Problem Solving

Problem solving is the cornerstone of mathematical thinking and represents the ability to apply mathematical knowledge to unfamiliar situations, real-world contexts, and multi-step challenges. In the Cambridge Primary Mathematics curriculum, problem solving through critical thinking develops students' capacity to analyze, reason, and find creative solutions rather than simply applying memorized procedures. This foundational skill prepares young learners to become independent mathematical thinkers who can tackle complex challenges with confidence.

Critical thinking in mathematics involves more than just calculating answers—it requires students to understand what a problem is asking, identify relevant information, select appropriate strategies, and evaluate whether their solutions make sense. Students must learn to question assumptions, recognize patterns, make connections between different mathematical concepts, and communicate their reasoning clearly. These skills extend far beyond mathematics classrooms, providing essential life skills for decision-making, logical reasoning, and analytical thinking.

Throughout the Cambridge Primary years (ages 5-11), problem-solving skills develop progressively, beginning with simple one-step problems in early years and advancing to complex, multi-step challenges requiring the integration of various mathematical concepts. Mastering critical thinking in problem solving enables students to approach mathematical challenges systematically, persist through difficulties, and develop the resilience necessary for academic success across all subjects.

Problem Solving: The process of finding solutions to difficult or complex mathematical questions by applying knowledge, skills, and reasoning strategies to unfamiliar situations.

Critical Thinking: The objective analysis and evaluation of information to form reasoned judgments, involving questioning, analyzing, and synthesizing mathematical ideas.

Strategy: A planned approach or method used to solve a problem, such as drawing diagrams, working backwards, or finding patterns.

Multi-step Problem: A mathematical challenge requiring two or more operations or procedures to reach the solution, demanding careful planning and organization.

Reasoning: The process of thinking logically about mathematical relationships, making connections, and justifying conclusions with evidence.

Mathematical Vocabulary: Specific words and phrases used in mathematics (such as sum, difference, product, quotient) that help students understand what operations or concepts are required.

Representation: The various ways mathematical information can be displayed, including pictures, diagrams, symbols, tables, graphs, or written descriptions.

Estimation: Making an educated guess about a numerical answer before calculating precisely, used to check if solutions are reasonable.

Verification: The process of checking whether a solution is correct by reviewing calculations, trying alternative methods, or testing the answer against the original problem.

Context: The real-world or story situation in which a mathematical problem is presented, requiring students to extract relevant numerical information.

Variable: An unknown quantity represented by a letter or symbol in mathematical expressions and equations.

Pattern: A regular, repeated sequence of numbers, shapes, or operations that can be identified and extended to solve problems.

Understanding the Problem

The first critical stage in problem solving involves comprehension—fully understanding what the question asks. Students must develop the habit of reading problems carefully, often multiple times, identifying key information and what needs to be found. This involves recognizing mathematical vocabulary that signals specific operations: words like "total," "altogether," and "combined" suggest addition; "difference," "fewer," and "left" indicate subtraction; "times," "each," and "groups of" point to multiplication; while "share," "split," and "per" suggest division.

Young learners should practice highlighting or underlining important numbers, units of measurement, and question words. They must distinguish between relevant information needed for solving the problem and extraneous details included as distractors. For example, in a problem stating "Sarah has 3 red balloons, 5 blue balloons, and 2 yellow balloons. Her friend is 7 years old. How many balloons does Sarah have in total?" students must recognize that the friend's age is irrelevant information.

Problem-Solving Strategies

Successful problem solvers possess a toolkit of strategies they can apply flexibly. The draw a diagram or picture strategy helps visualize relationships between quantities, particularly useful for problems involving measurement, geometry, or comparing amounts. Making a table or chart organizes information systematically, especially valuable for problems with multiple data points or when looking for patterns. The guess and check method involves making reasonable estimates, testing them, and refining based on results—teaching students that mistakes are valuable learning opportunities.

Working backwards proves effective when the final result is known but the starting point must be determined. For instance, if "Tom had some marbles, gave away 8, and now has 15," students work backwards: 15 + 8 = 23 original marbles. The find a pattern strategy involves identifying sequences or rules that can be extended to solve problems, fundamental for algebraic thinking. Using simpler numbers helps students understand problem structure before tackling complex calculations—replacing difficult numbers with easier ones, solving, then applying the same method to the original problem.

