You weren’t bad at maths you just weren’t looking at it right | Junaid Mubeen | TEDxNorwichED

Summary

This script explores the diverse reactions to mathematics, from fear to fascination, and argues that traditional teaching methods obscure the true essence of the subject. It highlights the historical shift from human computation to technology, emphasizing the importance of creative problem-solving over rote calculation. The script celebrates the beauty of mathematical proofs and the joy of discovery, urging us to embrace the creative potential within us all and to see math as a powerful tool for thinking and exploration.

Takeaways

• 🧩 Mathematics elicits varied responses, from fear to pleasure, suggesting the existence of different types of math rather than a division of 'math people' and 'non-math people'.
• 🌟 The four-color theorem, proven in 1976, highlights the use of computers in mathematical proofs, showing a shift from manual to computational verification.
• 🤖 The historical role of 'human computers' has shaped the perception of math as calculation-heavy, overshadowing its creative aspects.
• 📚 The public image of math is often a complex array of symbols, which can intimidate those who struggle with symbolic manipulation.
• 🧠 The human brain is not optimized for speed or precision in calculation, which is why tools and technologies have been developed to assist with these tasks.
• 📉 The slide rule and logarithm tables are historical examples of efforts to reduce the burden of calculation, emphasizing the importance of creative problem-solving over routine computation.
• 🎓 Gauss's method of summing numbers illustrates the elegance and creativity inherent in mathematical thinking, contrasting with the rote execution of formulas taught in schools.
• 📐 The Pythagorean theorem is a classic example of a mathematical curiosity that has been explored and proven in over 350 ways, showcasing the depth and variety of mathematical thought.
• 🌐 The exploration of mathematical ideas extends beyond traditional contexts, with questions about raising exponents or considering triangles on a sphere, indicating the expansive nature of mathematical inquiry.
• 🌟 The four-color theorem is an example of a problem that is puzzling for its own sake, unrelated to practical applications, and a testament to the value of pure mathematical curiosity.
• 🌈 The speaker advocates for embracing recreational mathematics as a way to foster creativity and problem-solving skills, emphasizing that math is about ideas and arguments, not just symbols and formulas.

Q & A

What is the four-color theorem mentioned in the script?

The four-color theorem is a mathematical statement that asserts that any map in a plane can be colored using four colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. The problem dates back to the 1850s and was famously proven in 1976 by Kenneth Appel and Wolfgang Haken.

How did Appel and Haken's proof of the four-color theorem rely on computers?

Appel and Haken's proof was the first to depend on a computer. They reduced the problem from infinitely many maps to around 2,000 specific configurations, which were then checked by a computer to ensure no two neighboring regions shared the same color. This computational task would have taken a human years to complete with no guarantee of accuracy.

What is the common misconception about mathematics that the script addresses?

The script addresses the misconception that mathematics is solely about calculation and manipulation of symbols. It argues that mathematics is actually about ideas, arguments, and creative problem-solving, which is often obscured by the focus on routine calculation in traditional education.

Why do some people struggle with mathematics according to the script?

Some people struggle with mathematics because they associate it with calculation, which is not a core strength for humans. The script suggests that the problem is not with the individuals but with the way mathematics is often taught, focusing on rote memorization and procedure execution rather than on understanding and creativity.

What historical tool did the script mention to illustrate the effort to ease calculational burden?

The script mentioned the slide rule as a historical tool that was used to ease the burden of calculation. It allowed users to perform multiplication and other complex operations by simple alignment and reading, reducing the need for manual computation.

Who introduced logarithm tables, and what was their purpose?

John Napier introduced logarithm tables around 400 years ago. The purpose of these tables was to simplify complex mathematical operations like multiplication to simpler ones like addition, thus reducing the difficulty and effort involved in mathematical calculations.

What is the significance of the story of Gauss and his teacher in the script?

The story of Gauss and his teacher is used to illustrate the power of creative thinking in mathematics. Instead of performing the tedious computation his teacher expected, young Gauss rearranged the numbers to simplify the problem, demonstrating that there is often more than one way to approach a mathematical problem.

What is the script's stance on the use of formulas in mathematics education?

The script criticizes the use of formulas in mathematics education when they are taught as rigid procedures without understanding. It suggests that this approach can obscure the elegance and underlying mechanisms of the mathematical concepts, stifling creativity and understanding.

What does the script suggest about the nature of mathematical proofs?

