# The Order in Which Arithmetic Operators are Taught

Should multiplication be taught before addition and subtraction?

The obvious answer for most people is 'no'. However, I think there are a few valid points that could change the way students approach and respond to higher level mathematical concepts.

1. The human brain is inherently logarithmic. We perceive light and volume in orders of magnitude, i.e. candela and decibel. Sources include Weber Fechnel law, and https://www.scientificamerican.com/article/a-natural-log/

This would help students gain an intuitive sense of numbers, which would help them gain a deeper understanding of advanced concepts.

Multiplication rules are easy. . It is easy for students to learn that anything times 1 is itself, everything times 0 is 0, everything times 10 is the same thing followed by a 0 (moved a decimal place over), etc. In contrast, when 100 is added to something it must be added to the hundreds place, which is further abstraction.

Though children are taught that multiplication is iterated addition, there are other methods of teaching it such as geometry or counting rectangular arrays.

1. When students reach algebra, logarithms could have more meaning. Rather than being taught at the end and hardly understood, logarithms could be a valuable tool to teaching concepts such as order of operations, exponents, powers, and more.

2. The skills and intuition developed by multiplication tend to be more used in daily life (when am I going to us this) and are more beneficial as a whole.

However, there are reasons why not to. For example, addition can be done on fingers so it is simple and then addition can be used to help teach multiplication (though I dislike this because students use this as a crutch through secondary education instead of actually memorizing the dang multiplication table, though I don't know if this would make multiplication any less labor intensive or accessible)

Let me know your thoughts, just be sure to include examples and backing instead of isolated opinions :)

• I'm not sure what kind of answer you are looking for here -- this looks like a post that you would use to start a forum discussion, but this is a Q&A site, so the best questions will prompt a certain type of answer. Are you hoping for research articles talking about teaching primary students multiplication before addition? If so, I can add a reference-request tag. Jul 17 '17 at 0:25
• I suppose it is more of an open ended question, which would better suited for a forum. I am interested to see if this idea has been implemented so a reference request tag would be appropriate
– user8490
Jul 17 '17 at 0:35
• "... everything times 10 is the same thing followed by a 0" -- I've had a few community-college students (high school graduates) who were shocked and surprised to be informed of this. Jul 17 '17 at 2:13
• @DanielR.Collins When I was tutoring in a math lab, I watched a college student type '1+3' and '2*2' into their calculator. It blows my mind that people can spend thousands of hours doing math without ever learning it.
– user8490
Jul 17 '17 at 2:24
• FWIW, from a purely algebraic standpoint, addition on the real numbers is a more primary operation than is multiplication. Jul 17 '17 at 15:09

One thing you have to keep in mind is the size of numbers. You don't start your first graders off with five digit numbers, no, you start with one digit, then increase to numbers below $20$, then slowly increase to one hundred. Here, you will run into problems with multiplication, as you can only build the (nontrivial) products $2 \cdot 2$, $2 \cdot 3$, $2 \cdot 4$ and $3 \cdot 3$ at first, while you haven't introduced multiple digits yet.

If we look a little further ahead and discuss the rules taught for adding or multiplying big numbers, you may notice that most multiplication rules use addition, so you will also get into trouble teaching kids how to multiply big numbers, if they don't know how to add them.

Last, but not least, let's talk about the brain. Even if we are assuming that we generally think in a logarithmic scale, there is something far more natural and more important: counting. The first thing taught to kids is counting to do basic addition of small numbers. And I really don't see how multiplication can be taught in any way that is more intuitive than counting fingers (and toes, once the numbers get too big :) ).
On the other hand, we do teach logarithmic scale by looking at the number of digits, but we don't start off with it before doing addition.

Oh, and by the way, I don't think that one single ancient Amazon tribe would be enough to change such fundamental basics of teaching...

• Good thoughts. Counting must be taught and understood first, otherwise numbers have no meaning. It is my thought that they will develop the idea of addition independently and thus have a stronger understanding of it. The Amazon tribe was merely an example of innate human comprehension of numbers and sizes, the same idea had been found in other cultures lacking the formal definitions of math.
– user8490
Jul 17 '17 at 18:30
• I strongly assume (unless you give me convincing source for other claims) that the "logarithmic scale" is similar to our understanding of "size" of numbers by looking at the number of digits. So that would mean two things: 1) We already work with a logarithmic scale and 2) It only gets relevant once the numbers get big - so not something to teach even before addition.
– Dirk
Jul 18 '17 at 9:11
• Regarding "counting fingers": You can teach children to count from 0 through 99 on their fingers, using Roman numerals. The fingers on the right hand are Is, the right thumb is a V, the fingers on the left hand are Xes, and the left hand is an L. Jul 19 '17 at 4:52
• @Jasper: With ten fingers, you could even count to 1023 using binary representation... Most kids develop some skills to count beyond ten just with their fingers, but I think most of them agree that at some points it helps to write things down.
– Dirk
Jul 19 '17 at 8:39
• @DirkLiebhold The social problem of counting binary on fingers often arises at the numbers 4, 128, and 132.
– user8490
Jul 19 '17 at 20:37

I will argue that no such sequence can be logically coherent. The standard definition of multiplication in natural numbers is based on addition being defined first (e.g., Peano axioms for a formal treatment).

In comments, the OP has been asked a few times for an alternative definition for multiplication, and hasn't been able to provide one. Exercises, e.g., skip counting, are equivalent to addition, just under a different name, with which the OP agrees in comments.

• A formal definition of multiplication is irrelevant to the purpose of this question. Although the standard definition uses addition, children don't need to be taught addition before multiplication. This doesn't mean they can't and won't understand addition. The purpose of this question is education-based, not trying to redefine arithmetic. You don't have to know how to add to count a rectangular array of objects even though you can argue it is just addition. Math is about finding patterns, but children's introduction to it is memorizing addition facts and counting on their fingers.
– user8490
Jul 19 '17 at 20:20
• @Bryce: I'll politely but firmly disagree. Definitions matter. Jul 19 '17 at 21:51
• Let's agree to disagree. Definitions are important. Lack of rigorous definitions led to Russell's paradox. But I personally value an intuitive understanding more. I guess in that regard I'm more like a physicist.
– user8490
Jul 19 '17 at 22:49