My mother-in-law telephoned me early in the morning recently. She lives alone, and during the long nights, she worries. Her concern on this particular morning was whether her granddaughter (my niece) would be safe during her visit to New York City. My mother-in-law—who lives in Otley, England—had just seen the news that 49 people had been murdered by a gunman attacking a nightclub in Orlando.
I tried to use mathematics to reassure her. I explained that Orlando is 1,740 kilometres from New York, and Otley is 1,740 kilometres from Helsinki, Finland. So choosing not to go to New York because of an attack in Orlando is equivalent to friends not visiting Otley because of a threat in Finland.
This argument didn't convince her, so I tried statistics. I pointed out that 12.3 out of every 100,000 deaths is caused by unintentional poisoning; 9.6 of every 100,000 deaths by accidental falls, and 10.7 deaths of every 100,000 by a traffic accident. This compares with 3.5 out of every 100,000 deaths being caused by assault with a firearm (Centers for Disease Control and Prevention, 2016). So in this country, you're three times more likely to die in a car accident than by being shot. Again this didn't help (although the idea of a car accident added to her worries).
My mother-in-law's real concern was many citizens' continued intolerance to difference, which in extreme cases leads to violence. My niece is gay, and the Orlando attack targeted the gay community. I accepted that sometimes mathematics alone doesn't hold the answer; we must use our understanding of mathematics to work for a more socially just society that embraces difference.
From Dewey to Brexit
Intolerance and international misunderstandings aren't new concerns. One hundred years ago, John Dewey (1916) wrote about the contradiction between those human activities that transcend national boundaries and engage individuals across nations in interdependent activity—like science, business, and artistic endeavours—and the reality of nation states living in "a state of suppressed hostility and incipient war with [their] neighbours."
Dewey was writing in the aftermath of a war in which Europe tore itself apart. His plea to educators to play their part in building a safer world has never seemed more pertinent.
Our students today live between two worlds. Much of their time is spent within their local communities, but they also exist in a global world made smaller by their access to the World Wide Web. Many young people now travel for work or pleasure with few limits, and those fortunate enough to do so return enriched by their experiences. People use technology to meet across cultures and continents. The rapid growth of international schools around the world is educating many of our students between and across cultures.
There's even a term, third culture kid, for children who do part of their growing up in a country different from their parents' country, so that they don't completely identify with either their parents' culture or the host culture. Many observers suggest that such kids build relationships to many cultures without having full ownership of any. Perhaps in third culture kids we're seeing the beginnings of global citizens, individuals who are citizens of many nations without exclusively belonging to any single one.
It's worth asking: Could we use math instruction to guide today's students to consider global issues—and the implications of certain policies—in more depth? Here are two activities toward that end.
Activity 1: Democracy in Action?
Last June, I visited the United States with my wife just after the United Kingdom held a referendum (known as Brexit) asking citizens whether or not they wished to remain a part of the European Union (EU). The final tally resulted in a decision to leave the EU. In the days following the referendum, many Americans—hearing our English accents—asked us, "What have you done?"
As a mathematics educator, part of my internal answer was to reflect on what mathematics teachers might do to build a world of connected global citizens, rather than the intolerant, suspicious world Dewey described. I began thinking about the referendum voting process itself, and I designed a mathematics activity that might shed light on how that process works in various places.
First, students think about voting by considering policies in Switzerland, a country that often uses referendums to make political decisions. For the results of any Swiss referendum that affects its constitution to take effect, a "double majority" is required. That means that in addition to a majority of voters supporting the change, a majority of the voting regions from which voters come must also support the change.
Teachers might guide learners to use data analysis to reflect on the nature of a democratic vote in three ways. The first approach would apply the Swiss notion of a double majority to data on the Brexit vote. If the Swiss rule had applied to this U.K. referendum, the double majority requirement would have been satisfied because 9 of the 12 regions of the country voted to leave. But another way of thinking about this would be to use the age groups of British voters as the second majority. That is, for the "leave" side to win, in addition to a majority of citizens voting for leaving, a majority of citizens in each major age group would also have to vote that way. The data in Figure 1 show that the Brexit vote split along age lines, with the majority of British people under 50 voting to remain and most of those older than 50 voting to leave.
Figure 1. How Age Groups Voted in Brexit
A third way to ponder how the referendum might have resulted in a different outcome is to consider British legislation related to trade unions, which requires that, for a strike in "important public services" to be legal, 40 percent of eligible voters must agree that the strike should happen. If such requirements applied to the Brexit vote, 18.6 million "leave" votes (representing 40 percent of eligible voters) would have been required for the "leave" campaign to succeed. The campaign would have been 1.2 million votes short.
To think about the Brexit results in a different way, teachers might guide students in an extrapolation exercise. Students would look at the data on the percentage of all British citizens who (1) voted in the referendum, (2) didn't vote, (3) voted to leave the EU, and (4) voted to remain. Students should apply those four percentages to the total number of students at their own school. I did this myself, pretending that the 1,430 students in my local high school represented the United Kingdom's population, and I got this data:
Voted: 1,032Did not vote: 398Voted "leave": 536Voted "remain": 496Vote carried by: 40
Students might then display the percentages they extrapolated and discuss questions like, Would you agree to a major change being made in your school on the basis of a majority of 40 pupils? Why or why not? What other rules might you put in place for a referendum voting? It's possible that this sort of discussion might encourage students to engage in the political process. I invite all mathematics teachers to model aspects of the democratic process using similar examples, to encourage a critical understanding of the nuances of this process.
Activity 2: Examining Refugee Journeys
What Makes a Global Citizen?
To further explore the contribution mathematics might make to developing global citizens, let's draw on the learner profile developed by the International Baccalaureate (IB), which offers a core curriculum to international schools around the world with a mission to "create a better world through education." The learner profile on the IB website states that all students should become
- Inquirers who develop their natural curiosity.
