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E-grāmata: Computer Aided Assessment of Mathematics

(Senior Lecturer, School of Mathematics, University of Birmingham)
  • Formāts: PDF+DRM
  • Izdošanas datums: 02-May-2013
  • Izdevniecība: Oxford University Press
  • Valoda: eng
  • ISBN-13: 9780191635854
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Computer Aided Assessment of MathematicsPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 02-May-2013
  • Izdevniecība: Oxford University Press
  • Valoda: eng
  • ISBN-13: 9780191635854
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Assessment is a key driver in mathematics education. This book examines computer aided assessment (CAA) of mathematics in which computer algebra systems (CAS) are used to establish the mathematical properties of expressions provided by students in response to questions. In order to automate such assessment, the relevant criteria must be encoded and, in articulating precisely the desired criteria, the teacher needs to think very carefully about the goals of the task. Hence CAA acts as a vehicle to examine assessment and mathematics education in detail and from a fresh perspective.



One example is how it is natural for busy teachers to set only those questions that can be marked by hand in a straightforward way, even though the constraints of paper-based formats restrict what they do and why. There are other kinds of questions, such as those with non-unique correct answers, or where assessing the properties requires the marker themselves to undertake a significant computation. It is simply not sensible for a person to set these to large groups of students when marking by hand. However, such questions have their place and value in provoking thought and learning.

This book, aimed at teachers in both schools and universities, explores how, in certain cases, different question types can be automatically assessed. Case studies of existing systems have been included to illustrate this in a concrete and practical way.

Recenzijas

It is a well written and well referenced text with attention to detail and practicalities ... of value for professionals in mathematics education of all levels. * Francis McGonigal, Mathematics Today * The detailed discussion of the nature of mathematics and its assessment, which is necessary to discuss the computer aided assessment of the subject, is interesting in itself and challenges even non-computer assessment users to consider how and why they assess their studentsThe author makes good use of historical references to enliven the text, and which also demonstrate the timelessness of many of the problems that users of such systems face today. * Tim Lowe, Open University * Automatic computer-aided assessment in mathematics has the potential to enhance learning in a serious way...This book provides an important contribution here, and will help to improve assessment in mathematics in many ways. * Mika Seppälä, Florida State University and University of Helsinki *

List of Figures viii
1 Introduction 1(8)
1.1 Multiple-choice questions
2(2)
1.2 Assessment criteria
4(3)
1.3
Chapters
7(1)
1.4 Acknowledgements
8(1)
2 An assessment vignette 9(10)
2.1 The student's perspective
9(5)
2.2 Assessing answers to simple questions
14(2)
2.3 Further integrals
16(2)
2.4 Discussion
18(1)
3 Learning and assessing mathematics 19(18)
3.1 The nature of mathematics
19(2)
3.2 Terms used in assessment
21(1)
3.3 Purposes of assessment
22(1)
3.4 Learning
23(2)
3.5 Principles and tensions of assessment design
25(8)
3.6 Learning cyder and feedback
33(2)
3.7 Conclusion
35(2)
4 Mathematical question spaces 37(16)
4.1 Why randomly generate questions?
38(1)
4.2 Randomly generating an individual question
39(3)
4.3 Linking mathematical questions
42(2)
4.4 Building up conceptions
44(2)
4.5 Types of mathematics question
46(3)
4.6 Embedding CAA into general teaching
49(2)
4.7 Conclusion
51(2)
5 Notation and syntax 53(20)
5.1 An episode in the history of mathematical notation
54(2)
5.2 The importance of notational conventions
56(4)
5.3 Ambiguities and inconsistencies in notation
60(1)
5.4 Notation and machines: syntax
61(4)
5.5 Other issues
65(1)
5.6 The use of the AiM system by students
66(1)
5.7 Proof and arguments
67(1)
5.8 Equation editors
68(2)
5.9 Dynamic interactions
70(1)
5.10 Conclusion
71(2)
6 Computer algebra systems for CAA 73(29)
6.1 The prototype test: equivalence
75(1)
6.2 A comparison of mainstream CAS
76(2)
6.3 The representation of expressions by CAS
78(4)
6.4 Existence of mathematical objects
82(4)
6.5 'Simplify' is an ambiguous instruction
86(2)
6.6 Equality, equivalence, and sameness
88(3)
6.7 Forms of elementary mathematical expression
91(3)
6.8 Equations, inequalities, and systems of equations
94(2)
6.9 Other mathematical properties we might seek to establish
96(1)
6.10 Buggy rules
97(2)
6.11 Generating outcomes useful for CAA
99(1)
6.12 Side conditions and logic
100(1)
6.13 Conclusion
101(1)
7 The STACK CAA system 102(25)
7.1 Background: the AiM CAA system
102(1)
7.2 Design goals for STACK
103(3)
7.3 STACK questions
106(1)
7.4 The design of STACK's multi-part tasks
107(4)
7.5 Interaction elements
111(1)
7.6 Assessment
112(1)
7.7 Quality control and exchange of questions
113(1)
7.8 Extensions and development of the STACK system by Aalto
114(3)
7.9 Usage by Aalto
117(4)
7.10 Student focus group
121(4)
7.11 Conclusion
125(2)
8 Software case studies 127(35)
8.1 Some early history
127(2)
8.2 CALM
129(3)
8.3 Pass-IT
132(6)
8.4 OpenMark
138(2)
8.5 DIAGNOSYS
140(6)
8.6 Cognitive tutors
146(1)
8.7 Khan Academy
147(1)
8.8 Mathwise
148(2)
8.9 WeBWorK
150(4)
8.10 MathXpert
154(3)
8.11 Algebra tutors: Aplusix and T-algebra
157(3)
8.12 Conclusion
160(2)
9 The future 162(11)
9.1 Encoding a complete mathematical argument
162(4)
9.2 Assessment of proof
166(3)
9.3 Semi-automatic marking
169(1)
9.4 Standards and interoperability
170(2)
9.5 Conclusion
172(1)
Bibliography 173(10)
Index 183
Chris Sangwin is a Senior Lecturer in the School of Mathematics at the University of Birmingham in the United Kingdom. From 2000-2011 he was seconded half time to the UK Higher Education Academy "Maths Stats and OR Network" to promote learning and teaching of university mathematics. In 2006 he was awarded a National Teaching Fellowship. His learning and teaching interests include (i) automatic assessment of mathematics using computer algebra, and (ii) problem solving using Moore method and similar student-centred approaches. Chris Sangwin is the author of a number of books, including How Round is Your Circle (Princeton University Press, 2011), which illustrates and investigates many links between mathematics and engineering using physical models.
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