## Abstract

Quasi-Arithmetic Scoring Theory grew out of a desire to justify peer assessment as a meaningful educational measurement technique. Peer assessment is used to evaluate group work and other collaborative learning activities. Students evaluate each other’s participation to joint (project) assignments. The evaluations are used to split an overall team score into individual student scores to inform student grading. So far, a formal treatment of peer assessment methodology has been lacking.

Traditionally, peer assessment uses bounded scales (equivalent to the standard percentage scale) together with a single n-ary operation: the weighted arithmetic mean. This practice is rather weak and questionable. Quasi-Arithmetic Scoring Theory puts peer assessment and judgmental scales in general on a sound and strong mathematical foundation. The bounded scales will be equipped with two basic operations (different from their arithmetic counterparts) so that they get the structure of a module. Peer assessment is then modelled by bivariate module theory of one bounded scale of peer ratings acting on another bounded scale of student scores.

A bounded scale is an interval [푚푖푛, 푚푎푥] of scores 푠 such that 푚푖푛 ≤ 푠 ≤ 푚푎푥 together with quasiarithmetic operations of addition and scalar multiplication of scores, where scores may be multidimensional composites, i.e. consisting of subscores (items or criteria). Such bounded scales are mathematical structures called modules. Modules have the right structure for educational metrics and related statistics. The percentage scale [0,1] plays the role of standard bounded scale. All properties valid for the standard scale hold for all bounded scales, e.g. the popular Likert scales.

To model peer assessment, Quasi-Arithmetic Scoring Theory needs to be extended to bivariate modules, i.e., a bounded scale of peer ratings acting upon another bounded scale of student scores. The assessor sets a group score as the default student score. Also, he specifies a scoring rule to define the action of peer ratings on student scores. The action of peer ratings on scores can be constrained to a subscale around the group score so that unrealistic deviations from the group score will be avoided.

Moreover, the impact of peer ratings on scores can be made weaker or stronger

There exist three distinct types of scoring rules for peer assessment. Besides the traditional arithmetic type of scoring rule, there are two types of scoring rules using modules with addition, scalar multiplication and quasi-arithmetic mean as main operations. The traditional approach is deceptively simple requiring

nothing more than knowledge of arithmetic and its related concept of arithmetic average – but it is weak and inadequate, resting on questionable assumptions about the calculus of scores. The modern approach based on modules. distinguishes between quasi-additive and quasi-multiplicative scoring rules,

using either one of the two module operations. All three scoring rules come with several extensions which allow fine-tuning of the underlying scoring models to tutor preferences, course specifics, or university regulations.

Traditionally, peer assessment uses bounded scales (equivalent to the standard percentage scale) together with a single n-ary operation: the weighted arithmetic mean. This practice is rather weak and questionable. Quasi-Arithmetic Scoring Theory puts peer assessment and judgmental scales in general on a sound and strong mathematical foundation. The bounded scales will be equipped with two basic operations (different from their arithmetic counterparts) so that they get the structure of a module. Peer assessment is then modelled by bivariate module theory of one bounded scale of peer ratings acting on another bounded scale of student scores.

A bounded scale is an interval [푚푖푛, 푚푎푥] of scores 푠 such that 푚푖푛 ≤ 푠 ≤ 푚푎푥 together with quasiarithmetic operations of addition and scalar multiplication of scores, where scores may be multidimensional composites, i.e. consisting of subscores (items or criteria). Such bounded scales are mathematical structures called modules. Modules have the right structure for educational metrics and related statistics. The percentage scale [0,1] plays the role of standard bounded scale. All properties valid for the standard scale hold for all bounded scales, e.g. the popular Likert scales.

To model peer assessment, Quasi-Arithmetic Scoring Theory needs to be extended to bivariate modules, i.e., a bounded scale of peer ratings acting upon another bounded scale of student scores. The assessor sets a group score as the default student score. Also, he specifies a scoring rule to define the action of peer ratings on student scores. The action of peer ratings on scores can be constrained to a subscale around the group score so that unrealistic deviations from the group score will be avoided.

Moreover, the impact of peer ratings on scores can be made weaker or stronger

There exist three distinct types of scoring rules for peer assessment. Besides the traditional arithmetic type of scoring rule, there are two types of scoring rules using modules with addition, scalar multiplication and quasi-arithmetic mean as main operations. The traditional approach is deceptively simple requiring

nothing more than knowledge of arithmetic and its related concept of arithmetic average – but it is weak and inadequate, resting on questionable assumptions about the calculus of scores. The modern approach based on modules. distinguishes between quasi-additive and quasi-multiplicative scoring rules,

using either one of the two module operations. All three scoring rules come with several extensions which allow fine-tuning of the underlying scoring models to tutor preferences, course specifics, or university regulations.

Original language | English |
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Title of host publication | Proceeedings of the 12th Annual International Conference of Education, Research and Innovation, iCERi 2019 |

Number of pages | 11 |

Publication status | Accepted/In press - 29 Sep 2019 |

Event | 12th annual International Conference of Education, Research and Innovation - Seville, Spain Duration: 12 Nov 2019 → 13 Nov 2019 |

### Conference

Conference | 12th annual International Conference of Education, Research and Innovation |
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Country/Territory | Spain |

City | Seville |

Period | 12/11/19 → 13/11/19 |