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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.

Timothy
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    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 $2\times y=3$? – JRN Dec 25 '18 at 08:07
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    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. – guest Dec 25 '18 at 11:44
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    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.) –  Dec 25 '18 at 17:20
  • @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 – Timothy Dec 25 '18 at 18:12
  • to understand how it's possible that not all real numbers are rational. – Timothy Dec 25 '18 at 18:13
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    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. – Dave L Renfro Dec 25 '18 at 22:39
  • I guess I didn't make it clear in my question and it might be too late without invalidating an answer but I think one purpose in constructing real numbers that way before teaching fractions might be because some students if they're first taught how to make computations on fractional notation might later make the assumption that the rational numbers are the only numbers as a result. – Timothy Jan 07 '19 at 19:19
  • @Timothy Could I ask how much teaching experience you have, and to what age group? – Jessica B Jan 08 '19 at 07:07
  • @JessicaB I have no teaching experience. I just had some ideas so I posted this question. – Timothy Jan 09 '19 at 02:11
  • @Timothy Then let me assure you: most people do not understand things anything like as easily as you imagine. – Jessica B Jan 09 '19 at 07:39
  • This question had a down vote but it might actually have a good idea for people to research. The question https://math.stackexchange.com/questions/895076/irrational-numbers-in-reality shows that some people have an intuition that the rational numbers are the only numbers even after knowing how to deduce from some of the real number axioms that there are irrational numbers. Maybe without realizing it, it was unconsciously because they haven't proven that a complete ordered field exists. My answer to that question got an upvote. Maybe teaching real numbers as described in this question and that – Timothy Jan 15 '19 at 02:16
  • answer really is better than teaching them as described in the answer at https://math.stackexchange.com/questions/1127483/how-to-make-sense-of-fractions/1127557#1127557 because it doesn't lead to the misconception that the rational numbers are the only numbers and another misconception that contradicts that misconception that one of them can be squared to get 2. Researchers sometimes make mistakes. I now realize that if you insist on being so careful that you don't do an engineering job unless you can find a way that you can without any mistakes figure out a proof that that way of doing it – Timothy Jan 15 '19 at 02:25
  • will work, then maybe that job couldn't get done at all but the world probably also needs it to get done. Maybe theories you hear are not written in stone and are open to the possibility that new evidence that refutes it will come. Also, this website is still in the Beta phase. Maybe the assumption that the idea in this question is bad could be refuted. I also don't upvote my own questions and answers which might be why it has a score of -1. – Timothy Jan 15 '19 at 02:31
  • This may be a possible duplicate of https://matheducators.stackexchange.com/questions/15034/could-students-learn-a-lot-more-from-school-if-theyre-only-taught-number-theory. The dyadic rationals with those operations could probably be replaced with operations on the natural numbers with the same structure, and those operations could be defined entirely using number theory and proven using number theory so satisfy the same 5 laws as I said in that question that the natural numbers satisfy. Maybe this question is not a duplicate because it also refutes the idea in the other question of treating – Timothy Jan 15 '19 at 04:58
  • $\mathbb{R}$ like a proper class in the version of type theory that extends number theory rather than Zermelo-Fraenkel set theory. – Timothy Jan 15 '19 at 05:00
  • Maybe it's too late because it might invalidate the answers but I'm wondering if this question could have been made better if I had asked it differently in the right way which includes being like my answer at https://math.stackexchange.com/questions/1127483/how-to-make-sense-of-fractions/3068324#3068324 and linking https://matheducators.stackexchange.com/questions/15034/could-students-learn-a-lot-more-from-school-if-theyre-only-taught-number-theory to explain how in addition to delaying non-number theory so that they can master number theory first making them learn other stuff more easily – Timothy Jan 15 '19 at 05:12
  • later because their expertise at number theory may help them better understand non-number theory, it could also use knowledge of number theory and set theory to construct a complete ordered field and call it the real numbers. Maybe that question and this question could both be useful because it's also possible that it's better to give a simpler description of the real numbers and state the definition of +, $\times$, and $\leq$ and say it has been shown to satisfy certain properties. I don't feel like everybody properly learned what a real number really is after high school or what structure – Timothy Jan 15 '19 at 05:20
