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Modeling Individualization in a Bayesian Networks

Implementation of Knowledge Tracing

Zachary A. Pardos1, Neil T. Heffernan

Worcester Polytechnic Institute

Department of Computer Science

zpardos@wpi.edu, nth@wpi.edu

Abstract. The field of intelligent tutoring systems has been using the well

known knowledge tracing model, popularized by Corbett and Anderson (1995),

to track student knowledge for over a decade. Surprisingly, models currently in

use do not allow for individual learning rates nor individualized estimates of

student initial knowledge. Corbett and Anderson, in their original articles, were

interested in trying to add individualization to their model which they

accomplished but with mixed results. Since their original work, the field has not

made significant progress towards individualization of knowledge tracing

models in fitting data. In this work, we introduce an elegant way of formulating

the individualization problem entirely within a Bayesian networks framework

that fits individualized as well as skill specific parameters simultaneously, in a

single step. With this new individualization technique we are able to show a

reliable improvement in prediction of real world data by individualizing the

initial knowledge parameter. We explore three difference strategies for setting

the initial individualized knowledge parameters and report that the best strategy

is one in which information from multiple skills is used to inform each

student’s prior. Using this strategy we achieved lower prediction error in 33 of

the 42 problem sets evaluated. The implication of this work is the ability to

enhance existing intelligent tutoring systems to more accurately estimate when

a student has reached mastery of a skill. Adaptation of instruction based on

individualized knowledge and learning speed is discussed as well as open

research questions facing those that wish to exploit student and skill

information in their user models.

Keywords: Knowledge Tracing, Individualization, Bayesian Networks, Data

Mining, Prediction, Intelligent Tutoring Systems

1 Introduction

Our initial goal was simple; to show that with more data about students’ prior

knowledge, we should be able to achieve a better fitting model and more accurate

prediction of student data. The problem to solve was that there existed no Bayesian

network model to exploit per user prior knowledge information. Knowledge tracing

1 National Science Foundation funded GK-12 Fellow

Pardos, Z. A., Heffernan, N. T. (2010) Modeling Individualization in a Bayesian Networks

Implementation of Knowledge Tracing. In Proceedings of the 18th International Conference on User

Modeling, Adaptation and Personalization. pp. 255-266. Big Island, Hawaii.

2 Zachary A. Pardos, Neil T. Heffernan

(KT) is the predominant method used to model student knowledge and learning over

time. This model, however, assumes that all students share the same initial prior

knowledge and does not allow for per student prior information to be incorporated.

The model we have engineered is a modification to knowledge tracing that increases

its generality by allowing for multiple prior knowledge parameters to be specified and

lets the Bayesian network determine which prior parameter value a student belongs to

if that information is not known before hand. The improvements we see in predicting

real world data sets are palpable, with the new model predicting student responses

better than standard knowledge tracing in 33 out of the 42 problem sets with the use

of information from other skills to inform a prior per student that applied to all

problem sets. Equally encouraging was that the individualized model predicted better

than knowledge tracing in 30 out of 42 problem sets without the use of any external

data. Correlation between actual and predicted responses also improved significantly

with the individualized model.

1.1 Inception of knowledge tracing

Knowledge tracing has become the dominant method of modeling student knowledge.

It is a variation on a model of learning first introduced by Atkinson in 1972 [1].

Knowledge tracing assumes that each skill has 4 parameters; two knowledge

parameters and two performance parameters. The two knowledge parameters are:

initial (or prior) knowledge and learn rate. The initial knowledge parameter is the

probability that a particular skill was known by the student before interacting with the

tutor. The learn rate is the probability that a student will transition between the

unlearned and the learned state after each learning opportunity (or question). The two

performance parameters are: guess rate and slip rate. The guess rate is the probability

that a student will answer correctly even if she does not know the skill associated with

the question. The slip rate is the probability that a student will answer incorrectly even

if she knows the required skill. Corbett and Anderson introduced this method to the

intelligent tutoring field in 1995 [2]. It is currently employed by the cognitive tutor,

used by hundreds of thousands of students, and many other intelligent tutoring

systems to predict performance and determine when a student has mastered a

particular skill.

