I happily provide the following link to Statistics Done Wrong, a free e-book illustrating pitfalls when using statistical inference. From its description:

If you’re a practicing scientist, you probably use statistics to analyze your data. From basic t tests and standard error calculations to Cox proportional hazards models and geospatial kriging systems, we rely on statistics to give answers to scientific problems.

This is unfortunate, because most of us don’t know how to do statistics.

Statistics Done Wrong is a guide to the most popular statistical errors and slip-ups committed by scientists every day, in the lab and in peer-reviewed journals. Many of the errors are prevalent in vast swathes of the published literature, casting doubt on the findings of thousands of papers. Statistics Done Wrong assumes no prior knowledge of statistics, so you can read it before your first statistics course or after thirty years of scientific practice.

http://www.refsmmat.com/statistics/index.html

Statistical errors and their tendency to mislead

As a follow-up to yesterday’s post, here’s a recent article in the scientific journal Nature about the slippery nature of P-values, including a history about how reliance on P-values has evolved in the past 100 years or so: http://www.nature.com/news/scientific-method-statistical-errors-1.14700

While I’m personally familiar with many of the pitfalls mentioned this article, I have to admit that a couple of the issues raised are brand new to me. So I’ll refrain from editorializing until I’ve had some time to reflect more deeply on this article.

Reducing pathogen transmission in a hospital setting. Handshake verses fist bump: a pilot study

A friend of mine shared this article from the professional literature, dryly commenting that he wishes that he had thought of this idea. Medical lingo aside, this is essentially a simple example of a two-sample t-test.

http://www.journalofhospitalinfection.com/article/S0195-6701%2813%2900318-6/fulltext

Binary

Source: http://www.xkcd.com/953/

Fun with Bases

Something I noticed last month:

10 in base ten…

is 101 in base three…

and 1010 in base two.

Notice that the representation changed by just adding an extra digit. Also, this is the maximum number of times that this could happen for 10; base two is the smallest base, while a two-digit representation is the shortest that could have a pattern like this (since 10 is represented as A for any base higher than ten).

Naturally, I’m curious if there’s another number like this. 64 comes close:

10 in base sixty-four…

100 in base eight…

1000 in base four…

1 000 000 in base two.

But the pattern breaks for base three.

Accommodating Blind Students Taking Mathematics

This was another interesting pedagogical article from the February/March 2014 issue of FOCUS, published by the Mathematical Association of America. I had never heard of the Nemeth code, the braille for mathematical notation.

http://digitaleditions.walsworthprintgroup.com/display_article.php?id=1639571

Flipping to Offer Low-Enrollment Courses

I just read a very interesting article about how an instructor at Ohio Dominican University is simultaneously teaching several upper-level mathematics courses with low enrollment by flipping the classroom (using the catchy title One-Room Schoolhouse). Here’s the article: http://digitaleditions.walsworthprintgroup.com/display_article.php?id=1639570

Approximating pi

I was recently interviewed by my city’s local newspaper about $\pi$ Day and the general fascination with memorizing the digits of $\pi$ . I was asked by the reporter if the only constraint in our knowledge of the digits of $\pi$ was the ability of computers to calculate the digits, and I answered in the affirmative.

Here’s the current state-of-the-art for calculating the digits of $\pi$ . Amazingly, this expression was discovered 1995… in other words, very recently.

$\pi = \displaystyle \sum_{n=0}^\infty \frac{1}{16^n} \left( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac{1}{8n+5} - \frac{1}{8n+6} \right)$

Because of the term $16^n$ in the denominator, this infinite series converges very quickly.

Proof: If $k < 8$ , then we calculate the integral $I_k$ , defined below:

$I_k = \displaystyle \int_0^{1/\sqrt{2}} \frac{x^{k-1}}{1-x^8} dx$

$= \displaystyle \int_0^{1/\sqrt{2}} x^{k-1} \sum_{n=0}^\infty x^{8n} dx$

$= \displaystyle \int_0^{1/\sqrt{2}} \sum_{n=0}^\infty x^{8n+k-1} dx$

$= \displaystyle \sum_{n=0}^\infty \int_0^{1/\sqrt{2}} x^{8n+k-1} dx$

$= \displaystyle \sum_{n=0}^\infty \left[ \frac{x^{8n+k}}{8n+k} \right]^{1/\sqrt{2}}_0$

$= \displaystyle \sum_{n=0}^\infty \frac{1}{8n+k} \left[ \left( \frac{1}{\sqrt{2}} \right)^{8n+k} - 0 \right]$

$= \displaystyle \sum_{n=0}^\infty \frac{1}{2^{k/2}} \frac{1}{16^n (8n+k)}$

We now form the linear combination $P = 4\sqrt{2} I_1 - 8 I_4 - 4\sqrt{2} I_5 - 8 I_6$ :

$P = \displaystyle \sum_{n=0}^\infty \left( \frac{4\sqrt{2}}{2^{1/2}} \frac{1}{16^n (8n+1)} - \frac{8}{2^{4/2}} \frac{1}{16^n (8n+4)} - \frac{4\sqrt{2}}{2^{5/2}} \frac{1}{16^n (8n+5)} - \frac{8}{2^{6/2}} \frac{1}{16^n (8n+6)} \right)$

$P = \displaystyle \sum_{n=0}^\infty \frac{1}{16^n} \left( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac{1}{8n+5} - \frac{1}{8n+6} \right)$

Also, from the original definition of the $I_k$ ,

$P = \displaystyle \int_0^{1/\sqrt{2}} \frac{4\sqrt{2} - 8x^3 -4\sqrt{2} x^4 - 8x^5}{1-x^8} dx$ .

