# Children's counting problems: Is this question phrased correctly?

Look at the following example:

Which picture has four apples?

A

B

C

D

B is the expected answer but should not the correct answer be BCD? Technically if a set has exactly $$m$$ elements, then it has $$k$$ elements if $$k\leq m$$. This is also how we talk in everyday language:

"Do you have three dollars?" "Yes."

The second speaker is not indicating he has exactly three dollars. He simply indicates that he has at least three dollars.

So I am wondering if we are teaching children correct logic here. Shouldn't the original question be rephrased as "which picture has exactly four apples"?

• I'm not a native speaker and might very well be wrong, but the question "Which picture has four apples?" does sound slightly off to me. Is this asked exactly like this somewhere, or is it a translation?
– quid
Apr 25 at 20:02
• The only thing I found confusing was that it was hard to tell which letter was visually grouped with which cluster of apples.
– user507
Apr 25 at 21:54
• This is like "Who has a car with one wheel?" vs. "Who has a car with only one wheel?" While I don't have any problem with your apple question in the way it's written, if you're in doubt, using the word "only" should help. Apr 25 at 22:33
• I see only one picture, with 17 apples… Apr 26 at 9:32
• Instead of "Which picture has four apples?," say "Which is a picture of four apples?" Apr 26 at 11:01

I don't think there's anything wrong with the wording; it's clear what is being asked. Your example with the three dollars is also not always the way we speak in everyday language. If you ask someone with three children if they have two children, they're unlikely to say "yes" and leave it at that.

Getting more silly, a bicycle isn't a unicycle despite the fact that bicycles have at least one wheel. The root words for unicycle are "one" and "wheel," but a unicycle is defined to have exactly one wheel even though no root word for "exactly" appears. Exactly one wheel is just the more natural interpretation, just as exactly four apples is the more natural interpretation.

• In case this answer seems snarky or dismissive, I've seen much worse examples of bad wording. One teacher I know repeatedly says that "fractions have to be equal," and I recently saw a worksheet that claimed that ten dimes equals 100. Not 100 cents, just 100. Compared to stuff like that, the apple question is wonderful! Apr 25 at 21:32
• The unicycle example is not relevant, because unicycle means exactly one wheel. But the kids example is very helpful. Apr 26 at 14:29
• I think the new, clarified unicycle example is much better. Apr 27 at 18:59
• Agreed, but I'd point out that on exams (and maybe any work where a student is being graded) it's important to avoid technicalities and close loopholes wherever possible Apr 27 at 21:30
• @SueVanHattum My dictionary says that a unicycle is "a vehicle with one wheel", which seems very relevant to me. Apr 28 at 18:28

When we describe counts in natural language, there's almost always an implicit "exactly" when phrasing like this. We use phrases like "at least 4" when we want a more general description. Most children who have reached a development level where this quiz would be reasonable will probably already have learned this.

In fact, this is why there's a common joke:

Q: How many months have 28 days?
A: Just one, February.
Q: No, they all do.

The punch line works because we normally don't treat "have 28 days" as meaning "have 28 days, and possibly others", but when someone points it out we can see the potential ambiguity. But some people probably still won't get the joke, because the implicit "exactly" is so pervasive.

Perhaps "shows" instead of "has". If you asked me to show you 4 apples, I can't think of a logical argument in favor of me grabbing 5 apples and smiling smugly.

Nearly every test like this includes instructions to choose the "best answer" to cover exactly this scenario. This looks like it's part of a test of basic counting skills, and in that context, the best answer is B. While one could make an argument for either C or D, I can't imagine an argument for either of those being the best answer when B is present.

• Well... If you were told "Pick any basket with 4 apples, and you get all of them for free", the "best" basket would be basket D. I guess "best" is all about context. In such a simple test, presumably aimed at very young children, I don't see the point not to be fully accurate, you don't want to "trick" them unwillingly. On the other hand, if I was given this exercise in a job interview, I would certainly look for any indication of what the expected answer may be, and probably even elaborate on possible answers and/or how the question could be less ambiguous. Apr 27 at 23:50
• Being "fully accurate" with young children may not be as simple as just adding the word "exactly" though, because then you will need to explain what "exactly" means. And to explain what "exactly" means, you will probably need to explain what "at least" means. And now you've entangled teaching comparisons with teaching counting whole numbers. This may work well for some children but confuse others; there may not be a good one-size-fits-all approach. That's what led me to mention the "best answer" instruction -- it's meant to help find the correct answer when there are multiple candidate answers Apr 28 at 13:59

