Transcript MAT 2013 Workshop Mulligan 11-5-13

Developing Rich Tasks in Pattern and
Structure in the Early Years:
Workshop
Associate Professor Joanne Mulligan
Faculty of Human Sciences
Macquarie University, Sydney, Australia
[email protected]
Mathematical Association of Tasmania Conference
Expanding Your Horizons 10-12 May Penguin High School
Overview
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Promoting a pattern and structure approach: PASMAP
Pattern-eliciting tasks
Pattern and Structure Assessment (PASA) Interview
Analysing structural development
What is a Rich Task?
• Address a range of learning outcomes, skills, strategies
• Encourage deep thinking, reflecting and creativity
• Allow for demonstration of a range of concepts, skills and
strategies
• Allow students to show connections between concepts
• Allow for differentiation; all students can engage with the task
• Need assessment strategies (rubric)
The Pattern and Structure Mathematics
Awareness Program Approach
• Highlight and model pattern and structure
• Draw attention to mathematical features – “sameness” and
“difference”
• Explicit focus on one aspect of structure at a time
• Make connections between components of pattern and structure
• Visual memory activities
• Measurement and spatial structuring as a basis for number
concepts
• Explain and justify thinking
• Translate and generalise pattern and structure
Tasks gradually become more complex and link to other concepts
Border Patterns
• Provide 6 x 6 template (or other sizes) and two/three different
sets of coloured multilink cubes.
• Predict how many cubes you will need of each colour to
complete a pattern.
• How many pattern “chunks” can you make?
• If the borders have a different number of squares can you always
complete a pattern? Why? Why not?
• Extend the border task to include other units of repeat and
different sizes and shapes
Using gift bags to investigate pattern and
structure
• Collect a range of small gift bags with attached identical small gift
tags
• Examine the gift bag and the identical smaller card attached
• What is the same? What is different? What patterns do you see
• How many cards fit into the large picture/pattern on the gift bag?
• What do you think we could do to make the small card by shrinking
the large picture on the bag?
• How much smaller is the small picture than the large picture?
• How do you know?
Staircase Patterns
• Staircase pattern produced by the sum of the first two
consecutive odd numbers e.g
• 1+3=4
• 1+3+5=9
• 1 + 3 + 5 + 7 = 16
• Is there anything special about these numbers?
• What shape/s can you make to show this pattern?
• How is the pattern of L shapes related to the structure
of squared numbers?
Re-arrange pattern as
L shape of odd numbers
• Make a pattern of L shapes 1, 3, 5, 7, 9 then rearrange as
an array.
• What is the pattern for the sum of consecutive odd
numbers ?
• Can the student see the relationship between the number
pattern and building the array?
Make a Photo Frame
Make a square photo frame with a border of
small square tiles
Each small square of the border measures 2cm
in length.
The length of one side of the inside frame
(where the photo fits) is 8cm.
Use the large cardboard square and small
squares (as tiles). (Rulers are not available for
this task).
How many tiles do you think you will need?
How did you work this out?
Children’s Responses
How many tiles do you need?
Students’ Strategies:
S1: Four corners and four tiles on each side (4
+ 4 x 4= 20)
S2: Six on one side so its 4x6=24 (incorrect use
of multiplication)
S3: You don’t count the corners twice so its
5+5+5+5 =20
S4: You go 4 sides x 6 take away the corners.”
It doesn’t matter how big the frame is you
always take away 4 from the number of tiles
when you multiply side x side (emergent
generalising)
S5: (side x 4sides take away 4 corners) s x 4–4
or 4(s–1)
Key Questions
• What are the key concepts and processes that you are
assessing?
• How do children use pattern and structure?
• How could you extend the task?
• “If you made the the square tiles larger how many different sized
frames could you design?”
• “If you only knew the area of the frame how could you work out
the number of tiles that you would need for the border?”
PASA Interview Assessment Items
17 items (6 verbal, 6 modeled and 5 drawn responses):
1, 2. Patterning: simple & complex
repetitions
10. Quotition / division
3. Subitising
12. Transformation
4. Counting by twos and threes
13. Visual memory: dot pattern
5. Recognising ten as a unit
14. Functional thinking
6. Fractions: halves and thirds
15. Time: clockface
7. Visual memory: 3x4 grid
16. Area: unitising
8. Volume: units / spatial
17. Volume: unit comparison
9. Combinatorial: multiplication
11. Analogical reasoning
PASA Extension items
7 items (2 verbal, 3 modeled and 2 drawn responses):
1.
2.
3.
4.
5.
6.
7.
Composite unit
Tens frames
Hundreds charts
Pattern of squares
Commutativity
Equivalence
Measurement
Pattern of Squares
What comes next in the pattern so the pattern gets bigger?
Volume: Units/ visualisation 2D-3D
Show one cube and the net of an open box.
Look how this cube fits into this shape.
Place one cube in one of the central four squares.
Imagine this shape folded up to make a box.
