Roles of assessment in learning in statistics and mathematics[评估在学习统计学和数学的作用](-39)

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1、*,*,,,,,,,,,,Click to edit Master title style,,Click to edit Master text styles,,Second level,,Third level,,Fourth level,,Fifth level,,,Roles of assessment in learning in statistics and mathematics,,Helen MacGillivray,,School of Mathematical Sciences, QUT,,Director, QUT Maths Access Centre,,Australi

2、an Carrick Senior Fellow, 2007,,President-elect, IASE (International Association for Statistics Education),,Visiting Fellow, UK CETL in university-wide maths & stats support,Queensland University of Technology,,1,Juggling hats?,Assessment also requires balancing,“Students learn only for assessment!”

3、,,Naturally,,Students think what is assessed must be what is of value,,If we value learning, assessing must be for learning,,2,,,,Engagement of students?,,3,,This presentation,Overall comments on assessment,,General HE & stats educ lit,,Criteria, standards, objectives,,,4 “stories” illustrating as

4、pects of assessment in different contexts,,Intro data analysis,:,service & core, teaching data investigation,,Components of assessment balanced over different objectives,,Intro prob & dist’nal modelling,:,core unpack, analyse, extend, link with data,,Assessment for learning problem-solving,,2,nd,ye

5、ar linear algebra,:,core looks towards industry problems, applied research, computational maths,,,Balance of continuous & tests, theory, applications & computing,,2,nd,year engineering unit,:,½ data analysis; ¼ dist’ns; ¼ comp maths,,Balancing components, workload, objectives,,,Comments throughout

6、on: alignment with objectives, balance, integrated assessment & learning packages, group work, collaborative work, plagiarism,,Like mathematical proofs, end product doesn’t reflect evolution to it,,4,,From general HE literature …..,If learning really matters most, then our assessment practices shoul

7、d help students develop .. skills, dispositions, and knowledge…..,,Angelo, T., 1999, Doing assessment as if learning matters most. Bulletin of the American Association for Higher Education.,,Students study more effectively when they know what they are working towards….. Students value assessment tas

8、ks they perceive to be ‘real’,,,James, R., McInnis, C., Devlin, M., 2002, Assessing learning in Australian universities. Melbourne: The University of Melbourne Centre for the Study of Higher Education,Objectives of learning & assessment must be clear,,5,,…..reflected in statistics education literatu

9、re,Care & indepth consideration of objectives, goals, contexts, content. Hogg (1991), Vere-Jones (1995), Moore (1997),,Emphasis on data, statistical literacy & reasoning Cobb (1999), delMas (2002), Garfield et al (2002)….,,In a survey of US statistics educators,,of all areas of statistics education,

10、 assessment practices have undergone the least reform,Garfield et al (2002),,Calls for,statistics educators to assess what they value,(Chance, 2002),,,Explicit aligning of assessment with objectives,features in both the general higher education (James et al, 2002) and statistics education literature

11、 (Gal and Garfield, 1998).,,6,,Aligning of assessment with objectives,This needs identification of,,Purpose of the learning,,What the cohort are bringing to their learning,,How the students manage their learning,,The students’ perception of its roles for them,Like mathematical proofs (& this present

12、ation!), an iterative process,,Components of assessment objectives to produce an assessment, teaching & learning package that is,,integrated, balanced, developmental, purposeful, with structured facilitation of student learning across the student diversity,For wide range of backgrounds, prog

13、rams, motivations, study skills……,,7,,Recent pressures for staff in tertiary assessment,Seeking balances & paths amongst:,,Formative, summative, flexible, continuous, rich, authentic,,Generic graduate capabilities,,Work-integrated learning,,Criteria & standards referenced assessment,,HE fads, genera

14、lisations & arbitrary rules,,,Plus challenges of:,,Avoiding over-assessment,,Politics of pass rates & attrition & standards,,Increasing diversity of student cohorts,,Instant gratification generation,,Workloads – students & staff,,8,,Criteria & standards referenced assessment,in criteria & standards-

15、referenced assessment it is the,configuration,,(Kaplan, 1964),,or pattern of performance,’,Sadler (1987),which is used for ranking or reporting a level of achievement ….,,,Good packages have inbuilt configuration or pattern of performance,,Configuration comes from,,construct of formative & summative

16、 assessment aligned with objectives & learning across cohort,,construct of timing, types & weights of tasks,,Exemplars help to identify characteristics of each component of assessment, with verbal descriptors for salient criteria,The term ‘criteria-referenced assessment (CRA)’ is often interpreted a

