Assessment in CAS and DGS Environment

1. Assessment in CAS and DGS Environment dr. Vlasta Kokol-Voljč University of Maribor, Slovenia

2. 1. New technology –a threat for assessment? • 25 years ago logarithm tablesenvironment: Calculate the value of ! scientific calculator environment: Sketch the graph of the following function: or: Derivate the following function…

3. 1. New technology –a threat for assessment? The use of scientific calculators caused shiftingthe assessment emphasis from learning and performingnumeric algorithms towards symbolic

4. 1. New technology –a threat for assessment? With the use of scientific calculators… … enough symbolic algorithms remained to be taught

5. 1. New technology –a threat for assessment? • Today CAS and DGS - perform all kinds of algorithmic operations (even symbolic!) - draw all kinds of functions and geometricobjects What remains to be taught if we allow CAS and DGS? What remains to be asked in theexams?

6. 1. New technology –a threat for assessment? The reaction: • teachers refuse or avoid the use of CAS and DGS in their teaching • school authoritiesban CAS and DGS from exams

7. 2. Use of CAS and DGS in assessment CAS and DGS can be great didactical tools for teaching mathematics(CADGME: C. Laborde 2007, E. Schneider 2007) • great demonstration tools • most valuable experimentation and exploration tools • can support general mathematics education Use of CAS and DGS shift focus from teaching procedural to teaching conceptual(Kadijevic 2007)

8. 2. Use of CAS and DGS in assessment use of DGS: concept development (trough problem solving) use of CAS: concept development + challenges and dilemmas in case of assessment

9. 2. Use of CAS and DGS in assessment two major ways of using CAS in assessment: • CAS as a tool for designing exam questions and for evaluating them • CAS as a tool for answering exam questions

10. 2.1. CAS and DGS as a tool for designing exam questions and for evaluating them- a challenge for the teachers and exam designers/writers and evaluators • Computer Aided Assessment (CAA) … packages which enable automatically generated (assigned) questions, evaluate responses and give automatically generated feedback and evaluations of responses. The motivation training of mathematical activities automatically performing exams

11. 2.1. CAS and DGS as a tool for designing exam questions and for evaluating them The squareroot of 9 is 4. • True. • False. • Who cares? (Glasbergen 2007)

12. 2.1. CAS and DGS as a tool for designing exam questions and for evaluating them • Computer Aided Assessment (CAA) - early experiments: background is numeric - with functionalities of CAS the possibilities of CAA systems increased • symbolic algebra can be tested in addition to numerics • tailored feedback • test not only syntactic but also the semantic correctness

13. 2.1. CAS and DGS as a tool for designing exam questions and for evaluating them Examples of CAA: • Computer Aided Assessment and Learning of Mathematical Methods - CalMæth(Stone&Scott 1996, Judd 1996) - one of the earliest CAA systems using CAS • Open-Software-Projects:Alice Interactive Mathematics -AIM(Klai, Kolokolnikov&Van den Bergh 2000, Sangvin 2003) and Computer Algebra Based Learning and Evaluation- CABLE(Naismith&Sangwin 2004) • Commercial products:MapleT.A.™(Schramm 2004),Questionmark™ Perception™(Etchells 2006) • CAME 07, CADGME 07: STACK (Sangwin), T-algebra (Prank)

14. 2.1. CAS and DGS as a tool for designing exam questions and for evaluating them • Computer Aided Assessment (CAA) • limitations and demands - technical (Heck 2005, Riegler 2005) - pedagogical (Monaghan CAME 07) • valuable educational implementations of CAS in teaching and learning mathematics • CAS helps the teachers to assess and to validate the student's knowledge.

15. 2.1. CAS and DGS as a tool for designing exam questions and for evaluating them CAS is a toolwhich enables automatic control of the answers provided by the students … to CONTROL the presented mathematical knowledge

16. 2.1. CAS and DGS as a tool for designing exam questions and for evaluating them DGS • DGS features are of a different character than those of CAS - basically they are not designed to automate operations • basically DGS are not used for controlling performed operations … not so much in demand in CAA

17. 2.2. CAS and DGS as a tool for answering exam questions 315 +204 DOWNLOADING ANSWER. ESTIMATED TIME REMAINING 00:07:18

18. 2.2. CAS and DGS as a tool for answering exam questions- a dilemma for the teachers and exam designers/writers and evaluators • Use of CAS in assessment in the hand of students?

19. 2.2. CAS and DGS as a tool for answering exam questions Research: • about the possible effects of the introduction of CAS into mathematics examinations(Stacey 1997, Drijvers 1998, Kissane 2000) • impact of the introduction of CAS into mathematics education and examinations(Meagher 2000, Monaghan 2000, Flynn& McCrae 2001, Brown 2001) • analyses of content of exam questions in terms of the impact that CAS can have in solving them - and in comparison with the traditional non-CAS-questions (Kokol-Voljc1999, MacAogáin2000,Lokar&Lokar 2000, Flynn&McCrae2001) • the use of CAS in examinations from the students’ perspective (Flynn & Asp2002, Ball & Stacey2003)

20. 2.2. CAS and DGS as a tool for answering exam questions teachers: “How to ask a questions in order to get the desired feedback about the intended content?" school authorities: "How to organize exams?”

