Abstract:

This study aims to propose a system that helps learners learn transformation, and teachers/learners scrutinize and analyze learners' behavior based on their study logs stored in the system. Transforming math expressions requires learners to know the individual parts of the transformation process, such as relevant formulae or theorems. A pilot experiment was conducted to grasp the behavior of individual users, including their trial-and-error attempts for learning analytics.

Introduction:
In learning mathematics, learners are frequently required to transform math expressions. What sometimes occurs to learners is that the expressions are too complex for them to find ways to transform. Reaching an impasse might cause learners to lose their motivation, too. Hence, we propose a web application that recursively displays a list of possible transformations for a designated region of math expressions by the learner. This application allows learners to solve the given problems through trial and error.

Proposed system:
Our system is based on the matching technique of math expressions proposed by (Watabe & Miyazaki, 2015). The system has an equation database (DB) consisting of a number of formulae and theorems. When an expression is entered that matches the left or right side of an equation in the DB, the other side of the equation is presented as a possible transformation. The user selects one expression from the displayed candidates to proceed to the next step of the transformation. The system also provides algorithms to partially match given expressions (e.g., transforming (sin x)^2 + (cos x)^2 + x to 1 + x by matching only (sin x)^2 + (cos x)^2 (equal to 1)) and to perform simple calculations recursively. Equations can be selected from the DB, allowing teachers on-site to adapt the system to suit their curriculum. This study also investigates the use of learners' study logs. For learning analytics, two modules are provided: one to show each user's transformation history and the other to present the types of transformation users are uncomfortable applying. The first module dynamically generates tree diagrams, allowing learners to view transitions and reduce excessive trial-and-error steps promptly. The system also has a function to play phase-by-phase by pushing the "Next" and "Previous" buttons while toggling between the transformation pane and the history pane.

Experiment and results:
Twelve university students were asked to solve three problems, each using the proposed system and in a paper-based (traditional) way. In short, the experiment results showed that the subjects made more varieties of transformations using the system rather than the traditional way. The system provided sufficient data from users' logs to guess when they hit an impasse in solving a problem. Specifically, it was found that the branch points of tree diagrams and math expressions remained untransformed for a certain period synchronized with points at which impasses took place at a high rate.

References:
[1] Watabe, T. & Miyazaki, Y. (2015). Graphical User Interface for Search of Mathematical Expressions with Regular Expressions, The 17th International Conference on Human-Computer Interaction (HCI2015), (published in Human-Computer Interaction: Design and Evaluation, Springer International Publishing, 9169, 438-447).

Keywords:

Mathematical expression, transformation, learning analytics, trial-and-errors, web application.