## Engaging Students in Mathematics: Why Context is Critical

#### As Billy Connolly says, *“Why would I want to learn Algebra? I never plan to visit there.”*

#### In this article we explain why providing real-world context can be critical to engaging students in mathematics and a key catalyst for understanding. We then provide five practical tips, linked to useful classroom resources, for helping teachers develop mathematical context in their lessons.

In his recent book “The Human Universe”, Dr Brian Cox explains to readers how the use of a fairly simple algebraic equation derived from Newton’s Law of Gravitation and his Second Law of Motion could be used to perform a “back of the envelope calculation”. This arithmetical expression can be used to fairly accurately determine the orbital velocity of the International Space Station in preparation for the return of resident cosmonauts back to Earth. Cox follows his explanation with applause for the wonder of applied mathematical physics: “The natural world is orderly and simple and can be described with great economy by a simple set of laws… it is nothing short of wonderful that we can calculate the orbital velocity of the International Space Station together in a few lines of a popular book.”

Calculating the speed of space stations as they orbit the earth is probably not the type of calculation most young people will need to perform every day – but it does demonstrate the power and the enabling capacity of mathematics. Without it, the human race would have remained a simian species of hunter-gatherers, incapable of developing even the most fundamental tool-making, agricultural or mechanical technologies.

Mathematics is important because of its real world context. On the inverse, without context, it is often hard to identify with the beauty and power of mathematics.

This point is essential to make when seeking to engage our students in the discipline. Sadly, too many children – and adults – immediately think only of slogging over a lead pencil and graph paper in a dreary classroom on a wet Monday when prompted by the word ‘mathematics’. Linking maths to the inspiring realm of space exploration, the fascinating field of micro-biology, the exciting pursuit of sports performance or the compelling virtual worlds of gaming and digital media seems to come less instinctively. However, as Bertrand Russell said, “Mathematics, when rightly viewed, possesses not only truth, but supreme beauty.”

During a recent school visit I was sitting in the staffroom when the head maths teacher entered, having just finished a class with some Year 12 Advanced Mathematics students. They were investigating an area of Cartesian geometry and trigonometry using complex numbers (an area of mathematics which I will readily admit is beyond my expertise). The teacher had been asked by her students: *“What are the practical applications for multiplying and dividing complex numbers expressed in their polar form?” * Feeling overwhelmed with dealing with the complexities of trigonomic ratios and the quirks of ‘imaginary numbers’, her students needed to know that at the end point of their mathematical exertions lay a place ‘in the real world’ in which these difficult concepts had some utility. Fortunately, we were able to defer the question to a colleague of mine who was able answer the question, hopefully to the students’ eventual satisfaction.

Context in maths for children and young adults is especially critical. As educators we are all too familiar with the catchcry from our students, “Why are we even learning this stuff, Miss / Sir?” Very often, objections of this nature indicate some underlying difficulty with content. The real question underlying their grumbling: “If I’m going to be bothered persevering with this, is it really going to worthwhile at the end?” As with any endeavour in life, slogging away at a difficult, confusing or outwardly onerous mathematical problem without the confidence of a satisfactory outcome is tantamount to digging holes and filling them up again.

Added to this mental impediment to engagement for children is the highly abstract nature of pure mathematics. For many, the language and notation of maths appears as nonsense; as Billy Connolly says, *“Why would I want to learn Algebra? I never plan to visit there.” *

Connolly’s witticism masks a salient point. Without the understanding that mathematics is a *tool *with which we explore the world around us (rather than simply a set of abstracted intellectual objects that are an end in themselves), mathematical processes beyond simple number calculations can seem like ‘gobbledygook’.

To take a familiar example, “e = mc²” means nothing at all until we understand that the ‘e’ (for ‘energy’), the ‘m’ (for the mass of an object) and the ‘c’ (the speed at which light travels) have *real values* in the *real world*. The *application* of Einstein’s General Law of Relativity (expressed by the abstraction “e = mc²”) to engineering, medical research, transportation technologies, chemistry and energy generation (to name but a few) has in fact enormous practical utility and potential. Besides its power and beauty as an elegant piece of mathematical physics, it is eminently *useful* for solving problems and improving lives. Without these applications, the mathematical derivation and expression of Einstein’s Law would be interesting to a few but meaningless to most.

