An Exercise and Two Problems

This post concerns an exercise and two problems. The exercise is from Alexandr Borovik and Tony Gardiner’s book, The Essence of Mathematics Through Elementary Problems:

Tanya said: “I have three more brothers than sisters”. How many more boys are there in Tanya’s family than girls?

The first problem is the problem that many teachers and authorities would now have with the above exercise, to the extent of refusing to use it or forbidding its use.

The second problem is that the first problem is insane.

61 Replies to “An Exercise and Two Problems”

  1. I guess you’re getting at some PC reason, but I can’t see any. I think brother = boy and sister = girl are still ok to use, right? Or am I missing something?

    1. Does Tanya have parent(s)?

      Does “boy” mean the same in this context as “human male”?

      Is a brother over the age of 18 still a boy, or do we have to say “man”?

      1. I’m not sure if you’re being serious, but I don’t think any of those questions amount to problems with the exercise that would prevent it from being used.

        It doesn’t mix up gender with biology. You’re trying to do that by the second question, but it isn’t part of the exercise. They only use gendered words, so it isn’t problematic in my view.

        For the first and third questions, I don’t think it matters. You can also invent issues by asking: Does family include people living there that aren’t brothers or sisters? What about estranged uncles, long-term exchange students, maybe refugees from a war-torn land?

        Seems pretty pointless to me (and also not a reason to stop us from using the exercise).

        At this point I feel like it is worth asking: Have you (anyone reading) seen an exercise like this one, or hopefully exactly this one, be rejected by an authority, or teachers prevented from using it?

        1. I am assuming that Marty is aware of at least one instance where a question such as this has been declared problematic.

          I don’t see the problem. I’m trying to imagine what problems anyone could have with this question…

          …and failing.

    2. I cannot promise you that many authorities would refuse use of this question, although I strongly suspect that this is true. I can promise you that at least one authority would.

  2. Tanya has at least 2 sisters.

    [because – sisters: more than one sister].

    She has 5 brothers [+3].

    That’s the floor of the exercise. [the fewest siblings of Tanya].

    Or if we had to multiply and Tanya had 3 sisters and 6 brothers…
    [or 6 sisters and 9 brothers]

    1. 1. How could you do that?

      2. Why the hell should anybody be obligated to try?

      The issue isn’t the wording. The issue is anybody who has an issue with the wording.

        1. Hi Terry!

          Please help me understand how the question is problematic? I just don’t see it. I’m genuinely curious about what the issue(s) are with it.

          1. Maybe ask “Me”. They appear to think my objection to such censorship makes me a snowflake. (I’ll wear “old person”.)

    1. I never use the term “woke”. But, for someone who places “woke” in scare quotes, the phrase “snowflake old people” doesn’t exhibit a hell of a lot of thought.

    2. Hi!

      Please help me understand how the question is problematic? I just don’t see it. But I’d like to know. I’m genuinely curious about what the issue(s) are with it.

  3. Using names this way as a shorthand for gender is not always a good idea, because it might not easily be understandable across cultures. My friend tells me of a similar problem from a math textbook she was assigned in class when she was in school, for which none of her classmates was able to get the correct answer. It turns out that the problem used the name “Kim”, which was intended to refer to a boy. Now we may say that this particular problem was badly written (it certainly was), and we can also blame political correctness (it’s almost certain that the name “Kim” for a boy was used because it’s a name that exists in some Asian cultures), but even names that we may think of as unambiguously gendered (like Tanya) may not be perceived as such by all students.

    1. Oh Christ. The problem isn’t “Tanya”. Change the introduction to,

      Tanya said to her father, …

      The people who regard the original exercise as “problematic” will still think so, probably more so.

  4. I do not regard the question as problematic. After all, I solved it without difficulty. However, I have learnt from experience that, in this day and age, some people might see it that way. So I often anticipate these issues by setting myself the task of re-writing such problems to avoid issues.

    In a previous career, I worked in cancer statistics. This was a great source of mathematical problems from the “real world”. However, a teacher once reminded me that, unbeknown to me, there was a student in the class whose mother was dying from cancer, and cancer problems may distress this student. Ever since, I have not used examples from this context because it is highly likely that some students will be closely related to someone with cancer. Similarly I avoid problems to do with body weight and other possibly sensitive topics.

    A classical mathematical problem is the Josephus problem. A more modern problem arises in discussing the marriage problem. I recall a lecturer getting into trouble for discussing this not so long ago.

    So, if the problem is an applied problem, I think hard (maybe too hard) about the context. BTW, high school students are acutely aware of these issues too.

    I prefer problems in pure mathematics.