The script suggests that mathematical proofs are not just about confirming observations but about elevating them to irrefutable truths. It uses the example of Pythagoras' theorem and its numerous proofs to illustrate how different arguments can illuminate the same truth.

Why does the script argue that the world needs more creative thinkers and problem solvers?

The script argues that the world needs more creative thinkers and problem solvers because they can approach problems from new angles, explore beyond the known, and innovate. It suggests that embracing the creative aspects of mathematics can help cultivate these skills.

What is the script's final message about the potential of humans in mathematics?

The final message of the script is that humans have a vast creative potential in mathematics that is often untapped due to a focus on calculation. It encourages individuals to cast aside the misconceptions about mathematics and awaken the mathematician within, emphasizing that everyone is more than just a calculator.

Outlines

00:00

🔢 The Dichotomy of Mathematics Perception

The first paragraph discusses the contrasting emotions people have towards mathematics, ranging from fear to pleasure, and argues that this is not due to a division between 'math people' and others, but rather the type of mathematics encountered, particularly the kind taught in schools. The four-color theorem is introduced as an example of a complex problem that was first proven with the assistance of a computer, highlighting a shift in the role of humans in mathematics from calculation to more creative problem-solving. The paragraph emphasizes the historical efforts to reduce the burden of calculation, from the use of slide rules to modern computational tools, and encourages a shift in focus from routine calculation to creative thinking.

05:00

🎓 The Creative Side of Mathematics

The second paragraph delves into the creative aspects of mathematics, using Carl Friedrich Gauss's method of quickly summing numbers as an example of how creativity can simplify complex problems. It criticizes the teaching methods that focus on formulas and procedures rather than understanding and creativity. The paragraph then discusses the Pythagorean theorem, questioning why it holds true and encouraging exploration beyond verification. It presents a proof of the theorem that uses geometric rearrangement to demonstrate its truth universally. The speaker calls for an embrace of 'recreational mathematics' as a way to foster creative thinking and problem-solving, emphasizing that mathematics is about ideas and exploration, not just symbols and formulas.

Keywords

💡Mathematics

Mathematics is a field of study that deals with the properties and relationships of numbers, quantities, shapes, and patterns. In the video, it is portrayed as both a source of fear and pleasure for different individuals, highlighting the varying perceptions of the subject. The video argues that the essence of mathematics is often obscured by its association with calculation, which is not the core strength of the discipline.

💡Four-Color Theorem

The Four-Color Theorem is a mathematical theorem stating that any map can be colored using four colors so that no two adjacent regions share the same color. The video discusses this theorem as an example of a problem that was famously proven in 1976 using computer assistance, illustrating the intersection of human ingenuity and computational power in advancing mathematical knowledge.

💡Calculation

Calculation refers to the process of performing mathematical operations to arrive at a result. The video script emphasizes that while calculation is a part of mathematics, it has overshadowed the more creative aspects of the discipline. It suggests that the public often equates mathematics with calculation, leading to a misconception of what mathematics truly is.

💡Cognitive Labor

Cognitive labor in the context of the video refers to the intellectual effort involved in problem-solving and creative thinking. The video contrasts cognitive labor with the more mechanical task of calculation, highlighting the importance of the former in the advancement of mathematical knowledge, as seen in the proof of the Four-Color Theorem.

💡Slide Rule

The slide rule is an historical calculating tool that was used to perform multiplication and division by aligning scales. The video mentions the slide rule as an example of how mathematicians have historically sought to reduce the burden of calculation, allowing them to focus more on creative problem-solving.

💡Logarithm Tables

Logarithm tables, introduced by John Napier, were used to simplify complex mathematical operations into simpler ones, like addition. The video uses logarithm tables as an example of an early effort to ease the computational burden on the human mind, paving the way for modern calculating tools.

💡Human Computers

Human computers were individuals who performed routine mathematical calculations by hand. The video references the historical role of human computers in tasks such as creating accounting ledgers and navigational tables, emphasizing the evolution from human to machine computation.

💡Problem-Solving

Problem-solving is the process of finding solutions to complex issues or questions. The video emphasizes the importance of problem-solving in mathematics, suggesting that it is a more accurate representation of the discipline than mere calculation. It encourages viewers to embrace the creative aspects of problem-solving in mathematics.

💡Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that states the relationship between the sides of a right-angled triangle. The video discusses the theorem as an example of a mathematical curiosity that has been explored and proven in various ways, illustrating the depth and creativity involved in mathematical exploration.