17、s meaning verbal descriptors of standards,,Not so,,9,,Statistical Data Analysis 1,,science, maths, surveying, educ….: approx 500 pa,,Theme is basic statistical data concepts and tools & using them in real data investigations.,,Separate phases – tools & building blocks of procedures, concepts and pro

18、cedural skills,,Synthesis – choosing, using, interpreting, combining in whole data investigations,,Structure, examples & learning experience built around real data investigations from first ideas through to report,,,Planning, collecting, handling, graphing, summarising, commenting on …. data,,Catego

19、rical data – chisq tests; principles of testing hypotheses; p-values,,Revision of normal; standard errors; confidence intervals and tests for 1 & 2 means, proportions, variances. Tolerance intervals,,ANOVA & exp’tal design (via software): interaction (2-way), multiple comparisons, checking assumptio

20、ns. Unbalanced data,,Multiple & polynomial regression (via software): interpretation, diagnostics, re-fitting,,10,,Learning & assessment package,Computer-based practicals on datasets from past student projects,,Worksheets with full solutions,,Fortnightly quizzes of fill-in-gaps out Sunday, in by Fr

21、iday: best 5 out of 6 contribute 10%,,Workfolder,containing their ongoing work on the worksheets and their marked (collected) quizzes,:,3%,,Whole semester group project in planning, collecting, analysing & reporting data investigation in context of group choice: 20%,,In-semester

22、test (similar to quizzes 1-4): 10%,,End of semester exam (similar to quizzes 1-6, more on 5, 6): 57% *,Quizzes, test, exam: exemplars + exemplar processes,Quizzes exam summative Assistance given for quizzes – most important aspect is DOING them,*For a few years also an optional essay

23、 on how statistics revolutionised science in the 20th century: 10% if improved overall result. Dropped because (i) almost never improved result (ii) attracted students who could least afford the time. Objective not worth student & staff effort,,,11,,Research on numeracy/maths & statistical reasoning

24、 of cohort,Numeracy/maths on entry: highly diverse –,,see Wilson & MacGillivray,,Counting on the basics: mathematical skills amongst tertiary entrants,, (2007) IJMest 38(1), 19-41,,,General statistical reasoning on entry,: Wilson and MacGillivray,,Numeracy and statistical reasoning on entering unive

25、rsity,, 7th International Conference on Teaching Statistics (2006),,,Numeracy & level of maths stood out as most important predictors of general statistical reasoning,,Fish question greatest discriminator between core & advanced maths preparation,,A farmer wants to know how many fish are in his dam.

26、 He took out 200 fish and tagged each of them. He put the tagged fish back in the dam and let them get mixed with the others. On the second day, he took out 250 fish in a random manner, and found that 25 of them were tagged. Estimate how many fish are in the dam.,,12,,Own choice group project,Te

27、aches & assesses data investigation & synthesis of procedure choice & interpretation,,Other assessment can focus on operational knowledge & skills - tools & building blocks of procedures, concepts and procedural skills,,Group because task needs a group,,Guidelines & descriptors of 3 criteria with st

28、andards given,(,MacGillivray,,Criteria, standards and assessment in statistical education,, Proceedings International Statistical Institute, 55th Session, 2005),,,Feedback on proposal + ongoing help; they propose – we advise,,Use of past datasets in class demonstrations and practicals,,Access to pas

29、t projects, including assessments, and model reports,,Each group receives a written assessment report with comments & marks for the 3 criteria,Criteria, standards & exemplars. Formative & summative,,13,,Own choice group project,Criteria,,(i) Identifying context and issues; planning and collecting o

30、f data; quality of data and discussion of context/problems,,(ii) handling, processing, preparing exploring and commenting on features of the data,,(iii) using statistical tools for statistical analysis and interpretation of the data in the context/issues,,Group problems?,,They form groups, we help

31、as necessary using pracs,,Dropouts after week 8 can cause some problems but solvable,,Plagiarism?,,Projects retained & designated “published”. To copy = 0,,Contributions balance? Not a problem with right emphasis on project as learning experience,,Almost never in (i),,Seldom in (ii); allocation of

32、tasks helps in (ii) ,,Leaders in (iii) tend to learn more, need less revision for exam & do better…..and learn by helping others,,14,,Just a few recent titles,Still time for play?,,How long can you suck?,,Talking your ear off,,Gym junkies,,Gifted hands,,Ah McCain you’ve done it again,,An analysis of

33、 alcohol induced loquaciousness,,Investigation into student internet usage,,Maritime museum usage,,pH of river,,Optical illusions,Voluntary student unionism: to join or not to join,,We love muffins,,Human curiosity,,Holding breath,,Usage of the 15 min workstations in the GP library,,Strength of our

