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Computer Aided Assessment
                 (CAA) for mathematics

                Chris Sangwin & Simon Hammond




Copyright c
Last Revision Date: June 1, 2009
2

Introduction.

... NOT multiple choice questions ...

   • Computer aided assessment (CAA)
   • CAA with computer algebra
   • Practical issues
     Implementations
   • Pedagogical issues
   • Future directions
3

JEM - Joining Educational Mathematics

eContentPlus Thematic Network
http://jem-thematic.net/
Founder members (15):
Universitat Politecnica de Catalunya, Helsingin Yliopisto, Tech-
nical University, Jacobs University, Universiteit van Amster-
dam, University of Birmingham, FernUniversitt Hagen, Maths
for More, NAG Ltd, Liguori Editore, ISN Oldenburg GmbH,
RWTH Aachen University, Univ. Nacional de Educacin a Dis-
tancia, Universitat Oberta de Catalunya, Universidade de Lis-
boa.
4

Use of objective tests

Consider the following question:
Example question 1
Determine the following integral:

                         cos(x) sin(2x)dx.


As a multiple choice question:
        ◦   (2/3) cos3 (x) + C
        ◦   −(2/3) cos(x) + (2/3) sin3 (x) + C
        ◦   −(2/3) cos(x) + (1/3) sin(x) sin(2x) + C
        ◦   Don’t know.
How do we know the students don’t differentiate the
candidate solutions to check?
5

Computer algebra marking

Computer algebra systems can be used to mark work.
This checks for algebraic equivalence.

                    (x + 1)2 ≡ x2 + 2x + 1

Useful for marking many routine problems.
6

Fundamental idea

if simplify(sa-ta) = 0 then
mark := 1 else mark := 0
7

STACK

System overview

The STACK system:
  • internet based CAA system,

  • uses very simple
    Maxima (computer algebra), and
    L TEX (type setting)
    A

  • All components open source (e.g. GPL).
8

Demonstrating the STACK system

http://www.stack.bham.ac.uk/
9

In learning and teaching

We are assessing a student provided answer.
This is an objective test.
This is
   • not Multiple Choice Question;
   • not string/regex match.
Other tests for the form of an answer.
10

Input of mathematics

This is a fundamental but unsolved problem.
There are a number of options

   1.   Strict CAS syntax.               eg. 2*(x-1)*(x+1)
   2.   “informal” linear text syntax.   eg. 2(x-1)(x+1)
                                         x(t-1)    ?
   3.   Graphical input tool.            eg. equation editor.
   4.   (Pen-based input ?)
   5.   (Geometry applet ?)

Not all groups of students are equal.
11

Syntax innovations

Difficult to achieve!

Babbage 1830’s

“a profusion of notations [...] which threaten, if not duly cor-
rected, to multiply our difficulties instead of promoting our progress”
Babbage, C. (1827)


                       sin2 (x)   sin−1 (x)



                       sin sin x = sin2 x

(composition)
12

Structure in random problem sets

In practice, the numbers often do not matter.




Tuckey, C. O., Examples in Algebra, Bell & Sons, London, (1904)

Too much randomization destroys structure.
An underlying question space.
13

Workshop task

Option A:
Context: end of first calculus course. (Age 18)
Write 6 questions which test whether a student can differentiate
elementary functions.
E.g. Differentiate cos(3x) with respect to x.
Option B:
Context: age 11.
Write 6 questions which test whether a student can add frac-
tions.
14

Randomization
  1. What could you randomize?
  2. What would you randomize?
  3. What are some likely incorrect answers?
  4. What feedback would you like to provide?

... with a view to implementing these questions live.
15

Issues
   • Well-posed questions.
   • Fair questions.
   • Structure in question sets.
     Schemes of work, vs isolated questions.
   • Algebraic form of answers as a goal.
16

Feedback

One third of feedback interventions decreased performance.
Kluger, A. N. and DeNisi, A., Psychological Bulletin (1996).

The nature of feedback determines its effectiveness.
17

Processing answers

Test for algebraic equivalence if simplify(sa-ta) = 0 then
 mark := 1 else mark := 0
Using mainstream CAS
   • Get a lot very quickly,
     Great for calculus and beyond.
   • Elementary algebra can be a problem.
Maxima seems to be more suitable than most.
18

Every CAS is different!

