There is increasing concern among educators, parents, and the public worldwide that technology might be impeding students' ability to think mathematically. A recent UK survey reported by The Guardian found that secondary school teachers observed pupils using artificial intelligence losing their critical thinking skills. Two-thirds of the teachers reported noticing this decline among children who no longer feel the need to spell out words because of voice-to-text technology. “Students are losing core skills – thinking, creativity, writing, even how to have a conversation,” one teacher told the National Education Union poll. This situation in the UK mirrors what is happening in Ghana.

Walk into any Ghanaian secondary school classroom today, and you will see students reaching for calculators to solve problems that previous generations handled mentally. Observe a child shopping — they will pull out a phone to calculate change rather than do simple mental arithmetic. These observations raise a genuine concern: is the very tool designed to assist students’ learning actually hindering the development of essential mathematical thinking skills? This question deserves careful consideration, especially as Ghana advances its digital transformation in education. However, to answer it honestly, we must first recognise that the relationship between technology and mathematical thinking is far more complex than it appears.

A Historical Perspective: Technology Has Always Been Present

Let us begin by questioning the notion of a "pure" era of mathematical thought untouched by technology. The slide rule, which served engineers and scientists for generations, was a top-notch tool in its time. Logarithm tables simplified complex calculations into easy-to-use reference guides. The abacus, thousands of years old, remains a highly developed calculating tool. Counting boards with jetons—physical tokens moved across marked surfaces—made place value and arithmetic operations visible and tangible. Mesopotamian accountants used clay tokens as early as 4000 BCE. The Inca recorded numbers using knotted quipu cords. The Japanese and Koreans calculated with sangi rods. Mathematics has always been shaped by technology. Among these historical tools, several merit particular attention for what they reveal about the relationship between tools and thinking.

Napier's rods (or Napier's bones), invented by the Scottish mathematician John Napier in 1617, are an early example of technology designed to aid mathematical calculation. These were physical devices—usually made of ivory, wood, or metal—inscribed with multiplication tables in a cleverly arranged grid. To multiply, a student would select rods representing the digits of the number being multiplied, place them side by side, read across the rows to find partial products, and then add them to reach the final result. The rods displayed multiplication facts openly—the 7×8=56 was visible to all, not hidden inside a microchip. The student needed to understand place value to align the digits correctly. They also needed to know when and why to add the partial products. Furthermore, they had to estimate whether the final answer was reasonable. The rods made the distributive property visible and tangible; a learner could literally see how 432 × 5 breaks down into (400 × 5) + (30 × 5) + (2 × 5).

The geoboard, invented in the 1950s by Egyptian mathematician Caleb Gattegno, is deceptively simple: a flat board with pegs arranged in a grid, on which students stretch rubber bands to form geometric shapes. Yet this humble tool embodies deep mathematical thinking. When a student stretches a rubber band around pegs to create a square, they are not merely seeing a square — they are experiencing its properties. The right angles become tangible. The equal sides are visible and verifiable. Area becomes something you can count by the squares enclosed. Perimeter becomes something you can trace with your finger. The geoboard makes abstract geometric concepts concrete in a way that even sophisticated dynamic software struggles to match. It demands active participation — you cannot passively experience a geoboard.

The protractor has a rich history spanning abstract mathematics, applied sciences, and education. Students using protractors must understand the concepts of angles, estimation, and accuracy—the tool reveals rather than conceals. Graph paper, now so common that we forget it is a form of technology, was promoted by mathematician Eliakim Hastings Moore (1862–1932) in the early twentieth century as part of a 'lab method' of teaching mathematics, making calculations easier and demonstrating the links between abstract principles and real-world applications. Geometric models—physical three-dimensional objects—enabled students to see and handle abstract forms. Cube root blocks facilitated understanding of root extraction through physical manipulation.

What is instructive about all these historical tools is not just their existence but also how they functioned and what they required of their users. They were not passive answer-makers. They were active partners in thought. They were transparent — users could see how they worked because the mechanisms were visible and the operations demanded active participation. The tool acted as a thinking aid, not a thinking replacement.

Modern tools often behave differently. They act as "black boxes", where students input numbers and receive answers without understanding the underlying process. A calculator does not show its workings. A maths app does not explain why the answer is correct. This contrast—transparency versus opacity, active versus passive involvement—lies at the heart of concerns about technology and mathematical thinking. 

