E Assessment in Mathematics

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Published on September 30, 2008

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E-assessment in Mathematics : E-assessment in Mathematics Cliff Beevers Heriot-Watt University Assessment in MSOR : Assessment in MSOR CAA reliable, robust, easy to use formative and summative display /input of maths equivalently right answers. Match assessment to Learning Outcomes for knowledge, basic skills, understanding and problem solving. Heavy demands for marking of coursework Formative Assessment : Formative Assessment Assessment is the measurement of learning. What should be measured? Inform on strengths/weaknesses Motivate - review and consolidate Enable demonstration of knowledge, skills and understanding Prepare students for summative exams Maximise performance in such exams. Early Milestones : Early Milestones 1985 start of CALM = CAL in Mathematics 1992 UK Mathwise assessment centre 1993 Partnership Teaching prize for CALM 1995 Bank of Scotland prize for HE 1997 Interactive PastPapers using Authorware The CALM Formula : The CALM Formula Summary of theory from lectures; Worked examples to illustrate method; Motivating examples to bring subject to life; and Test section for: students to assess progress teachers to monitor class. Test Section : Test Section Prompt for mathematical expressions as answers Use of random parameters Delivery in a variety of feedback modes Student choice of test difficulty Crude but effective reporting tool Feedback messages sent to students The Mathwise Years : The Mathwise Years Fifty modules created Each module covered 5/6 learning units Self-testing scatted throughout the materials More formal testing at module end Grading experiment performed in 1993 Student issues Included copying from screens, mathematical input & partial credit. Tests used to grade at some universities Role of the Computer in Assessment : Role of the Computer in Assessment Diagnostic tests - identify strengths Self-test - give fast feedback Monitor - continuous assessment Summative assessment - grades that count. Exploitation? : Exploitation? Interactive Assessment at Heriot-Watt Diagnostic - MCQ test in year 1, Service Maths Monitor - CALM Calculus tests each week (www.calm.hw.ac.uk) Self-test - SCHOLAR in Scotland Grading - up to 40% assessed by computer from 1994. Early Research : Early Research Report on first grading experiment C Beevers et al, (1995) Comp. in Educ. 25, 123. About Mathwise R Harding, D Quinney, (1996) Active Learn. 4, 53. Partial credit issues C Beevers et al, (1999) ALT-J vol. 7, 26. Interactive PastPapers : Interactive PastPapers Randoms in questions Input tool Multiple entry - vectors, factors, coordinates Special answers: scientific notation & sig figs Partial credit, key parts and differential marks Multiple saved tests On-line context sensitive help Flexible methods for question selection Demo at www.calm.hw.ac.uk Recent Milestones : Recent Milestones 1998 CUE migrates to the WEB with Delphi 2000 SCROLLA established 2001 CUE provides formative testing in SCHOLAR 2002 PASS-IT project on summative assessment What is CUE? : What is CUE? CUE developed as a collaboration between C = CALM U = Univ. of Cambridge Local Exam Syndicate E = EQL in Livingston, West Lothian The Assessment System : The Assessment System Uses content MathML and XML Has an editor for authoring/amending questions and tests  Includes randoms Possesses the educational features of key parts and steps for partial credit Has an increasing number of part types The Assessment System : The Assessment System Has Integrated multimedia Provides for test delivery in a variety of modes to cater for wide spectrum of students Is JAWS accessible Results gathered incrementally Results recorded in database for reporting Part Types : Part Types Information Only Numerical answers Algebraic answers – very extensive MCQ / MRQ – variety of marking schemes Multiple Hotspots Word and Phrase match Essay types Steps for partial credit Includes MM elements: flash/sound/video Tangent - steps : Tangent - steps Find the equation of the tangent to the curve f defined as f(x)=x3+4 at the point where x=1. KP1. Equation of tangent is y=? 3x+2 Steps 3 marks possible with STEPS available Tangent - steps : Find the equation of the tangent to the curve f defined as f(x)=x3+4 at the point where x=1. 1.1 f’(x)=? 