Vertical Acceleration

Vertical acceleration is a new approach to teaching mathematics, not only for students with special needs but for all students.  A mathematics concept is introduced at a very simple level and then developed through all year levels
using the same model. For example with the concept of multiplication, we
begin at the lowest level and then proceed to explore this concept at all levels:
one digit by one digit, two digits by two digits, multiplication of fractions,
decimals and then finally multiplication of binomial expressions. When we take
a concept such as multiplication we introduce it with a model that can be taken vertically through all levels, in this case the area model, (see video Numeracy
4: Multiplication.) The concept and its procedures are connected at different
levels and it becomes easier to develop complex concepts in an efficient
manner in a shorter period of time.  The task we select, the resources we use,
the teaching strategies we adopt and the quality and quantity of questions we
ask are all aimed at taking a concept from simple to very complex, from the
earliest level to the algebraic manifestations of that concept. 

Similarly in order to teach the concept of fractions, (see video, Numeracy3:
Teaching Fractions) we   develop lessons on the same model, called a
fraction machine. In this case the fraction machine is used to introduce the
concept of fractions and then operations are performed on numerical fractions
and then on rational fractions (algebraic fractions).

The use of different models of instruction at each grade level leads to
fragmented teaching and students not making the connections between
different components of the concept.  They end up learning a collection of procedures that are not connected and are not built upon each other.
Students think that mathematics is nothing but a collection of isolated “tricks.”
They resort to memorization of these procedures without understanding.

This phenomenon of disconnected models across the curriculum is very
common.  For example, when teachers begin multiplication of whole numbers
in the second or third- grade level, they introduce it as simple way of adding.
They use repeated addition as a model.   

In the case of multiplication of fractions they find the product by shading areas
in circles and pizzas or just give the usual procedure of multiplying the
numerator by numerator and the denominator by the denominator.  Whereas,
in the case of multiplication of decimals, they teach procedures where students
pay attention to the number of digits after the decimal in the two factors is the
way for determining the digits after the decimal in the product.  And, then we perform the multiplication of binomial expressions by just telling students to
use the “foil method.”

Figure 1

These examples show schemas for multiplication as a concept, which change
from one number system to another. In such an approach, students do not
see the overall picture of the concept, do not have conceptual understanding
of multiplication procedures and think mathematics is fragmented.

Vertical acceleration, using the same model, takes the concept through all
number systems, fractions, decimals, algebra. e.g.

by using the area model of multiplication, it takes the concept of multiplication through all number systems, explaining the reasons behind different
procedures and multiplication algorithms.