What Is Mathematical Thinking? How to Build and Develop It in Children Mathematical thinking is not just about calculating quickly or getting the right answer. In this article, Westlink helps parents understand what it really means, see how it differs from "math thinking" enrichment programmes, and recognise five key competencies. We also suggest age-appropriate ways to build these skills, so you have a clearer basis for supporting your child's maths learning. What is mathematical thinking? Understanding it correctly to teach it well Before exploring how to build this skill, parents need to understand what mathematical thinking actually is. Starting from the wrong idea can easily lead you to the wrong method for your child. An internationally recognised definition Mathematical thinking is the ability to think in a structured way when approaching a problem. A child who has it can analyse information, spot relationships, and express a solution using mathematical language. According to the National Council of Teachers of Mathematics (NCTM), learning maths is not just about calculations and formulas. Students need to develop competencies such as problem solving, reasoning, communication, making connections, and representation. These are essential components of learning maths properly. The International Baccalaureate (IB) is a global education system that focuses on building thinking skills, competencies and the ability to learn independently. Within the IB, students don't just absorb knowledge — they also practise how to learn, through skill groups covering thinking, communication, research, self-management and social skills, developed continuously across their studies. In simple terms, a child doesn't just need to know the answer. They need to understand why that answer is correct, whether the method makes sense, and whether the problem could be solved another way. Why being "good at maths" doesn't always mean strong thinking skills Many parents judge their child's maths ability by how quickly they finish a worksheet. Proficiency in mental arithmetic, speed, or high test scores often leads to the superficial assumption that a child is "good at maths". But these signs alone aren't enough to conclude that a child has a strong thinking foundation. Fast mental maths is just one operational skill. A child might multiply two-digit numbers very quickly without truly understanding what multiplication means. When a problem shifts into a real-world context, that same child may struggle to reason through it. Correct results alone do not equate to robust cognitive foundations either. Some students memorise a problem type and apply a formula by reflex. When the format of the question changes, they can get stuck because they never understood the underlying concept. A child with strong mathematical thinking can usually explain their method. They know why they chose that approach, which pieces of information mattered, and whether the result makes sense. So a student who works more slowly but explains clearly may actually have a more solid thinking foundation. Mathematical thinking vs logical thinking vs "math thinking" programmes These three concepts are often used interchangeably when parents research learning programmes for their child. Understanding the difference helps you choose the right approach, rather than choosing based on a name alone. Criteria Mathematical thinking Logical "Math thinking" programmes What it is The ability to solve problems using mathematical language The ability to reason in an orderly way, across many fields A term used by some maths enrichment centres or products Scope Tied to numbers, shapes, data, models and proof Broader than maths — applies to everyday life Depends on each centre's own curriculum How it shows up The child can analyse, reason and explain their solution The child reasons soundly and spots contradictions The child works through pre-designed question formats How it's assessed Through the solving process, presentation and application Through logic tests or reasoning scenarios Through the programme's own internal tests Example A child solves an open-ended problem in several different ways A child reasons through a real-life debate A child follows a weekly "math thinking" curriculum In this article, the concept is used in its academic sense. The focus isn't on exam tricks — it's on how a child thinks with maths. The five competencies of mathematical thinking This isn't a single skill. It's made up of several smaller parts, which you can observe in how a child learns, explains and works through problems. Five competencies that help children learn maths more systematically and deeply. Reasoning – structured argument Reasoning is the ability to move from information to a conclusion through logical steps. A child doesn't just give a final answer — they can explain why that result is correct. For example, when solving a geometry problem, a child first needs to identify what information the question gives. They then use that information to work towards a conclusion. Every step in the solution should have a clear justification. A sign of strong reasoning is a child who often asks "why." They're also comfortable showing several steps, rather than just jumping to the answer. This is an important foundation for learning proof in later years. Problem-solving – tackling open-ended problems Problem-solving is the ability to approach a question that has no ready-made formula. The child has to understand the problem themselves, choose an approach, try it out, and adjust as needed. For example, a class of 28 students wants to split into equal groups, each with at least 4 students. There's more than one valid way to divide them, so the child has to set their own conditions and rule out options. This is different from a question with an obvious template. A child can't just look at the numbers and slot in a familiar formula — they need to analyse the situation before they calculate. Strong problem-solving skills help children stay calmer when they meet an unfamiliar question. They also come to understand that a single problem can have more than one valid approach. Modelling – representing real-world situations Modelling is the ability to turn a real-life situation into a mathematical form. A child might use calculations, diagrams, data tables, charts or equations to represent the problem. For example, a child wants to work out how much paint a room needs. They measure the wall area, subtract the doors, and estimate the amount of paint based on that area. This is a real-world problem, not just a textbook exercise. Modelling helps children see that maths connects directly to everyday life. They stop seeing maths as numbers detached from