## 4 Results  
### Results for Research Objective One: Design of Open-Text Items for Assessing \"Selecting dLMs\"  
To evaluate the designed instruments' scientific quality, we used the criteria for psychometric instruments: objectivity, reliability, and validity [70]. Data were based on a sample of \(n=164\) mathematics pre-service teachers in Study 1 who worked on dLM1-dLM4. To evaluate the objectivity of the coding and scoring of the items, two coders independently categorized the participants' responses. The open-text responses to item 3 were evaluated and scored on a scale from zero to three based on the selections made in items 1 and 2 (see Figure 2). The open-text item 4--representing TCK, TPK, and TPACK reasoning--was coded according to the coding model in Table 1 and scored on a scale from \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Study** & **RQs** & **Size of** & **Sample Size Per University** & **dLMs** & **Mean Processing Time** \\  & & **Sample** & **(Austria/Germany) 1** & **Used in Study** & **of Task in Minutes** \\ \hline
1 & RQ 1.1 & 164 & 61/103 & dLM1-4 & 9.84 \\
2 & RQ 2.x & 395 & 55/324 & dLM1 & 4.29 2 \\ \hline \hline \end{tabular}  
* In Study 1, we included all responses to evaluate the coding and scoring approach’s robustness and to establish a lower time limit for carefully processing the items (2.50 min). In Study 2, we excluded all responses that did not meet the established minimum processing time. 2 The lower mean processing speed for an individual dLM in Study 1 (dLM1–dLM4), compared to Study 2 (dLM1 only), may be attributed to the task’s repetitiveness and the participant’s increasing familiarity with the items.  
\end{table} 
Table 3: Initial and adjusted sample sizes for both studies. zero to four (see Figure 2). Differences in the coding and scoring were resolved by mutual agreement, and an agreed coding and scoring was used. During this process, the examples of the coding, scoring (Tables A1 and A2), and coding model (see Figure 2 and Table 1) were used as guidelines. To assess the reliability of the items, we computed [PERSON]'s \(\alpha\), as items 1-4 were used four times, each time in the context of a different dLM (dLM1-dLM4). This denoted a repetition of the measurement \"selecting dLMs\". Using the scores established for items 1-3 across the four dLMs (dLM1-dLM4) yielded a value of \(\alpha=0.83\), and for item 4, a value of \(\alpha=0.91\). These coefficients indicate good to excellent reliability of the instruments' items. Content validity is provided since the designed instrument is based on scientifically tested and widely accepted frameworks and models, namely the TPACK framework by [PERSON] and [PERSON] (2016) and the competence model by [PERSON] et al. (2014). In addition, localized studies and [PERSON]'s core teaching practices ([PERSON] et al., 2015) support the coding model (see Figure 2 and Table 1), reflecting a more practice-oriented and Anglo-American perspective. In contrast, DigCompEdu (2016) was also employed to establish the coding model (see Figure 2 and Table 1) and describes a research-based, theoretical, and European framework for digital competence, further strengthening the approach's validity. Moreover, content validity was enhanced by engaging peers, teacher educators, and in-service teachers in developing the items. Additionally, content validity is supported by the conducted pilot ([PERSON], 2016; [PERSON], 2016) and interview studies, which examined the reasoning behind the selection of digital resources in preparation for the items' genesis and scoring. Results for Research Objective Two: Empirical Investigation of Pre-Service Mathematics Tachers Using the New Instrument to Assess \"Selecting dLMs\"  
To further examine the items (1-4) and the insights they provide regarding their generalizability, we applied them in an online test with a larger sample of pre-service mathematics teachers in Study 2 (\(n=379\)).