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Dropping Lowest Grades

Dropping Lowest Grades. What score(s) should be dropped to maximize a students grade? Dustin M. Weege Concordia College 2008 Secondary Mathematics Education. Not always what you think it to be. The Plan. Natural ideas for dropping grades Flaws

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Dropping Lowest Grades

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  1. Dropping Lowest Grades What score(s) should be dropped to maximize a students grade? Dustin M. Weege Concordia College 2008 Secondary Mathematics Education Not always what you think it to be

  2. The Plan • Natural ideas for dropping grades • Flaws • Mention of other possible methods for determining the best scores to drop • Optimal Drop Function

  3. “Natural” ideas for dropping grades • If the teacher is basing the final grade on the student’s raw score • Drop the lowest score earned • Notice the percentages x

  4. “Natural” ideas for dropping grades Also Consider: • Drop the lowest score earned • Notice the percentages x

  5. “Natural” ideas for dropping grades • Drop the lowest percentage • What should be dropped? • Drop 3 (lowest percentage)? • Total percentage of • Drop 2? • Total percentage of x

  6. What do we know? • Highest percentage will always be in the optimal retained set. • Reason: if S has grades that are less than the largest percentage, then the average will be less than the largest percentage. • The reverse is not necessarily true • The grade with the smallest score does not necessarily appear in the optimal deletion set • Ex. Beth

  7. Dropping more than one grade • Conflict arises depending on the number of scores dropped • Remove 1 score: • quiz 4 would be best to drop  63.4% • Remove 2 scores • quizzes 2 & 3 would be best to drop  74.6% x x x

  8. How can we find the best set? • Brute Force • Try all possibilities • Flaw: Can take too long especially with large quantities of scores • Greedy Algorithm • Do the best in each situation • Flaw: ex. Carl • Drop 4 & 3 • Yields a total score of 100+42=142 out of 191  74.3% • Drop 2 & 3 • Yields a total score of 100+3=103 out of 138  74.6% • Compare : (2&4 73.5%; 1&4  38.4%) x x x

  9. Can’t we come up with something? • You know it. • Leave it up to a MIT student and a Professor who received his BA in Mathematics at U of M-Duluth Not actually Daniel M. Kane, but this showed up in a Google Images search Jonathan M. Kane

  10. Terminology - also found on handout • kK – Total number of assignments; s.t. K • j – quiz # (1,2,3,…k) s.t. k>0 • mj – earned points on quiz • nj – possible points on quiz • r – score(s) dropped/”deletion set” • “optimal deletion set” – set of quiz(zes) dropped in order to yield the highest possible grade • k-r – Number of scores counted/“retained” • S – retained grades (*note: SK) • Sbest – Optimal retained set • q – the average score in S • qbest – best possible value for q • “Optimal Drop function” -

  11. Optimal Drop Function • q – Defined as: • qbest – q is defined s.t. the S score is maximized • Define for every j: • By substitution we get: Equation 1 Equation 2 Equation 3

  12. Optimal Drop Function • q is the average score in S • is a linear, decreasing function • is also a linear, decreasing function • Example from Carl • iff

  13. Optimal Drop Function • F(q) is the max of the sum of linear, decreasing functions, • F(q) must be a piecewise, linear, decreasing function Equation 4

  14. Optimal Drop Function • F(q) is the max of the sum of linear, decreasing functions, • F(q) must be a piecewise, linear, decreasing function Equation 4

  15. Optimal Drop Function • Now, find the rational number q, so that • Recall Equation 4

  16. Optimal Drop Function • Next, we need to find the line that yields the highest possible q value s.t. . • From this we are able to determine • composed of the top k-r fj(q) values.

  17. Optimal Drop Function • Tasks: • Evaluate each for each j • Identify the k-r largest from the values • S is the set of j values from the largest k-r values • Calculate

  18. Optimal Drop Function – Ex. Carl Drop 2 scores: • Possibilities for S: 1&2 1&3 1&4 2&3 2&4 3&4 • Estimate: q to be .75

  19. Relevance • Determining the best set of scores to drop • Understanding Computer gradebooks • Determine cuts that are necessary to be made in a company based on several assessments

  20. Sources • Daniel M. Kane and Jonathan M. Kane Dropping Lowest GradesMathematics Magazine, (2006) 79 (June) pp. 181-189. • Daniel Kane's Homepage http://web.mit.edu/dankane/www/ • Jonathan Kane Home Page http://faculty.mcs.uww.edu/kanej/kane.htm • http://www.ams.org/news/home-news-2007.html • Daniel Biebighauser’s brain

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