- Knowledgeable through exploring concepts that have local and global significance.
- Thinkers who exercise initiative in thinking critically and creatively.
- Communicators who understand and express ideas and information confidently.
- Principled, acting with integrity and honesty.
- Open-minded, understanding their own cultures and personal histories.
- Caring, showing empathy, compassion, and respect to others.
- Risk-takers who approach unfamiliar situations and uncertainties with courage and forethought.
- Balanced, understanding the importance of intellectual, physical, and emotional balance.
- Reflective, giving thoughtful consideration to their own learning and experience.
I often share this list of attributes with mathematics educators and invite them to audit their current practice against it by thinking through mathematics lessons they've recently taught and considering which of these attributes they've developed through these lessons (and which attributes they've sidelined). I subject my own practice to this same audit.
Calculating a Journey's Cost
An extended activity I've developed—which a colleague who teaches drama and I recently demonstrated at the Association of Teachers of Mathematics (ATM) conference in England—builds these attributes by engaging students in a task that dramatizes the journey that thousands of Syrians are currently forced to make.
I opened this article by suggesting that it's never been easier for some of us to travel. However, by accident of birth, it's almost impossible for some people living in the most perilous situations to travel to safety. This activity helps learners grasp that reality. Note that it's best to initiate this activity after you've built a trusting classroom community.
When I lead people in this activity, I first have participants use dramatic games that explore issues of traveling far from home. Then, in small groups, students make 100 miniature people out of modelling clay (100 is the total created by all the groups together). Specify that each clay person should be one-twentieth of the average size of the individual members of each group. Leave the definition of average to each group. You'll see students interpret the term in a variety of ways, with much evidence of measurement and division.
While students are working on this task, pose this additional problem for them to calculate: How long would it take you to walk 4,320 kilometres? I've found it effective to not offer, at this stage, a context for this problem (or the information that 4,320 kilometres is the distance from the Syrian border to Calais, France).
As the groups finish modelling, bring students together and show them this information:
From Syria to the Turkish border with Greece = 1,500 kmFrom Syria to the Greek border with Macedonia = 2,000 kmFrom Syria to the Austrian border with Germany = 3,100 kmFrom Syria to the German border with France = 3,700 kmFrom Syria to Calais, France = 4,320 km
Students then use masking tape to create on the floor a line that represents the relative distances from Syria to the various countries listed. They mark on this line where the border to each country would be reached if one were traveling to these countries from Syria, placing marks so the distance from Syria to each "border" is proportionately equal to the real-life distances. At this point, the context for the question about how long it takes to walk 4,320 kilometres becomes clear. Discuss the groups' answers. When I led this activity with colleagues, we agreed that the journey would take a minimum of three months if we included sleeping breaks and difficulties crossing borders.
Let students know that each model person represents one percent of the people displaced from Syria and within Syria as a result of the war. Share this data with students:
The population of Syria is 22.8 million.It's estimated that 6.6 million people have been displaced within the country and 4.8 million displaced to other countries. So approximately 50 percent of the population has been displaced—29 percent internally and 21 percent to other countries. Estimates suggest that refugees are now living in the following countries:Turkey: 2.8 millionLebanon: 1.5 millionJordan: 1.5 millionGermany: 500,000Macedonia: 400,000France: 10,000UK: 8,000Dunkirk and Calais (refugees in camps there): 6,000.
Next, learners place all 100 model people in the area along the line close to the various countries' symbolic borders. Students calculate what percentage of the total of Syrian refugees each country is hosting, and then place the number of figures showing that percentage near that country; for instance, if Turkey harbors 60 percent of the displaced Syrians, students would place 60 figures near the border of Turkey. As learners place the figures onto the path, the atmosphere changes. There's a real sense of not wanting to put the figures in dangerous areas.
Finally, students place on the map something (like a strip of cloth) representing the English Channel, calculating its proportional width. When the teachers at the ATM conference put our English Channel down, one participant looked back at the image of a journey we'd created and came to a realization: "If you'd come all that way, of course you are going to try and get over the English Channel."
This activity lets learners see what a small proportion of refugees from Syria are actually reaching the borders of the United Kingdom or other western European countries. When I tried this with the teachers at the conference, the 6,000 Syrian refugees living in camps at Dunkirk and Calais in France were represented by 0.1 percent of a person (an arm). One clay leg represented the portion of refugees who had safely reached the United Kingdom. At least two teachers who tried this activity with me have since carried it out successfully with their students.
So how would this simulation fare if we audited it to determine whether it develops the skills the International Baccalaureate hopes to build? The spirit of inquiry was evident throughout. Our introductory drama games involved problem solving (like finding the smallest area five people could take up), and the open question about the 4,320-kilometre journey aroused participants' curiosity. Participants developed knowledge (of local and global significance) about the refugee crisis in Europe. The tasks led them to be creative thinkers and to communicate their ideas and new information thoughtfully. We purposefully built trusting relationships within the groups, which allowed participants to take risks and be open minded, and participants treated one another and the context within which we worked with care.
At the activity's end, the participants spent time reflecting on their responses to this dramatization of an intense human problem—something I recommend teachers who try this activity build in. There were many emotional responses, testifying to the depth of participants' engagement.
A Call to Action
I opened with a story about my mother-in-law. I cannot convince her or anyone that the world isn't becoming an increasingly dangerous place, even though history (and applying mathematics to certain questions) indicate that this isn't the case. However, that doesn't remove mathematics teachers' responsibility to teach in ways that instill in students the attributes described earlier and that help create a more just, safer global society. So my question for you is both a call for reflection and a call to action: What have you done? More important, what can you do?
