Math word problems: the routine that stops the freeze

Your child can rattle off 7 × 8 and still sit frozen in front of a four-line story about vans and a school trip. Math word problems fail for a different reason than sums fail, and another round of arithmetic drill does nothing for them. What helps is a routine the child runs the same way every time.

Why math word problems are harder than the same sum written out

A word problem is harder than the bare sum because the arithmetic comes last, not first. Before your child can compute anything, they have to read the text, work out what is happening, and decide which operation matches the situation. The calculation 24 − 18 is easy; deciding that the problem calls for 24 − 18 is the hard part.

Two jobs at once

Reading load and calculation compete for the same attention. A child who reads slowly spends most of their working memory on decoding the sentences and has little left over for the mathematics. That is why a strong calculator can look weak on a word problem, and why reading the problem out loud often unlocks it.

The US standards expect this work early. Grade 2 asks children to use addition and subtraction within 100 to solve one- and two-step word problems involving adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions (2.OA.A.1).

What goes wrong when your child freezes

Freezing on a word problem usually has one of three causes, and weak arithmetic is not among them. Your child may not picture the situation, may not be sure what the question asks, or may be hunting for a trigger word instead of reading. The third cause is the one children pick up fastest and keep longest.

The key word trap

Key words promise a shortcut that breaks on ordinary problems. Children are told that altogether means add and that left means subtract, and both rules fail on problems a third grader will meet this year.

• Altogether with subtraction. Ana has 12 stickers altogether, and 5 of them are stars. How many are not stars? The answer is 12 − 5 = 7.
• Left with addition. Ben gave away 3 marbles and had 4 left. How many did he start with? The answer is 4 + 3 = 7.
• More with subtraction. Julie has three more apples than Lucy, and Julie has five apples. How many apples does Lucy have? The answer is 5 − 3 = 2.

The third example comes straight from Table 1 of the Common Core glossary, which sorts addition and subtraction situations into adding to, taking from, putting together or taking apart, and comparing. The writing team's progression for Operations and Algebraic Thinking names these cases as misleading language that suggests the wrong operation, and lists them among the hardest subtypes in grade 1.

A routine for solving word problems, step by step

A routine works because it puts the arithmetic last. One problem carries the whole routine below: a class of 24 students goes on a trip, each van holds 6 students, three vans are already full, and the question asks how many students are still waiting.

1. Read it twice. The first read is for the story, the second for the numbers. Ask your child to cover the numbers with a finger on the first pass.
2. Say what is happening in your own words. Your child retells the situation with no numbers at all: some children are going on a trip in vans, some are already seated, and the rest are waiting.
3. Name what is being asked. The question asks for students still waiting, not for vans and not for empty seats. Write that phrase at the top of the page before anything else.
4. Draw it. Three full vans become three boxes of 6, and the whole class becomes a bar of 24 above them. The gap at the end of the bar is the answer.
5. Estimate. Three vans hold 18 seats, which is most of 24, so the answer should be small. Anything over 10 is wrong before your child works it out.
6. Calculate. 3 × 6 = 18 students are seated, and 24 − 18 = 6 students are waiting.
7. Check the answer against the question. The question asked for students, the answer is 6 students, and 6 sits inside the estimate.

Step 3 is the step the standards ask for by name: grade 3 children solve two-step word problems using the four operations and represent them with equations using a letter for the unknown quantity (3.OA.D.8). A few rounds of solve for x make that letter feel ordinary long before algebra.

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The step most families drop

Estimating before calculating is the step that disappears first from homework, and it is the one that catches wrong answers. The What Works Clearinghouse practice guide Improving Mathematical Problem Solving in Grades 4 Through 8 puts monitoring and reflecting on the problem-solving process among its strongest-evidence recommendations, supported by prompts a child asks themselves at each stage.

The same grade 3 standard asks children to assess the reasonableness of answers using mental computation and estimation, rounding included. An estimate spoken out loud costs five seconds and rules out every answer of the wrong size.