        Input               Maple             Maxima             Axiom
     (numbers)
      0.5-1/2                 0.0
                              √                  0.0               0.0
      4^(1/2)                  √4                 2                 2
                             1                    1                 1
      4^(-1/2)                   4
                             √
                             4                    2               √2
      -4^(1/2)                 −4                2i              2√−1
      sqrt(-4)                2i                 2i              2 −1
      (indices)
     a^n*b*a^m              an bam             an+m b            bam an
    (a^(1/2))^2              √a                   a               √a
    (a^2)^(1/2)                 a2               |a|                a2
  (collecting terms)
      1+x^2-2*x           x2 − 2x + 1        x2 − 2x + 1      x2 − 2x + 1
    x/3+1.5*x+1/3      1.833x + .333 · · ·   1.833x + 1
                                                      3
                                                           1.833x + 0.333 · · ·
                              5                  5x               5
     3*x/4+x/12               6
                                x                 6               6
                                                                    x
                              5 1                 5                 5
  3/(4*x)+1/(12*x)            6 x                6x                6x
19
      Input          Maple        Maxima          Axiom
    (brackets)
    -1*(x+3)        −x − 3         −x − 3         −x − 3
     2*(x+3)         2x + 6       2(x + 3)        2x + 6
                                 2x−1
(2*x-1)/5+(x+3)/2   9
                    10
                       x  + 13
                            10     5
                                        + x+3
                                            2
                                                   9
                                                  10
                                                         13
                                                     x + 10
 (x-1)^3/(x-1)      (x   − 1)2    (x   − 1)2    x2 − 2x + 1

                    x2 −2x+1      x2 −2x+1
(x^2-2*x+1)/(x-1)      x−1           x−1
                                                   x−1
                    1 9x2 +3x      9x2 +3x
(9*x^2+3*x)/(3*x)   3    x            3x
                                                  3x + 1
     (other)
    log(x^2)         ln x2        2 log(x)       log x2
    log(x^y)         ln (xy )     y log(x)       log (xy )
  log(exp(x))        ln (ex )           x           x
  exp(log(x))              x            x           x
     cos(-x)         cos(x)        cos(x)         cos(x)
20

Issue: technical problems
  • Mixed data types in polynomials
    x/3 + 0.5?
  • Unary minus (no simplification).
                        1       −1       1
                   −       ,       , or     .
                       1−x     1−x      x−1

  • Display,
      1. Implicit multiplication, (xy, x · y, x × y)
      2. i vs j,
         √        1
      3. x vs x 2 .
21

Language

Do we have a way to talk about these fine details?
Unhelpful phrases:
   • simplify,
           1         1000
     e.g.22 = 4 or 22     = · · ·?
   • “move over”
22

Checking for properties

To mark

Example question 2
Give an odd function.

1. calculate f (x) + f (−x),
2. simplify,
3. check equality to zero.
23

Creating examples/instances

Some questions ask for examples of objects.
They require higher level thinking.
Such questions are rare. (11.5 questions from 486 ≈ 2.4%)
Pointon and Sangwin, 2003


Perhaps because they are time consuming to mark.
STACK may mark some questions of this style.
Exemplar questions
24

Students’ answers

Students show great variety in their answer, and method.
For example, 190 students were asked for two functions that
satisfy f (1) = 0.
Their answers were marked automatically.
25

The students (N = 190) gave 93 ‘different’ answers.




                          Frequency




                                                     Frequency
            1st Answer                2nd Answer
            x2 − 2x       45          x3 − 3x        29
            x2
             2 −x         31          x2 − 2x        10
            x3                        x3
             3 −x         11           3 −x          9
            x2 − 2x + 1   7           (x − 1)2       8
                                      x4
            x2 − 2x + 3   7            4 −x          8
            (x − 1)2      5           x4 − 4x        5
            2x2 − 4x      5           ex−1 − x       1
            x3   x2
             3 − 2        5           ex−1 + e−x+1   1
            0             4           ln(x) − x      1
26

Two strategies emerged:

        JL: Ok, just take the parabola and shift it one.
        ···
        B: I said, x − 1 = 0, then integrated it.
27

These problems can be used to generate (short) discussions.

