1. Introduction
One of the advantages of an Intelligent Tutoring System (Nwana, 1990) and massive open online courses (Kaplan and Haenlein, 2016)
is that they can potentially benefit from monitoring and tracking student activities in an adaptive learning environment, where learner modeling comes into play. A learner model provides estimates for the students’ state, and includes two interconnected aspects: domain modeling and knowledge modeling
(Pelánek, 2017).A domain model (Pelánek, 2017) studies the structure within a domain of problems (For example, it finds out which skill a problem is related to: “1+2=?” to “addition of integers” and “1.3+2.5=?” to “addition of decimals”). Another task of a domain model is to discover the structure of a skill domain, which can be performed either manually or automatically (Pelánek, 2017). On the other hand, a knowledge model (Anderson et al., 1990), in an abstract sense, traces students’ knowledge while they are solving problems. Knowledge has been described in various forms by the name of knowledge state, which has no universal definition yet. In this paper, the term knowledge state has been used as a state that can describe a student’s general level of attainment of skills. Often, domain modeling and knowledge modeling are viewed to be separate; however, we tried to provide approaches for both where problemsolving records of students can be important input features for finding the latent structure of a skill domain.
KT is a research area which analyzes student activities and studies knowledge acquisition, where its main task is to describe a student’s knowledge. To elaborate, consider a student who solved a sequence of problems. Then the student’s data is given by the temporal sequence of tuples, each of which is consisted of the skill that the problem at each time step is related to and the binary correctness that indicates whether or not the student gave a right answer. By calling such tuple student response, the KT problem is formulated as follows: 1. given the student responses up to the time step , describe the student’s knowledge state at the current time step , and 2. given the skill at the next time step , predict the correctness by modeling the interaction between the student’s knowledge state at time and the skill at time , which we will call knowledge interaction. Note that the knowledge state refers to the dynamic state of a student accumulated from the student responses while a skill indicates a particular ability that needs to be learned by a student to solve a problem.
Therefore, the quality of a KT model is measured by its ability to describe the knowledge state of a student and its accuracy of predicting correctness. Additionally, since modeling the knowledge interaction is to describe how a student’s knowledge state responds to different skills, it is desirable if a KT model can explain the relationship between skills that can be inferred from the knowledge interaction. For example, we can say that “addition of integers” is independent of “subtraction of integers” if a model observes that a student does not learn the latter while learning the former. Similarly, they are dependent if the change in a student’s knowledge state of one skill affects the knowledge state of the other. We believe that modeling such skill relationship can lead to further exploration of the latent structure of the skill domain, which is the subject of domain modeling.
However, existing KT models provide limited definitions of either knowledge state, or knowledge interaction, or both. For example, Bayesian Knowledge Tracing (BKT) (Anderson et al., 1990) imposes a binary assumption on the knowledge state, which is too restrictive to be intuitive, and Deep Knowledge Tracing (DKT) (Piech et al., 2015) does not give an explanation of knowledge interaction. In this paper, we propose a new neural network KT model called Knowledge Query Network (KQN) to generalize the knowledge state and explain the knowledge interaction more descriptively. The central idea is to use the dot product between a knowledge state vector and a skill vector to define the knowledge interaction while leveraging neural networks to encode student responses and skills into vectors of the same dimensionality. Additionally, we introduce a novel concept called probabilistic skill similarity
which relates the cosine and Euclidean distances between the skill vectors to the odds ratios for the corresponding skills. Based on those distances, we explore the latent structure of a skill domain with cluster analysis. Lastly, we show that KQN is stable in predicting correctness and learning skill vectors by comparing prediction accuracy and the distance matrices of the skill vectors when the vector space dimensionality is varied.
2. Related Work
Item Response Theory (IRT) is a framework for modeling the relationship between problems and correctness (Hulin et al., 1983)
. In its simplest form, it uses a logistic regression model by estimating student proficiency and skill difficulty. However, it assumes the proficiency to be constant and does not explain any structure for problems. To overcome those limitations, Bayesian extensions of IRT have been proposed to have a hierarchical structure over items (HIRT) and temporal changes in a student’s knowledge state (TIRT)
(Wilson et al., 2016). Still, HIRT assumes constant student proficiency while TIRT lacks the ability of domain analysis.