This point was reinforced for me in a recent conversation that took place in our office. A colleague asked a question that, on the surface, appeared to be nonsensical: *“If today is zero degrees Celsius, and tomorrow will be twice as cold as it is today, what would the temperature be tomorrow?”* We had a chuckle at the intended absurdity of the question – until another colleague pointed out that if you define temperature in its *scientific* rather than simply in its *representative mathematical* context, the question actually has an answer (and a surprising one at that)

Degrees Celsius is an arbitrary mathematical measuring system we have applied to the state of being relatively ‘hot’ or ‘cold’. I have used the words ‘arbitrary’ and ‘relative’ intentionally here. The Celsius scale is based on heat or cold, relative to the freezing point of water. H2O is a molecular arrangement that is common on planet Earth and in certain other unique environments in the cosmos. It is, however, by no means the most abundant molecular arrangement in the universe – this status is enjoyed by H2, or hydrogen. As organic beings on our small blue-green planet, we like water as it is important to us. The Celsius temperature scale is thus essentially anthropomorphic; when humans’ ‘favourite liquid’ freezes, we assign it a temperature value of 0º. Above this on the degrees Celsius scale we assign positive representative mathematical values to the real-world state of being ‘warmer’ than frozen water, and below it we assign negative (representative) mathematical values.

However, outside this arbitrary and relative measure, what actually *is* ‘temperature’? Scientifically speaking, temperature is the amount of thermal energy emitted by atoms. There is another mathematical measure of temperature that reflects this scientific (‘real world’) understanding of temperature, that is, the scale devised by Lord Kelvin in 1848. The ‘Kelvin Scale’ (K) describes temperature in terms of the thermodynamic properties of atoms. Thus, when atoms emit no thermal energy (or more correctly, when atoms register no thermal motion), temperature is said to be at ‘absolute zero’ or ‘0º K’.

When we convert zero degrees Kelvin into degrees Celsius, we find that 0ºK = – 273.15ºC. Accordingly, at 0ºC, the atoms in water molecules are actually registering thermal motion which we measure at 273.15 ºK.

Now, let’s reconsider our seemingly nonsensical mathematical question in its ‘real world’, scientific context: *“If today’s temperature was measured at 273.15 ºK, and tomorrow was going to be twice as cold, how cold will it be tomorrow?”*

Of course, we first need to unpack what we mean by ‘twice as cold’. If we are describing an atmospheric temperature of 273.15 ºK (0ºC) as ‘cold’, this would imply that ‘twice as cold’ would mean ‘half the measurable amount of thermal motion’ in the atoms of the atmosphere at that point of measurement. Applying the mathematics, this simply means we take 273.15ºK and divide by 2, giving us **136.58 ºK**. How cold is this in terms of degrees Celcius? The answer is **-136.58 ºC **– very cold indeed! (Presumably, for this to occur, the Earth might have been hit by an asteroid and pushed way outside it’s comfortable ‘Goldilocks Zone’ orbit around the Sun!)

Labouring through this illustrative point (if having fun with maths and science like this could ever be called ‘labour’) is instructive as it shows us and our students that maths is, at its heart, a tool. With that tool, however, we can describe, explore, discover and predict things in the world around about us in ways and at levels of precision that words or pictures alone could never allow.

It also shows us that without a **context** based in the real world – such as that of science (or finance, or statistics, or technology…), mathematics often *appears* nonsensical – in the same way as would a foreign language or a word for which we do not yet know the meaning. That appearance, however, should never let us be discouraged from persevering with mathematics, because by doing so we are unlocking wonderful new ways of understanding our world and solving problems.

The following provides some practical tips on ways in which teachers can help provide context for students in mathematics and so seek to more fully engage them in the discipline.

**Five Tips for Providing Context in Mathematics. **

**1. Start new topics with a Story (or Video, or Visit…). **Commencing new topics in mathematics with an anecdote from your own experience, a Youtube clip or film excerpt or a short talk from a friend or community member, during which the ways in which the topic area in question is applied in the ‘real world’ is made evident, can often pre-empt the glassy-eyed student refrain, “When will we ever use this?” An example might be starting a new unit on ‘Fractions & Decimals’; watching the short Youtube clip, ‘Occupations that Use Fractions‘, or asking a local bank manager in to talk about how fractions and decimals are used to calculate loan and savings interest rates for 15 minutes, might ‘warm up’ students to the topic and give them a mental picture of where their learning might lead down the track.

