CHERI Conference 2005: Learning about Learning Difficulties Understanding Young Children's Difficulties in Mathematics Learning Joanne Mulligan Macquarie University, Sydney Sample ‘Pattern and Structure’ Assessment (PASA) Tasks Number Counting: multiples Count aloud in ones 1, 2, Count aloud by twos 2, 4 … Count aloud by twos. Put a tick on the numbers (Use numeral tracks to 20, 30 etc) Repeat process backwards. Use pattern 3s. Space Unitising Someone has started to draw in some squares to cover this shape. Finish drawing the squares exactly the same size here Measurement Clockface Draw a clockface with everything you remember about a clockface drawn on it. Record various times e g 3 o’clock on it. Empty Number Track Use wide strip of blank paper. Imagine this is a number track. If this end is zero and this end is 10 show me where 5 would be with the clip. Child allowed to adjust clip. .Record numbers 1­10 on the track. Does the child ‘benchmark’ 5 in the centre and use equal spaced units? Reformulation of Fair Share There are 6 counters and 2 teddies. How many counters will each teddy get if we share fairly? Another teddy comes along. So there are now 3 teddies and 6 counters. How many counters will each teddy get? Partitioning This is a picture of one block of chocolate. Tell me how you would share all the chocolate fairly among 4 children so they get the same Border Pattern Complete the square border pattern of 20 tiles using two different colours. Use one colour for the corners. How many tiles do you need to complete the edges (4 on each side of the square)? Does the child count by ones each side? Identify corners? Recount corners? Unitising/partitioning How many of these small triangles will fit exactly inside this shape with no gaps or overlaps (Show large triangle) Make a Ruler Imagine that you have to make a ruler for your friend so they can measure the length of small thing like your book. What would you draw on the ruler so they could measure? Draw as many things on the ruler as you need. Square Units How many of these small squares (two) will fit exactly inside this shape with no gaps or overlaps? (Show large rectangle) Mass/ Volume Show three balls and ask the student to order them from lightest to heaviest. Ask for verification. Can a small ball be heavier than a large ball?. Explain. Array: Multiplication I’m going to show you this card very quickly. How many dots are there? Open card, cover one row with hand and place in front of student. How many dots are under my hand? 2D/3D Visualisation/ Volume Imagine this shape folded up to make a box without a lid. How many cubes would fit in this box (without any spaces left)? Length/ Fractions Show me a half of this whole ribbon length. If we cut the ribbon how many pieces do we have? If we cut the ribbon into three pieces how many cuts do we have? Four pieces?
Length Units Place the sticks along the ribbon to show how long it is. How many sticks did you use? Does it matter where you start and finish? Does it matter if the sticks are different sizes? Mulligan, J. T. (2005).Understanding young children’s difficulties in mathematics learning. Paper presented to CHERI Conference Learning about learning difficulties. September 1­2, Westmead Hospital, Sydney. Sample ‘Pattern and Structure’ Program (PASP) Tasks Triangular Pattern Tasks: Reproduce patterns in triangular formation. Alternatively, pattern of squares, rectangles, simple grids of varying sizes. v Child explains their initial inaccurate image (Intuitive justification) v Teacher shows pattern produced correctly by another student with coloured discs (Modelling) v How can we make your pattern the same as this one? Tell me why we are making it the same? (Focus on ‘sameness’) v Child’s attention is drawn to shape, size colour,equal sized spaces. (Focus on spatial or numerical structure) v Screen each row or side of triangle successively then child reproduces with counters. (Successive screening) v Child justifies that the pattern is the same (Justification) v Child must reproduce from memory (Visual memory) v Task repeated regularly (Repetition Staircase Patterns v Make a staircase pattern with unifix or multilink cubes such as one, two, three, four five, four three two one. Copy and continue a staircase pattern with cubes going up and back down. Predict next step in the staircase. v Describe the pattern using numerals. What pattern can you see of you look at the pattern of cubes in horizontal layers: one, three, five, seven, nine as a triangle? v Place and record staircases on a grid sheet preferably one that fits the size of the cubes. Repeat the process and record the patterns from memory. Border Patterns v Use coloured tiles or unifix or multilink cubes to build linear pattern borders for example in two colours in different sized rectangular frames. v Record the pattern using symbols of numbers. Complete a partially completed pattern. v Describe the pattern using ordinals eg “Every third block is blue so I have x blue blocks in my pattern”. v Join the cubes at each corner to make a tall tower or horizontal strip of cubes. Can the student see that the pattern remains the same? v Repeat the process and challenge students to record from memory. Hundreds’ Charts v Identify a number on a hundred’s chart. Screen. Reproduce the number in a blank frame. v Complete a partially completed or empty hundreds’ chart frame. v Draw a hundred’s chart from memory. Colour the various patterns of adding ten in a vertical sequence. v An example of integrating number and spatial patterns is using a basic cardboard hundreds chart where it is cut up into halves 50/50 or quarters 25/25/25/25 which make a pattern of 25s showing four rows or segments. The segments can be turned over on the blank side to show the equal sized parts of a puzzle. Similarly the activity can focus on other number patterns and spatial arrangement such as 5 lots of 20. References Clarke, D. An issue in teaching and learning subtraction: what’s the difference. Australian Primary Mathematics Classroom, 8(3), 4­12. Gervasoni, A. (1999). Using visual images to support young children’s number learning. Australian Primary Mathematics classroom, 4 (2), 23­ 28. Karp, K. & Howell, P. (2004). Building responsibility for learning in students with special needs. Teaching Children Mathematics, 11(3), 118­ 126. Mulligan, J.T., Prescott, A. & Mitchelmore, M. C., (2003). Taking a closer look at young children’s visual imagery. Australian Primary Mathematics Classroom, 8(4), 23­27. Mulligan, J.T., Prescott, A. & Mitchelmore, M. C., & Outhred, L. Taking a closer look at young children’s images of area measurement. Australian Primary Mathematics Classroom, 10(2), 4­8. NSW Department of Education and Training. (2002). Teaching measurement Early Stage 1/ Stage 1. Sydney: NSW Department of Education & Training Curriculum Directorate. NSW Department of Education and Training. (2003). Teaching measurement Stage 2/ Stage 3. Sydney: NSW Department of Education & Training Curriculum Directorate. NSW Department of Education and Training (1998). Developing efficient numeracy Strategies Stage 1. Sydney: NSW Department of Education and Training Curriculum Directorate NSW Department of Education and Training (2003). Developing efficient numeracy Strategies Stage 2. Sydney: NSW Department of Education and Training Curriculum Directorate NSW Department of Education and Training (2003). Fractions: Pikelets & Lamingtons. Sydney: NSW Department of Education and Training Curriculum Directorate NSW Department of Education and Training. (2003). Count Me In Too: A professional development package. Sydney: NSW Department of Education and Training Curriculum Directorate Wright, R.,Martland, J. & Stafford, A. (2000). Early Numeracy: Assessment for teaching and intervention. London: Paul Chapman Publishing. Wright. R. J., Martland, J.M., Stafford, A. K. & Stanger, G. (2002). Teaching Number: Advancing children’s skills and strategies. London: Paul Chapman Publishing http://www.aamt.edu.au/home.html http://www.lego.com http://www.curriculumsupport.nsw.edu.au/maths/countmein/