Challenges — Concept — Research Project
Maths is a beautiful and important language, meant to no less but to enable us to understand and to describe our world. Longitudinal studies underline how crucial it is to master this language, i.e. acquire a true basic conceptual understanding.
The challenge is that maths is an abstract language, while children children start from their concrete world and only gradually build their bridge over to abstract thinking.
The good thing is that all children have an inborn desire to learn this language, making it part of their natural development. It is our responsibility to provide an environment enabling children make the experiences they need and help them structure.
All maths experiences need to be embedded in children’s play – because playing is their way of learning. And this learning is a very individual process, based on each child’s individual experiences, needs, and associations.
Neuroscience and developmental psychology tell us that, when learning something new, we all build on what is already there, on our experiences, knowledge, associations as well as emotions. This is especially true for young children. They are self-centred and live in an emotional world, where everything is alive. They have a black-and-white view and a magical thinking. Anything new is associated with what they already learn – and how they feel about it.
So we need to provide an appropriate environment that helps them making the relevant experiences, structure them, and let the good things happen!
The complexity of numbers
We as adults tend to forget how complex the concept of numbers is and that children need enough time and many experiences to grown into them. Here is a long list of the many important facets:
— Cardinal aspect A number represents a quantity (3 cars, 3 children … 3 is always 3)
— Ordinal aspect Numbers have a defined order (1, 2, 3 …). They also stand for a rank (first, second, …)
— Invariancy Quantities stay the same, no matter how they are placed
— Partitioning Numbers can be split into other numbers/ quantities (Crucial for becoming able to work with numbers beyond 10)
— Subitising Being able to grasp quantities at a glance
— One-to-one-correspondence Each element counts only once, and one elements of a set relates to exactly one of another set
— Calculating A number is the result of an operation (2 + 3 = 5)
— Reversibility Turning 2 + 3 = 5 back by 5 – 3 makes 2 again.
— Operator A number as the multiple of an activity (clap 4 times, hop 3 times)
— Encoding A number encodes a quantity, but also the rank in an order, a house, a birthday …
— Geometric aspect Numbers relate to geometrical shapes
— Musical aspect Patterns of rhythm, beats
— Cultural aspect The meaning of numbers in stories and in cultures
— Measuring Numbers in measures for lengths, volume, weight, time, money.
The Numberland Concept
‘Let’s visit Numberland’ was developed to appropriately address the evolving interest of small children in counting, sets, numbers, comparing, and geometric shapes. It supports children to build the bridge between their inborn understanding of maths (‘1, 2, many’) to the formal language of maths we developed to describe our world and to handle things. In Numberland, children acquire a profound, well structured conceptual understanding of the number range 0 to 10 (building the crucial fundamental basis to their mathematical understanding) and the numerals to 20.
Yet the children benefit far beyond the matter itself: Language, self-esteem, concentration and cognition, motivation and creativity, social, motor and music skills are enhanced as well.
The edge of “Let’s visit Numberland” lies within the embracing approach: The concept allows a child to experience the entire aspects of numbers with the entire self, i.e. body, mind and spirit. This way, children in their daily play develop a multi-faceted, positive image and a deep, lasting understanding which they can build on.
‘Numberland’ is not an academic training but rather a central theme that triggers children to play in this carefully designed environment, and helps them structure their experiences. The specific elements encourage both children and staff to bring in own ideas. It can be easily adapted to individual needs and age brackets. Popular and established, relevant activities can be brought in and ‘Numberland’ can be a trigger for the further exploration of topics, e.g. bees with their six legs and their hexagonal combs.
It is very natural for children to travel to a country where numbers live: For a while, children regard regard everything around them as being alive. Anything magic strongly appeals to small children and going with that is a strong emotional thinking.
In addition, children of that age need tangible experiences of the inherent abstract character of mathematics.