34、athletes,,Where are all the single people?,,Seed germination,,The big news about breakfast,,Music and the people,,15,,Graphs 2006 sem 1,Low +ve relationship,,Moderate +ve relationship,,High +ve relationship,,,16,,Statistical modelling 1,All maths programs, maths electives, maths educ: approx 120,

35、Builds,skills and foundations in concepts & thinking in,,Intro probability, conditional arguments, distributional and stochastic modelling for,,applications in a wide range of areas, from communication systems and networks to traffic to law to biology to financial analysis,,Analyzes,prior understand

36、ing/ misunderstandings,,Links with data,, observation and simulation,,Links with and,consolidates 1,st,yr calculus & algebra skills,,,Whole approach is,,problem-solving & modelling,Statistics education “reform”: “more data & concepts, less theory, fewer recipes” (Cobb, 1992).,,It’s time to a

37、pply this in teaching probability & distributions,,17,,Formative components of assessment,Initial general probability reasoning questionnaire (PRQ) to seed thought & discussion (introduced 2004),,Class activities, simulations, selected computer modules, worksheets with unlimited help,,Each topic has

38、 preliminary experiences or exercises or discussion points (development completed 2005),,prior knowledge, foundations & seeds,,perceive, unpack, analyse, extend,“Using what we already knew to learn other stuff was really good and helped us learn other stuff”,A student definition of constructivism pe

39、rhaps?,,,18,,Formative/summative & summative components: all oriented to problem-solving,Four assignments based on class activities, examples and worksheets, with problems in authentic contexts,20% before 2006; 16% in 2006,,(Assistance available. Collaboration – yes; straight copying rare),,Group p

40、roject. 2 everyday processes that could be Poisson (free choice); data collected; Poisson-ness investigated by combination of tests and graphs,10%.,,,End of semester exam. Problem-solving based on activities, worksheets, assignments; ranging from simple to slightly complex in life-related authentic

41、contexts. Students design & bring in own summaries (4 A4 pages),70% before 2006; 66% in 2006,,19,,Some examples from group projects,Australian Rules (football) grand final,,Time spent on phone,,Pedestrian traffic in mall,,Time to be served icecream,,Occurrences of “Harry” per page in a Harry Potter

42、book,,Traffic on a pedestrian bridge,,Distribution of leaves on tiles,,Behaviour of ants,,Arrivals & service at library,,Distances between coffee shops,,Service in “fast” supermarket checkout,,Time between customers wearing high heels.,,Time between changes of a baby’s nappy,,20,,New assessment comp

43、onent in a problem-solving environment,Problem-solving environment Gal et al (1997),,“an emotionally and cognitively supportive atmosphere where students feel safe to explore, comfortable with temporary confusion, belief in their ability and motivation to navigate stages,.”,,,Formative assessment &

44、assignments designed for managed optimal learning but students needed greater persuasion to learn through trying (’ave a go ….),,,Some topics identified as most needful of immediate involvement of students in active problem-tackling in an environment that maximises engagement & learning,,,21,,Tutori

45、al group exercises, 2006,4 practicals structured for immediate “hands-on” learning.,,Groups allocated; different groups for each practical.,,No compulsion to complete exercise; credit for participation.,,Assistance available as required.,,Full collaborative work required, with groups ensuring that e

46、xplanations were shared within the group.,,Participation in each of these four special tutorials contributed 2% to the overall assessment.,,22,,Evaluation of new component,Qualitative,,Tutors and students voted experiment success.,,The tutorials were buzzing, and early departures were practically no

47、n-existent.,,Student opinion was that four was the ideal number.,,Other tutorials benefited significantly.,,Quantitative,,Assignments provide exemplars for exams,,Data support that assignments most important in predicting exam (as desired!),,In 2005, assignments score depended on group project & PRQ

48、 score,,In 2006, assignments score depended,only,on tut group exercises score for participation – strategy worked!,,23,,2,nd,year linear algebra unit,,maths+others e.g. maths educ, physics, eng – approx 80-90,Mixed student cohorts with often bimodal results,,Balance of theory and practice?,,Some cha

49、nges in continuous assessment – did they help or impede student learning?,,Challenge of student engagement,,Interface of first and second level courses,,first level courses respond to school/tertiary interface,,first year units – which are best predictors?,,,The examples and learning experiences in

50、unit are motivated by higher level needs in mathematics generally & particularly computational mathematics, & by applications based on experience with industry problems.,,24,,Assessment package, 2003 & 2005,2003,,21% continuous assessment,,3 Maple group assignments totalling 21%,,mid-semester exam 1

