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High stakes automatic assessments: developing an online linear algebra examination Chris Sangwin School of Mathematics University of Edinburgh August 2018 Chris Sangwin (University of Edinburgh) Exams August 2018 1 / 28 Introduction Can

SLIDE 1

High stakes automatic assessments: developing an online linear algebra examination

Chris Sangwin

School of Mathematics University of Edinburgh

August 2018

Chris Sangwin (University of Edinburgh) Exams August 2018 1 / 28

SLIDE 2

Introduction

Can we automatically mark current paper examinations?

To what extent is this equivalent to a paper exam?

Do we want to keep the current systems, and what are the alternatives?

Chris Sangwin (University of Edinburgh) Exams August 2018 2 / 28

SLIDE 3

Introduction

Can we automatically mark current paper examinations?

To what extent is this equivalent to a paper exam?

Do we want to keep the current systems, and what are the alternatives?

Chris Sangwin (University of Edinburgh) Exams August 2018 2 / 28

SLIDE 4

Introduction

Can we automatically mark current paper examinations?

To what extent is this equivalent to a paper exam?

Do we want to keep the current systems, and what are the alternatives?

Chris Sangwin (University of Edinburgh) Exams August 2018 2 / 28

SLIDE 5

The problem

Large and growing university mathematics courses. Demands for more assessment & feedback.

Chris Sangwin (University of Edinburgh) Exams August 2018 3 / 28

SLIDE 6

The problem

Large and growing university mathematics courses. Demands for more assessment & feedback.

Chris Sangwin (University of Edinburgh) Exams August 2018 3 / 28

SLIDE 7

Solution: online assessment

Chris Sangwin (University of Edinburgh) Exams August 2018 4 / 28

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STACK Demo

Chris Sangwin (University of Edinburgh) Exams August 2018 5 / 28

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Chris Sangwin (University of Edinburgh) Exams August 2018 6 / 28

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STACK

Computer algebra at the core. Separate validity from correctness.

◮ Students know what is required. ◮ Students not penalised on a technicality. ◮ Increases robustness of marking.

Include CAS calculations within feedback. Formative feedback. Answer tests: Test if this is in factored form. Support for scientific units. Flexible multi-part questions. Question blocks. Reasoning by equivalence. Unit testing of questions.

Chris Sangwin (University of Edinburgh) Exams August 2018 7 / 28

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STACK

Computer algebra at the core. Separate validity from correctness.

◮ Students know what is required. ◮ Students not penalised on a technicality. ◮ Increases robustness of marking.

Include CAS calculations within feedback. Formative feedback. Answer tests: Test if this is in factored form. Support for scientific units. Flexible multi-part questions. Question blocks. Reasoning by equivalence. Unit testing of questions.

Chris Sangwin (University of Edinburgh) Exams August 2018 7 / 28

SLIDE 12

STACK

Computer algebra at the core. Separate validity from correctness.

◮ Students know what is required. ◮ Students not penalised on a technicality. ◮ Increases robustness of marking.

Include CAS calculations within feedback. Formative feedback. Answer tests: Test if this is in factored form. Support for scientific units. Flexible multi-part questions. Question blocks. Reasoning by equivalence. Unit testing of questions.

Chris Sangwin (University of Edinburgh) Exams August 2018 7 / 28

SLIDE 13

STACK

Computer algebra at the core. Separate validity from correctness.

◮ Students know what is required. ◮ Students not penalised on a technicality. ◮ Increases robustness of marking.

Include CAS calculations within feedback. Formative feedback. Answer tests: Test if this is in factored form. Support for scientific units. Flexible multi-part questions. Question blocks. Reasoning by equivalence. Unit testing of questions.

Chris Sangwin (University of Edinburgh) Exams August 2018 7 / 28

SLIDE 14

STACK

Computer algebra at the core. Separate validity from correctness.

◮ Students know what is required. ◮ Students not penalised on a technicality. ◮ Increases robustness of marking.

Include CAS calculations within feedback. Formative feedback. Answer tests: Test if this is in factored form. Support for scientific units. Flexible multi-part questions. Question blocks. Reasoning by equivalence. Unit testing of questions.

Chris Sangwin (University of Edinburgh) Exams August 2018 7 / 28

SLIDE 15

STACK

Computer algebra at the core. Separate validity from correctness.

◮ Students know what is required. ◮ Students not penalised on a technicality. ◮ Increases robustness of marking.

