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tl;dr: Why do so many students have poor intuition of numbers, and what can be done about it?

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I've always been good with numbers. As a maths tutor, one of the things I notice is how poor the intuition and techniques that some of my students have when it comes to approximating numbers (amount of things) and fractions of things.

Example

Quickly approximate how many lines are on the following page, just by looking at it:

enter image description here

When I ask this question to my students, they will either count the number of lines "One, two, three, four...", which takes ages and moreover is tedious, or will just stare at the page and guess a random number. Some will guess like "seventeen" or "fifty", or "no idea", which worries me, as it shows their subitising skills aren't up to par. That said, not all are bad at this and some students will make a reasonable approximation of twenty to forty.

One method I use is to approximate $10$ lines by: Scrolling my eyes from line $1$ to line $3$ while counting "One, two, three" in my head. Then keep counting in my head at the same speed, scrolling my eyes at the same speed, and then stop scrolling my eyes once I get to the number $10$ in my head. This will take all of about $2$ to $3$ seconds for me to do. Now I know that I won't have counted exactly $10$ lines, but that's besides the point: I'm trying to get an approximation/estimation.

enter image description here

Once I have this "length of $10$", I'll then make copies of the "length of $10$ lines", using my imagination. Basically, doing this in my mind's eye:

enter image description here

Although in reality what I think I do is I try to look at a line in line the bottom of the first arrow, then look at a line in line with the bottom of the second arrow, then look at a line in line with the bottom of the third arrow, and notice that the bottom of the third arrow is very near the line closest to the bottom of the page.

This technique of "close-ish" approximation is useful because now I can try and do things with the information I have, without having spent $30$ seconds figuring out how many lines there actually are, which I don't care about. I can do the above process very quickly (in less than $5$ seconds) due to using it so often. This shortcut saves lots of time and therefore I think if students did this, they would lose less focus in maths because using these estimations help get close to the right answer quickly. They should use these shortcuts when approximating to see if their answers make sense.

Now let's say I'm writing an essay with the above paper and I know I can write on average $8$ words per line. The teacher requires the essay be minimum 600 words. I have 10 pages left in my paper pad. Do I have to buy a new one on the way home from school or not? Now a lot of my students will struggle with this question, spend $5-10$ minutes on it, as if it's a really difficult problem-solving exercise, and then some will give up, some will give the wrong answer, and some will give the right answer, assuming the right answer is "yes there is enough paper left because $8\times 30\times 10 = 2400 > 600.$ "

But here is my thought process: $8\times 30\times 10 = 2400,\ $ so you only need $\frac{600}{2400} = \frac{1}{4}$ of your $10$ pages (i.e. $2\frac{1}{2}$ pages), but since you're going to be making notes and drafts on your essay, you might need to use $7$ or $8$ pages total. Plus there's the other homework and I need more paper for school tomorrow and I don't want to buy it from the shop tomorrow morning so, hmmm... so yes I really do need to buy another book pad now. If I had $30$ pages rather than $10$ pages left in my pad, I would then think, "hmmm... how much paper do I need for homework tonight? Maybe like 15 pages total. And I'm pretty sure I can get away with $30-15=15$ pages at school tomorrow, so I'm not going to bother going to the stationary shop after school today".

So I think it's this fluent usage of maths within a self-conversation that improves my intuition. It is realistic that I would have actually thought these things on the way home from school, whereas I imagine students give up on this thought process and just think, "I'm going to get a new pad from the stationary shop now", or "I'll borrow paper from James at school tomorrow", or most likely they won't think about it at all. In order to have a thought process similar to my one, one must have enough confidence in their own thought process to begin with.

But I think it's exactly this confident application of arithmetic in self-conversations assisting in everyday life things that helped people including myself get really good at doing arithmetic quickly and well. So we maybe have a kind of a chicken-and-egg thing: students won't do this unless they have confidence, but they won't ever be confident unless they start doing it.

Now, my thought processes outlined above will not be a revelation to anyone here (I would hope). But I think lots of students don't use this and other similar methods to estimation-check their working, or solve everyday mini-problems. So I guess my over-arching questions are:

Q1) what can be done to help students improve these important quick estimation skills?

