Deep understanding of early Math concepts can make or break your child’s future. But getting that deep understanding in a classroom of 30, in a system that favours discovery learning, can be hard.
Why Early Concepts Matter So Much
Mathematics is built like a tower. If the lower levels are shaky—basic facts, number sense, fractions—everything built on top will wobble.
Concepts like place value, multiplication, division, and fractions are the foundation for algebra, geometry, and calculus. If a student is guessing their multiplication facts or half-understands what a fraction is, algebraic manipulation and equations will feel mysterious and fragile instead of logical and predictable.
International assessments like PISA have repeatedly shown that students in higher-performing systems tend to have much stronger mastery of these core building blocks, and that this mastery is closely linked with later success in more complex problem-solving. When students have that solid base, they can focus on new ideas rather than constantly fighting gaps from earlier years.
Confidence Comes From Deep Knowledge
Children don’t become confident in math because adults tell them they’re “smart.” They become confident because they know what they’re doing—and can prove it to themselves.
A student who deeply understands multiplication enters algebra feeling secure: distributing, factoring, and manipulating expressions feel like natural extensions of what they already understand. But a student who only half-remembers multiplication facts or procedures has to work twice as hard in algebra, because every new step rests on something uncertain.
This weak grasp of underlying concepts leads to fragile procedural fluency. Students might “get through” a topic one week, only to forget it the next, because they’re leaning on patterns, tricks, and guesswork instead of understanding. That fragility is exactly what erodes confidence when they face unfamiliar problems or higher-level courses.
By contrast, when students experience math as a connected network of ideas—each one clearly understood and securely stored in memory—they approach new material with calm curiosity rather than anxiety.
Why Explicit Instruction Works Best
A major body of research shows that explicit, well-guided instruction is the most effective way to build that kind of deep understanding.
Analyses of PISA data by organizations such as McKinsey and the OECD have found that structured, teacher-led explanations—combined with opportunities for students to practice and apply concepts—are consistently associated with higher mathematics achievement. These findings support what cognitive scientists have been saying for decades: students learn new, complex ideas best when an expert first makes those ideas clear.
Educational psychologist Paul Kirschner and colleagues famously argued in 2006 that “minimally guided instruction” (approaches where students are expected to discover key ideas largely on their own) is significantly less effective than guided methods. Their review of more than fifty years of research concluded that unguided or lightly guided approaches overload students’ working memory and ignore how long-term understanding is actually built.
In other words, throwing students into discovery tasks without first equipping them with clear explanations and worked examples is not just inefficient—it’s unfair. It advantages the few who can piece things together quickly and leaves many others feeling lost.
The Power of 1:1 Tutoring
Within the world of explicit, teacher-led instruction, one particular format stands out: one-to-one tutoring.
In 1984, Benjamin Bloom published his famous paper on the “2 Sigma Problem.” He showed that students who received one-to-one tutoring, with regular feedback and checks for understanding, performed on average two standard deviations better than students in conventional classroom instruction. That means the typical tutored student outperformed 98% of students taught only in a standard class.
Later work has reinforced this finding in real-world school settings. For example, research associated with economists such as Roland Fryer, Jon Guryan, and colleagues evaluated intensive, in-school math tutoring for struggling high school students. Programs like Saga Education’s daily tutoring produced learning gains equivalent to one to two extra years of math in a single school year, as well as large improvements in grades and pass rates.
These results underline a simple truth: when an expert can respond immediately to a student’s confusion, adjust explanations, and ensure each step is understood before moving on, learning accelerates dramatically.
What Makes Great Tutoring: Atomised Explanation
Not all tutoring is created equal. The method the tutor uses matters as much as the 1:1 ratio.
One of the most powerful approaches within tutoring is atomisation—a term developed and clarified by educator Kris Boulton. Atomisation means breaking down a cognitive routine (a process the student will eventually carry out fluently) into its smallest meaningful units, or “atoms.” Each atom is a concept, fact, or sub-routine that can be understood, practiced, and secured.
Boulton emphasizes that atomisation is not just “taking small steps.” Simply chopping a process into smaller pieces doesn’t guarantee understanding. True atomisation exposes the underlying structure and relationships between ideas so that the student can see how each piece fits into a coherent whole.
For example, instead of teaching solving linear equations as one big procedure, an atomised approach might separate:
- Understanding equality as “balance.”
- Recognizing inverse operations.
- Practicing undoing a single operation on one-step equations.
- Extending to two-step equations while narrating each action.
- Connecting symbolic equations to word problems and concrete situations.
Each of these is taught clearly, with explanation, examples, and deliberate practice before they are recombined into the full routine.
You can read more about Kris Boulton’s explanation of atomisation here:
https://unstoppablelearning.substack.com/p/what-is-atomisation
What Explicit, Atomised Tutoring Looks Like in Practice
In a high-quality, atomised tutoring session, you’d see several consistent features:
- The tutor starts by explaining a concept clearly, using simple language and well-chosen examples.
- They introduce one “atom” at a time, checking that the student can explain it back in their own words.
- They use guided practice: doing a problem together, then gradually handing more responsibility to the student.
- They ask targeted questions to reveal misconceptions early, then correct them with clear, direct feedback.
- Only once each atom is secure do they connect multiple atoms into a longer routine or more complex problem.
This method respects how the human brain actually learns: by attaching new ideas to stable prior knowledge, strengthening those connections through practice, and gradually automating them so working memory is freed up for higher-level thinking.
Why This Matters for Your Child
For your child, the implications are huge:
- Early, explicit teaching of core concepts gives them a rock-solid foundation, so later topics feel logical rather than overwhelming.
- Deep understanding fuels real confidence; they know they can handle new challenges because their knowledge is secure.
- One-to-one, atomised tutoring can close gaps quickly and open up higher pathways—like advanced middle school math, high school calculus, and ultimately STEM degrees—without the stress and frustration so many students experience.
Rather than leaving your child to “discover” key ideas on their own and hoping they don’t miss something important, explicit, atomised tutoring makes every step visible, graspable, and purposeful.
Jamie Martin is a 2x education founder, former Special Adviser to the UK Secretary of State for Education, and an education specialist at Boston Consulting Group.
To learn more about how explicit, atomised 1:1 tutoring can transform your child’s relationship with math, visit www.primemath.io.