When school pupils encounter numbers that cannot be conceptualised as quantities in an obvious way, clarity is required in terms of the language used by both teachers and textbooks. One significant risk is that the explanatory models and metaphors used by teachers are overgeneralised by pupils or create contradictions.
In her thesis, Cecilia Kilhamn shows how the difficulties experienced by pupils in understanding negative numbers – numbers that are less than zero – are similar to the difficulties that mathematicians have experienced historically. This suggests that better knowledge about the history of mathematics would provide a better understanding of pupils’ problems.
“A reluctance to accept negative numbers is closely linked to our desire to be able to concretise that which is abstract and understand negative numbers in terms of concepts such as debts, lifts or temperatures,” explains Kilhamn.
However, many of the concrete explanatory models used in school mathematics cannot deal with subtraction, multiplication or division using negative numbers. A transition to a clearer mathematical language is therefore needed when the number domain is expanded from natural numbers to signed numbers, i.e. positive and negative numbers.
The study in question is a longitudinal case study in which pupils in a school class were followed over a period of three years. The results show that pupils’ ability to accept and make sense of negative numbers depends on how well developed their sense of natural numbers is.
Insights such as being able to visualise zero as a number and not just a representation of nothing, understanding how subtraction works and being able to deal with the number line are important prerequisites for negative numbers. Another crucial factor is how clear teachers and textbooks are in their explanations. Numbers can be seen metaphorically as quantities, points, distances or operations, as constructed objects and as relations.
“But no individual metaphor for numbers can make negative numbers fully comprehensible,” continues Kilhamn. “It is therefore important that the deficiencies and limitations of these metaphors are also made clear in teaching, and that logical mathematical reasoning is used in parallel with concretised models.”
Her study also highlights a number of problems relating to the fact that the mathematical language used in Swedish schools is a little ambiguous or inadequate. For example, no distinction is made between subtracting the number x and the negative number x if both are referred to as “minus x”. There is also no word in the Swedish language corresponding to the English term “signed number”.
“Swedish textbooks introduce negative numbers without making it clear that all the natural numbers change at the same time and become positive numbers,” she adds. “Another difficulty is the size of negative numbers, which have two contradictory properties that are distinguished in mathematics by separating absolute value (magnitude) from real value (position). A large negative number has a smaller value than a small negative number. This distinction also needs to be made clear to pupils.”
The thesis was completed within the framework of the Graduate Research School in Educational Sciences at the Centre for Educational Sciences and Teacher Research (CUL) at the University of Gothenburg.