In BKT, a student’s knowledge state is viewed as a set of binary latent variables, one for each skill, with two possible states, known and unknown (Anderson et al., 1990)
. Then a set of observable variables, each of which corresponds to correctness per skill, are conditioned on the set of the binary variables. For example, let us say we have a running example of student Ben throughout this paper, who has records as shown in Table
1. Accordingly, there will be two knowledge states and correctness variables for skills 1 and 2, a total of four variables with two independent BKT models with input data as shown in Table 0(b). Then the knowledge acquisition in BKT is modeled with a Hidden Markov model (HMM), where knowledge interaction is controlled by a set of interpretable equations. Since BKT lacks the ability to forget and individualize, a number of extensions have been proposed
(Yudelson et al., 2013; Khajah et al., 2016). However, most importantly, BKT’s independence assumption on different skills is considered to be highly constrained and not effective, where the model cannot leverage the whole data. For example, Ben’s history of responses on skill 1 cannot tell his responses on skill 2 since there should be two separate models for each skill.Learning Factors Analysis (LFA) (Cen et al., 2006) models a student’s knowledge state as a set of binary variables, one for each skill. Correctness at each time step is predicted with a logistic regression model which has covariates related to students, skills, and a summary statistic of student responses, i.e., the number of past opportunities for each skill. Performance Factors Analysis (PFA) (Pavlik et al., 2009) extends LFA by separating ‘the number of opportunities per skill’ into ‘the number of correct answers per skill’ and ‘the number of incorrect answers per skill’ with the others the same, e.g., in Ben’s case, the input data are preprocessed as shown in Table 0(c) and Table 0(d).
Since an estimate for correctness is explained with student covariates, and skillspecific covariates without variable interactions in LFA and PFA, the two models do not describe how a student’s knowledge state with respect to one skill is affected by that with respect to another skill; instead, a student parameter, which is also called student proficiency, is the only factor that relates the knowledge state for different skills. Moreover, a skill is explained by the regression coefficients for the skillspecific covariates from which we cannot tell the structure of a skill domain directly.
As the first neural network KT model, DKT (Piech et al., 2015)
, given the student’s responses as input, encodes a student’s knowledge state as a summarized vector calculated from a recurrent neural network (RNN), which is a renowned neural network technique for modeling temporal data. However, DKT does not define the knowledge interaction directly. In detail, a student response at each time step is formed as a tuple
, where and refer to the problem ID and correctness, respectively. For example, Ben’s original data in Table 0(a) is expressed as at. The input is then passed to an RNN layer, where Long ShortTerm Memory (LSTM)
(Hochreiter and Schmidhuber, 1997) was used in the original paper (Piech et al., 2015), and its output hidden state is passed to a logistic function after an affine transformation, i.e., , where is the output hidden state of an LSTM layer and is an elementwise logistic function. Finally, the th element of is used to predict correctness at the next time step given that the next problem ID is . Despite its superior prediction performance over the existing classical methods, DKT has been criticized by other papers (Wilson et al., 2016; Khajah et al., 2016) for its lack of practicality in educational applications. This is because the output hidden state is inherently hard to be interpreted as the knowledge state, and the model does not give insights into the knowledge interaction.To make a neural network KT model more interpretable, Dynamic KeyValue Memory Networks (DKVMN) (Zhang et al., 2017) have been introduced by extending memoryaugmented neural networks (MANNs) (Graves et al., 2014; Weston et al., 2014). Like DKT, DKVMN uses the original data as input. DKVMN accumulates temporal information from student responses into a dynamic matrix, or the value memory, while embedding skills with a static matrix, or the key memory
. DKVMN defines the interaction between value vectors and key vectors using attention weights calculated with cosine similarity, and predicts correctness by passing a concatenation of a weighted sum of value vectors and an embedded skill vector to a multilayer perceptron (MLP). Even though the prediction accuracy of DKVMN has proven to be higher than that of DKT, the use of an MLP for the output of the model still makes it hard to explain the knowledge interaction.
3. Our Proposed Model
3.1. Motivation
To generalize the knowledge state while describing the knowledge interaction intuitively, we suggest a model that projects a student’s knowledge and skills into the same vector space of embedding dimensionality . An important constraint is to contain the skill vectors on the
dimensional positive orthant unit sphere, i.e., they have unitlength and positive coordinates. The logit of a probability estimate for correctness is given by the dot product between the current knowledge state vector and the skill vector of the next problem. This is only possible because both knowledge state and skill vectors lie in the same vector space.
Now, we illustrate why skill vectors are set to unitlength and constrained to a positive orthant: the former makes the logit only dependent on the direction of the related skill vector while the latter assures that learning on one skill does not decrease learning on another. For example, in a 2D vector space, suppose that Ben has knowledge state at after two responses while there are three skill vectors, and for the skills 1 and 2, and for a third “imaginary” skill as shown in Figure 1. At , his knowledge state may change to as he answers correctly for skill 1. Then the logit with respect to increases from to while that with respect to skill 2 remains the same as . However, the logit with respect to skill 3 would decrease from to , which would then decrease the probability estimate for the correctness of skill 3. This is counterintuitive since the datasets we are dealing with have a set of skills within the same area, e.g., mathematics.
3.2. Objective
Let the skill of a problem and the correctness at time be and , respectively. The correctness at each time step is viewed as a Bernoulli variable given the history of responses and the skill at time step
. Then the objective of a KT model is to find out the parameter of the Bernoulli distribution at each time step as follows:
3.3. Architecture Overview
KQN consists of three components: knowledge encoder, skill encoder, and knowledge state query. The knowledge encoder converts the temporal information from student responses into a knowledge state vector while the skill encoder embeds a skill into a skill vector. The two vectors are then passed to the knowledge state query to provide the prediction for correctness given the current knowledge state and the provided skill. The network architecture of KQN is shown in Figure 2.
3.4. Inputs
The model takes two inputs: a student response at the current time step and a skill at the next time step. Each of the student responses is onehot encoded and given as input
to an RNN layer as the following:where is the number of skills, and is the skill at time step . Similarly, the skill at time is onehot encoded to , where the th element is 1 and the other elements are 0’s. In Ben’s case, his response at and the skill for the problem at are encoded to and , respectively.
3.5. Knowledge State Query
Let the knowledge state vector and the embedded skill vector , both dimensional, be the two vectors encoded by the knowledge encoder and the skill encoder, respectively. Then the knowledge interaction is defined by the inner product of the two vectors. The logit and the corresponding prediction probability are calculated as follows:
where is a logistic function and refers to the inner product. In this way, knowledge interaction is welldefined for the following reasons:

