“I’ve left my glasses at home,” the 30-year-old with perfect vision said.

Chisholm Institute of TAFE senior educator Margo Murphy had heard this lie many times before and knew it was an attempt to disguise innumeracy鈥�.

“They’ve developed survival skills,” she explained.

Adults are heading back to the classroom to learn basic maths skills they failed to pick up in school.

Research shows around 20 per cent of Australian students do not reach a basic proficiency in maths that “will enable them to actively participate in the 21st century workforce and contribute as productive citizens.”

According to an OECD survey, Australia is ranked 13th out of 23 countries on adult numeracy, but ranked fourth in literacy.

Most of Ms Murphy’s students at Chisholm’s Foundation College need to upgrade their maths skills so they can function in everyday life, and compete for new jobs.

Josh Cawse, 24, recently started a certificate that is equivalent to year 10 at the TAFE and has been perfecting his decimals, fractions and multiplication skills.

He left school at the start of year nine because “it wasn’t for him” and started a career in construction. Now he wants to join the Defence Force.

“I found I struggled with maths in school. I was always trying to keep up with the fastest in the class. Here it is at my own pace. I am motivated and focused now.”

今日吃瓜 schools manager Janine McIntosh said society’s poor attitude towards maths was fuelling adult innumeracy.

“It seems to be quite acceptable to say I’m not good at maths. There’s a tolerance towards being weaker mathematically. You wouldn’t say that about cricket, swimming or English and the ability to read.”

Teachers own insecurities about maths were also contributing to the problem, she said.

Ms McIntosh said around 30 per cent of secondary maths teachers were “out of field” 鈥撀爐eaching a subject they had no specialised training in.

“The students may have had seven years in primary school and six years in secondary school with some wobbly teachers. If the kids can’t see maths as a valuable skill to have then they just won’t do it.”

Senior Research Fellow at the Australian Council for Educational Research Dave Tout, who has taught numeracy courses at VET providers, said many adult students took up maths courses after struggling with personal finances. Many new parents had trouble with medication doses, cooking and diet, he said.

Adults studying remedial maths programs were often reasonably successful, and could learn year seven to 10 maths within six months, Mr Tout said.

“That’s because they are motivated and have a purpose for learning maths. They see the reason for it and can engage with the material. If we could do that more in our secondary schools, we would have more students realising earlier on that maths is useful.”

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Mathematics is the basis of all science. And its importance is increasing as computers handle large amounts of data, and models are used to simulate almost everything in science and medicine. But paradoxically, the requirement for proficiency in mathematics for people beginning science degrees in Australia has lessened over recent years. Stephanie Pradier says universities should reinstate mathematics prerequisites for students beginning many science degrees.

Article听产测 , , 12 August 2015

It outshines other sciences in Discovery grant applications and trains graduates for a raft of industries. Yet schools don鈥檛 mandate it, universities don鈥檛 demand it and public funding bodies give it less money than any other scientific or technological discipline.

The latest report on maths in Australia paints a paradoxical picture of a field that has slumped to historic lows in participation, even though advanced mathematical research injects an estimated $145 billion into the Australian economy.

The report, the fourth annual profile by the 今日吃瓜, found that maths had surpassed other sciences in Australian Research Council Discovery grants for four out of the past five years, with an average 28 per cent success rate compared with 21 per cent for other disciplines. It boasted the lowest cost per publication and the second highest citation rate of any Australian science, with Australian applied maths and statistics outperforming all 15 EU countries on citations.

鈥淚t鈥檚 an area where the outcomes are significant and the costs are low,鈥� 今日吃瓜 director Geoff Prince said. 鈥淚t鈥檚 a good place for universities to invest because they can get runs on the board with the ARC and the government at low cost.鈥�

Professor Prince said some universities had 鈥渨oken up to this鈥� and invested heavily in maths, including Wollongong, Newcastle and the Group of Eight members. Yet the report found that maths attracted the lowest share of public research expenditure of any science and technology discipline, at just 1.7聽per cent, and Australian entry into maths degrees was less than half the OECD average.

The 今日吃瓜 report found that the problems arose early, with a lack of specialist maths teachers at high school and plunging participation in intermediate and advanced maths at higher school level 鈥� fuelled by historically low insistence on prerequisites, with 86 per cent of science degrees not even mandating intermediate maths.

