Götz, Gerhard

Pinkernell, Guido

Schönwälder, David

English

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Relevant aspects of proficiency in secondary school
arithmetic for a successful start in STEM subjects
David Schönwälder, Guido Pinkernell, Gerhard Götz
To cite this version:
David Schönwälder, Guido Pinkernell, Gerhard Götz. Relevant aspects of proficiency in secondary
school arithmetic for a successful start in STEM subjects. Eleventh Congress of the European Society
for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. ￿hal-
02459868￿
  
Relevant aspects of proficiency in secondary school arithmetic for a 
successful start in STEM subjects 
David Schönwälder
1
, Guido Pinkernell
2
 and Gerhard Götz
1 
1
Cooperative State University Baden-Württemberg Mosbach, Germany; 
david.schoenwaelder@mosbach.dhbw.de, gerhard.goetz@mosbach.dhbw.de 
2
University of Education Heidelberg, Germany 
pinkernell@ph-heidelberg.de 
Keywords: Secondary school mathematics, STEM education, basic knowledge, arithmetic 
Introduction and research interest 
According to recent studies, high dropout rates in STEM subjects at university can be explained at 
least in part by a lack of basic knowledge and comprehension of secondary school mathematics 
(Blömeke, 2016), among which specific secondary arithmetic areas like fractions, and especially 
other areas such as powers or logarithms need to be subsumed. Therefore, many universities 
provide so-called “bridging courses” for students beginning a STEM subject at their institution. 
Sometimes, the tasks for diagnostic and supporting measures rely on theoretical frameworks outside 
mathematics (e.g. Bloom’s taxonomy), sometimes the theoretical base is not made explicit but 
seems to follow expectations as expressed in local coursebooks. 
Hence, for diagnosing basic knowledge and comprehension in secondary school arithmetic, a 
comprehensive yet concise overview of the important aspects of this area is needed. While there is 
extensive educational literature on the formation of basic arithmetic skills in elementary operations 
or fractions, we do not focus on the formative, but on a summative view on how students should 
master the various fields of secondary arithmetic at the transition from school to university. Thus, 
our research question is: Which central aspects of knowledge and comprehension in secondary 
school arithmetic can be identified in relevant literature and how can these be summarized for 
diagnostic and supportive measures at university entry level? 
The methodological base is a systematic literature review (Durach, Kembro, & Wieland, 2017). 
Hereby, relevant didactic literature as well as formative literature about formation of concepts in 
secondary school arithmetic as mentioned above is considered. A qualitative content analysis of the 
literature found and subsequent clustering gives the aspects of the frame of reference.  
Following Pinkernell, Düsi, & Vogel (2017) two a priori settings are given: contents (answering the 
question: Which are the basic elements of secondary school arithmetic?) and understanding. 
Understanding is subdivided into knowing in the sense of declarative knowledge (Anderson, 1996) 
and acting, broken down into structuring as a meaningful reading of area-specific expressions 
(Musgrave, Hatfield, & Thompson, 2015: substitutional equivalence), transforming as restructuring 
the area-specific expressions into equivalent forms (ibid: transformational equivalence) and 
interpreting as coherent change between different forms of representation or possible extra-
mathematical contextualization of the same mathematical object (Duval, 2006).  
  
A theoretical framework for analyzing and constructing tasks in secondary 
school arithmetic – a first draft 
The outcome of this research takes on a tabular form, in which the various findings in literature 
have been clustered into nine aspects of knowledge and comprehension of secondary school 
arithmetic. Out of these we mention two here. As contents or basic elements of secondary school 
arithmetic numbers and quantities as well as terms arise in the frame of reference. Variables appear 
too, but are only used in the sense of the generalized number. 
- “to change within the same numerical representation” – A numeric number or quantity is 
converted into another, equivalent numeric number or quantity.   
- “to switch between different forms of representation” – A coherent change between 
numerical and other inner-mathematical forms of representation as well as verbal 
representations. 
The model is still subject to further development. As it is planned for the German project optes+ 
(www.optes.de) it will – in its final stage – serve as both a reference for analyzing existing tasks and 
constructing new ones and a conceptual base for discussion about what is considered necessary for 
successful studies in STEM subjects at university entry level.  
References 
Anderson, J. R. (1996). Kognitive Psychologie (2nd ed.). Heidelberg: Spektrum Akad. Verl. 
Blömeke, S. (2016). Der Übergang von der Schule in die Hochschule: Empirische Erkenntnisse zu 
mathematikbezogenen Studiengängen. In A. Hoppenbrock, R. Biehler, R. Hochmuth, & H.-G. 
Rück (Eds.), Lehren und Lernen von Mathematik in der Studieneingangsphase: 
Herausforderungen und Lösungsansätze (pp. 3–13). Wiesbaden: Springer Spektrum.  
Durach, C., Kembro, J., & Wieland, A. (2017). A new paradigm for systematic literature reviews in 
supply chain management. Journal of Supply Chain Management, 53(4), 67–85. 
Duval, R. (2006). A Cognitive Analysis of Problems of Comprehension in a Learning of 
Mathematics. Educational Studies in Mathematics, 61(1–2), 103–131.  
Musgrave, S., Hatfield, N., & Thompson, P. (2015). Teachers’ meanings for the substitution 
principle. In T. Fukawa-Connelly, N. Engelke Infante, K. Keene, & M. Zandieh (Eds.), 
Proceedings of the 18th Meeting of the MAA Special Interest Group on Research in 
Undergraduate Mathematics Education (pp. 801–808). Pittsburgh: Mathematical Association of 
America. 
Pinkernell, G., Düsi, C., & Vogel, M. (2017). Aspects of proficiency in elementary algebra. In T. 
Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Mathematical 
Society for Research in Mathematics Education (pp. 464–471). Dublin, DCU Institute of 
Education & ERME. 
 
 


Relevant aspects of proficiency in secondary school arithmetic for a successful start in STEM subjects

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Relevant aspects of proficiency in secondary school arithmetic for a successful start in STEM subjects

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