# Wason Selection Task: Part 3

I recently read about a simple but clever logic puzzle, known as the “Wason selection task,” which is often claimed to be “the single most investigated experimental paradigm in the psychology of reasoning.” More than 90% of Wason’s subjects got the answer wrong when Wason first studied this problem back in the 1960s, and this result has been repeated time over time by psychologists ever since.

Here’s the puzzle: You are shown four different cards, showing a 5, an 8, a blue card, and a green card. You are told that each card has a number on one side and a color on the other side. You are asked to test the truth of the following statement:

If a card has an even number on one side, then its opposite side is blue.

Question: Which card (or cards) must you turn over to test the truth of this statement?

Interestingly, in the 1980s, a pair of psychologists slightly reworded the Wason selection puzzle in a form that’s logically equivalent, but this rewording caused a much higher rate of correct responses. Here was the rewording:

On this task imagine you are a police officer on duty. It is your job to make sure that people conform to certain rules. The cards in front of you have information about four people sitting at a table. On one side of the card is a person’s age and on the other side of the card is what the person is drinking. Here is a rule: “If a person is drinking beer, then the person must be over 19 years of age.” Select the card or cards that you definitely must turn over to determine whether or not the people are violating the rule.

Four cards are presented:

• Drinking a beer
• Drinking a Coke
• 16 years of age
• 22 years of age

In this experiment, 29 out of 40 respondents answered correctly. However, when presented with the same task using more abstract language, none of the 40 respondents answered correctly… even though the two puzzles are logically equivalent. Quoting from the above article:

Seventy-five percent of subjects nailed the puzzle when it was presented in this way—revealing what researchers now call a “content effect.” How you dress up the task, in other words, determines its difficulty, despite the fact that it involves the same basic challenge: to see if a rule—if P then Q—has been violated. But why should words matter when it’s the same logical structure that’s always underlying them?

This little study has harrowing implications for those of us that teach mathematical proofs and propositional logic. It’s very easy for people to get some logic questions correct but other logic questions incorrect, even if the puzzles look identical to the mathematician/logician who is posing the questions. Pedagogically, this means that it’s a good idea to use familiar contexts (like rules for underage drinking) to introduce propositional logic. But this comes with a warning, since students who answer questions arising from a familiar context correctly may not really understand propositional logic at all when the question is posed more abstract (like in a mathematical proof).

# Wason Selection Task: Part 2

I recently read about a simple but clever logic puzzle, known as the “Wason selection task,” which is often claimed to be “the single most investigated experimental paradigm in the psychology of reasoning.”

Here’s the puzzle: You are shown four different cards, showing a 5, an 8, a blue card, and a green card. You are told that each card has a number on one side and a color on the other side. You are asked to test the truth of the following statement:

If a card has an even number on one side, then its opposite side is blue.

Question: Which card (or cards) must you turn over to test the truth of this statement?

The answer is: You must turn over the 8 card and the green card. The following video explains why:

Briefly:

1. Clearly, you must turn over the 8 card. If the opposite side is not blue, then the proposition is false.
2. Clearly, the 5 card is not helpful. The statement only tells us something if the card shows an even number.
3. More subtly, the blue card is not helpful either. The statement claim is “If even, then blue,” not “If blue, then even.” This is the converse of the statement, and converses are not necessarily equivalent to the original statement.
4. Finally, the contrapositive of “If even, then blue” is “If not blue, then not even.” Therefore, any card that is not blue (like the green one) should be turned over.

If you got this wrong, you’re in good company. More than 90% of Wason’s subjects got the answer wrong when Wason first studied this problem back in the 1960s, and this result has been repeated time over time by psychologists ever since.

Speaking for myself, I must admit that I blew it too when I first came across this problem. In the haze of the early morning when I first read this article, I erroneously thought that the 8 card and the blue card had to be turned.

# Wason Selection Task: Part 1

I recently read about a simple but clever logic puzzle, known as the “Wason selection task,” which is often claimed to be “the single most investigated experimental paradigm in the psychology of reasoning,” in the words of one textbook author.

Here’s the puzzle: You are shown four different cards, showing a 5, an 8, a blue card, and a green card. You are asked to test the truth of the following statement:

If a card has an even number on one side, then its opposite side is blue.

Question: Which card (or cards) must you turn over to test the truth of this statement?