21. 2.2. CAS and DGS as a tool for answering exam questions • model for organizing exams: “two-tier exams”(Kutzler 2000) - feedback about basic skills - feedback about abilities in the appropriate use of mathematical concepts

22. 2.2. CAS and DGS as a tool for answering exam questions … projects: • experience of the VCAA: pilot study (2001 – 2005) (Lancaster 2003, Flynn 2007) • comparative analysis of two pilot projects: DMW and VCAA(Brown 2003) • How do examination writers respond to the introduction of CAS? • What is the role of CAS within the examinations? • IB experience with two-tier exams (Brown, Flynn & Monaghan CADGME 07)

23. 2.2. CAS and DGS as a tool for answering exam questions CAS in assessment in the hand of students is a tool which helps the students to create answersby combining their mathematical knowledge with the power of the tool (to automate operations) … … to DEMONSTRATE the existing mathematical knowledge

24. 2. Use of CAS and DGS in assessment Two ways of using CAS in assessment: • CAS as a tool for designing assessment and for evaluating them: CAS enables automatic control of the answers- e.g.to CONTROL the presented knowledge. • CAS as a tool for answering exam questions: CAS helps to create answers - e.g.to DEMONSTRATE the existing knowledge.

25. 2. Use of CAS and DGS in assessment For evaluating the knowledge it is still decisivehow the students create their answer: • using the available tool (CAS, DGS,…) • or producing it "manually"

26. 2. Use of CAS and DGS in assessment CAS and DGS as a tool for answering exam questions: • The use of CAS causes much bigger confusion regarding assessment than the use of DGS Why did use of CAS irritate teachers in exams more than use of DGS? What is the core of the problems in assessment with CAS?

27. 3. What will be asked in exams? ?

28. 3. What will be asked in exams? “Algebra class will be important to you later in life because there is going to be a test six weeks from now.” (unknown math teacher)

29. 3. What will be asked in exams? Example (high school exams): 1. Given is the function: a) Find the domain and sketch the graph of the function b) Compute the integral of the function. ( http://www.emath.de/Mathe-Abituraufgaben-und-Loesungen/ ) 2. Solve the equations: ( http://www.ric.si/splosna_matura/predmeti/matematika/ ) 3. Calculate the integral. Give the exact result. ( http://www.ric.si/splosna_matura/predmeti/matematika/ ) 4. Solve the following system of linear equations:

30. 3. What will be asked in exams? • most of the traditional mathematics lessons are centered on the craftsmanship of learning and executing algorithms traditional exam questions very often assess the performing of algorithms. • standardized questions • … algorithmic teaching algorithmic exams teacher adjust their teaching

31. 3. What will be asked in exams? If CAS … …algorithmic exam questions become redundant! Is the performing of different algorithms really what we intend to achieve in mathematics teaching? ?

32. 3. What will be asked in exams? “All you have to know, to reach the subject goals.”

33. 3. What will be asked in exams? • …it depends on the GOALS. • Our students know intuitively, that the CONTENT is not the same as the GOALS. • The content is not the (only) goal.

34. 4. Through the glasses of teaching goals The teaching goalsare essential for mathematics teaching. The assessment (examinations and exam questions) are an indispensable part of mathematics teaching, the teaching goals are essential also for the assessment.

35. 4. Through the glasses of teaching goals • The goals of mathematics teaching: • - development of understanding the theoretical meaning of mathematical concepts(enabling to see mathematical concepts as general models) • development of abilities and competencies in applying mathematical knowledge (enabling to apply the general mathematical models in real situations)

36. 4. Through the glasses of teaching goals Example : Solving systems of linear equations: performing of the algorithm for solving systems of linear equations is not the main goal of the topic for 17-19 year old students. The content in most cases is only the "tool" to reach the actual teaching goal. Technology is only the tool which helps to reach theteaching goals. 

37. 4. Trough the glasses of teaching goals • Example: • Solve the following system of linear equations: • …solving strategy • the algorithm for solving system is not the only goal • recognize the concept • use the concept

38. 4. Trough the glasses of teaching goals c´)Find the intersection of two lines p1 and p2 given by the equations: and c´´) Ann and Tom have a party. Tom bought 6 chocolate muffins and 3 pieces of fruit tart and paid 639 SIT while Ann bought, in the same store, 8 chocolate muffins and 11 pieces of fruit tart and paid 1293 SIT. How much does one chocolate muffin cost and how much one piece of fruit tart? c´´´) A company makes two products (X and Y) using two machines (A and B). Processing time on machine A for each unit of X is 6 minutes and for each unit of Y 3 minutes. Processing time on machine B for each unit of X is 8 minutes and for each unit of Y 11 minutes. Machine A is available for 639 minutes per day and machine B for 1293 minutes per day. How many products A and B can be produced to use the complete processing time of A and B. 

39. 4. Through the glasses of teaching goals Example : Solving systems of linear equations: performing of the algorithm for solving systems of linear equations is not the main goal of the topic for 17-19 year old students. The content in most cases is only the "tool" to reach the actual teaching goal. Technology is only the tool which helps to reach theteaching goals.

40. 4. Through the glasses of teaching goals • … not decisive • - if exam questions are designed and evaluated:manually or automatically, using CAS or DGS, • - and how the students are assumed to create their answers: ("manually" or using CAS or DGS) • The decisive question is: • "What will be asked in exams?“ • … for the students and also for the teachers! • What needs to be asked in order to get the feedbackabout to what extent the intendedteaching goals have been achieved?

41. 5. "All roads lead to Rome" • The exams have to be organized appropriately (the examquestions have to be set appropriately) so that they enable to assess all teaching goals • "All roads leadthroughteaching goals."

42. 6. Conclusion Use of CAS and DGS in assessmenthelps to review our(perhaps too muchalgorithmically oriented?) mathematics teachingthrough the glasses of the teaching goals … help to get back to the teaching goals guided route of mathematics teaching.