Include CAS calculations within feedback. Formative feedback. Answer tests: Test if this is in factored form. Support for scientific units. Flexible multi-part questions. Question blocks. Reasoning by equivalence. Unit testing of questions.

Chris Sangwin (University of Edinburgh) Exams August 2018 7 / 28

SLIDE 16

STACK

Computer algebra at the core. Separate validity from correctness.

◮ Students know what is required. ◮ Students not penalised on a technicality. ◮ Increases robustness of marking.

Include CAS calculations within feedback. Formative feedback. Answer tests: Test if this is in factored form. Support for scientific units. Flexible multi-part questions. Question blocks. Reasoning by equivalence. Unit testing of questions.

Chris Sangwin (University of Edinburgh) Exams August 2018 7 / 28

SLIDE 17

STACK

Computer algebra at the core. Separate validity from correctness.

◮ Students know what is required. ◮ Students not penalised on a technicality. ◮ Increases robustness of marking.

Include CAS calculations within feedback. Formative feedback. Answer tests: Test if this is in factored form. Support for scientific units. Flexible multi-part questions. Question blocks. Reasoning by equivalence. Unit testing of questions.

Chris Sangwin (University of Edinburgh) Exams August 2018 7 / 28

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Integration with other systems

LTI API + YAML.

Chris Sangwin (University of Edinburgh) Exams August 2018 8 / 28

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Integration with other systems

LTI API + YAML.

Chris Sangwin (University of Edinburgh) Exams August 2018 8 / 28

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Community

Over 800 Moodle sites. Every university in Finland uses STACK. Abacus materials consortium: https://abacus.aalto.fi/ ILIAS community (Germany)

Chris Sangwin (University of Edinburgh) Exams August 2018 9 / 28

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Community

Over 800 Moodle sites. Every university in Finland uses STACK. Abacus materials consortium: https://abacus.aalto.fi/ ILIAS community (Germany)

Chris Sangwin (University of Edinburgh) Exams August 2018 9 / 28

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Community

Over 800 Moodle sites. Every university in Finland uses STACK. Abacus materials consortium: https://abacus.aalto.fi/ ILIAS community (Germany)

Chris Sangwin (University of Edinburgh) Exams August 2018 9 / 28

SLIDE 23

Research questions

Can we automatically mark current paper examinations?

To what extent is this equivalent to a paper exam?

Chris Sangwin (University of Edinburgh) Exams August 2018 10 / 28

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Research questions

Can we automatically mark current paper examinations?

To what extent is this equivalent to a paper exam?

Chris Sangwin (University of Edinburgh) Exams August 2018 10 / 28

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Method: background

Add online practice exam to a university course. Introduction to Linear Algebra (ILA). Year 1, semester 1. 20/120 credits. > 600 students, (578 took the written exam). Grade: 80% exam, 20% coursework including STACK quizzes. Students requested a practice exam. Only one week between end of teaching and exam. Paper exam takes 35 person-days to mark.

Chris Sangwin (University of Edinburgh) Exams August 2018 11 / 28

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Method: background

Add online practice exam to a university course. Introduction to Linear Algebra (ILA). Year 1, semester 1. 20/120 credits. > 600 students, (578 took the written exam). Grade: 80% exam, 20% coursework including STACK quizzes. Students requested a practice exam. Only one week between end of teaching and exam. Paper exam takes 35 person-days to mark.

Chris Sangwin (University of Edinburgh) Exams August 2018 11 / 28

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Method: background

Add online practice exam to a university course. Introduction to Linear Algebra (ILA). Year 1, semester 1. 20/120 credits. > 600 students, (578 took the written exam). Grade: 80% exam, 20% coursework including STACK quizzes. Students requested a practice exam. Only one week between end of teaching and exam. Paper exam takes 35 person-days to mark.

Chris Sangwin (University of Edinburgh) Exams August 2018 11 / 28

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Method

Conditions: Practice exam likely to be taken seriously. No contribution to grade = no incentive to cheat. Written exam “open book" Materials: Took oldest 2 past exams. Implemented as many questions as possible exactly. Justification vs answer?

Chris Sangwin (University of Edinburgh) Exams August 2018 12 / 28

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Method

Conditions: Practice exam likely to be taken seriously. No contribution to grade = no incentive to cheat. Written exam “open book" Materials: Took oldest 2 past exams. Implemented as many questions as possible exactly. Justification vs answer?

