Mathematics and Dyscalculia Professor Amanda Kirby
Amanda Kirby copyright Dyscovery Centre 2011
You had a tax bill to pay on January 29th for ÂŁ3500 which you failed to pay. For every day after that you have to pay 5% of this amount on top. It is now February 11th.
Estimate how much you need to pay all together. Amanda Kirby copyright Dyscovery Centre 2011
Which contains more milk?
Amanda Kirby copyright Dyscovery Centre 2011
PDDNOS
DCD
Specific language impairment Conduct disorder
ODD
Pragmatic language impairment Amanda Kirby copyright Dyscovery Centre 2011
We all have a bit of “it”
Amanda Kirby copyright Dyscovery Centre 2011
How do children learn mathematics?
Amanda Kirby copyright Dyscovery Centre 2011
Learning about maths in the real world • • • •
Shape concepts Number correspondence Ordinality Money
Amanda Kirby copyright Dyscovery Centre 2011
Mathematics • Abstract subject • Need for symbolic understanding • Estimation
Amanda Kirby copyright Dyscovery Centre 2011
Are there more difficulties today?
Amanda Kirby copyright Dyscovery Centre 2011
• Less opportunity for early multisensory experiences • More emphasis on mental mathematics • Alternatives to doing tasks- parents may find it harder to support Amanda Kirby copyright Dyscovery Centre 2011
Where do you use maths in your life?
Amanda Kirby copyright Dyscovery Centre 2011
Difficulties extend to other areas: • Poor management of money • Errors doing simple calculations • Difficulty understanding a series of commands- 2 to the left/1 to the right • Difficulty keeping score in a game • Harder to play strategic games like chess • Estimation – cooking, measuring Amanda Kirby copyright Dyscovery Centre 2011
What is dyscalculia?
Amanda Kirby copyright Dyscovery Centre 2011
Dyscalculia • Specific difficulties in numeracy skills • 6% • …but overlaps with Dyslexia and DCD
• Is this children who are working at Level 1 at age 7, or Level 3 at age 11, have some degree of mathematical difficulty? Amanda Kirby copyright Dyscovery Centre 2011
Characteristics • Same strategies as younger children but are error prone • Slow at calculating and counting • Difficulty retrieving number facts • Do not know their tables • Poor at monitoring their counting • Problems switching between different strategies when completing mathematical problems Amanda Kirby copyright Dyscovery Centre 2011
Difficulties despite: • Adequate • Appropriate Teaching and exposure Compared with age and cognitively matched peers.
Amanda Kirby copyright Dyscovery Centre 2011
Different terms used : 315.1 Mathematics Disorder As measured by a standardized test that is given individually, the patient's mathematical ability is substantially less than you would expect considering age, intelligence and education. This deficiency materially impedes academic achievement or daily living. If there is also a sensory defect, the mathematics deficiency is worse than you would expect with it. DSM1V (APA) Amanda Kirby copyright Dyscovery Centre 2011
Commonest maths problems • • • •
memory for arithmetical facts difficulty include word problem solving representation of place value the ability to solve multi-step
Amanda Kirby copyright Dyscovery Centre 2011
What are you doing when you are looking at this?
Amanda Kirby copyright Dyscovery Centre 2011
Need for : • • • • •
Prior knowledge Estimation Understanding Visual skills Time to do it
Amanda Kirby copyright Dyscovery Centre 2011
Some children have more than one problem as well .. • • • • •
Working memory difficulties Spatial representation Language Motor Executive functioning- planning and checking
Amanda Kirby copyright Dyscovery Centre 2011
What is a persistent problem? • The children with mathematical difficulties use almost exclusively counting- based strategies, while those without such difficulties children were more likely to use retrieval or derived fact strategies
Amanda Kirby copyright Dyscovery Centre 2011
What are core maths concepts?
