# Semantic difficulty

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 1374404.  Sat Feb 13, 2021 7:28 am That's why I added "if describing a maths puzzle". If you have never encountered any such puzzle, yes, I can see why you'd need clarification as to what was meant.

1374405.  Sat Feb 13, 2021 7:44 am

 Dix wrote: That's why I added "if describing a maths puzzle". If you have never encountered any such puzzle, yes, I can see why you'd need clarification as to what was meant.

Even if it was describing a maths puzzle, I'd have assumed that "three hotels = 12" represented a picture of three hotels, followed by the symbols " = 12". The fact that the English words "three hotels" were used on the left-hand side, rather than any mathematical symbols, would have led me to that conclusion.

Why would someone write "three hotels" if it wasn't actually a picture of three hotels?

 1374406.  Sat Feb 13, 2021 7:48 am Why would anyone write 3 fours are twelve? It is exactly the same thing 3 x 4 =12= 4+4+4 =12 3 x hotel =12 = Hotel + hotel + hotel = 12

 1374407.  Sat Feb 13, 2021 8:25 am I meant as a description of the pictures and symbols in the puzzle, rather than as a mathematical statement. (This is an example of the "use-mention distinction" that I referred to at the top of the thread.) The following three are all different, graphically: (1) a picture of three hotels (2) a picture of a hotel, followed by a "+" sign, followed by a picture of another hotel, followed by a "+" sign, followed by a picture of another hotel (3) the symbols "3 x " followed by a picture of a hotel I would describe (1) as "three hotels", (2) as "hotel + hotel + hotel", and (3) as "3 x hotel". The fact that they may all represent the same numerical value is irrelevant - the actual pictures and symbols are different.

1374409.  Sat Feb 13, 2021 8:39 am

 Numerophile wrote: @Leith: As a matter of interest (going back to the point at which this discussion sprouted off), are they expecting the answer 64 (left to right) or 34 (operator precedence)?

Good question. Having looked up the original site, I find that the given answer is actually wrong by either interpretation! They've gone for 44 having accidentally assigned 20 to the house on the last line. Perhaps not the best teaching resource.

[Edit: ah no, it's one of these devious ones that Dix mentions! They've snuck an extra tree in the last house]
[Edit 2: which I see Dix specifically points out too - sorry, was replying in a hurry from the phone without having caught up on the thread]
[Edit 3: seeing as all the highly numerate adults commenting on the puzzle have been either uncertain of the significance of the variation in the last house glyph or missed it altogether, I think I'm standing by the assertion that this is not a good kids' puzzle, though]

The working indicates they are expecting operator precedence to be applied, though.
The indicated target age range for the puzzle is 7 to 16. Is operator precedence often taught to seven year olds? Sounds a bit early to me.

It would be quite early for algebra, too, in my day, but I can imagine that kids who've grown up with phones might get the idea of abstraction with icons more readily than a child of my pre-digital generation might have done.

Here's the site for reference (doesn't seem to be explicitly commercial at quick glance) :
https://www.teachingideas.co.uk/problem-solving/emoji-maths-puzzles

1374411.  Sat Feb 13, 2021 9:06 am

 Leith wrote: The indicated target age range for the puzzle is 7 to 16. Is operator precedence often taught to seven year olds? Sounds a bit early to me.

The National Curriculum has operator precedence as a statutory Year 6 topic, meaning that it should be taught in the year in which pupils are 10-11.

The distributive law is actually mentioned in the notes to the Year 5 Curriculum, but here it is only expected that pupils will recognise that (eg) 23 x 7 = (20 x 7) + (3 x 7). In any case the notes are not statutory.

The notion of using symbols and letters to stand for numbers is also a statutory Year 6 topic, and pupils are expected to "express missing number problems algebraically".

(I can't link to my immediate source because it was the school Intranet, and you don't have a login for it. But it appears to be a very similar document - possibly the same document, but I haven't checked closely - as this.)