I’ll start discussing the answer to this puzzle in tomorrow’s post. If you’re impatient, you can click through the interactive video above or else read the article where I first learned about this puzzle: http://m.nautil.us/blog/the-simple-logical-puzzle-that-shows-how-illogical-people-are (I got the opening sentence of this post from this article).

# io9: “I Fooled Millions Into Thinking Chocolate Helps Weight Loss. Here’s How.”

Peer-reviewed publications is the best way that we’ve figured out for vetting scientific experiments and disseminating scientific knowledge. But that doesn’t mean that the system can’t be abused, either consciously or unconsciously.

The eye-opening article http://io9.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 describes how the author published flimsy data that any discerning statistician should have seen through and even managed to get his “results” spread in the popular press. Some money quotes:

Here’s a dirty little science secret: If you measure a large number of things about a small number of people, you are almost guaranteed to get a “statistically significant” result. Our study included 18 different measurements—weight, cholesterol, sodium, blood protein levels, sleep quality, well-being, etc.—from 15 people. (One subject was dropped.) That study design is a recipe for false positives.

Think of the measurements as lottery tickets. Each one has a small chance of paying off in the form of a “significant” result that we can spin a story around and sell to the media. The more tickets you buy, the more likely you are to win. We didn’t know exactly what would pan out—the headline could have been that chocolate improves sleep or lowers blood pressure—but we knew our chances of getting at least one “statistically significant” result were pretty good.

And:

With the paper out, it was time to make some noise. I called a friend of a friend who works in scientific PR. She walked me through some of the dirty tricks for grabbing headlines. It was eerie to hear the other side of something I experience every day.

The key is to exploit journalists’ incredible laziness. If you lay out the information just right, you can shape the story that emerges in the media almost like you were writing those stories yourself. In fact, that’s literally what you’re doing, since many reporters just copied and pasted our text.

And:

The only problem with the diet science beat is that it’s science. You have to know how to read a scientific paper—and actually bother to do it. For far too long, the people who cover this beat have treated it like gossip, echoing whatever they find in press releases. Hopefully our little experiment will make reporters and readers alike more skeptical.

If a study doesn’t even list how many people took part in it, or makes a bold diet claim that’s “statistically significant” but doesn’t say how big the effect size is, you should wonder why. But for the most part, we don’t. Which is a pity, because journalists are becoming the de facto peer review system. And when we fail, the world is awash in junk science.

# The Extent and Consequences of P-Hacking in Science

A couple months ago, the open-access journal PLOS Biology (which is a reputable open-access journal, unlike many others) published this very interesting article about the abuse of hypothesis testing in the scientific literature: http://journals.plos.org/plosbiology/article?id=10.1371%2Fjournal.pbio.1002106

Here are some of my favorite quotes from near the end of the article:

The key to decreasing p-hacking is better education of researchers. Many practices that lead to p-hacking are still deemed acceptable. John et al. measured the prevalence of questionable research practices in psychology. They asked survey participants if they had ever engaged in a set of questionable research practices and, if so, whether they thought their actions were defensible on a scale of 0–2 (0 = no, 1 = possibly, 2 = yes). Over 50% of participants admitted to “failing to report all of a study’s dependent measures” and “deciding whether to collect more data after looking to see whether the results were significant,” and these practices received a mean defensibility rating greater than 1.5. This indicates that many researchers p-hack but do not appreciate the extent to which this is a form of scientific misconduct. Amazingly, some animal ethics boards even encourage or mandate the termination of research if a significant result is obtained during the study, which is a particularly egregious form of p-hacking (Anonymous reviewer, personal communication).

Eliminating p-hacking entirely is unlikely when career advancement is assessed by publication output, and publication decisions are affected by the p-value or other measures of statistical support for relationships. Even so, there are a number of steps that the research community and scientific publishers can take to decrease the occurrence of p-hacking.

# Mathematical induction and blank space

I tried out a one-liner in class that I’d been itching to try all summer.

I was introducing my students to proofs by mathematical induction; my example was showing that

$1 + 3 + 5 + \dots + (2n-1) = n^2$.

After describing the principle of mathematical induction, I wrote out the $n = 1$ step and the assumption for $n = k$:

$n=1$: $1=1^2$, so this checks.

