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 work-

place 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 skillSo 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)

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)

Formative assessment

Types of formative assessmentLong-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 assessmentKey processesEstablishing where the learners are in their learningEstablishing where they are goingWorking out how to get there

ParticipantsTeachersPeersLearners

Where the learner is going

Where the learner is How to get there

TeacherClarify and share

learning intentions

Engineering effective

discussions, tasks and activities that elicit evidence of

learning

Providing feedback that moves

learners forward

PeerUnderstand and share learning

intentions

Activating students as learningresources for one another

LearnerUnderstand

learning intentions

Activating students as ownersof their own learning

Aspects of formative assessment

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 ideaUse 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? a) 16

, b) 23

, c) 13

, d) 12

.

Success rate 88%

Which fraction is the largest?

Success rate 46%; 39% chose (b)

a) 45

, b) 34

, c) 58

, d) 7

10.

[Vinner, PME conference, Lahti, Finland, 1997]

Draw an upside-down triangle…

Misconceptions

3a = 24

a + b = 16

Questioning in maths: discussionLook 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 evidenceKey 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: diagnosisIn which of these right-angled triangles is a2 + b2 = c2 ?

A a

c

b

C b

c

a

E c

b

a

B a

b

c

D b

a

c

F c

a

b

Lines of symmetry

AB

C

D E F

Constructing hinge-point questions

Discriminate incorrect cognitive rulesVersion 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

What is the area of this trapezium?

Discriminate correct cognitive rules

b

a

h

b

b

a

a

h2A = (a + b) x h

A = (a + b) x h

b a

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 Responses

A

B

C

D

Correct

Incorrect

Over- and under-generalization

B C DA

In which of the following diagrams, is one quarter of the area shaded?

Diagnostic item: mediansWhat is the median for the following data set?

38 74 22 44 96 22 19 53

A. 22B. 38 and 44C. 41D. 46E. 70F. 77G. This data set has no median

Diagnostic item: meansWhat 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

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: diagonalsWhich of the shapes below contains a dotted line that is also a diagonal?

Hinge-point questionsA 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 requirementsEvery 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

optionEach incorrect option response (distractor) should interpret a single

cognitive ruleCorrect option responses (keys) should interpret a single cognitive rule

Practical techniques: feedbackKey idea: feedback should

cause thinking provide guidance on how to improve

Comment-only gradingFocused gradingExplicit reference to rubricsSuggestions on how to improve

Not giving complete solutionsRe-timing assessment

(eg three-quarters-of-the-way-through-a-unit test)

Practical techniques: sharing learning intentionsExplaining 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 resourcesStudents assessing their own/peers’ work with rubricswith exemplars“two stars and a wish”

Training students to pose questions/identifying group weaknesses

Self-assessment of understandingTraffic lightsRed/green discs

End-of-lesson students’ review