Chris Sangwin (University of Edinburgh) Exams August 2018 12 / 28

SLIDE 30

Method

Conditions: Practice exam likely to be taken seriously. No contribution to grade = no incentive to cheat. Written exam “open book" Materials: Took oldest 2 past exams. Implemented as many questions as possible exactly. Justification vs answer?

Chris Sangwin (University of Edinburgh) Exams August 2018 12 / 28

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Marked exactly

5. Is it possible for A and B to be 3 × 3 rank 2 matrices with

AB = 0? True/False. Paper exam: 7 marks for answer and justification Online exam: 1 mark for answer. Students could provide typed free-text justifications. Not marked and no feedback. But, little partial credit.

Chris Sangwin (University of Edinburgh) Exams August 2018 13 / 28

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Marked exactly

5. Is it possible for A and B to be 3 × 3 rank 2 matrices with

AB = 0? True/False. Paper exam: 7 marks for answer and justification Online exam: 1 mark for answer. Students could provide typed free-text justifications. Not marked and no feedback. But, little partial credit.

Chris Sangwin (University of Edinburgh) Exams August 2018 13 / 28

SLIDE 33

Marked exactly

5. Is it possible for A and B to be 3 × 3 rank 2 matrices with

AB = 0? True/False. Paper exam: 7 marks for answer and justification Online exam: 1 mark for answer. Students could provide typed free-text justifications. Not marked and no feedback. But, little partial credit.

Chris Sangwin (University of Edinburgh) Exams August 2018 13 / 28

SLIDE 34

Marked exactly

5. Is it possible for A and B to be 3 × 3 rank 2 matrices with

AB = 0? True/False. Paper exam: 7 marks for answer and justification Online exam: 1 mark for answer. Students could provide typed free-text justifications. Not marked and no feedback. But, little partial credit.

Chris Sangwin (University of Edinburgh) Exams August 2018 13 / 28

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Result: extent of automation

240 marks available on the Dec-11 and Aug-12 papers. 109/240 marks 45% were automated.

Chris Sangwin (University of Edinburgh) Exams August 2018 14 / 28

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Result: extent of automation

240 marks available on the Dec-11 and Aug-12 papers. 109/240 marks 45% were automated.

Chris Sangwin (University of Edinburgh) Exams August 2018 14 / 28

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Results: students’ attempts

394 attempts in December 2017. (578 took the written exam = 68%). No significant technical problems. M = 47.9%, SD = 23.2%. Cronbach Alpha: 0.87. Paper exam grades were moderately correlated, r(374) = 0.593, p < 10−15.

Chris Sangwin (University of Edinburgh) Exams August 2018 15 / 28

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Results: students’ attempts

394 attempts in December 2017. (578 took the written exam = 68%). No significant technical problems. M = 47.9%, SD = 23.2%. Cronbach Alpha: 0.87. Paper exam grades were moderately correlated, r(374) = 0.593, p < 10−15.

Chris Sangwin (University of Edinburgh) Exams August 2018 15 / 28

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Results: students’ attempts

394 attempts in December 2017. (578 took the written exam = 68%). No significant technical problems. M = 47.9%, SD = 23.2%. Cronbach Alpha: 0.87. Paper exam grades were moderately correlated, r(374) = 0.593, p < 10−15.

Chris Sangwin (University of Edinburgh) Exams August 2018 15 / 28

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Results: practice vs paper exam scores

20 40 60 80 100 20 40 60 80 100 Paper Online Linear model: R2 = 0.3513 y = 0.727x + 0.727

Chris Sangwin (University of Edinburgh) Exams August 2018 16 / 28

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Free text responses

Students could type in justifications. Q4 276 (M=187, SD=167) Q6 211 (M=125, SD=117) Q8 242 (M=204, SD=155) Q10 249 (M=128, SD=124) Q12 220 (M=201, SD=194) Q15 227 (M=142, SD=145) Q22 213 (M=173, SD=145) Q26 194 (M=67, SD=69.4)

The number of free text responses. Mean number of characters M and the standard deviation of the response length.

Chris Sangwin (University of Edinburgh) Exams August 2018 17 / 28

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Discussion

A modest success. No technical problems; no students complained. Means scores were lower:

Low stakes → disengagement.

No partial credit implemented.

No assessment of students’ justification.

Time taken to write justifications?