It might strike the uninitiated as a surprise that the dominant method of modeling

student knowledge in intelligent tutoring systems, knowledge tracing, does not allow

for students to have different learn rates even though it seems likely that students

differ in this regard. Similarly, knowledge tracing assumes that all students have the

same probability of knowing a particular skill at their first opportunity.

In this paper we hope to reinvigorate the field to further explore and adopt models

that explicitly represent the assumption that students differ in their individual initial

knowledge, learning rate and possibly their propensity to guess or slip.

1.2 Previous approaches to predicting student data using knowledge tracing

Corbett and Anderson were interested in implementing the learning rate and prior

knowledge individualization that was originally described as part of Atkinson’s model

Modeling Individualization in a Bayesian Networks Implementation of Knowledge Tracing

3

of learning. They accomplished this but with limited success. They created a two step

process for learning the parameters of their model where the four KT parameters were

learned for each skill in the first step and the individual weights were applied to those

parameters for each student in the second step. The second step used a form of

regression to fit student specific weights to the parameters of each skill. Various

factors were also identified for influencing the individual priors and learn rates [3].

The results [2] of their work showed that while the individualized model’s predictions

correlated better with the actual test results than the non-individualized model, their

individualized model did not show an improvement in the overall accuracy of the

predictions.

More recent work by Baker et al [4] has found utility in the contextualization of the

guess and slip parameters using a multi-staged machine-learning processes that also

uses regression to fine tune parameter values. Baker’s work has shown an

improvement in the internal fit of their model versus other knowledge tracing

approaches when correlating inferred knowledge at a learning opportunity with the

actual student response at that opportunity but has yet to validate the model with an

external validity test.

One of the knowledge tracing approaches compared to the contextual guess and

slip method was the Dirichlet approach introduced by Beck et al [5]. The goal of this

method was not individualization or contextualization but rather to learn plausible

knowledge tracing model parameters by biasing the values of the initial knowledge

parameter. The investigators of this work engaged in predicting student data from a

reading tutor but found only a 1% increase in performance over standard knowledge

tracing (0.006 on the AUC scale). This improvement was achieved by setting model

parameters manually based on the authors understanding of the domain and not by

learning the parameters from data.

1.3 The ASSISTment System

Our dataset consisted of student responses from The ASSISTment System, a web

based math tutoring system for 7th-12th grade students that provides preparation for

the state standardized test by using released math problems from previous tests as

questions on the system. Tutorial help is given if a student answers the question

wrong or asks for help. The tutorial help assists the student learn the required

knowledge by breaking the problem into sub questions called scaffolding or giving

the student hints on how to solve the question.

2 The Model

Our model uses Bayesian networks to learn the parameters of the model and predict

performance. Reye [6] showed that the formulas used by Corbett and Anderson in

their knowledge tracing work could be derived from a Hidden Markov Model or

Dynamic Bayesian Network (DBN). Corbett and colleagues later released a toolkit [7]

using non-individualized Bayesian knowledge tracing to allow researchers to fit their

own data and student models with DBNs.

4 Zachary A. Pardos, Neil T. Heffernan

2.1 The Prior Per Student model vs. standard Knowledge Tracing

The model we present in this paper focuses only on individualizing the prior

knowledge parameter. We call it the Prior Per Student (PPS) model. The difference

between PPS and Knowledge Tracing (KT) is the ability to represent a different prior

knowledge parameter for each student. Knowledge Tracing is a special case of this

prior per student model and can be derived by fixing all the priors of the PPS model to

the same values or by specifying that there is only one shared student ID. This

equivalence was confirmed empirically.