      1. There have been similar issues in major examinations. In the LANTITE test, we were asked questions on our comprehension of a slab of text. The text contained some Latin. This might confuse a candidate who knew no Latin. If you knew some Latin, you would realise that it had no bearing on any of the questions. I wondered at the time: how would I feel if the text contained some Chinese?

    1. Hi Terry!

      Ok so you think other people might take issue with the exercise. That’s good enough — why? What in the exercise makes you think this way?

      I don’t think the rest of your comment (while tangentially related) is relevant to this particular exercise, right? I can’t imagine any reason to avoid all mention of the concepts of brothers, sisters, and families.

  5. I have a general question about problems in mathematics. Are there copyright issues associated with mathematical problems/puzzles?

    If you read enough about mathematical problems/puzzles, you encounter the same problem many times without any reference to other sources. Now I know that there are no copyright restrictions on chess problems, although it is regarded as good manners to cite the source and name of the composer, if they are available. Hence my question about mathematical puzzles/problems.

    Any ideas?

    1. Good question. Now, back to my question.

      Terry, you have a habit of writing nonsense, and, when you’re called on it, simply ignoring that you’ve been call on it.

      1. Let me try to answer the last question:

        Q: Why should anybody be obligated to try?.

        A: I am not so obligated. I let others speak for themselves.

  6. The thread of comments is particularly clear.

    Allow me to toss in my 7 cents (adjusted for cost of living) and scratch the surface for “the first problem”:

    “Gender identity is how a person feels and who they know them self to be when it comes to their gender. There are more than two genders, even though in our society the genders that are most recognized are male and female.”

    I think this is why Marty makes the comment “I cannot promise you that many authorities would refuse use of this question, although I strongly suspect that this is true. I can promise you that at least one authority would.”

    I may be wrong, I may be right. It’s what I think. Only Marty knows what he thinks (*)

    @Terry: You successfully dodged Glen’s request.

    * It reminds me of an anecdote by Isaac Asimov:

    Asimov came upon a group of fans at a convention. A man was speaking to them at length about one of Asimov’s stories going on and on about subtext and allegories and … so forth.
    Asimov cuts in and points out that it’s simply a story about (whatever the tale was).
    The speaker tut-tuts him and explains how it may SEEM so on the surface but, if one goes between the lines, one can find a richness and depth to the story which would be unsuspected to the untrained reader.
    Asimov retorted that the Speaker was full of it. It was a simple, straightforward story about […] nothing more.
    Speaker: Sir, I am an English professor who has been teaching literature for years. What makes you think you know more about it than I?
    Asimov: Because I, sir, am the story’s author.
    Speaker: So? Just because you wrote it, what makes you think you know anything about it?
    Both Asimov and the professor are correct, of course.

      1. What makes you think it’s about sex and not gender?
        I’m not goading, I think the ‘problem’ would be sociopolitical. But that’s only what I think. (You might be thinking something different. I think we can both be right). But …

        maybe the problem is tied up with what level is meant by “Elementary” and who are the “authorities”.

          1. In that case the problem is using family titles that are not gender free. Not goading, simply alluding to a minefield. Maybe you’re thinking of a different minefield.

            1. The minefield, any minefield, is entirely manufactured. The idea that “brothers” and “sisters” is, to quote Terry, “problematic” is completely nuts. And obviously so. It is so nuts that some commenters on this thread didn’t get the point. Others got the point, but, inside, realised the point was so nuts they couldn’t get themselves to write it. But everyone knows it’s nuts.

              The policing of such questions is nothing but pandering, and it is poisonous.

  7. I have no broader opinions on this topic, so if it looks like I’m pushing an agenda, please be assured that this is not the case, but you are free to believe what you want.

    Tanya can identify as whatever gender she wants, and in doing so makes it possible that there are (at least) three different ways of answering the question. Maybe. I don’t think there is anything wrong with the question and I don’t see it pushing any woke/anti-woke agenda at all.

    If you want to criticize the question because it doesn’t specify that Tanya identifies as a girl… go ahead, this is your right in a space (this blog) that encourages debate and allows commenters a lot more freedom than other forums.

    But to refer to someone as a snowflake for disagreeing with your criticism and not being willing to backup said claim… seems a bit pointless.

    1. The sex of Tanya is entirely a red herring. I don’t think there’s anything remotely ambiguous about it, but if someone does, it is trivial to reword the question: “Tanya thinks to herself, …”.

      1. Uh oh … “Tanya thinks to \displaystyle themself, …”
        Not goading, just sayin’. The minefield (and I don’t deny that many are manufactured) has a lot of mines in it (and many people waiting/itching for someone to step on one).

        In fact, you probably need the following to be safe:

        \displaystyle Phoenix thinks to \displaystyle themself, …”

        (Themself? Themselves? I’d rather have the grammar police than the PC police on my back).

