Key Concepts in Mathematics - Generalising
If these concepts are not fully developed students’ will find it difficult to engage
meaningfully with core aspects of the Number, Algebra and Functions strands in later
years
Generalising Claiming that something is always true
How does the concept develop?
‘‘Generalisation is a heartbeat of mathematics. If the teachers are unaware of its presence, and are
not in the habit of getting students to work at expressing their own generalisations, then mathematical
thinking is not taking place’’ Mason (1996) (p. 65).
Students begin to make generalisations when they begin to address the question Does this always
work? When they begin to justify their own generalisations, they tend to use diagrams, concrete
objects and words to do so.
As their statements become more complicated they begin to need other
ways to point at ‘the first number’, or ‘the bigger number’. This is the beginnings of what later
becomes conventional algebraic notation. As they move from particular numbers and actions to
patterns of results, they start viewing numbers and operations as a system. This reasoning about
operations rather than the notation is part of the work of the bridging period in algebra. Looking for
pattern and generalising it, the other area of focus during this period.
Students are ready to engage with the learning outcomes associated with generalisation when they
can
deal with equivalent forms of expressions
recognise and describe number properties and patterns
work with the complexities of algebraic text
Difficulties may arise if students
do not have an understanding of equality as a relationship between number sentences
have limited access to multiplicative thinking and proportional reasoning
Reasoning about mathematics is an objective of the syllabus and students can begin to show formal
reasoning by generalising patterns to fit various situations. In the bridging period we want students to
be able to do the following:
Reason about a problem
Extend what they already know
Make a conjecture
Provide a convincing argument
Refine their thinking
Defend or modify their arguments
For many students, this will not be formal proof, but it will help them be better prepared to use proof in
a more formal context later in post primary school. More importantly, as students become more adept
in explaining and justifying their thinking, the mathematics they are learning will make sense which is
what mathematics should be for all students – sensible and reasonable.
Read the case studies and tasks for ideas on how you can support and track your students’
development of the concept of Generalising and their Understanding of equality
