Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now
81

I've been using "pets" and "owners" (as in: possible pet-shelter adoptees) in recent years.


74

In the stable marriage problem, you can introduce the problem as it is. But then you ask your students how things change if you assume there are not only heterosexual but also gay and lesbian people (assuming that a heterosexual person will never marry a person of the same gender, and a gay or lesbian person will never marry a person of the opposite gender). ...


62

A few possibilities off the top of my head: Students and chairs. How many ways are there for $n$ students to sit in $k$ chairs. The game of musical chairs might be fun to play around with. One can also consider natural restrictions, such as myopic students who need to sit near the front. Replace "men" and "women" with faculty from different departments. ...


54

Anscombe's quartet is pretty good: All four of these sets have almost identical mean and variance for both x and y coordinates, correlation, and best-fit linear regression. But they're obviously very different!


52

Here are two well known examples: If someone tests positive for a rare disease (say its prevalence is 1 out of 100,000) with a test that has a 1% false positive rate, it is tempting to say that we are 99% sure they have that disease. This isn't true if you go through the numbers; they probably don't have that disease and are a false positive. (Bayes) If you ...


40

A book I remember has the title "the egg-laying dog". The titular dog enters a room where we placed 10 sausages and 10 eggs. After a while the dog leaves the room, and we observe, that the percentage of eggs relative to the sausages increased, so we conclude that the dog must have produced eggs. It's easy to spot the mistake in the above example, because ...


38

Sally Clark (http://en.wikipedia.org/wiki/Sally_Clark) was convicted in the UK of murdering both her infant sons, when in fact it is much more likely that they died of natural causes. The case against her was largely based on invalid statistical reasoning. The Royal Statistical Society made a statement about at at the time, which begins as follows: In ...


34

The issue is not making problems about heterosexual married couples. The issues are: Implicitly making the assumption that all married couples are heterosexual. Making problems about heterosexual marriages but not about other kinds of couples. Both points are unengaging for people following other types of marriage, but they can easily be solved while ...


33

Simpson's paradox: see http://en.wikipedia.org/wiki/Simpson%27s_paradox. To summarize the Berkeley Admissions example: in 1973, 43% of men applying to graduate school at Berkeley were admitted, but only 35% of women. But, broken down across the six departments, women either did better than men, or the difference was not significant. The paradoxical result ...


32

Try objects that often occur in pairs but are distinct from each other: forks and spoons (or forks and knives), left and right shoes, salt and pepper shakers, and so on (where each fork has an obvious partner spoon, perhaps sharing the same color or design, and so on).


22

Percentages are a source of many, many, many common mistakes. One that is very common is believing that percentages can be added. An example: one of our presidents increased its salary by 172%; the next president decreased the presidential salary by 30%. It was commented that compared to the salary before the raise, it was still a 142% increase. Another ...


21

Multiple hypothesis testing is a common one. Let's say you run a study where you try to link some genetic marker to cancer rates. You look at perhaps 80 different genes and see if any of them have a correlation with occurrence of cancer. Lo and behold, one does! With p-value = 0.03! You conclude that there is a strong correlation (and seek to prove ...


20

Just two (now three, see below), to whet the appetite. Stating the mistakes: "Correlation implies Causation": it doesn't. The finding of statistical correlation between two variables may strengthen a pre-existing theoretical/logical argument of existing causal links. But it may also reflect the existence of an underlying third variable that affects both ...


20

When I taught a class about the stable marriage problem last week, I replaced "men" and "women" with "medical students" and "hospitals": the classical instance in which the Gale-Shapley algorithm is used in real life. In addition to the gender issues already mentioned, this has the benefits that: We avoid envisioning a dystopian future where everyone's ...


17

This happened to me two days ago! The "What is it that you teach then?" conversation. I first said everyone's different and has different brains, and that a bunch of us who did PhDs together decided being a PhD mathematician was a mild form of mental disease, with a lot of common symptoms (that I won't need to list to anyone who's spent a lot of time ...


15

Sometimes extreme sample bias. Here is an example (numbers made up, but realistic): In some country with a population of 100 million people, every year 100 people are bitten by poisonous snakes and 50 of these die. Every year 50 people are given treatment against snake bites, and 10 of these die (40 die without getting treatment). Your chances of dying ...


15

I tutored a student who came with a kind of problem I had never seen before and found quite refreshing. It was something like: A child is being pushed on a swing by their father, reaching a maximum height of 4 feet. The father stops pushing, and the maximum height of the swing decreases by 15% on each successive swing. I don't remember the question ...


12

Protons and electrons (form hydrogen atoms) Or cations and anions (form salts), e.g. Na+ and Cl- Pens and pen-caps Bottles and bottle caps, etc. Textbooks (for the course being taught) and students Light bulbs and light sockets Power cords and electrical outlet sockets Cars and parking spaces Seats and attendees, e.g. students in the classroom and ...


10

If you looked at those other topics, you saw the links to the books page at my blog, Math Mama Writes. There are a number of books there that would work for a teen. One of my favorites is Carry On, Mr. Bowditch, by Jean Lee Latham, the (slightly fictionalized) biography of Nathaniel Bowditch, who modernized navigation in the 1700s. But maybe that has too ...


9

If you succumb to the temptation of ejecting, say, a 5-sigma outlier from a n=10 sample taken from what you believe to be a normally distributed source, then you are discarding 50% of the sample's information content. Not so harmless. EDIT: I'll give it a go: Low-probability events carry more information (a.k.a. surprisal) than high-probability events. E.g....


9

If a coin is biased to land heads with probability $p$ and $(a,b)$ is a $95\%$ confidence interval for $p$ then $p$ is in $(a,b)$ with probability $95\%$. Added in edit - While it is often argued that the difference is only philosophical, this distinction is of huge practical importance because the latter phrase is usually interpreted as if $p$ were the ...


9

Unfortunately, it is in German, but the book Angewandte Statistik: Eine Einführung für Wirtschaftswissenschaftler und Informatiker by Kröpfl, Peschek and Schneider contains many typical mistakes that you can make. My favorite example is that you can show a strong geographic correlation in Germany between the number of stork nests and the number of newborn ...


9

You've touched on why it's problematic for educators to only talk about proving in the context of formal proofs. Students need to be accustomed to mathematical reasoning and justification well before they ever see a formal proof. This is the reason that Common Core State Standards Mathematical Practice 3 exists: CCSS.MATH.PRACTICE.MP3 Construct viable ...


9

My go-to response is: I used to be "not a math person" as well, but I stuck with it, and eventually I learned what math was all about and I fell in love. My response may make me seem like a bit creepy -- after all who "falls in love" with math but creepy people? -- but it has the benefit of inviting follow-up questions that are positive in nature. "Oh ...


9

I might say: A lot of kids learn math from teachers who don't make it interesting. Here in the US, kids usually get their first instruction in math (aside from learning to count) in public schools, from teachers with multiple-subject credentials. Some of these folks are wonderful at teaching math, but many of them limped through their own math coursework ...


9

Taking a break from a convoluted computation... To me the key limiting factor is space. I can spread out several sheets over the table and have various bits and pieces directly visible. I cannot do this to the same extent on my tablet or my computer. Therefore I also use loose sheets not something that is bound together. One might argue I could set up a ...


9

An idea I find interesting is to use abstract objects: "given X squares and Y circles, in how many ways it's possible to pair one circle with one square?" If the students are at kindergarten or primary school level, maybe you can give the children squares and circles made with paper, plastic or wood, each with a different color, so they can try these ...


9

I would recommend activities that can be done at least once a day (instead of once a month). When setting the dinner table (especially when guests are present), request the child to help get the dishes and cutlery, asking "How many dishes will we need? How many forks will we need?" If each guest is to be given, say, three pieces of dessert, then how many ...


8

Problems with interpreting $P$-values are discussed by Regina Nuzzo, "Statistical Errors," Nature 504 (13 February 2014), 150-152. [link] In a nutshell: "Most scientists" [sic] interpret a $P$-value as the probability of being wrong in concluding the null hypothesis is false. The $P$-value is often valued more highly than the size of the effect. "$P$-...


8

The following two things do not match exactly the request but are perhaps close enough: Writing an $n$ as a difference of two square is useful as it yields a typically nontrivial factor of $n$, as $n = x^2 - y^2 = (x+y)(x-y)$. This is called Fermat's factorization method and is still relevant as algorithm, in fact SQUFOF which is a development of this ...


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