The work approaches theoretical and implementation issues of a framework aimed at supporting human knowledge acquisition of mathematical concepts. We argue that personalization support can be achieved from problem statement parameters, de ned/set during the creation of Learning Objects (LOs) and integrated with the skill level of learners and problem solution di culty. The last two are formally de ned here as algebraic expressions based on fundamental principles derived from extensive consultations with experts in pedagogy and cognition. Our implemented prototype framework, called ADAPTFARMA, includes a collaborative authoring and learning environment that allows short- and long-term interactions. We present our ongoing research about student modeling to support personalization. Finally, we draw conclusions about the suitability of the claims and brie y direct the reader's attention to future research.
The personalization in computer-based learning systems can range from simple student preferences to motivational state detection. Besides, the system is expected to adapt to speci c learning needs, including di erent assessment mechanisms. In Algebra, the student's expertise is usually developed by solving problems that require a set of assessed skills. This is done in both conventional education schools and by applying advanced learning technologies, such as Intelligent Tutoring Systems (ITS). Normally, human teachers detect students' misconceptions when marking tests and exercises. Depending on how much the answer of a question departs from its correct version, two students that missed the same question could be scored di erent grades for that speci c question.
Another aspect that can be used to compose the score is how di cult the
question is. The di culty degree of an exercise can be measured by the number of
students that have skipped or made a mistake in that exercise. Thus, a student
who nds the correct answer of a question that many missed, probably has
more skills than others and the score should re ect that. Conversely, a student
who makes a mistake in a question that many were successful to answer, might
possess fewer skills. A student error can basically be used as a guideline for
two actions: simply to assess the student or to detect misconceptions towards a
more e ective pedagogical practice. In the latter sense, recently, there has been
increasing interest for direct use of errors as a source of teaching material, in
order to learn more deeply about the content of the domain and thus develop
metacognitive skills [
Another desirable aspect in ITS is in predicting or prospecting whether a learner will be able to answer a question correctly or not before it is actually showed to him or her, allowing a more e ective personalization support. Usually this kind of feature requires that questions be previously calibrated according to their di culty and matched to the assessed student's skills. 2
Segedy et al. [
A manner to support personalization is by implementing algorithms that
generate di erent content sequencing according to a learner's needs. In this sense,
Champaign and Cohen propose an algorithm [
Ravi and Sosnovsky [
Rating systems are frequently used in games to measure the players skills and to rank them. Usually, the rating is a number in a range [minRank; maxRank] such that it is very unlikely that a player falls on the extremes. Inspired by game rating systems and taking the performance of other learners, this study proposes Equation 1 to assess iteratively a student's ability.
The following guidelines were adopted: (1) each question is scored a di culty degree with a Real value in the range [0::10] and the student is rated a number in the range [1::10] to express his or her expertise level in the subject matter; (2) the easier the question, the greater the likelyhood that student will answer it correctly (in this case, a student's rating should have just a small increase if he or she enters the correct answer and should have a large decrease in the case of failure); (3) students that are successful in the rst attempt to solve a question are scored a higher increment in their expertise level compared to those who need several attempts; (4) skipped questions are considered wrong.
Consider Equation 1. The details of its parameters are as follows: (1) (2) RJq = RJq 1 + Ak1 (10 9TJq ) q Tmed
Ek2 { RJq : student J 's rating after answering question q. RJ0 = 5:5 (initial rating); { A = 1 and E = 0 for successful in answering q, otherwise A = 0 and E = 1; { T q: number of unsuccessful attempts of student J to answer question q; J q { Tmed: median of wrong attempts on question q during classroom time; { = N1aq and = N1eq are weight factors to increase and decrease the rating respectively (Naq and Neq are the number of students that were successful and unsuccessful answering question q, respectively); { k1 and k2: multiplier factors of rating increase and decrease, respectively, calculated by k1 = 1 R1Jq0 1 and k2 = RJq101 1 .
Although there is no limit to the number of attempts a student can make to answer a question, for calculation purposes, 10 trials is considered the maximum. Factors k1 and k2 avoid results of the expression in Equation 1 to reach upper and lower bounds of the range [1::10].
