Solving a word problem asks a child to decode mathematics from a situation. Posing a word problem reverses the direction: the learner starts with a mathematical relationship and builds a situation that fits it.
That reversal is powerful. A child who can solve 47 + 28 = 75 may still not understand what kinds of stories addition can represent. When asked to create a valid problem for the equation, the learner must coordinate quantities, roles, language, units, unknown position and the relationship between them. Problem posing therefore exposes understanding that routine answer-getting can hide.
This guide develops Primary 2 problem posing through creation, rewriting and reverse engineering. Students create word problems from equations, diagrams, graphs, answers and conditions; change which quantity is unknown; repair incomplete or impossible questions; compare near-identical stories with different operations; and explain why the constructed problem matches the mathematics.
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When a learner can build a problem from a relationship, the relationship is becoming something they own rather than something they merely recognise.
Why This Guide Exists
NRICH explicitly treats question posing as part of young learners’ mathematical curiosity and uses data investigations to begin from students’ own questions. The Singapore Mathematics framework places problem solving, reasoning, communication, connections, applications and modelling at the centre of the curriculum. Problem posing sits naturally beside those aims because it asks students to construct and communicate a mathematical situation rather than only consume one.
This guide is distinct from the existing Open-Ended Problems guide. Open-ended work asks students to search across several valid answers or methods. Problem posing asks them to design the problem itself and control whether the mathematical structure is valid, complete and solvable.
The Problem-Posing Cycle
| Stage | Student job | Core question |
|---|---|---|
| Start | Choose an equation, model, graph, answer or context. | What mathematics must the problem contain? |
| Assign | Give each quantity a role. | What does each number mean? |
| Construct | Write a situation and question. | Does the wording preserve the relationship? |
| Solve | Work the posed problem. | Does it produce the intended mathematics? |
| Check | Test units, conditions and completeness. | Is there enough information and no contradiction? |
| Rewrite | Change unknown, context or condition. | What changes in the solution structure? |
1. A Word Problem Is a Mathematical Structure Wearing a Context
“There are 28 red balloons and 19 blue balloons. How many balloons altogether?” is not fundamentally about balloons. Its structure is combine: two known parts form an unknown whole. The context gives meaning, but the relationship determines the operation.
2. Start From an Equation
Given 32 + 17 = 49, students can write: “A shelf has 32 storybooks and 17 science books. How many books are on the shelf altogether?” The quantities and operation match the equation.
3. One Equation Can Support Different Story Structures
32 + 17 = 49 can represent combine, change-increase or comparison where a smaller amount and difference produce a larger amount. Problem posing helps students see that the same arithmetic sentence can model several relationships.
4. Start From a Subtraction Equation
For 63 − 28 = 35, students might write a take-away story, a missing-part story, a comparison-difference story or a change-decrease story. The posing task should name which structure was intended.
5. Start From a Multiplication Equation
For 4 × 6 = 24, a valid equal-group story is: “There are 4 bags with 6 marbles in each bag. How many marbles are there altogether?” Each number has a role: number of groups, group size and total.
6. Start From a Division Equation
For 24 ÷ 6 = 4, students can create a sharing story: 24 items shared among 6 groups gives 4 per group. Or a grouping story: 24 items arranged 6 per group gives 4 groups. The same division equation supports two different unknown roles.
Problem posing is strongest when students can explain not only that the equation fits, but which role each number plays.
7. Start From a Bar Model
Give a part-whole bar with parts 27 and 36 and an unknown whole. Ask students to create a story that matches. Then give two aligned comparison bars and ask for a different story structure using the same numbers.
8. Start From a Number Line
A number line showing 45 → 65 with a +20 jump can become: “A jar had 45 beads. Twenty more were added. How many beads are there now?” Ask students to identify start, change and result before writing the context.
9. Start From an Array
A 3-by-5 array can inspire rows of chairs, trays of buns, packets of stickers or windows in a grid. The story should preserve equal groups. A context with uneven groups would not match the array.
10. Start From a Picture Graph
Give a graph where one symbol represents two pupils. Students can pose questions about totals, differences, greatest/least categories or combined categories. This turns a finished representation into a generator of mathematical questions.
11. Start From an Answer
“Create a problem whose answer is 36” is open, but not uncontrolled. Students must produce a solvable question and verify that the intended answer really is 36.
12. Start From a Unit
Ask students to create a question whose answer is 12 litres, 18 metres, $4.50 or 35 minutes. The unit constrains the kind of situation that makes sense and reinforces quantity meaning.
