The design of See and Learn Numbers
The design of See and Learn Numbers is informed by what is known about how typically developing children learn about numbers and early mathematics, and by our understanding of the development of number skills for children with Down syndrome.
Important foundations for understanding mathematics are established in the preschool years as children explore their physical world and start to learn to count. Young children begin to develop an understanding of shapes, sizes, positions, patterns and quantity in play and daily activities before they learn to count. When children begin to learn to count they can link what they have learned about quantity, size and position to help them understand the number system.
There is evidence that early experience with numbers is fundamental for acquiring more complex maths concepts and skills. For children with learning difficulties, additional early practice may be particularly important for establishing a secure foundation for developing later number and maths skills.
Learning early number skills
Learning to count and to calculate is a challenge for many children, not just those with identified learning difficulties.
To master early maths skills, children must learn a number of basic procedures and concepts. Researchers have described these developmental steps in increasing detail in recent years[ 1-3 ] and this evidence informs current teaching recommendations in the UK and USA.[ 3-6 ]
These steps include:
- learning number words - learning to say the number word list - a list of words that must be kept in the correct order
- learning numerals - learning to link spoken number words to written numerals
- linking quantities to numbers - learning that number words and numerals represent quantities
- learning to count - using number words in the correct order to count objects
- learning "how many" - that we count to find out how many items we have and that when we count all of the items the last number word we say tells us how many
- learning the cardinal principle - learning to give a smaller quantity from a bigger set
- understanding equivalence - that if we share items evenly into two sets and then count the items in one set, this also tells us how many items are in the second set
- learning ordinality - that each number's position in the counting sequence is fixed and that each next number is one more equal unit
- understanding the uniqueness of numbers - that each number represents a specific quantity
- recognizing the relative sizes of numbers - for example, that 9 is bigger than 5 and that 4 is twice as big as 2
- learning quantity words and concepts and applying them to numbers - understanding the words used for the comparisons of sets - for example, same/different, more/less, bigger/smaller
- adding items using a count all strategy - for example, if calculating 5 + 2, counting out 5 blocks, counting out 2 blocks, and then counting all 7 blocks from 1 - "1, 2, 3, 4, 5, 6, 7"
- adding items using a count on strategy - for example, if calculating 5 + 2, counting out 5 blocks, counting out 2 blocks, and then counting on from 5 - "5, 6, 7"
- learning the inversion principle - that adding is the inverse of subtraction - for example, if you take away 2 and then add 2 back, then you have the same number of items you started with
Factors influencing progress
Research suggests that children's experiences of counting and number games at home influences progress. In particular, parents' number talk involving counting sets of objects with their children including sets larger than the child can count alone has been shown to accelerate children's understanding of cardinality.[ 7 ] Progress is also influenced by children's language, phonological awareness (ability to hear sounds in words), working memory, attention and motor skills.[see 8 ]
Intervention studies indicate a number of teaching strategies that help typically developing children learn more effectively, including the following:
- number and color words are learned faster when they are the last word spoken - for example, children learn the new concepts faster if we say "balls, there are two" rather than "there are two balls" and "the ball is red' rather than "it is a red ball"[ 9 ]
- playing a simple board game - a specific game with the numbers 1 to 10 can improve children's understanding of numbers, counting and addition[ 10 ]
- teaching children to count and calculate on their fingers in the preschool years can improve their early adding and subtracting skills in kindergarten[ 11 ]
- systematic daily teaching in small steps with repetition and practice can accelerate progress for children who are finding learning number difficult[ 12 ]
- computer games designed to teach children counting and cardinality can accelerate progress[ 13 ]
Evidence-based recommendations[ 3-6 ] also include:
- teaching counting using identical counters - at first using counters that are all the same size, shape and color
- presenting numbers and objects to be counted in a horizontal line
- teaching counting and cardinality with the numbers 1 to 5 first
- teaching number skills following developmental steps and recording progress - providing developmentally appropriate instruction for learning different skills based on the child's mastery of prerequisite skills, and recording progress to establish when to move on[ 6 ]
Number learning for children with Down syndrome
Surveys report that many (but not all) people with Down syndrome find counting and calculating difficult and these are the aspects of maths most studied. Many teenagers and adults have not mastered sufficient number skills to be able to work out change when using money or to calculate using numbers up to 100. However, most studies include only small samples with wide age ranges, and provide no information on the teaching provided. It is therefore difficult to draw reliable conclusions about the potential abilities of people with Down syndrome to learn number skills.