Employ the substitution $x = y/\sqrt{2}$ :

$P = \displaystyle \int_ 0^1 \frac{4\sqrt {2} - 2\sqrt {2} y^3 - \sqrt {2} y^4 - \sqrt {2} y^5}{1 - y^8/16}\frac {dy} {\sqrt {2}}$

$P = \displaystyle \int_ 0^1 \frac{16 (4 - 2 y^3 - y^4 - y^5)}{16 - y^8} dy$

$P = \displaystyle \int_0^1 \frac{16(y-1)(y^2+2)(y^2+2y+2)}{(y^2-2)(y^2+2)(y^2+2y+2)(y^2-2y+2)} dy$

$P = \displaystyle \int_0^1 \frac{16y-16}{(y^2-2)(y^2-2y+2)} dy$

Using partial fractions, we find

$P = \displaystyle \int_ 0^1\frac{4 y}{y^2 - 2} dy - \int_ 0^1 \frac{4 y - 8}{y^2 - 2 y + 2} dy$

The expression on the right-hand side can be simplified using standard techniques from Calculus II and is equal to $\pi$ .

So that’s the proof… totally accessible to a student who has mastered concepts in Calculus II. But this begs the question: how in the world did anyone come up with the idea of starting with the integrals $I_k$ to develop an infinite series that leads to $\pi$ ? Let me quote from page 118 of J. Arndt and C. Haenel, $\pi -$ Unleashed (Springer, New York, 2000):

Certainly not by chance, even if luck played some part in the discovery. All three parties [David Bailey, Peter Borwein and Simon Plouffe] are established mathematicians who have been working with the number $\pi$ for a considerable time… Yet the series was not discovered through mathematical deduction or inference. Instead, the researchers used a tool called Computer Algebra System and a particular procedure called the “PSQL algorithm” to generate their series. They themselves write that they found their formula “through a combination of inspired testing and extensive searching.”

The original paper that announced the discovery of this series can be found at http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P123.pdf.

Common Core, subtraction, and the open number line: Part 4

The following picture has been making the rounds lately.

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

The solution of this particular question.
The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

This is Part 4 of this series of posts: the importance of engaging parents and caregivers when unorthodox techniques of mathematics are taught.

I’ve only witnessed the implementation of the Common Core from afar, but there’s absolutely no doubt that the professional development of teachers who have been asked to teach math in a new way has left a lot to be desired. Ditto for explaining these new approaches to parents and caregivers who want to help their children.

True story: I personally did not know about the open number line until the “Meet the Teachers” night that was held for parents near the start of the school year. The teachers explained that they would be doing math a little differently and did a couple of examples using the open number line. I could feel many eyes in the room looking back at me (people know I’m a math professor) with facial expressions saying “Is this stuff really going to work for our kids?”

As this was my first exposure to the open number line, I was skeptical (I would have preferred using base-10 kits) but I held my tongue and listened carefully to the presentation. After the presentation, I was convinced that this was a completely legitimate way of teaching the subject and that the teachers had the requisite depth of understanding to teach arithmetic using this technique. After the presentation, I told anyone who’d listen that this technique was mathematically sound and pedagogically sound, even if it was different than the way that “it’s always been taught.”

Parents generally bought into the technique that evening. I’m not sure that they would have bought into it if its rationale had not been carefully explained to them.

And, as a reminder, Texas is not a Common Core state.

The failure to explain to parents and caregivers unorthodox but correct ways of teaching mathematics has been perhaps the greatest failure of the roll-out of the Common Core. It’s unacceptable that children are crying over their math homework and parents feel powerless to help (a common theme that I’ve heard over and over again from my friends).

Teachers and parents ought to be natural allies in wanting children to have a greater depth of understanding of arithmetic that will prepare them for algebra later. However, because the strategies of teaching the “why”s of mathematics have generally not been carefully explained to parents, they naturally feel somewhat helpless when trying to help their children with their homework.

My own field of research is not mathematics education. So, at professional conferences, I’ve asked friends and colleagues the same question over the years:

Letting children use their own natural curiosity to get at they “why”s of mathematics is good. Letting children use their own natural curiosity and also having the support of parents at home is better. So what research has been done on strategies on successfully engaging parents with how mathematics is currently taught versus how it was taught a generation ago (or, more accurately, what parents remember of their own experiences from elementary school)?

To my surprise, people that I greatly respect did not have an immediate answer to my question. So I’m guessing that while there’s been a lot of research into successful strategies for teaching mathematics in the classroom, there hasn’t been a lot of research into how these strategies can be supported when children are away from the classroom and asking their parents for help on their homework.

I’ll repeat the close of yesterday’s post: I won’t defend the indefensible way that the Common Core has been rolled out. Voters will be more than justified in voting out anyone who supports the Common Core if its implementation isn’t fixed in the very near future.

Month: April 2014

Password strength and combinatorics

Statistics Done Wrong