Fig. 1. The topology and parameter description of Knowledge Tracing and PPS

The two model designs are shown in Figure 1. Initial knowledge and prior knowledge

are synonymous. The individualization of the prior is achieved by adding a student

node. The student node can take on values that range from one to the number of

students being considered. The conditional probability table of the initial knowledge

node is therefore conditioned upon the student node value. The student node itself

also has a conditional probability table associated with it which determines the

probability that a student will be of a particular ID. The parameters for this node are

fixed to be 1/N where N is the number of students. The parameter values set for this

node are not relevant since the student node is an observed node that corresponds to

the student ID and need never be inferred.

This model can be easily changed to individualize learning rates instead of prior

knowledge by connecting the student node to the subsequent knowledge nodes thus

training an individualized P(T) conditioned upon student as shown in Figure 2.

Modeling Individualization in a Bayesian Networks Implementation of Knowledge Tracing

5

Fig. 2. Graphical depiction of our individualization modeling technique applied to the

probability of learning parameter. This model is not evaluated in this paper but is presented to

demonstrate the simplicity in adapting our model to other parameters.

2.2 Parameter Learning and Inference

There are two distinct steps in knowledge tracing models. The first step is learning the

parameters of the model from all student data. The second step is tracing an individual

student’s knowledge given their respective data. All knowledge tracing models allow

for initial knowledge to be inferred per student in the second step. The original KT

work [2] that individualized parameters added an additional step in between 1 and 2

to fit individual weights to the general parameters learned in step one. The PPS model

allows for the individualized parameters to be learned along with the non-

individualized parameters of the model in a single step. Assuming there is variance

worth modeling in the individualization parameter, we believe that a single step

procedure allows for more accurate parameters to be learned since a global best fit to

the data can now be searched for instead of a best fit of the individual parameters after

the skill specific parameters are already learned.

In our model each student has a student ID represented in the student node. This

number is presented during step one to associate a student with his or her prior

parameter. In step two, the individual student knowledge tracing, this number is again

presented along with the student’s respective data in order to again associate that

student with the individualized parameters learned for that student in the first step.

3 External Validity: Student Performance Prediction

In order to test the real world utility of the prior per student model, we used the last

question of each of our problem sets as the test question. For each problem set we

trained two separate models: the prior per student model and the standard knowledge

tracing model. Both models then made predictions of each student’s last question

responses which could then be compared to the students’ actual responses.

6 Zachary A. Pardos, Neil T. Heffernan

3.1 Dataset description

Our dataset consisted of student responses to problem sets that satisfied the following

constraints:

Items in the problem set must have been given in a random order

A student must have answered all items in the problem set in one day

The problem set must have data from at least 100 students

There are at least four items in the problem set of the exact same skill

Data is from Fall of 2008 to Spring of 2010

Forty-two problem sets matched these constraints. Only the items within the

problem set with the exact same skill tagging were used. 70% of the items in the 42

problem sets were multiple choice, 30% were fill in the blank (numeric). The size of

our resulting problem sets ranged from 4 items to 13. There were 4,354 unique

students in total with each problem set having an average of 312 students ( = 201)

and each student completing an average of three problem sets ( = 3.1).

Table 1. Sample of the data from a five item problem set

Student ID

1st response

2nd response

3rd response

4th response

5th response

750

0

1

1

1

1

751

0

1

1

1

0

752

1

1

0

1

0

In Table 1, each response represents either a correct or incorrect answer to the

original question of the item. Scaffold responses are ignored in our analysis and

requests for help are marked as incorrect responses by the system.

3.2 Prediction procedure

Each problem set was evaluated individually by first constructing the appropriate

sized Bayesian network for that problem set. In the case of the individualized model,

the size of the constructed student node corresponded to the number of students with

data for that problem set. All the data for that problem set, except for responses to the

last question, was organized into an array to be used to train the parameters of the

network using the Expectation Maximization (EM) algorithm. The initial values for

the learn rate, guess and slip parameters were set to different values between 0.05 and

0.90 chosen at random. After EM had learned parameters for the network, student

performance was predicted. The prediction was done one student at a time by entering

,as evidence to the network, the responses of the particular student except for the

response to the last question. A static unrolled dynamic Bayesian network was used.

