In this papers:
1. Students are indexed by $s \in [1, S]$
2. Items: $j \in [1, J]$
3. Skills: $k \in [1, K]$
4. Timestamps: $t \in \R^{+}$ (in days)
5. Correctness of the answer: $Y_{s, j, t} \in {0, 1}$
6. Logistic function: $\sigma = \frac {1} {1 + exp(-x)}$



General (DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills)

Adaptive spacing algorithms using user's study history make decisions about which question to ask. There are several adaptive spacing applications like Anki, Supermemo, Mnemosyne. 

In order to design spaced repetition adaptive schedulers, several researches have focused on modeling human memory statistically and the item which is recommended, should have a value closest to some predefined fixed value. 

There are other approaches as well in which the researches don't rely on memory models. In this cases deep learning algorithms were used and they were tested on 
simulated students.

In this paper the authors follow the second approach of the above-mentioned ones. 

Even thought traditional adaptive spacing algorithms make use of spacing retrieval  strategy, they still lack the ability to adapt to learning and memorization of skills.  The authors want to work on this part. They want to extend adaptive spacing algorithms to memorization of skills. 

Adaptive Spacing Algorithms (DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills)

As it was written in the paper, the are two main approaches to modeling student learning namely: **knowledge tracing** and **factor analysis**.

**Knowledge tracing** as it can clear from the name of this approach, here we trace student's knowledge and we model the development of student knowledge to predict the sequence of answers. Bayesian Knowledge Tracing model is the most famous model of knowledge tracing. 

**Factor Analysis** compared to knowledge tracing doesn't care about the order of observations. 

1. **Item Response Theory** (IRT) is factor analysis model:

$$ P(Y_{s, j} = 1) = \sigma(\alpha_s -  \delta_j) $$

Here $\alpha_s$ denotes student's  ability and $\delta_j$ item difficulty. (it is assumed that student ability is unchangeable, it is fixed). At first glance this might seem very simple model but it is able to learn better the **Knowledge Tracing** architectures like Deep Knowledge Tracing.

2. **Multidimensional Item Response Theory** (MIRT) extension of IRT:

$$ P(Y_{s, j} = 1) = \sigma(<\alpha_s, \delta_j> + d_j) $$

Here $alpha_s$ and $\delta_j$ denote exactly the same with only one difference, here we have multidimensional vectors and $d_j$  captures the easiness of the item j. 

In recent works, student history was also used in factor analysis and the following models where created: **Additive Factor Model** (AFM) and **Performance Factor Analysis** (PFA). 

The formula for the AFM looks as:

$$P(Y_{s. j }= 1) = \sigma (\sum_{k \in KC(j)}\beta_k + \gamma_k \alpha_{s,k} )$$

here $beta$ denotes bias for skill, $\alpha$ is for the number of attempts and the $\gamma$ is  for the bias 

$$ P(Y_{s,j} = 1) = \sigma (\sum_{k \in KC(j)}\beta_k + \gamma_k c_{s,k}  + \rho_k f_{s,k})$$

c (f) is for the number of correct (wrong) answers.

Knowledge Tracing Machine (KTM) incorporates IRT, MIRT, AFM and PFA. The probability of correctness is expressed as:

$$ P(Y_t = 1) = \sigma (\mu + \sum_{i=1}^N w_i x_{t,i} + \sum_{1\le i \le l \le N } x_{t, i} x_{t, l} <v_i, v_t>)$$

(**NOTE** this model was proposed in the paper: Knowledge Tracing Machines: Factorization Machines for Knowledge Tracing)


Here $\mu$ is a global bias, N  -  the number of abstract features, ( for instance, item parameters)., $x_t$ - a sample gathering all features collected at time t: which student answers which item, and information regarding prior
attempts (x is a sparse vector), $w_i$is the bias  of feature i  (w is the vector of biases) and $v_i \in \R^d$  its embedding of dimension d. In the prediction will contribute only those $x_t$ that are greater than 0.  The features involved in a sample $x_i$ are typically in sparse number, so this probability can be computed efficiently. 

DASH(Difficulty Ability and Student History) bridges the gap between factor analysis and memory models. Here both learning and forgetting processes are taken into account.  

The authors plan to extend DASH so that it can account for both multiple skills tagging and memory decay.

