Changing Role of Mathematics in a Digital Economy

Sarah@outnumber.app, Research Editor, 15 Feb 2021 Mathematical Freedom, Digital Economy, Transforming Learning

Mathematics education radically changed in the 15th century with the introduction of the printing press¹. Before this, mathematics textbooks looked a lot different than those used today. In his 2019 piece ‘How technology has changed what it means to think mathematically’, mathematician Keith Devlin noted that in place of symbol-rich examples and explanations, textbooks were once filled with long written prose. Describing the mathematical operations without using symbols minimised the likelihood of errors when learners copied the text for themselves. Margins were the place for notation, bringing ‘mathematical life’ to the words on the pages¹. Devlin recognised that the printing press facilitated a radical transformation for mathematics. With less time needed to copy and interpret text-heavy books, more focus went to the subject matter itself. Branches of algebra, calculus and geometry bloomed from this notable time in the subject’s history¹. Fast forward 600 years and we’re still benefiting from its inception. However, during this time we’ve also witnessed the rise of the digital age. Could it be that with ever-evolving technology, we’re on the cusp of a similarly radical transformation for mathematics education?

The Discrepancy Between Technology & Mathematical Curriculums

With technology changing the way we perform real-world maths, it’s imperative that we not only teach our children how to be technologically literate, but also how mathematics integrates with the technology around them. However, the traditional mathematics curriculum is currently falling short, failing to teach skills that are transferable and similar to those used in the workplace², where technology is commonplace. These shortcomings have been recognised by governments and policymakers, who appear to be taking measures to close gaps and increase competitiveness in the knowledge sector. For example, in the United Kingdom the Welsh government introduced a numeracy GCSE (secondary qualification) paper in 2015, alongside the standard Mathematics GCSE, seeking to help students leave school with the numerical skills required for future employment³. By posing ‘real-world’ questions, the aim is for pupils to strengthen their contextual mathematical capabilities. However, are these papers truly representing the types of mathematical problems faced in the workplace? Whilst there is a definite focus on problem solving and reasoning - skills highly prized by employers - it could be argued that the numeracy papers aren’t truly addressing the gap between the curriculum and essential modern day mathematical skills. With outcomes based on scoring marks, could these timed papers detract from the process of individualised and creative problem solving by continuing to place value on speed and accuracy? While the numeracy paper appears to address some of the current curriculum shortcomings, the lack of technological innovation is another caveat. Like all other exams, the numeracy paper allows for, at best, the use of a calculator. Will pupils be able to recognise the links between technology & mathematics if their perceived proficiency in the subject comes down to their ability to solve problems via pen and paper? Although implementing technology into the examination process may pose some challenges, ensuring its presence within lessons could go some way to teaching these skills to pupils. However, whilst technology may be more commonly used in classrooms today, it often plays a similar role to the traditional textbook¹. Digital presentations simply replace chalkboards and written questions are substituted for online quizzes. So, whilst the printing press accelerated mathematics learning, it seems that the digital age is currently not impacting the subject with the same magnitude.

“In a balanced mathematics program, the strategic use of technology strengthens mathematics teaching and learning (Dick & Hollebrands, 2011). Simply having access to technology is not sufficient.⁴”

Given the ease of access to technology in advanced economies, why is it that it’s not being used strategically to accelerate, facilitate and support mathematics learning? This can partly be attributed to a resistance to change. Teachers who are devoted to tried and tested methods are reluctant to introduce technology in a way that differs from traditional teaching practices. Policymakers and parents are also wary, lacking the understanding of how mathematics has evolved from their school days¹.

Mathematical Freedom, Digital Economy, Transforming Learning

When we look at certain topics on the curriculum, it’s clear to see how they often fail to take into account how technology has altered their real-world application. Take bus and train schedules for example. Being able to read and interpret them is still a prized skill in the classroom, but one that no longer resonates with children. Electronic signs, complete with line numbers, stops, anticipated arrival times and any alterations to the usual service have rendered the traditional timetable redundant. In the absence of those, smartphones can provide us with all the information needed to successfully get from A to B. Children are savvy and acutely aware of how to access information. However, continuing to teach skills no longer applicable to the present only works to further disillusion students into the thinking that mathematics, as it’s currently taught, is not applicable or relevant to them.

