- At the outset, we wish to state unequivocally that we hold Abacus, Kumon and other educational institutions in the highest regard for their contributions to students’ learning and development.
- This article is in no way intended to criticise, belittle or undermine these programs or the valuable work undertaken by the educators associated with them. We recognise that different educational programs serve different purposes and contribute to the development of students’ knowledge and skills in their own ways.
- The sole purpose of this article is to offer an educational perspective on the different mathematical skills required for school mathematics, advanced mathematics and selective entry examinations, and to help parents better understand why students may excel in one learning environment while experiencing challenges in another.
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Understanding the difference between calculation skills, mathematical understanding and problem-solving ability.
Many parents invest considerable time, effort and resources in developing their children’s mathematical abilities from an early age. Supplementary learning programs such as Abacus and Kumon are popular choices, and some students achieve impressive results, progressing through advanced levels and earning distinctions or other certifications.
Naturally, these achievements give parents confidence in their child’s mathematical abilities.
However, an interesting question sometimes arises: Why does a child who has achieved advanced proficiency in Abacus or Kumon still experience difficulties with mathematics at school or with the mathematical and quantitative reasoning questions encountered in Selective Entry examinations?
At Melbourne Tutorials, we have encountered students who have achieved advanced levels and distinctions in supplementary mathematics programs such as Abacus and Kumon, yet continue to experience difficulties with school mathematics, advanced mathematical problem-solving and Selective Entry preparation.
Interestingly, this observation is not confined to our own teaching experience. Through discussions with directors of various other educational and tutoring centres offering advanced mathematics and Selective Entry preparation programs, we have found that similar experiences have been reported elsewhere. Students with impressive achievements in supplementary mathematics programs do not always demonstrate the same level of proficiency when confronted with broader school mathematics concepts, unfamiliar mathematical problems or questions requiring higher-order reasoning.
These observations raise an important educational question: Why does demonstrated proficiency in one form of mathematical learning not always translate into comparable performance in school mathematics and competitive entrance examinations?
The answer lies, at least in part, in understanding the different dimensions of mathematical proficiency and recognising that computational fluency, conceptual understanding and mathematical reasoning, although closely related, are not necessarily developed to the same degree.
Mathematics Is Much More Than Calculation
Mathematics is often associated with numbers, formulas and calculations. While these are essential, mathematical proficiency encompasses a much broader range of abilities.
A mathematically proficient student should progressively develop the ability to:
- Perform calculations accurately and efficiently.
- Understand mathematical concepts and the principles behind them.
- Recognise patterns and relationships between numbers and quantities.
- Apply mathematical knowledge to unfamiliar situations.
- Interpret and solve multistep word problems.
- Think logically, evaluate different approaches and justify solutions.
- Connect mathematical concepts across different areas of the curriculum.
These abilities are interconnected, but they do not necessarily develop at the same rate.
A student may be exceptionally quick at mental arithmetic but struggle to interpret a problem involving fractions or proportional reasoning. Another student may confidently solve algebraic equations yet find it difficult to determine which equation is needed to solve a real-world problem.
Mathematical fluency, conceptual understanding and mathematical reasoning are related, but they are not interchangeable.
Understanding What Abacus and Kumon Develop
Abacus programs commonly focus on developing mental arithmetic, numerical visualisation, concentration, speed and calculation accuracy.
Kumon adopts a different approach, using structured, progressively challenging worksheets and independent practice to develop mathematical fluency and learning skills. Its mathematics curriculum can extend into advanced algebra and calculus.
Both approaches can offer valuable educational benefits.
However, the skills assessed through these programs may not fully represent the breadth of mathematical knowledge and reasoning expected in school or competitive entrance examinations.
A student who performs exceptionally well in a particular learning environment may have developed strong proficiency in the types of questions and procedures emphasised within that environment.
The challenge arises when the student encounters unfamiliar questions that require a different combination of skills.
For example, being able to calculate percentages quickly does not automatically mean a student understands proportional relationships or can solve an unfamiliar percentage problem.
Similarly, completing advanced algebraic exercises does not necessarily demonstrate that a student can independently construct an algebraic model from a complex word problem.
This does not diminish the value of either program. Rather, it reinforces the importance of evaluating mathematical development across a broader range of skills.
Why Can These Students Struggle with School Mathematics?
School mathematics is designed to develop mathematical understanding across multiple curriculum areas.
In Australia, students study number, algebra, measurement, space, statistics and probability. Mathematical reasoning and problem-solving are integral to learning across these areas.
