1. Formative assessment in mathematics: opportunities and challenges
Dylan Wiliam
Seminar at Teachers College, Columbia University
October 2013

2. A research agenda for formative assessment
Definitional issues
Domain-specificity issues
Effectiveness issues
Communication issues
Implementation issues
Adoption issues

3. Definitional issues

4. The evidence base for formative assessment
Fuchs & Fuchs (1986)
Natriello (1987)
Crooks (1988)
Bangert-Drowns, et al. (1991)
Dempster (1991, 1992)
Elshout-Mohr (1994)
Kluger & DeNisi (1996)
Black & Wiliam (1998)
Nyquist (2003)
Brookhart (2004)
Allal & Lopez (2005)
Köller (2005)
Brookhart (2007)
Wiliam (2007)
Hattie & Timperley (2007)
Shute (2008)

5. Definitions of formative assessment
We use the general term assessment to refer to all those activities undertaken by teachers—and by their students in assessing themselves—that provide information to be used as feedback to modify teaching and learning activities. Such assessment becomes formative assessment when the evidence is actually used to adapt the teaching to meet student needs” (Black & Wiliam, 1998 p. 140)
“the process used by teachers and students to recognise and respond to student learning in order to enhance that learning, during the learning” (Cowie & Bell, 1999 p. 32)
“assessment carried out during the instructional process for the purpose of improving teaching or learning” (Shepard et al., 2005 p. 275)

6. “Formative assessment refers to frequent, interactive assessments of students’ progress and understanding to identify learning needs and adjust teaching appropriately” (Looney, 2005, p. 21) “A formative assessment is a tool that teachers use to measure student grasp of specific topics and skills they are teaching. It’s a ‘midstream’ tool to identify specific student misconceptions and mistakes while the material is being taught” (Kahl, 2005 p. 11)

7. “Assessment for Learning is the process of seeking and interpreting evidence for use by learners and their teachers to decide where the learners are in their learning, where they need to go and how best to get there” (Assessment Reform Group, pp. 2-3)
“Assessment for learning is any assessment for which the first priority in its design and practice is to serve the purpose of promoting students’ 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 that teachers and their students can use as feedback in assessing themselves and one another and in modifying 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, Harrison, Lee, Marshall & Wiliam, 2004 p. 10)

8. Theoretical questions
Need for clear definitions
So that research outcomes are commensurable
Theorization and definition
Possible variables
Category (instruments, outcomes, functions)
Beneficiaries (teachers, learners)
Timescale (months, weeks, days, hours, minutes)
Consequences (outcomes, instruction, decisions)
Theory of action (what gets formed?)

9. Formative assessment: a new definition
“An assessment functions formatively to the extent that evidence about student achievement elicited by the assessment is interpreted and used, by teachers, learners, or their peers, to make decisions about the next steps in instruction that are likely to be better, or better founded, than the decisions that would have been taken in the absence of that evidence.”

10. Unpacking formative assessment
Where the learner is going
Where the learner is
How to get there
Engineering effective discussions, tasks, and activities that elicit evidence of learning
Providing feedback that moves learners forward
Teacher
Clarifying, sharing and understanding learning intentions
Peer
Activating students as learning
resources for one another
Activating students as owners of their own learning
Learner

11. Definitional issues: potential research
How can formative assessment be defined and what are the consequences of different definitions, for psychometrics, for communication, and for adoption?

12. Domain specificity issues

13. Pedagogy and didactics
Some aspects of formative assessment are generic
Some aspects of formative assessment are domain-specific
There is a continuing debate about what aspects of formative assessment are generic (pedagogy) and which are domain-specific (didactics)

14. Clarifying, sharing and understanding learning intentions

15. A standard middle school math problem…
Two farmers have adjoining fields with a common boundary that is not straight.
This is inconvenient for plowing.
How can they divide the two fields so that the boundary is straight, but the two fields have the same area as they had before?