If two skills are independent, their corresponding vectors are orthogonal to each other. Accordingly, an increase or a decrease in the logit with respect to one vector does not affect the logit with respect to the other vector.

If two skills are similar from the probabilistic perspective, then an increase in a logit with respect to one vector would lead to an increase in the logit with respect to the other vector, and vice versa.
Note that from the definition of the knowledge interaction above, it is implied that there can be at most mutually independent skills. For different values of , whether or not KQN learns the pairwise relationships between skills represented by pairwise distances was tested in experiments and shown in later sections.
As a result, KQN is approximating the parameter of the Bernoulli distribution at each time step as follows:
3.6. Knowledge State Encoder
Given input , the knowledge state encoder produces a knowledge state vector with the following equations:
where , , is an RNN layer, and is the state size of . In KQN, LSTM (Hochreiter and Schmidhuber, 1997)
and Gated Recurrent Units (GRU)
(Cho et al., 2014) have been tested as RNN variants. Also, to avoid overfitting, dropout regularization (Srivastava et al., 2014; Zaremba et al., 2014) has been used for the RNN output layer as was used for DKT in a previous work (Xiong et al., 2016).3.7. Skill Encoder
3.8. Optimization
At each time step, the crossentropy error given the probability estimate and the target correctness is calculated, and the error terms for are summed to give the total error.
The gradients of the total error with respect to the model parameters have been computed with backpropagation to be used by an optimization method.
4. Probabilistic Skill Similarity
Based on the architecture of KQN, we hereby introduce a novel concept called probabilistic skill similarity to measure the distance between skills from the probabilistic perspective.
4.1. Distance Measures for Skill Vectors
For any two skill vectors learned from KQN, the cosine distance differs from the squared Euclidean distance by only a factor of 2 since they are constrained to as follows:
4.2. Distances and Odds Ratios
Next, we show how a pairwise distance between two skill vectors is related to the logarithm of their odds ratio. Given a knowledge state vector and a skill vector , the probability estimate for correctness and the corresponding odds are calculated as follows:
Then for any two skill vectors , the logarithm of the odds ratio is characterized by their distance as follows:
where . Therefore, we say that two skills are probabilistically similar if they are ‘close’ enough based on the distance between their corresponding vectors.
5. Experiments
KQN has been tested for four tasks: correctness prediction, knowledge interaction visualization, skill domain analysis, and the sensitivity analysis of the dimensionality of the vector space. For correctness prediction, the performance of KQN was compared to that of other models on four public datasets: one synthetic and three realworld ones which are available online. Then for a sample student, knowledge interaction was visualized with a heat map to demonstrate the knowledge state query with respect to different skills. Next, the skill domain was explored with clustering based on skill distances. Finally, pairwise distances of the skill vectors in one dimensionality were compared to those in other dimensionalities to conduct the sensitivity analysis of the vector embedding dimensionality.
5.1. Datasets
The following four datasets have been used to evaluate models: ASSISTments 20092010, ASSISTments 2015, OLI Engineering Statics 2011, and Synthetic5. To make a fair comparison of the correctness prediction task, we used the ones provided by the DKVMN source code available online^{1}^{1}1https://github.com/jennyzhang0215/DKVMN. The statistics of the datasets are shown in Table 2.
