# Could students learn a lot more from school if they're only taught number theory until way later?

According to https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html, when students get taught a concept when they're so young, they're less likely to be able to learn it properly. For example, some students really struggle to learn what fractions are and some people also can't understand how it's possible that irrational numbers exist after they learned how to represent all rational numbers and make calculations on them. Maybe if they avoid teaching that so early, they can use the time to teach the students stuff they actually can understand such as number theory and go quite far with it and then later, they can teach about real numbers so much more easily as a result of the knowledge about number theory they learned earlier. For example, they could completely avoid teaching them about the concept of cardinality of a finite set and teach only the definition of a natural number as an ordinal number because some of them might not have yet thought of their own proof that distinct finite ordinal numbers always correspond to different cardinal numbers.

Maybe they could teach them how to write a formal proof in a certain weak system of number theory after giving them the resources from which they can form an intuition that everything it proves is true where that system only talks about natural numbers. They might later test them on their ability to write a formal proof of the statement they're asked to write a formal proof of. If the student writes an intuitive reason why a certain statement is obviously true, the teacher will not progress with the material until after the student learns that the task was to write a formal proof the way they were asked to to show that they're able to figure out how to write a formal proof of it and masters the skill of figuring how how to write formal proofs of different statements when they're asked to write formal proofs of them. By waiting until all the students in the class master the skill of figuring how how to write a formal proof of a new statement when they're asked on a test to write a formal proof of that statement before moving on, they might gain the ability to think and figure stuff out really well and understand other material they're getting taught later.

Without being told ahead that these are some of the statements they're going to be asked to write a formal proof of so that they won't collaborate with another student to figure out the formal proofs of those statements ahead of time, here are some of the statements they could be asked to write a formal proof of except that those statements in their current form aren't actually describable in the system so they will be given strings of characters that represent a different but similar statement. They could be told on the test that by definition

• $$\forall x \in \mathbb{N} x + 0 = x$$
• $$\forall x \in \mathbb{N}\forall y \in \mathbb{N} x + S(y) = S(x + y)$$
• $$\forall x \in \mathbb{N} x \times 0 = 0$$
• $$\forall x \in \mathbb{N}\forall y \in \mathbb{N} x \times S(y) = (x \times y) + x$$

without any prior being introduced to the concept of addition and multiplication and then asked to write a formal proof of the following statements.

• $$\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} (x + y) + z = x + (y + z)$$
• $$\forall x \in \mathbb{N}\forall y \in \mathbb{N} x + y = y + x$$
• $$\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} (x \times y) \times z = x \times (y \times z)$$
• $$\forall x \in \mathbb{N}\forall y \in \mathbb{N} x \times y = y \times x$$
• $$\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} x \times (y + z) = (x \times y) + (x \times z)$$

It could be even later that they're taught the inductive definition of the decimal notation for each natural number which goes as follows

To get to the successor follow the following rules

• If the last digit is not 9, increase that digit by 1
• If the sting of characters is 9, change it to 10
• If the last digit is 9 but is not the only digit, change that digit to a 0 and change the string of all digits before it to the string that represents the successor of that string

They can also be left to figure out on their own how to multiply numbers in decimal notation mentally and to know how to divide two numbers when ever the division problem is defined as well as learning how to check whether it actually is defined and write undefined when it's not defined instead of only knowing how to form a division strategy that always works when the given problem is defined and being the type of person who would write 8 as the answer to 14 $$\div$$ 3.

They might not get a feel that real numbers other than natural numbers actually exist but later, they could be taught how to become a savant at number theory who can discover results without reliance on a calculator, pencil and paper, a computer program, or the internet. They might later work in stronger systems of number theory dealing with more complex properties of the natural numbers the stronger system describes and when they read a book that uses the phrase real number, they might not understand what it's saying but will be so good at noticing patterns that they notice a certain connection between the text in the book and what they discovered about number theory and decide that when they talk about the real number $$\sqrt{2}$$, they're really informally describing a certain property of the natural numbers.

One down side of that is that not all real numbers correspond to a property of the natural numbers the system describes so the person might say that the real numbers it does describe are the only real numbers. They might later create their own version of type theory which is weaker than the already existing type theory to only extend number theory and not be at all an extension of ZF where Godel's first incompleteness theorem on proof systems of number theory is a meaningful statement and they don't assume unrestricted comprehension on all properties of the natural numbers anymore than normal type theory assumes unrestricted comprehension on all sets and will treat $$\mathbb{R}$$ like a proper class and not get along with mathematicans who work in Zermelo-Frankel set theory which does assert that there is a set of all subsets of the set of all finite ordinals and write formal proofs in it. Also, we can informally define ordinal numbers in terms of proof systems of pure number theory where stronger systems can define larger ordinal numbers. However, in that version of type theory, if we include the Church-Turing theris in the type theory, we could probably also derive that the ordinal numbers at and above the Church-Kleene ordinal don't exist at all. See https://mathoverflow.net/questions/133597/what-would-remain-of-current-mathematics-without-axiom-of-power-set/133598#133598

