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Due to the lockdown where I am students have taken exams at home. This obviously opens the easy possibility of cheating. While marking, I noticed two scripts had solutions and methods that were remarkably similar if not the exact same. I am thinking of how to gather statistical evidence of such cheating. My question will be: how can we use such evidence to be able to make a quantifiable judgement of potential cheating?

Consider the exam question: Prove that an odd number multiplied by an odd number equals an odd number. Let's say that many solutions followed such a method/template:

  1. Let a and b be some odd numbers,
  2. then a=2n+1 and b=2m+1 for some n and m
  3. then some algebra
  4. then the conclusion that we have an expression of the form 2p+1 which is some odd c.

While marking I would have some students freely use a,b,c,n,m,p as their parameters, others who would use x,y,z,p,q,r, others using a,b,c,x,y,z etc. Some might assume the two numbers need to be the same. Some might skip the question completely. Crucially, there is variation in the choices available to a student at this position in the exam.

Consider another test question: Find the inverse function of $f(x)=\frac{2x+1}{3-2x}$ and the solution could be of the form:

  1. Express with $y=$
  2. use a few algebraic steps
  3. swap x and y
  4. express as $f^{-1}(x)=$

Other students decided to swap first, or do the algebra steps in different ways, or to make particular algebraic errors, or to express the solution as $y=$. So again we have choices and variation.

This can be done for every question, one can collect the number of choices students actually made in their solutions at a particular position. So if the exam has 15 questions I would have potentially 15 (or more) different positions where there is variation (or perhaps none if they all happen to use the same choice). Let's say that the number of variations are $n_1,\ldots,n_{15}$.

Now it turns out that two students happened to have the exact same choice in say 12 of the 15 positions.

So then my question: What would be the best way to use this evidence of $n_1,\ldots,n_{15}$ choices/variations and that in the 12 positions {1,2,3,4,6,7,9,10,12,13,14,15} there was a match and in positions {5,8,11} there was no match, to support the claim of cheating?

I hope the question is on topic, appropriate and clear. Incidentally, when I mentioned this to my bosses, they simply shrugged their shoulders and said that we cannot prove cheating so there's nothing we can do. I want to challenge that statement.

Geoff
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    (1) The forum for 'Academia' might be a better place for this. (2) Students can be heavily influenced by previous similar problems as to order of steps and choice of variable names. This influence may have come before the exam in perfectly legitimate study sessions--especially if held online where there is a trail to re-visit. (3) The people who will decide what is 'proof' may not be sufficiently familiar with probability argument to be convinced by whatever may come from this site. – BruceET Jun 26 '20 at 20:12
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    (4) Over too many decades of teaching, I've encountered similar instances of possible cheating too many times. In my experience, if you wonder about copying instead of hints and collaboration, then format of the answers may hold the strongest clues. Where equations are broken. Specifically how words and equations are interspersed. How fractions are written. Number of places of accuracy carried throughout the argument, and how far. Capital vs lower-case letters. How math is written can be unusually individual. – BruceET Jun 26 '20 at 20:19
  • (5) Esp. now, with hastily arranged online courses, it's important to think ahead how to make cheating harder. (a) Multiple, essentially equiv, versions of Qs. (Just changing key nrs in Qs can make copying obvious. Binomial Q with $n = 5$ will be worked differently than w/ $n = 25;$ with $p=.3$ dif than with $p=1/3.)$ (b) Require tighter sched btw availability of exam and time due. (Not as strict for online as for in-class, but not so flexible that there's lots of time for illegal collaboration.) (c) Rather than forbid using Internet, refer to particular Internet pgs that might be relevant. – BruceET Jun 26 '20 at 20:35
  • I think convincing quantification will be difficult. People unacquainted with course won't know whether you always use $y = mx + b$ in class (so student use unsurprising) or whether you use $Y = \beta_0 + \beta_1x$ in which case $y = mx + b$ by several students might be a surprise. // But it's trying to make a probability argument to a dean who's a historian and a head of fairness cmte who's a musician is not something I'd want to undertake. // Confronting students with specific instances informally may have better effect than probability arguments up the line. – BruceET Jun 26 '20 at 20:48

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