Chris Sangwin (University of Edinburgh) Exams August 2018 18 / 28

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Discussion

A modest success. No technical problems; no students complained. Means scores were lower:

Low stakes → disengagement.

No partial credit implemented.

No assessment of students’ justification.

Time taken to write justifications?

Chris Sangwin (University of Edinburgh) Exams August 2018 18 / 28

SLIDE 44

Discussion

A modest success. No technical problems; no students complained. Means scores were lower:

Low stakes → disengagement.

No partial credit implemented.

No assessment of students’ justification.

Time taken to write justifications?

Chris Sangwin (University of Edinburgh) Exams August 2018 18 / 28

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Discussion

Other tools needed: MCQ! “pmatch” (short answers) Comparative judgement ... human marking of some parts. In practice we need to guard against Impersonation Plagiarism Online sharing/posting answers ... which have nothing to do with mathematics.

Chris Sangwin (University of Edinburgh) Exams August 2018 19 / 28

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Discussion

Other tools needed: MCQ! “pmatch” (short answers) Comparative judgement ... human marking of some parts. In practice we need to guard against Impersonation Plagiarism Online sharing/posting answers ... which have nothing to do with mathematics.

Chris Sangwin (University of Edinburgh) Exams August 2018 19 / 28

SLIDE 47

Discussion

Other tools needed: MCQ! “pmatch” (short answers) Comparative judgement ... human marking of some parts. In practice we need to guard against Impersonation Plagiarism Online sharing/posting answers ... which have nothing to do with mathematics.

Chris Sangwin (University of Edinburgh) Exams August 2018 19 / 28

SLIDE 48

Discussion

Other tools needed: MCQ! “pmatch” (short answers) Comparative judgement ... human marking of some parts. In practice we need to guard against Impersonation Plagiarism Online sharing/posting answers ... which have nothing to do with mathematics.

Chris Sangwin (University of Edinburgh) Exams August 2018 19 / 28

SLIDE 49

Discussion: we didn’t change anything

5. Is it possible for A and B to be 3 × 3 rank 2 matrices with

AB = 0? True/False. Opportunity for the future... .... if true give examples. .... if false explain why not. [human marking?] We don’t ask some questions ONLY because they are hard to mark.

Chris Sangwin (University of Edinburgh) Exams August 2018 20 / 28

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Discussion: we didn’t change anything

5. Is it possible for A and B to be 3 × 3 rank 2 matrices with

AB = 0? True/False. Opportunity for the future... .... if true give examples. .... if false explain why not. [human marking?] We don’t ask some questions ONLY because they are hard to mark.

Chris Sangwin (University of Edinburgh) Exams August 2018 20 / 28

SLIDE 51

Discussion: we didn’t change anything

5. Is it possible for A and B to be 3 × 3 rank 2 matrices with

AB = 0? True/False. Opportunity for the future... .... if true give examples. .... if false explain why not. [human marking?] We don’t ask some questions ONLY because they are hard to mark.

Chris Sangwin (University of Edinburgh) Exams August 2018 20 / 28

SLIDE 52

Discussion: we didn’t change anything

5. Is it possible for A and B to be 3 × 3 rank 2 matrices with

AB = 0? True/False. Opportunity for the future... .... if true give examples. .... if false explain why not. [human marking?] We don’t ask some questions ONLY because they are hard to mark.

Chris Sangwin (University of Edinburgh) Exams August 2018 20 / 28

SLIDE 53

Comparison with school exams?

(Nadine Köcher & Chris Sangwin, 2014)

International Baccalaureate examinations in STACK? # marks (i) Awarded by STACK (2014) exactly 112 18% (ii) Final answers and implied method marks 227 37% (iii) Reasoning by equivalence 218 36% Total of max of (ii) and (iii) per question 376 61% Scottish Highers Repeat analysis with STACK in 2018: Scottish Highers papers from 2015. # marks (i) Awarded by STACK (2018) exactly 47 36% (ii) Of which reasoning by equivalence 35 27% The most important single form of reasoning in school mathe- matics is reasoning by equivalence.

Chris Sangwin (University of Edinburgh) Exams August 2018 21 / 28

SLIDE 54

Comparison with school exams?