3x2 1.2 Tangent equation has y-b=m(x-a), m is called? gradient/slope 1.3 Value of m for tangent at x=1 is ? 3 1.4 Value of b when x=a=1 is ? 5 KP1. Equation of tangent is y=? 3x+2 3 marks but partial credit is possible here. This allows more students to progress. Tangent - steps The SCHOLAR Programme : The SCHOLAR Programme In 1999 HW decided to create distributed learning materials for: Advanced Higher and Higher courses at school; Higher National courses in FE Colleges; and first year courses at university. in Chemistry, Computing, Mathematics, Physics and Biology. How was it done? And with what effect? Successful Ingredients : Successful Ingredients Teams in each of five subjects - experience /technical knowledge XML technology for on-line/printed resources Mature automatic assessment technology Initial partnership with 4 Education Authorities and 6 FE Colleges Comprehensive staff development programme All 32 Education Authorities are now members with 40,000 students on-line Pilot programme now planned for England. What does SCHOLAR have? : What does SCHOLAR have? HW have created interactive on-line learning and assessment resources for delivery: into schools for Higher and Advanced Higher into FE for HN provision; and as first year courses on campus. Resources include: Written materials interactive animations and assessments discussion areas and frequently asked questions. URL: scholar.hw.ac.uk Further Research : Further Research Considers the role of the medium in Mathematics G McGuire and M Youngson, (2002) LTSN Maths/Stats Website Role of steps in partial credit G McGuire et al, (2002) Sixth International CAA Conf, Lboro. A Description of CUE J Paterson, (2002) MSOR Connections, Vol 2. Bloom’s Taxonomy of Educational Objectives : Bloom’s Taxonomy of Educational Objectives B S Bloom et al, Taxonomy of Educational Objectives: the classification of educational goals volume 1&2 (Longman, London) (1956/64).   Level 1: Knowledge Level 2: Comprehension Level 3: Application Level 4: Analysis Level 5: Synthesis and Level 6: Evaluation. The PASS-IT Project : The PASS-IT Project Scottish Executive have invested £ 1 M for a pilot project to deliver NABs directly into schools. A consortium of SCROLLA, LTScotland, Scottish FE Unit, Scottish Qualifications Authority and the BBC are preparing automatic assessment material to test lower order skills . Schedule : Schedule Phase1: August 2002 – October 2003 Maths: Higher, Advanced Higher; Chemistry: Higher, Advanced Higher; Computing: Higher National; Phase 2:November 2003 – October 2004 Maths: 5 – 14, Access, Intermediate; English/Social Subject: Higher PASS-IT Progress : PASS-IT Progress http://www.pass-it.org.uk Inter agency staff development Traditional examiners work with CAA experts Schools/Colleges contacted Educational experiments defined and performed Pilots completed Analysis underway PASS-IT Experiments : PASS-IT Experiments Role of the medium in Chem/Comp Use of steps Feedback in summative tests Accessibility issues Human/Computer marked free text Random display in MCQ Tangent – summative 1 : Tangent – summative 1 Find the equation of the tangent to the curve f defined as f(x)=x3+4 at the point where x=1. KP1. Equation of tangent is y=? 3x+2 3 marks possible. No partial credit. No steps available Tangent – summative 2 : Find the equation of the tangent to the curve f defined as f(x)=x3+4 at the point where x=1. KP1. f’(x)=? 3x2 KP2. Tangent equation has y-b=m(x-a), m is called? gradient/slope KP3. Value of m for tangent at x=1 is ? 3 KP4. Value of b when x=a=1 is ? 5 KP5. Equation of tangent is y=? 3x+2 3 possible marks. Allows partial credit. But, is it too prescriptive? Tangent – summative 2 Tangent – summative 3 : Find the equation of the tangent to the curve f defined as f(x)=x3+4 at the point where x=1. 1.1 First find f’(x) - part mark lost 1.2 Tangent equation has y-b=m(x-a), m is called? gradient/slope 1.3 Value of m for tangent at x=1 is ? 3 1.4 Value of b when x=a=1 is ? 5 KP1. Equation of tangent is y=? 