  • they have. I don't even feel like I ever got taught it myself and instead came up with my own construction of the real numbers as described at https://matheducators.stackexchange.com/questions/7718/effects-of-early-study-of-advanced-books/15078#15078. I wasn't taught enough about them to be able to deduce that it is a complete ordered field such as the completeness and the associative law of real number addition. They just left me to figure out which extra properties it's obvious it ought to be defined to have. Maybe people should be taught my explicit construction of all real numbers and – Timothy Jan 15 '19 at 05:26
  • their operations and be asked to write a proof that the real numbers also satisfy the laws I wrote in my other question that the natural numbers satisfy. I'm not sure teaching rational numbers makes it any easier to teach real numbers later but it does go the other way, teaching real numbers makes it easier to teach rational numbers later. According to https://matheducators.stackexchange.com/questions/14831/how-to-explain-fractions-to-7-year-old-kid, some people still struggle with fractions even in high school. They may have some idea of a real number but those who don't jump to the – Timothy Jan 15 '19 at 05:34
  • conclusion that those 5 laws of natural numbers apply to real numbers also and every nonzero real number has a multiplicative inverse might be less likely to make the mistake that $(a + b)^2 = a^2 + 2ab + b^2$ applies to the quaternions also but also be unable to figure out that when you add, multiply, and divide rational numbers in the form of an integer numerator and denominator the way they were taught, you get a notation that represents the same number when the given expressions represent the same number regardless of how they were represented and insist not only on knowing how to do that – Timothy Jan 15 '19 at 05:43
  • but also on understanding why it works because they're the type of person who refuses to perform a calculation without understanding why it works. If the teacher says because of the distributivity of multiplication over addition, the student might learn the wrong message "I finally tried making the assumption that those laws that I know apply to the natural numbers apply to the real numbers also and then I started getting higher marks on tests after I tried it so it must be right." Then they may later incorrectly assume that for the quaternions as a result. I feel like jumping straight into – Timothy Jan 15 '19 at 05:52
  • constructing the real numbers with those operations after they're learned the basic skills of number theory and set theory will make it ever so much easier for them to be taught rational numbers and how to perform calculations on them. I feel like people claim that people become too old to learn stuff. That's true for some situations but it's not a universal rule. Some material students can learn better when they're older. Maybe for some students, they actually keep struggling and failing because they haven't mastered the earlier knowledge it relies on because school tried to rush in that – Timothy Jan 15 '19 at 05:57
  • material before they claim the student would become too old to learn it. Maybe they're not making the definite assumption that students learn everything less well when they're older but rather don't want to take the risk of changing the system to delay teaching certain material. It's too hard to teach them to deduce properties from all 17 axioms of real numbers I wrote in my answer until way later so I think it's better to explicitly construct a system which turns out to satisfy all of them most of which they will not yet use or get taught and might figure out on their own later. Maybe I could – Timothy Jan 15 '19 at 06:04
  • ask another question that's like this question except that it explains all that. I would love to get feedback on whether I should write one and ideas of what I should include in that question. – Timothy Jan 15 '19 at 06:05
  • This question has a score of -2. I don't see why it would deserve that score. According to https://matheducators.stackexchange.com/questions/14831/how-to-explain-fractions-to-7-year-old-kid/14833#14833, some students 14-18 really struggle to learn fractions even. On the other hand, the dyadic rationals and their addition and multiplication operations are so easy to describe in a way students can understand. They can teach that by convention, in any system where addition and multiplication were already defined, subtraction means addition of the multiplicative inverse and division means – Timothy Apr 04 '19 at 03:53
  • multiplication by the multiplicative inverse. Now the rest of the real numbers are directly constructed from those Dedekind cuts of the dyadic rational numbers where the lower part has no maximal element nor does the higher part have a minimal element and the addition and multiplication operation and inequality again defined in the intuitive way. It is now easy to show that for any integer x, $1 \div x$ exists in this system. They can teach that by convention for any integer a and nonzero integer b, $\frac{a}{b}$ means $(1 \div b) \times a$. Then they can be taught to figure out on their own – Timothy Apr 04 '19 at 04:00