(Nadine Köcher & Chris Sangwin, 2014)

International Baccalaureate examinations in STACK? # marks (i) Awarded by STACK (2014) exactly 112 18% (ii) Final answers and implied method marks 227 37% (iii) Reasoning by equivalence 218 36% Total of max of (ii) and (iii) per question 376 61% Scottish Highers Repeat analysis with STACK in 2018: Scottish Highers papers from 2015. # marks (i) Awarded by STACK (2018) exactly 47 36% (ii) Of which reasoning by equivalence 35 27% The most important single form of reasoning in school mathe- matics is reasoning by equivalence.

Chris Sangwin (University of Edinburgh) Exams August 2018 21 / 28

SLIDE 55

Comparison with school exams?

(Nadine Köcher & Chris Sangwin, 2014)

International Baccalaureate examinations in STACK? # marks (i) Awarded by STACK (2014) exactly 112 18% (ii) Final answers and implied method marks 227 37% (iii) Reasoning by equivalence 218 36% Total of max of (ii) and (iii) per question 376 61% Scottish Highers Repeat analysis with STACK in 2018: Scottish Highers papers from 2015. # marks (i) Awarded by STACK (2018) exactly 47 36% (ii) Of which reasoning by equivalence 35 27% The most important single form of reasoning in school mathe- matics is reasoning by equivalence.

Chris Sangwin (University of Edinburgh) Exams August 2018 21 / 28

SLIDE 56

Nature of the subject

Polya 1962: Mathematical Discovery: on understanding, learning and teaching problem solving. Patterns of thought for solving problems the pattern of two loci superposition recursion Cartesian pattern Legitimate patterns of thought → an acceptable proof.

Chris Sangwin (University of Edinburgh) Exams August 2018 22 / 28

SLIDE 57

Nature of the subject

Polya 1962: Mathematical Discovery: on understanding, learning and teaching problem solving. Patterns of thought for solving problems the pattern of two loci superposition recursion Cartesian pattern Legitimate patterns of thought → an acceptable proof.

Chris Sangwin (University of Edinburgh) Exams August 2018 22 / 28

SLIDE 58

Nature of the subject

Polya 1962: Mathematical Discovery: on understanding, learning and teaching problem solving. Patterns of thought for solving problems the pattern of two loci superposition recursion Cartesian pattern Legitimate patterns of thought → an acceptable proof.

Chris Sangwin (University of Edinburgh) Exams August 2018 22 / 28

SLIDE 59

Nature of the subject

Polya 1962: Mathematical Discovery: on understanding, learning and teaching problem solving. Patterns of thought for solving problems the pattern of two loci superposition recursion Cartesian pattern Legitimate patterns of thought → an acceptable proof.

Chris Sangwin (University of Edinburgh) Exams August 2018 22 / 28

SLIDE 60

Nature of the subject

Polya 1962: Mathematical Discovery: on understanding, learning and teaching problem solving. Patterns of thought for solving problems the pattern of two loci superposition recursion Cartesian pattern Legitimate patterns of thought → an acceptable proof.

Chris Sangwin (University of Edinburgh) Exams August 2018 22 / 28

SLIDE 61

Nature of the subject

Polya 1962: Mathematical Discovery: on understanding, learning and teaching problem solving. Patterns of thought for solving problems the pattern of two loci superposition recursion Cartesian pattern Legitimate patterns of thought → an acceptable proof.

Chris Sangwin (University of Edinburgh) Exams August 2018 22 / 28

SLIDE 62

Cartesian pattern

Descartes’ Rules for the Direction of the mind.

Reduce any kind of problem to a mathematical problem.

Reduce any mathematical problem to algebra.

Reduce any algebra problem to a single equation & solve. Polya: “The more you know, the more gaps you can see in this project” (Note for later: solving the equation is only the last step...)

Chris Sangwin (University of Edinburgh) Exams August 2018 23 / 28

SLIDE 63

Cartesian pattern

Descartes’ Rules for the Direction of the mind.

Reduce any kind of problem to a mathematical problem.

Reduce any mathematical problem to algebra.

Reduce any algebra problem to a single equation & solve. Polya: “The more you know, the more gaps you can see in this project” (Note for later: solving the equation is only the last step...)

Chris Sangwin (University of Edinburgh) Exams August 2018 23 / 28

SLIDE 64

Cartesian pattern

Descartes’ Rules for the Direction of the mind.

Reduce any kind of problem to a mathematical problem.

Reduce any mathematical problem to algebra.

Reduce any algebra problem to a single equation & solve. Polya: “The more you know, the more gaps you can see in this project” (Note for later: solving the equation is only the last step...)

