Representation and Mathematical Structure
Representation and Structure
NCETM, 5 Big Ideas of Teaching for Mastery
5 Big Ideas of Teaching for Mastery
Mathematical Structure is central to the 5 Big Ideas of TfM
Research and Evidence
PUPILS’ AWARENESS OF STRUCTURE ON TWO NUMBER/ALGEBRA QUESTIONS
Dietmar Küchemann and Celia Hoyles Institute of Education, University of London
Item A1 (which was familiar to English pupils) was asked in Yrs 8 and 10 (a different but parallel item was used in Yr 9). It is a standard number/algebra item involving a tile pattern, and was designed to test whether pupils could discern and describe a structure (assessed by carrying out and explaining a numerical calculation).
Spotting number patterns, no structure;
Some recognition of structure (incomplete or draws & counts);
Recognition and use of structure: specific (correct answer, eg showing 60+60+3+3);
Recognition and use of structure: general (correct answer and general rule, eg x2, +6)
Recognition and use of structure: general, with use of variables (correct answer and general rule, with naming of variables in words or letters).
ARCHITECTURE OF MATHEMATICAL STRUCTURE
HAMSA VENKAT, MIKE ASKEW, ANNE WATSON, JOHN MASON
structure ⇔ mathematical relationship between elements
Emergent structure (involving analyzing/forming/ seeing local relationships)
Mathematical structure (involving analyzing/forming/ seeing general relationships)
This grid presents several formats of symbolic, iconic, graphical representations of multiplicative relationships. It also enables the user to juxtapose these to produce formats for encountering some of the structure of multiplication in the field of real number (Mason, Watson, Askew & Venkat, 2018).
Appreciating Mathematical Structure for All
We start from the position that mastering procedures is an important component of taking advantage of opportunities to make mathematical sense, but that it is of little value to learners if it is simply a procedure, because as the number of procedures increases, the load on memory and retrieval becomes more and more burdensome. When procedures are accompanied by even a minimal appreciation of the mathematical structures which make them effective and which provide criteria for appropriateness, learning shifts to focusing on re-construction based on re-membering (literally) rather than relying totally on photographic or rote memory.
The underlying theoretical frame being used here is a distinction between different forms, states, or structures of attention (Mason, 2003, Mason & Johnston-Wilder, 2004):
• Holding wholes (gazing),
• Discerning details (making distinctions),
• Recognising relationships (among specific discerned elements),
• Perceiving properties (as generalities which may be instantiated in specific situations),
• Reasoning on the basis of identified properties.
Designing Questions to Probe Relational or Structural Thinking in Arithmetic
Max Stephens* The University of Melbourne
Making Connections in Mathematics
The Connective Model
Understanding mathematics involves identifying and understanding connections between mathematical ideas. Haylock and Cockburn (1989) suggested that effective learning in mathematics takes place when the learner makes cognitive connections. Teaching and learning of mathematics should therefore focus on making such connections. The connective model helps to make explicit the connections between different mathematical representations: symbols, mathematically structured images, language and contexts.
Derek Haylock and Anne Cockburn (2008), Understanding Mathematics for young children
The CPA approach is often seen as a linear path from concrete to abstract which is not always appropriate. Haylock and Cockburn suggest that students are engaged in mathematical activity when they are involved in manipulating one or more of the four components.