Unit 3 Emotional Intelligence and Spiritual Intelligence.pdf
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
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)
9.
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
12.
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
16.
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
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’?
25.
26.
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
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).
30.
31.
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.
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
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
Notes de l'éditeur
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.