There is some evidence to suggest that at the early stages of number development (learning to count and to understand cardinality to the stage of giving a smaller number from a larger set), children with Down syndrome can acquire similar skills to typically developing children at the same non-verbal mental age level.[ 17 ] This longitudinal study also reports that the children with Down syndrome had mastered a shorter number word list than the typically developing children with similar counting and cardinality abilities. Learning the number words in order will be affected by being able to say them and to remember the list - both areas of difficulty for children with Down syndrome.
One study suggests computer-based instruction may be helpful.[ 18 ] Another study suggests that young children with Down syndrome may learn the written symbols for numbers (numerals) before they master reciting the number words, and that we should focus on understanding and using numbers 1 to 5 before moving on.[ 19 ]
There is no research looking in any detail at the next stages of number development. It is therefore not clear why many older children with Down syndrome seem to only acquire quite limited maths abilities. One possibility is that the early teaching they received did not ensure that they understood the basic concepts on which the development of later maths skills depend.
There have been some intervention studies examining the teaching of children with Down syndrome - mostly looking at counting - and these have been recently reviewed. The reviewers conclude that the evidence suggests intervention may improve progress but there is insufficient data to inform precise guidelines beyond suggesting that teaching should take account of the children's profile of strengths and weaknesses.
At the present time, therefore, the evidence available suggests we should teach number skills to children with Down syndrome following an evidence-based developmental progression with adaptations for the particular difficulties that the children usually experience.
Learning early concepts
The maths curriculum in school includes geometry and measurement. Children learn about size, shape, color, quantity, order and pattern in the early years through play and structured teaching.
Initially, children learn some basic attributes (characteristics) of objects, animals and people, including:
- size - big, little, tall, short
- shape - circle, square, triangle
- color - red, blue, green, yellow
Children then learn that size, color and shape are category words and that they can classify (sort or group) items based on an attribute such as color, shape or size. They then go on to learn more complex classification - sorting by 2 or more attributes (for example, big red, small red, big blue, small blue items)
Sequences and patterns
Young children also learn to make patterns and sequences. For example they learn to thread beads in a red, blue, red, blue, red, blue sequence or a more complicated red, blue, blue, red, blue, blue sequence.
Children also learn concepts that are not fixed attributes of an item (such as color) but depend on comparisons between items. There are many important comparative concepts for quantity, size, order and position, including:
- quantity - same, more, less
- size - big, bigger, biggest, bigger than, smaller than
- order - first, second, third …last, before, after
- position - in, on, under, in front, behind, next to
These concepts are important for understanding numbers and calculations.
There is little research into how children learn these basic concepts. There is a small amount of research examining how children learn about shapes and early geometry. This suggests that as children move beyond identifying the basic circle, square and equilateral triangle and begin to learn about quadrilaterals, rectangles, and a range of triangles, they progress in a developmental sequence and need to be taught to recognize the essential distinguishing features for each shape.
There is no research we know of looking at how children with Down syndrome learn these concepts. Many will learn some of the simpler concepts for size, color, shape and position during the preschool years as they occur in the first 500 words that most children learn. However, most children will learn about the more difficult comparative concepts during their primary/elementary education.
Teaching children with Down syndrome
While all children with Down syndrome experience learning difficulties and developmental delays, not all aspects of cognition and development are equally affected. In general, there is a pattern of relative developmental strengths and weaknesses that is common among young people with Down syndrome and which can inform more successful teaching approaches.
Some of the key developmental characteristics and adaptations to consider when teaching numbers skills are:
- hearing impairments and speech delays - likely to make it more difficult to discriminate and say all the sounds in number words
- language delay - may mean that children with Down syndrome need to be explicitly taught more of the vocabulary needed for number concepts
- verbal short-term memory difficulties - may make learning the number word list a significant challenge - this may be helped by using written numerals to provide visual prompts from the outset when learning number words and the counting sequence (therefore, earlier than is usual for typically developing children)
- motor delay - slower progress with fine finger and hand control may make pointing, picking up and moving objects while counting more difficult - large, light and easy-to-handle objects may be helpful
- attention and motivation - short activities, fast-paced sessions, and modeling and prompting for errorless learning are likely to help maintain the children's interest
- small steps and plenty of practice - regular activities and opportunities for repetition are likely to help to consolidate learning
- sign language - can help children struggling with speech to communicate while continuing to learn and practice spoken words
The design of See and Learn Numbers
We have designed See and Learn Numbers to follow the developmental progressions identified by research into number and maths learning and as recommended in good practice guidelines. We have adapted teaching activities to accommodate the specific needs of children with Down syndrome. These adaptations and design features include:
- small developmental steps - providing explicit instruction at each stage, starting with learning the number words and proceeding in small developmental steps through early number skills and concepts
- clearly-defined progression - keeping records of progress to determine when to move on and progressing when the prerequisite skills for the next step have been learned
- practice and repetition - activities that can hold attention and be regularly repeated
- clear and consistent visual representations - early use of written numerals, large black counters (kits) and simple black counters with distraction-free screens (apps), consistent horizontal 1-5 counter arrangement
- simplified language - minimal, clear spoken prompts with key words last, explicit vocabulary teaching
- Sarnecka, B.W. & Carey, S. (2008) How counting represents number: what children must learn and when they learn it. Cognition , 108, 662-674.