51、5%,,final examination 64%.,,,Lecturer’s observations plus feedback:,,Maple group assignments too heavy for 7%,,Students needed more structured help with their learning,,,2005,,40% continuous assessment,,2 Maple group assignments totalling 24%,,,3 “homework” quizzes totalling 16%,,final examination 6

52、0%.,,Similar in style, format and level to 2003,,25,,Analysis of data: assessment components,For both continuous assessment programs, a test-type component and a Maple group assignment component combined as best predictors of exam,,,Exam has applications but no actual Maple use, providing support of

53、 the claims in the literature, that both theory and practice contribute to overall learning and understanding in linear algebra,,,Reassurance that the change in the continuous assessment program is not detrimental to performance, and appears to assist in learning,,Lecturer’s concerns about high mark

54、s in the 2005 continuous assessment program are reflected by only 25% of the variation in exam marks being explained,,but the challenge of how to grade the continuous assessment can be tackled with confidence in the program’s facilitation of student learning across the theory and practice components

55、 of the unit,,,26,,Analysis of data: 1,st,year predictors,Formal prerequisites,,1st level calculus unit and,,1,st,level introductory linear systems and analysis unit, with the brief synopsis,,linear systems and matrices; vector algebra; coordinate systems; introduction to abstract algebraic systems;

56、 complex numbers; first and second order differential equations.,,,,Entry to 1,st,yr units via advanced mathematics in senior school or equivalent 1,st,yr unit.,,Alternative prerequisites 1,st,yr engineering maths,,Other compulsory 1,st,year units for maths degree are an introductory unit in computa

57、tional mathematics, Statistical Data Analysis 1 & Statistical Modelling 1.,,27,,Analysis of data: 1,st,year predictors,,Data are complex because of different pathways. But best single predictor amongst 1,st,yr units, of performance in 2,nd,yr linear algebra in 2003 & 2005* was Statistical Modelling

58、1.,,Synthesis of techniques and problem-tackling with new contexts, theory and applications appears to be the common thread linking these unlikely partners,,* Note: changes in the 1,st,year units since then have probably changed this,,,28,,2,nd,year engineering maths unit,,all engineering programs -

59、 approx 450-520,Unit “new” in 2007 but composed of sections common across previous engineering units,,,Content in 2007:,,Statistical data investigations & analysis (1/2 unit),,As in Statistical data analysis 1; as given in all eng programs since 1994,,Introductory numerical analysis (1/4 unit),,Intr

60、oduction to random variables & distributional modelling,,,including linear combinations of normals, goodness-of-fit & introduction to reliability,,(1/4 unit),,29,,Level of unit,First year work in Science and Maths,,Statistical data analysis 1,,Numerical component extract from 1,st,yr unit,,Intro rv’

61、s & distributions extract from Statistical Modelling 1,,But,,It’s different,,It’s not straight calculus/algebra & any of these that are needed must be at fingertips in new contexts because of amount of material,,The statistics (both parts) full of new concepts & new ways of thinking,,30,,‘tis always

62、 thus in Australian eng courses,Because,,of the philosophy of Australian eng courses (whether new, old or middling),,engineering needs the most technical maths faster than any other discipline,,AND,,engineering needs the most maths generic skills faster than any other discipline,,,Advantages of stat

63、s being in 2,nd,year eng are….,,they’re 2,nd,years in some ways & they have better maths thinking than most other disciplines,,they start reflecting on their studies (I’ve been listening to & observing 2,nd,(or 3,rd,) year eng students for over 30 years),,Disadvantages of stats being 2,nd,or 3,rd,ye

64、ar eng are….,,they think they’re 2,nd,years in every way,,it’s stats & they’re eng students,,many tend to think it’s less important than other units,,31,,Learning & assessment package,focus is on learning by doing,Computer-based practicals on datasets from past student projects,,Weeks 1-6 on statist

65、ical data analysis,,Worksheets with full solutions,,For all sections: 15 worksheets in total,,,Stats (¾): five quizzes of fill-in-gaps & short response type 14%,,Whole semester group project in planning, collecting, analysing & reporting data investigation in context of group choice 20%,,As

66、 for all eng since 1995 & as in Stat data Analysis 1,,Numerical analysis (¼ ): assignment 6%,,End of semester exam (based on quizzes & w’sheets): 60%,,Ensures overall coverage correctly proportioned,Quizzes, assignment, exam: exemplars + exemplar processes Project: criteria, standards & exemplars,,,Quizzes, project & assignment formative & summative,Assistance given for quizzes – most important aspect is DOING them,,Exam summative,,32,,Assessment data: stats quizzes,Stats quizzes