The Promise: How Technology Can Enhance Mathematical Thinking

When properly implemented, modern technology offers remarkable opportunities to foster the very mathematical thinking skills we fear losing. The key is to select tools that, like Napier's rods and the geoboard, reveal mathematical structures rather than hide them.

Dynamic Visualisation Makes Abstract Concepts Concrete

One of the biggest challenges in mathematics education is helping students grasp abstract concepts. Dynamic geometry software such as GeoGebra and the Geometer's Sketchpad has revolutionised this aspect of teaching. These tools enable students to manipulate mathematical objects in real time, observing how parameter changes affect outcomes.

For instance, when exploring quadratic functions, students can use sliders to adjust coefficients and immediately observe how the parabola shifts, narrows, or flips. This interactive approach fosters what researchers call "geometric imagination" and "functional thinking"—two essential aspects of mathematical intelligence. Students are not merely memorising that "a" affects the curve's width; they are actively experiencing the relationship and developing intuition that leads to a deeper understanding. Software such as the geoboard makes the abstract more tangible.

A thorough review of research on the Geometer's Sketchpad shows that its dynamic visualisation features significantly enhance students' geometric imagination and spatial reasoning. The software helps students understand spatial relationships by manipulating figures through transformations, rotations, and projections. This is not passive learning; it involves active cognitive engagement with mathematical structures, similar to how manipulating Napier's rods revealed patterns in multiplication. These software programmes should be accessible in Ghanaian classrooms to support learning and make the study of mathematics easier.

Developing Computational Thinking

Beyond visualisation, modern technology can foster computational thinking—a problem-solving approach that has become vital in the twenty-first century. Computational thinking involves breaking down complex problems, recognising patterns, developing algorithms, and analysing solutions. When students use tools such as GeoGebra or Scratch to create programmes that solve mathematical problems, they engage in reasoning that reinforces and extends their mathematical understanding. They must think like both mathematicians and computer scientists, identifying the logical steps required to reach a solution. This mirrors what students using Napier's rods did implicitly—they followed an algorithm (select rods, read rows, add partial products) while understanding why each step was necessary.

Recent research has shown that incorporating computational thinking into maths lessons through dynamic software produces positive outcomes. Students learn to approach problems methodically, troubleshoot their thinking when solutions fail, and develop algorithmic reasoning that benefits both maths and wider problem-solving contexts. These are higher-order thinking skills, not replacements for them. 

Immediate Feedback and Scaffolded Learning

Technology also offers immediate feedback. When students work on paper, mistakes may go unnoticed until a teacher marks their work days later. Digital tools can provide instant alerts when something is wrong, enabling students to recognise and correct misunderstandings in real time.

However, research offers a cautionary note. A study of automated feedback in GeoGebra found that although pop-up notifications helped many students identify and correct errors, in some cases the feedback steered students towards specific strategies rather than fostering independent problem-solving. At times, it even obstructed task completion. This highlights an important point: the tool matters less than how it is used—a lesson that Napier's rods and the geoboard's transparent design naturally illustrated. 

Personalised Learning Pathways

The integration of artificial intelligence into mathematics education presents new opportunities. AI-powered platforms can tailor support to each student's needs, providing extra practice where they struggle and enabling quicker progress through concepts they have already understood. This personalisation was not possible in traditional classrooms, where teachers had to maintain a single

pace for all students. Research indicates that effective strategies for AI integration include using these tools for personalised learning while ensuring that technology supports, rather than replaces, teacher-student interactions. The human element remains essential, just as a teacher's guidance was vital when students used slide rules, geoboards, and logarithm tables.

The Peril: How Technology Can Undermine Mathematical Thinking

Although technology has potential, it can also impede mathematical thinking if misused. Recognising these risks helps us avoid them. I introduce you to some potential dangers.

The Black Box Problem

The greatest risk is the "black box" phenomenon. When students use calculators or computer algebra systems without understanding the underlying processes, they may become adept at finding answers while remaining mathematically illiterate.