Amanda Kirby copyright Dyscovery Centre 2011
Core primary mathematical competencies Numerosity Ability to accurately determine the quantity of sets up to 3 or 4 items, or events, without counting Ordinality Implicit understanding of “more than” & “less than” for comparison of sets of 3 to 4 items Counting Nonverbal system for enumerating small sets of items & implicit knowledge of counting principles (1 to 1 correspondence) Amanda Kirby copyright Dyscovery Centre 2011
Simple arithmetic Sensitivity to increases (addition) and decreases (subtraction) in the quantity of small sets of items
Estimation Inexact estimation of relative quantity, magnitude, or size
Geometry manipulating shapes, visualising Geary (2007): Child Dev. 78(4) Amanda Kirby copyright Dyscovery Centre 2011
Which lessons and how will mathematics be involved?
Amanda Kirby copyright Dyscovery Centre 2011
Difficulties can impact on different lessons Science- weights, measures, recording, using tools, estimation, calculations Sport- sequences of instructions, direction, timing, understanding the plan/teams Geography- map reading, recording, orientation, graphs
History- time lines
CDT- measuring, tool usage, planning, language
Amanda Kirby copyright Dyscovery Centre 2011
Doing a mathematics problem John walked down the street to the shops to buy a DVD . He had £20. The decided to buy one for £11.99 and then buy an apple for 50p. How much change will he have. He needs £7.20 for the train home, will he have enough money?
Amanda Kirby copyright Dyscovery Centre 2011
How do you do this? What skills do you need?
Amanda Kirby copyright Dyscovery Centre 2011
Mathematical anxiety? • Many people develop anxiety about mathematics, which can be a distressing problem in itself, and also inhibits further progress in the subject (Fennema, 1989; Hembree, 1990; Ashcraft, Kirk and Hopko, 1998). • This is rare in young children (Wigfield and Meece, 1988) and becomes much more common in adolescence. Amanda Kirby copyright Dyscovery Centre 2011
These all overlap? ADHD
SLI difficulties
EF
reading
Receptive language
Dyslexia
DCD
Visual processing
Auditory processing
spelling
Amanda Kirby copyright Dyscovery Centre 2011
maths
Motor difficulties
writing
Dyscalculia commonly overlaps with: • DCD • Dyslexia • Specific language difficulties
Amanda Kirby copyright Dyscovery Centre 2011
Dyscalculia and….. • Dyslexia-Miles (1993) found that 96% of a sample of 80 nine-to-twelve-year- old dyslexics had were unable to recite the 6x, 7x and 8x tables without stumbling. • Dyspraxia/DCD-spatial and copying (Piek) • Speech and language disorders- verbal concepts- counting-related concepts, such as the fact that the last item in a count sequence indicates the number of items Amanda Kirby copyright in the set. Dyscovery Centre 2011
What are the underlying cognitive skills needed for mathematics?
Amanda Kirby copyright Dyscovery Centre 2011
Vision and perception
Language and comprehe nsion
Sufficient time
Focus and attention
Sequencing
Working memory
Fine motor skills
Gross motor skills
Amanda Kirby copyright Dyscovery Centre 2011
What are the underlying cognitive skills needed for mathematics? • • • • •
Vision- acuity, perception, sequencing Hearing-hearing, filtering, perception Language skills -receptive, inference Motor skills –for rulers/protractors Planning skills - EF; working memory, focus and attention Amanda Kirby copyright Dyscovery Centre 2011
Lets go through each of these components for learning....
Amanda Kirby copyright Dyscovery Centre 2011
Focus and inattention as rated by classroom teachers, is a significant of Gr. 1 & Gr. 3 children‟s in three key areas • Fact fluency • Computation • Story problems
Amanda Kirby copyright Dyscovery Centre 2011
Working Memory is: A limited-capacity cognitive system that allows us to hold and manipulate information “on-line� for a few seconds.
Amanda Kirby copyright Dyscovery Centre 2011
WM required for.. Keeping track of a conversation • Who said what..to whom • who asked what… Keeping track of a game • Whose turn is next… • What cards have • already gone… • What stage is the game Amanda Kirby copyright Dyscovery Centre 2011 • at….