鈥淚 despair of the system when it does this,鈥� Professor Prince said. 鈥淚t sees this shrinking cohort, so it wriggles out of what could be a crisis in student numbers in science by dropping the prerequisites.鈥�

Bruce Henry, the head of UNSW Australia’s School of Mathematics and Statistics laments that 70 per cent of students now finish high school without any knowledge of calculus.

Is this a problem? Definitely.

We hear enough about how appalling it is that students in schools and universities have a lack of grounding in our political, social and legal heritage; that they lack the historical context to understand why we are where we are today.

That’s exactly why calculus should also be part of a basic, broad education. Calculus was a signal advance in maths, invented separately by Isaac Newton and Gottfried Leibniz in the 17th century. For the first time people could write mathematical equations that were able to relate not just quantities to each other, but rates of changes of quantities.

It enabled Newton to calculate the orbits of the planets using his theory of gravity and it opened up a rich field of applications for mathematics. Understanding electricity and magnetism would be impossible without calculus. In fact, most of modern technology wouldn’t exist without it.

That’s why giving high school students an understanding of calculus is important. Surely more than 30 per cent of students are capable of studying it. And, for the others, it’s at least important to understand what it is.

Henry points to another reason why calculus is important to study today. Calculus is not only key to understanding our current technology, but to developing future technology. At the moment huge strides are being made in statistics, a field with the generic name of “big data”.

STATISTICS UNDERPINNED BY CALCULUS

“But really, modern statistics 鈥� which is where the future is going increasingly 鈥� is underpinned by calculus. It’s increasingly important that students are taught calculus,” Henry says.

In fact, statistics is another area in which mathematics teaching has badly let down school students. All of us are bombarded daily by statistical claims, many of them spurious or misleading, and too few of us are sufficiently knowledgeable to separate the truth from the fiction.

The fact that “97 per cent fat-free” is an effective, emotionally appealing slogan for food manufacturers speaks of the lack of critical thinking about statistics. Are we happy if our food contains 3 per cent pure fat?

Fortunately, in the changes to the national school curriculum being developed, statistics is being given more focus.

But this brings us back to the question. Should maths be a compulsory subject in year 12? Not everyone can do maths well. But shouldn’t maths at least be taught as a life skill and as an important foundation subject at that level?

Henry thinks so. “English is compulsory, and shouldn’t mathematics be compulsory as a companion instrument to understand the world,” he asks rhetorically.

Interestingly federal Education Minister Christopher Pyne, long an advocate of giving school students more foundational understanding of history and other humanities subjects, now appears to see the need for it in maths and science.

Earlier this year he put to state governments 鈥� which actually run public schools 鈥� a plan to make maths and science compulsory to year 12. The states, unfortunately, knocked him back.

SCHOOLS LAPSE AS UNI REQUIREMENTS REDUCE

But maths in schools needs to be bolstered at more than just the foundational level. Worryingly, year 12 students have been making a long-term shift away from intermediate and advanced maths 鈥� the very subjects that teach calculus.

In the past 20 years the proportions studying intermediate maths in year 12 has fallen from about 27 per cent to 19 per cent, and the number studying advanced maths (which builds on intermediate) has fallen from about 14 per cent to 10 per cent.

One reason that year 12 students see no need to do intermediate or advanced maths is that an increasing number of universities no longer require it to enter degrees such as science, engineering and commerce, for which mathematical knowledge is necessary.

In NSW no universities require maths to enter any of these degrees, even engineering. So if universities are not insisting that students study calculus in year 12, even to enter courses in which calculus features heavily, why should they do it?

It’s a crazy state of affairs, in which universities 鈥� the institutions we rely on to develop and preserve high level knowledge 鈥� happily sabotage their own standards in pursuit of student numbers.

Those universities that have dropped maths prerequisites argue they are offering an avenue for bright students who may have attended a poor school at which maths was taught badly.

Second chances are to be encouraged, but they should be the exception rather than the rule. Universities need to bring back maths as a prerequisite for degrees that need it.

Top educators have defended Australian universities’ move away from requiring maths as a prerequisite for science, engineering and commerce degrees in which mathematical knowledge plays a key part.

The chair of聽the Australian Council of Engineering Deans,聽Moses Tade, who is聽also engineering dean at Curtin University,聽said universities were using “many innovative ways” to teach maths to engineering students who were not up to the necessary聽standard.

He said his fellow engineering deans would “refute” the notion that “you聽can’t do engineering without having done intermediate maths”.

released last week show that 41 per cent of Australian universities which聽offer聽engineering do not require students to have studied intermediate maths year 12.