• Jan 3 '19 at 7:29
• A student with no experience with ordinal numbres certainly will not have "yet thought of their own proof that distinct finite ordinal numbers always correspond to different cardinal numbers." In general notions such a quantifiers and abstract variables are far, far more difficult than is basic arithmetic, and the elementary school student capable of giving formal proofs as in the question - moreover before learning how to add fractions - is a very rare beast - I would say a unicorn if I weren't able to imagine a young Grothendieck. Jan 3 '19 at 14:25
• "Students" as a generalization don't EVER need number theory. Less than 1% will study it at all (late or early, doing well or poorly). Jan 3 '19 at 17:00
• I'm sorry, but this all seems to me a non-starter solution to some imaginary problems. To take just one example, why is it important that school children know there exist irrational numbers and how (in your experience) has this lack been a problem? Surely general number sense is way more important and age-appropriate. However, I do think some exposure to proof at this age can be good, but by "proof" I don't mean the same thing you mean, but rather things like three $17$'s is three $10$'s and three $7$'s, as can be seen from $(10+7)+(10+7)+(10+7).$ Jan 3 '19 at 18:27
• This question begs for an answer that discusses Piaget's theory of cognitive development, particularly with respect to concrete vs. abstract thinking. Jan 3 '19 at 19:05

The way I understand the question is: If students are not taught fractions, but instead formal deductive proofs of properties of natural numbers, would they learn mathematics better?

I find it unlikely.

1. Students struggle with fractions in primary school. Students usually struggle with proofs in gymnasium or university. It seems unlikely that primary school students would master proofs (a very abstract concept), when even the older students with more experience struggle with the matter. Furthermore, everyone attends primary school (around here, at least), while around half the population attends gymnasium (around here) and even fewer take university mathematics classes with proofs. Hence, there is some amount of filtering on who currently learns proofs; not everyone faces it, at the moment, and around here.

2. Fractions are useful outside formal mathematics and even engineering. Deductive proofs of properties of natural numbers are much less so. This argues for teaching fractions to everyone, rather than deductive proofs of properties of natural numbers. As far as I understand, evidence for the transfer effect (of studying mathematics helping with reasoning in general) is weak, so it is a good idea to teach content that is useful in and of itself, especially in primary school.

3. Many teachers would have an awfully hard time understanding deductive proofs of properties of natural numbers; in my teacher education, many future middle school teachers had serious problems with proofs and advanced mathematics, and the effect would likely be even more pronounced for elementary school teachers. They already teach mathematics that not all of them understand very well, which might be one cause of mathematics anxiety among the students. Asking them to teach more abstract mathematics would likely make the problem worse.

I understand the question in the same way Tommi Brander did but I would like to give my own answer. Maybe I misunderstood the question or the age group you meant, but I assumed it to be children who are about 12-13 or under and who are beginners at learning fractions (for instance).

First off, I’m not sure where you get your premise that “learning something young means that that thing is more likely not to be learned properly”.

The article you linked mostly cites studies about Denmark kids. In particular, it says at the end that “In Denmark […] there is universal access to decent pre-kindergarten. Relatively few American cities and towns offer the same thing.” Put that way, of course you would expect kids who are put through better preparation would do better later in life.

The article cites one study about OECD countries, including data from Canada and the United States which says that kids who started school later are more likely to attend University.

I can’t conclude that “not attending University” means that I probably learned fractions wrong. So I’ll remain skeptical with the premise.

Second, even if we were to assume that the studies are right, the thing to do would be to postpone the age at which we start learning (at school), not replace the current content by other content. If kids are truly “getting fractions wrong” right now because they learn too soon, then they would learn number theory wrong instead…

Third, you seem to think that learning fractions is harder than doing number theory proofs. Imagine for one minute that you are a kid and you see this:

$$\forall x\in\mathbb{N}x+S(y)=S(x+y)$$

and this:

$$2-\frac{5}{6}$$

which do you think is easier to understand, or at least would take less time to understand properly?

You also suggest that kids who give “an intuitive reason” to “a certain statement that is obviously true” would be told to start again until “a formal proof” is written and that they can’t “collaborate with another student”. The whole group stays "locked" in that spot until everybody gets it “right”. This is so disgustingly behaviorist and so far off from modern socioconstructivism that I don’t think this would be applicable in the near future.

To clarify, there is currently a shift in paradigm in teaching (well, it depends on the country but I think you'll understand). Essentially, it isn't about "how should we teach this", it's about "how should they learn this".

I'll try to explain it (grossly) here.

Before, the prerogative was that students were "entirely ignorant" and "wrong" until they got the same answer as the teacher. In other words, if a student does something that doesn't meet the teacher's standards, it's rubbish. So the teacher's job was simply to "punish" a student if he had different answers and "reward" if he had the same answers as the teacher. (By "punish", I mean things like saying "that is wrong", or marking 0, etc.) There is basically no regard to what goes on in the students mind, it's all about changing their behavior until they "learn" to do it right (like the teacher). This is, essentially, behaviorism.