5.1.1. ASSISTments 20092010
It is a dataset^{2}^{2}2https://sites.google.com/site/assistmentsdata/home/assistment20092010data/skillbuilderdata20092010 collected by the ASSISTments online tutoring systems (Feng et al., 2009). It was gathered from skill builder problem sets, where students work on the problems to achieve mastery, a certain level of performance, working on similar questions. During the preprocessing, those records without skill names have been discarded. After a problem with duplicate records had been reported by a paper (Xiong et al., 2016), the dataset has since been corrected by the ASSISTments system. Therefore, those results reported by a number of previous papers are not compared in this paper.
5.1.2. ASSISTments 2015
Compared to ASSISTments 20092010 which has 110 distinct skill tags, this dataset^{3}^{3}3https://sites.google.com/site/assistmentsdata/home/2015assistmentsskillbuilderdata contains 100 distinct ones with more than twice the number of student responses. Data records with invalid correct values that are not in have been removed.
5.1.3. OLI Engineering Statics 2011
This dataset^{4}^{4}4https://pslcdatashop.web.cmu.edu/DatasetInfo?datasetId=507 was gathered from a college level statics course in Fall 2011 (Koedinger et al., 2010). The concatenation of a problem name and a step name has been labeled as a skill tag. Note that the number of skills is much larger than those of other datasets.
5.1.4. Synthetic5
It is a dataset^{5}^{5}5https://github.com/chrispiech/DeepKnowledgeTracing originally generated by the authors of the DKT paper (Piech et al., 2015). Each student response was generated using skill difficulty, student proficiency, and the probability of a random guess set to a constant based on IRT (Hulin et al., 1983). The dataset consists of a number of subdatasets, and those with five concepts from version 0 to version 19 have been used, i.e., a total of 20 subdatasets from the original dataset were used.
Dataset  Students  Skills  Size  Max Steps 

ASSIST2009  4,151  110  325,637  1,261 
ASSIST2015  19,840  100  683,801  618 
Statics2011  333  1,223  189,297  1,181 
Synthetic5  4,000  50  200,000  50 
Dataset  Test AUC (%)  
IRT+  BKT+  DKVMN  DKT  DKT+KQN  KQN  
ASSIST2009  77.40    81.570.1  80.530.2  82.050.04  82.320.05 
ASSIST2015      72.680.1  72.520.1  73.410.02  73.400.02 
Statics2011    75  82.840.1  80.200.2  80.270.22  83.200.05 
Synthetic5    80  82.730.1  80.340.1  82.580.01  82.810.01 
5.2. Setup and Implementation
All the program codes for the implemented KQN and DKT were written in TensorFlow 1.5
^{6}^{6}6https://www.tensorflow.org/versions/r!1.5/. For the data splits of each dataset, we used the same ones^{\getrefnumberdkvmndata}^{\getrefnumberdkvmndata}footnotemark: dkvmndata used by DKVMN for a fair comparison of prediction accuracy.5.2.1. Correctness Prediction
All the sequences of student responses were preserved in their original length without truncation. Each dataset except Synthetic5 has been split into training, validation, and test sets with 8:2 and 7:3 for training to validation and (training+validation) to test ratios, respectively. For Synthetic5, the corresponding ratios of 8:2 and 5:5 have been set. Hyperparameters have been gridsearched with holdout validation with earlystopping. Note that no earlystopping was used in the testing phase. The number of epochs was set to 50 and 200 during the validation and testing phases, respectively. During the testing phase, KQN was run for five times, and the mean and standard deviation of the performance metric results have been reported.
The hyperparameters of KQN and their candidate values have been set as follows:

Type of the RNN layer in the knowledge state encoder: LSTM, GRU.