This enabled individual inferences of knowledge and performance to be made about

the student at each question including the last question. The probability of the student

answering the last question correctly was computed and saved to later be compared to

the actual response.

Modeling Individualization in a Bayesian Networks Implementation of Knowledge Tracing

7

3.3 Approaches to setting the individualized initial knowledge values

In the prediction procedure, due to the number of parameters in the model, care had to

be given to how the individualized priors would be set before the parameters of the

network were learned with EM. There were two decisions we focused on: a) what

initial values should the individualized priors be set to and b) whether or not those

values should be fixed or adjustable during the EM parameter learning process. Since

it was impossible to know the ground truth prior knowledge for each student for each

problem set, we generated three heuristic strategies for setting these values, each of

which will be evaluated in the results section.

3.3.1 Setting initial individualized knowledge to random values

One strategy was to treat the individualized priors exactly like the learn, guess and

slip parameters by setting them to random values to then be adjusted by EM during

the parameter learning process. This strategy effectively learns a prior per student per

skill. This is perhaps the most naïve strategy that assumes there is no means of

estimating a prior from other sources of information and no better heuristic for setting

prior values. To further clarify, if there are 600 students there will be 600 random

values between 0 and 1 set for for each skill. EM will then have 600 parameters to

learn in addition to the learn, guess and slip parameters of each skill. For the non-

individualized model, the singular prior was set to a random value and was allowed to

be adjusted by EM.

3.3.2 Setting initial individualized knowledge based on 1st response heuristic

This strategy was based on the idea that a student’s prior is largely a reflection of their

performance on the first question with guess and slip probabilities taken into account.

If a student answered the first question correctly, their prior was set to one minus an

ad-hoc guess value. If they answered the first question incorrectly, their prior was set

to an ad-hoc slip value. Ad-hoc guess and slip values are used because ground truth

guess and slip values cannot be known and because these values must be used before

parameters are learned. The accuracy of these values could largely impact the

effectiveness of this strategy. An ad-hoc guess value of 0.15 and slip value of 0.10

were used for this heuristic. Note that these guess and slip values are not learned by

EM and are separate from the performance parameters. The non-individualized prior

was set to the mean of the first responses and was allowed to be adjusted while the

individualized priors were fixed. This strategy will be referred to as the “cold start

heuristic” due to its bootstrapping approach.

3.3.3 Setting initial individualized knowledge based on global percent correct

This last strategy was based on the assumption that there is a correlation between

student performance on one problem set to the next, or from one skill to the next. This

is also the closest strategy to a model that assumes there is a single prior per student

that is the same across all skills. For each student, a percent correct was computed,

8 Zachary A. Pardos, Neil T. Heffernan

averaged over each problem set they completed. This was calculated using data from

all of the problem sets they completed except the problem set being predicted. If a

student had only completed the problem set being predicted then her prior was set to

the average of the other student priors. The single KT prior was also set to the average

of the individualized priors for this strategy. The individualized priors were fixed

while the non-individualized prior was adjustable.

3.4 Performance prediction results

The prediction performance of the models was calculated in terms of mean absolute

error (MAE). The mean absolute error for a problem set was calculated by taking the

mean of the absolute difference between the predicted probability of correct on the

last question and the actual response for each student. This was calculated for each

model’s prediction of correct on the last question. The model with the lowest mean

absolute error for a problem set was deemed to be the more accurate predictor of that

problem set. Correlation was also calculated between actual and predicted responses.