In a nutshell
On their imaginative, cheerful journey to ‘Numberland’, the children meet the numbers 0 to 10 as nice, living characters who motivate them to deal with mathematical aspects and coherences. Number Lane, starting at 0, shows the way to Number Town where each number lives in its characteristic, geometrically shaped garden. The children decorate these gardens with houses, towers, flowerbeds, and other things going with the number.
There is a mischievous (but not scary) little character messing around and a nice fairy/princess sorting things out. No matter how these two characters are named, children love the good vs. bad. Tales and music, lots of active games and other things children enjoy complement the experiences.
A neighbourhood full of maths –
Let’s build homes for our number friends!
While building a neighbourhood with gardens and homes for their living number friends, the children experience all vital aspects of numbers, shapes, operations. It triggers a lot of communication, reasoning, helping each other and can be used to deliberately address an abundant choice of content. They are the starting point for cross curricular activities and support the transition to further, more abstract maths experiences.
All elements are designed to trigger children as well as to transport max mathematical content – structured but still flexible and different.
“Imagine, someone invited me to a very special place …”
— Number puppets turn abstract symbols into living and lovable characters with distinct characteristics to play with and to talk to. They live in Number Town as happy neighbours.
— A Number Lane leads us into and out of Numberland (from 0 to 10, then 20). So many fun active games we can play on and with it to ensure we always find our way in and out of Numberland.
— Gardens define the property of each number and correspond to the geometric aspect. We decorate these gardens and experience all number aspects at a glance. Access is strictly limited and, therefore, a lot of communication, detailed cognition and reasoning is triggered: “I wonder in which garden this little toy cow may graze?”
— Houses because also numbers need a house to live in, with a matching set of windows and a house number on top of the chimney. Once we understood the concept 1 to 5, we can easily build those houses 6 to 10.
— Towers built with single blocks for the beautiful view in each garden. Much to explore, also when we put them next to each other as an impressive staircase.
— Flowerbeds and a large Meadow for those gardening enthusiasts who love picking / planting / buying flowers or vegetables, exploring numbers, patterns, symmetry, even money.
— Hodgey Podgey and Forgetmenot These two imaginative characters account for our magic thinking. We particularly love cheeky Hodgey Podgey who ever so often messes around in Numberland, so that we have to spot and correct the mistakes! Fortunately, we can also call kind Forgetmenot for help.
— Stories like that of poor Four who is sick! To cure her, a tea is prepared from four bits of four different herbs out of her square herb garden. After four minutes brewing time, four spoons of honey are added and stirred four times – curing immediately…
Or that of Nine who needs to be rescued by his number friends …
— Number Songs because music, rhymes, and singing have the highest value for our body, mind, and soul. A refrain for the beginning as well as for the end of a visit to Numberland make these visits special. In addition, there are songs for each number the beat of which and number of tones corresponds to the number as well as the content does to the number tales. Popular nursery rhymes or relating songs can be included as well. It is fun to explore the patterns in rhythms and to sing, dance, and make music alongside Numberland.
— My Book of Numbers where I can collect my creative treasures around Numberland, maybe some activity sheets, definitely colouring pages of the number characters and a table to stamp the visits to Numberland. Highly treasured by children and a good documentation of a child’s development.
— Plennnty of room for creativity, own ideas and needs, for free play, for cross-curricular activities – and to connect with things us children love and need.
— Games and Activities Each visit to Numberland can be the basis for lots of activities around the specific Number of the Week. Fun active games, related songs or rhymes, designing or moulding, exploring specific questions, sports or outdoor activities – whatever is part of the term anyway or an objective can be brought in.
Let’s visit Numberland started as a research project in Germany from 2003 to 2005. The objective was to find out whether combining findings from brain research, developmental psychology, elementary pedagogics, and maths into an emotional and imaginative, open and playful concept had significant effects on learning.
And it had: Friedrich/Munz proved that within only ten weeks children aged 3 to 6 gained the mathematical competence and understanding they would have normally achieved only within one year. The same was true for language skills and for under-privileged children with partly very poor understanding of the German language.
The University of Heidelberg confirmed the findings in a second study from 2005 to 2009.
