Could students be taught the concept of rational numbers the same way as in the Formal construction section of the Wikipedia article rational number?

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.

• Formalism should come after understanding. How else would you know which formalism to choose? – Adam Apr 9 at 3:58
• Paragraph breaks and a link to the Wikipedia article would improve the question. – Tommi Brander Apr 9 at 7:27
• @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 9 at 12:19
• 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 at 13:47
• @AndreasBlass So, as per the joke, Serge Lang? – Adam Apr 10 at 21:44

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).

• 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 9 at 3:46
• 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 9 at 15:34
• 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 9 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 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 at 18:34