18. How many rectangles?

19. Engineering effective discussions, activities, and classroom tasks that elicit evidence of learning

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

21. Diagnostic item: medians
What is the median for the following data set? 22
38 and 44 41 46 70 77
This data set has no median Q
Key: C
(Mis)conceptions: median has to be a number in the data set, cannot calculate median with an even number of elements
(B) And (F) would not be included – stick out as longer – neither correct

22. Diagnostic item: means
What can you say about the means of the following two data sets?
Set 1:
Set 2:
The two sets have the same mean.
The two sets have different means.
It depends on whether you choose to count the zero. Q
Key: B
(Mis)conception – added a zero to data set does not impact the mean – A and C variations of each other

23. Providing feedback that moves learners forward

24. Getting feedback right is hard
Response type
Feedback indicates performance…
falls short of goal
exceeds goal
Change behavior
Increase effort
Exert less effort
Change goal
Reduce aspiration
Increase aspiration
Abandon goal
Decide goal is too hard
Decide goal is too easy
Reject feedback
Feedback is ignored

25. Activating students: as learning resources for one another as owners of their own learning

26. +/–/interesting: responses for “+”
I got that ball-park estimates are supposed to be simple
I know that you have to look at it and say “OK”
I know that when I am adding the number I end up with must be bigger than the one I started at
I get most of the problems
It was easy for me because on the first one it says 328 so I took the 2 and made it a 12
I know that we would have to regroup
I know how to do plus and minus because we have been doing it for a long time
I get it when you cross out a number and make it a new one
I know that when you can’t – from both colomes you go to the third colome and take that from it
I know that when my answer is right the ball park estimate is close to it

27. +/–/interesting: responses for “–”
I am still a tiny bit confused about subtraction regrouping
I am a little bit confused about ball park estimates
I get confused because sometimes I don’t get the problem
I am confused when you subtract really big numbers like 1,000 something
I’m still a little bit confused about regrouping
Minus is confusing when you have to regroup twice
Minus is a little bit hard when you have to regroup
I don’t understand when you borrow which colome you borrow from when both are 0
I am still confused about showing what I did to solve the problem
I am a little confused about when you need to subtract

28. +/–/interesting: responses for “interesting”
Carrying the number over to the next number
It’s interesting how some people go to the nearest hundred while some go to the nearest ten
It’s interesting how some have to regroup twice
It’s pretty interesting about how you have to work really hard
I am interested in borrowing because I didn’t just get it yet. I want to really get to know it
I find it weird that you could just keep going from colome to colome when you need to borrow
On the ball park estimate it is easy but sometimes hard
I really think that regrouping is pretty amazing
It is cool how addition and subtraction regrouping is just moving numbers and you could get it right easily

29. Domain-specificity issues: potential research
How much domain-specific knowledge does a teacher need in order to be able to implement high-quality formative assessment routines consistently?
Can domain-specific formative assessment tools be independent of a particular curriculum?

30. The effectiveness issue

31. Effects of formative assessment
Standardized effect size: differences in means, measured in population standard deviations
Source
Effect size
Kluger & DeNisi (1996)
0.41
Black &Wiliam (1998)
0.4 to 0.7
Wiliam et al., (2004)
0.32
Hattie & Timperley (2007)
0.96
Shute (2008)
0.4 to 0.8

32. Understanding meta-analysis: “I think you’ll find it’s a bit more complicated than that” (Goldacre, 2008)

33. Understanding meta-analysis
A technique for aggregating results from different studies by converting empirical results to a common measure (usually effect size)
Standardized effect size is defined as:
Problems with meta-analysis
The “file drawer” problem
Variation in population variability
Selection of studies
Sensitivity of outcome measures

34. The “file drawer” problem

35. The importance of statistical power
The statistical power of an experiment is the probability that the experiment will yield an effect that is large enough to be statistically significant.
In single-level designs, power depends on
significance level set
magnitude of effect
size of experiment
The power of most social studies experiments is low
Psychology: 0.4 (Sedlmeier & Gigerenzer, 1989)
Neuroscience: 0.2 (Burton et al., 2013)
Education: 0.4
Only lucky experiments get published…

36. Variation in variability

37. Annual growth in achievement, by age
A 50% increase in the rate of learning for six-year-olds is equivalent to an effect size of 0.76
A 50% increase in the rate of learning for 15-year-olds is equivalent to an effect size of 0.1
Bloom, Hill, Black, and Lipsey (2008)

