Early Mathematics for the Better
As someone who has been immersed in early childhood and in particular, early mathematical development, for over 30 years, I have often been frustrated at the direction of travel taken by England’s educational policy makers. This has come to a head in the latest iteration of the EYFS (DfE 2020). Although there have been some changes for the better regarding the mathematics area of learning in this document, for example a focus on numbers to ten; others are at the least very questionable, i.e. the inclusion of the automatic recall of number bonds without reference to aids, and the removal of shape, space and measures as an ELG. Despite both these being stoutly challenged by the mathematics community at all stages of the consultation, they remain in this latest Framework. The case against these is available elsewhere, see for e.g. https://www.tes.com/news/why-government-has-got-maths-elgs-wrong
What are we, as practitioners, to make of all this? In this short blog I focus on the child and how to best explore important mathematical experiences informally, in situations where the child has an opportunity to control what plays out.
We have learned a lot about what matters in early mathematics over the last ten years. We need to better understand the exact relationship between spatial skills and maths knowledge, especially before children begin school. Research is clear on the role played by the development of early spatial reasoning in predicting later and wider mathematical achievement (Gunderson et al., 2012; Cheng and Mix, 2014).
The overwhelming predictive evidence suggests we would be foolish not to try to teach children spatial visualisation. Hawes and Ansari (2020: 466) cite Mix & Cheng (2012):
“The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related.”
What does this mean and how can we support the development of spatial skills in everyday mathematics? Spatial reasoning relates to our understanding of how things (including ourselves) move and interact in relation to the surrounding physical. Children engage
spatially all the time, from a baby reaching for a toy, to a six-year-old judging how much paper to cut to successfully wrap a present. There are several aspects to spatial reasoning, all of which are inter-related:
· Understanding language: hearing and then using a range of descriptive and positional words and phrases.
· Understanding relationships: learning how shapes fit together and the connections between them.
· Spatial memory: remembering where things are and how to get to and from places.
· Spatial representations: creating mental images, reading models and diagrams.
It is not really surprising that spatial knowledge is influential for number understanding when we think about how we rely on visual and spatial imagery in order to understand relationships between numbers and manipulate numbers successfully. The manipulatives children use that represent numbers are spatially designed, for example base-10 blocks and Numicon
There are many everyday activities that children engage in joyfully that encompass all these aspects of spatial understanding.
Taking things apart and putting them together again
Taking things apart and later, reassembling them is basic baby play. Jigsaw play is significant in supporting young children’s spatial language, especially when supported by an adult talking about what they are both doing (Ribiero et al 2020); for e.g. “You have chosen that one with the straight side” “I’m going to try this large piece with the corner.”
How about making your own jigsaws? Cut a birthday card or familiar photograph into strips – can we put it back together again? Which piece shall we start with? Why? Now what are we looking for? It can’t be that piece because…
Develop this into more complex cut-ups where you can look at how the shapes of the pieces fit together. Children often like cutting up their own but be prepared for lots of pieces!
Tangrams are more formalised, and ancient, paper puzzles made by cutting a square into seven pieces; five triangles and two quadrilaterals image.
The aim is to make pictures using all seven pieces.
In order to successfully complete a tangram picture, a child has to visualise ‘what might happen if..’ they place a shape here or here, this way or that; manipulating shapes by turning, flipping and rotating these. This book (Rinck and van der Linden 2017) is a beautiful exploration of some creatures that can be made, children often enjoy playing with the pieces and creating their own.
As well as the obvious opportunities to discuss similarities and differences, where and why items are stored together or separately, tidying up is a marvellous opportunity to discuss where things fit best and whether there is enough space for something. Giving children the responsibility of stacking the blocks away carefully in order they fit together safely is as much of a maths activity as building with them. Time to discuss the decisions they make and why, is valuable in extending the learning.
Tidying up also provides opportunities for some purposeful counting as children to check whether all the items are there or some are missing. Children often enjoy the search for the missing pencil or block! Labelling how many felt tips belong in this pot or large bikes in this garage with children in control of checking these, can gradually lead to rich number conversations; here is an example from a Nursery:
Collecting up scissors Rahim said: “We need to find six altogether … see it says ‘6’ here.” “We have three we need to find three more cos three and three makes six you know.” Once all scissors had been retrieved Rahim encouraged his friend Imran to count all the scissors. “Six,” said Imran. (Davenall 2015)
Tidying up is a problem solving situation that makes perfect sense and over time, can become a context where even the youngest children can develop and articulate their mathematical reasoning through judicious adult interaction.
