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Some students really struggle to learn fractions. Not only that but also, once they've mastered an understanding of real numbers, they can learn about fractions so much faster and more efficiently later. Maybe teaching real numbers renders teaching rational numbers totally unnecessary. In addition to that, teaching rational numbers too early might cause some people to form the misconception that rational numbers are the only numbers. I think so because I read on the internet about people who don't understand how irrational numbers exist. This question and this page seem to support my theory. Also when they omit unnecessary material, they can fit in more other material while still moving slowly enough that all the students can actually keep up with learning what school is trying to teach them.

It may seem wierd but it might be better to use the base 2 notation for the fractional part of a real number and the base 10 notation for the integer part because of they way they construct the real numbers. Real numbers could be taught as follows. First we can define a natural number as a finite ordinal number. Next, we invent the negative numbers and then redefine +, $\times$, and $\leq$ on them. Next, since each odd number $x$ is not a solution to $2 \times y = x$ in the integers, we invent a solution to that equation in $I$. I know that's how I say it but for them, it's better not to introduce a variable and just say that none of them get you that number when you multiply 2 by it. Let's call each invented solution a half integer. Each half integer $y$ is still not a solution to $2 \times z = y$ so we can again invent a solution to each of them. Now it's easier to define +, $\times$, and $\leq$ on this system than it is to teach fractions and how to multiply and divide them and determine which of two is greater. Some people may quickly figure out that not all numbers can be gotten by multiplying a number by 3 in this system and get confused but the teacher might just have to explain that that's how the system was defined and that they will later teach them a different system where there is a solution to $3 \times x = 1$. They can later be taught the concept of my definition of a Dedekind cut of the dyadic rationals which is not the real definition and my definition is a subset of them that has the property that it is not empty and its complement is not empty and for any dyadic rational in the subset, all smaller dyadic rationals are in the subset. They might start to notice that for some cuts, that cut has a maximal element and for some cuts, its complement has a minimal element and for some cuts, it has no maximal element nor does its complement have a minimal element. Now there's on obvious one-to-one correspondence from the cuts between 0 and 1 to all the functions from $\mathbb{N}$ to {0, 1}. However, we want to invent a new number for the cut only when there isn't already a maximal element of the cut or a minimal element of its complement. Now this gives an obvious binary notation for the fractional part of each real number but that notation forbids a string of trailing 1's just like some authors forbid a string of trailing 9's.

We can then redefine +, $\times$, and $\leq$ on this system. - and $\div$ in this system are just defined in terms of + and $\times$ in this system. We can show that multiplication can be defined in that way in that system and that in that system, multiplication by any nonzero number is bijective and squaring restricted to the nonnegative numbers is also bijective on the nonnegative numbers. Now that they already know a lot of the laws of real numbers used as some of the defining criteria for a complete ordered field, all we need to do is teach them how to divide any real number by any nonzero real number and then say a real number is defined to be a rational number if and only if for some integer $p$ and nonzero integer $q$, it is $p \div q$. Then they might quickly figure out so many properties of rational numbers that some students are struggling to learn.

