CMTP: Changes in Mathematics Teaching Project

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How did teachers help students deal with complex tasks?

There was a clear distinction between use of tasks which required the discernment of one feature only and tasks in which students had to discern two interconnected variables and perhaps their relationship. Nearly all the core teachers offered some kind of complexity during every lesson. A further noticeable factor was whether attention was given to special cases that showed up particular features of an idea (such as x=2 being a straight line) or only 'generic' cases (which show all features). All teachers used connectionist, transmissional and discovery methods in various proportions. In this project we have found that looking at mathematical complexity has enabled us to distinguish between teaching.

Methods of simplification
A few teachers, particularly during year 8, simplified mathematics for the PLAS by using images, representations, examples or approaches which are limiting, and may even be confusing. Research shows that simplifying complex mathematics is not associated with high attainment.

• Breaking a question into smaller steps
• Only using binary operations with numbers and algebra, no squaring etc.
• Always offering examples which are based in small positive integers (e.g. with enlargements)
• Measuring limited to angles of 90 degrees
• Reducing tasks to calculation questions (TAs were often seen to do this)
• Students advised to choose the simplest task
• Using obvious letters: "c for chocolates", "a for apples" without reference to number
• Using empirical approaches (cases, numbers, measuring) for situations which are mathematically based on structure and reasoning.
• Giving shapes only in usual orientation, mirror lines for reflection only on axes of symmetry, etc.

Complex tasks given to PLAS
The majority of teachers never simplified the questions they had posed for PLAS. In several cases we saw students presented with ideas that were deliberately awkward. adFor example:

• Division with awkward decimals, so that students need a method
• Enlargement being done with numbers that avoided 'doubling' and additive reasoning
• Avoid using small positive integers which encourage ad hoc methods
• Using negative numbers in a worked example, again so that students need a method
• Use multiple representations; students have to think to connect representations
• Provide exercises that gradually became harder; students choose where to start
• A first lesson about 'like terms' included negatives, squares and 2pq³ to avoid students developing simplistic methods

Scaffolding to achieve more than they could do on their own

• Develop thinking:
• • Teacher 'models' by thinking about maths out loud during the lesson
  • Ask questions that cannot be guessed
  • Teacher gives a typical wrong answer, or wrong reasoning, and asks for criticism
  • How are we going to...?


• Develop repertoire
• • Compare scope and uses of different methods; discuss consequences of choice
  • Use multi-stage problems as context for revision and consolidation in whole class discussion
  • Habitually representing proportions as fractions, decimals and percentages whenever possible (or similar) to emphasise relationships
  • Sort some questions according to the methods you might have to use
  • Think of another, and another, and another...


• Develop confidence with challenges
• • Harder ideas and examples are always available and offered
  • Explicit discussion of difficulties (e.g. it took people 200 years to accept negative numbers - why?)
  • Give formats, grids, layouts, writing frames, to help organise work
  • Offering 'convert to something you know about' as a strategy
  • Try on own, discuss with neighbour, try on own again, then participate
  • Get students to make up hard or multistage examples


• Develop ability to understand structure
• • Use small numbers to show how division is inverse of multiplication, and to compare different layouts, then give harder numbers to do
  • Increasing parameters or constraints from one, to two, to three
  • Using very big numbers to help students shift from calculation to algebra
  • Giving lots of examples that all emphasise a main idea with little irrelevant features
  • Using stories to develop sense of connected ideas
  • Use colour and/or font size to emphasise particular bits of diagram or algebra or calculation