Chris Sangwin (University of Edinburgh) Exams August 2018 23 / 28

SLIDE 65

Cartesian pattern

Descartes’ Rules for the Direction of the mind.

Reduce any kind of problem to a mathematical problem.

Reduce any mathematical problem to algebra.

Reduce any algebra problem to a single equation & solve. Polya: “The more you know, the more gaps you can see in this project” (Note for later: solving the equation is only the last step...)

Chris Sangwin (University of Edinburgh) Exams August 2018 23 / 28

SLIDE 66

Cartesian pattern

Descartes’ Rules for the Direction of the mind.

Reduce any kind of problem to a mathematical problem.

Reduce any mathematical problem to algebra.

Reduce any algebra problem to a single equation & solve. Polya: “The more you know, the more gaps you can see in this project” (Note for later: solving the equation is only the last step...)

Chris Sangwin (University of Edinburgh) Exams August 2018 23 / 28

SLIDE 67

Two strands of mathematical activity

(1) The use of routine techniques. recognition standard form & algorithm accuracy (2) Problem-solving. novelty creativity struggle ... ... satisfaction? CAA is probably most useful for (1). Current exams mostly test (1).

Chris Sangwin (University of Edinburgh) Exams August 2018 24 / 28

SLIDE 68

Two strands of mathematical activity

(1) The use of routine techniques. recognition standard form & algorithm accuracy (2) Problem-solving. novelty creativity struggle ... ... satisfaction? CAA is probably most useful for (1). Current exams mostly test (1).

Chris Sangwin (University of Edinburgh) Exams August 2018 24 / 28

SLIDE 69

Two strands of mathematical activity

(1) The use of routine techniques. recognition standard form & algorithm accuracy (2) Problem-solving. novelty creativity struggle ... ... satisfaction? CAA is probably most useful for (1). Current exams mostly test (1).

Chris Sangwin (University of Edinburgh) Exams August 2018 24 / 28

SLIDE 70

Two strands of mathematical activity

(1) The use of routine techniques. recognition standard form & algorithm accuracy (2) Problem-solving. novelty creativity struggle ... ... satisfaction? CAA is probably most useful for (1). Current exams mostly test (1).

Chris Sangwin (University of Edinburgh) Exams August 2018 24 / 28

SLIDE 71

Skills vs concepts

... possibly a false dichotomy? To be trained is to be prepared against surprise. To be educated is to be prepared for surprise. Carse, Finite and Infinite Games (1986) Conceptual development needs expertise in the technique. Any student studying STEM at university needs strong algebraic technique. (And the arguments are > 400 years old.....)

Chris Sangwin (University of Edinburgh) Exams August 2018 25 / 28

SLIDE 72

Skills vs concepts

... possibly a false dichotomy? To be trained is to be prepared against surprise. To be educated is to be prepared for surprise. Carse, Finite and Infinite Games (1986) Conceptual development needs expertise in the technique. Any student studying STEM at university needs strong algebraic technique. (And the arguments are > 400 years old.....)

Chris Sangwin (University of Edinburgh) Exams August 2018 25 / 28

SLIDE 73

Skills vs concepts

... possibly a false dichotomy? To be trained is to be prepared against surprise. To be educated is to be prepared for surprise. Carse, Finite and Infinite Games (1986) Conceptual development needs expertise in the technique. Any student studying STEM at university needs strong algebraic technique. (And the arguments are > 400 years old.....)

Chris Sangwin (University of Edinburgh) Exams August 2018 25 / 28

SLIDE 74

Skills vs concepts

... possibly a false dichotomy? To be trained is to be prepared against surprise. To be educated is to be prepared for surprise. Carse, Finite and Infinite Games (1986) Conceptual development needs expertise in the technique. Any student studying STEM at university needs strong algebraic technique. (And the arguments are > 400 years old.....)

Chris Sangwin (University of Edinburgh) Exams August 2018 25 / 28

SLIDE 75

Skills vs concepts

... possibly a false dichotomy? To be trained is to be prepared against surprise. To be educated is to be prepared for surprise. Carse, Finite and Infinite Games (1986) Conceptual development needs expertise in the technique. Any student studying STEM at university needs strong algebraic technique. (And the arguments are > 400 years old.....)

Chris Sangwin (University of Edinburgh) Exams August 2018 25 / 28

SLIDE 76