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: $\sigma = \frac {1} {1 + exp(-x)}$



San Diego State University

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

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

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:

Learn Before

Related