The act it out strategy, particularly effective for younger students, involves using concrete objects, role-playing, or physical movement to represent problem situations. Breaking complex problems into smaller steps makes challenges manageable, with students solving one part at a time and combining results. Teaching students to write number sentences or equations translates word problems into mathematical symbols, bridging verbal and symbolic representation.

Reasoning and Justification

Critical thinking demands students not only find answers but explain their reasoning. This involves stating which strategies they used, why they chose particular operations, and how they know their answers are correct. Students should learn to use mathematical language precisely: "I know this is correct because..." or "I chose to multiply because the problem asks for groups of equal size." This metacognitive awareness—thinking about thinking—strengthens understanding and helps identify errors in reasoning.

Logical reasoning requires students to make connections between concepts, recognize relationships, and draw conclusions based on evidence. For example, understanding that if 4 + 7 = 11, then 11 - 7 = 4 (inverse operations) demonstrates logical thinking. Students develop the ability to generalize from specific examples to broader rules, such as recognizing that adding zero to any number leaves it unchanged, or that multiplication is commutative (order doesn't matter).

Checking and Verifying Solutions

A crucial but often overlooked aspect of problem solving is verification—confirming answers make sense. Students should habitually ask: "Is my answer reasonable?" This involves estimation before calculating (rounding numbers to check approximate size of the answer) and reviewing calculations for computational errors. Verification might include using inverse operations (checking 45 ÷ 9 = 5 by computing 5 × 9 = 45), solving the problem using an alternative method, or substituting the answer back into the original problem context.

Students must develop number sense—intuition about the relative size and relationships between numbers. If a problem asks how many sweets 6 children receive when 30 sweets are shared equally, an answer of 50 should immediately seem unreasonable. Similarly, understanding measurement contexts helps verification: a person's height shouldn't be 200 centimeters when they're described as young, or a journey time shouldn't be 2 minutes when traveling great distances.

Multi-Step and Complex Problems

As students progress through Cambridge Primary, they encounter increasingly complex multi-step problems requiring integration of multiple concepts and operations. These problems demand careful planning—determining the sequence of steps before beginning calculations. Students might create a solution plan listing each step: "1) Find total cost of items, 2) Calculate change from money given, 3) Check if there's enough for an additional purchase."

Complex problems often involve combining different mathematical topics—for example, a problem might require understanding both fractions and measurement, or geometry and data handling. Students must recognize which mathematical knowledge applies to each part of the problem. Word problems with multiple questions require reading all parts before beginning, as sometimes later questions depend on earlier answers, or the same calculated value might be needed multiple times.

Real-World Problem Solving

Mathematics becomes meaningful when applied to authentic contexts. Real-world problems help students understand why mathematics matters and develop critical thinking about everyday situations. These might involve money and shopping (calculating costs, change, and budgets), time and scheduling (planning activities, calculating durations), measurement in cooking or construction, or data from sports, weather, or school contexts.

Real-world problems require students to make assumptions and decisions not always explicit in the problem statement. For example, when calculating how many cars are needed to transport people, students must recognize that partial cars aren't possible—you can't have 3.5 cars, so must round up to 4. This develops practical mathematical thinking beyond pure calculation.

Collaborative Problem Solving

While students must develop individual problem-solving abilities, learning to work with others enhances critical thinking. Mathematical discussion allows students to hear different approaches, challenge each other's reasoning, and refine their own understanding. When students explain their thinking to peers, they must articulate reasoning clearly, strengthening their own comprehension. Encountering alternative solution methods broadens students' strategic repertoire and demonstrates that mathematical problems often have multiple valid approaches.

Worked Example 1: Multi-Step Addition and Subtraction Problem

Problem: A school library has 245 books. On Monday, 37 new books are added. On Tuesday, 58 books are borrowed by students. On Wednesday, 29 more books are borrowed. How many books are in the library now?

