Transcript Inside the black box: Raising standards through classroom

Improving the learning of
numeracy through formative
assessment
Dylan Wiliam
National Numeracy Conference
Edinburgh, March 2009
www.dylanwiliam.net
Raising achievement matters…
Which of the following categories of skill is disappearing from the workplace most rapidly?
1. Routine manual
2. Non-routine manual
3. Routine cognitive
4. Complex communication
5. Expert thinking/problem-solving
…but what is learned matters too…
Autor, Levy & Murnane, 2003
The only 21st century skill
So the model that says learn while you’re at school, while you’re young, the
skills that you will apply during your lifetime is no longer tenable. The skills
that you can learn when you’re 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 they’re faced with situations for which they
were not specifically prepared.
(Papert, 1998)
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)
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
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
Aspects of formative assessment
Teacher
Peer
Learner
Where the learner is
going
Where the learner is
How to get there
Clarify and share
learning intentions
Engineering effective
discussions, tasks and
activities that elicit
evidence of learning
Providing feedback that
moves learners forward
Understand and share
learning intentions
Activating students as learning
resources for one another
Understand learning
intentions
Activating students as owners
of their own learning
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)
…and one big idea
Use evidence about learning to adapt teaching and learning to meet
student needs
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
Eliciting evidence of
student achievement
Kinds of questions: Israel
Which fraction is the smallest?
1
2
1
1
a) , b) , c) , d) .
6
3
3
2
Success rate 88%
Which fraction is the largest?
4
3
5
7
a) , b) , c) , d) .
5
4
8
10
Success rate 46%; 39% chose (b)
[Vinner, PME conference, Lahti, Finland, 1997]
Draw an upside-down triangle…
Misconceptions
3a = 24
a + b = 16
Questioning in maths: discussion
Look at the following sequence:
3, 7, 11, 15, 19, ….
Which is the best rule to describe the sequence?
A. n + 4
B. 3 + n
C. 4n - 1
D. 4n + 3
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
Questioning in maths: diagnosis
In which of these right-angled triangles is a2 + b2 = c2 ?
A
b
a
B
a
c
C
b
a
b
D
c
c
b
c
E
c
a
a
b
F
b
c
a
Lines of symmetry
A
D
C
B
E
F
Constructing hingepoint questions
Discriminate incorrect cognitive rules
Version 1 (Hart, 1981)
If e+f = 8, then e+f+g =
A. 9
B. 12
C. 15
D. 8+g
Version 2
If f+g = 8, then f+g+h =
A. 9
B. 12
C. 15
D. 16
E. 8+h
Discriminate correct cognitive rules
What is the area of this trapezium?
a
h
b
a
b
h
b
a
A = (a + b) x h
h
b
a
2A = (a + b) x h
A = h x (a + b)
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
9:20 and arrives in Oldtown at
10:55. The second flight from
Newtown leaves at 2:15. 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
9:05 and arrives in Oldtown at
10:55. The second flight from
Newtown leaves at 2:15. At what
time does the second flight arrive
in Oldtown? Show your work.
Cognitive Rules
Correct
Incorrect
Responses
A
B
C
D
Over- and under-generalization
In which of the following diagrams, is one quarter of the
area shaded?
A
B
C
D
Diagnostic item: medians
What is the median for the following data set?
38
74
A.
B.
C.
D.
E.
F.
G.
22
44
96
22
22
38 and 44
41
46
70
77
This data set has no median
19
53
Diagnostic item: means
What can you say about the means of the following two data sets?
Set 1:
Set 2:
10
10
12
12
13
13
15
15
0
A. The two sets have the same mean.
B. The two sets have different means.
C. It depends on whether you choose to count the zero.
Diagnostic item: diagonals
Which of the shapes below contains a dotted line that is
also a diagonal?
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
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)
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
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