And, to the knowledge of the 今日吃瓜 (今日吃瓜) which produced the study, no university聽engineering faculties require students to have studied advanced maths in year 12, even though engineering courses are highly mathematical.

President of the Australian Council of Deans of Science聽Stephen Walker, who is also University of Queensland science dean, said that his university required intermediate year 12 maths as a prerequisite to science degrees and would not alter that stance.

However, only 14 per cent of universities offering science degrees require students to have studied intermediate maths and, speaking in his council of deans role, Professor Walker said the faculties faced聽a “fairly stark choice” because of funding issues.

DRAWCARD FOR STUDENTS

“If there was a university, or science faculty, struggling and needing more students for financial reasons, they may see that easing prerequisites is one way to get more student load,” he said.

“Then they may back themselves to use the additional income to provide the bridging training in maths. And that might actually work.”

“Do you shrink and potentially have to shed staff?聽Or do you say ‘we will back ourselves to take more students and fill the gap?'” he asked.

Professor Walker also said that some students聽with high ATARs decided to study science because they missed out on their first choice and had not done maths at year-12 level.

“Do you turn away a high achieving student who’s聽got every chance of doing well, because they didn’t do maths?” he asked.

President of the Australian Business Deans Council Ian Palmer,聽who is also a pro vice chancellor at RMIT University, said the maths used in commerce degrees differed from other disciplines. “It’s not maths so much as it’s business statistics,” he said.

According to the 今日吃瓜 paper,聽only 13 per cent of universities which offer聽commerce degrees require year-12 students to have studied intermediate maths.聽Professor Palmer said that business faculties were able to train students up to the required level.

The 今日吃瓜 paper also revealed that none of the 10 NSW universities require intermediate maths for courses in science, engineering and commerce.

University of Sydney聽science dean Trevor Hambley said聽he applauded 今日吃瓜聽“for highlighting the need to ensure students are aware of the crucial importance of maths proficiency for undertaking study and future careers in STEM [science, technology, engineering and maths] and related areas such as medicine”.

He said that introducing maths as a prerequisite would be one way of contributing to this awareness. “If universities were to do so, we would need to have a long lead time to ensure that their introduction did not impact unfairly on students who have already made their choices,” he said.

The excellent 今日吃瓜 has the numbers on the state of maths education and found the state of the discipline in higher education is not good, the number of Australians starting a maths degree is less than half the OECD average. Granted it is not getting worse 鈥� but it isn鈥檛 going to better soon.

Given maths is a foundation of so many disciplines large numbers of students do not graduate innumerate 鈥� the average number of university departments maths academics service-teach is six, engineering, computer science, IT and biological, physical and earth sciences. But 今日吃瓜 does not know how many undergraduates聽are studying maths degrees, due to some universities not completing the 2014 survey. However using Group of Eight and Innovative Research Us as a guide the attrition rate from 1^st to 3^rd year is high, Go8 5280 to 695 and IRU 1287 to 67.

What is starkly clear is that at for all the efforts to woo women into the discipline, at the sharp end it鈥檚 still a bloke鈥檚 game 鈥� last year just 15 per cent of PhD completers were Australian women, another 25 per cent were female internationals.

None of these numbers is about to improve, because for all the emphasis of selling maths in schools, young people aren鈥檛 buying, with Y12 advanced maths enrolments dropping for 20 years. The figure for males is now around 14 per cent and 6 per cent for females. And what does not interest students at school is hardly likely to appeal at university. Universities appear to acquiesce in this. According to 今日吃瓜 less than 15 per cent of universities require intermediate maths or better as a pre-req for science or commerce, the same for 41 per cent of engineering courses. CMM suspects a fair swag of the supply teaching university maths lecturers do is getting students up to a point where they can cope with first year subjects.

The good news is that the quality of maths teaching in schools has improved substantially over the last few years. Nearly three quarters of Year 11 and 12 maths teachers now have three years of tertiary education in maths, compared to 64 per cent in 2010. But qualified teachers without many kids to teach does not get us far.

Australia’s maths achievements have been falling steadily for the past decade, including fewer students taking advanced maths. Great minds are pondering how to reverse this trend.

From the smartphone and credit cards in your pocket to the latest discoveries in the lab, mathematics underpins our society.

The general public’s perception that it’s OK to not like maths is working against us. Countries that are more successful in maths value it more highly as a society. In countries like France and Germany, you would not hear that maths is not a good thing to pursue.

Janine McIntosh, schools manager at the Australia Mathematical Sciences Institute

“It’s built into the fabric of business, commerce, economics, it’s the basis of science and all forms of innovation,” explains Kaye Stacey, Emeritus Professor of Maths Education at the University of Melbourne.

