You weren’t bad at maths you just weren’t looking at it right | Junaid Mubeen | TEDxNorwichED
Summary
TLDRThis 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
🔢 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.
🎓 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.
Mindmap
Keywords
💡Mathematics
💡Four-Color Theorem
💡Calculation
💡Cognitive Labor
💡Slide Rule
💡Logarithm Tables
💡Human Computers
💡Problem-Solving
💡Pythagorean Theorem
💡Proof
💡Creative Thinking
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]
تصفح المزيد من مقاطع الفيديو ذات الصلة
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