1374416.  Sat Feb 13, 2021 9:53 am

 Leith wrote: Having looked up the original site, I find that the given answer is actually wrong by either interpretation! They've gone for 44 having accidentally assigned 20 to the house on the last line. Perhaps not the best teaching resource. [Edit: ah no, it's one of these devious ones that Dix mentions! They've snuck an extra tree in the last house] [Edit 2: which I see Dix specifically points out too - sorry, was replying in a hurry from the phone without having caught up on the thread] [Edit 3: seeing as all the highly numerate adults commenting on the puzzle have been either uncertain of the significance of the variation in the last house glyph or missed it altogether, I think I'm standing by the assertion that this is not a good kids' puzzle, though]

I must hold my hand up and admit that I hadn't spotted the minor difference in that last line. But in that case it's a very poor puzzle. Even if the picture were clear, there's a lot in it apart from a tree or trees; why should doubling just the tree be expected to double the whole value, as if the remainder of the picture counted for nothing? It's not like Dix's example of a bunch of five bananas rather than four.

My reaction (if I had spotted it) would be either (a) this is an error, and the pictures are meant to be the same, or (b) the problem is undefined, since insufficient information has been given to deduce a value for the first picture in the bottom line.

[Edit: in fact, I see from a larger version that as well as the two trees in the last picture, there are also eight tulips, whereas there were only two in the earlier pictures. So why doesn't it count as 40 rather than 20? I conclude that the problem is unanswerable; knowing that 1 house + 1 tree + 2 tulips = 10 doesn't allow you to deduce any value for 1 house + two trees + 8 tulips.]

 Leith wrote: The working indicates they are expecting operator precedence to be applied, though.

Thank goodness for that!

Last edited by Numerophile on Sat Feb 13, 2021 10:06 am; edited 1 time in total

1374417.  Sat Feb 13, 2021 10:04 am

Brock wrote:
 Dix wrote: That's why I added "if describing a maths puzzle". If you have never encountered any such puzzle, yes, I can see why you'd need clarification as to what was meant.

Even if it was describing a maths puzzle, I'd have assumed that "three hotels = 12" represented a picture of three hotels, followed by the symbols " = 12". The fact that the English words "three hotels" were used on the left-hand side, rather than any mathematical symbols, would have led me to that conclusion.

Why would someone write "three hotels" if it wasn't actually a picture of three hotels?

Because all details aren't always given in an informal description?
Maybe the writer didn't actually remember precisely what the puzzle looked like, maybe the details were thought unimportant because it was just a description of "that kind of maths puzzle that has drawings in it".
Why would anyone waste time on describing all details correctly if it was just meant to give a broad indication?

It's not as if you were asked to solve a particular puzzle based on the description.

1374420.  Sat Feb 13, 2021 10:53 am

 Numerophile wrote: [Edit: in fact, I see from a larger version that as well as the two trees in the last picture, there are also eight tulips, whereas there were only two in the earlier pictures. So why doesn't it count as 40 rather than 20? I conclude that the problem is unanswerable; knowing that 1 house + 1 tree + 2 tulips = 10 doesn't allow you to deduce any value for 1 house + two trees + 8 tulips.]

After looking at the pictures carefully, I note that there are actually four tulips in the first picture; and while there may appear to be only one house in the second picture, there are two chimney-pots sticking up, which I presume is meant to suggest a second house behind the first one. So the logic appears to be:

1 house + 1 tree + 4 tulips = 10
2 houses + 2 trees + 8 tulips = 20

How on earth anyone is supposed to guess that from a cursory inspection of the puzzle, I've no idea.

Furthermore, it casts doubt on the previous contention that the symbols are meant to act as algebraic variables. How is it possible to superimpose one algebraic variable on another one?

1374425.  Sat Feb 13, 2021 11:34 am

Brock wrote:
 Numerophile wrote: [Edit: in fact, I see from a larger version that as well as the two trees in the last picture, there are also eight tulips, whereas there were only two in the earlier pictures. So why doesn't it count as 40 rather than 20? I conclude that the problem is unanswerable; knowing that 1 house + 1 tree + 2 tulips = 10 doesn't allow you to deduce any value for 1 house + two trees + 8 tulips.]

