08/26/2009 | Maggie Martin Connell
In other words, we are in a position to change the debilitating perception of mathematics as a collection of marks to be manipulated.
Now I have your attention! “How so?” you ask. Read on.
Over time, brilliant mathematicians have developed progressively clever ways of using the mark to communicate other profound ideas related to quantity. Each new discovery allowed us to better cope with our increasingly complex world.
To bring us any one of these powerful tools, they had to have understood that this science we call Mathematics follows patterns and that, if the same steps are applied to other situations sharing a similar idea, it ‘works’ every time! What does this have to do with your classroom tomorrow? Everything!
The problem is, somewhere between the birth of this symbol for ‘three-ness’ and today, we forgot our history. In the name of efficiency, mathematics became the art of computing, using established procedures that represent someone else’s understanding. For the general population, learning those procedures became more important than the original ideas they represented; the medium became the message! A ‘math culture’ evolved where the symbols that were supposed to represent the math became the ‘math’ itself.
The importance of precision digit-dancing outstripped that of understanding... even if you didn’t know where that dance was leading you!
But this story isn’t just about 3. It’s about the entire collection of marks representing mathematical ideas.
We have learned to record them, say the words associated with them and manipulate them to arrive at ‘answers’. But do we know why we are doing it... other than because the teacher said so?
Have we considered the BIG ideas represented by those digits? Can we tell someone why each step in the ‘digit-dance’ makes sense?
Consider a symbol we have seen since the very beginning of our school math experiences, the lowly, often tak-en-for-granted = symbol.
In a small survey 74 students from six different middle-level schools were asked what = means. Out of the 74 students, just eight made reference to the idea of balance:
- “It only works if each side has the same amount.”
- “To have the same value. For example, $1 equals 100 cents.”
- “You have to have the same on both sides or it’s wrong.”
Of the 74 students, 68 said “equals.” When asked to elaborate, they gave a variety of responses, some simply repeating the word in a different tone.
Other responses included:
- “That’s what it makes.”
- “It means two numbers combine to make a bigger one.”
- “It’s sort of like a question mark.”
- “It’s when you know the problem is over.”
- “It’s what you have to say before you give the answer.”
- “It means a certain thing (probably a number) is equal to the next possible number.”
In the vast majority of cases, students interpreted the = symbol as a ‘command’ to give an answer!
Clearly, the idea behind the mark (balance) has been missed for the majority of these students.
Furthermore, many of them have adopted the misconception that the purpose of the = sign is about the positioning of the answer.
But what if we taught this idea inside out? Instead of teaching the symbol and how to make it perform, what if we set up an investigation with a pan balance where kids could explore what balances a given number sentence and what does not? A good investigation question will uncover a variety of possibilities and create the need to record trials. Enter meaningful symbols that actually represent ideas. Enter reasoning. Imagine the difference!
Such an investigation would also generate an authentic need for a mark that communicates the antithesis of balance, imbalance (=). The imbalance symbol is not traditionally taught in the early years, but I have yet to discover a good reason for that. If taught as an idea rather than an isolated symbol, it makes perfect sense to introduce it along with the = symbol. Think about it… your back must have a front; to know ‘up’ you have to understand ‘down’; for the inside of a cup to be useful there has to be an outside. Reason gives us the logical sequel — to know balance you must understand imbalance, AND you need to have a way to record that distinction.
And so it goes for any symbol; for real understanding to blossom, learners of all ages must engage in rich investigations that allow personal discovery of ideas and they must come to appreciate each symbol (mark) as a tool that allows them to efficiently communicate a specific mathematical idea. We as teachers have the ability to influence that by teaching ideas instead of digit dances, then using marks (symbols) as a way to record our thinking. In other words, we are in a position to change the debilitating perception of mathematics as a collection of marks to be manipulated. But only if we are willing to believe that the mark is not the math.
Every single one of us, including you, has the right to understand the ideas behind the marks and reap the rewards and the excitement of that understanding.
Three or 300 or 33, we need to very seriously consider this important question: “Are we teachers of marks or teachers of math?”
Next issue we’ll discuss Full Contact Math.