Developing Rich Tasks in Pattern and Structure in the Early Years:

WorkshopAssociate Professor Joanne Mulligan

Faculty of Human SciencesMacquarie University, Sydney, Australia

joanne.mulligan@mq.edu.au

Mathematical Association of Tasmania Conference Expanding Your Horizons 10-12 May Penguin High School

Overview

• 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 structureTasks 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 ResponsesHow 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 =20S4: 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

1, 2. Patterning: simple & complex repetitions

3. Subitising

4. Counting by twos and threes

5. Recognising ten as a unit

6. Fractions: halves and thirds

7. Visual memory: 3x4 grid

8. Volume: units / spatial

9. Combinatorial: multiplication

10. Quotition / division

11. Analogical reasoning

12. Transformation

13. Visual memory: dot pattern

14. Functional thinking

15. Time: clockface

16. Area: unitising

17. Volume: unit comparison

17 items (6 verbal, 6 modeled and 5 drawn responses):

PASA Extension items

7 items (2 verbal, 3 modeled and 2 drawn responses):

1. Composite unit2. Tens frames3. Hundreds charts4. Pattern of squares5. Commutativity6. Equivalence7. 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 No response / Guess.Emergent Attempts to count all the squares in the net.Partial-structural Unsuccessful attempt to visualise the folding.Structural Four squares in the middle, with no

explanation.Advanced 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?

Pre-structural No response or two or more cuts.Emergent Points to one place near end of the strip.Partial-structural Points to one place towards the middle but a

poor estimate. No effort is made to consider equal sections.

Structural Points to a place near the middle of the strip. Makes two approximately equal sections.

Advanced Folds the strip in order to locate the middle.

Levels of structural responses (halving)

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.

Emergent Partial

Structural Advanced

Levels of structural responses

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 Makes errors on both the number of ears on 4 dogs and the number of legs on 3 dogs.

Emergent Correctly predicts number of ears on 4 dogs, but not the number of legs on 3 dogs.

Partial-structural Completes both parts correctly, using addition for either part.

Structural Completes both parts correctly by counting in 2’s and 4’s.

Advanced Completes both parts correctly using a multiplication fact in either part.

Structural Functional Thinking

“1 dog – 4 legs2 dogs – 8 legs3 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.

Pre-structural Emergent

Partial structural Advanced

Levels of structural responses

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. Measurement

Ext PASA Q7. Levels of structural responses

Pre-structural

Emergent

Partial

Structural Advanced

References

• 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.