- Sarnecka, B.W. & Wright (2013) The idea of exact number: children's understanding of cardinality and equinumerosity. Cognitive Science , 38 1-14.
- Nunes, T., Bryant, P. & Watson A. (2009) Key Understandings in Mathematics Learning. London: Nuffield Foundation . https://www.nuffieldfoundation.org/key-understandings-mathematics-learning
- Clements, D.H. & Sarama, J. (2009) Learning and Teaching Early Math: the learning trajectories approach . New York: Routledge.
- Krasa, N. & Shunkwiler, S. (2009) Number Sense and Number Nonsense . Baltimore: Brookes.
- Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children: A practice guide (NCEE 2014-4005). Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=18
- Gunderson, E.A. & Levine, S.C. (2011) Some types of parent number talk count more than others: relations between parents' input and children's cardinal-number knowledge. Developmental Science. 14, 1021-1032
- Lemons, C.J., Powell, S.R., King, S.A., & Davidson, K.A. (2015) Mathematics interventions for children and adolescents with Down syndrome: a research synthesis. Journal of Intellectual Disability Research. doi:10.1111/jir.12188
- Ramscar, M., Dye, M., Popick, H.M. & O'Donnell-McCarthy. (2011) The enigma of number: why children find the meanings of even small number words hard to learn and how we can help them do better. Plos One , 6 e22501.
- Ramani G.B. & Siegler, R. S. (2011) Reducing the gap in numerical knowledge between low- and middle-income pre-schoolers. Journal of Applied Developmental Psychology . 32, 146-159.
- Jordan, N.C., Kaplan, D. Ramineni, C & Locuniak, M.N. (2008) Development of number combination skill in the early school years: when do fingers help? Developmental Science . 11, 662-668.
- Bryant, D.P., Bryant, B.R., Roberts, G., Vaughn, S. Pfannenstiel, K.H., Porterfield, J. & Gersten, R. (2011) Early numeracy intervention program for first-grade students with mathematical difficulties. Exceptional Children . 78, 7-23.
- Praet, M. & Desoete, A. (2014) Enhancing young children's arithmetic skills through non-intensive, computerised kindergarten interventions. Teacher and Teacher Education , 39, 56-65.
- Faragher, R. & Clarke, B. (2014) Mathematics profiles of the learner with Down syndrome. In R. Faragher & B, Clarke (Eds.) Educating Learners with Down syndrome . pp 119-145. Oxford: Routledge.
- Brigstocke, S., Hulme, C. & Nye, J. (2008) Number and Arithmetic skills in children with Down syndrome. Down Syndrome Research and Practice . https://library.down-syndrome.org/reviews/2070/
- de Graaf, G, van Hove, G. & Haveman, M. (2013) More academics in regular schools? The effect of regular versus special school placement on academic skills in Dutch primary school students with Down syndrome. Journal of Intellectual Disability Research , 57, 23-38.
- Nye, J. Fluck, M. & Buckley, S. (2001) Counting and cardinal understanding in children with Down syndrome and typically developing children. Down Syndrome Research and Practice , 7, 68-78. https://library.down-syndrome.org/reports/116/
- Ortega-Tudela, J.M. & Gomez-Ariza, C.J. (2006) Computer-assisted teaching and mathematical learning in children with Down syndrome. Journal of Computer Assisted Learning , 22, 298-307.
- Faragher, R. & Clarke, B. (2014) Developing early number concepts for children with Down syndrome. In R. Faragher & B, Clarke (Eds.) Educating Learners with Down syndrome . pp 146-162. Oxford:Routledge.
- Clements, D.H. & Sarama, J. (2000) Young children's ideas about geometric shapes. Teaching Children Mathematics , 6, 482-88.