Measurement is often an underused opportunity to explore number, in particular, comparisons between the size of different units. The Russian psychologist Davydov believed that abstract number teaching should be based on a sound foundation of measure and comparison experiences, with comparisons in measures underpinning the comparison of
quantities (Venenciano and Dougherty 2014). Filling and emptying often engages young children and seeing how many of something fit into a larger something else is a natural development to this interest. The following game, using different lengths of stick, draws children’s attention to the available space and visualising what will best fill it. As with any game, it requires playing a few times before mathematical discussion emerges about the decisions being made.
A game with sticks
Firstly, go out and collect a pile of all sorts of length of stick. You also need chalk or similar to make a mark on the ground.
This game is for two or more players. Mark a ‘Start’ line on the ground. Now mark a ‘Finish’ line. Between you, you are going to make a straight line of sticks to reach the Finish line.
Decide who starts.
The first person lays a stick with one end on the Start line.
Take it in turns to choose a stick and lay it down so one end touches the end of the last stick laid. keep taking it in turns until you reach the Finish. The winner is the person who lays the stick that is nearest to the Finish. The object is to choose your sticks carefully to get as close to the Finish line as possible – even touching it!
How many sticks did it take to reach from Start to Finish?
How are you choosing your sticks each time?
Over time, we can design opportunities, whilst playing with water, mud or sand, where we keep what is to be filled constant – providing one size of container to fill rather than a multitude – and fill this using uniform, non-standard units. Changing the unit size from small (spoonfuls) to large (buckets) we meet head-on the mathematical fact that the smaller the unit the more you need and visa versa. Over time this can lead to children making decisions about which unit is best to use and more accurate predicting of how many will be needed.
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Listen to Helens Podcast by clicking below, all will become revealed ….
Dr Helen J Williams is an independent educational consultant who specialises in the learning and teaching of Early Years and KS1 mathematics. She has a special interest in working alongside colleagues developing effective playful curricula opportunities.
She tweets as: @helenjwc
And her blog ‘Ponderings on maths education’ is at: https://info125328.wixsite.com/website
Cheng Y. and Mix K.S. (2014). Spatial training improves children’s mathematics ability. Journal of Cognition and Development 15(1): 2–11
Davenall, J. (2015). Developing Number Through Tidying Up: Age 3 to 7. Cambridge: Nrich. https://nrich.maths.org/11528
Department for Education (2020). Statutory Framework for the Early Years Foundation Stage: Setting the standards for learning, development and care for children from birth to five. EYFS Reforms Early Adopter Version. London: HMSO https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/896810/EYFS_Early_Adopter_Framework.pdf
Early Childhood Mathematics Group (2020). Why the government has got the maths ELGs wrong. London: TES https://www.tes.com/news/why-government-has-got-maths-elgs-wrong
Gunderson, E.A., Ramirez, G., Beilock, S.L. & Levine, S.C. (2012). The relation between spatial skill and early number knowledge: The role of the linear number line Developmental Psychology 8(5) 1229-1241
Hawes, Z., and Ansari, D. (2020). What explains the relationship between spatial and mathematical skills? A review of evidence from brain and behavior. Psychonomic Bulletin & Review 27: 465–482 https://www.researchgate.net/publication/338720447_What_explains_the_relationship_between_spatial_and_mathematical_skills_A_review_of_evidence_from_brain_and_behavior
Ribeiro, L.A., Casey, B., Dearing, E., Berg Nordahl, K., Aguiar, C. & Zachrisson, H. (2020). Early Maternal Spatial Support for Toddlers and Math Skills in Second Grade, Journal of Cognition and Development, 21:2, 282-311, DOI:10.1080/15248372.2020.1717494
Rinck, M., and van der Linden, M. (2017). Tangram Cat. The Netherlands: Lemniscaat
Venenciano, L. and Dougherty, B. (2014). Addressing priorities for elementary school mathematics. For the Learning of Mathematics 34 (1). 1
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