reality, and instead learn to use it to make decisions and solve problems. Abstraction Abstraction is the ability to spot a general pattern across several specific examples. A child starts from real observations, then draws out a general model or formula. For example, a child arranges matchsticks into a series of connected triangles. The number of sticks used is 3, 5, 7, 9 and 11. From this, the child notices that each new shape adds 2 sticks. Once a child understands the pattern, they can predict the next shape. This is the first step towards algebraic thinking. This skill becomes very important once children start learning expressions, functions and proofs. Children who are weak in abstraction often struggle from secondary school onward. They can complete specific exercises but find it hard to grasp a general formula. That's why this skill needs to be built early. Mathematical communication – expressing thinking in mathematical language Mathematical communication is the ability to present mathematical thinking clearly. A child knows how to use the right terms, structure their solution, and explain it so others understand. Many students can calculate correctly but can't explain how they did it. This suggests they haven't fully mastered the thinking process yet. When asked to explain, they can get flustered or skip an important step. Parents can observe this skill quite simply. Ask your child to explain a maths problem to a younger sibling, or to you. If they can walk through it step by step, their maths learning is on the right track. An age-by-age path for building mathematical thinking There's no single method that works for every age. A 5-year-old needs a different approach from a 13-year-old. Pushing a child to learn at the wrong stage can unintentionally make them afraid of maths. Each age group needs a different, age-appropriate way of building maths skills. Ages 3–6: patterns and meaningful counting At ages 3–6, children don't yet need to learn complex calculations. The focus should be on recognising patterns, sorting objects, and counting using real objects. Parents can have their child sort toys by colour or shape. They can also count fruit in a bowl, find round objects, or arrange beads following a pattern. These activities help children understand quantity through real experience. They're not just reciting numbers from memory — they understand how many objects a number actually represents. A common mistake is pushing children to learn addition and subtraction tables too early. When a child doesn't yet understand the concept of quantity, calculation becomes pure rote learning. This can easily make maths feel dry and stressful. Ages 6–11: from real objects to symbols The primary years are a key period for building a foundation in mathematical thinking. Children need to understand what operations, fractions, geometry and measurement actually mean before moving on to more complex symbols. An effective approach moves from real objects, to pictures, and only then to symbols. A child touches an object first, sees a model next, and only writes the formula last. For example, when learning fractions, a child might cut a cake into four equal pieces. They then draw the divided cake. Only at the end do they write the symbol 1/4. If a child jumps straight to symbols, learning easily becomes mechanical memorisation. They might correctly write 1/4 without understanding that it means one part out of four equal parts. Ages 11–16: reasoning, proof and real-world modelling At secondary level, the foundation of thinking needs to shift towards a more abstract level. Children start learning proof, algebra, functions and multi-step reasoning problems. This is when reasoning and abstraction become especially important. Children need to understand why a formula is correct, not just how to apply it. They also need to learn how to check whether a solution makes sense. Real-world problems should also appear more often at this stage. For example, a child might build a spending table, analyse data, or model a measurement scenario. A good sign is a child who's willing to ask "why is this formula correct?" They also become less afraid of unfamiliar questions, because they're used to analysing a problem before solving it. Three evidence-based methods for building mathematical thinking There's plenty of advice online about building mathematical thinking. However, not all of it is backed by a clear process. Parents should prioritise methods with a solid educational foundation. Polya's four-step process Polya's process is a well-known framework for teaching problem-solving. It helps children avoid diving straight into calculation, and instead work through a problem step by step. Polya's process helps children solve problems through clear, sequential steps The first step is understanding the problem. The child reads the question, identifies the given information, and works out what's being asked. If needed, they can draw a diagram or underline key information. The second step is making a plan. The child tries to connect it to a problem they've solved before, chooses a suitable formula, or thinks about how to break the problem into smaller parts. This step stops them from solving on instinct alone. The third step is carrying out the plan. The child works through it step by step, checking their progress as they go. If an approach isn't working, they can go back and adjust. The final step is looking back at the solution. The child checks whether the answer makes sense, and whether there's another way to solve it. This step is very important but often gets skipped. When your child gets the right answer, don't stop there. Ask: "Is there another way you could do this?" This question helps children build deeper thinking skills. Concrete – Pictorial – Abstract Concrete – Pictorial – Abstract (CPA) is a teaching approach that moves from real objects, to pictures, and finally to symbols. It helps children understand the underlying concept before they memorise a formula. At the Concrete stage, children learn using real objects — counters, sweets, pieces of paper or building blocks to observe an operation, for example. At the Pictorial stage, children turn real objects into pictures, diagrams or models. They start to see the relationships between quantities. At the Abstract stage, children finally use mathematical symbols. By this point, the formula is no longer unfamiliar, because they've already understood it through earlier experience. This approach is often associated with Singapore Math. In PISA 2022, Singapore ranked first in the global mathematics assessment. The OECD also found that 41% of Singaporean students were top performers in maths, compared with an OECD average of 9%. This