The naughty nought: students' difficulties with zero

Abstract It has been known for decades that the number zero causes problems to many students at several education levels. In this paper, we ask the question of what specific characteristics of the number zero give rise to students’ difficulties in coming to grips with the notion, properties and roles of this particular number. We provide an extensive literature review of previous studies related to this issue and propose a categorisation of the main aspects and roles of zero from a mathematical point of view. Based on this, we have devised a test consisting of 20 questions addressing students’ conceptual understanding of and operations with the number zero that was administered to 214 Danish upper secondary school students (grade 10, approx. 15–17 years of age). Of these 20 questions, we analyse, in this paper, the answers to 14, organised into five different categories. Main findings include the following: Many students do experience serious difficulties in dealing with zero in various mathematical contexts. Many are unable to reconcile their everyday real world understanding of zero as signifying “nothing” in abstractum with its mathematical properties. Thus, several students give zero as the answer to questions that do not involve zero, e.g. the value of a fraction, whose numerator and denominator is the same non-zero number. This suggests that in-depth, systematic and understanding-focused teaching of zero as a mathematical object needs to be designed and implemented if the observed difficulties are to be counteracted.