The problem with Number Sense -- By Axelle Faughn

Did you read in the recent News? A group of American scientists created a fluid with negative mass... now that would be one beautiful selfie to illustrate the notion of negative numbers, wouldn't it?  The fact is, Number Sense is one area of our work with selfies where participants have a truly hard time finding concrete representations of the seemingly simple concepts at stake... why is that?
In a previous post I listed the categories we use during workshops for Number Sense treasure hunts, here is the list again (with a few extras):

Treasure Hunt Category:  Number Sense

1. Illustrate types of numbers (whole numbers, integers, rational, irrational, real)
2. Illustrate place value or base 10 notation
3. Illustrate the Concept of number operations
4. Illustrate Properties of number operations
5. Illustrate the concepts of measurements and units
6. Illustrate geometric thinking
7. Illustrate a number line
8. Proportional reasoning
9. Definition of Fractions
10. Illustrate concept of fraction operations
11. Decimal
12. Percents
13. Prime Numbers
14. Prime factorization
15. Least Common Multiple & Greatest Common Factor

Typically teams will request the Number Sense list thinking they won't have any trouble with the mathematical concepts, and hoping this will make the task of finding selfies easier. However it is often the case that after a few minutes of reflection on how one might represent these numerical concepts, groups end up switching to a different list (such as geometry or functions). Let us remember that counting is one of the oldest human mathematical activity, along with playing games, organizing objects, and representing shapes. We are therefore faced with a paradox: why does it seem so difficult to find a visual representation for ideas that seem so "basic"? Is it that workshop participants are so removed from the concrete meanings behind numbers? Have the daily abstract manipulations of numbers severed the link to tangible objects and situations in the world around us?

Picture 1: Decimals and Money

Nevertheless, I do have some remarkable Number Sense Selfies that I would like to share with you today so we can all realize that Number Sense is indeed all around us if we stop to pay attention. In a previous post I commented on number line representations (see Number Line). These are fairly common and usually the first ones to be photographed by participating teams.

Moving on, a typical school representation for decimal numbers is to use money as in the vending machine example collected in North Carolina during a Middle School teacher workshop (Picture 1).

Picture 2: Weight machine place value

Similarly, the weight machine (Picture 2), also collected during the same teacher workshop, can illustrate skip counting (multiplication),  as well as provide a visual representation for place value. As we add more weight, higher digit face values, and eventually a greater number of digits, are needed.




Moving to two sets of pictures collected in South Africa, let us consider fraction representations.

Picture 3: Wheel fractions

The wheels in Picture 3 illustrate the definition of fractions, as the students in this secondary methods class noted: "Fractions are numbers which indicate that one number is being divided by another". The partitioning of the wheels on two different cars offers the visual for it. Taking it a step further, one can notice that although this image can provide a geometric representation of fractions (parts of a whole area), students referred back to a numerical definition using number operation in their description, possibly losing some of connections that could be established.

More explicitly, and using discrete representations, the following two examples hinge on the connections between fractions, ratio, and probability. 

Picture 4: Fraction switches

Assuming the five switches in Picture 4 represent the whole, the students explained that this situation offered a good illustration for the "fraction 1 out of 5". Using this image one could imagine playing around with changing the value of the whole and expressing other kinds of fractions, including non-proper ones. 

Picture 5: Mailbox fractions and multiplication

The Math Club facilitators who found the mailbox selfie (Picture 5) offered two explanations for why it was a good example of Number Sense. First they noticed that it showed "Definition of Fractions, as we can assume that a particular number of the post boxes are full out of the total of 75", therefore looking at the chance that something happens, meanwhile getting to the core of probability. Then to calculate the number of possible events during the whole group discussion, they realized that the array of mailboxes also provided a good illustration for the area model of multiplication (5x15).

Picture 6: Window operations

And since we are now looking at illustrating number operations, let us finish with one more array which offers considerations a bit more rich and complex than the mailbox example. The set of windows in Picture 6 was put forward as an addition and multiplication of whole numbers selfies with the caption: "Lower windows ADDITION 18 plus 16 equals 34, upper windows MULTIPLICATION 15 times 3 equals 45". One could argue that it could therefore also serve as a subtraction and division selfie, these being the corresponding inverse operations, and requiring only a change of perspective. Further discussions on this particular selfie actually opened up a window (!) on many other operations, including their properties (associativity, commutativity, and even distributivity), that could be visualized and turned into a game among participants of guessing how one might represent different multiples of two, three, five, six, eight and fifteen.

          While this post helps us be more mindful of the wealth of number representations that surround us, comparing the examples above to the categories listed shows that we are still missing a lot of concepts that have yet to be illustrated.  Here is where I pitch a call to you reader: please if you feel inclined to do so, submit your own number sense selfie in the comment section, and we can open up a new discussion.