38. Variation in variability
Studies with younger children will produce larger effect size estimates
Studies with restricted populations (e.g., children with special needs, gifted students) will produce larger effect size estimates

39. Selection of studies

40. Feedback in STEM subjects
Review of 9000 papers on feedback in mathematics, science and technology
Only 238 papers retained
Background papers 24
Descriptive papers 79
Qualitative papers 24
Quantitative papers 111
Mathematics 60
Science 35
Technology 16
Ruiz-Primo and Li (2013)

41. Classification of feedback studies
Who provided the feedback (teacher, peer, self, or technology-based)?
How was the feedback delivered (individual, small group, or whole class)?
What was the role of the student in the feedback (provider or receiver)?
What was the focus of the feedback (e.g., product, process, self-regulation for cognitive feedback; or goal orientation, self-efficacy for affective feedback)
On what was the feedback based (student product or process)?
What type of feedback was provided (evaluative, descriptive, or holistic)?
How was feedback provided or presented (written, video, oral, or video)?
What was the referent of feedback (self, others, or mastery criteria)?
How, and how often was feedback given in the study (one time or multiple times; with or without pedagogical use)?

42. Main findings Characteristic of studies included Maths Science
Feedback treatment is a single event lasting minutes
85%
72%
Reliability of outcome measures
39%
63%
Validity of outcome measures
24% 3%
Dealing only or mainly with declarative knowledge
12%
36%
Schematic knowledge (e.g., knowing why) 9% 0%
Multiple feedback events in a week
14%
17%

43. Sensitivity to instruction

44. Sensitivity of outcome measures
Distance of assessment from the curriculum
Immediate
e.g., science journals, notebooks, and classroom tests
Close
e.g., where an immediate assessment asked about number of pendulum swings in 15 seconds, a close assessment asks about the time taken for 10 swings
Proximal
e.g., if an immediate assessment asked students to construct boats out of paper cups, the proximal assessment would ask for an explanation of what makes bottles float
Distal
e.g., where the assessment task is sampled from a different domain and where the problem, procedures, materials and measurement methods differed from those used in the original activities
Remote
standardized national achievement tests.
Ruiz-Primo, Shavelson, Hamilton, and Klein (2002)

45. Impact of sensitivity to instruction
Effect size
Close
Proximal

46. Effectiveness issues: potential research
Under what kind of conditions does the implementation of formative assessment practices in classrooms lead to student improvement?
What kinds of increases in the rate of student learning are possible?

47. Communication issues

48. Dissemination models Gas-pump attendant FedEx IKEA Sherpa Gardener
PhD supervisor

49. So much for the easy bit…
Theorization
Ideas
Products
Evidence of impact
Advocacy

50. Communication issues: potential research
How can the vision of effective formative assessment practice be communicated to teachers?

51. Implementation issues

52. Hand hygiene in hospitals
Study
Focus
Compliance rate
Preston, Larson, & Stamm (1981)
Open ward
16%
ICU
30%
Albert & Condie (1981)
28% to 41%
Larson (1983)
All wards
45%
Donowitz (1987)
Pediatric ICU
Graham (1990)
32%
Dubbert (1990)
81%
Pettinger & Nettleman (1991)
Surgical ICU
51%
Larson, et al. (1992)
Neonatal ICU
29%
Doebbeling, et al. (1992)
40%
Zimakoff, et al. (1992)
Meengs, et al. (1994)
ER (Casualty)
Pittet, Mourouga, & Perneger (1999)
48%
36%
Pittet (2001)

53. Implementation issues
What are the practical obstacles to the introduction of formative assessment practices, and how can they be overcome?
What kinds of tools and supports can be provided for teachers, and what needs to be developed locally?

54. Adoption issues

55. The story so far…
Review of research on formative assessment
Face-to-face implementations with groups of teachers
Attempts to produce faithful implementations at scale
Creating the conditions for implementations at scale

56. Adoption issues: potential research
How can we support leaders in prioritizing changes that make the most difference to student outcomes?

57. Comments? Questions?