But there is a qualifier to the left. It’s “apples”. Apples are > X means apples are greater than X. Apples are < X means that apples are less than X. If “X” is “1 lb”, then “apples are > 1 lb” means that the apples are greater than 1 lb.
People also use it without any numbers. Like, chocolate > vanilla. Or the non-mathematical, but rather, political, 44>45.* It’s an expression of comparison, and you need to know the difference between the symbols to correctly read the sentence.
*I’m not intending to inject politics into this. I literally googled for examples, saw that, and thought it was clever, because of course, it is false if you read it mathematically.
I remember that. There were some who thought there was no inherent “bigness” on the wide side and that the whole thing was completely arbitrary and needed to be learned by rote.
I feel like it’s quite intuitive but it’s been a long time since I learned it so maybe it wasn’t as simple as I think now.
I was a pretty precocious kid math-wise. Through about calculus I was mostly asleep as the poor teacher struggled to ram the trivial lesson du jour into the rest of the class.
But at first these two symbols and the idea behind them totally buffaloed me. I can remember to this day my angry frustrated outburst that really floored the teacher:
Little LSLKid can’t be having a tantrum over this? Can he? Wow, he is! Whodathunkit!
Somehow I’d totally missed the idea that “three is less than five” is reflexively equivalent to “five is greater than three”. So I knew there had to be 4 symbols since the bigger one could be on either side, and the smaller one could be on either side. So the glib explanation repeated over and over that the big end of the symbol went on the big number side was useless. It was an answer to only half the possibilities and nobody but me could see it. Waah!!!
Why yes, in my younger days I had always struggled with the idea that everybody else is out of step, not me. Why do you ask? 
So, you thought there could be statements where the big number was on the left, and the small number was also on the left? Did anyone ever ask you for examples of such a statement, and what they would look like?
(Sorry, I’m not trying to single you out. But understanding student misconceptions is a big part of being a teacher, and understanding yours (years later) might help in understanding some other student.)
I think it’s going to be one of those weird tricks of perception where something familiar seems very obvious, but to people who didn’t learn it, it’s either nearly invisible, or visible but inscrutable. Like for example when I look at Simplified Chinese (or, for that matter any unfamiliar script such as Hebrew or Arabic), to me, a lot of the characters look very similar to one another. Of course I know they must be distinct, but my brain has never received the conditioning to perceive them properly.
Glad to (try to) help. I wouldn’t have told the story if I felt it was some shameful dark secret. It’s odd the things that stick with you.
As best I can recall I was probably having several competing and mutually incompatible conceptions at once, kind of scintillating back and forth. So I didn’t have a stable mental picture I could have explained as you suggest. Which BTW would have been a very skillful way to dispel my confusion if it had come to the teacher’s attention before I wigged out. (And no, I wasn’t one of those emotional problem kids who routinely wigged out at stuff. This was memorable precisely because it was so out of my usual character.)
I just had swirling chaos ruling within. Which might have been a mix of some of these idea. This occurred before I first encountered computer programming, so the comparisons below are hindsight.
- Teacher said 'It’s an arrow pointing towards the small side". My internal counter. “OK, but what if the small side is the other side?”
- In computer programming we recognize the distinction between “=” as assignment and e.g. “==” as a pure comparison, and “==” as an assertion. At that stage of sorta-algebra I was scintillating between thinking of it as a statement of fact “three is in fact less than five” and as an assertion that might fail if the numbers were put in the wrong way round. Fact? Assertion possibly true? Fact? Assertion possibly false? Does not Compute!
- Continuing that idea, in a programming language an assignment statement like “var1 = var2” is at first glance ambiguous; which way does causality run? Is var2 copied to var1 or vice versa? Yes, any practical computer language has a convention for this, but it’s conventional precisely because it’s arbitrary. Math equality is reflexive (A=B implies B=A). I may have gotten stuck trying to be reflexive with the > and < operators which aren’t individually reflexive but are collectively so.
- As I said originally, some (mistaken) concept that there were 4 possibilities but only 2 symbols so TILT!
- I may also have been half-paying attention before that just long enough that teacher had covered some preliminary concept I missed. With the result that when I tuned back in I was plopped unannounced far enough beyond my understanding that I couldn’t find my footing and was lost and flailing in the big blue sea.
When all was said and done I had my 2 minutes of tantrum, teacher said “just let it go and we’ll talk later”, and by the end of class I had cleared the chaos and it was easy.
