## Wednesday, January 14, 2009

### The Abstraction of Number

As my department was looking over the new state learning results, I came across an interesting sentence:
"It is expected that when working with measurements students... understand that a number without a unit is not a measurement, and that an appropriate unit must always be attached to a number to provide a measurement." (From Maine Learning Results 2007 Mathematics Grades 9-Diploma, Standard B)
I immediately wondered why is this so important that it needs to be listed in concepts that high schoolers should grasp? Yes, many students forget to write units at the end of answers, but how does the proper usage of units show more sophisticated thinking?

These questions brought me back to why man first started using numbers. Our first evidence of counting comes from the Ishango Bone, found in the Congo dating back to 20,000 B.C.E. When man first started using numbers, they represented a one-to-one correspondence to elements in a set of physical objects. In fact, even as we began to use negative numbers and fractions, they still had meaning in the physical world, such as the area of land owned or how many bags of grain were owed in a business transaction.

Eventually, the study of numbers and patterns became a pursuit of its own that lost its connection to the real world. A number, such as 3, could be thought of without associating an image of three goats or a length of three cubits. The number existed on its own, holding its own special properties such as being odd and prime and that adding 1 to 3 made 4. The change from having numbers exist only with units to an abstract thought of its own took mankind thousands of years to develop.

Considering how long it took our species to make this leap, it is a major leap for a child to make and realize in their understanding of mathematics. How many high school students still use their fingers to add and subtract? They do not have an abstract idea of number and still see numbers as representing how many objects belong to a set. Their understanding of mathematics is prohibited by their inability to abstract numbers without units. It is obvious why these same students flounder in algebra, an abstraction of the abstract idea of number.

My question now is not why this is important for high schoolers, but why is this standard not included earlier? A further question is how do we foster this deeper understanding of numbers with students that have gotten this far without the abstraction?