Nothing in math is "easy" and that's good news
Last updated
Last updated
My aim in this post is not to discourage you, but to do the opposite - encourage you by helping you realize that your difficulties with mathematics were never your fault or weakness. This subject just happens to be that rich. If you struggled with something, it was because the curriculum was too fast and natural questions at foundational stages were not given sufficient attention.
Suppose you ask the janitor to work from 8am to 11am, and clean floor numbers 8 to 11. Simple - it's one floor per hour, right?
Nope! There are 4 floors to clean (8, 9, 10 and 11) but only 3 hours to work (8 to 9, 9 to 10, and 10 to 11).
Whoa! we count floors and hours differently? You bet. But somehow, if you had said "Clean floors 8 to 11 on April 8th to 11th", everything would be okay!
It turns out natural numbers have (at least) 2 interpretations: A count of number of items and a span between numbers.
Realizing there are two types of counting is a big mental shift. We have two possible choices when "counting" from a to b:
Distance from a to b: b - a [regular subtraction]
Number of items from a to b: b - a + 1 (span touches extra element)
This is especially difficult for measurement of time. Check a few examples:
Working Days: I worked April 8th to April 11th. How many days did I work? Well, that's a span of 3 (11-8), but we "touch" 4 days: April 8, 9, 10, 11. So I worked 4 days.
Hours: Hours are like spans. Working from 8 to 11 means you are covering the spans 8-9, 9-10, and 10-11. 8 to 11 means a "timespan" of 3 hours.
Seconds: I start a race, and the start time at 12:01:08 (12 hours, 1 minute, 8 seconds). It ends at 12:01:11. How long did it go? 11 - 8 = 3 seconds. (Short race)
Interesting, eh? Some units of time are measured with spans (hours, seconds) and others are items to be counted (days).
The measuring type depends on the context. We see shorter units of time as "instants" and want the duration between those instants, not the "number" of instants we touched.
We see days as a large fuzzy blob covering a time period (9am-5am) -- and we want to know how many blobs we covered. Saying you worked April 8 to April 9th implies you worked a timespan of 9am-5pm on two days.
The counting type depends on the context - but at least you know why we count them differently.
This is also known as the fencepost error:
Now if this fencepost was not a line with two ends but a circle with no ends instead, how many posts would n sections require?
So, Counting is not easy. In fact, this directly connects to the most frequently seen type of bug in programming, which even has its own wikipedia page: Off-by-one error. The other reason why we make such bugs is the confusion over Zero-based numbering.
Not so fast. "Pause and ponder" should be our mantra in math.
If you have never heard this argument before, it should blow your mind:
Which set is bigger? Natural numbers (0, 1, 2, 3, 4, ... forever) or even numbers (0, 2, 4, 6, ... forever)?
It seems perfectly reasonable to think that even numbers are completely included in the natural numbers. Because Natural numbers include both Even numbers AND Odd Numbers. The "And" here refers to the mathematical idea of "union of two sets".
So you might think there are more natural numbers than even numbers. But now think this: Is it not true that for every natural number, we can get a unique even number?
In math lingo, this process of mapping to "one and only one" element is called finding a "bijection".
In fact, if you think about it, this is what Counting really is: Finding a bijection.
If there exists a bijection between two sets, their sizes must be equal.
Well our idea of "forever" (which I used above as "...") is not easy or simple as we had thought. It turns out that we must be a bit careful when "counting" infinite sets.
I can't believe I went 25 years before discovering this. How did we manage to screw up the teaching of Counting?? By rushing. By not pausing enough. By not listening enough to students.
If we move on from counting to addition, there is a lot more complexity as addition has 3 interpretations as geometric transformations:
And I still hear from many that they were never taught "addition tables" before "multiplication tables". I posted about this in our community chat (you should join):
So, if we want to be consistent, we must accept that there are as many even numbers as there are natural numbers!!
Nothing in Math is "easy" or "obvious". But that's exactly what makes it so interesting and fun. So, everyone, please be patient, both with yourself and your children/students. Remember the 5 principles of extraordinary math teaching.
