Multiple Entry Points (Van de Walle, 2006)

Differentiation Using Multiple Entry Points

Van de Walle (2006) recommends using multiple entry points, so that all students are able to gain access to a given concept. These entry points are based on the five representations, which are considered to be tangible ways of representing knowledge.

  • concrete – materials are used to model a concept
  • contextual – situations to engage student interest
  • pictorial – diagrams represent a problem or record understanding
  • verbal – students talk and write about their learning
  • symbolic – students use symbols to record understanding

Teaching Student-Centered Mathematics: Grades 5-8.
Van De Walle, John A., and Louann H. Lovin.
Boston, USA: Pearson Education Inc, 2005.


Log or Linear? Distinct Intuitions of the Number Scale …

Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures
Science 30 May 2008: 320(5880), pp. 1217-1220
Stanislas Dehaene, et al.

The mapping of numbers onto space is fundamental to measurement and to mathematics.
Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education?

At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting.

This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic.
The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.