- Within ten weeks only, the Numberland children gained the mathematical competence and understanding they normally would have achieved within one year; the control group more or less remained the same.
- The same was true for the linguistic competence, both passive and active, of the children.
- The socio-economic background of the children was irrelevant for the gain of competence.
- Both sexes benefit, the girls even a little bit more than the boys.
- The age specific results show a striking development of the children regarding both math and language skills: Naturally the younger children had a much lower base level than the older (red dots). Yet the tasks were all taken from tests evaluating whether or not a child aged 5+ may enter school. So obviously the older children achieved much higher scores (green triangles). But after only 10 weeks of Numberland the under 4 year old children scored on average higher than the base level of the one year older! The same was true for the development of the 4 to 5 year olds compared to the base level of the over 5 year olds who again made significant progress.
- The teachers felt motivated and competent to creatively apply Numberland.
- The second study by Sabina Pauen with a similar test design included 4 to 5 year olds and confirmed the positive effects of “Let’s visit Numberland”. It additionally pointed out that the concept increased the confidence of the staff in teaching math and that they felt very comfortable with the concept.
Description of the Friedrich / Munz research study
During 10 weeks 46 children aged 3 to 5+ traveled to Numberland once a week. Each week they visited one number for approximately 50 to 60 minutes. Whenever suitable, the respective number of the week was picked up during the daily routine (circle time, creative play, sports, meals etc.). The children had access to the material for free play. In parallel, a control group of 46 comparable children carried on with their normal routine and had no access to Numberland. In a second run, this setting was repeated with underprivileged children, many of them having severe language problems and lacking other skills. The charts depict the results of that second run.
Project and control groups were tested before and after the 10 weeks in which the project took place. The tests comprised tasks such as conceiving and building sets of numbers, colors and shapes, remembering numbers, cognition of details, general reasoning powers. Regarding active and passive language skills, the children made a story out of some pictures, were asked what they remembered from a story that had been read out to them, and they had to carry out tasks of differing complexity. The effects of ‘Let’s visit Numberland’ both on math relevant thinking and on language have been tested twice with significant results.
Gerhard Friedrich, Horst Munz (2006). Förderung schulischer Vorläuferfähigkeiten durch das didaktische Konzept “Komm mit ins Zahlenland”.
Published in the peer-reviewed journal: Psychologie in Erziehung und Unterricht, Vol. 53, S. 134-146. München und Basel: Ernst-Reinhardt-Verlag.
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Sabina Pauen, Viktoria Herber (Ed.) (2009). Offensive Bildung: Vom Kleinsein zum Einstein. Cornelsen Verlag Scriptor 2009
Second study 2005 – 2009
Please email email@example.com for English translations of the articles as pdf (So far unable to integrate on this website).
Further English Publications
- Gerhard Friedrich, Barbara Schindelhauer (2015). Let’s visit Numberland: A highly emotive and efficient reason for reasoning.
In: Mathematical Association, Primary Mathematics, volume 19, issue 1, pp. 13-16. Mathematical Association, www.m-a.org
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- Gerhard Friedrich, Barbara Schindelhauer (2014). Let’s visit Numberland: An Emotive, Story-based Contribution to Numeracy and Literacy Development.
In: Jörg F. Maas, Simone C. Emig, Carolin Seelmann (eds.). Prepare for Life! Raising Awareness for Early Literacy Education. Results and Implications of the International Conference of Experts 2013. Stiftung Lesen 2014 (pp. 79-85)
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- NCETM National Centre for Excellence in the Teaching of Mathematics (2011). Focus on…Numberland.
Early Years Magazine, Issue 25/2011 (online resource). www.ncetm.org.uk
- Gerhard Friedrich, Viola de Galgóczy, Barbara Schindelhauer (2010). Let’s visit Numberland – playfully discover the world of numbers.
Self-published handbook, ifvl Waldkirch.
For more publications (in German) and articles on other subjects please refer to the German homepage www.ifvl.de