Abstract:

One of the difficulties we find in our pupils when doing maths is their understanding of a number as sum or product of others. In order to find a solution route, we have studied which countries are the best performers in mathematical knowledge measuring international programmes, and chosen Singapore, which has been ranking in the first places for decades.

Since the beginning of their mathematical instruction, Singaporean children work with a model called Number Bonds, in which, in a very visual way, each number is shown as sum of another two numbers. This understanding, tied in with logic, brings along great calculus capacity. Even though the result can be somewhat confusing for children, some commercial media (vendors of didactic material and/or educational software) have extended this model to product, out of the initial use for which it was designed.

Gattegno (1971) developed a method that allows to realize the number as product via Cuisenaire rods, also known as numbers in colour. This method also allows the comprehension of diverse concepts such as product, power, divisor, multiple, square root or logarithm. Despite the enormous potential of this material, it also suffers from limitations. For instance, only numbers from 1 to 10 can be represented. In addition, availability of materials for all the students is necessary. Moreover, specially when working with groups that show concentration difficulties, the rods can be more distracting than helpful.

The model we present here as innovation in didactics of mathematics, named Tower Model, is inspired in Singapore's Number Bonds and Bar Model Method, retaining their graphical and visual properties and can be applied to a wide variety of problems. Likewise, this model inherits from Cuisenaire rods some of its applications although being a graphical model provides great versatility and can be used in the classroom without other requisites than writing tools and a surface to write in (whether paper and pencil, blackboard, tablet computer...).

The Tower Model conceives numbers as representing a quantity as well as product of several factors. One of the advantages presented by this model is that, not being based on physical materials, it can be used for representing any number, no matter how big the magnitude of its factors may be; it allows working not only with whole numbers, but also decimals, rationals and irrationals. The Tower Model is designed so that pupils easily make associations between a number and a product of factors. This leads to a better comprehension of product and division, and makes it easy to develop calculus strategies when using these two operations. With the Tower Model we can work on many concepts, such as prime and compound numbers, divisor and multiple, or prime factor decomposition. Also, using this model all properties of powers can be shown. One of the most interesting applications is square and cubic roots and their transformations to fractional powers, as well as factor extraction from radicals. Also, just as with the method of Gattegno (1971), the concept of logarithm can be introduced.
Summarising, what we present here is an innovative model that, being easy to understand and apply in classroom, contains a huge potential in didactics of mathematics and to improve the performance of our pupils.

Keywords:

Didactic model, product, Singapore, educational innovation, didactics of mathematics.