Berkshire Mathematics

Four major principles for teaching

1. Use of Appropriate Concrete Models

For early mathematical concepts, it is important that a child experiences mathematics through appropriate and efficient learning models.
Cuisenaire rods, base 10 materials and the Invicta Balance provide
appropriate models for these concepts.

2. Levels of Knowing Mathematical Ideas

Every activity in this programme represents 6 levels of knowing. Each lesson follows the order of these levels of knowing.

intuitive
every new fact is introduced as an extension of something
the child already knows
concrete
each new fact is presented through a concrete model
pictorial
the model of the new fact may be sketched or illustrated

Before abstract recording is asked for, a lot of oral and mental arithmetic
activity is necessary.

abstract
the new fact is recorded in symbolic form. e.g. 3 + 4 = 7
applications
the child is able to form a number story using the fact
communications
the child is able to explain the strategy.
e.g. Since 3 + 3 = 6, I know 3 + 4 is one more, so 3 + 4 = 7

3. The Three Components of a Mathematical Idea

Every mathematics concept consists of three components: linguistic,
conceptual and procedural.

The linguistic component is the language (vocabulary, syntax, and translation from native language to mathematical language, and vice versa) used in understanding, conceptualising and communicating mathematical information.
For example, to understand the concept of lowest common multiple, one
needs to know the meaning of the individual words in the expression and
their relationship to each other. In that sense mathematics is a language.

The conceptual component is the mathematical idea itself. Modelling the idea
(the concept) with concrete materials and manipulating these materials
develops conceptual understanding. The child will then have a mental image
of the concept to refer to. For example, conceptually what do the expressions
9 x 3 or ½ x 1/3, or (a)(b) mean? Each one of these expressions creates a physical or conceptual model in our minds.

Finally, the procedural component is the algorithm or the method emanating
from the use of the concept, for example, the division algorithm or the
procedure for adding fractions.

Acquisition of the conceptual and linguistic components of mathematics will
help the child to think mathematically, otherwise he/she will remain at a very procedural level. Mathematics will be a collection of tricks and remembering
these tricks will become the learning of mathematics. Children will see mathematics as merely a collection of procedures if this aspect is emphasised without the use of concrete models and language to clarify the concept.

Children very often forget the procedural aspect, but once the conceptual
and language model is developed, it is difficult to forget the concept.
Therefore, conceptual and linguistic understanding lies at the heart of
learning mathematics.

4. The Questioning Technique

For the development of concepts, the teaching process must engage the
child by asking key questions. Appropriate questioning is important for the introduction of a concept, for reinforcing it and for helping the child to
memorise facts.

Proper questioning is the key because we know that:

questions instigate language…
language instigates models…
models instigate thinking…
thinking instigates understanding…
understanding instigates competent performance…
competent performance produces long-lasting self-esteem…
self-esteem is a strong motivator for learning…

Mahesh Sharma is Professor of Education at Cambridge College,
Cambridge, Massachusetts, USA .

He is the Director of the Center for Teaching/Learning of Mathematics.
He edits Focus on Learning Problems in Mathematics, an international and interdisciplinary journal dealing with the learning and teaching of
mathematics, in particular with issues dealing with learning problems
in mathematics such as: dyscalculia, acalculia, mathematics anxiety
and specific learning disabilities in mathematics.
He also writes Math Notebook, a newsletter for teachers and parents.

His Center for Teaching and Learning of Mathematics is affiliated to
Berkshire Mathematics in the U.K. run by Patricia Brazil.
She organises Prof. Sharma’s lectures and courses, and produces videos / DVDs, which are for sale along with the U.S. publications.

Tel: 0118 947 4864 Fax: 0118 946 1574
Email: info@berkshiremathematics.com