Imagine a student solving a quadratic equation. If they enter numbers into an app and get solutions, they learn nothing about factoring, completing the square, or what solutions truly mean. The tool has thought of them. When these students face problems that do not fit neatly into a calculator’s capabilities, they become helpless.                                                                                                                        

This sharply contrasts with Napier's rods or the geoboard. A student using the rods could not find an answer without understanding the process, as the rods required active engagement at each step. A student using a geoboard cannot discover geometric properties without actively constructing and manipulating shapes. The tools were transparent—you could see the multiplication facts, trace how the partial products appeared, and feel the right angles under your fingers. Modern tools, by design, lack this transparency, increasing the responsibility on educators to ensure that students engage with the underlying concepts.

This issue goes beyond calculators. Many digital maths programmes present content in isolated segments, causing students to switch from fractions one day to geometry puzzles the next without developing a coherent understanding. Traditional textbooks were carefully organised so that each concept naturally built on the previous one. Some digital alternatives lack this logical sequence, leaving students with fragmented knowledge.

Superficial Engagement

Gamification—the use of game elements such as points, badges, and levels—can motivate some learners, but it also risks replacing genuine understanding with superficial engagement. Students may rush through problems to earn rewards without considering why answers are correct or incorrect. The focus shifts from accumulation to understanding. Entering answers into digital boxes is not the same as working through multi-step problems on paper. Written work requires organisation, encourages students to demonstrate their reasoning, and creates a record that can be reviewed and discussed. Without this practice, students miss opportunities to develop mental stamina and precision—both essential for mathematical mastery. A student who has only ever tapped screens has never felt the resistance of a pencil on paper, the satisfaction of constructing a geometric proof by hand, or the discipline of organising thoughts in writing. 

Infrastructure and implementation challenges

In contexts like Ghana, additional challenges exacerbate these issues. Research from neighbouring Nigeria identified key barriers to effective technology implementation in mathematics education: inadequate infrastructure, limited access to computers, insufficient teacher training, and unreliable electricity and internet connectivity.

These challenges mean that even when technology is accessible, it may not be used effectively. Teachers without adequate training cannot guide students in using tools thoughtfully. Unreliable connectivity hinders learning. Limited access prevents students from developing fluency with tools that could support their thinking.

The Crucial Difference: Active versus Passive Technology Use

The evidence emphasises a crucial distinction that determines whether technology supports or hampers mathematical thinking: Is the technology being utilised actively or passively?

Active use of technology engages students in thinking. They manipulate variables and observe outcomes, forming hypotheses about mathematical relationships. They write programmes that reflect mathematical processes, requiring them to think algorithmically. They use immediate feedback to identify and correct mistakes, learning from errors. The technology acts as a partner in thinking, not a replacement for it. This is what Napier's rods required, what the geoboard needs, what slide rules anticipated, and what logarithm tables presumed.

Passive use of technology hampers critical thinking. Students present problems and receive answers without engaging with fundamental concepts. They work through gamified exercises focused on earning points rather than understanding. They imitate AI solutions without analysing the reasoning behind them.

Historical tools required active use because they demanded understanding to be used effectively. Modern tools lack this requirement; they can be used passively with apparent success, concealing the absence of genuine learning. This is not an argument against modern technology but a plea for deliberate intent in its use.

Appropriating Technology for Mathematical Thinking in Ghana

How can Ghanaian educators, parents, and policymakers ensure that technology enhances rather than undermines mathematical thinking? The key is in careful appropriation—intentionally shaping how technology is used to achieve our educational goals, while learning from the wisdom embedded in historical tools such as Napier's rods and the geoboard.

Invest in Teacher Development

The most crucial factor in effective technology integration is the teacher. Research consistently underscores the importance of teacher training and ongoing professional development. A teacher who understands both mathematics and technology can guide students in using tools thoughtfully, ask insightful questions that prompt reflection, and plan activities that foster active engagement.

Investing in teacher development should take precedence over investing in hardware. Providing computers without training teachers to use them effectively is like giving textbooks in a language teachers cannot read.

Choose Transparent Tools

Where possible, choose tools that reveal rather than hide mathematical processes. Dynamic geometry software such as GeoGebra, which allows students to see and manipulate mathematical objects, promotes understanding more effectively than black-box calculators. Tools that show their workings, require input at several stages, and make mathematical structures visible should be prioritised over those that simply provide answers.

GeoGebra's additional benefit is that it is free and broadly accessible, even in resource-limited settings. Its growing adoption in mathematics education research demonstrates its effectiveness in promoting mathematical thinking across various contexts.