Table 2. Prediction accuracy and correlation of each model and initial prior strategy

Most accurate predictor (of 42)

Avg. Correlation

P(L0) Strategy

PPS

KT

PPS

KT

Percent correct heuristic

33

8

0.3515

0.1933

Cold start heuristic

30

12

0.3014

0.1726

Random parameter values

26

16

0.2518

0.1726

Table 2 shows the number of problem sets that PPS predicted more accurately than

KT and vice versa in terms of MAE for each prior strategy. This metric was used

instead of average MAE to avoid taking an average of averages. With the percent

correct heuristic, the PPS model was able to better predict student data in 33 of the 42

problem sets. The binomial with p = 0.50 tells us that the probability of 33 success or

more in 42 trials is << 0.05 (cutoff is 27 to achieve statistical significance), indicating

a result that was not the product of random chance. In one problem set the MAE of

PPS and KT were equal resulting in a total other than 42 (33 + 8 = 41). The cold start

heuristic, which used the 1st response from the problem set and two ad-hoc parameter

values, also performed well; better predicting 30 of the 42 problem sets which was

also statistically significantly reliable. We recalculated MAE for PPS and KT for the

percent correct heuristic this time taking the mean absolute difference between the

rounded probability of correct on the last question and actual response for each

student. The result was that PPS predicted better than KT in 28 out of the 42 problem

sets and tied KT in MAE in 10 of the problem sets leaving KT with 4 problem sets

predicted more accurately than PPS with the recalculated MAE. This demonstrates a

meaningful difference between PPS and KT in predicting actual student responses.

The correlation between the predicted probability of last response and actual last

response using the percent correct strategy was also evaluated for each problem set.

The PPS model had a higher correlation coefficient than the KT model in 32 out of 39

problem sets. A correlation coefficient was not able to be calculated for the KT model

in three of the problem sets due to a lack of variation in prediction across students.

Modeling Individualization in a Bayesian Networks Implementation of Knowledge Tracing

9

This occurred in one problem set for the PPS model. The average correlation

coefficient across all problem sets was 0.1933 for KT and 0.3515 for PPS using the

percent correct heuristic. The MAE and correlation of the random parameter strategy

using PPS was better than KT. This was surprising since the PPS random parameter

strategy represents a prior per student per skill which could be considered an over

parameterization of the model. This is evidence to us that the PPS model may

outperform KT in prediction under a wide variety of conditions.

3.4.1 Response sequence analysis of results

We wanted to further inspect our models to see under what circumstances they

correctly and incorrectly predicted the data. To do this we looked at response

sequences and counted how many times their prediction of the last question was right

or wrong (rounding predicted probability of correct). For example: student response

sequence [0 1 1 1] means that the student answered incorrectly on the first question

but then answered correctly on the following three. The PPS (using percent correct

heuristic) and KT models were given the first three responses in addition to the

parameters of the model to predict the fourth. If PPS predicted 0.68 and KT predicted

0.72 probability of correct for the last question, they would both be counted as

predicting that instance correctly. We conducted this analysis on the 11 problem sets

of length four. There were 4,448 total student response sequence instances among the

11 problem sets. Tables 3 and 4 show the top sequences in terms of number of

instances where both models predicted the last question correctly (Table 3) and

incorrectly (Table 4). Tables 5-6 show the top instances of sequences where one

model predicted the last question correctly but the other did not.

Table 3. Predicted correctly by both

# of Instances

Response sequence

1167

1 1 1 1

340

0 1 1 1

253

1 0 1 1

252

1 1 0 1

Table 4. Predicted incorrectly by both

# of Instances

Response sequence

251

1 1 1 0

154

0 1 1 0

135

1 1 0 0

106

1 0 1 0

Table 5. Predicted correctly by PPS only

# of Instances

Response sequence

175

0 0 0 0

84

0 1 0 0

72

0 0 1 0

61

1 0 0 0

Table 6. Predicted correctly by KT only

# of Instances

Response sequence

75

0 0 0 1

54

1 0 0 1

51

0 0 1 1

47

0 1 0 1

Table 3 shows the sequences most frequently predicted correctly by both models.