Impact of poor working memory Difficulty with: Mental arithmetic • will tend to use „finger counting‟, • need visual representation Retrieval of math facts in word problems will tend to use • immature strategies („count all‟, „count on from largest”) Arithmetic procedures, such as „carrying‟ &„borrowing‟ Ignoring irrelevant information in word problems Amanda Kirby copyright Dyscovery Centre 2011
Working memory processes are important predictors of performance on national curriculum achievement scores: Verbal WM • Literacy: vocabulary, reading comprehension,written expression Quantitative literacy: math Visual-Spatial WM • Science • Quantitative literacy: math (Gathercole & Pickering, 2000; Jarvis & Gathercole, 2003) Amanda Kirby copyright Dyscovery Centre 2011
Signs of visual/VP difficulties • • • • • • • • • •
Copying from the board Needing to reread text Avoiding reading out in class Moving in a moving environment Poor writing – spacing , not on the lines Words bouncing Spelling errors- longer words Not getting their “abc ” Geometry Not being able to select key words and sense easily and quickly Amanda Kirby copyright Dyscovery Centre 2011
Planning and executive functioning difficulties
Amanda Kirby copyright Dyscovery Centre 2011
EF and Specific Learning Difficulties • ADHD – all ADHD children have EF impairment to varying degrees (Barkley 2001) • ASD – Pennington and Ozonoff (1996) found children performed 1 SD below control group on EF tasks • DCD – children impaired on tests of working memory (Alloway & Temple, 2007) • Dyslexia – studies have found WM deficits that compound their phonological problems (Wolf 2010) • Dyscalculia – Askenazi & Henik (2010) found evidence of specific EFDs in university students with „pure‟ dyscalculia Amanda Kirby copyright Dyscovery Centre 2011
Components – Activation- organising , prioritising tasks, time estimation, initiation.. procrastinate – Focus-sustaining and shifting.. Reading over and over – Effort-regulating alertness..completing tasks, sleep pattern (can’t shut off) Amanda Kirby copyright Dyscovery Centre 2011
Components – Emotion-managing frustrations and modulating emotions..keeping things in perspective – Memory- using working memory and accessing recall.. what has just been said, remembering a sequence – Action- monitoring and regulating self action..impulsive, not considering the context, can’t adjust pace Amanda Kirby copyright Dyscovery Centre 2011
Seeing the process as well as the product • Do a problem on a Tablet PC using software like Camtasia, and then upload the process to a website- see the process in action
Amanda Kirby copyright Dyscovery Centre 2011
Visual perceptual difficulties ... In mathematics • • • • •
Estimation Manipulation Orientation Sequencing Problem solving
Amanda Kirby copyright Dyscovery Centre 2011
e.g. Using a number line • To compare integers, plot the points on the number line, the number farther to the right is the larger number • Compare 1 and -3 ___________ *_______*_______________ -8 8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
Since 1 is to the right of -3, 1 > -3 or Kirby 1, copyright Since -3 is to the Amanda left of -3 < 1 Dyscovery Centre 2011
5
6
7
Time concepts/estimation When do you need to know about time passing?
Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Linguistics
Amanda Kirby copyright Dyscovery Centre 2011
The linguistic difficulties Word problem
Problem type
(Tannock) Arithmetic operation
John had three marbles and then Nina gave him six more. How many does he have?
change
addition
John had some marbles. Then he gave six marbles to Nina. Now John has three marbles. How many marbles did John have in the beginning?
change
addition
John has three marbles. Nina has six marbles. How many marbles do they have altogether?
combine
Addition
John and Nina have nine marbles altogether. John has three marbles. How many marbles does Nina have?
combine
subtraction
John has nine marbles. Nina has six marbles. How many marbles does John have more than Nina?
compare
subtraction
John has nine marbles. He has six more marbles compare than Nina. How many marbles does Nina have? Amanda Kirby copyright
subtraction
Dyscovery Centre 2011
Fine motor difficulties How will this impact on mathematics?
Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Need to reread it several times before starting Not sure of the process- where to start
Too many words Not sure what calculation to do
Donâ&#x20AC;&#x;t understand the words Too little time
Know the calculation but make an error Canâ&#x20AC;&#x;t show workings Misread it Work very untidy
Amanda Kirby copyright Dyscovery Centre 2011
How can you help?
