Teaching Math: Why “Obvious” Beats Memorized Rules

Teaching math well means teaching students to ask “why” until an idea feels inevitable, not just memorizing rules. A math teacher’s central goal is to help students see that mathematical statements have reasons behind them, unlike arbitrary language rules. When students grasp this, they stop treating math as glorified long division and start treating it as an extension of common sense.

Key Takeaways

• The Commutative Property of Addition (3 + 7 = 7 + 3) can be shown as obvious using a simple apple-counting example rather than memorized as a rule.
• A grid model of three rows of seven squares versus seven columns of three squares demonstrates why 3 × 7 = 7 × 3 for any pair of numbers, not just as a coincidence.
• Unlike foreign language rules such as French verb conjugations (“Je vais, tu vas, il va”), which are historical conventions, math statements like why you cannot take the square root of a negative number have underlying logical reasons.
• Long division style repetition in early math education, divide, multiply, subtract, bring numbers down, can leave students believing math is procedural rather than reasoned.
• Lockhart’s Lament and Dan Meyer’s TEDx talk are two resources that argue for engaging students in formulating math themselves rather than following procedures.

What is the main goal when teaching a math class?

The primary goal in teaching a math class is to convey a way of thinking about math, not simply to transfer facts and techniques. Many students find this approach difficult and frustrating at first because it differs from how they were taught previously. Over time, though, many students begin to see math differently, and that shift in perspective counts as a major success even if they eventually forget specific details, facts, and techniques.

How does the Commutative Property of Addition illustrate “obvious” math?

On the first day of class, the statement 3 + 7 = 7 + 3 goes on the board. This is formally called the Commutative Property of Addition, but a simpler word applies: obvious. If you have three apples and someone else has seven, or if you have seven apples and they have three, the total between you is the same either way.

Why is 3 × 7 = 7 × 3 not just a coincidence?

The statement 3 × 7 = 7 × 3 means three groups of seven items each (7, 14, 21) produce the same total as seven groups of three items each (3, 6, 9, 12, 15, 18, 21). Both calculations land on 21, but that alone does not prove it will always work for every pair of numbers, such as 6 × 9 or 12 × 137.

A grid picture makes the reasoning obvious: three rows of seven squares can be read as 7 + 7 + 7, or seven columns of three squares can be read as 3 + 3 + 3 + 3 + 3 + 3 + 3. Counting the grid either way produces the same number of squares. This visual proof extends to any pair of numbers, including 12 × 137, without needing to count them individually or take anyone’s word for it.

What does “obvious” mean in this context?

The word “obvious” here does not mean “anyone would figure it out quickly.” Sometimes making something obvious to oneself takes hours of work. Succeeding means reaching the point of saying, “Of course, it couldn’t possibly be any other way.” That moment of realization is what mathematical understanding is about.

Why is comparing math to a foreign language misleading?

Math is frequently compared to a foreign language, and some math teachers describe math as “just another language.” This analogy is misleading because it confuses the vocabulary of math with the substance of math itself.

Math does have its own language. Words like “root” carry a different meaning than in everyday speech, and words like “polynomial” appear almost exclusively in mathematical contexts. But these words express the math; they are not the math itself. If the word “root” were replaced with “fizbot,” the underlying statement, that you cannot take the square fizbot of a negative number, would remain equally true.

What math shares with language is the need to memorize how components fit together, whether that is French verb conjugation (“Je vais, tu vas, il va”) or the quadratic formula (“negative b plus or minus the square root of b-squared minus…”). Applying these rules incorrectly leads to wrong answers in both cases.

What is the key difference between math rules and language rules?

Asking a French teacher why verb conjugation follows the pattern “Je vais, tu vas, il va” typically produces a historical answer. The pattern is not meant to “make sense” in a deeper logical way; it simply reflects how the language evolved over time.

Math statements, by contrast, have reasons behind them. There is a reason you cannot take the square root of a negative number, and a reason why 12 ÷ 4 produces a smaller result than 12 while 12 ÷ (1/4) produces a larger result than 12. That reason goes deeper than the rule “when dividing by a fraction, flip and multiply.” Exploring what division actually means, using an example like dividing pizza, can lead to the realization that 12 ÷ (1/4) must equal 48 and could not logically be anything else.