These happen to also be among the top 5 occurring sequences overall. The top

occurring sequence [1 1 1 1] accounts for more than 1/3 of the instances. Table 4

shows that the sequence where students answer all questions correctly except the last

question is most often predicted incorrectly by both models. Table 5 shows that PPS

10 Zachary A. Pardos, Neil T. Heffernan

is able to predict the sequence where no problems are answered correctly. In no

instances does KT predict sequences [0 1 1 0] or [1 1 1 0] correctly. This sequence

analysis may not generalize to other datasets but it provides a means to identify areas

the model can improve in and where it is most strong. Figure 3 shows a graphical

representation of the distribution of sequences predicted by KT and PPS versus the

actual distribution of sequences. This distribution combines the predicted sequences

from all 11 of the four item problem sets. The response sequences are sorted by

frequency of actual response sequences from left to right in descending order.

Fig. 3. Actual and predicted sequence distributions of PPS (percent correct heuristic) and KT

The average residual of PPS is smaller than KT but as the chart shows, it is not by

much. This suggests that while PPS has been shown to provide reliably better

predictions, the increase in performance prediction accuracy may not be substantial.

4 Contribution

In this work we have shown how any Bayesian knowledge tracing model can easily

be extended to support individualization of any or all of the four KT parameters using

the simple technique of creating a student node and connecting it to the parameter

node or nodes to be individualized. The model we have presented allows for

individualized and skill specific parameters of the model to be learned simultaneously

in a single step thus enabling global best fit parameters to potentially be learned, a

potential that is prohibitive with multi step parameter learning methods [2,4].

We have also shown the utility of using this technique to individualize the prior

parameter by demonstrating reliable improvement over standard knowledge tracing in

0

200

400

600

800

1000

1200

1400

1600

1 1 1 1

0 0 0 0

0 1 1 1

1 1 0 1

1 0 1 1

1 1 1 0

0 1 0 0

0 0 0 1

0 0 1 1

1 1 0 0

1 0 0 0

0 1 1 0

1 0 0 1

0 1 0 1

0 0 1 0

1 0 1 0

Frequency of response sequences

Student response sequences

Response sequences for four question problem sets

actual

pps

kt

last

response

Modeling Individualization in a Bayesian Networks Implementation of Knowledge Tracing

11

predicting real world student responses. The superior performance of the model that

uses PPS based on the student’s percent correct across all skills makes a significant

scientific suggestion that it may be more important to model a single prior per student

across skills rather than a single prior per skill across students, as is the norm.

5 Discussion and Future Work

We hope this paper is the beginning of a resurgence in attempting to better

individualize and thereby personalize students’ learning experiences in intelligent

tutoring systems.

We would like to know when using a prior per student is not beneficial. Certainly

if in reality all students had the same prior per skill then there would be no utility in

modeling an individualized prior. On the other hand, if student priors for a skill are

highly varied, which appears to be the case, then individualized priors will lead to a

better fitting model by allowing the variation in that parameter to be captured.

Is an individual parameter per student necessary or can the same or better

performance be achieved by grouping individual parameters into clusters? The

relatively high performance of our cold start heuristic model suggests that much can

be gained by grouping students into one of two priors based on their first response to

a given skill. While this heuristic worked, we suspect there are superior

representations and ones that allow for the value of the cluster prior to be learned

rather than set ad-hoc as we did. Ritter et al [8] recently showed that clustering of

similar skills can drastically reduce the number of parameters that need to be learned

when fitting hundreds of skills while still maintaining a high degree of fit to the data.

Perhaps a similar approach can be employed to find clusters of students and learning

their parameters instead of learning individualized parameters for every student.

Our work here has focused on just one of the four parameters in knowledge

tracing. We are particularly excited to see if by explicitly modeling the fact that

students have different rates of learning we can achieve higher levels of prediction

accuracy. The questions and tutorial feedback a student receives could be adapted to

his or learning rate. Student learning rates could also be reported to teachers allowing

them to more precisely or more quickly understand their classes of students. Guess

and slip individualization is also possible and a direct comparison to Baker’s

contextual guess and slip method would be an informative piece of future work.

