Maths in the Staff Room – Short Professional Development Meetings
This section provides suggestions and resources for a professional development meeting for teachers that can be led by the maths subject leader or another person with responsibility for developing mathematics teaching and learning in the school. You can find previous features in this series here.
- Understand the importance of using visual and concrete representations to develop conceptual understanding of mathematical ideas
- The importance of using representations to help primary pupils give meaning to numerical concepts Tony Harries, David Bolden, Patrick Barmby, Durham University (UK)
- National Curriculum Programme of Study
- 10s frames, Numicon (if available in the school), coins, straw bundles, Dienes (place, value), place value counters, Cuisenaire/coloured rods staircase, number lines, 100 squares, dice, dominoes, etc
1. Setting the scene: Why should we use representations?
Prior to the staff meeting ask the teachers to read this short article by Harries et al, which explores some reasons for using representations, supported by research evidence.
Share the aim of the professional development meeting. Emphasise that there are many occurences of representing number in the new programmes of study. Ask colleagues to scan through the programme of study to identify where these occurrences are, including drawing attention to this statement from the Purpose of Study:
Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. (p3)
- identify and represent numbers using objects and pictorial representations including the number line
- identify, represent and estimate numbers using different representations, including the number line
- identify, represent and estimate numbers using different representations
- identify, represent and estimate numbers using different representations;
- read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of zero and place value.
- read Roman numerals to 1000 (M) and recognise years written in Roman numerals.
Ask the teachers to draw or make visual and concrete representations for 5. How many different representations of 5 have been found?
e.g. 1 hand, coin, clock-face, 5 counters, 5 Numicon shape, 5 interconnected cubes, the numeral, dice arrangement of 5 etc.
Discuss which representations are currently used most in their classes.
2. Understanding the mathematical properties of different representations
This session draws from the work of John Mason (2005, cited in the Harries et al paper that was read prior to the meeting).
Mason suggests interrogating a representation in the following ways:
- gazing (looking at the whole)
- discerning details,
- recognising relationships,
- perceiving properties,
- reasoning on the basis of the properties
These might be seen as hierarchical because it is unlikely that pupils are unable to reason about a property of a representation unless they have had time to identify that property in the given representation through gazing/attending to the representation.
Allow teachers time to compare the range of representations that were identified, by considering the following set of questions.
What mathematical ideas do these representations help pupils to attend to? This might be any of the following:
- cardinal or ordinal property of 5
- 5’s odd-ness
- part/part/whole relationships (i.e. parts that combine to make 5)
- place value (i.e. 5 tens in straw bundles/Dienes/place value counters).
Then focus on these questions to interrogate the representations:
- What is the same about the different representations?
- What is different about the representations?
- What are the particular characteristics of the various representations?
- How might we move from one representation to another?
- What are the most useful characteristics of a particular representation?
- When would you use certain representations/ when wouldn’t you? Why?
3. Conclusion and Reflection
Consider which representations will form the core representations across the school to ensure consistent exposure by pupils of these and when they should be used to ensure that the representations are meaningful in drawing out the properties you wish pupils to attend to.
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