Step 1: Understand the problem

• Starting amount: 245 books
• Books added: 37
• Books borrowed (removed): 58 on Tuesday, 29 on Wednesday
• Find: Number of books remaining

Step 2: Choose a strategy I'll work through each day's changes step by step, keeping track of the running total.

Step 3: Solve

• Start: 245 books
• After Monday (adding 37): 245 + 37 = 282 books
• After Tuesday (removing 58): 282 - 58 = 224 books
• After Wednesday (removing 29): 224 - 29 = 195 books

Alternative method (combining operations):

• Total added: 37 books
• Total borrowed: 58 + 29 = 87 books
• Final amount: 245 + 37 - 87 = 282 - 87 = 195 books

Step 4: Check Estimate: 245 + 37 is approximately 280, minus about 90 (60 + 30) gives approximately 190. My answer of 195 is reasonable.

Answer: The library now has 195 books.

Worked Example 2: Problem Requiring Pattern Recognition

Problem: Emma is saving money for a bicycle. In Week 1, she saves £3. In Week 2, she saves £5. In Week 3, she saves £7. If this pattern continues, how much will she save in Week 6? How much will she have saved in total after 6 weeks?

Step 1: Understand the problem

• Identify the pattern in weekly savings
• Find Week 6 savings amount
• Calculate total saved over 6 weeks

Step 2: Identify the pattern

• Week 1: £3
• Week 2: £5 (£3 + £2)
• Week 3: £7 (£5 + £2)
• Pattern: Each week, Emma saves £2 more than the previous week

Step 3: Extend the pattern

• Week 4: £7 + £2 = £9
• Week 5: £9 + £2 = £11
• Week 6: £11 + £2 = £13

Step 4: Calculate total savings Week 1: £3 Week 2: £5 Week 3: £7 Week 4: £9 Week 5: £11 Week 6: £13

Total: £3 + £5 + £7 + £9 + £11 + £13 = £48

Step 5: Verify

• Pattern check: The differences between consecutive weeks are all £2 ✓
• Addition check: (3 + 5 + 7) + (9 + 11 + 13) = 15 + 33 = 48 ✓

Answers: In Week 6, Emma will save £13. After 6 weeks, she will have saved £48 in total.

Worked Example 3: Problem Involving Multiple Operations and Real-World Context

Problem: A baker makes 8 trays of cupcakes. Each tray holds 12 cupcakes. She decorates 3/4 of the cupcakes with sprinkles. The remaining cupcakes are decorated with chocolate chips. She sells the cupcakes in boxes of 6. How many boxes can she fill?

Step 1: Understand what's being asked

• Need to find: Total number of boxes
• Given: 8 trays, 12 cupcakes per tray, boxes hold 6 cupcakes
• Extra information about decoration types (not needed for this question)

Step 2: Break into steps

1. Find total number of cupcakes made
2. Divide by cupcakes per box

Step 3: Calculate

• Step 1: Total cupcakes = 8 trays × 12 cupcakes per tray

  • 8 × 12 = 96 cupcakes

• Step 2: Number of boxes = 96 cupcakes ÷ 6 cupcakes per box

  • 96 ÷ 6 = 16 boxes

Step 4: Check the answer

• Estimation: 8 × 12 is approximately 8 × 10 = 80, but we need to add 8 × 2 = 16 more, giving 96 ✓
• Inverse check: 16 boxes × 6 cupcakes = 96 cupcakes ✓
• Real-world sense: 16 boxes from 96 cupcakes seems reasonable ✓

Answer: The baker can fill 16 boxes with cupcakes.

Note: The information about sprinkles and chocolate chips was irrelevant to this particular question—an important critical thinking skill is identifying what information is necessary.

Question Type 1: Word Problems Requiring Operation Selection

Example Question: "A toy shop has 156 action figures. They receive a delivery of 89 more action figures. They then sell 124 action figures. How many action figures does the shop have now? Show your working."

How to approach:

• Underline key numbers: 156 (starting amount), 89 (added), 124 (sold)
• Identify operations: "receive" and "more" indicate addition; "sell" indicates subtraction
• Create a plan: Add delivery, then subtract sales
• Calculate step by step:
  • 156 + 89 = 245 (after delivery)
  • 245 - 124 = 121 (after sales)
• Write clear working: