Percentage change inquiry
Mathematical inquiry processes: Interpret; apply in different contexts. Conceptual field of inquiry: Percentage of a number; percentage increase and decrease; reverse percentage to find an initial amount.
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 the teaching of mathematics 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.
Amanda Kirby devised the prompt for a year 8 (grade 7) class that was new to inquiry. The students' initial questions and comments (above) link the bars to percentages, decimals and fractions, as well as trying to understand their meaning in relation to financial terms. Amanda then structured the inquiry by requiring students to consider the link between statements and equations using multipliers, such as:
'increase 400 by 20%' and '400 x 1.2'
'decrease 400 by 5%' and '400 x 0.95'
Students explained the links by using bar models similar to those in the prompt. Amanda also asked the students to use a calculator when necessary, which led to supplementary questions about the percentage button on the calculator. While Amanda chose not to use the regulatory cards to direct the inquiry at a whole-class level, she left the pack of six on the desks to prompt the students' thinking during the inquiry.
Amanda evaluated the impact of the inquiry on the class: "The obstacles with this class are they have low self-esteem on their own ability in mathematics, which means they aren't willing to make mistakes or try things out and generally want 'spoon feeding' by being told how you do something. I wanted them to take some ownership of the inquiry and lead it to what they needed to know. I was actually surprised they came up with the equivalent fractions and decimals at the start. The idea of 120% often leads to arguments that you can't have more than 100% but because we talked about this in real life they were able to come up with justifications as to what 120% meant."
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).
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.