$n =k$: Assume that $1 + 3 + 5 + \dots + (2k-1) = k^2$.

Then, for the inductive step, I had my students tell me what the left- and right-hand side would be if I substituted $k+1$ in place of $n$. I wrote the answer for the left-hand side at the top of the board, the answer for the right-hand side at the bottom of the board, and left plenty of blank space in between the two (which I would fill in shortly):

$n = k+1$:

$1 + 3 + 5 \dots + (2k-1) + (2[k+1]-1) =$

$~$

$~$

$~$

$~$

$~$

$~$

$= (k+1)^2$

So I explained that, to complete the proof by induction, all we had to do was convert the top line into the bottom line.

As my class swallowed hard as they thought about how to perform this task, I told them, “Yes, this looks really intimidating. Indeed, to quote the great philosopher, ‘You might think that I’m insane. But I’ve got a blank space, baby… and I’ll write your name.’ “

The one-liner provoked the desired response from my students… and after the laughter died down, we then worked through the end of the proof.

And, just in case you’ve been buried under a rock for the past few months, here’s the source material for the one-liner (which, at the time of this writing, is the second-most watched video on YouTube):

# Engaging students: Graphs of linear equations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Nada Al-Ghussain. Her topic, from Algebra: graphs of linear equations.

How could you as a teacher create an activity or project that involves your topic?

Positive slope, negative slope, no slope, and undefined, are four lines that cross over the coordinate plane. Boring. So how can I engage my students during the topic of graphs of linear equations, when all they can think of is the four images of slope? Simple, I assign a project that brings out the Individuality and creativity of each student. Something to wake up their minds!

An individualized image-graphing project. I would give each student a large coordinate plane, where they will graph their picture using straight lines only. I would ask them to use only points at intersections, but this can change to half points if needed. Then each student will receive an Equation sheet where they will find and write 2 equations for each different type of slope. So a student will have equations for two horizontal lines, vertical lines, positive slope, and negative slope. The best part is the project can be tailored to each class weakness or strength. I can also ask them to write the slop-intercept form, point slope form, or to even compare slopes that are parallel or perpendicular. When they are done, students would have practiced graphing and writing linear equations many times using their drawn images. Some students would be able to recognize slopes easier when they recall this project and their specific work on it.

Example of a project template:

Examples of student work:

How has this topic appeared in the news?

Millions of people tune in to watch the news daily. Information is poured into our ears and images through our eyes. We cannot absorb it all, so the news makes it easy for us to understand and uses graphs of linear equations. Plus, the Whoa! Factor of the slopping lines is really the attention grabber. News comes in many forms either through, TV, Internet, or newspaper. Students can learn to quickly understand the meaning of graphs with the different slopes the few seconds they are exposed to them.

On television, FOX news shows a positive slope of increasing number of job losses through a few years. (Beware for misrepresented data!)

A journal article contains the cost of college increase between public and private colleges showing the negative slope of private costs decreasing.

Most importantly line graphs can help muggles, half bloods, witches, and wizards to better understand the rise and decline of attractive characters through the Harry Potter series.

How can this topic be used in your students’ future courses in mathematics or science?

Students are introduced to simple graphs of linear equations where they should be able to name and find the equation of the slope. In a student’s future course with computers or tablets, I would use the Desmos graphing calculator online. This tool gives the students the ability to work backwards. I would ask a class to make certain lines, and they will have to come up with the equation with only their knowledge from previous class. It would really help the students understand the reason behind a negative slope and positive slope plus the difference between zero slope and undefined. After checking their previous knowledge, students can make visual representations of graphing linear inequalities and apply them to real-world problems.

References:

http://www.hoppeninjamath.com/teacherblog/?p=217

http://walkinginmathland.weebly.com/teaching-math-blog/animal-project-graphing-linear-lines-and-stating-equations

http://mediamatters.org/research/2012/10/01/a-history-of-dishonest-fox-charts/190225

http://money.cnn.com/2010/10/28/pf/college/college_tuition/

http://dailyfig.figment.com/2011/07/13/harry-potter-in-charts/

https://www.desmos.com/calculator

# Multiplication and Animaniacs

One of my guilty pleasures in the 1990s, when I was far too old to be watching children’s cartoons, was the fantastic show “Animaniacs.” In the clip below, Yakko Warner describes how to multiply 47 and 83.