Hidden state size of the chosen RNN layer: 32, 64, 128.

Hidden state size of the MLP layer: 32, 64, 128.

Dimensionality of the vector space in which the knowledge state and skills are embedded: 32, 64, 128.
The retention rate of 0.6 for the RNN dropout and the batch size of 128 were set to default. The Adam optimization method (Kingma and Ba, 2014) was used to minimize the total error .
Additionally, DKT was run with the skill vectors learned by KQN to evaluate their quality for the correctness prediction task. Specifically, at each time step, input given to DKT was set to the concatenation of two vectors: onehot encoded correctness and the learned skill vector corresponding to the original skill . We denote DKT with such a setup as DKT+KQN. The hyperparameters of DKT+KQN have been searched in the same way as explained previously for KQN with the same dropout rate of RNN and the batch size. Meanwhile, LSTM was used for the RNN layer following past works (Piech et al., 2015; Xiong et al., 2016).
5.2.2. Knowledge Interaction Visualization
Throughout a student’s responses, prediction estimates for correctness with respect to the skills the student solved were calculated with the knowledge state query followed by the logistic function. A sample from the test set of ASSISTments 20092010 was used for the task. Then those estimates were visualized with a heat map to evaluate their changes as the student solved the problems.
5.2.3. Skill Domain Analysis
Skill distances have been used for clustering on the four datasets. To decide the linkage and the type of distance measures to use, we first performed flat clustering on Synthetic5 with the number of clusters fixed to 5, the ground truth number of clusters. Then the quality of clustering or partitioning with respect to the original cluster labels was measured with the Adjusted Rand Index (ARI) (Hubert and Arabie, 1985). It has a maximum value of 1 when the clusters are formed to match the original partitioning perfectly, and a minimum value of 0 when they are randomly partitioned. Since there are 20 subdatasets for Synthetic5, 20 ARI scores were averaged. The linkage between clusters and the type of distance measures have been set to hyperparameters as follows:

Cluster linkage: average, centroid, complete, median, single, ward, weighted

Type of distance measure: cosine, Euclidean
After deciding which linkage and distance measure to use, the number of clusters has been explored. First, for different values of , the skills of ASSISTments 20092010 were clustered based on the distances computed from the skill vectors learned by KQN. Then DKT was used to quantify the quality of those clusters as follows: First, all the original skill IDs were substituted with the assigned cluster labels, where data splits remained the same as those for the correctness prediction task. Then, DKT was run five times. Finally, the average and the standard deviation of the test AUCs of DKT were reported.
5.2.4. Sensitivity Analysis of the Vector Space Dimensionality
Let be the dimensionality of the vector space in KQN and be the optimal values of obtained in the correctness prediction task previously. KQN was trained on the four datasets by varying to and with the data splits kept the same, and the other hyperparameters set to the optimal values. To analyze the effect of on the correctness prediction task and the learning of skill vectors, prediction accuracy was reported, and the three distance matrices of the skill vectors for each of were compared.
6. Results and Analysis
6.1. Correctness Prediction
2 Area Irregular Figure  41 Finding Percents  74 Multiplication and Division Positive Decimals  55 Divisibility Rules 