Embrace low-tech tools alongside High-Tech.

The geoboard teaches an important lesson: advanced thinking does not need high-tech tools. In Ghanaian classrooms, simple tools should be valued alongside digital resources. They are affordable, require no infrastructure, and encourage active participation. A geoboard can be made from scrap wood and nails. Rubber bands are inexpensive and reusable. Graph paper costs only a few pence. Protractors last for many years. These tools suit any classroom, regardless of weather or time.

They are not competitors to digital technology but complements to it.

Teach the History of Mathematical Tools

Introducing students to the origins of our tools adds value. A lesson that progresses from counting tokens to Napier's rods, geoboards, and dynamic software helps students understand that technology has always supported mathematics, albeit in different forms. They can learn why historical tools required understanding and carry that appreciation into their use of modern tools. This historical awareness fosters an active mindset that avoids passive use of technology.

Design for Active Engagement

Educators should develop learning activities that promote active thinking. Instead of instructing students to "use GeoGebra to graph y = x²," encourage them to "discover what happens to the graph when you change the coefficient." Rather than allowing AI to solve problems for them, have students generate commands with AI, then implement and test these commands, explaining why they work. Instead of simply accepting a calculator's answer, ask students to estimate first and assess whether the answer makes sense. Instead of merely showing students geoboard configurations, challenge them to design their own and explain their discoveries.

A model combining ChatGPT with GeoGebra shows that AI tools are most effective when they help students develop and refine their own methods rather than offering complete solutions. The goal is to encourage collaboration between human thinking and technological ability.

Maintain Balance

Technology should assist, not replace, traditional mathematical methods. Students still need to practise mental arithmetic, work through problems on paper, and clearly explain their reasoning, both verbally and in writing. These activities enhance cognitive skills distinct from those developed through technology-based learning, and all are vital.

The Mathematical Association of Ghana (MAG) and other organisations interested in mathematics should emphasise that effective mathematics learning requires conceptual understanding, consistent practice, and human feedback—elements that technology alone cannot provide. Just as Napier's rods supplemented, but did not replace, understanding of multiplication, modern technology should support, but not substitute, fundamental mathematical thinking.

Address Infrastructure Realistically

Ghanaian schools face significant infrastructure challenges. Rather than waiting for ideal conditions, teachers can apply methods that work within existing constraints. Computational thinking activities can be taught "unplugged"—without computers—through hands-on tasks that develop the same reasoning skills. Geoboards, graph paper, and physical models require no electricity. Simple tools, when used effectively, can be more beneficial than complex tools used poorly. A classroom with well-trained teachers and basic resources will always outperform one with advanced technology and unprepared teachers.

Conclusion: Thinking with Technology, Not Through It 

The question "Has technology destroyed mathematical thinking?" contains a hidden assumption—that technology and thinking are opponents in a zero-sum game. The evidence suggests otherwise. When used appropriately, technology becomes a partner in thinking, extending and enhancing human cognitive abilities rather than replacing them.

Napier's rods did not undermine the mathematical thinking of seventeenth-century students because they required understanding to use effectively. The geoboard does not undermine today's students' mathematical thinking because it demands active engagement. Slide rules, logarithm tables, protractors, graph paper, and geometric models—all these technologies have served mathematics by revealing its structures and inviting active participation. Modern technology carries the same potential. The difference lies not in the tools themselves but in how we choose to use them.

For Ghanaian educators and parents, the way forward involves carefully selecting tools — both ancient and modern, both low- and high-tech — investing in teacher development, designing activities that require active participation, and maintaining balanced approaches that blend technological and traditional practices. It means teaching students to think critically with technology rather than through it.

The aim of mathematics education remains unchanged: to develop young people who can think mathematically, creatively, and inventively, and who can find new solutions to novel problems. From counting tokens to GeoGebra, and from Napier's rods to artificial intelligence, well-chosen technology supports this aim. It is our duty to ensure that it does so.

Kwamina Arhin is a Senior Lecturer at the University of Skills Training and Entrepreneurial Development (USTED) and principal consultant at Arc Educational Consult. His deep understanding of assessment practices and mathematics instruction is grounded in years of academic study, applied research, and hands-on work on key issues in educational assessment and pedagogy.