Similarly, general film clips such as ‘Donald Duck in Mathmagic Land’ (an oldie, but a goodie!), or the Slideshare set ‘11 Damn Lies We Tell Our Kids About Maths‘ (a bit long, but suitable for older secondary students, particularly in higher levels of maths) can help stimulate students’ thinking about how mathematics overall relates to life and the world around them at the start of a school year.

**2. Take students on a “Maths Walk”.** This may be done at the beginning or at the end of a topic. Pick your maths topic focus and have students take a stroll with you around the school or the local neighbourhood / community. Arm them with pencils and a clipboard or a tablet and get them to note down anything they see that might relate back to their understanding of the topic. For example:

**‘2D Shape’ or ‘3D object’ at Year 2 level:** ask students to draw, write down or snap pictures of places or objects in which they recognise particular shapes;

**Coordinate Geometry at Year 7 level:** ask students to find examples on their walk of situations where the use of coordinates and/or the use of linear equations might come in handy (locating a particular store or landmark on a street map; describing the path from one point to another in equation form…); or

**Linear and Non-Linear Relationships at Year 10 level:** ask students to consider examples of occupations (from buildings, stores or industries they might walk past) in which an understanding of quadratic equations and / or parabolic functions might assist with problem solving in that occupation (eg. a farmer estimating the amount of materials needed to build several intersecting fences; the angle and velocity at which a ball might need to be kicked to score a goal in footy; the amount of sales needed per hour for a retail store to achieve a given level of profit or revenue after a given time…).

**3. Integrate or relate mathematics into other subject areas.** I often refer to this with my students as “making the maths visible” in other subjects, and it often only requires drawing attention to, or providing a verbal explanation of, how mathematics is used across in other academic disciplines as and when you are teaching them. Explicitly referencing maths outcomes in planning and programming documents across disciplines is useful if you are using integrated units of work in a primary classroom, but a simple discussion with students about the application of maths during a geography topic or art activity is all it takes.

If you are teaching at secondary level, start your maths class by asking students what other classes they have attended during the day or week, and have a discussion that ‘makes the maths visible’ in these areas. Alternately, brainstorm with your students at the start of a new topic ways in which the mathematics might have applications in other subjects they are studying at school.** **

Science topics are obviously fertile fields for pointing out where mathematics is useful and essential (the importance of maths in applying the ‘scientific method’ is a great place to start), however, it doesn’t take much contemplation to begin to recognise the footprints of mathematics in all other disciplines: paragraph and syllable density in Literature, meter in Poetry, pattern in Dance, symmetry and shape in Art, fractions in Music, coordinate geometry, demography and statistics in Geography, time and dating calculations in History, graphing of growth rates and performance indicators in Sport and Personal Development… The world is beautiful, fascinating and mathematical!

A note of caution: in cases where we are unsure about the real world context of a mathematical concept, it is important that we don’t try to ‘force it’ or conjure up a tenuous context where it really doesn’t exist *(“Errm, I think they use polynomial equations to calculate the weave density in Scandinavian basket-weaving contests…”) *If you’re unsure, tell students you don’t know, go away, do some research and then get back to them. There’s few things worse than a transparent lack of credibility to reinforce the damaging belief in the minds of students that mathematics is a load of senseless bunk.

**4. Involve local businesses or industry partners in your mathematics program. **Mathematics can be enriched and contextualised for many students by exposing them to the wider world beyond the classroom. This can be achieved through excursions, class talks or activity days held in partnership with local business people or industry partners. Having students develop surveys and statistical summaries with local businesses; involving local banking or credit union representatives in lessons on financial maths and literacy; or allowing a local engineering or architectural firm to help set and mark a mathematically challenging design task, are all ways in which you can enliven students’ conceptualisation of mathematics.

A 2013 Norwegian study of school – local industry partnership programs specifically investigated the extent to which the inclusion of ‘outside-school mathematics’ in maths teaching facilitated pupils’ participation in problem solving, encouraged communication of ideas and promoted discussions about strategy use and solutions. From the examples studied, the researchers concluded that *“the source of relevant mathematical models in the classroom is enhanced by the partnership with a company.”* Their observations included that these real-world connections may be helpful in engaging pupils and students in discussions that help ‘mathematise’ their understanding of real-world problems and help to foster ‘critical democratic competence’ and ‘functional understanding’ in mathematics.