How do math rules differ from arbitrary agreements?

If all English speakers agreed tomorrow that “thurple” meant “lunch,” they would be correct by definition simply because they agreed on it. But if all mathematicians agreed that 3 to the 7th power equals 7 to the 3rd power, they would be wrong regardless of consensus. Engineers and the structures they build rely on mathematical facts that no one gets to decide by agreement; these facts are discovered, not invented, and anyone can rediscover them through reasoning.

Why do many students resist thinking through math problems?

Many students respond to open-ended math questions with “Just tell us the answer,” rather than attempting to reason through a problem. Early schooling bears much of the responsibility for this pattern. Few elementary teachers enter the profession because they love math specifically; more often they are drawn to teaching because they love working with children, telling stories, playing games, or because it offered a stable career.

When a teacher learns rules from a textbook and teaches strictly from that book, a student’s question like “Why do we need a common denominator to add fractions but not to multiply them?” can leave the teacher without an answer. Rather than risk appearing uncertain, the teacher may respond, “Because that’s how we do it.”

Students absorb the lesson that thinking is unnecessary and that learning rules is what matters. Math becomes reduced to repetitive procedure: divide, multiply, subtract, bring numbers down, repeated across pages of exercises through high school. This pattern leaves many students concluding that math is pointless and boring.

What should students take away from a math class years later?

Common Core students will likely forget the laws of logarithms within five years, and calculus students will likely forget the quotient rule. That forgetting is not a concern, since specific formulas can be looked up when needed.

What matters more is remembering that math is not merely a foreign language, a set of rules to memorize, or glorified long division. Math is the most perfect elaboration of common sense, building on reasoning that everyone already possesses. Whenever someone states “this is the way it is” in math, there is both a right and an obligation to ask why, continuing until the answer feels like, “Of course. Now that I see it this way, it’s obvious.”

Further Reading

Two resources expand on this approach to teaching math. Their specifics differ, but both align with the idea that actively engaging students in formulating math and applying their own thinking matters more than having them follow procedures to reach a correct answer.

• Lockhart’s Lament, a longer essay worth reading in full; the opening “musician’s nightmare” section is a strong starting point if time is limited.
• Dan Meyer’s TEDx talk, which has been shown directly to students as a complement to classroom discussion.

Frequently Asked Questions

What is the Commutative Property of Addition?

The Commutative Property of Addition states that 3 + 7 equals 7 + 3, meaning the order of numbers being added does not change the sum. This can be understood intuitively: if you have three apples and someone else has seven, or the reverse, the combined total is identical either way.

Why does 3 × 7 equal 7 × 3?

A grid of three rows by seven columns contains the same number of squares whether counted as three groups of seven (7 + 7 + 7) or seven groups of three (3+3+3+3+3+3+3). Both counting methods total 21 squares, and this same grid logic applies to any pair of numbers being multiplied.

Is math really just another language?

Math has its own vocabulary, with words like “root” and “polynomial” carrying specific meanings, but that vocabulary is separate from the math itself. Unlike language rules, which are historical conventions, mathematical statements have underlying logical reasons that can be explored and understood rather than simply memorized.

Why can’t you take the square root of a negative number?

There is a specific mathematical reason behind this rule rather than it being an arbitrary convention. Exploring what square roots and negative numbers actually represent leads to understanding why no real number, when multiplied by itself, produces a negative result.

Why do some students think math is boring or pointless?

Many students are taught by elementary teachers who entered the profession for reasons other than a love of math and who may not fully understand the reasoning behind the rules they teach. When students ask “why” and receive answers like “because that’s how we do it,” they learn to follow procedures without thinking, reducing math to repetitive steps that feel meaningless.

What should students remember from math class in the long run?

Specific formulas and rules, such as the laws of logarithms or the quotient rule, are often forgotten and can be looked up when needed. What matters more is understanding that math is an elaboration of common sense built on reasoning, and that any mathematical claim can and should be questioned until it becomes clear why it must be true.