# A Solution to the Measurement Problem in the Idiographic Approach Using Computer Adaptive Practicing

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## Abstract

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## 1. Introduction

#### 1.1. Computer Adaptive Practice

#### 1.2. Math Garden

## 2. Scientific Analysis

#### 2.1. Intraindividual Analysis

## 3. Study 1: Learning Analytics

- Response probabilities of the last two responses.
- Transition probability matrix of correct and incorrect responses.
- Coefficients of a logistic regression model.

#### Results

## 4. Study 2: The Problematic Assumption of Unidimensionality

#### Results

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

IRT | Item response theory |

CAT | Computer adaptive testing |

## References

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Sample Availability: Interested researchers can contact the first author or last author to request parts of the data for research purposes. | |

1 | Given the current large number of Math Garden players item ratings converge within about a week. |

2 | For many results in this section, formal significance tests could be provided. However, we refrain from doing so because: (1) no clear a priori hypothesis could be formulated and (2) due to the large data sets very small and uninteresting results become significant in a null-hypothesis testing framework. For example, the 35% did significantly differ from the expected 36% according to a simple proportion test (${X}^{2}$(1) = 58.60, p = 1.936 ×${10}^{-14}$) |

3 | For the permutation test we used 5000 replications where in each replication all items were randomly distributed between two sets. For each set we calculated each player’s average slopes (i.e., mean learning speed) and the correlation between both of these averages values. |

**Figure 2.**One player’s development in learning to solve multiplication problems correctly. The colors refer to correct (green), incorrect (red), question mark (blue) or responses that were too late (yellow). The minimum number of responses for each time-series was 5. The items are sorted by item difficulty (low = easy and high = difficult). Plots for other players, providing different patterns, are available on www.abehofman.com/papers. The arrows along the Y axis indicate the items that are further described in the text (black arrows illustrate different response patterns; grey arrows indicate different trajectories of mastery).

**Figure 3.**The distribution of the transition probabilities of switching from incorrect to correct responses (

**top**) and remaining at a correct response (

**bottom**) for addition (

**left**) and multiplication (

**right**) for all collected time-series. For, example, the transition (learning) probability $(0\to 1)$ of 0.7 indicates that 70% of incorrect responses are immediately followed by a correct response. The bar at 0.7 in the upper-left panel indicates that this is the case in about 10% of all the collected series in the addition game.

**Figure 4.**An example of three different developmental patterns of responses to different items by the same player. The left panel shows a time-series with a clear increase in the probability of a correct response. The middle and right panel respectively show a series of a previously learned item and a series that indicates no learning.

**Figure 5.**The distribution of the estimated slope parameters, i.e., indicator of learning speed, for both data sets (

**left**-panel), and the relation between the length of the series and learning speed (

**middle**-panel) and the probability of switching from an incorrect to a correct response (

**right**-panel).

**Figure 6.**The distributions of the estimated correlations between the local skills in the addition, multiplication and simulated data.

**Figure 7.**A heatmap based on the correlation matrix of the local skills estimated on a simulated data set based on the unidimensional model. Gray squares indicate missing values in the correlation matrix, resulting from adaptive item selection. The figure can also be found on www.abehofman.com/papers allowing for more detailed inspection.

**Figure 8.**A heatmap based on the correlations matrix of the local skills of 200 multiplication items. Gray squares indicate missing values in the correlation matrix, resulting from the adaptive item selection. The figure can also be found on www.abehofman.com/papers allowing for more detailed inspection.

**Figure 9.**A heatmap based on the correlations matrix of the local user abilities of 200 addition items. Gray squares indicate missing values in the correlation matrix. See www.abehofman.com/papers for a downloadable version.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hofman, A.D.; Jansen, B.R.J.; De Mooij, S.M.M.; Stevenson, C.E.; Van der Maas, H.L.J. A Solution to the Measurement Problem in the Idiographic Approach Using Computer Adaptive Practicing. *J. Intell.* **2018**, *6*, 14.
https://doi.org/10.3390/jintelligence6010014

**AMA Style**

Hofman AD, Jansen BRJ, De Mooij SMM, Stevenson CE, Van der Maas HLJ. A Solution to the Measurement Problem in the Idiographic Approach Using Computer Adaptive Practicing. *Journal of Intelligence*. 2018; 6(1):14.
https://doi.org/10.3390/jintelligence6010014

**Chicago/Turabian Style**

Hofman, Abe D., Brenda R. J. Jansen, Susanne M. M. De Mooij, Claire E. Stevenson, and Han L. J. Van der Maas. 2018. "A Solution to the Measurement Problem in the Idiographic Approach Using Computer Adaptive Practicing" *Journal of Intelligence* 6, no. 1: 14.
https://doi.org/10.3390/jintelligence6010014