Katrin Klingbeil

Understanding the values that variables can take is a central challenge in students’ early encounters with algebra. While previous research has documented a range of numerical preconceptions related to variable values, much less is known about how they co-occur and develop. This study examines German Year 7/8 students’ (mis)understandings of variable values before and after four weeks of regular algebra teaching. Data were collected from 2205 students who completed the SMART ‘Values for Letters’ test. The test targets well-documented numerical preconceptions, including difficulties with substituting given values, alphabet-based value assumptions, treating multiple occurrences of the same letter as referring to independent values, and assuming that different letters must represent different numbers. To capture qualitative differences in students’ response patterns and their development over time, we applied multilevel Latent Transition Analysis (LTA), accounting for students nested within teachers. The analysis identified four latent classes representing distinct ways of reasoning about variable values. Although overall response rates changed only modestly, the LTA revealed substantial movement between latent classes. Alphabet-based value assumptions were rare, whereas treating multiple occurrences of the same letter as referring to independent values emerged as a persistent, intermediate bottleneck in students’ learning. Moreover, the assumption that different letters must refer to different values appeared as an overgeneralization rather than a foundational difficulty, occurring primarily among students in the highest-performing latent class. These findings show how LTA reveals learning trajectories invisible in aggregate analyses and supports diagnostic work in algebra instruction. Limitations include attrition and the short period between timepoints.

Multiple-choice (MC) items play a crucial role for any assessment as they can be quickly administered and automatically analysed, allowing a quick insight into students’ current understanding. While they are commonly used in the context of summative assessments, they are also suitable for formative assessments aimed at enhancing mathematics teaching and learning. However, such MC items need to be carefully designed based on research and validated empirically. Drawing on scientific findings from both educational psychology and mathematics education, this chapter aims to provide possible guidelines on how to design and validate MC items using an example from the digital formative assessment tool SMART. Furthermore, different scenarios are described, illustrating how well-designed MC items, especially when combined with digital technology, can be utilised for, on the one hand, diagnosing and enhancing students’ understanding and, on the other hand, providing teachers with important knowledge about possible misconceptions and relevant facets of a mathematical concept, thereby enhancing their pedagogical content knowledge.

When students first encounter variables in algebra, they often develop the Letter-as-Object misconception, interpreting letters as labels for objects rather than references to numerical values. This study analysed the response patterns of 2220 German Year 7 and 8 students (aged 12–14) using the SMART test ‘Meaning of Letters’, designed to assess various subtypes of this misconception. Employing multilevel Latent Transition Analysis, we identified five latent classes and tracked shifts in students’ classifications over 4 weeks of algebra instruction. These classes revealed persistent patterns in how the Letter-as-Object misconception and its subtypes manifest. While the majority of students exhibited this misconception, their responses varied depending on the equation type in the test items. Encouragingly, the class of students with mostly correct responses tripled in size after instruction, though it remained small, with a majority of students staying in their original classes. This study highlights the potential of multilevel Latent Transition Analysis to track misconceptions and conceptual changes over time and is among the first to account for the multilevel data structure of students nested within teachers in a Latent Transition Analysis.

Assessing students’ (mis)conceptions is a challenging task for teachers as well as for researchers. While individual assessment, for example through interviews, can provide deep insights into students’ thinking, this is very time-consuming and therefore not feasible for whole classes or even larger settings. For those settings, automatically evaluated multiple-choice (MC) items could be a solution. However, it is a challenge to design those items and to adapt them for other countries in a way that they adequately reveal students’ (mis)conceptions. In this article, we investigate the question whether it is valid to use a German adaption of a multiple-choice test developed in Australia for formative assessment of the letter-as-object misconception in Germany. For this, first semi-structured interviews with five German Year 8 students were conducted, and second, 616 students were asked for short written explanations. These data were analysed with regards to the students’ (mis)conceptions and compared with their automatic online diagnosis. In general, a high concordance between online SMART test results and students’ explanations was observed, confirming that useful diagnoses of student misconceptions can be obtained from such a short well-designed MC test.

In this crossover experiment, we investigated the impact of a statement bank, enabling the reuse of previously written feedback (SA condition), on 45 math teachers' feedback for 60 completed linear equation tests, compared to traditional pen‐and‐paper feedback (PP condition). In the SA condition, teachers were encouraged to use atomic feedback, a set of formulation requirements that makes feedback items significantly more reusable. A previous study found that significantly more feedback was written in the SA condition but did not investigate the content of the feedback. To address this gap, we employed a novel approach of combining text mining with qualitative methods. Results indicate similar wording and sentiments in both conditions. However, SA feedback was more elaborate yet general, focusing on major and minor strengths and deficits, while PP feedback was shorter but more concrete, emphasising main issues. Despite low feedback quality in both conditions, the statement bank led to less effective diagnostic activities, implying that teachers' careless use of statement banks, although convenient, might lead to lower‐quality feedback. Practitioner notes: What is already known about this topic High‐quality feedback should strike a balance between the volume and focus on the main issues, as more feedback does not necessarily equate to better feedback. Feedback should analyse a student's solution whenever possible: interpreting mistakes and communicating that interpretation as feedback. Text mining identifies meaningful patterns and new insights in text using computer algorithms. When teachers can reuse already given feedback using a software tool (statement bank), they tend to write more feedback instead of saving time. What this paper adds Feedback is compared when teachers could use a tool to reuse already given feedback (referred to as ‘statement banks’) versus a scenario without such a tool. Both approaches observed similar word frequencies, sentiments and amounts of erroneous, descriptive and corrective feedback. However, feedback with a statement bank tended to be more elaborate yet less specific to individual student solutions. In contrast, feedback without the tool was shorter but more concrete, focusing on main issues. Overall, the tool for reusing feedback directed teachers towards less effective diagnostic activities. The paper introduces a novel methodological approach by combining text mining with qualitative techniques in educational research. While text mining provides an overall understanding of differences and similarities in feedback approaches, qualitative methods are essential for in‐depth analysis of content characteristics and feedback quality. Implications for practice and/or policy Statement banks can support teachers by giving more feedback, but in order to improve feedback quality, further measures are necessary (eg, improving pedagogical content knowledge). Teachers may not confuse handiness with quality: statement banks can help, but when used carelessly, teachers tend to describe and correct students' work instead of analysing underlying (mis‐)conceptions using it. Continued attention to feedback quality remains necessary when using such tools.