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    $\begingroup$ How will you teach these "half-integers"? What notation and terminology would you use? For example, how would you name and write the half-integer $y$ so that $2\times y=3$? $\endgroup$ – Joel Reyes Noche Dec 25 '18 at 8:07
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    $\begingroup$ I think facility with fractions is difficult. But the intuitive concepts are not hard at all (split the cake in three pieces). And are foundational. This is another, why don't we change math question. I am skeptical that it benefits kids to make the change you want. And also if you teach them that weird system, they are going to have issues later unless you propose to change the whole world also. Good luck. $\endgroup$ – guest Dec 25 '18 at 11:44
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    $\begingroup$ You seem to be starting from some false premises. (1) That children at age 8 reason abstractly. (2) That a number system has to be treated constructively rather than axiomatically. To the extent that anything is formalized at all in elementary education, number systems are treated axiomatically. Euclidean geometry is also effectively an axiomatic formalization of the reals, although it is not presented that way these days to children. (Instead the standard approach seems to be to describe the reals as existing separately from geometrical objects, so a length is a measure of a line segment.) $\endgroup$ – Ben Crowell Dec 25 '18 at 17:20
  • $\begingroup$ @BenCrowell I'm not sure that will work either. They also sometimes stubbornly insist on their own false assumption that a real number is defined by a decimal notation so 0.999... $\neq$ 1. That assumption contradicts the assumption that $(\mathbb{R}, 1, 0, +, \times, \leq$ is a complete ordered field. Some of them might think it can be explained by the fact that the set of all numbers falls into the hyperreal number system and not see a problem with the fact that the hyperreal number system is not complete. Also, introducing fractions first might lead later to their lack of ability $\endgroup$ – Timothy Dec 25 '18 at 18:12
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    $\begingroup$ They can later be taught the concept of my definition of a Dedekind cut of the dyadic rationals --- It's very unclear to me who your intended students are. I suspect fewer than 0.1% of elementary school students will encounter anything like this anytime later in their life. And I realize you're speaking to math knowledgeable people here, but you do realize that you can't use terms like "field", "ordered field", "maximal element", "complement", "axiomatic", "dyadic", etc. in elementary school classes? In fact, few elementary school teachers will know what you're talking about. $\endgroup$ – Dave L Renfro Dec 25 '18 at 22:39
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I appreciate you raising the question of improving the teaching of fractions which is certainly needed. I don't feel optimistic about your suggestion. I taught gifted elementary math students for over 25 years. I have a degree in math from MIT. I have no sense of how I would present your suggestions to my very gifted students. The elementary classroom teachers I worked with have problems with each change in the math curriculum and very little math background. I don't see how your ideas could work in the regular classroom.

Children bake with fractions, measure with fractions, and make sense of decimal arithmetic using their understanding of fractions. When I introduce decimal multiplication and division, they understand it only because of the work we've done with fractions. My feeling is teaching fraction first has a great value. However, I agree with you that students struggle with many aspects of fractions and it would be great if they could be taught better by everyone.

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  • $\begingroup$ Maybe some new students who have not yet learned fractions will be really hard to teach fractions to so well that they won't struggle. If it's from the axioms of real numbers, they might later get so much experience with it that they can't believe not all real numbers are rational. I feel that it's easier to explain how to fill in the holes if you use the set of all numbers terminating in base 2 rather than the set of all numbers terminating in base 2 rather than the set of all numbers terminating in base 10 and then later show that it doesn't have infinitesimal numbers and that you can divide $\endgroup$ – Timothy Dec 27 '18 at 15:40
  • $\begingroup$ any number by 10 as many times as you want in that system. I feel like it was not until I was 18 that I managed to think of my own explicit definition of a real number after not quite knowing what one was and it was by constructing the dyadic rationals, the numbers that are terminating in base 2, then filling the spaces between the Dedekind cuts that didn't already have a maximal element in the lower part or a minimal element in the higher part and I feel like I previously didn't quite know what I real number was. I now realize that didn't quite work because I also defined natural numbers in $\endgroup$ – Timothy Dec 27 '18 at 15:45
  • $\begingroup$ terms of real numbers. That's why I asked the question at math.stackexchange.com/questions/2437893/…. Now I realize that you can define a natural number from scratch as a finite ordinal number. $\endgroup$ – Timothy Dec 27 '18 at 15:47
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    $\begingroup$ The point of my answer was to frame the difficulties of introducing your ideas at the elementary level. I stand by my opinion that this just won't work at the elmentary level. $\endgroup$ – Amy B Dec 27 '18 at 19:22
  • $\begingroup$ I feel like when I was in elementry school, I didn't understand how it was possible that there were no infinite or infinitesimal numbers. I think the truth is that it's impossible to teach all students to understand real numbers properly before they're finished elementry school at the moment. May in the future, researchers will solve the problem of how to do so. $\endgroup$ – Timothy Dec 27 '18 at 20:34
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I'll give a brief defense of the somewhat conventional view.

