Computer aided assessment of mathematics: the current state of the art and a look to the future. Chris Sangwin School of Mathematics University of Edinburgh October 2018 Chris Sangwin (University of Edinburgh) STACK October 2018 1 / 30
Question level Computer algebra systems (CAS). Chris Sangwin (University of Edinburgh) STACK October 2018 2 / 30
Question level Computer algebra systems (CAS). Include CAS calculations within feedback. Formative feedback. Chris Sangwin (University of Edinburgh) STACK October 2018 2 / 30
Question level Computer algebra systems (CAS). Include CAS calculations within feedback. Formative feedback. Test for properties, e.g. is this factored? Chris Sangwin (University of Edinburgh) STACK October 2018 2 / 30
Question level Computer algebra systems (CAS). Include CAS calculations within feedback. Formative feedback. Test for properties, e.g. is this factored? Multi-part questions. Chris Sangwin (University of Edinburgh) STACK October 2018 2 / 30
Question level Computer algebra systems (CAS). Include CAS calculations within feedback. Formative feedback. Test for properties, e.g. is this factored? Multi-part questions. Student “enters steps”. Chris Sangwin (University of Edinburgh) STACK October 2018 2 / 30
Question level Computer algebra systems (CAS). Include CAS calculations within feedback. Formative feedback. Test for properties, e.g. is this factored? Multi-part questions. Student “enters steps”. Unit testing of questions. Chris Sangwin (University of Edinburgh) STACK October 2018 2 / 30
Question level Computer algebra systems (CAS). Include CAS calculations within feedback. Formative feedback. Test for properties, e.g. is this factored? Multi-part questions. Student “enters steps”. Unit testing of questions. Special libraries: e.g. support for scientific units. Chris Sangwin (University of Edinburgh) STACK October 2018 2 / 30
Question level Computer algebra systems (CAS). Include CAS calculations within feedback. Formative feedback. Test for properties, e.g. is this factored? Multi-part questions. Student “enters steps”. Unit testing of questions. Special libraries: e.g. support for scientific units. But every system is still different. Chris Sangwin (University of Edinburgh) STACK October 2018 2 / 30
Student Interface (1) Typed expressions: validity/correctness. Chris Sangwin (University of Edinburgh) STACK October 2018 3 / 30
Student Interface (1) Typed expressions: validity/correctness. Validity: Students know what is required. Students not penalised on a technicality. Increases robustness of marking. Chris Sangwin (University of Edinburgh) STACK October 2018 3 / 30
Student Interface (1) Typed expressions: validity/correctness. Validity: Students know what is required. Students not penalised on a technicality. Increases robustness of marking. (2) Handwriting recognition Chris Sangwin (University of Edinburgh) STACK October 2018 3 / 30
Integration with other systems Chris Sangwin (University of Edinburgh) STACK October 2018 4 / 30
Integration with other systems LTI API Chris Sangwin (University of Edinburgh) STACK October 2018 4 / 30
Material banks High quality materials are expensive. Chris Sangwin (University of Edinburgh) STACK October 2018 5 / 30
Material banks High quality materials are expensive. Design level: Freudenthal institute Chris Sangwin (University of Edinburgh) STACK October 2018 5 / 30
Material banks High quality materials are expensive. Design level: Freudenthal institute Options Community led ◮ WebWork (open) ◮ Abacus (bring to the party) Commercial, via publishers Chris Sangwin (University of Edinburgh) STACK October 2018 5 / 30
Line by line reasoning Chris Sangwin (University of Edinburgh) STACK October 2018 6 / 30
Chris Sangwin (University of Edinburgh) STACK October 2018 7 / 30
Expert systems and AI Excellent examples. Chris Sangwin (University of Edinburgh) STACK October 2018 8 / 30
Expert systems and AI Excellent examples. BUT still require large expert projects. Bespoke areas (not general systems). Chris Sangwin (University of Edinburgh) STACK October 2018 8 / 30
The nature of the subject Is 2 log( x ) ≡ log( x 2 )? Chris Sangwin (University of Edinburgh) STACK October 2018 9 / 30
The nature of the subject Is 2 log( x ) ≡ log( x 2 )? ( 0 , ∞ ) � = R / { 0 } . Chris Sangwin (University of Edinburgh) STACK October 2018 9 / 30
