### Question: Why teach mathematics?

Mathematics is the study of patterns and, in the realm of these patterns, mathematicians guess, zigzag around, conjecture, refine with counter examples and then seek to prove an equation or relationship (Boaler 2010, pp.20-26). Mathematics revolves around the subatomic to the super-galactic level; the universe is immersed in patterns. However, mathematics is arguably the most universally despised subject taught at school. Stephen Hawking’s publishers told him that every time he included an equation it would halve the sales of his brilliant book A Brief History of Time (Stewart 2013, p.viii). Society, it would seem, sees mathematics as a torturer locking them in a dungeon of digits. So why do we teach mathematics?

This essay posits that mathematics should be taught because it has an intrinsic beauty and cultural value, it advances humanity, it creates the skills that citizens need to function in society and obtain jobs, it develops the brain and it aligns perfectly with a powerful learning approach, that is, active learning. This essay will discuss each of these benefits in turn and conclude that mathematics must be taught at secondary school but the curriculum should be adapted to maximize its effectiveness.

Let us begin by exploring the view that mathematics should be taught because of its beauty. Ancient mathematicians like Euclid, Archimedes and Thales investigated mathematics because they discovered this beauty and appreciated the systems of patterns, logic and symmetry (Green 1977, p.27). Mathematicians marvel, in particular, at the beauty and ingenuity of Euler’s identity. Mathematics can be thought of as an art rather than a science; it reveals to us the innermost secrets of the universe. Nevertheless, mathematics’ intrinsic beauty may not be discernible to the majority of people, it may only accrue to a handful of students (Green 1977, p.27). Furthermore, the narrow requirements of the national curriculum and the uninspiring nature of GCSE mathematics topics may make it difficult for students to enjoy the subject (Rice 1998, p.23).

Mathematics is an artefact of humanity, it holds cultural value and is, perhaps, our curriculum’s oldest subject. It is part of our common heritage and should not be withheld from future generations (Heiede 1992, p.151). Apart from tallying, the earliest known number notation was discovered on clay envelopes by Mesopotamian accountants in 8000 BC (Stewart 2013, p.23). These numbers became a series of written symbols and formed the origin of all later number symbols and, perhaps, writing (Stewart 2013, p.23). Students should be taught mathematics to appreciate the subject itself, its role in history and society and the diversity of human thinking and accomplishment. But, if we teach mathematics for this reason then it may end up like Latin and disappear from the curriculum (Green 1977, p.28).

Another benefit of teaching mathematics is that it helps humanity advance. Mathematics is woven into our history, it pulls the strings behind our progression; the development of the human race has been redirected time and time again by equations (Stewart 2013, p.viii). Ancient societies in Mesopotamia and Greece advanced our understanding of numbers, accounting, geometry and surveying; medieval Europe and the Middle East augmented our knowledge of algorithms, commerce, trigonometry, navigation and mechanics; and the modern era has increased our skills with agriculture, medicine and digital computing (Ernst 2005, p.28). In particular, Pythagoras’ theorem1 has been key to the development of our modern world and has shown us our place in the universe. Pythagoras’ theorem bridges algebra with geometry, helps us to build, create maps, use GPS and is the bedrock of theories of space, time, gravity and special and general relativity.

Mathematics has, in particular, been of great assistance to science. All the great scientists were extremely able mathematicians. Think, for instance, of Newton and the proverbial apple or Coulson and the atom (Green 1977, p.27). Mathematics has pointed to the existence of many phenomena and problems and subsequently provided models and answers for them. For example, mathematics lead to the search for the ‘God’ particle in physics and opened up philosophical debates that resulted in the formulation of Schrodinger’s cat. Conversely, mathematics also plays a part in the destruction and downfall of our planet. Consider, for instance, the dread that nuclear weapons brought upon humanity or the role of the Black-Scholes equation in the 2007 global financial meltdown.