As students progress through primary and secondary school, mathematics becomes increasingly conceptual.
In the early years, students spend considerable time developing number sense and basic arithmetic. Later, they must apply this knowledge to fractions, ratios, percentages, algebra, geometry, probability and increasingly complex problems.
A student who has developed strong calculation skills may still experience difficulties if their conceptual foundations are incomplete.
- Gaps in Foundational Understanding
Mathematics is cumulative. Later concepts often depend on earlier knowledge.
For example, understanding fractions is important for learning percentages, ratios, proportions, algebraic relationships and probability.
A student may be able to perform complex calculations yet have difficulty explaining why dividing by a fraction is equivalent to multiplying by its reciprocal.
If foundational concepts have not been fully understood, difficulties may become more noticeable as school mathematics advances.
- Difficulty Applying Knowledge
Students may perform well when questions follow a familiar structure but struggle when the same concepts are presented differently.
For example, a student might confidently calculate:
25% of 240 = 60
However, consider this question:
A shop increases the price of an item by 25%. The new price is $240. What was the original price?
The correct answer is $192.
Although both questions involve percentages, the second requires a deeper understanding of the relationship between the original amount and the increased amount.
A student who relies primarily on familiar calculation procedures may find this more difficult.
- Mathematical Language and Comprehension
School mathematics also requires students to interpret written information.
Words such as difference, consecutive, proportional, equivalent, remainder and at least have specific mathematical meanings.
A student may possess the calculation skills needed to solve a problem but struggle to understand what the question is asking.
This becomes increasingly important in secondary school, where mathematical questions often involve several pieces of information and multiple steps.
- Differences Between Practice and Assessment
Students often become proficient at the types of exercises they practise regularly.
However, school assessments may require them to select methods independently, explain reasoning or apply several concepts together.
A student who performs well on familiar exercises may therefore find broader curriculum assessments more challenging.
It is also important to recognise that mathematical difficulties can arise for many reasons, including gaps in prior learning, inconsistent instruction, mathematical anxiety, comprehension difficulties or other individual learning needs.
They should not automatically be attributed to participation in a particular supplementary program.
Why Is Selective Entry Mathematics Even More Challenging?
Selective Entry examinations place particular demands on students because they assess mathematical knowledge and the ability to reason through unfamiliar problems under time constraints.
Students preparing for Victorian Selective Entry examinations need to develop both sound mathematical foundations and the flexibility to apply those foundations in challenging situations.
A question may involve familiar mathematical concepts but present them in a way that students have not previously encountered.
For example, consider the following two questions.
Question A — Routine Algebra
Solve:
3x + 7 = 25
A student familiar with linear equations can readily determine that x = 6.
Question B — Mathematical Reasoning
A group of students is participating in a competition. Three-fifths of the students are girls. When 12 additional boys join, the number of boys becomes equal to the number of girls.
How many students were originally participating?
The correct answer is 60 students.
Although the second question uses relatively elementary mathematics, it requires students to interpret relationships, organise information and develop an appropriate solution strategy.
The challenge lies not in performing the calculations but in determining how to approach the problem.
This is an important distinction.
Selective Entry mathematics is not simply about knowing more advanced mathematics. It is also about using mathematical knowledge intelligently, flexibly and independently.
Mathematical Reasoning: The Missing Link for Some Students
Mathematical reasoning is the ability to analyse information, recognise relationships, make logical deductions and develop justified solutions.
It is central to both successful school mathematics learning and competitive examination preparation.
Consider a student who can solve numerous algebraic equations accurately.
When presented with a word problem, however, the student must first determine:
- What information is given?
- What needs to be calculated?
- Which mathematical relationships are relevant?
- What strategy should be used?
- Does the final answer make sense?
These decisions require more than procedural knowledge.
Students who have mainly practised familiar question types may initially find unfamiliar reasoning problems difficult, even when they possess the mathematical knowledge needed to solve them.
Developing reasoning skills requires regular exposure to varied problems, opportunities to discuss different solution strategies and encouragement to explain why an approach works.
It also requires students to learn from mistakes rather than simply memorise the correct procedure.
Advanced Mathematics Does Not Always Mean Strong Mathematical Foundations
Parents sometimes assume that a child studying mathematics several years ahead of their school grade must be mathematically advanced in every respect.
However, mathematical acceleration and mathematical depth are not necessarily the same.
A student may have encountered advanced algebra while still experiencing difficulties with foundational concepts such as fractions, measurement, geometry or proportional reasoning.
Likewise, a student who has completed advanced mental arithmetic training may not have developed the same proficiency in interpreting mathematical information or constructing logical solutions.
