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I am from Taiwan (ROC). When I was in elementary school, as I remember, instructors and textbooks notated division (of real numbers, for sure) as "÷", which (as I just read Wikipedia) I know is called the obelus. But at that time I read many advanced math book myself, and learned that people use "/", the slash, to notate division (and fraction, if you see this way).

Strangely, when I was in the first year of middle school (equivalent to the 7th grade in the US), the math textbook announced that, from now on, the obelus is to be discouraged and deprecated, and we shall always use fraction, and when the space does not allow, use a slash.

Except elementary school textbooks (and exercise books), I have never seen the obelus. I don't claim I read widely about math, but I have basic familiarity with like analysis, algebra, geometry. Still, I have nowhere seen the obelus. Neither does any scientific publication use it, as far as I recall. I may have had a calculator that has an obelus on it, but I can't find it now.

Why is the obelus taught in the elementary school, though it virtually never occurs elsewhere? I see no advantage in teaching kids one very rare (if ever used) notation -- obelus -- in the beginning, and abandon it, and then enforce the use of another notation -- slash. It is also very mysterious why obelus stands for division, which Wikipedia doesn't say. On the contrary, a slash reminds me it is a fraction "lying down" due to limited space, and is completely intuitive.

From Wikipedia: (lest it be edited away)

Although previously used for subtraction, the obelus was first used as a symbol for division in 1659 in the algebra book Teutsche Algebra by Johann Rahn. Some think that John Pell, who edited the book, may have been responsible for this use of the symbol.

My guess is that the obelus was first used to stand for division. No wonder 17th century people still used strange notation borrowed from Greek text criticism (see above). The convention of algebra equations was still being formed, and the Principia Mathematica's equations were mainly written verbally!

Afterwards, the slash was probably gradually widely used, but elementary school textbooks, somehow, did not follow the trend. The appearance of the obelus in elementary school textbooks, then, is probably a relic of old notation. Do you agree my guess? Is there any source that confirm this, or claim otherwise?

Related posts:

Where else is ÷ used?

Wikipedia: Obelus

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    $\begingroup$ You may also want to try asking this on hsm.stackexchange.com $\endgroup$ – Wraith1995 Aug 9 '17 at 17:11
  • $\begingroup$ Oh, sorry, I don't even know the existence of that site! $\endgroup$ – Aminopterin Aug 9 '17 at 17:13
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    $\begingroup$ Possibly you might also consider opening a question on matheducators.stackexchange... though the best question there they could probably answer (or at least bring up to debate) is about the question of whether elementary (or younger) students could/should be taught using the obelus or the slash to denote division and the current reasoning as to why they are both used... $\endgroup$ – JMoravitz Aug 9 '17 at 17:22
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In elementary school, it's hard for kids to "switch between notation", as we have done later on in our lives.

We start with $\div$, then we go to $/$, and now we do something like $\frac{a}{b}$.

In elementary, students are not familiar with "fractions", or just know the very basics about them. Hence the $\frac{a}{b}$ ratio doesn't make sense ... not until you actually use fractions in middle school and so on.

It's important to realise that $'/'$ the slash, refers to fraction. When we write $(a/b)$, it's because we are lazy and do not want to write $\frac{a}{b}$. So technically it's the same thing as fraction.

This is why students in elementary just use the regular $\div$ symbol. It's just a symbol. No fractions or anything complicated just yet. It's the same reason why kids in elementary use $\times$ to indicate multiplication and not the dot, $\cdot$, as we use when we are older. They do not deal with complex numbers, and so these symbols that we use as adults are really after we see the use of math. In elementary, it's very basic, hence basic symbols.

