Maths to share - CPD for your school
Mathematics Subject Knowledge - fractions, decimals and percentages
The concepts of fractions, decimals and percentages are traditionally considered hard for children and adults to understand. Analysis of Key Stage 2 tests acknowledges that questions relating to fractions, decimals and percentages pose particular problems for children, especially those working at level 3. It is well known that children and adults often do not appreciate that fractions, decimal and percentages are equivalent ways of writing the same quantity and that they are different ways of expressing related ideas.
At the meeting
Questions from the paper to consider for discussion:
- were the children’s misconceptions highlighted in the paper familiar
- what explanations could be offered for the misconceptions?
- were you surprised children were more successful in division problems? Is this the case in your class?
- how do you use children’s experiences to teach the different aspects of fraction?
- do you include opportunities, sharing as situations, in which meaning of fraction can be explored by pupils?
Ensure that teachers are aware that this is an opportunity to develop their subject knowledge and address misconceptions – no judgements will be made.
10-minute activity – you may need to provide some mathematical resources, eg. multilink, number lines, etc. to facilitate thinking and discussion.
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Ideally, the teachers should work in groups of three or four. Give each group a large sheet of paper with a different fraction written on it, e.g. 1/3, 4/5, 3/8, 5/6, 1/12, (not 3/4). Ask the groups to write or draw as many different aspects of the fraction as they can. Allow 10 minutes to make a spider diagram. Model this example for 3/4 to get them started:
After ten minutes, ask one person from each group to share some of their different representations. This is a good opportunity to discuss and address any misconceptions. You may also want to discuss whether this would be a good activity to use in class.
Here is an opportunity to explain and illustrate the different meanings of fraction notation as both names of numbers and as operators:
This is a good opportunity to encourage colleagues to self-evaluate their subject knowledge further in this area using the NCETM Self-evaluation Tool.
- part of a whole unit
- comparisons between part of a set and the whole set
- a point between two whole numbers
- result of a division operation
- comparing the sizes of two
- sets of objects
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Misconceptions in fractions, decimals, and percentages
This area of mathematics is fraught with misconceptions. The next activity (approximately 20 minutes) will provide colleagues with a chance to identify and rectify common misconceptions in fractions, decimals and percentages.
Pre-prepare the questions from Checking your Subject Knowledge: Fractions Decimals and Percentages by cutting them up individually and placing them in an envelope.
Ask colleagues to work in pairs. One person should take one misconception from the envelope. They must then identify what the misconception is and then teach it correctly to their partner. They should identify any vocabulary and consider what visual images would best support the explanation.
Here are some of the misconceptions and areas that have been addressed:
- fractions are always parts of one, never bigger than one
- inaccurate recognition of the ‘whole’
- fractions are parts of shapes and not numbers in their own right
- a fraction such as 3/4 is always seen as 3 lots of 1/4 without recognition that it can also be a 1/4 of 3
- I want the biggest half! Inaccurate division into equal parts
- 1/2 is smaller than 1/3
- common misconceptions when performing fractional operations e.g. 2/4 +1/4 =3/8.
- decimals with more digits are larger - ordering 0.25 is bigger than 0.3 (remember significant digits!)
- 0 as a place holder 3/100 is 0.03 not 0.3
- reading 0.11 as nought point eleven
- misaligning digits when calculating using vertical columns
- moving the decimal point when multiplying by 10.
- percentages can never be bigger than 100%
- not understanding that percent means out of 100.
Other examples can be found in two excellent texts:
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- Children’s Errors in Mathematics. Alice Hansen (Ed). Learning Matters (2008)
- Children’s Mathematics 4-15. Julie Ryan and Julian Williams. McGraw Hill (2007)
Moving towards greater understanding of fractions, decimals and percentages
Remind your colleagues that children need physical objects and pictures and social activity to 'hook into' on their journey towards a better understanding of mathematical concepts and ideas. A range of models, forms and representations are paramount to good teaching of any subject, not least to support different learning styles. The movement between concrete, visual and symbolic representations of a problem is often quite complex, and the connections between the different forms and the implications of using different representations need to be explored. You might find it helpful to share the excellent work on models and representations from the NCETM primary CPD module with the teachers. There is also a comprehensive list of mental images for fractions, decimals, percentages, ratio and proportion (FDPRP) on the ATM website.
This final section provides a link to further CPD that will enable your colleagues to reflect on their understanding of models and representations and the use of them within their teaching.
In small groups, solve this problem adapted from a problem in Cohen, S. (2004), Teachers' professional development and the elementary mathematics classroom: Bringing understandings to light, Mahwah, New Jersey, Lawrence Erlbaum Associates.
Sheila works in a café where baked potatoes are a speciality, the most popular filling for which is cottage cheese. Each potato uses three fifths of a tub of cottage cheese. Sheila has four tubs of cottage cheese in the fridge. How many portions can she serve?
Allow colleagues time to solve the problem.
Ask colleagues to look at Session 2 of the Primary Module ‘Models and Representations’ on the NCETM portal.
Ask them to consider how their models compared with those on the video clips. Ask them to bring with them their thoughts about use of models and images for a five-minute feedback session at the next meeting.
Effective mathematics teaching
What makes some teachers of mathematics more effective than others?