Collaborating between primary, secondary and higher education: The case of a project on fractions
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Collaborating between primary, secondary and higher education: The case of a project on fractions
- 1. Collaborating between primary, secondary and higher education: The case of a project on fractions Clare Hill, Twynham School, Dorset Christian Bokhove, University of Southampton
- 2. Introduction
- 3. ‘Knowledge of fractions at age 10 will predict algebra knowledge and overall mathematics achievement in high school at age 16’ Early Predictors of High School Mathematics Achievement R Siegler et al. 2012 Related research
- 4. Other aspects involved.. • Role of a fraction – Quotient – As a number – Operator • Discrete and continuous • Useful tool? Bar models?
- 5. For example… The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities Yoonkyung Jeong Catholic, Susan C. Levine & Janellen Huttenlocher
- 6. The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities Yoonkyung Jeong Catholic, Susan C. Levine & Janellen Huttenlocher
- 7. • The model method can be seen as the ‘Pictorial’ part of Singapore’s Concrete- Pictorial-Abstract pedagogy (roots in Bruner’s work) • The model method provides a visual way to help pupils understand and use the four basic operations through meaningful problem situations and language. • Engage these pupils in active thinking using concrete and pictorial representations. What is the bar model method? Professor Kho
- 8. Pictorial representation can lead to numerous strategies for solving. Southampton University MOOC Lee, Yeong, Ng, Venkatraman, Graham & Chee (2010)
- 10. ‘We are not advocating imposing the bar on students as ‘the right way to think about fractions’ Instead we believe that the bar should arise naturally out of good problems that are easily pictured as a linear model’ Using bar representations as a model of connecting concepts for rational numbers. Middleton, van den Heuvel-Panhuizen, Shaw
- 11. The current project • Participating institutions – Three primary schools from the south of England. Two primary school teachers per school, so six in total. – One secondary school from the south of England
- 13. • a capacity to work flexibly and efficiently with an extended range of numbers (for example, larger whole numbers, decimals, common fractions, ratio and percent) • an ability to recognise and solve a range of problems involving multiplication or division including direct and indirect proportion • the means to communicate this effectively in a variety of ways (for example, words, diagrams, symbolic expressions and written algorithms). Multiplicative Reasoning
- 14. Materials • Lesson Study • NCETM materials – Baguette (parts of a whole) – Cheddar cheese – Brie – Ribbon
- 15. a) Draw a picture to show how you would share out the sandwiches in group A b) Write down how much each teacher in group A will get c) Find a different way to share out the sandwiches in group A. (Again draw a picture and write down how much each teacher will get) From: NCETM KS3 Multiplicative Reasoning Project Unit 2 Lesson 2a Baguette task
- 17. What happened? • The primary teachers really deconstructed children’s understanding of fractions (and possibly their own) in a way they had never before considered • Extensive discussions around the importance of moving thinking past halves and quarters • Extensive discussions around the importance of children being able to mathematically justify their answer
- 18. Discuss the possible misconceptions that may have arisen from this task. T
- 23. Cheddar cheese task From: NCETM KS3 Multiplicative Reasoning Project Unit 2 Lesson 2a T
- 24. What happened? • Children felt comfortable splitting rectangular shaped cheese into equal proportions and were able to use their knowledge of times tables to split the cheese into equal parts • ‘When the given information was placed at the end of the bar, children were unwilling to extend into bigger numbers ‘ • Relationship with ‘bar model’?
- 27. This piece of brie weighs 1200 grams. Show where to cut off 900 grams. This piece of brie weighs 900 grams. Show where to cut off 100 grams. From: NCETM KS3 Multiplicative Reasoning Project Unit 2 Lesson 2a Brie task
- 28. Use your pieces of brie to demonstrate potential misconceptions T
- 29. What happened? • Faux pas of cutting the nose of the brie (cultural capital?) • Using a strategy that worked for Cheddar but not for Brie • Even if student does correctly, a new ‘misconception’ can prop up. • Also shows importance of sequencing the materials (here: a counter-example is generated).
- 32. What happened? • The children weren’t taking into consideration the quantity of cheese at either end. They therefore assumed each portion was the same amount, and disregarded the shape . With some prompting, the more able children changed the direction of the cut and applied portions horizontally. • Children found concept of cutting cheese easier than the ribbons.