The nature of the subject Is 2 log( x ) ≡ log( x 2 )? ( 0 , ∞ ) � = R / { 0 } . We are still arguing about what algebraic steps are legitimate! Chris Sangwin (University of Edinburgh) STACK October 2018 9 / 30
The nature of the subject Is 2 log( x ) ≡ log( x 2 )? ( 0 , ∞ ) � = R / { 0 } . We are still arguing about what algebraic steps are legitimate! Online assessment can (and should) drive this argument. Chris Sangwin (University of Edinburgh) STACK October 2018 9 / 30
Current State of Proof Assessment {} Chris Sangwin (University of Edinburgh) STACK October 2018 10 / 30
Current State of Proof Assessment {} Professional automatic reasoning systems. (COQ) Chris Sangwin (University of Edinburgh) STACK October 2018 10 / 30
Current State of Proof Assessment {} Professional automatic reasoning systems. (COQ) But professional mathematicians use L A T EX. Chris Sangwin (University of Edinburgh) STACK October 2018 10 / 30
Current State of Proof Assessment {} Professional automatic reasoning systems. (COQ) But professional mathematicians use L A T EX. Educational purpose. 1 Professional practice. 2 Chris Sangwin (University of Edinburgh) STACK October 2018 10 / 30
Babbage and the Analytical Engine Chris Sangwin (University of Edinburgh) STACK October 2018 11 / 30
Technology which looks back Babbage set out to print log tables! Chris Sangwin (University of Edinburgh) STACK October 2018 12 / 30
Technology which looks back Babbage set out to print log tables! Knuth set out to replicate movable type! Chris Sangwin (University of Edinburgh) STACK October 2018 12 / 30
What we teach and why The “big three”. Factor Integrate Solve Currently teach ad hoc methods. Chris Sangwin (University of Edinburgh) STACK October 2018 13 / 30
What we teach and why The “big three”. Factor Integrate Solve Currently teach ad hoc methods. CAS don’t use these! ... foundations for 1 ... special cases (e.g. Gauss elimination) 2 ... direct methods not guess and check. 3 Chris Sangwin (University of Edinburgh) STACK October 2018 13 / 30
19th Century methods! Chris Sangwin (University of Edinburgh) STACK October 2018 14 / 30
Solving quadratics x 2 + 4 x − 12 = 0 x 2 + 4 x = 12 x 2 + 4 x + 4 = 16 ( x − 2 ) 2 4 2 = ( x − 2 ) 2 − 4 2 = 0 ( x − 2 − 4 )( x − 2 + 4 ) = 0 ( x − 6 )( x + 2 ) = 0 Both “solve” and “factor” are direct methods. Chris Sangwin (University of Edinburgh) STACK October 2018 15 / 30
Proof: Assessment of whole argument Will require a sea-change in how we write mathematics. Chris Sangwin (University of Edinburgh) STACK October 2018 16 / 30
Proof: Assessment of whole argument Will require a sea-change in how we write mathematics. “Those who cannot remember the past are condemned to repeat it.” (George Santayana) Chris Sangwin (University of Edinburgh) STACK October 2018 16 / 30
Better interface Pell (1668) (see Stedall (2002)) Chris Sangwin (University of Edinburgh) STACK October 2018 17 / 30
Better interface Pell (1668) (see Stedall (2002)) The constraints of the interface focus thought. Chris Sangwin (University of Edinburgh) STACK October 2018 17 / 30
Better interface Pell (1668) (see Stedall (2002)) The constraints of the interface focus thought. The constraints of algebraic symbolism focus thought. Chris Sangwin (University of Edinburgh) STACK October 2018 17 / 30
Reasoning by equivalence demo Chris Sangwin (University of Edinburgh) STACK October 2018 18 / 30
Chris Sangwin (University of Edinburgh) STACK October 2018 19 / 30
STACK and RE (2018) Working Polynomials, rational expressions ± , √ Single variable inequalities over R Simultaneous equations Equating coefficients Not RE: “Let” (previous line only...) Not RE: Calculus operations Chris Sangwin (University of Edinburgh) STACK October 2018 20 / 30
STACK and RE (2018) Working Polynomials, rational expressions ± , √ Single variable inequalities over R Simultaneous equations Equating coefficients Not RE: “Let” (previous line only...) Not RE: Calculus operations Future | x | Systems of inequalities Chris Sangwin (University of Edinburgh) STACK October 2018 20 / 30
STACK and RE (2018) Working Polynomials, rational expressions ± , √ Single variable inequalities over R Simultaneous equations Equating coefficients Not RE: “Let” (previous line only...) Not RE: Calculus operations Future | x | Systems of inequalities Distant future Trig (requires infinite solution sets) Chris Sangwin (University of Edinburgh) STACK October 2018 20 / 30
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