Improving the learning of numeracy through formative assessment

1. Improving the learning of numeracy through formative assessment Dylan Wiliam National Numeracy Conference Edinburgh, March 2009 www.dylanwiliam.net

2. 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

3. …but what is learned matters too… Autor, Levy & Murnane, 2003

4. 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)

5. 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)

6. 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

7. 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

8. Aspects of formative assessment

9. 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)

10. …and one big idea • Use evidence about learning to adapt teaching and learning to meet student needs

11. 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

12. Eliciting evidence of student achievement

13. 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]

14. Draw an upside-down triangle…

15. Misconceptions 3a = 24 a + b = 16

16. 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

17. 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

18. b c A B a a c b a c C D b b c a a b E F c c b a Questioning in maths: diagnosis • In which of these right-angled triangles is a2 + b2 = c2 ?

20. Lines of symmetry C A B F D E

21. Constructing hinge-point questions

22. Version 1 (Hart, 1981) If e+f = 8, then e+f+g =  9 12 15 8+g Version 2 If f+g = 8, then f+g+h =  9 12 15 16 8+h Discriminate incorrect cognitive rules

23. Discriminate correct cognitive rules What is the area of this trapezium? a h b

24. b a h a b h a b • 2A = (a + b) x h • A = (a + b) x h • A = h x (a + b)

25. 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. Discriminate between incorrect and correct cognitive rules

26. Correct Incorrect

27. Over- and under-generalization In which of the following diagrams, is one quarter of the area shaded? A B C D

28. 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

29. 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.

30. Diagnostic item: diagonals Which of the shapes below contains a dotted line that is also a diagonal?

31. 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

32. 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)

33. 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

34. 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