        But then everything is so ambiguous that the question can’t even be answered and so it gets deleted. Which will make some people(?) happy.

        Channeling my inner Terry:

        X said: I have 3 more X’s than Y’s. How many more X’s than Y’s are there?
        (\displaystyle Reductio \displaystyle ad \displaystyle absurdum).

        In fact, to be very safe, let’s just have maths questions that aren’t embedded in artificial ‘real life’ contexts. (No criticism of the above question and I’m sure you know the direction I’m heading. Even a minefield might achieve some good).

        We live in an age where TV land has added the disclaimer:

        “This program contains outdated cultural depictions. Viewer discretion is advised.”

        to old TV shows. Is it a bridge too far to imagine such disclaimers placed in older textbooks (or just books) in the not-to-distant future…? Not goading, just speculating.

  8. The sex and gender points are not relevant to this particular exercise, as far as I can tell. Yes, there are multiple answers depending on how the reader perceives the gender of Tanya, but there are other questions with multiple answers out there that haven’t been “forbidden”. Furthermore you may just do what Marty suggests and place a gender qualifier somewhere (i.e. saying “her” when referring to Tanya).

    The exercise has boys, girls, brothers, sisters, and family in it. I don’t see mention of male or female. Boys are brothers and girls are sisters. If a brother transitions to become a woman, then they would be referred to as a sister. The exercise doesn’t care about that, and it doesn’t make a comment on it. If it instead said something like “I had two brothers but then one of them transitioned, how many brothers do I have now?” it would be a different story.

    I started off curious and I asked multiple times what the actual problem is with this exercise. But now I’m just becoming tired repeating the question. Some people indicated that they can “perceive” of a problem with it. PLEASE explain what you are “perceiving” here. I don’t understand, and I do like to understand where possible.

    1. My concern was that I can imagine that others would raise non-mathematical issues with the question. So I was seeking simply to find a non-controversial formulation of the same mathematical problem.

      1. These non-mathematical issues, which you still refuse to declare, are absurd. And, attempting to address these non-mathematical issues to placate the whiners is even more absurd.

    2. Yes, you are correct. I got distracted by BiB’s gender bullshit. Even pretending that girls can turn into boys, the question is mathematically fine.

      1. So there’s “a problem that many teachers and authorities would now have with the above exercise, to the extent of refusing to use it or forbidding its use”. But:

        1. “the question is mathematically fine.”
        2. What BiB has said about gender (entirely in the context of giving their opinion on what they see could be the problem) is distracting “bullshit”.
        3. “The sex of Tanya is entirely a red herring.”

        BiB did also offer the possibility that “maybe the problem is tied up with what level is meant by “Elementary” and who are the “authorities”.” But that went through to the keeper.
        In the context of what has turned into an increasingly frustrating game of Cluedo, this seems as reasonable as Professor Plum, in the Dining Room, with the candlestick.

        I don’t know if I’m Martha or Arthur. Or Professor Plum.

        In the meantime, Terry still refuses to give a straight (or queer) answer.

        That’s enough from me. I’m taking my candlestick and retiring to the dining room.

          1. I think BiB is trying to say that they guessed at what might be a reason for the question to be “a problem”, and they are now unable to think of any more reasons, and are not going to attempt any longer.

            Seems like it is down to you, Marty!

                  1. The exercise doesn’t say (Not brother = sister) or (Not boy = girl). There could be other non-binary family members. The question doesn’t say there are or aren’t any.

                    I do not believe it is implicit or suggested. I thought about the possibility of non-binary zisters and zuthers but it doesn’t make any difference when doing the exercise.

                    If a reader thinks it is implied, then I would suggest they consider if it is their own view or bias at work.

                    If an exercise implicitly assumes two genders, then that’s political, sure. I don’t see this one doing that. That’s my view. If you think I’m wrong, please tell me where in the text of the question it is implied.

                    1. I’m telling you questions referring to brothers and sisters, boy and girls, and so on, are frowned upon or forbidden in certain quarters. Even when these questions don’t preclude some imaginary third sex.

  9. I tried to solve the original problem using a technique that I see all too often.

    Cut and paste the original question into Google (Tania said … ) – no quotation marks – students seem not to know about them. And bingo, you get many sites that provide the answer.

        1. You’ve very eloquently put the point that I was going to make, I think. Even as a brainwashed radical, I don’t see how the question presents anything that contradicts the neoprogressive view on sex and gender.

          1. Neither do I. But who knows what gets thought out on the fringe? There’s always somebody ready to take offence about something. We’re led to believe there’s a non-mathematical issue, so I put on the black hat and said what I think.

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