What does the research say?
You have surely heard many of your students read the number sentence ‘7 – -4’ as ‘7 minus minus 4’ and wondered what sense they made of the calculation and symbols.
- What does ‘minus minus’ mean to them?
- Do they think of the first minus as an operation or not?
- Do they notice the different size ‘minus signs’ you wrote on the board?
- What did they write in their books?
- Is the minus sign different from the negative sign?
- Is minus minus 4 the same as 4 or (+4)?
- Why write +4 and not just 4?
These questions make it easy to understand why working with negative numbers is confusing and difficult for students to master. This step provides a brief overview of some of the literature which focuses on the implications of language and notation when working with negative numbers.
What does the negative sign mean?
Vlassis (2008) argues that the negative sign can take on at least three meanings in mathematics, and as such there are three main categories of students’ difficulties related to the numerical system: the meaning of the mathematical operations and the meaning of the minus sign. These difficulties arise from students’ lack of awareness of the functions of the minus sign, namely the unary, binary, and symmetric functions.
- In the unary function the minus sign acts as guidance to the reader that the number is indeed negative, as in -10 being ‘negative 10’.
- The binary function refers to any situation where the minus sign shows that the operation is a subtraction (an operation sign). This is perhaps school students’ most common interpretation, i.e. a negative sign interpreted as an action such as taking away or subtraction.
- With its symmetric function the minus sign is an indicator of the number being the opposite of its related number, as in the example 5 and -5. To exemplify, this means that for the expression -(-8), the first negative sign would signify the operation of taking the opposite of -8, while in 2a – (4a – 3b), the first minus in the expression indicates the opposite of 4a-3b.
Students should understand the minus sign in these three senses: unary, binary, and symmetric.
Promoting dialogue around the different meanings
However, researchers have also shown that students’ application of these different and often ambiguous conceptualisations can lead to incorrect reasoning in calculations. They also suggest that students must be given opportunities to discuss and develop a sense of the different meanings of the minus sign in order to be flexible, develop a robust concept image of the sign, and promote a symbol sense.
In line with the three concept dimensions of negative number mentioned above, Bofferding (2014) recommends that:
- Use of the term ‘minus’ should generically refer to the ‘-‘ symbol;
- The subtraction sign refers to the binary meaning of the minus sign; and
- The negative sign refers to the unary meaning of the minus sign.
Teachers should carefully consider the implications of the language and notations around negative numbers! They should model helpful language, especially with younger learners, for example:
- Read -3 as ‘negative 3’, rather than ‘minus 3’;
- Read 6 - 5 as ‘6 subtract/take away 5’ rather than ‘6 minus 5.
This is a very big pedagogical and conceptual challenge. Referring to -5 as negative 5, rather than minus 5, may help on the context of the classroom (Beswick 2011). However, the students will have heard negative temperatures read as ‘minus 5’, so the question remains whether and to what extent the recommendations are helpful.
In the following activity you are presented with a number of scenarios involving students’ work related to negative numbers. We invite you to engage with the research reviewed here to gain an insight into why students make such mistakes and how you can help them.
Beswick, K. (2011). Positive Experiences with Negative Numbers: Building on Students in and out of School Experiences. Australian Mathematics Teacher, Vol. 67 (2,), 31-40.
Bofferding, L. (2014). Negative integer understanding: Characterizing first graders’ mental models. Journal for Research in Mathematics Education, 45(2), 194-245.
Vlassis, J. (2008). The Role of Mathematical Symbols in the Development of Number Conceptualization: The Case of the Minus Sign. Philosophical Psychology, 21(4), 555-570