Let us move on and discuss the view that mathematics should be taught because it is at the heart of everything we do; linearity, for instance, or rectangularity. 21st Century society assumes its members have a basic mathematical understanding and can function as numerate critical citizens. Every citizen should be able to perform simple calculations and possess a concept of numbers, decimals and fractions. For example, if you want to drive 30 miles and you have 2 gallons of petrol in your tank will you make the journey or do you need to stop at a petrol station? Moreover, society expects us to apply our mathematical knowledge in social and political realms for the furtherance of both the self and democratic society (Ernst 2005, p.28). The news, for example, is an amalgamation of budgets, profits, inflation and weather probabilities all grounded in graphs, percentages and charts (Boaler 2010, pp.8-9). The individual must identify, interpret and evaluate the mathematics embedded in social, commercial and political systems and, where needed, reject spurious claims. Individuals must utilize these skills if ideals like democracy and values of a civilized society are to be protected (Ernst 2005, p.28).

Mathematics is continually becoming more deeply and invisibly embedded in all realms of everyday life. Society is becoming more complex and creating social pressure for us to subconsciously use mathematical models and numbers. For instance, discussions of football matches are conducted in the domain of a model concerning probabilities of success and relegation, percentages and proportions (Green 1977, p.29). These models are not externalized, they are not clearly identifiable as models of mathematics, they do not utilize differentiation or matrices, they are “shrouded in language and distorted by common usage and shared understandings, they exist and are fundamental to interpreting the world around us” (Green 1977, p.29). However, it is argued that the mathematics curriculum focuses too much on computational manipulations that will not help students to apply mathematical reasoning to solve real world problems (Boaler 2010, p.10). Also, where do algebra, geometry and calculus fit into solving everyday life issues? These topics are not a prerequisite to understand and interpret the news or to plan household budgets. Much of what students are taught in mathematics classes like the quadratic formula are never used again in the real world.

Another area of debate concerns the argument that mathematics must be taught at secondary school because it is a vocational necessity. Many university courses and jobs in the UK require applicants to possess a certain standard of numeracy, usually a mathematics GCSE. Modern industrialized mechanical life, and the resultant jobs that come with it, is founded on a mathematical basis and its progression requires the constant application of mathematical principles (Jackson 1940, p.338). Additionally, mathematics is required to prepare students for jobs in science and engineering. Boaler (2010, p.7) claims that an estimated 20 million more jobs will be provided in the future for those who are mathematical problem-solvers. On the other hand, not all students need to learn advanced mathematics to prepare for future careers. The option of further mathematical courses should be available for those who need to pursue them but they should not dominate or distort the curriculum for all students (Ernst 2005, p.28).

The brain is like a processor that knits together mathematics with literature and it is important to develop both aspects to boost the power of the brain. The brain works in a mathematical manner without us even realizing. Weber-Fechner’s law, for instance, suggests that the brain operates using logarithms at a subconscious level (Stewart 2013, pp.33-34). A person goes to the gym to build their arm and leg muscles; likewise, the brain is a muscle and must also be put to work. Mathematics possesses some of the deepest and most exciting ideas created by the human race and exercises your brain (Ernst 2005, p.28). Learning algebraic and geometric equations boosts your brainpower; even if you never use those equations later in life they help you to optimize your ability to make complex decisions.

Moving on to the next issue in the debate, we could contend that mathematics should be taught because it is a prime subject to interweave with active learning. Active learning requires the learner to take on part of the responsibility to develop activities, emphasizing that successful learning entails a sense of ownership and personal involvement (Capel et al 2013, p.328). Active learning works within the sphere of constructivism where students learn through experience, that is, by selecting the tools they believe are needed to answer a question rather than being told what to do. A major benefit of active learning is the development of higher-order thinking skills because it focuses on metacognitive, formal and heuristic approaches to learning (Capel et al 2013, p.341). By discussing and problem-solving, students access higher-order thinking skills like analysis, synthesis and evaluation, the tip of Bloom’s taxonomy pyramid. One of the main ideas behind active learning is to encourage students to become autonomous learners. Mathematics, problem-solving especially, is a natural subject to work in the active learning framework because it allows students to take control of their own learning and use a creative approach to solve tasks (Ernst 2005, p.28). Mathematical problem-solving allows students to build connections between concepts, view problems through different lenses, modify and reflect on their approach and make use of cognitive abilities such as symbolic thinking, temporal-sequential organization and verbal memory (Triphathi 2009, p.168).