13. Start From a Condition
“Create a word problem that uses two even numbers and has an answer less than 50.” Conditions force deliberate number choice and give students a way to check whether the posed problem satisfies the task.
14. Change Only the Unknown Position
For a combine structure, compare:
- 27 red + 36 blue = ? total.
- 27 red + ? blue = 63 total.
- ? red + 36 blue = 63 total.
The numbers may stay the same while the unknown changes position, producing a different solving demand.
15. Change the Story but Keep the Structure
Take a comparison problem about cards and rewrite it using money or length while keeping larger, smaller and difference roles unchanged. This develops transfer beyond surface vocabulary.
16. Change the Structure but Keep the Numbers
Using 42 and 18, create one combine story, one change story and one comparison story. Students should explain why the same numbers can require different operations depending on their roles.
17. Rewrite Keyword-Trap Problems
Create two problems containing the phrase “more than”: one requiring addition and one requiring subtraction. This directly challenges the idea that a keyword uniquely determines the operation.
18. Create Near-Miss Pairs
Write two almost identical problems where only the unknown changes. Example: “Lina has 12 more than Mei” with Lina unknown versus Mei unknown. Ask classmates to explain why the operations differ.
19. Reverse-Engineer a Worked Solution
Give the working 58 − 23 = 35 and ask students to invent a problem that would reasonably produce this calculation. Then ask for a second problem with a different structure but the same equation.
20. Reverse-Engineer a Two-Step Solution
Give:
- 42 + 17 = 59
- 59 + 42 = 101
Ask students to construct a situation where the first answer is an intermediate quantity and the second answers the final question. This requires state tracking and coherent dependency between steps.
21. Problem Posing From Fractions
Show 3/8 and ask students to write a story identifying the whole and eight equal parts. A valid story must preserve equality of the parts and make clear why three of them are selected.
22. Problem Posing From Money
Give $10, a cost of $6.40 and an unknown change. Ask for a shopping story. Then change which quantity is unknown: payment unknown, cost unknown or change unknown.
23. Problem Posing From Time
Start time 2:35 pm, duration 45 minutes, end time unknown. Students write a context such as a lesson, journey or activity. Then rewrite it with the duration unknown while keeping start and end times fixed.
24. Problem Posing From Measurement
Give two lengths and a difference. Students create a comparison story using metres. The context should involve lengths, not masses or capacities, because the units constrain the quantity type.
25. Problem Posing From Data
After collecting class data, ask students to pose one total question, one difference question and one greatest/least question. This strengthens the link between representation and operations.
26. Create an Incomplete Problem Deliberately
Write: “A box has 28 blue beads and some red beads. How many beads are there altogether?” Ask classmates what information is missing and how the question could be repaired.
27. Create a Problem With Extra Information
Write a solvable problem that includes one irrelevant number. Then ask another student to identify which information is unnecessary and explain why it does not connect to the target unknown.
28. Create an Impossible Problem
At an appropriate level, students can intentionally create contradictory conditions such as “Find two even numbers that add to 15.” The class then explains why no solution exists. This develops condition checking.
29. Repair an Impossible or Ambiguous Problem
Problem posing includes editing. Change one condition, number, unit or phrase so that the question becomes clear and solvable. This shows that mathematical writing can be revised for logical completeness.
A good posed problem is not merely grammatical. It is mathematically coherent.
30. The Solvability Check
- Is the target unknown clear?
- Are the necessary quantities given or derivable?
- Are units compatible?
- Does the wording preserve the intended relationship?
- Do the conditions permit at least one solution?
- If one unique answer is expected, is the information sufficient to determine it?
31. The Realism Check
A mathematically solvable story can still be absurd. A pencil that is 8 metres long or a school bag weighing 300 kg should trigger revision. Realism is not always required in pure mathematics, but a real-world word problem should respect ordinary scale unless the unusual value is intentional.
32. The Language Check
Pronouns and comparison phrases should have clear referents. “She has 12 more” is incomplete unless the reader knows who “she” is and 12 more than whom. Mathematical language must locate the roles unambiguously.
33. The Unit Check
If the answer is meant to be 7 litres, the story should describe liquid volume. If it is 35 minutes, the problem should ask for duration. Units help validate the context.
34. Worked Example | Create From an Equation
Equation: 54 − 19 = 35.
Possible posed problem: “A basket held 54 oranges. Nineteen were taken out. How many oranges remained?”
Check: start 54, change decrease 19, result unknown. The story matches subtraction and produces 35.
35. Worked Example | Rewrite the Unknown
Original: “A basket held 54 oranges. Nineteen were taken out. How many remained?”