Here’s another oddity. When I wrote my post above I consciously chose to spell out the numbers and the operators to move the problem more into the conceptual comprehension = semantic domain rather than in the shorthand symbolic = syntactic domain. And in fact what I typed and posted was:
…Somehow I’d totally missed the idea that “three is greater than five” is reflexively equivalent to “five is less than three”.
Yup; I’d typed those relationships backwards. It even passed my proofreading before posting. Only after posting did I see what I’d done & edit it to turn the “greater” & “less” around.
I have no language dyslexia at all. I’m not aware of any number-oriented dyslexia-equivalent, whatever that’s called. But here and now I fell into a similar hole for at least a few minutes.
Human cognition is a very weird miracle.
Thanks, LSLGuy. Now I’ll share my cognition problem, to demonstrate how that which is so intuitive to one person might not be so to another.
I was around 6 or 7 years old, and all the kids in my class were given a routine eye exam. I looked into the machine, and he asked me, “What do you see?”
I answered, “The letter E.”
“Is it pointing right or left?”
At this, I was stumped. It was a totally normal, bland, sans serif, upper-case “E”. If there were arrows on it somewhere, answering the question would have been simple, but alas, there are no arrows on an E. All it has is one vertical line and three horizontals, and it was my job to figure out what “pointing” meant in that context.
My subconscious gnawed at me, telling me that the correct answer is “pointing to the right”. But to my conscious brain, that made no sense at all. To that 7-year-old comic book reader (and, actually, this convention is followed in many of the visual arts) motion is indicated by lines extending toward where the object used to be. If an object moves from right to left, the artist will indicate this by drawing lines from the object to the right. In other words, if someone would draw a letter “I” moving to the left, the result would look very similar to an “E”.
So I was stuck. My memory is a little fuzzy on what happened next. I may have been unable to explain my confusion, or it could be that I did explain myself but he was unable to answer, and despite my young age, I do remember being aware that any prompting he might offer could taint the results of the test. It was quite frustrating, and the test was halted at that point.
I have no idea whether, afterwards, I asked the other kids which way an E points, or if I was too embarrassed to ask.
Concepts like these are similar to seeing the arrow in the FedEx logo: invisible until it’s obvious, and forever after not unseeable.
For kids this stuff is novel and hard whereas for adults, most won’t even remember a time this was even a sensible question. Had you asked the adult to explain what they mean by “pointing” they may well have struggled to understand the question. Unless that’s a standard question kids ask when first exposed to these tests that they’d come to expect.
I had another thought that may be closer to what was going on in my confused head. Or at least is another aspect of it.
At that point in our math training we’d dealt with addition and multiplication that are both commutative. And knowing the fact of commutativity and appreciating at least some of its consequences was part of that.
Of course we’d been exposed to subtraction and division which are not. But I don’t think the point of their non-commutativity had been really called out as such.
And, abusing the terminology a bit, as I noted above, equality is “commutative” too.
So I was trying to force-fit < and > into a mental model that started with an assumption / demand that they be commutative, since after all they’re cousins of equality which is commutative. The < and > are the first operators of that class to be non-commutative.
It’s almost what I wrote above, plus the similarity to other elementary operators on the integers.
Ah, I see, so it was just a momentary confusion that cleared itself up by the end of the day (just, not before you’d tantrumed over it). That’s a different sort of beast from a misconception that gets itself fixed into one’s head, and not usually a cause of significant long-term difficulty.
Well, strictly speaking, “Five is less than three” is equivalent to “Three is greater than five”. They’re just both equivalently false statements.
Not the only one. When I was in a share house, I learned that optometry students have another set of symbols for use with children who don’t do right / left or ABC. “Which way is the duck walking? – Towards me or away from me?”
In elementary algebra (meaning, as taught in high school), < and > and = aren’t even taught as “operators” (that is, symbols representing operations that take operands and produce a result). The idea that these are operators first came to my attention in the context of computer programming.
I also had a rather delayed epiphany about the concept of “relations”. I had understood < and > and = to represent relations between numbers. But I had also learned (way back in Algebra 1) that a “relation” is simply a set of ordered pairs. Thus, the idea of a “relation” had two distinctly different meanings.
It was only much later that I came to realize that < and > and = do in fact describe sets of ordered pairs and thus are “relations” in that sense too.