Amanda Kirby copyright Dyscovery Centre 2011
â&#x20AC;&#x153;interventions that focus on the particular components with which an individual child has difficulty are likely to be more effective than those which assume that all children's arithmetical difficulties are similarâ&#x20AC;? (Weaver, 1954; Keogh, Major, Omari, Gandara and Reid, 1980)
.
Amanda Kirby copyright Dyscovery Centre 2011
General
Find specific areas
Practice sufficiently
Amanda Kirby copyright Dyscovery Centre 2011
Adapt
Help needs to be: • in different formats- multisensory • regular ( at least 3 times per week minimum) • reinforced • some people process very slowly.. So rote learning may be counterproductive
Amanda Kirby copyright Dyscovery Centre 2011
â&#x20AC;&#x153;the child's need is just as much to unlearn his incorrect rule as it is to learn the correct rule.â&#x20AC;?
Amanda Kirby copyright Dyscovery Centre 2011
TIE
TASK
Individual
Environment
Amanda Kirby copyright Dyscovery Centre 2011
Keep cognitive load at an optimal level When cognitive load is reduced &/or supported to a level that is within the individualâ&#x20AC;&#x;s capacity, that person will be better able to benefit from instruction
Amanda Kirby copyright Dyscovery Centre 2011
What works…. • Learning concrete operations to give confidence
• Metacognition – Knowing what one does not know – Reflection – Planning Amanda Kirby copyright Dyscovery Centre 2011
First.... create a resource kit • • • • • • •
Dominos Playing Cards Egg cartons Coins and buttons Different shapes Number cards Language of maths cards Amanda Kirby copyright Dyscovery Centre 2011
Rehearsal cards â&#x20AC;˘ mathematical facts (5 x 4 =20, 6X2=12;4x3=12)
â&#x20AC;˘ Each individual child is given a small set of cards to practice each day under adult supervision. Amanda Kirby copyright Dyscovery Centre 2011
Different children need different approaches â&#x20AC;˘ e.g clapping and dancing as the children chant or sing multiplication tables, with the aim of assisting those children who, for example, have better motor than verbal memories.
Amanda Kirby copyright Dyscovery Centre 2011
Different children need different approaches e.g. Learning numbers adding to 12....
Amanda Kirby copyright Dyscovery Centre 2011
Multiple methods to solve problems â&#x20AC;˘ Encourage multiple methods to carry out problems (e.g. to solve arithmetic problems with a number line, a calculator, concrete objects such as blocks, in written form,Numicon etc.)
Amanda Kirby copyright Dyscovery Centre 2011
Spatial Recognize sets of objects in patterned
1 and 2 more
Anchors
1 and 2 less
5 and 10
Part Part Whole
7
7
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1
2
3
4
1
2
3
4
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5
5
6
6
100
10
1
100
10
1
Amanda Kirby copyright Dyscovery Centre 2011
NUMICON Amanda Kirby copyright www.numicon.com Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Counting activities â&#x20AC;˘ These include general practice in counting objects. â&#x20AC;˘ They also include practice in grouping objects, especially into tens.
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HOW MANY?
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Playing with dice
Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
TURNING WORDS INTO PICTURES Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Specific sessions
Amanda Kirby copyright Dyscovery Centre 2011
Create a session • 2-3 minutes practicing counting skills. • 2 minutes revising individual known facts. • 10-12 minutes practicing derived fact strategies building on known facts. • 2 minutes playing with big numbers or working on a problem.
Amanda Kirby copyright Dyscovery Centre 2011
Visual clues
Amanda Kirby copyright Dyscovery Centre 2011
The understanding of maths Joan has 12 sweets. She has twice as many sweets as Leela. How many has Joan got?
Amanda Kirby copyright Dyscovery Centre 2011
The language of mathematics Highlight key words in a maths question Provide a glossary or checklist
Amanda Kirby copyright Dyscovery Centre 2011
Check the language of mathematics +, add, and, plus, in addition, more
-, take away, minus, less, subtract
Amanda Kirby copyright Dyscovery Centre 2011
Key Language Development: Terms being used Player
1st
2nd
More / Less /Same in their language is a benchmark for the teacher that 3rdchanges 4th over Finaltime.