figure doesn't mean that simply following Singapore Math will make a child good at maths. It does, however, suggest that teaching systems focused on understanding — not just procedure — have real value as a reference point. Inquiry-based learning Inquiry-based learning starts from a question, rather than starting from a formula. Children observe, experiment, predict, and draw their own conclusions. For example, a teacher doesn't need to say straight away, "a centimetre is the standard unit of measurement." Instead, they might ask: "Why did everyone get a different result measuring the table?" When children measure using a handspan, the results will differ. From that experience, they understand why a shared unit of measurement is needed. The concept of a centimetre then emerges naturally. A research review by Furtak and colleagues found that inquiry-based teaching needs to clearly distinguish the level of guidance a teacher provides. The approach works better when students aren't left entirely on their own, but are guided appropriately. So learning through questions doesn't mean leaving children to figure everything out alone. Teachers still need to ask open-ended questions, observe, and step in with support at the right moment. Three activities you can do with your child this week You don't need to be good at maths yourself to help your child build this skill. What matters is creating opportunities for them to think, explain, and try different approaches. Math talk at mealtimes Math talk means bringing maths into everyday conversation. Parents can weave it into mealtimes, trips to the supermarket, or playtime. For example, when sharing a cake, ask how many pieces each person will get. When looking at the dinner table, ask what shapes different objects are. When shopping, ask your child to estimate the total cost. These questions don't need to be complicated. What matters is that the child gets to think mathematically in a real situation — and learns to put their thinking into words. Don't rush to correct your child when they get something wrong. Instead, ask: "Why do you think that?" or "Can you check that again?" This kind of question helps children self-correct. One problem, many solutions Each week, pick a simple problem and encourage your child to find more than one way to solve it. For example, with 25 × 16, a child might multiply the usual way. They could also break it into 25 × 4 × 4. Another option is turning it into 100 × 4. The goal of this activity isn't finding the fastest method. It's helping your child see that a single problem can be approached in more than one way. When your child finds a new approach, praise the thinking, not just the answer. This helps them understand that the process matters just as much as the result. Connecting maths to everyday life Maths becomes easier to understand when children see it show up in daily life. Parents can have their child measure, estimate, compare and build simple tables. For younger children, ask them to estimate the length of a table or the height of a door, then check the result with a ruler. This helps them understand measurement through real experience. For older children, ask them to work out the cost of breakfast over a week. They can build a table, add up the total, and calculate the daily average. This is a way of bringing maths into everyday decisions. These activities help children build modelling skills naturally. They come to understand that maths isn't confined to a workbook. The role of the learning environment in nurturing mathematical thinking Many parents assume that enrolling their child in a "math thinking" enrichment class is enough. In reality, mathematical thinking is a habit that needs to be built through the everyday learning environment. Why two hours of tutoring a week isn't enough Two hours of tutoring a week can expose a child to new question types. However, that amount of time is unlikely to change how a child approaches maths as a whole. If a child spends most of their time learning by following templates, their thinking skills are unlikely to develop in a lasting way. They may handle familiar questions well but still feel uneasy with unfamiliar ones. This skill needs to be built across many different situations. Children need regular opportunities to be asked questions, to try and get things wrong, and to explain their reasoning. That's why the main learning environment matters so much. A good classroom doesn't just check answers — it pays attention to how students are thinking. Teaching method shapes learning outcomes Given the same amount of time spent on maths, outcomes can look very different. The difference comes down to how the teacher structures the lesson. If a teacher only explains a formula and hands out sample exercises, children tend to learn passively. They may complete familiar questions quickly but lack the ability to work through something new on their own. If a teacher asks open-ended questions, children get to think more. They have to explain, debate, and check their own solutions. This process helps build a stronger thinking foundation. Research on inquiry-based teaching also highlights the importance of the level of guidance provided. This shows that teachers need to design questions and offer support at the right moments. Why the IB framework suits the development of mathematical thinking The IB organises learning around the Approaches to Learning (ATL) framework. This covers five skill groups: thinking, communication, research, self-management and social skills. Of these, thinking skills relate directly to mathematical thinking. With this approach, Mathematics isn't just about formulas. Students are encouraged to observe, ask questions and explain their reasoning. This process helps them understand why an answer is correct. The IB framework also aligns well with the mathematical competencies covered earlier in this article — reasoning, problem-solving, modelling, abstraction and mathematical communication. Learning this way means children aren't just working through templates. At PYP level, the IB is designed for students aged 3 to 12. At MYP level, it's designed for students aged 11 to 16. At Westlink, parents can find out more about the PYP and MYP pathways to understand how the IB journey works at the school. Mathematical thinking needs to be built through observation, reasoning, modelling and explanation. Once parents understand the true nature of this skill and how it develops at each stage, it becomes much easier to choose the right way to support their child. If you'd like to explore a learning environment built around the IB framework, find out more about the PYP and MYP pathways at Westlink, or book a tour for tailored advice.