   • sorting the data,
   • methods used,
   • ‘exotic’ examples.
                                                           −1
         f1 (x) = 0,   f2 (x) = |x|(x − 2),   f3 (x) = e (x−1)2 .
28

Automatic feedback

Sophisticated automatic feedback may be provided by
computer algebra systems.
This
   • is immediate,
   • is based on properties of students’ answers,
   • could be positive and encouraging,
   • may be based on common mistakes,
   • may be based on common misconceptions.
29

Common misconceptions

Computer algebra can also test for a type of incorrect answer.
Misconceptions may be identified by
   • educational research,
   • previous teaching experience,
   • examining answers from previous students
30

Odd functions

On examining the odd functions given by students,
the majority of coefficients (= 1) are odd,
eg
                   3x5 , 5x7 , 7x5 − 3x.
Students’ concept image of an odd function requires odd coeffi-
cients.
Furthermore, f (x) = 0 is odd, but was absent.
31

Functions that are odd and even.

When asked for a function that was both odd and even
35% gave the correct answer (eventually),
35% failed to answer the question.
Incorrect answers revealed that 24% of the students added an
odd and even function.
Examples include

                x + x2 ,   x2 + x3 ,   x5 − x6 .

The computer algebra system can test for these misconceptions.
32

Student feedback

What do you like about the system? Did you have any difficul-
ties? If so please describe them.

Feedback & partial credit
     i like the way that you are given credit if your an-
     swer is partially correct and also given guidance on
     achieving the full mark for that question.



     I like the fact that feedback is immediate, but I do
     not like the fact that if I get an answer wrong I do
     not know where in my working I have made the error
33

Random questions
    The questions are of the same style and want the
    same things but they are subtly different which means
    you can talk to a friend about a certain question but
    they cannot do it for you. You have to work it all
    out for yourself which is good.




Syntax problems
    I feel the aim system is reasonably fair, however i
    have lost a lot of marks in quiz 3 for simple syntax
    errors
34

Give me an example...
    Recognising the turning points of the functions pro-
    duced in question 2 was impressive, as there are a
    lot of functions with stationary points at x=1 and
    it would be difficult to simply input all possibilities
    to be recognised as answers.
35

Authoring questions

In authoring, there is tension:
  1. Ability to use all features of CAS.
  2. Ease of writing questions.
     Not making question authors into programmers.
36

Conclusion

Some important questions
   • For what purposes is this tool useful?
   • What properties do we want?
      – Not “looks correct”.
      – Not “select the correct answer”.
   • What feedback should we give?

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Computer Aided Assessment (CAA) for mathematics