Rewritten: “A basket had some oranges. Nineteen were taken out and 35 remained. How many oranges were there at first?”
The context and numbers are related, but the unknown has moved from final amount to starting amount, changing the solving direction.
36. Worked Example | Create From a Comparison Model
Model: larger amount 72, difference 18, smaller amount unknown.
Possible story: “Noah has 18 more cards than Sara. Noah has 72 cards. How many cards does Sara have?”
Check: 72 − 18 = 54. The phrase “18 more than” describes the relationship; it does not force addition.
37. Worked Example | Create a Two-Step Problem
Desired working:
- 38 + 17 = 55
- 55 − 9 = 46
Possible story: “A box had 38 pencils. Seventeen more pencils were added. Then 9 pencils were given away. How many pencils remained?” The intermediate state after adding is 55; the second event acts on that updated quantity.
38. Common Error | Story Does Not Match the Equation
A student given 4 × 6 = 24 writes “Mina has 4 apples and buys 6 more.” That story describes 4 + 6, not equal groups. Repair by asking what the 4 and 6 must represent in a multiplication situation.
39. Common Error | Numbers Are Inserted Without Roles
Students may write a sentence containing all required numbers but never connect them mathematically. Require a brief role label: first part, second part, whole; start, change, result; groups, size, total; larger, smaller, difference.
40. Common Error | Question Is Missing
A story can describe quantities without asking what to find. Add a precise question whose unknown corresponds to one missing role in the structure.
41. Common Error | Too Much Information Accidentally Creates Ambiguity
Adding extra characters, dates and quantities can obscure which relationships matter. Primary 2 problem posing should favour clarity first, then deliberate complexity when students can control relevance.
42. A Problem-Posing Quality Checklist
| Check | Question |
|---|---|
| Structure | What relationship does the problem represent? |
| Roles | What does each number mean? |
| Unknown | What exactly must be found? |
| Completeness | Is enough information supplied? |
| Units | Do the quantities use sensible units? |
| Language | Are roles and referents clear? |
| Solvability | Does the problem have the intended solution? |
| Verification | Can the equation/model be recovered from the story? |
43. Repair Path
If students create stories that do not match equations, reduce the task. Give a labelled model and ask them only to replace the context nouns while preserving the mathematical roles.
44. Stabilise Path
Use the same numbers across several structures and ask students to explain what changed. Then vary unknown positions while preserving one structure.
45. Extend Path
Ask students to create a pair of near-miss problems designed to trick a keyword solver, or to create an incomplete problem that a partner must repair. Extension comes from controlling constraints and structure deliberately.
46. Parent Diagnostic Questions
- What does each number represent in your story?
- What relationship are you trying to show?
- Which quantity is unknown?
- Does your question contain enough information?
- Can you solve your own problem?
- Does the equation recovered from the story match the one you started with?
- How could you rewrite the problem so a different quantity is unknown?
47. Teacher Diagnostic Map
| Observed behaviour | Likely next diagnostic |
|---|---|
| Uses all numbers but wrong operation | Quantity-role understanding. |
| Can pose only combine stories | Range of addition/subtraction structures. |
| Multiplication stories use unequal groups | Equal-group concept. |
| Problem impossible to solve | Necessary-information awareness. |
| Cannot change unknown position | Reversibility and inverse relationships. |
| Story realistic but mathematically vague | Target unknown and explicit relationship. |
48. What Mastery Looks Like
A strong Primary 2 learner can create a coherent word problem from an equation, model, graph, answer or condition; assign meaningful roles to numbers; vary the unknown position; rewrite contexts while preserving structure; identify missing or extra information; solve and verify the posed problem; and explain why the final wording matches the intended mathematics.
Problem posing is mathematical authorship: the learner controls the relationship, the information and the question.
49. Primary 3 Bridge
Primary 3 introduces longer problems, more varied unknowns and a wider range of topics. Students who can already reverse-engineer relationships and write coherent problems are better equipped to recognise structure in unfamiliar questions because they understand how such questions are built.
Evidence and Scope Notes
NRICH highlights problem posing and learner-generated questions as part of primary mathematical curiosity. The Singapore Mathematics framework places problem solving, reasoning, communication, connections, applications and modelling centrally. This guide applies those principles to age-appropriate problem construction; it does not claim that a particular problem-posing routine has one universal effect size for all learners.
- NRICH: P Is for Posing
- MOE Primary Mathematics Syllabus
- EEF: Developing Problem Solving Strategies in Mathematics