  • how to perform calculations on rational numbers given in that notation from the complete ordered field axioms the real number system has already been proven to satisfy, and that just because the rational numbers are the only numbers those calculations can be performed on by that method doesn't mean all real numbers are rational. – Timothy Apr 04 '19 at 04:03
  • I see that this question now has a score of -3. I'm not sure it actually deserves a negative score because I later saw in https://nrich.maths.org/2550 that other people independently thought of the same idea as I proposed in this question and thought it was good. The only difference is they did not discuss how to construct the rest of the real numbers other than the ones that have a terminating notation in binary. – Timothy May 06 '19 at 20:12
  • Now I see that this question almost has an answer at https://matheducators.stackexchange.com/questions/186/good-definition-for-introducing-real-numbers/199#199. It introduces defining real numbers by a decimal expansion. Now I'm smarter and see that that question already covers what this question was asking. I think it may be a good idea to mark this question as a duplicate of that question. – Timothy Dec 15 '20 at 21:14

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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.

Amy B
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  • 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 – Timothy Dec 27 '18 at 15:40
  • 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 – Timothy Dec 27 '18 at 15:45
  • terms of real numbers. That's why I asked the question at https://math.stackexchange.com/questions/2437893/what-is-a-natural-number. Now I realize that you can define a natural number from scratch as a finite ordinal number. – Timothy Dec 27 '18 at 15:47
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    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. – Amy B Dec 27 '18 at 19:22
  • 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. – Timothy Dec 27 '18 at 20:34
  • Thinking back to when I first learnt about decimals, I'm pretty sure I was taught that $0.1$ means $=\frac{1}{10}$, $0.01$ means $=\frac{1}{100}$, etc. But I just thought "Oh, so it works just like with ordinary numbers then—moving to the right makes it a tenth as big" and didn't understand why other children found it hard. I'd say the two things that made decimals obvious were (i) the idea of multiplying by $10$ in one direction and dividing by $10$ in the other, and (ii) fractions to understand the numbers smaller than $1$. – timtfj Dec 28 '18 at 16:04
  • I guess that method won't work. If they knew only of the natural numbers, then the description of the real number $\sqrt{2}$ could be replaced by a description of a property of the natural numbers describable in a certain system and the statement that it is $\sqrt{2}$ could be replaced by the statement that that property has a certain property. However, elementry school students observe continuity all around them and may claim that the number $\frac{1}{2}$ already existed and when you construct the property of the natural numbers to replace the construction of the number itself, they may claim – Timothy Jan 16 '19 at 03:59
  • that the property of the natural numbers you constructed is a distinct object from the already existing number $\frac{1}{2}$. – Timothy Jan 16 '19 at 04:00
  • Once somebody claimed that it was impossible to fully understand quantum mechanics so they invented the expression "Shut up and calculate" to discover the results of the calculations on its assumptions without understanding what they mean. Maybe students can always learn the construction of the real numbers when they're older if they want to by not letting the unsolvability of the question of whether the undefined real numbers already exist and have the complete ordered field property cause them to waste their entire life trying to solve the impossible, and devote some of their time to – Timothy Jan 16 '19 at 04:09
  • actually learning the definable construction of a complete ordered field while leaving open the question of whether those constructed objects actually are the real numbers themselves or the real numbers already existed as undefinable objects and they constructed objects that play the role of the real numbers. – Timothy Jan 16 '19 at 04:11
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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.

James S. Cook
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  • 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 – Timothy Dec 25 '18 at 18:22
  • 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 – Timothy Dec 25 '18 at 18:27
  • 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 – Timothy Dec 25 '18 at 18:34
  • 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 – Timothy Dec 25 '18 at 18:40
  • 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. – Timothy Dec 25 '18 at 18:42