Nowadays, this kind of approach is less and less popular. One of the reasons is that we know that kids are capable of being creative and telling them to meet very specific standards essentially inhibits that creativity. (Instead of learning the content, they learn how to do what teachers' want). So if you try to teach a 12 year-old how to do proofs and you tell him to start over until he's doing it just like you want (which, if I understood you correctly, sounds like the work of a much more advanced student), he'll end up completely lost, discouraged and unmotivated. At beast, he'd be able to copy a proof that you wrote on the board, but without understanding very much.

But an even better reason not to use this approach (behaviorism) is because there are better ones out there now, that implement what we now know about metacognition (knowing how we learn), with the advent of research in psychology, mainly.

Here's the picture:

Students arrive in class with their own conception of reality (or "mathematical truth" in our case). They learn new things (what the teacher tells them, what they read in books, what they hear from other students). From this new knowledge, they modify and improve their original conception of reality; they construct a new version of what they think reality is. It might not be perfect yet, but the idea is that this new version is (in general) better than the old one. In another course, or later in life, that same knowledge might be improved further as the student experiences more things. That's constructivism. So this is a pedagogical approach where knowledge is not fixed in place, but rather, is something that is constructed throughout one's life.

Socioconstructivism is essentially constructivism made through interactions with other people (peers, the teacher). Activities like group work, debates and such are characteristic of socioconstructivism. For example, you ask students a question. They answer it on their own. Then they trade their work with a neighbour. If they disagree, they have to argue with them to try and convince them that their answer is better. If they agree, they try to see if the neighbour's work had any interesting differences. By doing this, the students might glean some insight on the content that they wouldn't have had if they had only listened to the teacher.

In case you are worried that "everyone constructs their own reality" is a dangerous view in mathematics, that is where the teacher comes in. His/her role is (among other things) to make sure that what is assimilated (through constructivist activities) is not erroneous.

So if you have a 12 year old who "proves" in a very naive or intuitive way some statement, he's basically using all of his previous knowledge to try and come up with something convincing enough. To expect anything more abstract is a bit too much, since he/she doesn't have any background knowledge on how to do a proper proof. You can start doing proofs when you have some background knowledge about math. But when you first learn math, your background knowledge is just a bunch of intuitions and ideas that are probably littered with falsehoods; so that background knowledge ought to be fleshed out a bit.

For example, when you first learn that $$1+1=2$$, do you have to know that you are using a binary operation that is closed in some field? No. But if you want to make proofs involving fields, you ought to know how $$+$$ works in some intuitive way.

So to conclude with the "paradigm shift" I was talking about, instead of focusing on the content we want to pass on, we should focus on how to pass on the content first and socioconstructivist strategies are much richer than behaviorist ones, for the learners. I'm not entirely against teaching proofs, but the way you presented it goes against the new paradigm, so I'm hoping this answers your question.

Children are not full-fledged mathematicians. Doing proofs requires some mathematical maturity that kids don’t have. It sounds like you want them to be able to do math the way a mathematician does, before they even start learning anything about math.

• @TommiBrander I've added some explanations but might edit further if I find a better way to explain myself. I hope this clarifies for the moment. Please tell me if you think there's anything else I should review. Jan 3 '19 at 16:03
• "Before, the prerogative was that students were "entirely ignorant" and "wrong" until they got the same answer as the teacher. In other words, if a student does something that doesn't meet the teacher's standards, it's rubbish." This has never been the "paradigm" in any healthy educational system. Jan 3 '19 at 16:35
• @DanFox I agree, educational systems were/are not always healthy. Jan 3 '19 at 21:41
• I know that that study doesn't mean it's definite that kids learn something less well if they learn it younger. However, it is reasonable for me to think it's probable that that's the case because of reading that page. Jan 4 '19 at 16:46

It does seem to me that this is the wrong way round. I remember, in primary school, being shown via a rectangular arrangement of $$20$$ dots that $$4$$ rows of $$5$$ was the same as $$5$$ columns of $$4$$ so $$5×4=4×5$$. That wasn't a formal proof—it was an explanation, in visible form, of why the order didn't matter. Neither was it presented so we'd learn what a proof was—it was there to help us understand how multiplication worked.

Proofs are abstract things because of needing to show that something is always true, not just true of the concrete example in front of you. So we were shown the example with a simple explanation of how it worked, not given a proof involving all $$m, n \in \mathbb N$$.

The natural progression is surely

• simple explanations of day-to-day concepts and procedures
• more detailed explanations of less simple ideas
• explanations that are general enough to amount to informal proifs
• formal proofs with all the non-obvious holes plugged.

In the early stages the purpose of the explanations is understanding of whatever concepts are essential to that stage of learning—understanding how explanations work and how they can be made rigorous comes later.

Similarly there's a progression in the ability to make arguments:

• understanding simple arguments (like the $$5×4$$ grid) but not coming up with them
• arguing in order to solve a simple problem (if $$3x=15$$, dividing both sides by $$3$$ keeps them equal to each other and gives us $$x$$)