Ms. Brooks Mr. Adams Game Facts: 1. Ms. Brooks won by 15 points. 2. In one quarter they scored the same number of points. 3. In another quarter, Mr. Adams scored 10 more points. 4. In yet another quarter, Mr. Adams was outscored by 5 points. Amanda Kirby copyright Dyscovery Centre 2011
Taller, Shorter Up down
Amanda Kirby copyright Dyscovery Centre 2011
Getting harder Harry Potter has asked his friend Hermione for a potion to turn them and their friend Ron into birds. (The flying car is in for repairs, so they need to make the trip to Diagon Alley as birds.) Diagon Alley is 9 miles away and a dose of Hermione's potion lasts 50 minutes. They only have enough potion for one dose each. If they can go 24 miles an hour as birds, and they start at 4:30 p.m., can they get to Diagon Alley and back to Hogwarts again before the potion runs out at 5:20 p.m.? If so, how much time will they be able to spend in the Alley? Amanda Kirby copyright Dyscovery Centre 2011
Breaking down the problem What exactly is the question you need to answer? • What do you need to know- what can you ignore • Can you draw a picture or rewrite the question in your own words • Is there a formula you need to answer this- have you the skills to do this • Have you checked your results • Have you been given enough time • Have you got the answer write but written it down wrong • Are you anxious…. Amanda Kirby copyright Dyscovery Centre 2011
45 minutes 5 minutes
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Estimation Visual skill/EFskills
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“I am thinking of a number between 25 and 72”
“I‟m thinking of a number between ….”
Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
If difficult in visualising or hearing it â&#x20AC;˘ Need to feel and see it
Amanda Kirby copyright Dyscovery Centre 2011
If difficult in visualising or hearing it â&#x20AC;˘ Need to feel and see it
Amanda Kirby copyright Dyscovery Centre 2011
Visual prompts
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Could you build an abacus?
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Amanda Kirby copyright Dyscovery Centre 2011
http://countdown.luc.edu/ Amanda Kirby copyright Dyscovery Centre 2011
Use poems etc as reminder tools
Find your number. Look right next door. 4 or less just ignore. 5 or more, add 1 more. Amanda Kirby copyright Dyscovery Centre 2011
Give rules e.g. number squares • • • •
1 more is 1 square to its right 10 more is 1 square down 1 less is 1 square to its left 10 less is 1 square up
Amanda Kirby copyright Dyscovery Centre 2011
If planning is an issue • Check the student understands the expected outcome • Providing worked model answers with the processes as examples ( e.g. Camtasia) • Provide steps to complete- a scaffold • Talk through what is required by the student- what skills are needed, how will points be apportioned in a test • Provide a glossary of terms to refer to Amanda Kirby copyright Dyscovery Centre 2011
http://www.readwritethink.org/files/resources/i nteractives/essaymap/
Amanda Kirby copyright Dyscovery Centre 2011
Using colour coding â&#x20AC;˘ The elements of a question â&#x20AC;˘ Signs being used
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The tools of maths
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Scissors
Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Recording maths â&#x20AC;˘ Use set symbols for science â&#x20AC;˘ Larger maths paper with bigger squares
Amanda Kirby copyright Dyscovery Centre 2011
Larger graph paper
http://www.mathsphere.co.uk/resources/MathSphereFreeGraph Paper.htm Amanda Kirby copyright Dyscovery Centre 2011
Amanda Kirby copyright Dyscovery Centre 2011
Offer a choice of pens and pencils
Amanda Kirby copyright Dyscovery Centre 2011
Functional mathematics â&#x20AC;˘ Handling money â&#x20AC;˘ Filling in a form
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How do you make maths real? What could you do with cookery to teach maths concepts? Amanda Kirby copyright Dyscovery Centre 2011
Using ICT and other resources
Amanda Kirby copyright Dyscovery Centre 2011
