ?

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(No Transcript)
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Lines of symmetry
C
A
B
F
D
E
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Constructing hinge-point questions
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Discriminate incorrect cognitive rules

• Version 1 (Hart, 1981)
• If ef  8, then efg  
• 9
• 12
• 15
• 8g

• Version 2
• If fg  8, then fgh  
• 9
• 12
• 15
• 16
• 8h

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Discriminate correct cognitive rules
What is the area of this trapezium?
a
h
b
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2A (a  b) x h A  (a  b) x h
A  h x (a  b)
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Discriminate between incorrect and correct
cognitive rules

• Version 1
• There are two flights per day from Newtown to
  Oldtown. The first flight leaves Newtown each day
  at 920 and arrives in Oldtown at 1055. The
  second flight from Newtown leaves at 215. At
  what time does the second flight arrive in
  Oldtown? Show your work.

• Version 2
• There are two flights per day from Newtown to
  Oldtown. The first flight leaves Newtown each day
  at 905 and arrives in Oldtown at 1055. The
  second flight from Newtown leaves at 215. At
  what time does the second flight arrive in
  Oldtown? Show your work.

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Correct
Incorrect
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Over- and under-generalization
In which of the following diagrams, is one
quarter of the area shaded?
B
C
D
A
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Diagnostic item medians

• What is the median for the following data set?
• 38 74 22 44 96 22
  19 53
• 22
• 38 and 44
• 41
• 46
• 70
• 77
• This data set has no median

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Diagnostic item means

• What can you say about the means of the following
  two data sets?
• Set 1 10 12 13 15
• Set 2 10 12 13 15 0
• The two sets have the same mean.
• The two sets have different means.
• It depends on whether you choose to count the
  zero.

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Diagnostic item diagonals
Which of the shapes below contains a dotted line
that is also a diagonal?
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Hinge-point questions

• A hinge question is based on the important
  concept in a lesson that is critical for students
  to understand before you move on in the lesson.
• Design requirements
• Every student must respond to the question within
  two minutes.
• You must be able to collect and interpret the
  responses from all students in 30 seconds
• Priorities (in order)
• In no case should correct and incorrect cognitive
  rules map ontp the correct option
• Each incorrect option response (distractor)
  should interpret a single cognitive rule
• Correct option responses (keys) should interpret
  a single cognitive rule

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Practical techniques feedback

• Key idea feedback should
• cause thinking
• provide guidance on how to improve
• Comment-only grading
• Focused grading
• Explicit reference to rubrics
• Suggestions on how to improve
• Not giving complete solutions
• Re-timing assessment
• (eg three-quarters-of-the-way-through-a-unit test)

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Practical techniques sharing learning intentions

• Explaining learning intentions at start of
  lesson/unit
• Learning intentions
• Success criteria
• Intentions/criteria in students language
• Posters of key words to talk about learning
• eg describe, explain, evaluate
• Planning/writing frames
• Annotated examples of different standards to
  flesh out assessment rubrics (e.g. reports of
  mathematical investigations)
• Opportunities for students to design their own
  tests

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Students owning their learning and as learning
resources

• Students assessing their own/peers work
• with rubrics
• with exemplars
• two stars and a wish
• Training students to pose questions/identifying
  group weaknesses
• Self-assessment of understanding
• Traffic lights
• Red/green discs
• End-of-lesson students review