This video and its comments got me thinking about how difficult it really is to read an analog clock:
There are multiple comments on TikTok sneering at how inane and unintelligent modern children must be to need more than ten minutes, let alone more than two days, to learn how to read a clock.
So let’s talk about that. Here’s a fairly standard analog wall clock, available on Amazon:
Here’s a learning clock from Oyster & Pop, also available on Amazon:
Here’s how a parent might see the idea of “Teaching a child to read an analog clock”: This is a basic life skill, fairly isolated from other life skills. On that level, given the ready presence of digital clocks in our environment, it might seem to some adults like educational time best spent on other topics.
The meme above deserves a separate essay reflecting the tension between teachers and students, but the relevant point is: If you don’t know how to read an analog clock, use a digital one.
However, here is how a mathematics teacher might see the idea of “Teaching a child to read an analog clock”: Analog clocks capture several fundamental ideas in mathematics, and hence teaching a student how to read a clock symbiotically teaches those ideas.
Which is to say: Reading a clock is much easier if we understand those basic ideas, and learning to read a clock teaches and reinforces those ideas.
Let’s consider the basic skills needed to read an analog clock. If you’re already adept, you might wonder what’s so difficult, but that’s true of so many complex things that we do every day: If you understand how to do it, you might well take it for granted.
An analog clock is a circular display generally consisting of a ring of numbers from 1 to 12, sometimes also a ring of 60 dots, and two or three rotating sticks. Some clocks lack some of these details.
On other objects, we might call these rotating sticks “arms”, but on clocks specifically, they’re called “hands”. The longer the stick, the shorter the time period it represents. If there’s a third one, it’s called the second one.
Also, as the Oyster & Pop clock shows, there is time-specific language to learn: “Quarter past”, “Half past”, “Quarter to”, “O’clock”.
So, just from a linguistic aspect, it’s a fairly unusual object. Let’s talk about the math.
An analog clock is often the first time students are exposed to numbers laid in a sequence that’s not a line. A number line just goes on forever; the one on the wall of an elementary school classroom goes on as long as the production company and the teacher feel like it. In contrast, the number ring around the clock is restricted.
This is the first time that many students are exposed (albeit not by name) to modular arithmetic, a somewhat advanced topic that is not discussed again (usually without that language) until trigonometry.
Unless you’re in the military, it’s never 13 o’clock, and even if you are, it’s never 25 o’clock. Time starts over perpetually, every twelve hours, every sixty minutes, every sixty seconds.
And why 12? Why 60? And why just half a day, not a whole day?
Hence, time is also an opportunity to explore number base systems. We strongly prefer base ten because we have ten fingers on our hands (while a clock is another place we use “hand” and it doesn’t involve base ten).
There are sixty seconds in a minute and sixty minutes in an hour because the Babylonians preferred base sixty. Sixty has the advantage of being a multiple of 2, 3, 4, 5, and 6, so working with fractions is more straightforward than in base ten, but it has the clear disadvantage of requiring far more distinct symbols.
Attempts to use a metric, base ten, alternative for time have largely failed, as discussed in Wikipedia.
The 12 hours may have been a parallel to the 12 months. Most years have 12 full moons (some have 13). Initially dividing only the sunlight hours makes sense as well for cultures that have low to little lighting at night. Also, the Egyptians apparently divided the day into ten working hours plus a twilight hour at either end, and then divided the night into twelve hours based on star movements.
There’s some science discussion to be had there, and cultural discussion as well. The cultural discussion includes conversation about how mathematical choices are based on what a culture values, both practically and aesthetically.
None of this is needed to be able to read a clock, but it’s useful to understand the decisions made behind the clock. Without the context, it just feels like a bunch of arbitrary decisions, and arbitrary decisions are harder to learn.
Let’s revisit the basic clock from earlier:
The clock says that it’s about 10:11. If you’re fluent at reading clocks, that might not be at all confusing, but let’s look at it from a newcomer’s perspective.
The shorter stick is pointing at the 10, so it makes sense that the time would likewise involve 10. The longer stick, though, is pointing at the 2. Here’s what you have to do:
- Look at the numbers the shorter stick is between. Take the floor function, that is, decide what number it has most recently passed. This is the hour.
- Look at the number the longer (or middle) stick is nearest to, and multiply that by five. That is the minute. Unlike the hour, this is generally rounded. If there are small dots, you can get more accurate by adding or subtracting the extra dots.
- If there’s a third stick (usually fairly thin, long, and a different color), you can read the number of seconds. In this case, it’s about 10:10:33.
We don’t normally teach elementary school children what a “floor function” is. Instead, we give them directions like “decide what number it has most recently passed”, which requires mathematical consideration of extrapolation and conjectures.
Another point on the mathematics: Why is ten minutes past noon not 0:10? If we think of it solely in terms of “decide what number it has most recently passed”, that makes sense, but if we think of it in terms of a floor function, it doesn’t. If we were strictly adhering to modular mathematics, in fact, there ought to not be 12:00 at all: The minute after 11:59 ought to be 0:00.
This mismatch is an opportunity, later, when discussing modular arithmetic to distinguish zero-based and one-based indexing and the different mathematical choices people make in different mathematical contexts.
This also applies to the mismatch between analog clocks and standard angles, which are measured starting at “3:00” and go counterclockwise. There are reasons for that which I can discuss in a separate item.
To summarize, then: Reading an analog clock is not a triviality, and teaching it as an isolated skill undermines the cultural and mathematical significance. Once you fully know a complex skill, it is often difficult to realize how complex that skill is.
Post-script: So why DO we call the third hand the second hand? The terms “minute” and “second” come from the Medieval Latin “pars minuta prima” (first small part) and “pars minuta secunda” (second small part), for dividing a circle first into 60 parts and then into 3600.