Using only the number of attempts, the di culty degree of a question q can be de ned by Equation 2 and its parameters are as follows:
Dq =
PJJ==0n T q
J Neq + Na q { Dq: di culty degree of the question q after an exercise session; { tThJqe: nnuummbbeerr ooff autntesumcpctesssifsuglraetatteemr pthtsanof1s0tutrdieanlst, Jthteon a1n0siwsetrakqeunesatsioTnJqq;. If { Neq and Naq are the same as in Equation 1 4
The ADAPTFARMA (Adaptive Authoring Tool for Remediation of errors with
Mobile Learning) prototype software tool is a modi ed version of FARMA [
For each question, the teacher-author must set a reference solution, which is the correct response to the question. ADAPTFARMA allows arithmetic and algebraic expressions to be entered as the reference solution. Under the learner's functioning mode, the tool deals automatically with the equivalence between the learners response and the reference solution.
An important feature of ADAPTFARMA is the capability of backtracking the teacher to the exact context in which the learner made a mistake. It allows the teacher to view a learner's complete interaction with the tool in the chronological order by means of a graphical timeline. In addition, he/she can perform a closer monitoring of problem solutions from other classroom students, as long as system permission is given through the collaboration mechanisms. Likewise, learners can backtrack to the context of any of their right or wrong answers in order to re ect about their own solution steps and nd new solution hypotheses. Additionally, on the collaborative side, it is possible for the teacher to carry out a review of students' responses and then provide them with non-automatic feedback, which can be done by exchanging remote messages through the system. 5
The ADAPTFARMA environment was designed such that di erent pedagogical strategies can be used and tested. In this study, we propose an algorithm for sequencing exercises, named Adaptive Sequencing Method (ASM), to be shown in ascending order of di culty, combined with a mechanism similar to numerical interpolation. We carried out an experiment with 149 highschool students, aging fteen to seventeen, including pre- and post-tests. The results demonstrate that there has been a signi cant increase between pre- and post-test scores of students that were subject to ASM (p-value = 0:0037). However, there has been no signi cant di erence in student score gains between ASM-determined and teacher-de ned sequencing methods.
A minimal sequence of exercises is de ned such that it always begins with the easiest exercise and nishes with the most di cult one. The intermediate-level exercises in the minimal sequence are distributed evenly among the easiest and most di cult exercises such that the number of exercises is l stepnsize m, where n is the total of exercises and the stepsize, set by the LO's author, refers to the number of exercises that may be skipped when the student is successful.
Initially, the algorithm presents the exercises in the minimal sequence order. If the number of attempts in an exercise reaches the average number of attempts obtained in the calibration phase,the next exercise presented to the student is of a mid range di culty, considering the last exercise correctly answered and the current one. Unlike the calibration phase, the student cannot skip exercises and if he/she continually misses the correct answer, the presentation becomes strictly sequential.
Our ongoing research related to student modeling, including the learner interaction and context, is based on problem statement parameters and ranking solution di culty in order to support personalization. During the creation of the LO, the teacher-author sets certain parameters that a ect the pedagogical strategy, as follow: { maximum number of retries (attempts) per question; { tips for each question; { remediation rules for each question; { prerequisites for the solution of the exercise, that can be topics, theoretical pages of the LO itself or other LOs in ADAPTFARMA; { di culty degree for each question in the range [1 10], such that [1 2] means very easy, [3 4] means easy, [5 6] means medium, [7 8] means di cult and [9 10] means very di cult; { exercises sequencing strategy, that can be di culty-biased, teacher-de ned or ASM-determined (presented in the previous section).
The student pro le is assembled from the previous parameters. By analysing the tips used and relating them to associated prerequisites, the system can provide feedback to both teacher and student on topics that should be further explored or even recommend other complete LOs to be inspected. In addition, the di culty degree of the questions and the student rating can be updated after each problem solving session has nished.
The personalization support in learning systems can include adaptive mechanisms of assessment and generation of di erent content sequencing. We proposed an automatic rating system that can be used as an additional tool to assess students. Depending on the number of attempts and the di culty degree of a question, di erent students can get di erent scores for the same solution. Also, we proposed an algorithm for sequencing exercises using a formalization of the intuitive notion of di culty degree combined with a mechanism similar to numerical interpolation. All that was implemented in the ADAPTFARMA environment, a web authoring tool for creating and executing LOs.
Future research concentrates in adding new features to ADAPTFARMA in
two ways. Firstly, we are working in a deeper approach to user adaptation that
includes more dimensions than just the matching between problem di culty and
student skill. One such new feature will be a function for generating problem
statements based on teacher-de ned problem template parameters as in [