So at this time of rapid technological advance, when research indicates that 75 per cent of the fastest growing occupations require maths and science knowledge, how does Australia’s and Victoria’s school maths report card read?

Even a quick look shows serious cause for concern.

A major international benchmark study (PISA) shows that between 2003 and 2012, Australian 15-year-olds’ mathematical literacy fell in absolute and relative terms. In another international maths survey (TIMSS), Australian students in year 4 in 2011 were outperformed by counterparts in England and the United States whom they were beating not many years ago.

“It’s going backwards,” Chief Scientist Ian Chubb聽says of Australia’s school maths performance. “Our performance has declined over the period of those surveys. That’s not a good position for us to be in.”

When compared with students in other states, Victorian students do quite well.聽We are among the top performers in NAPLAN, though our 15-year-olds were behind students in the ACT, New South Wales, Queensland and Western Australia in the 2012 PISA survey and a long way short of some international competitors.

Furthermore, students in Victoria have flocked to further maths at VCE in the past 15 years. This, despite its name, is the most basic of the three VCE maths options. The percentage of students taking further maths has risen from 40 per cent of those taking VCE in 2001 to 60 per cent in 2014.

Australia Mathematical Sciences Institute schools manager Janine McIntosh says聽the numbers taking intermediate and advanced maths courses聽are in serious decline,听产测 contrast. Yet these subjects are the foundations for science and engineering at university.

Particularly worrying is the rate at which girls are turning their backs on advanced maths. In Victoria, just 5.5 per cent of girls studied specialist maths at VCE in 2014.

So what’s going wrong in Australian and Victorian maths classrooms? Ask around and there are many suggestions for where the problem lies.

Some cite the very high numbers of teachers without a maths background teaching maths in our secondary schools. According to Ms McIntosh, 40 per cent of Australian students from years 7 to 10 are taught maths by a teacher without a maths background.

“Junior secondary is very poorly served,” says Professor Stacey. “You need a thorough understanding of what you are teaching, and pedagogical content knowledge in maths, but a very high number are teaching without having done maths at year 12 themselves.”

But the problem may start earlier still. Many of our primary teachers, who have often struggled with maths themselves at school, have “wobbly foundations” in maths, Ms McIntosh says. “They don’t always have good confidence in their own abilities and that comes across to the children.”

But Peter Sullivan, Professor of Science, Maths and Technology Education at Monash University, doesn’t think teacher qualification is the “critical issue” that some say it is. “If schools were doing collaborative learning and planning, I am not overly concerned by it,” he says. “The real issue is maths is difficult to learn. We have not found the optimal way for students to overcome the hurdles.”

He explains that research shows students like to engage in maths for themselves and be less reliant on the teacher. “Rather than finding the student who needs help and telling them what to do, teachers need to step back and find ways to facilitate conversations between students.” He explains that teachers need professional learning and support to do this.

Others point to the time allocated to maths in a crowded curriculum as an issue, while NAPLAN may also be a problem when it comes to inspiring students in maths. “The test is making it hard for teachers to find time to be engaging,” says Colleen Vale, Associate Professor in Maths Education at Deakin University, who notes that lots of students are getting to junior secondary school, “not fully engaged, challenged or interested in maths”.

She adds that relating maths to the real world and students’ interest takes planning, teamwork and time. “The time to sit down and work it out is probably not sufficient,” she says.

Then there’s the Australian attitude to maths and our low aspirations. “To say you can’t do maths seems acceptable in Australia,” says Ms McIntosh. “You would not say that about reading. The general public’s perception that it’s OK to not like maths is working against us. Countries that are more successful in maths value it more highly as a society. In countries like France and Germany, you would not hear that maths is not a good thing to pursue. People need to change their attitudes to maths.”

So what’s being done?

Maths education is clearly on policymakers’ minds. In October last year, the federal government announced $12 million for initiatives to boost students’ interest and competency in science, technology, engineering and maths (STEM), including funding for computer coding and maths summer schools. It has also announced that new primary teaching graduates will have a subject specialisation which could include maths.

In May, federal, state and territory ministers –聽though rejecting making science or maths compulsory to year 12 – agreed to develop a national STEM school education strategy. Meanwhile, the Chief Scientist recently commissioned research looking at schools across Australia that do significantly well in maths to see what they are doing right. He is also looking overseas at what other countries have been doing.

