“Mathematics, rightly viewed, possesses not only truth, but supreme beauty a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.” – Bertrand Russell
“Here’s my simple answer about whether math is real: No. Math is just a way to describe patterns. Patterns are real, but not math. Nonetheless, math is really, really useful stuff!”- Scott Beaver
You can use any or all of the resources linked below with attribution when appropriate
Web Resources on Maths
Handouts and Activities
This is a bit of a longer handout I made from the first chapter of the book Mathematics for the Nonmathematician. It helps lay out some basic concepts of what math is and why we use it but also discusses ideas relating to ways of knowing and the centrality of reason in Mathematics.
Math and Language
Kids have been “doing math” their entire school careers but rarely have they taken the opportunity to examine what math is, what numbers are, and what it means to “do math.” I like to ask kids to start by asking to what extent they think Math can be considered a language. Often, they hear that “Math is the universal language” but what does that mean? After that, I like for them to consider how many arbitrary choices have gone into our system of Math.
“Closer To Truth” Youtube Channel
This is truly an amazing resource. The interviewees discuss questions like “Is Math invented or discovered?” “Why the unreasonable effectiveness of math?” and “How is mathematics truth and beauty?” There are math other topics and areas discussed as well.
The page with the math topics is here:
Here is a link to one of the videos on youtube along with a playlist on the question:
Mayan Maths, Base 10 systems, and Different Ways of Counting
It is strange for kids to consider the fact that there is nothing inevitable about having 10 digits in our counting system and not all people thought of numbers this way and not everything we encounter thinks in these ways. Here is a good article that introduces these ideas.
What would our system be like if we had more or fewer digits? Depending on student interest you could have them count and do math with a base 6 system or even a base 11 by inventing a new digit. Ultimately what comes through is that there is nothing inherently “real” about having 10 digits other than the fact that humans have 10 fingers (digits) and so it feels right to count this way though not every culture did so.
Here is a handout I made to introduce the Mayan math system that allows them to do some simple calculations.
Here is a great interactive feature which talks about the Mayan math system and how it allows for complex math to be done in simple and intuitive ways.
What happens when you can’t count past four?
This is an adaptation of an article about an Amazonian tribe that does not have words for numbers past four. Does this affect the way they perceive quantities? This article goes over well with students and I would recommend reading. An important question to ask is “What are the implications of this article?”
This really depends on your audience but some students are fascinated by the idea of binary number systems and further fascinated by using fingers to count binary numbers. Here is a comic that explains the idea.
Math and Intuition
This is a topic that goes over really well with students. To get them started, I have them answer questions interpreting number lines and their own spatial concepts of math (on the worksheet linked below). This is meant to get them to ask why their intuition about math is “incorrect.” That leads them to a couple of different articles about how our intuition about math and numbers is logarithmic and not linear. A further question to ask is why?
Here is a short article explaining how our intuitive perception of numbers is logarithmic.
Here is a longer version that establishes similar points.
I originally got the idea of intuition and math from a Radiolab Podcast titled, Numbers.
You can download the episode here.
The Monty Hall Problem
The old game show, Let’s Make a Deal, featured a segment in which contestants could choose the prize behind one of three doors. “Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to take the switch?”
Here is an article explaining the concept though it may be unnecessary for an extended reading. What may be more effective is just to explain the scenario and let students answer. Here is a link to a simulator that lets students test their guesses and prove statistically whether it is better to switch doors than not to.
This site spells out the dilemma in further detail: https://www.businessinsider.com/the-most-controversial-math-problems-2013-3#-9
Math and Sports
One modern day controversy in modern sports has been between “old school” scouts who have watched and analyzed the game for decades and those who use statistics and advanced analytics to understand games. Moneyball by Michael Lewis popularized the concept in baseball to a wide audience. This analytical approach has taken over American football and basketball as well. I’ve done a few activities and readings highlighting this conflict.
“Hot Hands” Fallacy
To introduce the topic in the first place we look at the so called “hot hands” fallacy. I ask them:
“In basketball, there is a player who has a long track record, who makes 25% of his shots. He has made four consecutive shots in a game and his teammates want to make sure he takes the next shot since he is “hot.” Another player on his team, a career 50% shooter has missed his last four shots and his teammates avoid passing him the ball because he is “cold.” Is the decision to pass the ball to the “hot” shooter a sound decision? Why or why not?
Either you can explain that this is just an illusion or have them go on and read an article about it. The first article linked below explains the research that went into disproving it. The second article talks about some more recent research that may prove that “hot hands” actually exist.
Sports Analytics: Ruining the game?
Here are a few articles to use that explain and engage in the debate
For people really interested in the topic, this article breaks down the fact that “despite having more advanced data than ever, it sometimes feels as if we know less for certain than ever.”
Probabilistic Thinking and Breast Cancer
There is a famous case of doctors interpreting mammography data. The question, given proper probability data, asked of doctors was what are the odds that a woman who has a positive result in a mammogram actually has breast cancer? The actual answer, given the data, was 10% but only 21% of doctors accurately assessed this question. You can give the question to your student here: Download Assessing Medical Data Probabilistically
Here is an article that tells the story in greater detail.
