Same but Different!
“How is it that so many minds are incapable of understanding mathematics? Is there not something paradoxical about this? Here is a science which appeals only to the fundamental principals of logic ……………. And yet there are people who find it obscure and they are the majority.” Henri Poincare 1914
As a mathematics educator, this quote torments me. It runs around my mind demanding to know the reason that we seem to be no better off in 2012 than in 1914! I have some theories, and I’ll share one of them with you.
I would argue that the field of mathematics is not as black and white is it appears. There is an underlying assumption that in mathematics, if we just understand and learn a procedure, everything will fall into place nicely. Black is black and white is white. 2 + 2 will always equal 4. We move students through the educational system who become adults who fear and despise mathematics, because secretly they wonder why things aren’t always simple and easy.
But, here’s the truth that those that love and appreciate mathematics know: The field of Mathematics contains many shades of gray! Things that appear to be the same can also be different.
Let’s consider 4. True, 2 + 2 = 4, but we can also say 4 = 2 + 2. This is the same, but different. And, 4 can be thought of 1, 2, 3, 4 objects or a collective group of 4 objects considered all at the same time. 4 is also 3 + 1 or 1 + 3 or 1 + 1 + 1 + 1. And it goes on ………… so, the idea of 4-ness is quite complex. This is true of each and every number.
When we move out of the primary grades and begin to explore fractions, we encounter more shades of gray. Equivalence. We tell students that 2/4 = ½. They are equivalent. They are the same ............ but do we tell students that they are also different? If we create a visual image of the quantity represented as 2/4, we would show 2 of 4 equal shares. If we create a similar visual for a quantity represented as ½, we would show 1of 2 equal shares. If they are the same, then how can this be? We lose many students when we don't explicitly discuss such shades of gray that they are confronted with.
Students can have a very difficult time with the notion of same but different. They see the world in black and white. The ability to deal with this type of gray scale thinking is part of the work of the frontal lobe of our brain called executive functions. This part of our brain continues to develop into our twenties!
So, what do we do? We celebrate the fact that mathematics is NOT actually black and white. We encourage children to compose and decompose numbers in many different ways – such as in the example of 4. We pull apart numbers using properties based on place value in various ways, identifying 125 as 100 + 20 + 5 but also as 12 tens and 5 ones or even as 11 tens and 15 ones. We also avoid saying things like these are the same, without pointing out that they are also different. We think carefully about what it really means to be equal or equivalent. We explicitly talk about how things are the SAME, and how things are DIFFERENT.
We are closing in upon a century since Poincare made his statement about mathematics. Let’s work on changing things, one student at a time!
Happy Spring,
Sue Looney , president, LMC
PARCC Update:
PARCC has shared that their assessment system is going to be a 4-part system. If adopted, 2 of the parts will be required and 2 will be optional. The required parts will be a Performance-Based Assessment (PBA) that will focus on application of knowledge, and an End-of-Year Assessment (EOY) which will focus on math items that can be scored by a machine. The optional portions include a Diagnositc Assessment and a Mid-Year assessment. These are designed to be more formative in nature than the two mentioned above. For more information, visit PARCC at www.PARCConline.org