2 + 2 DOES NOT ALWAYS equal to 4

I think I unleashed my inner Young Sheldon recently. You know that awkward “intellectual flexing” of what you know even though there are a million other things you don’t know! I do not know a lot of things, but what I feel like I know a little about is mathematics education. I promise I had the best of intentions. My goal was to educate.

The other day I was sitting at lunch listening to a few of my “non-mathy” friends talk about their preference towards the ways they learned math over some of the “new” ways their children were learning (That is another story!) I was keeping to my quiet self until the most vocal of the bunch said something to the effect that “You can’t change math! Two plus two is ALWAYS four!”

At this moment I muttered a line that will more than likely limit my future invitations to aforementioned luncheons, “Did you know that two plus two DOES NOT ALWAYS equal four?” My Young Sheldon mind kicked in at this point. I proceeded to share the following explanation. I am sorry! I just could not help myself! That is what they get for inviting a math person to lunch! They were going to ask me to calculate the tip anyway so I might as well go all in on the explanation!

2 + 2 is not ALWAYS equal to 4, but it is SOMETIMES. For example, 2 dimes and 2 dimes is 4 dimes, 2 inches and 2 inches is 4 inches, and 2 fifths and 2 fifths is 4 fifths. But, what about 2 nickels and 2 dimes? Is it 4 of something? What about 2 feet and 2 inches? Or, 2 thirds and 2 fifths?

2 + 2 equals 4 when we are combining quantities of the same size of unit. Addition is a combining action that requires both quantities to be the same size of unit. The implied unit in 2 + 2 = 4 is 2 ones. This means the statement “two plus two is four” is not true if we are talking about quantities of different sizes of units. For example, 2 nickels and 2 dimes is not 4 of either unit. However, we can use an equal exchange (1 dime = 2 nickels) to show that while 2 nickels and 2 dimes cannot be combined 2 nickels and 4 nickels can be combined (2 nickels + 4 nickels = 6 nickels) as can 1 dime and 2 dimes (1 dime + 2 dimes = 3 dimes).

Likewise, 2 feet and 2 inches cannot be combined, but again utiilizing an equal exchange (1 feet = 12 inches) we can combine 24 inches and 2 inches (24 inches + 2 inches = 26 inches). And, 2 thirds and 2 fifiths cannot be combined but again using two equal exchanges (2 thirds = 8 fifteenths; 2 fifths = 6 fifteenths) we can combine 8 fifteenths and 6 fifteenths (8 fifteenths + 6 fifteenths = 14 fifteenths).

I define numbers as a tethering of quantities of sizes of units so 2 of some unit, whether those units are measurement (e.g. feet) or numerical (e.g. fifths) combined with 2 of that same unit will be 4 of that unit. Otherwise, we cannot combine 2 of different sizes of units without performing an equal exchange. In this case, as the examples show, it is no longer a quantity of 4, but a quantity dependent on the unit. So, 2 + 2 = 4 sometimes, but not always!