The + and - Symbols: Simple Enough for Elementary Students, Yet the World's Most Confusing Concept
Let me talk about the most challenging concept in the world—at least for me. It’s so simple that even elementary school students understand it. But believe me, it gets you when you least expect it. I’m talking about... + and -. And yes, there’s a twist. Actually, there’s a twist to the twist—so buckle up.
In my systems thinking lectures, I typically explain that when cause and effect move in the same direction, we use the + symbol and refer to it as a positive causal relationship. When they move in opposite directions, we use the - symbol and call it a negative causal relationship. I always emphasize firmly that "positive" and "negative" here are purely mathematical concepts—they have nothing to do with good or bad.
Elementary School Level
For example, when the birth rate increases, the total population also increases. Since both birth rates and population changes move in the same direction, we write:
Births → Population (+)
Conversely, when the death rate increases, the total population decreases. Since the death rate and population change in opposite directions, we write:
Deaths → Population (-)
This is an elementary-level explanation. But when we move to the high school level, the meaning of "same direction" and "opposite direction" becomes much deeper.
Advanced Level
Let's consider today's reality: decreasing birth rates and death rates
What happens to the total population when births decrease? If you think the population decreases, you've fallen into my trap—you'd essentially be killing off perfectly healthy, living people!
What happens to the total population when deaths decrease? If you think the population increases, you've fallen into my trap again—you'd essentially be resurrecting the dead!
To escape this logical trap, we need to think in terms of the bathtub model.
As shown in the diagram, whether you turn on faucet #1 (which controls births) wider or gradually close it, water still flows into the bathtub. In other words, total population increases in either case—only the rate of increase changes.
Similarly, whether you open or close faucet #2 (which controls deaths), water still drains out of the bathtub. Total population decreases in either case—only the rate of decrease changes.
So does this mean that the same direction (+) and opposite direction (-) notation is inappropriate for describing relationships between births and population, or deaths and population?
Not at all! Systems thinking is a crucial language for logically expressing "the relationship between changes in two variables" that influence each other. This language contains a hidden expression—specifically, the following:
Same direction influence means:
If cause (A) increases, the effect (B) increases above what it would otherwise have been, all else being equal.
If cause (A) decreases, the effect (B) decreases below what it would otherwise have been, all else being equal.
Opposite direction influence means:
If cause (A) increases, the effect (B) decreases below what it would otherwise have been, all else being equal.
If cause (A) decreases, the effect (B) increases above what it would otherwise have been, all else being equal.
When births decrease, total population decreases relatively, which means "same direction"—explained through the bathtub model as follows: Births → Population (+)
As shown in Scenario ③, when the birth rate decreases, the population changes, as illustrated in Scenario ④. While the population still increases in absolute terms, it decreases relatively below what it would otherwise have been, all else being equal. This is why we call it the "same direction." In other words, the concept reflects relative change.
Similarly, when deaths decrease, total population increases relatively, which means "opposite direction"—explained through the bathtub model as follows: Deaths → Population (-)
As shown in Scenario ③, when deaths decrease, the population changes, as illustrated in Scenario ④. While the population still decreases in absolute terms, it increases relatively above what it would otherwise have been, all else being equal—or put another way, it increases relatively. This is why we call it the "opposite direction." Again, the concept reflects relative change.
However, the biggest challenge comes when teaching kindergarteners and elementary students. This insight comes from the director of Moliip, the only institution in Korea that teaches systems thinking to pre-school children.
Why Do Addition/Subtraction and Causal Symbols Clash? A Difference in Directional Concepts?
There was a kindergartener who expressed "using air conditioning makes it cooler" as: Air conditioning use → Heat (-). This seemed reasonable because air conditioning use and heat have an opposite directional relationship. However, this child completely failed to understand that "reducing air conditioning use makes it hotter" could also be expressed as Air conditioning use → Heat (-) because it represents the opposite causal relationship. The child interpreted Heat (-) as subtraction. Moreover, the director realized this problem became even more serious when imagining a situation where parents would worry about their children getting confused between +/- symbols of systems thinking and addition/subtraction operations.
Exactly right. No matter how much teachers emphasized that +/- in systems thinking are not addition and subtraction, it was useless because young students are so accustomed to arithmetic operations. So I looked for ways to help them understand systems thinking +/- notation by building on their familiarity with addition and subtraction.
+5+2-4+1.
This seems easy enough at first.
-
In step ①, we start at zero and move five steps to the right. Naturally, we write +5.
So far, everything aligns—the “+” means movement in the same direction. No confusion.