46 Algebraic Solving  42 Pattern Finding  107 Parts of a Polynomial, Terms, Coefficient, Monomial, Exponent, Variable  57 Perimeter of a Polygon 
33 56 Reading a Ruler or Scale  43 Write Linear Equation from Situation  4 Table  71 Angles on Parallel Lines Cut by a Transversal 
63 Scale Factor  44 Square Root  12 Circle Graph  72 Write Linear Equation from Ordered Pairs 
67 Percents  47 Percent Discount  32 Box and Whisker  80 Unit Conversion Within a System 
78 Rate  54 Interior Angles Triangle  49 Complementary and Supplementary Angles  83 Area Parallelogram 
84 Effect of Changing Dimensions of a Shape Proportionally  62 Ordering Real Numbers  53 Interior Angles Figures with More than 3 Sides  91 Polynomial Factors 
85 Surface Area Cylinder  65 Scientific Notation  58 Solving for a variable  97 Choose an Equation from Given Information 
86 Volume Cylinder  76 Computation with Real Numbers  59 Exponents  101 Angles  Obtuse, Acute, and Right 
88 Solving Systems of Linear Equations  79 Solving Inequalities  68 Area Circle  104 Simplifying Expressions positive exponents 
44 92 Rotations  81 Area Rectangle  70 Equation Solving More Than Two Steps  20 Addition and Subtraction Integers 
93 Reflection  82 Area Triangle  75 Volume Sphere  31 Circumference 
96 Interpreting Coordinate Graphs  87 Greatest Common Factor  102 Distributive Property  34 Conversion of Fraction Decimals Percents 
1133 14 Proportion  89 Solving Systems of Linear Equations by Graphing  1 Area Trapezoid  60 Division Fractions 
66 Write Linear Equation from Graph  90 Multiplication Whole Numbers  6 Stem and Leaf Plot  77 Number Line 
1144 69 Least Common Multiple  98 Intercept  10 Venn Diagram  19 Multiplication Fractions 
11 3 Probability of Two Distinct Events  99 Linear Equations  11 Histogram as Table or Graph  25 Subtraction Whole Numbers 
5 Median  100 Slope  21 Multiplication and Division Integers  61 Estimation 
7 Mode  105 Finding Slope from Ordered Pairs  22 Addition Whole Numbers  109 Finding Slope From Equation 
44 8 Mean  106 Finding Slope From Situation  26 Equation Solving Two or Fewer Steps  24 Addition and Subtraction Fractions 
9 Range  108 Recognize Quadratic Pattern  33 Ordering Integers  50 Pythagorean Theorem 
13 Equivalent Fractions  110 Quadratic Formula to Solve Quadratic Equation  37 Ordering Positive Decimals  103 Recognize Linear Pattern 
2244 15 Fraction Of  16 Probability of a Single Event  38 Rounding  29 Counting Methods 
18 Addition and Subtraction Positive Decimals  45 Algebraic Simplification  39 Volume Rectangular Prism  95 Midpoint 
44 23 Absolute Value  73 Prime Number  40 Order of Operations All  64 Surface Area Rectangular Prism 
44 28 Calculations with Similar Figures  94 Translations  48 Nets of 3D Figures  36 Unit Rate 
22 30 Ordering Fractions  17 Scatter Plot  51 D.4.8understandingconceptofprobabilities  
35 Percent Of  27 Order of Operations +,,/,* () positive reals  52 Congruence 
Prediction accuracy was measured with the Area Under the ROC curve (AUC) during the testing phase. Note that the AUC of a model that guesses 0 or 1 randomly should be 50%. As representatives of nonneuralnetwork models, BKT, IRT, and their variants have been compared with KQN while DKT and DKVMN were compared to the stateoftheart neural network models. The AUC results for those models have been cited from other papers as follows: those of IRT and its extensions from (Wilson et al., 2016), of BKT and its variants from (Khajah et al., 2016; Xiong et al., 2016), and of DKT and DKVMN from (Zhang et al., 2017).
Test AUCs for all the datasets are shown in Table 3. Overall, KQN performed better than all the previously available KT models and showed a more stable performance with the lowest standard deviation values.
For ASSISTments 20092010, the test AUC of KQN was 82.32% beating the previous highest value by 0.75%. DKT+KQN showed the AUC of 82.05%, not only higher than the original DKT performance but also higher than all the others. Surprisingly, for ASSISTments 2015, DKT+KQN achieved the highest test AUC of 73.41%, even slightly higher than KQN. Both KQN and DKT+KQN performed better than all the previous results, which is promising in that KQN should be learning useful skill vectors that are transferable to other models and applications. For OLI Engineering Statics 2011, KQN achieved the highest value of 83.20%, higher than the previous highest by 0.34%. DKT+KQN showed a performance comparable to the vanilla DKT with the slightly higher test AUC of 80.27% and the standard deviation of 0.22%. Lastly, also for Synthetic5, KQN performed the best with the highest average 82.81% and the lowest standard deviation of 0.01%. Interestingly, the standard deviation of DKT+KQN was much lower than that of the original DKT, showing that the learned skill vectors should be contributing to the stable prediction of the model.
In summary, KQN showed the best performance for the correctness prediction task compared to all the previous models while it achieved the second best result of all the models as DKT+KQN had the highest score on ASSISTments 2015. In addition to the best mean test AUC scores, our model had much lower standard deviation values compared to other models. DKT+KQN also had low standard deviation values for all the datasets except for OLI Engineering Statics 2011. Therefore, we speculate that KQN is able to produce stable prediction estimates due to its ability to learn a meaningful latent structure of the skill vectors.
6.2. Knowledge Interaction Visualization
For a sample student from ASSISTments 20092010, prediction estimates for correctness in percentage are visualized in Figure 3 through the knowledge state query with respect to particular skills. On the xaxis, student responses with skill IDs and correctness values as tuples are marked while on the yaxis, all the skills that the student solved are sorted in ascending order from the top. The corresponding skill names can be found in Table 4.
Changes in probability estimates are mostly intuitive. For example, at , after the student solved a problem with skill 52 correctly, the probability estimate for skill 52 increased from 72% to 82%. However, it can also be observed that some changes are counterintuitive. For example, at , as the student solved a problem with skill 92 incorrectly, the corresponding estimate increased from 23% to 24% even though the change was only 1%. This problem has also been addressed for DKT in a previous work and is still an open problem to be improved (Yeung and Yeung, 2018).
Linkage  Distance  ARI 