In the end it's because of the type of math we do. Imagine writing something like this:

$$\int^{\dfrac{x}{5}}_{0}(x^3+\frac{x^2}{5}+\ln(\sin(\frac{x}{4})))dx$$

$$\int^{x\div 5}_{0}(x^3+(x^2 \div 5)+\ln(\sin(x\div 4)))dx$$

We even had to add extra parentheses to make it clear that $x^2\div 5$ is one thing. Heavier math = more compact symbols

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  • $\begingroup$ I still don't completely get it. Whether $\div$ or $/$, they are just a symbol. In themselves, neither is superior. You claim that $\div$ is unrelated to fractions, but $/$ is visually similar to it (as I understand), and thus $\div$ doesn't unnecessarily confuse elementary school students. But, if $/$ is used in the very beginning, and students are told that $/$ and fraction is the same thing, doesn't that even simplify matters further? $\endgroup$ – Aminopterin Aug 10 '17 at 17:00
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    $\begingroup$ It could be maybe $\div$ symbol gives children in younger grades a better conceptual understanding of division before "/". Certainly when I was in elementary, I thought of these "dots" and a "line through them" as placeholders for dividend and divisors. So it made more sense to me personally. Then as I got older I realized the "dots" are not needed, and then just have to "fraction line", as we have now. $\endgroup$ – K Split X Aug 10 '17 at 17:34
  • $\begingroup$ Wow, I have never see it this way, and I am surprised! Your comment makes me realize the first time in my life that $\div$ is a fraction with placeholders!! .... I suspect that many kids would not discover such connection if the instructors did not point out. $\endgroup$ – Aminopterin Aug 11 '17 at 4:39
  • $\begingroup$ @Aminopterin it isnt officially but that's how I always thought of it :P $\endgroup$ – K Split X Aug 11 '17 at 12:38
  • $\begingroup$ I read somewhere once that Euler did teach his elementary age students complex numbers along with fractions as a primitive concept. $\endgroup$ – DanielV Sep 10 '17 at 9:07
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That $a \div b = \frac{a}{b}$ is not trivial, and can be difficult for some students to grasp. For some students, $a \div b$ doesn't even mean anything unless $b|a$. In most of their experiences, $a \div b$ requires $a \ge b$ and $\frac{a}{b}$ requires $a \le b$.

Most students start out thinking of $x \div y$ as "I have $x$ things that I want to fairly allocate to $y$ people". They think of $\frac{x}{y}$ as "it takes $y$ of them to make $x$". Skipping over the lesson where "the process $x \div y$ results in the thing $\frac{x}{y}$" by conflating notation is not going to result in fewer confused children.

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  • $\begingroup$ This is a little confusing, I think. "unless b|a"? Why change from a,b to x,y at mid answer? Could you give numerical examples? "it takes y of them to make x" looks quite cryptic without one (or, to me it did) $\endgroup$ – Rolazaro Azeveires Sep 10 '17 at 10:09
  • $\begingroup$ For most children, division is "how do you divide 6 pieces of candy among 3 people" and they know "each gets 2". So asking them "what if you divide 7 pieces of candy among 3 people" they usually don't know until they are a bit older. This is $\div$. On the other hand, "1/4 of a cake" means they need 4 of them to make an entire cake. Anyone who thinks fractions and division are conceptually same thing probably either never had a conceptual understanding of them before the symbols were dumped on them, or they forgot the day they learned they are interchangeable. $\endgroup$ – DanielV Sep 10 '17 at 14:43
  • $\begingroup$ So why can we not teach them that: $\frac{6}{3}$ is "how do you divide 6 pieces of candy among 3 people"? And later, tell them that $\frac{a}{b}$ need not be integer, avoiding $\div$ altogether? (I don't say $\div$ must be extinguished, but just raise such possibility.) $\endgroup$ – Aminopterin Sep 10 '17 at 16:20
  • $\begingroup$ @DanielV. I thought so, but it took me some time to understand your point (specially the magic in it takes four to make one :-). So I wish you'd add that comment to the reply, it would be clearer. Well... it already helps, having it in comments. Thanks. $\endgroup$ – Rolazaro Azeveires Sep 10 '17 at 21:47
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    $\begingroup$ @Aminopterin. Maybe because we must walk before we can run. And run before we take large leaps. Think physics: we do not start with quantum physics or Einstein's relativity, we don't even start with Newton's laws, we start maybe at Galileo's relativity. I really do not recall how it was for me and I never taught that start up levels. But I note that all the basic operations are on liners: 1+2, 4-3, 4x5, 6÷3. One dimensional, there is nothing "above" nor "below". $\endgroup$ – Rolazaro Azeveires Sep 10 '17 at 21:58

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