“We need to pause, reflect, rethink, reposition and introduce programs and processes which will change the culture and get people to understand why maths is important and how it can be interesting,” Professor Chubb says. “It’s not difficult if it’s taught in the right, inspiring way.”

What may seem like an abstract study today may end up being part of the cure for cancer tomorrow or new wi-fi technology in five years.

On 29 June, Senator Bridget McKenzie will open the 11th annual 今日吃瓜 Winter School, at The University of Queensland in Brisbane, reminding us of the importance of theoretical mathematical research — that beautiful mathematics often turns out to be useful mathematics.

Mathematicians in the 1860s were not thinking about computer graphics when studying two-dimensional differential geometry. And in 1822, how could Joseph Fourier have known his research into heat flow would transform the way we process, store and transmit information. This led to a transformation in the way we live as profound as that caused by the Industrial Revolution. It has also resulted in huge advances in medical diagnostic therapies such as MRI and PET.

As in the 1800s, humans today cannot see into the future; we cannot begin to imagine the infinite possibilities discoveries in fundamental mathematics may have in centuries to come.

The famous astronomer and polymath Galileo Galilei said that the book of nature is written in the language of mathematics. So, by developing an understanding of symmetry, structure, geometry and other mathematical constructs we may be able to reveal the patterns of nature.

Einstein鈥檚 1915 theory of general relativity asserted that the presence of mass distorts the geometry of space and time in a way described by the mathematics developed by Bernhard Riemann sixty years earlier. A critical experimental test of this geometrical theory of gravity required the occurrence of a solar eclipse.

While the development of physics and mathematics may proceed along different paths, each fundamental theory in physics has a corresponding specific mathematical structure, for general relativity this is Riemannian geometry and for quantum mechanics it is the Hilbert space.

These descriptions of nature are works of mathematical beauty and affect our everyday lives. We couldn’t decode the human genome, build aeroplanes or have millions of people talking on their phones across the world simultaneously without mathematics.

A Winter School on Algebra, Geometry and Physics to grow tomorrow鈥檚 Einsteins.

The 今日吃瓜 Winter School gives Australian students the chance to expand their skills in the mathematical sciences and build collaborative networks with other students and early career researchers. They will also hear from leading international experts from USA and Canada as well as domestic experts from across the nation.

The school will also present a Women in Maths evening designed to highlight the contribution of women in mathematics and provide a forum for discussion of career paths.

— ends —

The University of Queensland will hold 今日吃瓜’s Winter School from 29 June – 10 July 2015
Further details:
Full speaker list:

WOMEN IN MATHS EVENT:
Thursday 2 July, Science Learning Centre 5-7pm

PUBLIC LECTURE:
The Glass Bead Game
Tuesday 7 July, The Edge, Queensland State Library, 6pm
Professor Arun Ram, University of Melbourne

What does connect-the-dots have to do with watching a Pixar film? More than you might think.

A connect-the-dots page starts with nothing but some labelled points. As each dot is joined to the next, however, a picture emerges. Each step is simple 鈥� just add a line segment between two points 鈥� but the resulting image can be extremely complex.

Toddlers can produce masterpieces this way, but so can computers. When a computer needs to draw a curve, it starts by connecting a sequence of points. Using just a few points might result in a zigzag with sharp corners, but increasing the initial number of points makes the resulting curve look smoother. With enough points and line segments, we can approximate even the most complicated curves.

Dots lines and curves

But maybe you’re more ambitious. Suppose you’re not interested in drawing curves, but rather, in constructing two-dimensional surfaces. Can you use an analogous approach to build a plane or a sphere or something more elaborate?

Triangles 鈥� the simplest two-dimensional objects 鈥� serve as building blocks for more complicated surfaces. Just as we can connect a pair of points with a line segment, we can connect three points via a triangle. And, just as we can make complicated curves by gluing lots of segments at their endpoints, we can make complicated surfaces by gluing lots of triangles along their edges. We can approximate extremely complex surfaces as long as we use enough triangles.

Mathematicians started thinking seriously about constructing surfaces from triangles in the late 19th century, hoping to classify surfaces. In particular, they wanted to understand when two surfaces could be deformed to look the same without cutting or gluing. They developed mathematical tools to study this question, and a century later, it became clear that they’d also laid the groundwork for an important technique in computer graphics.

Imagine trying to model moving cloth, perhaps a flag flapping in the breeze. Since the flag changes shape as it moves through space, this is a much more difficult problem than simulating the movement of a rigid object like a table. If the flag is approximated by triangles, however, modelling it becomes possible because the computer only needs to keep track of sets of three points. When the points move, they carry the triangles with them.