We have shown that choosing a prior per student representation over the prior per

skill representation of knowledge tracing is beneficial in fitting our dataset; however,

a superior model is likely one that combines the attributes of the student with the

attributes of a skill. How to design this model that properly treats the interaction of

these two pieces of information is an open research question for the field. We believe

that in order to extend the benefit of individualization to new users of a system,

multiple problem sets must be linked in a single Bayesian network that uses evidence

from the multiple problem sets to help trace individual student knowledge and more

fully reap the benefits suggested by the percent correct heuristic.

This work has concentrated on knowledge tracing, however, we recognize there are

alternatives. Draney, Wilson and Pirolli [9] have introduced a model they argue is

more parsimonious than knowledge tracing due to having fewer parameters.

12 Zachary A. Pardos, Neil T. Heffernan

Additionally, Pavlik et al [10] have reported using different algorithms, as well as

brute force, for fitting the parameters of their models. We also point out that more

standard models that do not track knowledge such as item response theory that have

had large uses in and outside of the ITS field for estimating individual student and

question parameters. We know there is value in these other approaches and strive as a

field to learn how best to exploit information about students, questions and skills

towards the goal of a truly effective, adaptive and intelligent tutoring system.

Acknowledgements

We would like to thank all of the people associated with creating the ASSISTment

system listed at www.ASSISTment.org. We would also like to acknowledge funding

from the US Department of Education, the National Science Foundation, the Office of

Naval Research and the Spencer Foundation. All of the opinions expressed in this

paper are those of the authors and do not necessarily reflect the views of our funders.

References

1. Atkinson, R. C., Paulson, J. A. An approach to the psychology of instruction.

Psychological Bulletin, 1972, 78, 49-61.

2. Corbett, A. T., & Anderson, J. R. (1995). Knowledge tracing: modeling the acquisition of

procedural knowledge. User Modeling and User-Adapted Interaction, 4, 253–278.

3. Corbett A. and Bhatnagar A. (1997). Student Modeling in the ACT Programming Tutor:

Adjusting a Procedural Learning Model with Declarative Knowledge. In User Modeling:

Proceedings of the 6th International Conference, pp. 243-254.

4. Baker, R.S.J.d., Corbett, A.T., Aleven, V.: More Accurate Student Modeling Through

Contextual Estimation of Slip and Guess Probabilities in Bayesian Knowledge Tracing. In:

Wolf, B., Aimeur, E., Nkambou, R., Lajoie, S. (Eds.) Intelligent Tutoring Systems. LNCS,

vol. 5091/2008, pp. 406-415. Springer Berlin (2008)

5. Beck, J.E., Chang, K.M.: Identifiability: A Fundamental Problem of Student Modeling. In:

Conati, C., McCoy, K., Paliouras, G. (Eds.) User Modeling 2007. LNCS, vol. 4511/2009,

pp. 137-146. Springer Berlin (2007)

6. Reye, J. (2004). Student modelling based on belief networks. International Journal of

Artificial Intelligence in Education: Vol. 14, 63-96.

7. Chang, K.M., Beck, J.E., Mostow, J., & Corbett, A.: A Bayes Net Toolkit for Student

Modeling in Intelligent Tutoring Systems. In: Ikeda, M., Ashley, K., Chan, T.W. (Eds.)

Intelligent Tutoring Systems. LNCS, vol. 4053/2006, pp. 104-113. Springer Berlin (2006)

8. Ritter, S., Harris, T., Nixon, T., Dickison, D., Murray, C., Towle, B.(2009). Reducing the

knowledge tracing space. In Proceedings of the 2nd International Conference on

Educational Data Mining. pp. 151-160. Cordoba, Spain.

9. Draney, K. L., Pirolli, P., & Wilson, M. (1995). A measurement model for a complex

cognitive skill. In P. D. Nichols, S. F. Chipman, & R. L. Brennan (Eds.), Cognitively

diagnostic assessment (pp. 103–125). Hillsdale, NJ: Erlbaum.

10. Pavlik, P.I., Cen, H., Koedinger, K.R. (2009). Performance Factors Analysis - A New

Alternative to Knowledge Tracing. In Proceedings of the 14th International Conference

on Artificial Intelligence in Education. Brighton, UK, 531-538.