- 33. Ribbon task From: NCETM KS3 Multiplicative Reasoning Project Unit 2 Lesson 2a
- 34. What happened? • ‘Some children found the concept of extending the bar (length of ribbon) difficult as they saw a definite end point.’ • ‘It was interesting that particularly the most able mathematicians were unwilling to extend beyond 60cm. The bar model seemed to restrict their thinking.’ • Important to choose useful numbers…but sequence towards more difficult ones: variation, scaffolding
- 38. Given information: 80 cm costs 80p 5p 10p 20p 40p 60p 70p 80p 90p £1 5cm 10cm 40cm 70cm 80cm 90cm 1m Student 1 1 1 1 1 1 1 1 1 1 1 1 1 Student 2 1 1 1 1 1 1 Student 3 1 1 1 1 1 1 1 1 1 1 Student 4 1 1 1 1 1 1 1 1 1 1 Student 5 Student 6 1 1 1 1 1 1 1 1 1 1 1 Student 7 1 1 1 1 1 1 1 1 1 1 Student 8 1 1 1 1 1 1 Student 9 1 1 1 1 1 1 1 1 Student 10 1 1 1
- 39. Given information: 40 cm costs 60p 5 10 20 30 40 50 60 65 70 80 100 105 120 7.5 15 30 45 60 75 90 90 97.5 105 80 Student 1 1 1 1 1 1 Student 2 1 1 1 1 1 1 1 1 1 1 1 1 Student 3 1 1 1 1 1 1 1 1 1 1 1 1 Student 4 1 1 1 1 1 1 1 1 1 1 1 1 1 Student 5 1 1 1 1 1 1 1 1 1 1 Student 6 1 1 1 1 1 1 1 Student 7 1 1 1 1 1 1 1 1 1 1 1 1 1 Student 8 1 1 1 1 1 1 Student 9 1 1 1 1 1 1 1 1 1 1 Student 10 1 1 1 1 1 1 1 1 1
- 40. What happened? • The children found it difficult to keep different units of measurement to one particular side of the bar leading to confusion in answers and understanding • ‘We repeated this lesson with the higher of the two sets, we introduced the activity using equivalent fractions as a mental starter. This highlighted to the children the concept of the proportions remaining the same on both the top and bottom of the bar’
- 42. Discuss • What, in your experiences, are the biggest challenges teaching multiplicative reasoning? T
- 43. Experiences from the different perspectives • Gain expertise and experiences; unpick ‘multiplicative reasoning’ in depth. Insight • Researchers can ‘unpick’ underlying literature and ‘translate’ to primary and secondary context, including in practice & methodology. • Primary school teachers learned more approaches to multiplicative reasoning: “what students need to learn, rather than what I need to teach them.” Lower achieving capable of greater challenge.
- 44. The project has opened our eyes to using more open ended problems to understand what the children need to learn rather than my preconceived ideas about what I need to teach them. In the future we would incorporate more opportunity to use concrete resources and real life scenarios in our mathematics teaching to help deepen understanding of concepts. St. Katharine’s Primary School, Bournemouth As primary school teachers, the experience has influenced our teaching and understanding of children’s misconceptions and will encourage us to use a more investigative approach in future. When faced with the challenges (which we would not have previously attempted in these particular sets), the children rose to the occasion and were freer to use and question their own mathematical knowledge. Stourfield Junior School, Bournemouth Some reflections from the primary schools involved
- 45. Working with a ‘knowledgeable other’ Some reflections from the secondary school (me!) Considering qualitative research methods to give insight to project Broadening my horizons
- 46. Conclusions • Substantive (fractions) and – Fractions as part of a whole, versus other areas of fractions. – Discrete versus continuous • Social (collaboration) findings. – Primary, Secondary and Higher Education could work together more – “To each his own”
- 47. References Middleton, James A.; van den Heuvel-Panhuizen, Marja; Shew, Julia A (1998) Using Bar Representations as a Model for Connecting Concepts of Rational Number, Mathematics Teaching in the Middle School, v3 n4 p302-12 Jeong, Y., Levine, S. and Huttenlocher, J. (2007). The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities. Journal of Cognition and Development, 8(2), pp.237-256 Siegler, R., Duncan, G., Davis-Kean, P., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. and Chen, M. (2012). Early Predictors of High School Mathematics Achievement. Psychological Science, 23(7), pp.691-697. doi:10.1177/0956797612440101
Notas del editor
- Our presentation will focus on the collaboration of three primary schools, Twynham School and the University of Southampton to jointly develop multiplicative reasoning skills and deeper understanding of fractions with primary colleagues through the process of lesson study. By combining expertise on teaching fractions from primary, secondary and higher education, we gained insight in the way fractions were taught at primary, but also how such a collaboration could influence teacher practice within the primar y school classroom in terms of both pedagogy and subject knowledge. We will report on both the substantive (fractions) and social (collaboration) findings. 5-10 mins intro and background literature 10-20 mins Baguette example 20-30 mins Cheddar 30-40 mins Brie 40-50 mins Ribbon 50-60
- CHI – background to project / funding from Jurassic hub / link with Rob Tait and Keith Jones – led to link with Christian
- I used the NCETM website as a base for the first face to face session on fractions.
