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### Percentage increase and decrease inquiry

The prompt was devised by Amanda Kirby (a secondary school mathematics teacher). It has been inspired by bar modelling, which has become more common in UK classrooms since the growth of interest in Singapore. As a tool, the bar model can be used to teach a variety of mathematical concepts from ratio to solving algebraic equations. However, Amanda does not take the conventional classroom approach of introducing a problem and then the tool with which to solve it. Instead, she has made the tool itself the starting point. In this way, students are expected to construct a conceptual understanding of the bar model, rather than simply accept it as part of a procedure that helps to get the correct answer. The inclusion of '100' invites students to link the bar models to percentages and multipliers used in percentage increase and decrease.

Singapore framework
The Singapore Mathematics Framework incorporates five inter-related components: concepts, skills, processes, attitudes and metacognition. The Ministry of Education explains that the components underpin problem-solving, which, it contends, is central to mathematics learning. Basing their definition on Polya's work in, for example, How to Solve It, the ministry defines problem solving as involving "the acquisition and application of mathematics concepts and skills in a wide range of situations, including non-routine, open-ended and real-world problems."

The components from Singapore are similar to the five "essential aspects for developing young mathematicians" mentioned in NRICH's post on mastery: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. At the risk of over-emphasising the similarities, it seems possible to match up the components and aspects (see table).
 Singapore components NRICH essential aspects concepts conceptual understanding skills procedural fluency processes adaptive reasoning attitudes productive disposition metacognition strategic competence
Inquiry maths seeks to promote these characteristics of mathematical thinking. In particular, 'adaptive reasoning' and 'strategic competence' are developed through, respectively, requiring students to bring their current knowledge to bear on understanding a prompt and encouraging them to take a role in directing the inquiry.

Resources
Prompt sheet