If I understand history correctly, part of what made Newton's advance in calculus possible was the (then) recent introduction of the decimal expansion of a real number. Calculation to arbitrary precision was something that everyone could both do and easily communicate with the introduction of decimal expansion. Lexographic ordering of decimal expansions is simple to see. In contrast, comparing rational numbers is far more complicated. Even if you limit the scope to a particular class of rationals. The manifest ordering of decimal expansions cannot be beat for kids. In addition, the ease of multiplication and division by 10 and the whole of scientific computation. One can learn all this and still fail to understand the construction of negative numbers, much less the construction of the real numbers.

In fact, the idea of a decimal representation is quite natural if the student has an intuition for length and/or direction. Surely many have such intuition. Therefore, I would argue, formulating real numbers as their decimal representations is a quick way to get very far. Furthermore, while questions of non-unique representation and convergence lurk under the surface, the problem of calculation in decimals is worthy of every student's attention. Rationals, negatives, the whole line is there. You just have to grapple with arithmetic of decimal expansions. Of course, the problem of justifying decimal expansions requires significant effort. But, that is the beauty. All that effort is wrapped up into something kids should and can learn.

So, in summary, I would say your idea is intriguing, but I would save it for an abstract algebra class. For children, the usual material is already quite good if it is taught well.

Incidentally, we come to another significant problem for your proposal. You'd have to find a way to convince elementary math educators to completely up-end their teaching and understanding of real numbers. Many of these teachers are quite poor at abstract algebra, as such getting them to undertake a reformulation which is abstract-algebraic at its core is a really hard sell.

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  • $\begingroup$ Teaching rational numbers might be good for not making them form the intuition that there are numbers infinitesimally close to 1 but also might lead later to their confusion about how it's possible that some real numbers are irrational. Defining a real number by a decimal notation might not work either because they might fail to understand why 0.999... $\neq$ 1. Maybe there is no solution. Maybe only when they're older, all of them are capable of learning the truth that the real number system is one of many possible systems and that system itself is a complete ordered field. Maybe they could $\endgroup$ – Timothy Dec 25 '18 at 18:22
  • $\begingroup$ start school at an older age like in Finland and learn the material better as a result. Maybe when they're younger, they could just be taught how to write a proof in a weak system of pure number theory and a lot of the time instead of being used to teach more material will instead be used to help them train themselves to be really good at solving puzzles and figuring out how to solve puzzles of writing complete formal proofs of statements such as the commutativity of natural number multiplication. They could teach only the recursive definition of finite ordinal addition and completely abandon $\endgroup$ – Timothy Dec 25 '18 at 18:27
  • $\begingroup$ the definition of finite cardinal number addition and subtraction because they might not have the maturity to figure out the difficult to prove theorem that it exactly corresponds to finite ordinal addition and subtraction. Then they will derive no contradictions and avoid any confusion over statements about continuity of positions in the real world because the system of number theory is a consistent subtheory of Naive set theory. Maybe later they can easily understand the construction of the real numbers considering a rel number describable in the system as an informal way to refer to a $\endgroup$ – Timothy Dec 25 '18 at 18:34
  • $\begingroup$ property of the natural numbers describable in the system. Having them learn only stronger systems of number theory might lead them to devise a weak version of type theory where Godel's incompleteness theorem on systems of number theory is a meaningful statement and treat $\mathbb{R}$ like a proper class and then find that ZF is an extension of number theory in a totally different direction so they will find ZF strong enough to be dubious. Maybe that doesn't matter as long as they know how to get along in a math career and show that the formal system of ZF is consistent and write formal proofs $\endgroup$ – Timothy Dec 25 '18 at 18:40
  • $\begingroup$ in it. I expect researchers not to blindly follow this idea if they haven't already heavily researched it and then decided it is a better way to teach or they found a way to teach that makes some use of this idea and modifies it in a way that works well. $\endgroup$ – Timothy Dec 25 '18 at 18:42

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