**5. Use ‘authentic’ and ‘rich’ tasks in the maths classroom.** Authentic tasks and questions are those drawn from problems or situations in which students might need to apply or use mathematics in the real world. The Australian Curriculum: Mathematics makes regular mention of ‘authentic situations’ in its levelled descriptions of the mathematical proficiency of ‘Problem Solving’.

Authentic tasks need not always be in the form of worded problems, however, they should definitely link a mathematical concept to a situation in which the student needs to consider practical, real-world contexts and responses.

Many authentic mathematical tasks or problems are ‘open-ended’ in that they are vaguely defined, have no single ‘correct’ answer (although they usually do have a range of ‘incorrect’ answers) and involve more than one area, application or operation of mathematical thinking to solve. Examples might be *“How can we use our knowledge of calculating area to design the landscaping for a backyard on a given budget?”* Or, *“In what ways might the distribution of income and wealth be considered unequal in Australia?”*

These types of investigations take time and the application of mathematical reasoning and not just understanding and fluency. They will tend to throw those of your students who are whizzes with their times tables but less comfortable with thinking ‘outside the box.’ Nevertheless, they do make very clear the importance of mathematics to solving real-world issues and problems, from the level of the personal (‘design my garden’) to the national and international (‘find a way of improving economic equality’).

The highly regarded ‘NRICH’ mathematics teaching website describes rich tasks as those “having a range of characteristics that together offer different opportunities to meet the different needs of learners at different times”, and lists in detail the features and characteristics of good ‘rich tasks’ in maths.

NRICH emphasise that what really makes a rich task “rich”, however, is the way in which it is presented, including the support and questioning used by the teacher and the ‘roles’ that students adopt within the task. They also provide a series of excellent professional learning resources on integrating rich learning tasks into the curriculum.\

[1] Cox, B. and Cohen, A. (2014),

__Human Universe__, pp.158-159. William Collins : London, UK. [2] Hana, Hansen, Johnsen-Høine, Lilland and Rangnes (2013

*), ‘Learning Conversation in Mathematics Practice – School-Industry Partnerships as Arenas for Teacher Education’*. In

__Educational Interfaces between Mathematics and Industry__, Volume 16, pp 147-155. Springer International Publishing : Switzerland. [3] See Felton, M. (2014),

*‘Mathematics and the Real World’*(Website article, posted July 7, 2014), on the National Council of Teachers of Mathematics – Mathematics in the Middle School. URL – http://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/Blog/Mathematics-and-the-Real-World/. Accessed 21/01/16.

**References:**

Coulson, D. (2011), ‘11 Damn Lies We Tell Our Kids About Maths’, on Slideshare, URL: http://www.slideshare.net/yaherglanite/11-damn-lies-we-tell-our-kids-about-maths-18606963. Accessed 21/01/16.

Cox, B. and Cohen, A. (2014), __Human Universe__, pp.158-159. William Collins : London, UK.

Disney (1959), ‘Donald Duck in Mathmagic Land’, published by TrueDisney on YouTube, URL: https://www.youtube.com/watch?v=U_ZHsk0-eF0. Accessed 21/01/16.

Felton, M. (2014), *‘Mathematics and the Real World’ *(Website article, posted July 7, 2014), on the National Council of Teachers of Mathematics – Mathematics in the Middle School. URL – http://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/Blog/Mathematics-and-the-Real-World/. Accessed 21/01/16.

Hana, Hansen, Johnsen-Høine, Lilland and Rangnes (2013*), ‘Learning Conversation in Mathematics Practice – School-Industry Partnerships as Arenas for Teacher Education’*. In __Educational Interfaces between Mathematics and Industry__, Volume 16, pp 147-155. Springer International Publishing : Switzerland.

Hurst, C. (2007), *‘Numeracy in Action: Students Connecting Mathematical Knowledge to a Range of Contexts’*, in __Mathematics: Essential Research, Essential Practice__, Volume 1, pp.440-449. Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia. MERGA : Sydney, Australia.

Piggott, J. Rich (2008),’Rich Tasks and Contexts’, article on NRICH – Enriching Mathematics website, URL http://nrich.maths.org/5662, Accessed 21/01/16. University of Cambridge : UK.

TeacherTube Math (2013), *‘Occupations that Use Fractions 2’*, on __Youtube__, URL https://www.youtube.com/watch?v=nU_kdjeGACI. Accessed 21/01/16.