After looking at the pictures carefully, I note that there are actually four tulips in the first picture; and while there may appear to be only one house in the second picture, there are two chimney-pots sticking up, which I presume is meant to suggest a second house behind the first one. So the logic appears to be:

1 house + 1 tree + 4 tulips = 10
2 houses + 2 trees + 8 tulips = 20

How on earth anyone is supposed to guess that from a cursory inspection of the puzzle, I've no idea.

Furthermore, it casts doubt on the previous contention that the symbols are meant to act as algebraic variables. How is it possible to superimpose one algebraic variable on another one?

They are deliberately trying to trick you. Not you specifically, of course, but all the would-be solvers. That generates discussion, and people share the puzzles trying to outsmart their friends. They have evolved from fairly straight problems where the only trap would be going from only addition and subtraction to also having multiplication (and operator precedence) in the last line to having ever more weird little details that "counts" (as it were).

Sometimes they (or other types of puzzles and quizzes) come with captions such as "only 5% of people will get this right".

What they're getting out of doing this isn't clear; maybe it ties on to some kind of advertisement revenue being generated by number of "likes" or the number of times it gets shared.

In any case, they're annoyingly popular. I just ignore them.

1374434.  Sat Feb 13, 2021 1:18 pm

 Brock wrote: After looking at the pictures carefully, I note that there are actually four tulips in the first picture; and while there may appear to be only one house in the second picture, there are two chimney-pots sticking up, which I presume is meant to suggest a second house behind the first one.

You are quite right. I see that it is in fact two copies of the first picture, stacked (almost) on top of each other. So I suppose we must allow their answer.

But how anyone would be able to make that out, if they were trying to do the puzzle on a phone, I don't know.

1374435.  Sat Feb 13, 2021 1:33 pm

Numerophile wrote:
 Brock wrote: After looking at the pictures carefully, I note that there are actually four tulips in the first picture; and while there may appear to be only one house in the second picture, there are two chimney-pots sticking up, which I presume is meant to suggest a second house behind the first one.

You are quite right. I see that it is in fact two copies of the first picture, stacked (almost) on top of each other. So I suppose we must allow their answer.

It was only after examining some of the other puzzles that I realized that two stacked copies of a picture represent double the value of a single copy. (I don't know if more than two stacked copies are allowed.)

So in this system, there are apparently two ways of representing multiplication; by stacking and by using the "x" operator. One is entirely graphical, the other uses a conventional mathematical symbol.

I can't help feeling that it's not a great way of teaching arithmetic or algebra. One would hope that their conventions would be consistent.

 Quote: But how anyone would be able to make that out, if they were trying to do the puzzle on a phone, I don't know.

Most of the other doubled graphics were clearer (e.g. one railway carriage behind another). That one was just confusing.

1374437.  Sat Feb 13, 2021 1:50 pm

Brock wrote:
I can't help feeling that it's not a great way of teaching arithmetic or algebra. One would hope that their conventions would be consistent.

 Quote: But how anyone would be able to make that out, if they were trying to do the puzzle on a phone, I don't know.

Most of the other doubled graphics were clearer (e.g. one railway carriage behind another). That one was just confusing.

I quite agree. That particular example and its ilk are deliberately designed with subtleties to catch out the unwary. That's fine for a mildly diverting internet puzzle (if better rendered than the example I posted), but doesn't seem at all helpful to me as a teaching aid (not that I have any particular expertise in teaching children).

I think the simpler version that Awitt describes, lacking the additional complexity of small variations in the glyphs could have more useful potential for teaching.

 1374440.  Sat Feb 13, 2021 2:59 pm I'd certainly expect a teacher to stick to the "regular" ones. As it is, since the original puzzle description was vague, we simply don't have that information.

1374601.  Mon Feb 15, 2021 10:11 am

 Dix wrote: As it is, since the original puzzle description was vague, we simply don't have that information.

Argh no!!!! We've gone all the way back to the start....

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