Four ways to draw a word problem

Drawings turn a story into a picture of how the quantities relate, which is the part your child is missing. The What Works Clearinghouse guide recommends teaching a few visual representations well rather than many badly, and names strip diagrams, number lines, tables and schematic diagrams as the working set.

Which model to pick

The problem's own shape picks the model. A problem about who has more wants a bar model, a problem about a journey or a clock wants a number line, and a problem with a repeating group wants a table. Your child should own two of them well before a third arrives.

Mixed practice keeps all four in play. A mixed worksheet for 4th graders puts the four operations on one page, so your child has to decide which one fits instead of repeating the same operation twenty times.

Free printable worksheetMixed Operations — 4th GradersWorksheet · PDF

Problem types worth naming at home

Naming the type of problem gives your child somewhere to start. The Common Core glossary sorts addition and subtraction situations into four families, each with the unknown sitting in a different position: add to, take from, put together or take apart, and compare.

Compare problems and change problems

Compare problems and change problems differ in whether anything moves. A change problem has something happening over time, such as Ben giving away 3 marbles. A compare problem has nothing happening at all: Julie and Lucy both keep their apples, and the difference between them is what the question wants.

The difference in a compare problem is invisible, and that is what makes it hard. The progression document makes the point plainly: the difference is not physically present in the situation and has to be constructed in a drawing. A bar model helps more here than anywhere else, and a round of comparing numbers is a useful warm-up before a set of compare problems.

Two-step problems arrive in grade 3 and never leave. The order of the two calculations matters, which is where order of operations starts to pay off, and division is the operation children misread most often because sharing and grouping sound alike in English. The method itself is covered in how to teach long division.

How to check your child has it

Five signals tell you whether the routine has landed, and none of them is a score on a worksheet. Watch what your child does in the first thirty seconds after reading a problem.

• Can they retell the problem with no numbers? A child who describes the situation in their own words has understood it. A child who repeats the sentences back word for word has not.
• Does the estimate come before the answer? An estimate spoken before any calculation shows your child is thinking about size, and it catches wrong-operation answers on the spot.
• Do they draw without being asked? A drawing that appears on its own, before the arithmetic, is the clearest sign the routine has become a habit.
• Can they say what the answer means? The number 6 is not an answer; 6 students still waiting is.
• Do they notice a wrong answer? A child who says an answer cannot be right, and explains why, is monitoring their own work.

A two-minute test at the kitchen table

One test shows all five signals at once: hand your child a problem and ask them to explain it before solving it. Pick a problem one grade below their level, so the arithmetic is never the obstacle. If the explanation is fluent and the estimate arrives unprompted, the routine is working.

Common mistakes and your word problem checklist

Five mistakes account for most wrong answers in grades 2 to 5, and each one has a fix that takes seconds.

• Grabbing the numbers in the order they appear. A child who writes 24 − 6 because 24 came first has skipped the situation entirely. The fix is step 2: retell the story with no numbers.
• Key-word hunting. Words such as altogether, left and more point at the right operation often enough to feel like a rule, and go wrong often enough to cost marks. The fix is to ask what is happening, then what is being asked.
• Answering the wrong question. A two-step problem often makes the first result look like the answer, and 18 seated students is not the number still waiting.
• Ignoring units. An answer of 6 with no label hides an error, because 6 vans and 6 students are different claims about the same arithmetic.
• Skipping the second step. Two-step problems in grades 3 and 4 hide the second step behind a comma, so the answer has to be checked against the question rather than against the last calculation.

One edge case deserves a sentence of its own. A problem can contain a number that is never needed, and no rule about key words will spot it. A child trained to use every number in the text will be caught by the first problem that carries a spare one.

Run this checklist at the next homework session:

• Read twice, covering the numbers on the first pass.
• Retell the story with no numbers.
• Write the question as a short phrase before starting.
• Draw one model, the same one for that problem type every time.
• Estimate out loud, then calculate.
• Read the answer back with its unit and compare it to the question.

Ten minutes an evening is enough, and the problems can be spoken rather than printed. Ideas for practicing together sit on the math games for parents and kids page, and fractions explained for parents covers the topic that word problems break on next.