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The system’s student model is the core of this thesis. Its aim is to represent the student’s knowledge in order to determine the student’s zone of proximal development (cf. subsection 2.1.1) and find out what exercise will be most beneficial to the student’s learning at any given moment.

In subsection 2.4.2 we presented an overview of student models, and noted that they often form an overlay over a domain model. We will therefore take the domain model presented in the previous subsection as the starting point, make an overlay that represents student’s knowledge, and then extend it further.

As already noted in this chapter, the interaction with the program cannot give certain evidence about what the user knows, and therefore we need a probabilistic model, such as a Bayesian network. We make a simplifying assumption that words and constructions selected as learning targets are the only parameters that determine the difficulty of a sentence. We also assume that the learning targets can be in one of only two states in the learner’s mind: known and unknown, and the probability is used to express incomplete knowledge about these states. We can now structure the core of the Bayesian network after the domain model. The edges can indicate the direction of influence: knowing all the learning targets that appear in an exercise has a positive influence on the probability of finishing the exercise. The information about the student’s vocabulary knowledge can be represented in the conditional probability tables (cf. subsection 3.2.1) of the nodes representing learning targets.

The network has a structure that may cause problems: there can be many learning targets in an exercise, and therefore an exercise node may have a large number of parents. We discussed this issue in subsection 3.2.3, and noted that a common solution to such a problem is to use noisy functional dependence. An exercise is likely to be successfully completed only when the user knows all the learning targets that appear in it. It is therefore a natural setting for using the noisy-AND function to create the CPT. One might try to differentiate the inhibition and substitution probabilities for different types of learning targets, depending for example on their word class or frequency. We will, however, make the assumption that all learning targets in all exercises share the same inhibition and substitution probabilities.

The learning target nodes are hidden – we cannot directly observe their values. We can, however, observe the user doing the exercises. Most importantly, we know the outcome of an exercise, whether it was successfully completed or not. We can therefore use Bayesian inference to modify CPT of the hidden nodes as the system gets information about exercise outcomes. The learning outcome can provide quite strong evidence about what the user knows: a successfully completed exercise implies that the user created a sentence with the learning targets related to the exercise, and they are more likely to be known. The lack of success, however, does not indicate which of the learning targets was problematic.

knowledge. As mentioned in subsection 3.1.4, the system can be integrated with a dictionary. A word look-up is an indication that the user does not know a particular word. Since we are concerned with English-to-Chinese translation exercises, the user will most likely look up English words in order to find their Chinese translations. The learning targets need therefore to be associated with lists of English words whose look-up indicates that the learning target is unknown. In order to integrate the evidence obtained from look-ups, we need to create additional nodes in the Bayesian network. We assume that several look-ups of the same word during one exercise count as one, but looking up the same word in different exercises are separate pieces of evidence. Therefore, for each learning target node, we will create look-up nodes, one for each exercise that contains this learning target.

We need to consider the question of prior probability, that is, the assumptions about what the user knows before the system gets any evidence. We could make all the learning targets have 50% probability of being known, but this would not be efficient. For most users, the system would have to separately collect evidence about each individual word. Learners do not learn words randomly, but they do it in a more or less predictable order. Therefore, we can divide learners into levels. While the system can never be sure whether a user at a particular level knows a particular word, some words are definitely more likely to be known by users at some levels, and unknown by users at other, lower levels.

Such defined levels are not directly observable, and we need an indirect way to assess them. We will use two methods: users’ self-assessments about their level of written and spoken Chinese, and a character recognition test. For languages such as English, the ability to tell real written words from made-up ones can be used to estimate the learner’s vocabulary, and indirectly indicate their general written proficiency. In the case of Chinese, there is no unambiguous definition of word, and therefore we will use characters instead, even though they usually represent morphemes rather than words. We can assume that self-assessments and the results of the test will be influenced by the user’s actual level. This, again, lets us use Bayesian inference to use observable data to assess the user’s level.

Figure 3.4 presents the structure of a student model with three exercises and five learning targets. White ovals represent hidden nodes, and grey ovals represent observable nodes. Note thatnumber of charsis a continuous variable, but in order to simplify the inference we may discretise it. The model can be used as follows. The system asks the user to make a self- assessment of written and spoken proficiency in Chinese. Then it asks the user to do a character recognition test, which provides an estimate of the number of character they know. Based on these three values the system performs the inference and calculates the probability distribution of the user’s level. Based on this distribution, for each learning target, it calculates the probability that the user knows that construction. During the session with the program, the evidence about look-ups and exercise outcomes becomes available, and is used to infer new probabilities that the user knows the learning targets.

LT1 LT2 LT3 LT4 LT5 ex1 ex2 ex3 level self-rating (written) self-rating (spoken) number of chars LT1ex1 look-up LT2ex1 look-up LT2ex2 look-up LT3ex2 look-up LT1ex3 look-up LT4ex3 look-up LT5ex3 look-up

Figure 3.4: Example student model represented by a Bayesian network

chapter 5 we will present how they have been estimated. The aim of the student model is to provide data that can be used to select the next exercise. Next subsection provides more information about how this is done.