average  cosine  0.3180 
Euclidean  0.3266  
centroid  cosine  0.0373 
Euclidean  0.0143  
complete  cosine  0.2898 
Euclidean  0.2898  
median  cosine  0.0368 
Euclidean  0.0071  
single  cosine  0.0703 
Euclidean  0.0703  
ward  cosine  0.3201 
Euclidean  0.3234  
weighted  cosine  0.2996 
Euclidean  0.3020 
Number of Clusters  Test AUC (%) 

5  79.770.03 
6  79.970.03 
7  80.100.03 
8  80.040.02 
9  80.140.03 
10  80.100.02 
11  80.230.05 
12  80.200.05 
13  80.550.03 
14  80.640.03 
6.3. Skill Domain Analysis
Distance  Dataset  

cosine  ASSIST2009  0.11  0.09  0.10  0.75  0.76  0.76 
ASSIST2015  0.11  0.09  0.10  0.70  0.69  0.69  
Statics2011  0.15  0.09  0.20  0.44  0.55  0.37  
Synthetic5  0.12  0.10  0.11  0.78  0.78  0.79  
Euclidean  ASSIST2009  0.09  0.07  0.09  1.22  1.23  1.23 
ASSIST2015  0.09  0.07  0.09  1.17  1.17  1.17  
Statics2011  0.16  0.11  0.21  0.92  1.04  0.84  
Synthetic5  0.10  0.08  0.09  1.25  1.24  1.25 
Average ARI scores for clustering with different linkage methods and distance measures are reported in Table 5. ARI was the highest when the linkage was set to average, and the distance measure was set to Euclidean for clustering. Not surprisingly, the average ARI scores did not differ much when the distance measure was set to either cosine or Euclidean.
After clustering the skills of ASSISTments 20092010 with the linkage and the distance measure set to average and Euclidean, respectively, and substituting the original skill IDs with those cluster labels, DKT was run five times. The test AUCs of DKT are reported in Table 6. They increased gradually as the number of clusters changed from 5 to 14. The lowest test AUC was 79.77% when , not differing much from the highest test AUC of 80.64%, which means that skills were clustered preserving useful information.
On ASSISTments 20092010, skill IDs and skill names were grouped by 14 clusters as shown in Table 4. Different clusters were separated by dashed lines while in each cluster, skills were sorted in ascending order based on their skill IDs. Since the skills must have been grouped based on probabilistic skill similarity, a number of intuitively similar skills were clustered together. For example, ‘30 Ordering Functions’ and ‘62 Ordering Real Numbers’ were assigned to the fourth cluster while ‘33 Ordering Integers’ and ‘37 Ordering Positive Decimals’ were assigned to the eighth cluster.
6.4. Sensitivity Analysis of the Vector Space Dimensionality
For the four datasets, the test AUCs of KQNs with the embedding dimensionality and are shown in Table 8, where refers to the optimal values chosen from the holdout validation for the correctness prediction task. We could observe only a little difference in the prediction accuracy as the values of were varied.
For each pair of , the average difference between the pairwise distances of the skill vectors of two different dimensionalities was calculated as follows:
where refers to the pairwise distance between two skill vectors and , and is the number of skills. is then compared to the average pairwise distance as follows:
In Table 7, the lowest values of are indicated in bold. As can be seen, is always lower than . From this, it can be inferred that KQN learned the skill relationships better when was set to a number high enough since controls the maximum number of mutually independent skill vectors. Also, the values of were relatively low compared to the corresponding values of . For example, was only when the Euclidean distance was used for ASSISTments 20092010 while the corresponding and were 1.22 and 1.23, respectively.
To further evaluate the distance matrices, we performed Mantel tests (Legendre and Legendre, 1998), which measure the similarity between two distance matrices with a correlation coefficient and a pvalue. has the same range as that of correlation coefficients in statistics while a pvalue indicates statistical significance. The Pearson correlation and the permutation number of 999 were set for the Mantel tests.
The results of the Mantel tests are reported in Table 9, where pvalues are omitted since they were in all cases, indicating that the values of are statistically significant. The fact that the values of were always the highest for and confirmed that there was the strongest positive correlation between the corresponding distance matrices. Specifically, was over for OLI Engineering Statics 2011 while it had the minimum value of for all the other datasets, proving strong positive correlations as well. Therefore, from Table 7, Table 8, and Table 9, KQN was shown to be stable in predicting correctness and learning the relationships between the skill vectors as the value of the vector space dimensionality was varied.
Dataset  Test AUC (%)  