Q2) How can we be encouraging students who are not naturally good at maths and arithmetic and fractions to do this on a day-to-day basis as part of their self-conversation? Or is it unethical or unrealistic to expect to be able to influence their self-conversation?

Originally my question also included "estimating fractions" (which is closely related to this question) to help improve intuition with fractions, but this question is already overly-long and a ramble (apologies for that), so maybe I will ask that as a separate question in the future.

Possible answer to my own question: I need to convey to my students the methods to help improve intuition of numbers outlined above (and various other methods) in a clear way. I think I already do this though. But maybe I need to do it more slowly and clearly? I'm not sure.

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    $\begingroup$ The book Math Semantics, by Edward MacNeal, addresses this question at length in a chapter on estimation. He talks about your web of knowledge (some landmark numbers it helps to know, like the population of the world and the U.S.), among other things. Reading it will give you some good ideas. bookshop.org/books/mathsemantics-making-numbers-talk-sense/… $\endgroup$
    – Sue VanHattum
    Apr 30, 2021 at 22:14
  • $\begingroup$ The book by Bentley, "Programming Pearls" (2nd edition 1999, Addison-Wesley) has a chapter on "back of the envelope" estimates. Quite enlightening. And yes, the phenomenon you note is quite widespread. $\endgroup$
    – vonbrand
    May 2, 2021 at 3:51
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    $\begingroup$ "I only have 30 pages left, so I'm going to get a new notebook now rather than wait for it to become a problem" sounds much more reasonable to me than "I roughly estimate that I need 15 pages for homework tonight plus 15 pages for the schoolday tomorrow, so there is no need to get a new notebook since I have exactly 30 pages left." $\endgroup$
    – Stef
    Jun 14, 2023 at 11:34
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    $\begingroup$ @Stef The results of a detailed calculation being superseded by other non-quantitative high-level considerations is probably the part of this problem that bears the closest resemblance to my lived personal experience in the working world. $\endgroup$
    – Steve
    Jun 14, 2023 at 11:37
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    $\begingroup$ @Stef, I have no idea that the page is 30 cm high. It would be 297 mm if it were A4, but there are many paper formats, and there is nothing to assess the scale. So, estimating a number of lines in an arbitrary section and then the number of sections is more generic and does not require a prior knowledge of paper formats. $\endgroup$
    – Rusty Core
    Jun 14, 2023 at 18:44

1 Answer 1

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Beyond the visual guesswork of scanning the looseleaf to estimate how many lines are there, the process described amounts to modeling the situation based on estimated values of a few important key factors. Most school mathematics does not encourage this type of thinking because problems are often aimed at exercising or testing specific math techniques rather than developing "reasonable" answers to more speculative questions. Further, most problems settings are not rich enough to permit this type of modeling; usually there is not enough contextual data provided to allow students to make choices about their approach, and there is not an opportunity to make estimates of factors based on measurements in the physical world or recollections of their own prior experience as was done with your example (estimating words/line based on experience, estimating lines/page based on a physical procedure). Most of the estimation that is emphasized is "sense-making", often limited to word problems based on familiar experiences (e.g., a human is not 100 ft tall, a trip in a car does not likely have an average speed of 1 cm/hr...).

I suspect it is too much to ask for the youngest children who are just learning to add and multiply to also grapple with these concepts at the same time, so I think one of the first opportunities for elementary students to consciously develop these skills would be in a science class like physical science (a combination of physics, chemistry, geology, and ecology often taught to middle schoolers in the US). One can talk about the use of data to grow understanding and make predictions, the development and testing of a hypothesis that assumes certain factors are important, and the reasonable precision to assume for estimated or collected data while giving students more practice setting up word problems and doing simple addition/multiplication/equations. Once a student has enough science background, you could introduce other meaningful physics or chemistry word problems in a math setting without too much back-tracking to discuss basic aspects of the subject matter. A statistics class is another natural opportunity to practice this thinking where it is clear that the aim is to specify a simplified model of a random process that meets desired accuracy and explainability criteria and that multiple models with different factors might be considered on par with one another.

Edit: This is not meant to draw lines between math and science but just to emphasize how closely connected they are and point out that richer problem contexts come up naturally in the science setting.

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