Dietmar Küchemann
This blog is written for mathematics teachers and others involved in mathematics education and provides a commentary on a series of tasks on Multiplicative Reasoning. The tasks themselves are aimed primarily at students in secondary school (Key Stages 3 and 4 in the UK).
The big ideas involved in multiplicative reasoning might be said to be ratio, proportion and rational number. However, the diversity of ways in which we meet and interpret these ideas indicates that multiplicative reasoning forms a rich and extensive conceptual field¹. In my view, there is no optimal learning trajectory through this field, indeed the notion of trajectory is itself problematic. As students engage with and develop ideas involving multiplicative reasoning they also develop, strengthen and reconstruct connectons with other ideas in the field. Over time, one hopes that the resulting network of ideas becomes more extensive and better connected so that students develop a richer and more secure understanding.
Put another way, students’ understanding of a concept such as ratio is not all or nothing. While students might be able to solve a ratio task set in a recipe context when there is a simple numerical relation between numbers in the same measure space (eg converting a recipe for 4 people into a recipe for 12 people), they might struggle if such a simple relation can only be found between measure spaces (eg being told 4 people require 12 pancakes and being asked to convert this to a recipe for 7 people). And students who recognise that a particular recipe task is multiplicative, might well treat a task involving the same numbers, but set in a geometric context, as additive².
The tasks in this blog have been designed to give students and teachers a fresh perspective on multiplicative reasoning. Some can be treated as puzzles and as jumping off points for further mathematical investigation. But the main purpose of the blog is to reveal the diversity of students’ multiplicative thinking, and it is hoped that this will help teachers to adapt their teaching in ways that acknowledge and build on students’ efforts at making sense. It is hoped too that the range of tasks, though in no sense exhaustive, will help us think about the nature and diversity of the field of multiplicative reasoning and what would make a more effective multiplicative reasoning curriculum.
Some of the tasks in this blog are more highly structured than they need to be when used ‘live’ - where one can adjust the activity in the light of students’ responses. I hope, therefore, that when teachers use them in the classroom, they will find ways of making some of them more open. By the same token, some tasks are sequenced in a fairly narrow way. There is a lot to be said for starting at the ‘deep end’ (see the ‘observation’ about task 09B), and while some of the tasks do this, I hope teachers will find other occasions where this could be productive.
The blog follows the same format as my two Algebra blogs, Algebradabra and Algeburble, which have since been published in book form by ATM (The Association of Teachers of Mathematics). It has the same format, of 100 tasks organised into 20 ‘weekly’ sets of 5 ‘daily’ tasks. This format should not be interpreted too literally, however. It is not intended that teachers should use one of these tasks every day or that they should necessarily be used in the given order. On the other hand, many of the tasks could be developed into a whole lesson and many of the weekly sets into a sequence of lessons. It is therefore for the teacher to decide which tasks to use in the classroom, how to adapt them, and how much lesson time to devote to any one of them.
Finally, I would urge the reader to work through a task before reading the commentary on it. This way, the reader will get more from the task, and the commentary will make more sense!
Finally, finally, as you work through some of the tasks, or try them with your students, please send feedback via this blog or through posts on Twitter (@ProfSmudge) or by email (google@mathsmed.co.uk).
Dietmar Küchemann
¹ Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes. Academic Press, New York.
² Küchemann, D. E. (2018). The influence of context and numerical complexity on the tendency to focus on scalar relations when solving missing-value ratio items. A study involving lower secondary school pupils and PGCE mathematics students. In Curtis, F. (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 38 (3).