“Mathematicians do not study objects, but relations between objects.” Henri Poincare
I was chatting with a few of our staff this week about the ways we help our students develop their ability to work with numbers, specifically when they are adding and subtracting in the early years. An important point that I often stress when talking with parents and teachers about mathematics is that the area that many of us believe to be the critical focus of mathematics (the operations of adding, subtracting, multiplying and dividing) is actually of secondary importance.
Simply put- we spend way to much time trying to force children to memorize or learn the operations and not nearly enough time helping them understand the numbers they are using. Put another way- think of the numbers as nouns and the operations as verbs; in math, as in life, there are way more nouns than verbs and they are much more interesting!
This media clip on Using Open Number Lines from Dr. Alex Lawson does a great job explaining how using a model like a number line can support children to think about the quantity value and relationships that exist between numbers in a mathematical situation. It also shows how using models as a precursor to what we call the standard algorithm is important at all stages of mathematical development. For most of us, this type of instruction was just not used when we were learning math in school and it is too bad, because it would have save many of us from a life of math phobia.
A lot of students and adults think that using the algorithms is the math- it’s part of it, but not nearly the most important part. In fact, the algorithm can most simply be described as a way of showing (or modelling) what has been done with the numbers. A student who uses an algorithm to solve a problem without understanding the relationships between the numbers is no better off than a student that uses a calculator- they both don’t really know what they have done.That’s why we are using models like the open number line- they allow the student to see the connections and relationships between the numbers and build a model of how they can solve a problem.
There are only a few mathematical operations but the numbers are (literally) infinite- the numbers are much more interesting than the operations- tools like the open number line help our students discover and harness this idea.