  • 1. Computer Aided Assessment (CAA) for mathematics Chris Sangwin & Simon Hammond Copyright c Last Revision Date: June 1, 2009
  • 2. 2 Introduction. ... NOT multiple choice questions ... • Computer aided assessment (CAA) • CAA with computer algebra • Practical issues Implementations • Pedagogical issues • Future directions
  • 3. 3 JEM - Joining Educational Mathematics eContentPlus Thematic Network http://jem-thematic.net/ Founder members (15): Universitat Politecnica de Catalunya, Helsingin Yliopisto, Tech- nical University, Jacobs University, Universiteit van Amster- dam, University of Birmingham, FernUniversitt Hagen, Maths for More, NAG Ltd, Liguori Editore, ISN Oldenburg GmbH, RWTH Aachen University, Univ. Nacional de Educacin a Dis- tancia, Universitat Oberta de Catalunya, Universidade de Lis- boa.
  • 4. 4 Use of objective tests Consider the following question: Example question 1 Determine the following integral: cos(x) sin(2x)dx. As a multiple choice question: ◦ (2/3) cos3 (x) + C ◦ −(2/3) cos(x) + (2/3) sin3 (x) + C ◦ −(2/3) cos(x) + (1/3) sin(x) sin(2x) + C ◦ Don’t know. How do we know the students don’t differentiate the candidate solutions to check?
  • 5. 5 Computer algebra marking Computer algebra systems can be used to mark work. This checks for algebraic equivalence. (x + 1)2 ≡ x2 + 2x + 1 Useful for marking many routine problems.
  • 6. 6 Fundamental idea if simplify(sa-ta) = 0 then mark := 1 else mark := 0
  • 7. 7 STACK System overview The STACK system: • internet based CAA system, • uses very simple Maxima (computer algebra), and L TEX (type setting) A • All components open source (e.g. GPL).
  • 8. 8 Demonstrating the STACK system http://www.stack.bham.ac.uk/
  • 9. 9 In learning and teaching We are assessing a student provided answer. This is an objective test. This is • not Multiple Choice Question; • not string/regex match. Other tests for the form of an answer.
  • 10. 10 Input of mathematics This is a fundamental but unsolved problem. There are a number of options 1. Strict CAS syntax. eg. 2*(x-1)*(x+1) 2. “informal” linear text syntax. eg. 2(x-1)(x+1) x(t-1) ? 3. Graphical input tool. eg. equation editor. 4. (Pen-based input ?) 5. (Geometry applet ?) Not all groups of students are equal.
  • 11. 11 Syntax innovations Difficult to achieve! Babbage 1830’s “a profusion of notations [...] which threaten, if not duly cor- rected, to multiply our difficulties instead of promoting our progress” Babbage, C. (1827) sin2 (x) sin−1 (x) sin sin x = sin2 x (composition)
  • 12. 12 Structure in random problem sets In practice, the numbers often do not matter. Tuckey, C. O., Examples in Algebra, Bell & Sons, London, (1904) Too much randomization destroys structure. An underlying question space.
  • 13. 13 Workshop task Option A: Context: end of first calculus course. (Age 18) Write 6 questions which test whether a student can differentiate elementary functions. E.g. Differentiate cos(3x) with respect to x. Option B: Context: age 11. Write 6 questions which test whether a student can add frac- tions.
  • 14. 14 Randomization 1. What could you randomize? 2. What would you randomize? 3. What are some likely incorrect answers? 4. What feedback would you like to provide? ... with a view to implementing these questions live.
  • 15. 15 Issues • Well-posed questions. • Fair questions. • Structure in question sets. Schemes of work, vs isolated questions. • Algebraic form of answers as a goal.
  • 16. 16 Feedback One third of feedback interventions decreased performance. Kluger, A. N. and DeNisi, A., Psychological Bulletin (1996). The nature of feedback determines its effectiveness.
  • 17. 17 Processing answers Test for algebraic equivalence if simplify(sa-ta) = 0 then mark := 1 else mark := 0 Using mainstream CAS • Get a lot very quickly, Great for calculus and beyond. • Elementary algebra can be a problem. Maxima seems to be more suitable than most.
  • 18. 18 Every CAS is different! Input Maple Maxima Axiom (numbers) 0.5-1/2 0.0 √ 0.0 0.0 4^(1/2) √4 2 2 1 1 1 4^(-1/2) 4 √ 4 2 √2 -4^(1/2) −4 2i 2√−1 sqrt(-4) 2i 2i 2 −1 (indices) a^n*b*a^m an bam an+m b bam an (a^(1/2))^2 √a a √a (a^2)^(1/2) a2 |a| a2 (collecting terms) 1+x^2-2*x x2 − 2x + 1 x2 − 2x + 1 x2 − 2x + 1 x/3+1.5*x+1/3 1.833x + .333 · · · 1.833x + 1 3 1.833x + 0.333 · · · 5 5x 5 3*x/4+x/12 6 x 6 6 x 5 1 5 5 3/(4*x)+1/(12*x) 6 x 6x 6x
