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Link to article cited in title, Wikipedia rational number.

According to this answer, some students 14-18 are still struggling to understand fractions. Maybe some students know how to perform the calculations on rational numbers given in fraction notation but don't understand why it works. If they're taught the method described in that section, they might actually understand why it works so well. At first, the teacher could teach the student how to do creative thinking and then ask the student to explain why addition and multiplication can be defined that way and then guide them through to come up with an answer showing that addition and multiplication of ordered pairs in the same class always gives you a result in the same class so two ordered pairs can be defined to represent the same rational number when they're in the same class. Later, they could say that by definition, subtraction is addition of the additive inverse and division is multiplication by the multiplicative inverse. Later so that the student will become smarter, the student could be left with the task to teach themself how to compute a division problem on rational numbers given in fraction notation to get a result expressed in fraction notation. That way, they not only will know how to perform the calculations but will have also been guided to show that they have a real understanding of what the calculations actually mean. Even later to make them even smarter, they could be guided to come up with a proof that $\mathbb{Q}$ is a field.

JTP - Apologise to Monica
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Timothy
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    Formalism should come after understanding. How else would you know which formalism to choose? – Adam Apr 09 '19 at 03:58
  • @Adam But even without it, some students 14-18 struggle to understand fractions. Maybe for some of them, they can learn how to perform the calculations but don't understand why they give the right answer because it's not obvious to them that the set of all real numbers is a complete ordered field, and of all those, the ones who want to gain an understanding could discuss the topic with the teacher in the form of research and the teacher could say $\frac{1}{2}$ doesn't exist until we invent it and then say we seek a way of inventing things and defining operations on them that satisfies certain – Timothy Apr 09 '19 at 04:11
  • properties. If for the same student, that doesn't work either and the student says it's obvious that $\frac{1}{61}$ already existed whether or not you invent it, they may not understand the construction either and there may be no solution to teach them yet. Maybe later when they're older, they can be taught that regardless of the way the world really works, you can use a mathematical system where you construct objects and define operations on them how ever you want and consider only the objects you already constructed even if you have not yet constructed $\frac{1}{61}$ and you find it obvious – Timothy Apr 09 '19 at 04:18
  • that $\frac{1}{61}$ exists in the real world. Later, they can be taught to understand that that's how addition and multiplication were defined in the formal construction of the rational numbers and will then see that it satisfies all the properties of a complete ordered field but completeness, and there also exists a complete ordered field with a subset of it isomorphic to the rational numbers the way they learned them and that the physical measurements actually correspond to the elements of the formal complete ordered field as described at – Timothy Apr 09 '19 at 04:23
  • https://blogs.scientificamerican.com/observations/brain-gain-a-person-can-instantly-blossom-into-a-savant-and-no-one-knows-why/. – Timothy Apr 09 '19 at 04:28
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    Paragraph breaks and a link to the Wikipedia article would improve the question. – Tommi Apr 09 '19 at 07:27
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    @Timothy That link is about savants,not rationals. Anyway, it is quite unlikely that, if a student cannot compute $\frac{2}{3} + \frac{1}{6}$, their problem is that they do not understand that $\mathbb{Q}$ satisfies the field axioms. – Adam Apr 09 '19 at 12:19
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    There doesn't seem to be an explicit question here. For the implicit question "will the suggested method work", I'm tempted to say that it will work if and only if the student is Nicolas Bourbaki (combined into one person). – Andreas Blass Apr 10 '19 at 13:47
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    @AndreasBlass So, as per the joke, Serge Lang? – Adam Apr 10 '19 at 21:44

1 Answers1

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  1. Bad idea. You are going in exactly the opposite direction. Weak students do not need more formalism. For instance, your Q field comment at end (WTF?). Or basically any Wikipedia article--which all assume formal proof or definitions are explanations--they are not.

  2. If your math explanations are anything like your communication here, you will make things worse, not better with students. The whole question itself is very hard to parse. No paragraph breaks. And one incredible run-on stream of consciousness sentence in the middle:

    "At first, the teacher could teach the student how to do creative thinking and then ask the student to explain why addition and multiplication can be defined that way and then guide them through to come up with an answer showing that addition and multiplication of ordered pairs in the same class always gives you a result in the same class so two ordered pairs can be defined to represent the same rational number when they're in the same class."

  3. Oh...I love the "first teach creative thinking" comment. As if this was a well understood spice to just pull out of the rack and add to the soup. No issue with how/what (and with weak students nonetheless).

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  • For those students who were going to continue struggling with fractions even when they're 14 years old, I'm not sure teaching that formal definition by adopting the creative thinking approach and discussing the topic with the student in the form of research won't work. – Timothy Apr 09 '19 at 03:46
  • The first and third points could be made relevant, but the second comment is not really addressing the question and is almost ad hominem (ditto the implied vulgarity in the first point). – kcrisman Apr 09 '19 at 15:34
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    This comment could be worded a lot more constructively but I also think it's essentially the correct answer, so I wasn't sure what to do. (from review) – Chris Cunningham Apr 09 '19 at 16:10
  • Maybe you're right. If the student thinks that the real number system with the operations of +, $\times$, and $\leq$ already existed and the only way to figure out what a sum of two numbers defined by a certain property is is to deduce the sum from some of the axioms of real numbers, and then you give a formal definition, either they will question why the sum and product are what they are or they will blindly accept that that's what the sum and product are, leading to overconfidence of untrue mathematical statements later. Maybe most elementry school students are too young to learn why you can – Timothy Apr 11 '19 at 18:29
  • invent objects and define operations on them what ever way you want. Maybe the next best thing to do is to teach students that the real number system satisfies certain requirements for a complete ordered field which they will find so intuitive and then define subtraction and division in terms of addition and multiplication and say something that means the same thing as "By definition for any integer x and nonzero integer y, $\frac{x}{y}$ is another way of saying $(1 \div y) \times x$ and then deduce from the axioms how to perform the calculations on numbers written in that notation to get a – Timothy Apr 11 '19 at 18:34
  • number written in that notation. Although it can be shown that doing so is pretty much the same as performing the calculations the way they are formally defined and that changing the fractional notations in both of the terms to one in the same class according to the formal definition always gets you a fractional notation in the same class, most students will probably blindly form the intuition of that fact so maybe it's worth leaving them thinking the way they naturally think. There may be other students who don't blindly accept the axioms of real numbers and I think the norm in United States – Timothy Apr 11 '19 at 18:38
  • may be to just tell them that the method of performing calculations the way they were taught to is the way it is done, but for the sake of those students who don't want to answer a math question such as what is $\frac{1}{3} + \frac{1}{2}$ because nothing can make them feel sure that the answer is $\frac{5}{6}$, it may be worth it for researchers to work towards United States adopting the Finnish education system which does a really great job of teaching and doesn't have standardized tests and not ordering the student to perform calculations the way they were taught when they don't want to. – Timothy Apr 11 '19 at 18:43
  • Doing so might teach them that they should answer questions the way they were taught because they were told to and lead them to apply the law $(a + b)^2 = a^2 + 2ab + b^2$ for matricies. If they are then told that they are wrong and will lose marks, then they might start living in fear because not answering a question might make them lose marks but answering a question also might risk them answering it wrong and losing marks. – Timothy Apr 11 '19 at 18:46