How to Help Your Child With Order of Operations

My fifth grader showed me his homework the other night, proud of it. The problem was 3 plus 4 times 2, and he had written 14, big and confident. I asked him how he got there and he said, plainly, "you go left to right, like reading." Then, a few problems down, a different one had parentheses, and there he stopped cold and looked up at me, because now the left-to-right rule he trusted was contradicting the fence of parentheses staring back at him. He was not being careless. He had two rules in his head that did not know about each other.

If your child rushes an expression left to right, or chants "PEMDAS" and still gets it wrong, or freezes the moment parentheses and multiplication show up in the same line, here is the short answer: order of operations is not a list of six ranked steps. It is four tiers of strength. Grouping symbols first, then exponents, then multiply and divide together, then add and subtract together, and the last two tiers are worked left to right. Teach it as a pecking order instead of a straight line and most of the mistakes stop. Here is how we got there at our kitchen table.

How to help my child with order of operations

Order of operations is the agreement that everyone evaluating the same expression gets the same answer. Without it, 3 plus 4 times 2 could be 14 or 11 depending on who is reading, and math would stop being reliable. To help your child, you teach the four tiers in order and, just as importantly, you teach that the bottom two tiers are pairs, not single steps. Multiply and divide share a tier and you work them left to right, whichever comes first. Add and subtract share a tier and you work those left to right too. Get the tiers straight, drill the "left to right inside the pair" part until it is automatic, and only then reach for the mnemonic, because the mnemonic is a memory aid for a structure your child can already see, not the structure itself. Tiers first, letters second. Do it in that order and the guessing stops.

Why PEMDAS backfires

Here is the uncomfortable part. PEMDAS, the mnemonic almost every parent learned, causes a huge share of the errors it is supposed to prevent. Spelled out as P, E, M, D, A, S, it looks like six steps in a strict line, and a child reads it exactly that way: parentheses, then exponents, then multiplication, then division, then addition, then subtraction, each one finished before the next begins. So they conclude that multiplication always comes before division and addition always comes before subtraction.

Neither of those is true, and the counterexamples are everywhere. Take 8 divided by 2 times 4. A child running PEMDAS literally multiplies first, 2 times 4 is 8, then divides, 8 divided by 8 is 1. The real answer is 16, because division and multiplication are the same strength and you go left to right: 8 divided by 2 is 4, then times 4 is 16. Same trap with subtraction: in 10 minus 3 plus 2, the literal reader adds 3 plus 2 to get 5, then 10 minus 5 is 5, when the honest left-to-right path gives 10 minus 3 is 7, plus 2 is 9. The mnemonic did not fail because your child forgot it. It failed because they remembered it too well.

This is why a lot of teachers now write GEMS or GEMDAS instead: grouping, exponents, then multiply-and-divide stacked as one step, then subtract-and-add. The stacking is the whole point. It tells the truth that PEMDAS hides.

Teach the four tiers, strongest to weakest

The fix is to replace the straight line in your child's head with a pecking order they can picture. Four tiers, and you always work the strongest one still on the board:

• Tier 1, grouping.Anything inside parentheses or brackets gets done first. The symbols are a fence that means "do me now, before anything outside touches me." In 20 minus (4 plus 2) times 3, the parentheses go first: 4 plus 2 is 6, and now the line is 20 minus 6 times 3.
• Tier 2, exponents. The powers come next. In 5 plus 2 to the third, the exponent is 2 times 2 times 2, which is 8, so the line becomes 5 plus 8. Exponents do not show up until sixth grade, so younger kids can skip this tier for now.
• Tier 3, multiply and divide, left to right. One tier, not two. Scan left to right and do whichever comes first. Back to 20 minus 6 times 3: the times is the strongest thing left, 6 times 3 is 18, and the line becomes 20 minus 18.
• Tier 4, add and subtract, left to right. Also one tier. 20 minus 18 is 2, and we are done.

The phrase to keep saying is strongest tier wins. A child who asks "which tier is strongest right now" before every move owns something a child who only chants PEMDAS does not: a way to know what to do next no matter how the operations are arranged.

Make the plan visible before any number moves

The habit that changed things for us costs nothing: underline the plan before computing. Have your child scan the whole expression and mark the tiers first, circling the grouping, then the exponents, then the multiply-divide, then the add-subtract, so the order is on paper before a single number changes. It slows them down for about a week and then it makes them faster, because they stop backtracking.

It also turns a wrong answer into information you can actually use. When the result is off, you do not have to hunt through the arithmetic. You look at the plan. Did they mark the tiers in the right order? Then the structure is solid and the slip is in the computing, probably the multiplication underneath, since the multiply-divide tier leans directly on the times tables. Did they mark a tier out of order? Then the arithmetic is fine and the rule is the gap. Different misses, different fixes, and the visible plan is what lets you tell them apart.

Practice without turning it into a fight

Here is where parents, me included, get it wrong. Drilling a page of four-operation expressions before the tiers are solid does real damage. The child guesses, gets a red mark, guesses again, and the lesson that sticks is that math is a coin flip with a grade attached. Fluency should grow out of the tiers, not get forced ahead of them.

So keep the reps short and pointed:

• Play "which tier wins?" Write one expression, and instead of solving it, have your child just name the first move and why. Ten seconds each. You are drilling the plan, not the arithmetic.
• Feed the trap on purpose. Give them 8 divided by 2 times 4 and 10 minus 3 plus 2, the exact expressions PEMDAS gets wrong, and let them catch that left-to-right beats the mnemonic. Kids remember the rule they discovered.
• Keep sessions to five minutes. Two short reps a day beats one long grind, and it does not sour the whole subject.

When your child gets one wrong, you do not have to supply the answer. Ask, "which tier is strongest here, and are those two the same strength?" and their reply tells you exactly which piece slipped. Reading an expression correctly is the same muscle as reading a math word problem, so this habit pays off well past this one topic. Carol Dweck's research on praising strategy over speed is worth remembering here: praise the plan your child made, not how fast they finished.

Figure out where the gap really is

The nice thing about teaching the tiers out loud is that a miss tells you where the gap sits. If your child cannot name the four tiers in order, the structure has not set yet, so go back to the pecking order. If they name the tiers fine but keep flattening multiply-and-divide into a strict left-of-right rule, the tier idea is solid and the "same strength, left to right" part is the fix. If the plan is right but the numbers are wrong, check the multiplication, because a shaky times-table shows up here in plain sight. Different gaps, different fixes, and you can only tell them apart by watching the reasoning, not the final number.

It is also worth getting an outside read now and then, because a single homework sheet is a small, noisy sample. Test My Kid is a free, eight-minute adaptive assessment for math and reading, grades K through 8, calibrated to the same NWEA MAP and iReady benchmarks schools use. It reports which topics are solid and which are shaky, so you can see whether order of operations is an isolated soft spot or part of a wider multiplication or expression-reading pattern. If it looks like more than this one topic, the signs your child is struggling in math are worth a read too.

My son still slows down on the long expressions with a bracket buried in the middle. But now the slowing down is him marking the tiers before he computes, not adding left to right and hoping. The rule never changed. He just stopped treating the operations as a sentence to read straight through and started seeing them as four tiers, strongest first, the way they were built to be read.