  • 19. 19 Input Maple Maxima Axiom (brackets) -1*(x+3) −x − 3 −x − 3 −x − 3 2*(x+3) 2x + 6 2(x + 3) 2x + 6 2x−1 (2*x-1)/5+(x+3)/2 9 10 x + 13 10 5 + x+3 2 9 10 13 x + 10 (x-1)^3/(x-1) (x − 1)2 (x − 1)2 x2 − 2x + 1 x2 −2x+1 x2 −2x+1 (x^2-2*x+1)/(x-1) x−1 x−1 x−1 1 9x2 +3x 9x2 +3x (9*x^2+3*x)/(3*x) 3 x 3x 3x + 1 (other) log(x^2) ln x2 2 log(x) log x2 log(x^y) ln (xy ) y log(x) log (xy ) log(exp(x)) ln (ex ) x x exp(log(x)) x x x cos(-x) cos(x) cos(x) cos(x)
  • 20. 20 Issue: technical problems • Mixed data types in polynomials x/3 + 0.5? • Unary minus (no simplification). 1 −1 1 − , , or . 1−x 1−x x−1 • Display, 1. Implicit multiplication, (xy, x · y, x × y) 2. i vs j, √ 1 3. x vs x 2 .
  • 21. 21 Language Do we have a way to talk about these fine details? Unhelpful phrases: • simplify, 1 1000 e.g.22 = 4 or 22 = · · ·? • “move over”
  • 22. 22 Checking for properties To mark Example question 2 Give an odd function. 1. calculate f (x) + f (−x), 2. simplify, 3. check equality to zero.
  • 23. 23 Creating examples/instances Some questions ask for examples of objects. They require higher level thinking. Such questions are rare. (11.5 questions from 486 ≈ 2.4%) Pointon and Sangwin, 2003 Perhaps because they are time consuming to mark. STACK may mark some questions of this style. Exemplar questions
  • 24. 24 Students’ answers Students show great variety in their answer, and method. For example, 190 students were asked for two functions that satisfy f (1) = 0. Their answers were marked automatically.
  • 25. 25 The students (N = 190) gave 93 ‘different’ answers. Frequency Frequency 1st Answer 2nd Answer x2 − 2x 45 x3 − 3x 29 x2 2 −x 31 x2 − 2x 10 x3 x3 3 −x 11 3 −x 9 x2 − 2x + 1 7 (x − 1)2 8 x4 x2 − 2x + 3 7 4 −x 8 (x − 1)2 5 x4 − 4x 5 2x2 − 4x 5 ex−1 − x 1 x3 x2 3 − 2 5 ex−1 + e−x+1 1 0 4 ln(x) − x 1
  • 26. 26 Two strategies emerged: JL: Ok, just take the parabola and shift it one. ··· B: I said, x − 1 = 0, then integrated it.
  • 27. 27 These problems can be used to generate (short) discussions. • sorting the data, • methods used, • ‘exotic’ examples. −1 f1 (x) = 0, f2 (x) = |x|(x − 2), f3 (x) = e (x−1)2 .
  • 28. 28 Automatic feedback Sophisticated automatic feedback may be provided by computer algebra systems. This • is immediate, • is based on properties of students’ answers, • could be positive and encouraging, • may be based on common mistakes, • may be based on common misconceptions.
  • 29. 29 Common misconceptions Computer algebra can also test for a type of incorrect answer. Misconceptions may be identified by • educational research, • previous teaching experience, • examining answers from previous students
  • 30. 30 Odd functions On examining the odd functions given by students, the majority of coefficients (= 1) are odd, eg 3x5 , 5x7 , 7x5 − 3x. Students’ concept image of an odd function requires odd coeffi- cients. Furthermore, f (x) = 0 is odd, but was absent.
  • 31. 31 Functions that are odd and even. When asked for a function that was both odd and even 35% gave the correct answer (eventually), 35% failed to answer the question. Incorrect answers revealed that 24% of the students added an odd and even function. Examples include x + x2 , x2 + x3 , x5 − x6 . The computer algebra system can test for these misconceptions.
  • 32. 32 Student feedback What do you like about the system? Did you have any difficul- ties? If so please describe them. Feedback & partial credit i like the way that you are given credit if your an- swer is partially correct and also given guidance on achieving the full mark for that question. I like the fact that feedback is immediate, but I do not like the fact that if I get an answer wrong I do not know where in my working I have made the error
  • 33. 33 Random questions The questions are of the same style and want the same things but they are subtly different which means you can talk to a friend about a certain question but they cannot do it for you. You have to work it all out for yourself which is good. Syntax problems I feel the aim system is reasonably fair, however i have lost a lot of marks in quiz 3 for simple syntax errors
  • 34. 34 Give me an example... Recognising the turning points of the functions pro- duced in question 2 was impressive, as there are a lot of functions with stationary points at x=1 and it would be difficult to simply input all possibilities to be recognised as answers.
  • 35. 35 Authoring questions In authoring, there is tension: 1. Ability to use all features of CAS. 2. Ease of writing questions. Not making question authors into programmers.
  • 36. 36 Conclusion Some important questions • For what purposes is this tool useful? • What properties do we want? – Not “looks correct”. – Not “select the correct answer”. • What feedback should we give?