Abstract? Applied? Both!

Mathematicians certainly weren’t thinking about computer graphics in the 1890s. They were studying abstract questions about two-dimensional geometry and developing beautiful mathematics. Nevertheless, the techniques they invented in order to state this question precisely and then answer it have turned out to be extremely useful. In fact, this theme recurs throughout human history: mathematics developed to solve abstract problems turns out to be useful. Maybe not always and definitely not quickly, but it happens over and over again.

The first mathematical objects most people meet are the counting numbers 1, 2, 3鈥� Most counting numbers are formed by multiplying smaller numbers, but not all of them. Some numbers have only 1 and themselves as factors, and these are known as prime numbers. For example, the numbers 2, 3 and 5 are prime, but 4 = 2 x 2 is not.

Prime numbers act as building blocks for the entire number system. Centuries ago, the mathematicians who studied primes didn’t think their efforts would defend a castle or build a better steam engine, but they were intrigued by the search for structure and patterns.

Some of the questions they posed continue to capture the mathematical imagination; today, the most famous unsolved problem in mathematics is the Riemann Hypothesis, which addresses how the primes are distributed among the counting numbers.

Pondering prime numbers might seem like an intellectual game that’s divorced from ‘real world’ concerns. But suppose you ask yourself, ‘Why is it safe to use my credit card to buy something online?’ (Or, maybe better: ‘Is it safe to use my credit card online?’) In fact, the basic techniques for sending data securely over the internet rely on what’s known about factoring a number into primes. Every time you enter your credit card number on a website and hit ‘send’, you have a number theorist to thank.

Likewise, Persian mathematicians began developing the subject we now call algebra in the Middle Ages. This field evolved over centuries, and today, it underpins the algorithms for internet search and Netflix recommendations.

Fourier analysis, which was developed as part of calculus in the late 1700s, provides the basic mathematical tools for signal processing in telecommunications and medical imaging.

Algebraic topology 鈥� a branch of mathematics that wasn’t even created until the 20th century 鈥� is being used in the 21st century to study artificial intelligence and cancer genomics.

The list goes on, but the striking thing in all these examples is that the original researchers couldn’t anticipate which applied questions would require their work. Plenty of important mathematics is specifically developed to solve real-world problems, but curiosity-driven research is as important today as it ever has been.

But back to triangles

New applications for old mathematics are exciting, but progress also comes in the form of new mathematics.

Computer graphics uses flat triangles to approximate smooth surfaces, but if you’re willing to allow triangles to bend a bit, then you can build any surface by gluing enough of them together. This is equivalent to saying that you can cut any surface into curved triangular pieces.

Sphere made of curved triangles

These curved triangles are an important tool for generalising what we know about geometry to higher dimensions – after all, mathematicians, scientists, and engineers don’t care only about surfaces.

Higher-dimensional spaces occur not only in pure mathematics, but also in nature as patterns in large data sets, as relationships between physical quantities, and in descriptions of the universe itself.

Mathematicians develop formal techniques to study them, compensating for the fact that a seven-dimensional space is harder to picture than a flag. Luckily, the idea of gluing triangles generalises to any dimension! In three dimensions, for example, the analogue of a triangle is a tetrahedron, and just as gluing triangles together builds surfaces, gluing tetrahedra together builds new three-dimensional objects. In higher dimensions, the analogue of a triangle is called an n-simplex, and gluing n-simplices together builds n-dimensional objects.

Tetrahedrons make up 3D objects

Since any surface can be cut into curved triangles, and it’s reasonable to ask if the analogous fact holds in higher dimensions: can any n-dimensional space be cut into n-simplices?

Mathematicians initially speculated that the phenomena they’d observed in dimensions one and two would generalise to all dimensions. This belief came to be known as the Triangulation Conjecture.

Decades of research failed to deliver a proof. Then, breakthroughs in the 1980s revealed examples of four-dimensional objects which can’t be cut into 4-simplices. But in 2012, the Triangulation Conjecture was finally proved false for all dimensions greater than four. The mathematical universe has some very strange shapes!

Disproving the Triangulation Conjecture is a triumph of curiosity-driven research, and the truth now sits on humanity’s bookshelf. It may rest there quietly. Then again, maybe some 26-dimensional space with no triangulations will lead to a cure for cancer.

You never know.

Director of the 今日吃瓜 explains the challenges of making maths compulsory for Years 11-12