This is in stark contrast to what Capel et al (2013, p.326) refers to as ‘atomistic learning’, that is, the rote memorization of equations and multiplication tables. To develop the cognitive abilities of children we must focus on active, rather than passive, learning. However, the mathematics curriculum has neglected the problem-solving aspects of the subject and therefore limited its effectiveness as a tool for active learning (Ernst 2005, p.28). The teaching of mathematics in schools focuses on marathon memory exercises rather than teaching logical thought (Jackson 1940, p.341).

Mathematics has been misrepresented by the national curriculum and, therefore, has made it harder to teach and squashed the positive impact it can have on students (Boaler 2010, p.6). If we return to our definition of mathematics we see that it is the study of patterns, but the curriculum focuses on a target culture where thinking and reasoning are neglected and performing calculations is championed (Boaler 2010, pp.20-33). Mathematicians are trained to think and reason but students are forced into a passive relationship where they must follow rules to answer questions (Boaler 2010, p.36). Real mathematics revolves around problem-solving, creating ideas, exploring puzzles and discussing methods (Boaler 2010, p.2). If students are taught this style of mathematics they will engage with the subject and be more successful (Boaler 2010, p.5). Rice (1998, p.24)

posits that the use of school-level brainteasers from Egyptian, Babylonian, Indian and Islamic mathematics can be used to show students that mathematics can be beautiful, inspiring and enjoyable and teaches them to reason and construct their own answers to questions. Moreover, the mathematics curriculum must be reorganized to allow students to problem-solve with real world topics like industry, commerce, personal health, use of leisure time and household budgeting (Hendrickson 1974, p.469). At the start of the industrial revolution, manufacture required men to calculate and measure accurately but now manufacture and information technology requires employees to use estimation skills, three dimensional geometry and the disposition to think through problems and blend visual with mechanical information (Boaler 2010, p.8). Mirroring this, the curriculum should change accordingly to develop these skills.

Reflecting on the issues discussed in this essay, we can conclude that mathematics must be taught at secondary school, but the way it is taught should be altered. To study and discover patterns is, undoubtedly, beautiful, and there can be no disputing the fact that mathematics holds cultural value. Moreover, from the advent of the abacas to the culmination of space travel and beyond, mathematics has evolved, and evolved with, the human race. Furthermore, it is necessary to teach mathematics to form students into functional numerate citizens and to build their job prospects. Additionally, the study of mathematics is essential in the development of the brain. By the same token, mathematics intertwines with active learning and can help students become autonomous learners. As this essay has shown, mathematics should be taught in secondary schools because its benefits range from the personal development of children to the functioning of society. However, as highlighted, for mathematics to benefit all children we must regenerate the curriculum to refocus on thinking and reasoning skills rather than rote memory or computational manipulations. Mathematics is, arguably, the most important subject to teach, as long as it is taught within the sphere of active learning to encourage higher-order thinking and autonomous learning.

#### Bibliography

Boaler, J., (2010), Elephant in the Classroom: Helping Children Learn and Love Maths, London: Souvenir Press.

Capel, S., Leask, M. and Turner, T., (2013), Learning to Teach in the Secondary School: A Companion to School Experience, New York: Routledge.

Ernst, P., (2005), Why Teach Mathematics, Mathematics in School, 34(1), pp.28-29.

Green, J., (1977), Maths: Why so Much?, Mathematics in School, 6(2), pp.26-28.

Heiede, T., (1992), Why Teach History of Mathematics?, The Mathematical Gazette, 76(475), pp.151-157.

Hendrickson, D., (1974), Why do We Teach Mathematics, The Mathematics Teacher, 67(5), pp.468-470.

Jackson, N., (1940), Why Teach Mathematics, School Science and Mathematics, 40(4), pp.338-343.

Rice, A., (1998), Doubling the Square or Why do I Teach Maths?, Mathematics in School, 27(4), pp.23-24.

Stewart, I., (2013), Seventeen Equations that Changed the World, London: Profile Books.

Triphathi, P., (2009), Problem Solving in Mathematics: A Tool for Cognitive Development, in Subramaniam, K. and Mazumdar, A., (eds), epiSTEME-3 Third International Conference on Review of Science, Technology and Mathematics Education, Mumbai, India: MacMillan.