💡Proof

A proof in mathematics is a logical argument that demonstrates the truth of a statement. The video script uses the Pythagorean Theorem as an example to explain how a proof can elevate a curious observation to an irrefutable truth, showcasing the power of mathematical reasoning.

💡Creative Thinking

Creative thinking refers to the ability to generate new ideas, concepts, or solutions. The video argues that mathematics is fundamentally about creative thinking and exploration, rather than just calculation. It encourages viewers to embrace the creative potential within themselves and to see mathematics as a field for imaginative problem-solving.

Highlights

Mathematics evokes contrasting emotions such as fear and pleasure among different individuals.

The division in people's perception of math is not due to the existence of 'math people' and 'non-math people', but rather the types of mathematics they encounter.

The four-color theorem, proposed in 1850s, was proven in 1976 using a computer, marking the first mathematical proof dependent on computational assistance.

The proof of the four-color theorem involved reducing an infinite problem to a finite set of configurations checked by a computer.

The collaboration between human mathematicians and computers in the four-color theorem demonstrates the division of cognitive labor in problem-solving.

Public perception of mathematics is often as a complex system of symbols, leading to the misconception that one must be adept at symbol manipulation to be good at math.

The historical role of 'human computers' has contributed to the association of mathematics with calculation, which is not its core strength.

The slide rule and logarithm tables are examples of tools developed to ease the burden of calculation for mathematicians.

Cognitive science confirms that the human brain is not optimized for speed or precision in calculation, suggesting a natural limitation in this area.

The development of modern calculating tools has shifted the focus from calculation to creative and non-routine thinking in mathematics.

Gauss's method of summing numbers demonstrates creativity over computation, showing a different approach to problem-solving.

The teaching of mathematics often focuses on formulas and procedures, obscuring the elegance and underlying mechanisms of mathematical concepts.

The Pythagorean theorem is an example of a simple formula with profound implications, which has been proven in over 350 different ways.

The essence of mathematics lies in ideas and arguments, not just symbols and formulas.

Mathematics is about exploring and reasoning, with computers playing a supportive role in problem-solving.

The four-color theorem is an example of a mathematical problem that is interesting for its own sake, unrelated to practical applications.

Recreational mathematics encourages creative thinking and problem-solving, which are essential skills in various fields.

The speaker advocates for embracing the creative potential in mathematics rather than focusing solely on calculation.

The speech concludes with an encouragement to see beyond the traditional view of mathematics and to awaken the mathematician within.