“-” matches both the direction (left) and the subtraction operation. No problem.
However, step ④ now trips us up.
-
We change direction again and move one step to the right. But how do we write that? It’s +1.
Here’s where it gets confusing—especially for young learners.
Even though we just changed direction, we still use the “+” sign, because in math notation, the sign refers to absolute position on the number line—not relative to the previous step.
So their brain says:
“Wait, I just turned around—shouldn’t that be a ‘−’?”
That’s where the logic of arithmetic and the logic of relative direction begin to collide.
Now let’s look at a different example:
-
In step ④, we start at zero and move five steps to the left.
Naturally, we write −5. No problem here.
In step ⑤, we move two more steps in the same direction—still left. But we still write −2, and this is where it begins to get confusing.
“Wait, we’re going in the same direction—shouldn’t it be a ‘+’?”
Then comes step ⑥:
We now switch directions and move four steps to the right.
And we write +4. But this is also confusing.
“I just changed direction—and I’m using a + sign again?”
This is where students start to lose the thread.
Finally, in step ⑦, we turn back to the left and move one step. This is marked −1, which feels natural again: Direction and subtraction align.
So far, we’ve seen that direction and arithmetic are not always aligned. But a deeper question now arises.
In systems thinking, what matters is not which variable changes first,
but how the change in one variable affects the direction of change in another.
The directionality of causal relationships (+ or -) is determined by whether changes between variables move in the same direction or opposite directions. That's why in systems thinking notation, we place the + or - symbol after the second variable or between the two variables. We never attach the symbol to the first variable.
So then why, in math, do we write +5 or −5 at the beginning of an expression?
Why assign direction at the very first step?
The reason lies in how traditional school math is taught:
It begins not with relationships, but with position—specifically, with a coordinate system.
In that system, numbers are seen as fixed values (scalars),
and the “+” or “−” sign refers to a number’s position relative to zero—not to its relationship with other numbers.
So when a child learns +5, they don’t learn “moving in the same direction as something else.”
They learn “five units to the right of zero.”
The concept is anchored in absolute position, not relational change.
As a result, children grow up thinking of “+” as just addition—
not as a symbol of directional influence.
This makes it incredibly hard for them to later understand how systems thinking uses the same symbols
to mean something fundamentally different.
If we try to solve this by introducing multiplication of signs
(e.g., “let’s multiply the new direction by the previous one”),
we only deepen the confusion.
And if we simply tell students, “+ in systems thinking is different from + in math—just memorize it,”
we’ve solved nothing.
We’ve replaced confusion with blind memorization.
So far, I've demonstrated that directional concepts and addition/subtraction concepts are not the same. But this explanation raises a fundamental question. In systems thinking, what matters more than which change occurred first is how that change influences other variables in a directional manner. The directionality of causal relationships (+ or -) is determined by whether changes between variables move in the same direction or opposite directions. That's why in systems thinking notation, we place the + or - symbol after the second variable or between the two variables. We never attach the symbol to the first variable.
So why does mathematics assign meaning by attaching symbols to the initial movements +5 and -5 from zero? The reason is that conventional mathematics education, based on coordinate systems, teaches students to understand numbers as "positions" rather than "relationships." In other words, school mathematics teaches students to perceive numbers as fixed positions (scalars), and coordinate values are understood as "absolute positions" relative to a reference point. Consequently, the systems thinking meaning of directionality and relationality is blocked for children who only learn +5 as "addition."
When we try to understand systems thinking +/- notation through addition and subtraction as shown above, massive confusion occurs in our minds. Trying to solve this by introducing multiplication concepts—multiplying previous signs to understand directional concepts on coordinates—doesn't resolve the problem either. Nor can we solve it by simply having students memorize that addition/subtraction and systems thinking +/- symbols are different.
This made me think more deeply about why such confusion fundamentally occurs. I had been explaining that systems thinking +/- symbols come from mathematical and engineering vector concepts and represent direction, not addition/subtraction. However, I should question whether this explanation is correct, because mathematical vectors include magnitude while systems thinking notation does not.
Mathematical vectors come in three main types:
- Position vector: magnitude and direction from the origin
- Displacement vector: magnitude and direction from one arbitrary point to another
- Free vector: expressing only magnitude and direction, such as "wind at 5km/h toward the northeast"
All these vectors include magnitude. Therefore, emphasizing systems thinking as a vector concept could inadvertently introduce misconceptions. This is precisely the symbol clash problem—the +/- symbols being used in different meaning systems, causing educational confusion. We must maintain mathematical accuracy while approaching this educationally and practically. In other words, we need to carefully design cognitive transfer regarding the meaning of +/- symbols. Using educational terminology, this could be referred to as "conceptual refinement" or "cognitive scaffolding."