- CB – can you talk to this one?
- We find that all age groups tested( 6-, 8- and 10-year-olds) performed better in the continuous condition than the discrete conditions. Our finding that children performed better on a proportional reasoning task involving continuous amounts than on a parallel task involving discrete quantities may help clarify a distinction made by Inhelder and Piaget (1958), almost a half century ago, when they wrote that children have a qualitative grasp of proportions before they are able to manipulate numerical proportions.
- Within RME, students are not handed ready-made models that embody particular mathematical concepts, but they are confronted with context problems, presented in such a way that they elicit modelling activities, which in their turn lead to the emergence of models During this process of growing understanding of percentage, the bar gradually changes from a concrete context-connected representation to a more abstract representational model that moreover is going to function as an estimation model, and to a model that guides the students in choosing the calculations that have to be made. This means that the model then becomes a calculation model. At the end of the trajectory, when the problems become more complex, it can also be used as a thought model for getting a grip on problem situations. However, the foregoing does not mean that separate stages in the use of the bar model can be distinguished, or that there is a strict order in which these different applications are learned; this is not the case. Indeed, though there is a kind of sequence laid down in the teaching units, the different interpretations of the bar model are accessible in all stages of the learning process. It all depends on how the students see and use the model.
- CH – how we recruited – our feeder primary schools initially
- CH –define multiplicative reasoning
- CH –we defined lesson study
- Why we chose this lesson It comes with a detailed lesson commentary BUT actually developing the teachers confidence to deliver this lesson was at the heart of our collaborative meetings – being able to use and practice the language in advance and being aware of misconceptions that might crop up and planning to address them.
- CH – CJS:Some children split three baguettes into halves and summarised that each person would get a half of the baguette and 1/5. They had to be prompted that this was 1/5 of the half, not the whole which would have been a half plus 1/10. Children were given a calculation for dividing by 5 and asked to come up with a scenario, lots of children found this hard to apply their knowledge of fractions. Some children changed the shapes of the baguettes into pizzas as they associated fractions in circles rather than oblongs Lower prior achieving students found it difficult to move past halves and quarters. St Ks :The task increased their ability to use pictorial representations of fractions (not in circles). Also at a deeper level it increased their understanding of the relationship of numbers within a fraction.
- The lesson allows students to think about what was fair and not fair in order to justify whether or not their drawings – so the language could be moved away from fractions and towards language that they were much more comfortable with in order to seek an initial solution and then back to fractions in order to arrive at a mathematical solution.
- This is the next lesson which moves closer to bar modelling. Not something the teachers were familiar with when we first ran this project. One primary school introduced dual scales –money and weight – this was additional to the task but led into the following task.
- My bold – we have been examining the preciseness of language in our maths lessons and now things like this strike me. Take away point: ‘future’ misconceptions will arise, for doing at top same as at bottom. This accentuates how primary and secondary curriculum content are linked.
- Discussion around this one
- Teachers noticed that they could move past just using 100g splits and recognised the connection with 50 g and 250 g
- The teachers made the second comment in heir report but didn’t elaborate They also commented that some children ‘squared’ the cheese but we have no evidence of this.
- Refer back to slide 20!
- Give them time to discuss the sequencing of this set of examples.
- We examined the number of students which tried to find a value and those that got it right. Needless to say the most successful version was 80 cm costs 80p
- We examined the number of students which tried to find a value and those that got it right. Needless to say the most successful version was 80 cm costs 80p
- Discussion here around sequencing – Discussion about direct proportion
- Highlight anxiety of teachers in not using lesson time to teach procedures – predominantly year 6.