Transcripts

mathematics does the strangest things to

us for some people it's induces fear and

anxiety for others unadulterated

pleasure how there's one subject lead

different people to such wildly

different extremes I don't think it's

because the world is divided into maths

people and nonmetal I believe it's

because there are different types of

mathematics and the particular brand of

maths that we experience at school that

put so many people off conceals the

essence of the subject it's time to lift

the lid we start in 1976 when two

mathematicians Kenneth Oppel and

Wolfgang Haken

and nouns that they cracked the

four-color theorem this problem goes

back to the 1850s it says you can color

any map with four colors in such a way

that no two neighbors share the same

color that's quite a claim because there

are infinitely many maps in conceivably

many and mathematicians labored for well

over a century to show that four killers

always suffice regardless of the map and

when the proof finally arrived there was

something rather unusual about it it was

the first mathematical proof to depend

on a computer Appel and Haken had

reduced the problem from infinitely many

maps down to around 2,000 really messy

configurations that would take a human

years to check through with no guarantee

of accuracy but calculating machines

don't tire like humans do and even in

the 70s they could plow through those

remaining configurations in a matter of

days without error now think about the

division of cognitive labor that went on

here

the human mathematicians crafted an

ingenious argument to reduce the problem

down to finitely many maps and they

outsource the most menial tasks of

checking through those remaining cases

calculation was the understudy to the

more creative act of problem solving and

that's all it ever should be

mathematics has been tainted by its

association with calculation this is the

public image of mathematics a sprawling

mess of symbols and when people declare

as they feel often do that they're no

good at maths what they really mean is

they're not able to shunt around symbols

or perform mental arithmetic with speed

and precision and the archetypal maths

genius is the number wizard the human

computer and for a time computers were

human in the 17th century it was human

computers who created accounting

Ledger's and navigational tables by

carrying out routine procedures by hand

in the 1960s the flesh-and-blood

calculations of NASA propelled us to the

moon so in its time the work of human

computers was profitable and even noble

but calculation has never been a core

strength for humans and when we look

through the history of mathematics we

find a deliberate effort among

mathematicians to ease their

calculational burden now I'm slightly

too young to have used the slide rule

but it really was the ultimate in

calculational hacks because instead of

multiplying two numbers you would simply

align your two rulers and read off the

values and the slide rule has its

origins in the logarithm tables that

john napier introduced around 400 years

ago and during his travels napier had

observed how merchants toiled with their

everyday arithmetic and the idea behind

his log tables was to reduce

multiplication and other more complex

operations to much simpler ones like

addition an AP has been over 20 years

complaining these tables and when he

dedicated his Canon to Charles the first

he wrote of how the log tables take away

the difficulty that heretofore have been

in mathematical calculations and is so

fitted to help the weakness of memory

while Napier's ahead of its time because

today cognitive science confirms that

the human brain isn't optimized for

speed or precision and that's perfectly

fine

because over the past 50 years that

slide rule has given way to all manner

of calculating tools those technologies

that shutters to the moon and back are

now available at our fingertips and

that's our invitation to stop fixating

on routine acts of calculation and

instead focus on our more creative non

routine thinking capacities so take a

look at this problem now it may not

strike you as creative after all what

does one do but some of those terms in

order well number crunching isn't the

only way to go

humans are drawn to patterns and

structure and the most natural thing we

can do is play with them as and unravel

their connections and that's exactly

what confident that I doused did when he

famously solved this problem as an

eight-year-old Gauss's teacher had

closed the problem expecting his young

student to work for each of those steps

but Gauss rejected the tedium of

computation and what he did instead was

creativity personified he folded over

the Sun so that the hundredth pairs with

the one the ninety-nine pairs with the

two the 98 with the three and so on and

he noticed that this rearrangement left

him with 50 copies of a hundred and one

so much easier to compute so much more

elegant and now consider how I taught to

solve those problems at school with

prescribed formulas just like this which

is actually Gauss's method in disguise

except now that elegance has been lost

and there's something rather unedifying

about executing procedures that we

scarcely grasp it's only when we

exercise the freedom to play as Gauss

did that we illuminate the mechanisms

that underpin those rules well this may

be the most famous or infamous rule of

them all it may take your mind back to a

certain bearded Greek mathematician

named Pythagoras although civilizations

as far back as the Babylonians have

puzzled over this curiosity and it is a

curiosity that such a simple formula

should hold for every right angle

Crangle small and large and at school we

were instructed to verify hundreds of

instances of this formula a most bizarre

ask of humans we should have been guided

to examine the bigger questions why does

this formula hold how can we be sure

that somewhere out there in the

mathematical universe there isn't a

right-angled triangle where it breaks

apart so we see an argument a proof that

elevates our curious observations to the

status of irrefutable truth and here's

one of those proofs that takes four

copies of our triangle arrange them in

two different ways now let this image

speak to you let it can meet you in the

most unequivocal terms that the sides of

a right-angled triangle are bound to one

another in this most permanent way one

image that carries an eternal truth for

every right-angled triangle and if

you're not yet persuaded by this

argument then rest assured there are

over 350 known proofs of Pythagoras out

there awaiting your examination that's

300 350 different ways of illuminating

the same truth this is the essence of

mathematics it was never about symbols

and formulas but ideas and arguments

that condition the mind to think and to

reason and to explore because we don't

have to stop there we can ask just

because we can what happens when we

raise the exponent or when we consider

triangles on a sphere rather than in a

flat plane and these are the questions

that eat away at mathematicians that

arouse our liveliest imaginations and

that fuel whole new areas of inquiry and

computers may play their paths as we put

them to task but it's we humans with

dream up problems and chart our while

there

explorations and those problems by the

way may be rooted in the rural world but

they don't have to be no cartographer

ever dwelled on the four-color theorem

there's no reference to it in any

historical atlas it's puzzling for its

own sake and if you believe as I do that

the world desperately needs more

creative thinkers and problem solvers

then this recreational brand of

mathematics is one we must all embrace

it's the most powerful thinking system

we've ever remembered it's also the most

inclusive because it plays to our core

human strengths and when we reduced it

to something as blunt as calculation we

squander our creative potential you are

more than a calculator so cast aside the

drudgery that you thought was

mathematics and awaken the mathematician

that lurks within all of us thank you

[Applause]