How Technology is Changing Mathematics Learning

Current pushback on further implementing technology into the mathematical curriculum often overlooks the important role it’s already played in shaping how the subject is learnt and practised today. Before the 1960s, mathematics lessons prized accuracy and speed. This changed somewhat with the introduction of the calculator. Executing error-free calculations was no longer a prerequisite for excelling in mathematics. This simple piece of digital equipment largely took the focus off of rote learning and instead placed it on understanding¹. From there, technology continued to shift the focus of mathematics away from refining manual calculations. Maths programs created space for mathematicians to delve deeper into their subject¹. Essentially, it’s not mathematics that has changed, it’s the way we use it¹.

But does technology create lazy mathematicians?

The rise of technology has prompted some to worry about our over-reliance on it, with many blaming the calculator for a loss of arithmetic prowess. However, being able to use technology in a mathematical sense still comes with its prerequisites. For example, when faced with a numerical reasoning problem, whilst a calculator may be used, the learner needs to have an understanding of what information should be extracted and how it should be processed to render an answer. Studies have shown that calculators improve mathematical understanding and fluency⁵. When implemented correctly, calculators work to further ‘develop number sense and skills of mental computation’⁶. Not only that, imaginatively employed technology can further develop advanced mathematical skills such as reasoning, justification and problem solving⁴. Students have also been found to enjoy and play a more active role in lessons when technology is involved². So it appears it’s not lazy mathematicians that technology creates, but ones that are more engaged and creative.

How Technology Paves the Way for Mathematical Freedom

Numerical skills form the base of a sound mathematical understanding. It’s these skills that teachers are trying to instil in pupils where a ruthless repetition of written calculations is encouraged. It’s also likely why many feel that the calculator hinders mathematical learning, believing that it removes the need to solidify understanding of the underlying concepts. However, traditional pen and paper practice is no longer the only way to facilitate this numerical groundwork.

“The goal of mathematics teaching today is not execution, it is understanding.¹”

As highlighted by Devlin, practise is still essential to establishing numerical prowess, but the goal today is to produce understanding, not speed or precision¹. Prior to the digital age, pupils were essentially trained to execute calculations in a similar fashion to those performed by a modern computer. This prescriptive method of learning creates linear-minded mathematicians; those that believe mathematics doesn’t allow for number flexibility⁷. Computers have become much more proficient than humans when it comes to algorithmic thinking. However, it's flexibility and creativity where computers fall short and we have the potential to flourish⁷.

Performance Mathematics Vs Mathematical Freedom

Maths that’s presented prescriptively through repetition of taught methods, valuing speed and accuracy, can be thought of as performance mathematics⁷. Mathematical freedom, on the other hand, opens the door to exploration within the subject. Instead of being teacher-centred, mathematical freedom places the focus back on the student. Rather than being shown how to do something, learners are encouraged to be creative with their approach. Different methods of problem-solving are praised in the mathematical freedom model of learning, highlighting to pupils that mathematical thinking isn’t linear⁷ - and that mistakes are a vital part of learning. Professor of mathematics education, Jo Boaler, talks about the power of visuals. By demonstrating seemingly abstract concepts through interesting and engaging imagery, students are better able to grasp the topics at hand. In fact, presenting maths in different ways has been shown to strengthen connections in the brain⁸. Just as technology can help to develop skills such as problem solving, justification and reasoning, so can the mathematical freedom approach to learning. It creates flexible thinkers, allowing students to explore mathematics in a way that resonates with them. Through technology, educators can provide rich learning resources and variety that is fundamental to mathematical freedom and move away from the performance mathematics model.

Bringing Technology Into the Student’s Mathematical Toolkit

Whilst technology could hold the key to radicalising mathematics learning, teachers play a vital role in orchestrating its use⁹. Unfortunately, the full potential of technology is not currently being put into play in learning environments. Instead of being used in a way to encourage and facilitate mathematical freedom, it simply takes the place of traditional performance mathematics tools and methods. However, technology can provide the flexibility and adaptability to develop the critical thinking skills of a creative mathematician when implemented with mathematical freedom in mind¹⁰. So, how can technology be used to enhance mathematical learning through the mathematical freedom approach?

❑ Early-stage visualisation

Using imagery, ahead of notation and symbols, brings context to mathematics, allowing students to connect with the concepts they’re learning. This is commonplace in Singapore, a country whose mathematical teaching practices are highly revered¹¹. For example, children are often expected to memorise times tables with little understanding of the multiplication process. However, technology can add visual and contextual dimensions, linking times tables to areas of shapes, or groups of objects. Interactive apps allow pupils to explore how different inputs affect outputs, building links in the learner’s mind between the imagery and numbers.

