Mathematical competence can play a crucial role in the successful transition from secondary school to university. This relevance does not only apply to STEM programs, but mathematics also gains importance in various non-mathematical academic fields (Advisory Committee on Mathematics Education, 2011). A study in Germany has shown that approximately 80 percent of all students in Germany are expected to possess basic mathematical competencies in their field of study (Rohenroth et al., 2024). Therefore, the role of preliminary mathematical knowledge as a precondition for academic success (Schneider & Preckel, 2017) is in focus. Among the different mathematical areas, elementary algebra plays a key role, as it provides access to both mathematical and non-mathematical domains like engineering and natural sciences (Stacey & Chick, 2004). However, algebra is also known to pose challenges for many students (MacGregor & Stacey, 1997). Hence, possessing algebraic competencies is a critical component of academic preparedness.

In this work, the construct of algebraic competencies refers to the context-specific cognitive performance dispositions (Koeppen et al., 2008) in elementary algebra. This term is closely related to concepts such as "functional mathematics" in Britain (Council for the Curriculum, Examinations and Assessment [CCEA], 2017, p. 8). Although the content of the competencies is usually taught in secondary education, they should still be sustainably available and without aid when entering higher education. Focusing on these basic competencies ensures long-term retention and facilitates a shared understanding of expectations, making it easier to align standards across countries and educational systems. Finally, the aim is to describe algebraic competencies at empirically and normatively defined levels (e.g., including a minimum level at the beginning of university) and thus to include and depict both procedural and conceptual activities (Hiebert & Lefevre, 1986) that are complementary to elementary algebra (Arcavi et al., 2017).

There are several approaches in the literature for structuring algebraic competencies. Kieran's (2007) model describes typical algebraic activities on three facets: the generative activity, in which learners recognize and generalize patterns; the transforming activity, which focuses on the transformation and manipulation of algebraic expressions; and the global-/meta-level activity, which includes an understanding of algebraic concepts, for example for argumentation and proof (in short: GTG model). Other models also contribute to structuring algebra, such as the Specification Standards for Mathematics in the UK (CCEA, 2017), which incorporate numbers and algebra as a unified subdomain, and the model of defining algebra as encompassing signs, variables, algebraic expressions, equations, relations and functions (Arcavi et al., 2017).

Some universities in Germany provide self-assessment-tests for students to check their preparedness in mathematical competencies. But these tests vary in the content areas tested and often lack theoretical and empirical foundation. Despite its importance, there is so far no valid and reliable test that assesses algebraic competencies for students entering higher education. The project’s aim was to create an instrument that records and describes the individual competence level of learners at the onset of university and thus aids targeted support where necessary. Therefore, a test and a competence level model for elementary algebra with six different levels were developed.

The test was a priori conceptualized covering three dimensions. First, elementary algebra was split into the four subdomains numbers, expressions, equations, and functions. Numbers were included since irrational numbers and exponent laws are taught alongside elementary algebra. The second dimension captured the three facets of the GTG model, ensuring a comprehensive representation of algebraic activities. The third dimension reflected increasing levels of difficulty, scaling tasks to differentiate between competency levels. Based on this pre-conceptualization, an item pool consisting of 163 constructed or selected response items was developed (31 items on numbers, 39 on expressions, 48 on equations, and 45 on functions). Content validity was ensured through alignment with the conceptual framework, as the GTG model comprehensively addresses core algebraic processes. Additionally, some items were adapted from international student achievement tests such as TIMSS to further strengthen validity. Subsequently, a competence level model was created based on responses to the test. The sample involved N=232 students enrolled in math-based degree programs in Germany, such as secondary teacher education (n=98), physics (n=118), and computer science (n=8), most of whom were in their first year of study. Each participant completed a random selection of 60 items from the entire pool of 163 items, maintaining variability and comprehensive coverage of all four subdomains. The online test was designed to last 60 minutes and was completed without any aid except paper and pen. Data analysis was conducted using item response theory by estimating a Rasch model. To assign items to an ability value on the logit scale, the response probability was set to two-thirds. To develop the competency level model, the scaled items were organized into an ordered item booklet format, which systematically arranged tasks by increasing difficulty. A description was written for each item to capture the specific algebraic competencies being assessed. These item descriptions were used to formulate the respective competency level descriptions, providing a detailed characterization of each level containing all four subdomains. The Common European Framework of Reference for Languages (Council of Europe, 2001), which categorizes proficiency levels from A1 to C2, was adopted as the structural basis for this model. Thresholds between levels were defined a priori. The levels were defined in intervals of 1 logit. A value of 0 logit was set as the criteria benchmark, fitting a normatively defined minimum standard of basic algebraic competencies at the B2 level, which students are expected to possess upon entering university.

As test development was central to the project, item statistics are reported in the results section. The average item difficulty was -0.68 logit (SE = 0.31 logit) varying from -2.98 to 5.19 logit with nine items exceeding three logits. The mean solution rate was 61.79%. The point-biserial correlation coefficients for item discrimination ranged from 0.06 to 0.69 (Mdn = 0.42). Three items fell below the threshold of 0.15 logit and 16 items below 0.25 logit. The EAP reliability of 0.92 indicates a high level of measurement precision in estimating participants' abilities (Bond & Fox, 2015). Infit mean square values ranged from 0.68 to 1.32. Two out of 163 items exceeded an infit value of 1.3 and one item fell below an infit value of 0.75. Hence, more than 90% of all items were used due to appropriate difficulty, discrimination values and model fit. A comprehensive competence model was developed describing each level of competence. For illustration, descriptions from level B2 are given: In the subdomain of numbers, participants are expected to calculate the square root of a perfect square greater than 100. In the subdomain of expressions, they must be able to simplify expressions with multiple variables and parentheses. For equations, they can solve a simple quadratic equation of the form x² + px + q = 0 by systematic trial and error or using a formula (e.g., by p-q formula or Vieta's theorem). Finally, in the subdomain of functions, they can construct linear equations for given intra- or extra-mathematical contexts. In conclusion, the test and competence level model provide a framework for classifying individuals based on elementary algebraic competencies. Repeated administration will increase sample size and enhance model accuracy. A follow-up study in upper secondary schools is planned to explore differential item functioning, extending the model’s applicability across educational settings.

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