Mathematician Friend's Sharp Critique: "Those Are Operators!"
When I showed this blog to my mathematician friend with a PhD, I received an unexpected pushback:
"Benjamin, you're completely confused. In +5+2-4+1, the + and - are operators. Each is an independent operation, and what happened before is completely irrelevant. What do you mean by 'related to previous direction'?"
He was absolutely right. From his perspective, my attempt to fit the systems thinking meaning of "direction" onto mathematical +/- symbols was uncomfortable.
Context 1: +/- are simply operators
- a + b → + is the addition operator
- a - b → - is the subtraction operator
Each operation is entirely independent. Regardless of what comes before, each operator simply performs its own role. The + (addition operator) silently performs only addition, while the - (subtraction operator) silently performs only subtraction.
Context 2: +/- are not operators but simply signs
- +5 → + is a positive sign
- -5 → - is a negative sign
When we're not performing calculations but simply referring to +5 or -5, these values are simply positions relative to zero as a reference point.
My friend's critique was mathematically completely accurate. But this revealed something fascinating.
Same Symbols, Completely Different Worlds
We actually live in at least three different +/- worlds:
๐ข The Mathematical World
- +5+2-4: Independent calculation operators, or when referring to +5, +2, or -4 individually, signs indicating position relative to zero
- Meaning of each symbol: Clearly defined operations or signs
- Relationships: Independent processing, unrelated to previous operations
๐ฏ The Systems Thinking World
- A→B(+), A→B(-): Same direction or opposite direction influence relationships
- Meaning of each symbol: Directionality of influence between variables
- Relationships: Causal relationships between variables are central
๐ถ Children's Cognitive World
- "Plus is good, minus is bad."
- Meaning of each symbol: Emotional, intuitive interpretation
- Relationships: Understanding through connection with existing experiences
Concrete Examples: How Three Worlds Collide
Let's interpret the same situation from three different perspectives:
Situation: Air conditioning use → Heat reduction
Mathematician's interpretation:
"If we define air conditioning usage as x and heat as y, this can be expressed as a linear function: y = -ax + b. Here, -a represents a negative slope coefficient."
Systems thinking expert's interpretation:
"Air conditioning use → Heat (-). When air conditioning use increases, heat relatively decreases, so this is an opposite direction influence relationship."
Kindergartener's interpretation:
"Air conditioner → Heat (-). Heat is minus? If we subtract heat, does it get cold? But when we turn off the air conditioner it gets hot—is that also minus?"
Why Does Confusion Occur?
The problem is that the same symbols are used in completely different logical systems:
| Perspective | +/- Symbol Meaning | Interpretation Basis | Confusion Point |
|---|---|---|---|
| Mathematical | Operators/Signs | Fixed formal rules | Children don’t know these rules yet |
| Systems Thinking | Direction of causal influence | Relational logic between variables | Identical notation to math symbols |
| Child Cognition | Emotional labels (good/bad) | Prior experience | Difficulty understanding abstract relationships |
Especially for children, these three worlds collide simultaneously:
- +/- learned in math class (operators, signs)
- +/- encountered in systems thinking (relationships)
- +/- felt in daily life (good/bad)
๐ Not Just Similar Symbols—Entirely Different Languages
Thanks to my mathematician friend's critique, things became much clearer. This is not about mathematics being wrong or systems thinking being wrong.
Each is a perfectly accurate system within its own domain. The problem is that they use the same symbols but operate by entirely different rules. +/- are just symbols, but the grammar of the languages in which these symbols are used is entirely different. Mathematics follows the grammar of calculation, while systems thinking follows the grammar of relationships.
What we need is a "symbols from another planet" recontextualization strategy. We must help children naturally accept that these are the same shapes, but expressed in different languages.
Plus means same way! (๐)
Minus means different way!! (๐)
Symbols from another planet!
Our very special symbols!
A goes up, B goes up too~
Same direction means plus! (๐)
A goes up, B goes down~
Opposite means minus!(๐)
Different from math symbols~
They show relationship paths!
Plus and minus! Symbol explorers!
๐ง Kids: "Plus!" (๐)
๐ฉ"Opposite direction?"
๐ง "Minus!" (๐)
๐ฉ "What are these?"
๐ง "Symbols from another planet!"
๐ฉ "Who uses them?"
๐ง "We do! Our special symbols!"
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