ASSIST2009  82.32  82.35  82.32 
ASSIST2015  73.40  73.38  73.40 
Statics2011  83.20  83.17  83.16 
Synthetic5  82.81  82.79  82.82 
7. Conclusions and Future Work
Distance  Dataset  

      
cosine  ASSIST2009  0.521  0.653  0.570 
ASSIST2015  0.582  0.661  0.609  
Statics2011  0.616  0.825  0.609  
Synthetic5  0.495  0.526  0.511  
Euclidean  ASSIST2009  0.531  0.660  0.583 
ASSIST2015  0.601  0.682  0.629  
Statics2011  0.620  0.816  0.607  
Synthetic5  0.508  0.536  0.523 
From the experiment results for the four tasks, we list the contributions of this paper as follows:

KQN performs better than all the previous models on the four datasets for the correctness prediction task.

KQN enables the knowledge state of a student to be queried with respect to different skills, which is helpful for interpreting the knowledge interaction through visualization.

KQN’s architecture leads to the concept of probabilistic skill similarity to relate the cosine and Euclidean distances between two skill vectors to the odds ratio for the corresponding skills as introduced previously in the paper. This makes the skill vectors and their pairwise distances useful for domain modeling, e.g., with cluster analysis.

KQN is robust to the changes in the dimensionality of the vector space for the knowledge state and skill vectors in that its prediction accuracy is not degraded and it learns strongly positively correlated sets of pairwise distances between the skill vectors as the value of the dimensionality is varied, or equivalently, KQN learns the latent relationships between skills stably.
Compared to other neural network models, KQN has more parameters to learn. For example, since it includes an MLP in the skill encoder in addition to an RNN in the knowledge state encoder, KQN is computationally heavier than DKT which only has an RNN for encoding student responses. Heuristically, more GPU memory was required for training KQN compared to DKT+KQN. Still, we believe that the advantages of KQN mentioned above are meaningful enough to compensate for the increase in space complexity.
KQN proposes an alternative approach to the KT problem by defining the knowledge state and skill vectors in the same vector space. It has a general form of the knowledge state and skills as vectors while defining the knowledge interaction clearly as the dot product between the two types of vectors. From the fact that the pairwise distances between skill vectors are interpreted as the logarithm of the corresponding odds ratios from the probabilistic perspective, those distances can become useful features for domain modeling to explore the latent structure of the skill domain, which can be a future direction of the KT research.
8. Acknowledgements
This research has been supported by the project ITS/227/17FP from the Innovation and Technology Fund of Hong Kong.
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