This is why advanced certificates, program levels and distinctions should be viewed as evidence of achievement within a particular program rather than as comprehensive measures of mathematical ability.
A balanced assessment should consider what the student understands, what they can calculate and how effectively they can apply their knowledge.
Why Simply Giving More Difficult Questions May Not Help
When students struggle with mathematics, a common response is to provide additional worksheets, more homework or increasingly challenging questions.
Practice is certainly important, but its effectiveness depends on whether the student understands the underlying concepts.
If a child has difficulty with proportional reasoning, repeatedly attempting advanced ratio problems without addressing that difficulty may produce frustration rather than improvement.
Similarly, memorising solutions to particular reasoning questions may improve performance on familiar examples without developing the flexibility needed for new problems.
Effective mathematical support should therefore begin by identifying the source of the difficulty.
Does the student lack the necessary foundational knowledge? Do they understand the concepts but struggle to apply them? Is the difficulty related to mathematical language, problem interpretation or selecting an appropriate strategy?
Different difficulties require different instructional responses.
What Should Parents Look for Beyond Certificates and Advanced Levels?
Parents should certainly celebrate their children’s achievements in supplementary learning programs. However, it is equally important to look beyond certificates, speed and progression levels.
Some useful questions include:
- Can my child explain why a mathematical method works?
- Can my child solve a question when its format is unfamiliar?
- Can my child apply mathematical concepts to practical situations?
- Can my child identify which mathematical strategy is appropriate?
- Can my child explain the reasoning behind an answer?
- Can my child recognise and correct mistakes?
- Is my child progressing confidently in school mathematics across different curriculum areas?
These questions help parents develop a more complete understanding of their child’s mathematical abilities.
A child who can solve unfamiliar problems thoughtfully, even if they take longer, may be demonstrating valuable mathematical reasoning skills that are not reflected in calculation speed alone.
The Melbourne Tutorials Approach to Mathematics Education
At Melbourne Tutorials, we believe that strong mathematics education must develop computational fluency, conceptual understanding and mathematical reasoning together.
Our teaching philosophy recognises that students need more than exposure to advanced content or repeated practice of familiar procedures.
They need to understand mathematical principles, connect concepts, interpret unfamiliar questions and develop independent problem-solving strategies.
This is particularly important in our Selective Entry preparation programs, where students encounter challenging mathematical and quantitative reasoning questions.
We encourage students to participate actively in lessons, ask questions, explain their thinking and understand the reasoning behind different solution methods.
We also recognise that students enter our programs with different educational experiences and levels of preparation.
Where foundational gaps exist, identifying and addressing those gaps is essential to meaningful progress.
Our objective is not simply to help students complete increasingly difficult mathematics exercises. It is to develop the mathematical understanding and confidence necessary for success at school, in competitive examinations and in their future learning.
A Balanced Approach to Mathematical Development
Abacus, Kumon, school mathematics and Selective Entry preparation need not be viewed as competing approaches.
They can serve complementary purposes.
Abacus can strengthen mental arithmetic and numerical fluency. Kumon can support independent learning and the development of mathematical procedures through structured progression.
School mathematics provides broad curriculum-based learning, while effective Selective Entry preparation develops the ability to apply mathematical knowledge to challenging and unfamiliar problems.
The important consideration is whether a child’s overall learning experience is developing a balanced range of mathematical abilities.
No single certificate, program level or examination result can provide a complete picture of mathematical proficiency.
Final Thoughts
A student may be exceptionally fast at calculations yet struggle with mathematical reasoning. Another may have studied advanced algebra but still experience difficulties with foundational concepts. A third may perform well in routine school exercises but find unfamiliar Selective Entry questions challenging.
These situations are not necessarily contradictory. They reflect the different skills involved in mathematical learning.
True mathematical proficiency is not simply about how quickly a student can calculate or how far ahead they have progressed through a syllabus. It is about how deeply they understand mathematics and how effectively they can apply that understanding.
For parents, the goal should be to nurture well-rounded mathematical development rather than focusing exclusively on speed, certificates or accelerated progression.
For educators, the responsibility is to build strong foundations, encourage independent thinking and develop students’ capacity to approach unfamiliar problems with confidence.
At Melbourne Tutorials, we believe that the real measure of mathematical learning is not merely whether a student can answer a question they have practised before, but whether they can understand, reason through and solve a problem they have never encountered.
Melbourne Tutorials — Building Strong Foundations, Mathematical Understanding and Independent Thinkers.