This is the example of activation of a knowledge tracing machine:

Modeling Student Learning and Forgetting (DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills)



1. General
2. Adaptive Spacing Algorithms
3. Modeling Student Learning and Forgetting

San Diego State University

1. Introduction
2. Related Work
3. Framing the Problem
4. Our Model DAS3H
5. Experiments
6. Conclusion and Future Work

DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills

Reference for DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills

It is very important for the students to efficiently manage their time so that they can learn new concept and retain the previously learned ones. Unfortunately, only few of the colleges and the universities can suggest students efficient structure which would make it possible to achieve both of these goals. In the 19th century testing and spacing methods were proposed and they make it possible to efficiently manage the learning time as well as improve long-term memorization. According to **spacing** method it is better distributing learning over time than massive learning in order to have long-term retention. **Testing** (_retrieval practice_) "basically consists in self-testing after being exposed to new knowledge instead of simply reading the lesson again."  The combination of these to can be referred as **spaced retrieval practice**  (temporally distributing tests after the first exposure to the knowledge). Nowadays, researcher have focused on using adaptive and personalized spacing schedulers to improve student learning. Spacing effects have been used in various areas and aren't limited to the vocabulary learning. In this paper the authors aim to create " student learning and forgetting for inferring skills knowledge state and memory dynamics. " As it was described, the authors propose to extend adaptive spacing framework where memorization will focus on the underlying skills. This means that the learner will have the chance to master the set of problems involving that particular skill. They think that this requires new student model which would involve forgetting and learning processes for the multiple skills. This model is called DAS3H .

Introduction (DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills)

Related Work (DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills)

Student masters skills by interacting with and adaptive spacing system. This system chooses each exercise per each iteration in a manner that it will optimize utility function l  which in turn rewards long-term mastery of Knowledge Components(KCs). After getting students answer, corresponding updates are made. 

As the authors claim: "In a nutshell, our present research goal is to maximize mastery and memory of a fixed set of skills among students during a given time interval while minimizing the time spent studying."

This assumptions are made:
1. information to learn and remember consists in a set of skills $k\in [1, K]$
2.  skill mastery and memorization of student s at time t is measured by the ability of s to answer an (unseen) item involving that skill, i.e. by their ability to generalize
to unseen material
3. students first have access to some theoretical knowledge about skills, but learning happens with retrieval
practice
4. items are tagged with one or multiple skills and this information is synthesized inside a binary q-matrix
5. students forget: skill mastery decreases as time goes by since last practice of that skill



Framing the Problem (DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills)

**DAS3H** stands for Difficulty, student Ability, Skill, Student Skill practice History. 
 DASH is the model which they extend and it showed better results than the Bayesian version of IRT (Item Response Theory). This model didn't handle multiple skills item tagging. and additionally, it is assumed that "the impact
of past practice on the probability of correctness does not vary across the skills."
In DAS3H DASH is extended in the following ways:

1. Handle multiple skill tagging: $h_{\theta}$ now takes into account multiple skills. 
2. For each item and skill easiness of the parameters are estimated.
3. Logistic regression is replaced with KTMs (Knowledge Tracing Machines)

DAS3H is formulated as (when embedding dimesion is 0):
(1)$$P (Y_{s, j, t} = 1) = \sigma (\alpha_s - \delta_j + \sum_{k \in KC(j)} \beta_k + h_{\theta} (t_{s, j, 1: l}, y_{s, j, 1: l-1}  )  $$

The formula for $h_{\theta}$ is:
(2)$$ h_{\theta}(t_{s, j, 1:l}, y_{s, j, 1: l-1}) = \sum_{k\in KC(j)} \sum_{w=0}^{W-1}\theta_{k, 2w + 1} \log (1 + c_{s,k, w}) - \theta_{k, 2w + 2} \log (1 + a_{s,k,w})$$


In the first formula probability of correctness for student s,on item j at time t, depends on student ability ($\alpha_2$), item difficulty ($\delta_j$), sum of easiness $\beta_j$ and on the temporal distribution and the outcomes of the past practice which is expressed using $h_{\theta}$.

In the second formula, w is the index of the time window, $c_{s, k, w}$  is the amount of correctly remembered Knowledge Component (KC) k by student in the time window w earlier and $a_{s, k, w}$ is for the total amount of encountering Knowledge Component (KC) k at time window w earlier.  


Our Model DAS3H (DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills)

1. Experimental Setting
2. Results
3. Discussion

Experiments (DAS3H: Modeling Student Learning and Forgetting for Optimally Scheduling Distributed Practice of Skills)

As the authors of the paper claim their work can be extended in various ways. Firstly, DAS3H and DASH can be compared to other models. Also in the real world items might include several skills at the same time. The interesting question to investigate is: how to select next item to maximize student memory ?

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