UNDER CONSTRUCTION – NOT YET COMPLETE
Children need coherence in the models/tools they meet, ensuring new models enhance their learning and understanding, as well as being developmental as they meet further mathematical content. The models are a key part of foundational knowledge, which all children need understand securely. A consistent development of models, avoiding tricks and algorithms are needed if equity for children is to be achieved, especially for those with additional needs.
Every representation used creates the potential for a different and deeper understanding, but also for further misunderstandings or misconceptions. As such, as with mathematical concepts and contents, carefully scaffolded learning sequences should be developed for representations and the rationale and justification for choice of representation must be clear to children.
North, Dalby, Wake – Primary Maths Autumn 2020
Below are examples for consideration when developing a mathematics curriculum and sequencing in planning.
Linearity is key in the Japanese Curriculum – Grade 1 (Y2 in UK)
Linearity
- addition (aggregation and augmentation)
- subtraction (take away – reduction and difference model)
Illustrating the physical gestures rather than relying on PPT slides with animations. Notice also that proportionality is maintained.
A practical example leading to the linear model for the whole and its parts is shown below, where the same linear model can used when there are more than 2 parts.
Array
In the case of multiplication, division and factors the array is the model
The array
Models to avoid
Cherry diagrams alone for Part, Part, Whole model – intended to be an animated model but frequently is not, does not evidence proportionality and is not connected with further mathematical content. If anything the model causes confusion when children are taught factor trees!
Factor Trees and Bugs
Factor Trees or Bug representations don’t provide insight into the mathematical structure of factors instead they provide a procedural method for finding factors.
North, Dalby, Wake – Primary Maths Autumn 2020
Typical bugs are ladybirds as they are regarded as ‘positive’ insects. The addition confusion that is created with using this model is ladybirds and most insects have 6 legs, yet the bug in the example had 8 legs! This is yet another example of where the science is not connected with the mathematics.
Factor Triangles where children are shown the triangle to represent multiplication. Does the largest number have to be at the top? What link does it have with the structure of multiplication? Multiplication is connected to an array and area model.
Speed Distance Time Triangles where children are taught the mnemonic. However evidence shows children frequently do not remember to put the letters in the correct position and therefore the triangle is not usable. Children do not have the understanding that a speed of 20m/s is actually relating the metres travelled in the number of seconds and therefore to recall the units of a measure would be more valuable. The SDT triangle is something highlighted as needing to avoid in the recent work in T&L of Mathematics and Science by NCETM.
This is not an extensive list of examples to use and avoid, but one that will extend with time and deeper research. However this quote stands as a good summary
Carefully chosen representations accurately reflect the mathematical structure under investigation, supported by familiar contexts that ground engagement with abstract mathematical concepts in familiar experiences, help children develop visual models on which to pin their understanding.
North, Dalby, Wake – Primary Maths Autumn 2020
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