Math and Assessing Risk
One interesting activity is to have students think about assessing the relative risks of different behaviors or factors and then reflect on the extent to which their assessments match reality. Further discussion is how we use math to understand reality. Here are some questions and activities to get them started thinking about these topics and an article that explains our disparities. The explanations for people’s misunderstanding also ties together well with cognitive biases.
Here is the article that accompanies the handout above.
Some different questions that get at the same point.
Here is yet another interesting example from Swedish physician Hans Rosling. He came up with a simple quiz assessing people’s knowledge of human development statistics. Even development experts were woefully unaware. You can take the quiz here:
Does the news reflect what we die from?
The short history of global living conditions and why it matters that we know it
From the same site an exploration of macro world trends regarding global living conditions and our misperceptions of them.
How to Lie With Statistics
Statistics and numbers come with an air of authority and certainty and passive recipients of that information can often be misled if they are not critical about the assumptions and language of those statistics, charts, or graphs.
Factfulness by Hans Rosling , in one chapter, shows a series of charts and graphs about Math SAT scores and gender. Each one gives a very different impression about the nature of the gap in scores. This works well if you display the graphs and ask students to evaluate the information being shown and to assess the performance gap. Each successive visualization gives a very different impression and is a good conversation starter on the issue of assumptions in graphs and data. All of this goes well with the book, How to Lie with Statistics
Illinois 2011 Tax Increase
This is an interesting case of the same event being reported using different statistical measures. I made a simple handout that I think demonstrates the profound impact of language choices in Math have.
A humorous exaggerated take on similar concepts
Math and Ethics
Now that we’ve established that humans have a poor intuitive sense of risk, probability, and numbers in general, we can ask the question whether this knowledge comes with ethical responsibility.
A great topic to help explore this is with the lottery. Should governments make money off of people’s poor understanding of math? Attached below is a worksheet to get the kids started thinking about the relevant ideas and then a reading that explores the questions.
Here are some more articles on the topic.
Math and Art
I don’t do much on this topic but here are some web resources I tagged with Math and Art.
Fun Math website
How to Life With Statistics by Darrell Huff
This book is a classic and should be required reading for all students.
“Darrell Huff runs the gamut of every popularly used type of statistic, probes such things as the sample study, the tabulation method, the interview technique, or the way the results are derived from the figures, and points up the countless number of dodges which are used to full rather than to inform.”
The Golden Ratio by Mario Livio
“Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887…This curious mathematical relationship, widely known as “The Golden Ratio,” was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed.The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God;”
Many of the following books get into the challenges and limitations of knowledge, particularly, mathematical knowledge. I think each of these books cover similar ground but they are all excellent.
The Drunkard’s Walk by Leonard Mlodinow
By showing us the true nature of chance and revealing the psychological illusions that cause us to misjudge the world around us, Mlodinow gives us the tools we need to make more informed decisions. From the classroom to the courtroom and from financial markets to supermarkets, Mlodinow’s intriguing and illuminating look at how randomness, chance, and probability affect our daily lives will intrigue, awe, and inspire.
The Signal and the Noise by Nate Silver
Drawing on his own groundbreaking work, Silver examines the world of prediction, investigating how we can distinguish a true signal from a universe of noisy data. Most predictions fail, often at great cost to society, because most of us have a poor understanding of probability and uncertainty. Both experts and laypeople mistake more confident predictions for more accurate ones. But overconfidence is often the reason for failure. If our appreciation of uncertainty improves, our predictions can get better too. This is the “prediction paradox”: The more humility we have about our ability to make predictions, the more successful we can be in planning for the future.
Fooled by Randomness by Nassim Taleb
This book is about luck–or more precisely, about how we perceive and deal with luck in life and business. Set against the backdrop of the most conspicuous forum in which luck is mistaken for skill–the world of trading–Fooled by Randomness provides captivating insight into one of the least understood factors in all our lives. Writing in an entertaining narrative style, the author tackles major intellectual issues related to the underestimation of the influence of happenstance on our lives.
Naked Statistics by Charles Wheelan
From batting averages and political polls to game shows and medical research, the real-world application of statistics continues to grow by leaps and bounds. How can we catch schools that cheat on standardized tests? How does Netflix know which movies you’ll like? What is causing the rising incidence of autism? As best-selling author Charles Wheelan shows us in Naked Statistics, the right data and a few well-chosen statistical tools can help us answer these questions and more.
Proofiness by Charles Seife
In Zero, Charles Seife presented readers with a thrilling account of the strangest number known to humankind. Now he shows readers how the power of skewed metrics-or “proofiness”- is being used to alter perception in both amusing and dangerous ways. Proofiness is behind such bizarre stories as a mathematical formula for the perfect butt and sprinters who can run faster than the speed of sound. But proofiness also has a dark side: bogus mathematical formulas used to undermine our democracy-subverting our justice system, fixing elections, and swaying public opinion with lies. By doing the real math, Seife elegantly and good-humoredly scrutinizes our growing obsession with metrics while exposing those who misuse them.