I showed this question to my three-year old son. His response - because he counted the apples one by one in each picture, passing "4" each time - was B, C and D. Hence, we need to take into account how children arrive at their conclusion, since they do not apply formal logic. The thought process is very different from the abstract approach a programmer might take when deciding whether his boolean expression is correctly formulated. I have worked with kids taking ability score tests, and sometimes even the very strictly standardised question sets with pictures, where the question itself was worded correctly, elicit unusual answers. But many children, if prompted, give a very compelling answer. "Odd one out" questions are especially problematic, and the question here is a variation of that. It does not matter if the formal logic is the correct one, linguistically or mathematically, if children can not apply the rules in a conscious way. But of course we want a specific answer - at least for grading - and not find out how the child's analytical capabilities work exactly. To limit the risk of such unusual reasoning, best remove ambiguities, albeit without introducing logical loops that frame other questions in a more complicated way.

"Which picture shows exactly 4 apples" is the most accurate phrasing, and it avoids having to put "at least" into other questions to keep the logic consistent (see the "how many children do you have" example given in another reply).

• This is perhaps preceding the area where someone will use the literal interpretation which suits what they want to do rather than what they know the asker meant, if they intend to be contrary, or otherwise explore the boundaries of the question without intending confrontation. Older people may be averse to the reactions to the exploration aspect. Apr 27 at 17:09
• This answer garnered my upvote for giving the result of an actual experiment and for suggesting that the age of the subject of the experiment could be a factor. Apr 27 at 17:13
• It's great that you asked your son. His answer was surprising to me (probably because I rarely worked with children less than 6). I am curious. If you repeat the experiment and said you want only 1 answer , what would he pick? Apr 27 at 18:39
• @AmyB He was adamant that it was all three options and then just chose one randomly when I asked again. I would expect an older child to be more decisive and have a better mental concept of amounts ;) I simply wished to illustrate that one does not need to misunderstand the question itself, on purpose or not, or have any grasp of linguistic traps to arrive at a different (unexpected) result based solely on their approach to solving the problem. Apr 28 at 9:16
• @Nardya Very interesting. Thank you for trying it and sharing. Perhaps - add this comment to your answer with an edit. Apr 29 at 6:45

Suppose the same maths test said "John has 2 apples and Lucy has 3 apples. How many apples do they have in total?"

By your logic, the child could say "1" and be entirely correct. If you have 5 apples and someone asks you "do you have an apple?", the answer of course is "yes". So clearly your proposal fails, because your logic is not internally consistent.

It is important to remember that what we say casually in everyday use is NOT mathematically sound, nor even necessarily logically or factually sound. Technically, if we ask "which set has 4 members?" (or perhaps more accurately, "which set has an ordinal of 4?"), then a set which has 5 members does NOT meet the question. For an even better example of how everyday usage does not match mathematical usage, look up the differences between Boolean AND/OR and how we might actually use "and" and "or" in conversation.

• "By your logic, the child could say "1" and be entirely correct. ". This seems false. The question was "How many apples do they have in total?" and not "Do they together have 1 apple?". Following OP's way of thinking, I assume that OP would say that the answer is "At least 5 apples". Apr 26 at 12:55
• To push this "logic" even further, the correct answer could be "they have no apples in total, because they refuse to share them with each other". Apr 26 at 16:40

I think a lot of it has to do with the age of the child, and what the goal of the question is. If this is for children just learning their numbers, like say 4 or 5 years old, then I think B is the correct answer as thy are not being asked to stretch their logical capabilities, but to simply recognize and call out the difference between 4 of something, or 5 of something, etc.

However if this was for older children who already have comfort with basic numbers concepts and the goal was to demonstrate more creative thinking, then I believe B, C, and D would be the correct answer.

This was not the question:

"Which pictures have four apples?"

This test is a multiple choice question that does not appear to have other than the 4 unique choices: a, b, c, d.

There is no all of the above or b-d choice which can be selected.

If the test was a fill in the blank write in, there is a bit of ambiquity.

More importantly though, examining the response with the student would be far better than pushing a black and white mindset.

• "This test is a multiple choice question" If you read the post, you will see that there is no indication that it is a multiple choice question. It is possible that it is a multiple response question, where more than one answer may be marked as correct. Apr 28 at 0:51
• @JoelReyesNoche multiple choice vs multiple response --> genius
– BCLC
Oct 10 at 18:44

It is wrong exactly for the reasons you have pointed out. There shouldn't be an issue with including the word 'exactly' as in 'exactly 4 apples'.