Abstract
Making use of a phenomenological stance which first and foremost values the lived experience of learners, six tasks are used to illustrate what it might mean for a mathematical task to be deemed worthy of being offered to learners. These take the form of encounters with, and opportunities to develop, pervasive mathematical themes, use of mathematical powers and experience of mathematical concepts and topics. Comments about how worthwhile mathematical tasks can evolve centre around developing the propensity, the habit of mind to extend, vary and generalise for oneself. Mathematical thinking is sustained by developing this disposition.
Highlights
The word worthwhile in the title begs questions such as worthwhile to whom, and to what end? Here worthwhile is taken to mean that a teacher considers it worthy of a learner’s time and attention, not because the topic is on the curriculum, but because consequent activity could serve to enrich and enhance learner appreciation and comprehension of the use of their natural powers and of ubiquitous mathematical themes, as well as of specific mathematical constructs, procedures or topics
In order that human life becomes sustainable on this planet, it is necessary that mathematical models are developed which allow us to analyse the human impact on the environment
In order to create and to critique effective mathematical models, it is essential to have developed the use of natural powers of sense-making in a mathematical context. This includes powers such as: Imagining, and Expressing what is imagined, in multiple modes; Specialising and Generalising, which includes trying particular examples, and using them to detect possible generalisations; Conjecturing and Convincing, which includes not believing the conjecture, and trying to convince oneself, a friend, a sceptic; Classifying and Characterising constructs by means of local and global properties; Stressing and Ignoring, which are actions of attention, and include discerning differences. Alongside these natural powers, which are fundamental to mathematical thinking, there are ubiquitous mathematical themes which draw upon these powers, and amongst which are: Doing and Undoing, which includes seeking inverses and bypasses [1]; Seeking and exploiting Invariance in the Midst of Change; Extending and Restricting meaning; Freedom and Constraint, as in placing increasing constraints or conditions in order to limit the number of objects satisfying the constraints; most routine exercises constrain to a single object to be found
Summary
The word worthwhile in the title begs questions such as worthwhile to whom, and to what end? Worthwhile is taken to mean that a teacher considers it worthy of a learner’s time and attention, not because the topic is on the curriculum, but because consequent activity could serve to enrich and enhance learner appreciation and comprehension of the use of their natural powers and of ubiquitous mathematical themes, as well as of specific mathematical constructs, procedures or topics. What matters most to me, certainly, is the lived experience of learners. After stating some initial assumptions, I offer some specific tasks which seem to me to illustrate and highlight qualities of worthwhile tasks, with comments about how they were generated
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