3x+2 3 marks available. Partial credit possible. Penalty for using steps. Tangent – summative 3 Smith's Mathematical Question Taxonomy : Smith's Mathematical Question Taxonomy G Smith et al, Int J Math Educ Sci Tech, 27(1), 65 (1996). Recall of factual knowledge Comprehension Routine use of procedures Information transfer Application in new situations Justifying and interpreting Implications, conjectures and comparisons Evaluation More Recent Work : More Recent Work Describes the introduction of AIM at Sheffield N Strickland, (2002) MSOR Connections, Vol 2, 27. Opportunities for encouraging higher level mathematical learning using CAA C Sangwin, (2002) http://ltsn.mathstore.ac.uk/articles/maths-caa-series/ Discusses follow through implications H Ashton and C Beevers, (2002) Sixth International CAA Conf, Lboro. Example 1 : Example 1 Consider a polynonial P(x) and the following properties: (a) P(x) is a cubic (b) P(a) = 0, (random: a = [1,..,5]) (c) P(b) = 0, (random: b = [1,..,4]+a) (d) P(0)=a*b, (generated by a×b) Give an example of a polynomial satisfying: 1. (a) 2. (a) and (b) 3. (a), (b) and (c) 4. (a) - (d) inclusive 5. only (b) - (d). Example 2 : Example 2 Given the complex number z =u +iv, (u,v: [1..10]) In each case find a nonzero complex number w =a+bi such that z+w is real. z.w is real. both z+w and z.w are real. Given a nonzero complex number z, is there a unique nonzero complex number w such that both z+w and z.w are real? [1] No, there are more than one, [2] Not unique if z is real, [3] Yes, always unique, [4] Don't know. Equation Type : Equation Type A straight line has a gradient of 2 and passes through the point x=0, y=1. Find the equation of this line. Express your answer in the form y=mx+c. Equation of line is y=? 2x+1 Rewording could change the learning objectives, Answer may not be in a familiar or most appropriate form (i.e. y-1=2x, y-2x-1=0 etc. are also correct) Equation Type : Equation Type A straight line has a gradient of 2 and passes through the point x=0, y=1. What is the equation of the line? y=2x+1 Would allow alternative form of answers (i.e. y-1=2x, y-2x-1=0 etc. are also correct) Already have implementation of common factors and simplification Requires implementation of Equation Rearranging Hidden Multiple Choice : Hidden Multiple Choice D Lawson, Computer Assisted Assessment in Relation to Learning Outcomes, http://ltsn.mathstore.ac.uk/articles/maths-caa-series/ Choices appear one at a time on screen Student chooses if correct or incorrect If choice is rejected it cannot be recalled Reduces the element of guessing Order of appearance of choices can be randomised Follow Through : Find the equation of the tangent to the curve f defined as f(x)=x3+4 at the point where x=1. 1.1 Gradient of tangent at x=1 is ? 3 (m) 1.2 Value of b when x=a=1 is ? 5 (b) KP1. Equation of tangent is y=? 3x+2 (m*x + b-m) Follow through is possible here. This ensures marking does not penalise correct method. Follow Through Question : Question A function f is defined by F(x) = (x2 - 3x + 18)/(x- 2) Express f in the form f(x) = g(x) + c/(x-2) Write down equations for the two asymptotes; Show that f has two stationary points; Sketch the graph of f; and State the range of values of k such that the equation f(x) = k has no solution. Question - Computer : Question - Computer F(x) = (x2 - 3x + 18)/(x- 2) f(x) = g(x) + c/(x-2), g(x)= ? x-1 f(x) = g(x) + c/(x-2), c= ? 16 Slant asymptote has equation y= ? x-1 Other asymptote has equ ? x=2 Stationary point with greater x value in form (x,y) is ? (6,9) Other stationary point in form (x,y) is ? (-2,-7) f(x)=k has no solution p < x < q, p = ? -7 f(x)=k has no solution p < x < q, q = ? 9 Question - Random : Question - Random F(x) = (x2 – {a+b}x + {a*b+n2})/(x- {b}) f(x) = g(x) + c/(x-{b}), g(x)= ? x-{a} f(x) = g(x) + c/(x-{b}), c= ? {n2} Slant asymptote has equation y= ? x-{a} Other asymptote has x= ? {b} Stationary point with greater x value is x=? {b+n} Other stationary point has x=? {b-n} f(x)=k has no solution p < x < q, p = ? {b-a-2n} f(x)=k has no solution p < x < q, q = ? {b-a+2n} The UMTC Tests : The UMTC Tests http://www.calm.hw.ac.uk/cue/caewebui.exe Login with “rhsjb” to find: UMTC Test A Tangent questions from the slides UMTC Test B Variable tangent; Constant tangent; Variable graph plot; and Constant graph plot. More general automatic test at: http://www.calm.hw.ac.uk/ Features of an Assessment Engine : Features of an Assessment Engine Non technical editor Randoms Marks algebraic and free text answers Handles partial credit Accessible Expandable - simulations, adaptive Different feedback Modes Reporting System Further Information : Further Information CALM www.calm.hw.ac.uk SCHOLAR scholar.hw.ac.uk SCROLLA www.scrolla.ac.uk PASS-IT www.pass-it.org.uk

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