Maths mania Topics covered include: • Numeracy with number questions on the four rules, decimals, fractions etc • Angles, what they are and how they differ • Telling the time, with time intervals of days, months and years • Measures includes length, volume and mass • Shape and Space Amanda Kirby copyright Dyscovery Centre 2011
“Maths Circus” Topics covered include: • Spatial awareness • 3-dimensional geometry • Addition and multiplication • Investigating direction and angle, compass bearings, time and vectors • Positive and negative numbers • Planning a sequence of moves in advance, and executing them later on
Amanda Kirby copyright Dyscovery Centre 2011
Math talk MathTalk & Scientific Notebook is a bolt on bundle for Dragon NaturallySpeaking â&#x20AC;˘ for creating, editing and typesetting mathematics and scientific equations by voice. â&#x20AC;˘ The program has learning modules for algebra, trigonometry, calculus, statistics and video demos demonstrating how to use features. Amanda Kirby copyright Dyscovery Centre 2011
Playing Bingo Bingo card maker : http://www.teach-nology.com/web_tools/materials/bingo/
Amanda Kirby copyright Dyscovery Centre 2011
Resources â&#x20AC;˘ http://www.teachingideas.co.uk/maths/cont ents02problems.htm ICT in Maths;Alison Clark-Jeavons :ISBN: 1 85539 191 0,Exciting Precision maths (http://www.johnandgwyn.co.uk/home.html) Useful links to other sites: http://www.mad4maths.com/parents/links/ Amanda Kirby copyright Dyscovery Centre 2011
â&#x20AC;˘ http://www.woodlandsjunior.kent.sch.uk/maths/index.html
Amanda Kirby copyright Dyscovery Centre 2011
Ten ticks http://www.10ticks.co.uk/s_help.aspx http://www.math.com/students/tools.html
Amanda Kirby copyright Dyscovery Centre 2011
CDROM Steve Chinn What do you do when you canâ&#x20AC;&#x;t learn times tables
Mental Maths Olympics Year 4. Amanda Kirby copyright Dyscovery Centre 2011
Reading materials â&#x20AC;˘ El-Naggar (1996) and Poustie (2001) on mathematical difficulties in general; â&#x20AC;˘ Kay and Yeo (2003) on mathematical difficulties associated with dyslexia National Numeracy Strategy such as Guidance to Support Pupils with Special Educational Needs in the Daily Mathematics Lesson (DfES 05451/2001) and Including All Children in the Literacy Hour and Daily Mathematics Lesson (DfES 0465/2002).
Amanda Kirby copyright Dyscovery Centre 2011
Further reference materials â&#x20AC;˘ http://www.dcsf.gov.uk/research/data/uplo adfiles/RR554.pdf
Amanda Kirby copyright Dyscovery Centre 2011
Bringing maths into other curricular areas • Fishing game- for numbers • Adding and subtracting- crawling to the end of the room • Cookery • Gardening • CDT
Amanda Kirby copyright Dyscovery Centre 2011
Conclusions • There are different reasons for someone finding mathematics difficult • It is necessary to “ tease” out the reasons to find the right approach to help • Sometimes the child needs confidence in order to become competent
Amanda Kirby copyright Dyscovery Centre 2011
Vision and perception
Language and comprehe nsion
Sufficient time
Focus and attention
Sequencing
Working memory
Fine motor skills
Gross motor skills
Amanda Kirby copyright Dyscovery Centre 2011
Big Math Programme components •
(1) use of numbers, involving counting procedures and principles, the use of numbers as labels (e.g. house numbers), and the different ways in which numbers may be represented
•
(2) shape, involving not only recognition and naming of shapes, but exploration of their characteristics (e.g number of sides and angles), symmetry, and ways of partitioning them into other shapes
•
(3) measurement, involving comparison, seriation, and iteration (repeated use of a measurement unit) with regard to a wide variety of quantities: length, weight, capacity, area, time, temperature and money
•
(4) working with numbers, including grouping of objects, adding and subtracting, and the relationships between sets and their subsets
•
(5) patterns, involving the systematic repetition of elements in the context of number, shape, colour, and sound (e.g. rhythm). Children copy patterns; extend them, e.g. adding 2 repeatedly to make 1, 3, 5, 7...); describe them; and create their own.
•
(6) spatial relationships, involving describing and mapping positions and routes Amanda Kirby copyright Dyscovery Centre 2011
.