A lesson without the opportunity for learners to generalise (mathematically) is not a mathematics lesson
A lesson without the opportunity for learners to generalise (mathematically) is not a mathematics lesson
Mathematics is fundamentally about becoming aware of and expressing generality.
Main Section
No-one expects young children to memorise all two-digit additions and subtractions. Rather learners are expected to reconstruct a collection of general methods which will enable them to carry out not only all two digit additions and subtractions, but any additions and subtractions. So even the youngest of children are expected to generalise.
Caleb Gattegno (ref.) once said that “something is mathematical only if it is shot through with infinity”. By this he meant that there is some generality present, however limited (even to finite cases!). [see Freedom & Constraint]
When several or many separate experiences or objects are subsumed under a single heading, there is a slight frisson of excitement. Human beings operate by classifying things under headings so that they don’t have to deal with a plethora of single and individual items. This is reflected in language, where each noun provides a label for a whole class of objects. Words like ‘cup’, ‘chair’, ‘number’ and ‘triangle’ can be used to describe a whole range of objects with certain properties. Through the use of labels to summarise properties, actual or potential infinity is tamed.
If learners are left on their own to generalise in a lesson, without considerable explicit exposure to useful mathematical generalisations, then they are likely to form generalisations about the teacher’s behaviour, and about their relationship to mathematics as a whole. For example, if tasks consistently use the same types of questions, learners are likely to form the impression ( a generalisation) that all of mathematics is similar, and so miss out on experiencing the creative and exploratory nature of mathematical thinking. [see asking mathematical questions mathematically]
The notion of an example space is one way to think about how learners might encounter generality, as they extend the range of examples to which they have access in the future.
Recognising, or recognising and expressing a generality deserves to be the yamada(the highpoint or central point around which the lesson is constructed).
Probes and Prompts
In what ways do you call upon learners to generalise?
Ask: what does this remind you of?
Taking Action
Discuss with colleagues different ways of being more explicit about getting learners to generalise in lessons.
Look at a teaching page of your textbook and ask yourself what implicit generalities are indicated or available, and presumably intended by the author.
Case Studies
Research Sources
See Also
Categories
Concepts, Pedagogy, Themes