❑ Mathematical exploration & manipulation

Technology provides students with opportunities to explore mathematical concepts. Graphing and geometry programs allow for hands-on learning. By manipulating parameters, learners can see firsthand the impact this has on the visuals in front of them.

❑ Gamification

Gamification can be used not only to add context and enjoyment to mathematics but also to develop number-sense and problem-solving skills, which further lend themselves to other topics within the subject¹. Technology provides the ideal medium through which gamification can be delivered to pupils. Research has shown that gamification of mathematics empowers students and has a meaningful impact on the learning process, providing opportunities for students to work collaboratively¹².

The variety of ways technology can be implemented in the maths curriculum and its ability to individualise learning demonstrates its effectiveness as a tool to improve mathematical literacy. However, a 2016 report by the OECD stated that ‘struggling students should receive instruction adapted to their ability and needs’ and found that socio-economic status directly impacts upon access to mathematics content, influencing mathematical literacy². Technology would be a useful tool to bridge these gaps, but a potential lack of access means that more disadvantaged students may be missing out on a vital tool. Not only does this impact their mathematical outcomes, but a lack of technological literacy concerning mathematics could lead them to fall short on the skills required to succeed in and secure future employment. This highlights a need for universal access to technological tools that implement progressive teaching methods, ensuring children are both engaged in the subject and prepared for adult life.

Is Technology the Key to Transforming Mathematics Learning?

It’s clear to see the potential technology has for mathematics education. Facilitating and working alongside the mathematical freedom model of teaching, technology acts as a reminder that maths is about more than numbers and equations. It’s problem-solving. It’s investigative. It’s about exploration. By imparting these qualities in our children, not only we will create better mathematicians, but more inquisitive humans in general. Still, technology’s potential is largely untapped in regards to mathematics education. But with increasingly more attention being turned to its role in the classroom, could it be that we are on the cusp of a digital age transformation? It’s important to remember that technology is just one piece of the puzzle. As we’ll explore in future articles, a world-class mathematics learning experience will combine three things: a curriculum that focuses on 21st-century skills, the application of knowledge from cognitive science and educational psychology, and the use of modern digital apps as learning tools which bring all these concepts together.

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References: 1. Devlin, K. (2019). How technology has changed what it means to think mathematically, in Danesi. M (Ed), Interdisciplinary Perspectives on Mathematical Cognition, New York, NY: Springer, 2019 2. OECD (2016), Equations and Inequalities: Making Mathematics Accessible to All, PISA, OECD Publishing, Paris, http://dx.doi.org/10.1787/9789264258495-en 3. WJEC, 2015. WJEC GCSE In Mathematics - Numeracy Specification. [online] WJEC CBAC Ltd, p.5. 4. National Council of Teachers of Mathematics. (2017). Strategic use of technology in teaching and learning mathematics. he report is available from: https://www.nctm.org/Standards-and-Positions/Position-Statements/Strategic-Use-of-Technology-in-Teaching-and-Learning-Mathematics/ 5. Hodgen, J., Foster, C., Marks, R., & Brown, M. (2018). Evidence for Review of Mathematics Teaching: Improving Mathematics in Key Stages Two and Three: Evidence Review. London: Education Endowment Foundation. The report is available from: https://educationendowmentfoundation.org.uk/evidence-summaries/evidencereviews/improving-mathematics-in-key-stages-two-and-three/ 6. Stacey, K. & Groves, S.: 1994, ‘Calculators in Primary Mathematics’, Paper presented at the Annual Meeting of the National Council of Teachers of Mathematics, Indianapolis. 7. Boaler, J. and Devlin, K., 2019. The Nature Of 21st Century Mathematics. 8. Boaler, J., 2019. Creative, Flexible Mathematics. 9. Drijvers, P., 2015. Digital Technology in Mathematics Education: Why It Works (Or Doesn’t). Selected Regular Lectures from the 12th International Congress on Mathematical Education, pp.135-151. 10. Idris, N. and Nor, N., 2010. Mathematical creativity: usage of technology. Procedia - Social and Behavioral Sciences, 2(2), pp.1963-1967. 11. Vasagar, J., 2016. Why Singapore’S Kids Are So Good At Maths. [online] Ft.com. Available at: <https://www.ft.com/content/2e4c61f2-4ec8-11e6-8172-e39ecd3b86fc> [Accessed 25 March 2020]. 12. Cunha, G., Barraqui, L. and de Freitas, S., 2018. Evaluating the use of gamification in mathematics learning in primary school children. 2018 IEEE Frontiers in Education Conference (FIE),.

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