How many cubes like this one (point to single cube) would
you need to fill the box to the top?
Levels of structural responses
Pre-structural
Emergent
Partial-structural
Structural
Advanced
No response / Guess.
Attempts to count all the squares in the net.
Unsuccessful attempt to visualise the folding.
Four squares in the middle, with no
explanation.
Explanation of the four central squares
creating the base of the box.
Stages of Structural Development
• Prestructural: representations lack evidence of relevant
numerical or spatial structure
• Emergent: representations show some relevant elements, but
their numerical or spatial structure is not represented
• Partial structural: representations show most relevant aspects
but representation is incomplete
• Structural: representations correctly integrate numerical and
spatial structural features
• Advanced: children show they recognise the generality of the
underlying structure
Categorising Responses for Stages of
Structural Development
• What are the structural features of the mathematical concept
e.g. equal grouping?
• What are the structural features of the mathematical representation
e.g. number line?
• What are the spatial structural features e.g. square or cube; array
or grid?
• Are there any patterns e.g. repetition, growing pattern, functional
relationship; numerical or spatial?
• Is there a relationship between the structural features and the
pattern?
• What evidence is there that the child has integrated or connected
pattern and structure?
Length: Halves and thirds
Show the student a strip of paper tape approximately 30cm long.
I want to cut this tape into 2 pieces exactly the same size.
Can you show me where to cut it?
Suppose I want to cut it into 3 pieces the same size.
Where should I cut it now?
Levels of structural responses (halving)
Pre-structural
Emergent
Partial-structural
Structural
Advanced
No response or two or more cuts.
Points to one place near end of the strip.
Points to one place towards the middle but a
poor estimate. No effort is made to consider
equal sections.
Points to a place near the middle of the strip.
Makes two approximately equal sections.
Folds the strip in order to locate the middle.
Visual Memory: Triangular Dot Pattern
I’m going to show you a pattern of dots but only for a short time.
The I want you to draw it. Are you ready?
Uncover the dot pattern for 2 seconds, then cover it again.
Now draw exactly what you saw.
Allow second attempt.
Levels of structural responses
Emergent
Partial
Structural
Advanced
Functional thinking
Look at these dogs.
Cover the picture with the sheet of card, but leave the first dog uncovered.
How many ears are there altogether on:
1 dog?
After the child has answered, uncover the second dog.
2 dogs? After the child has answered, uncover the third dog.
3 dogs? Do not uncover the fourth dog.
4 dogs?
Repeat for legs up to 3 dogs.
Levels of structural responses
Pre-structural
Emergent
Partial-structural
Structural
Advanced
Makes errors on both the number of ears on 4
dogs and the number of legs on 3 dogs.
Correctly predicts number of ears on 4 dogs,
but not the number of legs on 3 dogs.
Completes both parts correctly, using addition
for either part.
Completes both parts correctly by counting in
2’s and 4’s.
Completes both parts correctly using a
multiplication fact in either part.
Structural Functional Thinking
“1 dog – 4 legs
2 dogs – 8 legs
3 dogs – 12 legs
… You put 4 with each dog
… it’s four for every dog
you have no matter how
many”
Area: Unitising
Provide diagram on sheet, pencil, eraser.
Someone has started to draw some squares to cover this
shape. Finish drawing the squares here.
Point to the space.
Levels of structural responses
Pre-structural
Emergent
Partial structural
Advanced
Ext PASA Q7. Measurement
Provide a piece of paper with the outline of a ruler.
Imagine this is a ruler to measure with.
Draw things on the ruler so you can measure with it.
Ask.
Explain what you’ve drawn on the ruler.
Show me how you would measure your pencil with your ruler.
Ext PASA Q7. Levels of structural responses
Pre-structural
Emergent
Partial
Structural
Advanced
References
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Bobis, J., Mulligan, J. T., Lowrie, T. (2013). Mathematics for Children: Challenging
Children to Think Mathematically (4e). Sydney: Pearson Education Australia.
Mulligan, J.T., & Mitchelmore, M.C. (in press). The Pattern and Structure
Assessment-Early Mathematics (PASA-P;PASA-1,PASA1-X; PASA-2); Teacher
guide. ACER: Camberwell, Victoria.
Mulligan J. T., Mitchelmore, M. C., Kemp, C., Marston, J. & Highfield, K. (2008).
Encouraging mathematical thinking through ‘Pattern and Structure’: An intervention
in the first year of schooling. Australian Primary Mathematics Classroom, 13 (3),
10-15.
National Council of Teachers of Mathematics: Prekindergarten – Grade 2; Grades 3
-5, Principles and Standards for School Mathematics Navigations Series. Reston,
VA: National Council of Teachers of Mathematics
NSW Department of Education & Training (2005). Talking about patterns and
algebra. Ryde: NSW DET Curriculum Directorate
Papic, M. (2005). Turn and learn pattern activities. Knowledge Builder.