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).

The Mayan Activity: A Way of Teaching Multiple Quantifications in Logical Contexts

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight Kyeong Hah Roh & Yong Hah Lee (2011) The Mayan Activity: A Way of Teaching Multiple Quantifications in Logical Contexts, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21:8, 685-698, DOI: 10.1080/10511970.2010.485602

Here’s the abstract:

In this article, we suggest an instructional intervention to help students understand statements involving multiple quantifiers in logical contexts. We analyze students’ misinterpretations of multiple quantifiers related to the ϵ-N definition of convergence and point out that they result from a lack of understanding of the significance of the order of the quantifiers in the definition. We introduce the Mayan activity which is designed to cause and then to help resolve students’ cognitive dissonance. In particular, the Mayan stonecutter story in the activity is presented in an understandable and colloquial form so that students can recognize the independence of ϵ from N in the ϵ-N definition. Consequently, the Mayan activity can be regarded as a useful instructional intervention to study statements related to the ϵ-N definition of convergence.

The full article can be found here: http://dx.doi.org/10.1080/10511970.2010.485602

Axiom of choice

Source: http://www.xkcd.com/982/

Engaging students: Using a truth table

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 Bich Tram Do. Her topic, from Geometry: using a truth table.

Funny video to engage student that a university professor made in class.

Or another clip from the movie “Liar, Liar”

How can you tell if an argument is valid or invalid? In this lesson, we will learn about the truth table and technique to detect the validity of any simple argument.

A2. How could you as a teacher create an activity or project that involves your topic?

I could split students into a group of three students and hand each group 3 bags of different colors cards with printed statements on each one. For example:

Bag 1 has statements such as:

If you are a hound dog, then you howl at the moon.

Bag 2 contains conditions:

You don’t howl at the moon.

Bag 3 has conclusions:

Therefore, you aren’t a hound dog.

In each group, the teacher gives a poster/ construction paper that students must search for the correct responses, match them up, and paste them on the construction paper on the left side. On the right side of the paper, the students are asked to answer the question whether the arguments are valid or not and their reason by making a truth table.

Students will have total of five sets and given about twenty minutes to finish. When the students have all finished, I will ask each group coming up with a new example, state their reasons and present to the class. I might have the students volunteer to be 3 judges and vote for the group with the best example. The activity is fun and helps students to apply what they learned as well as their mastery of the materials.

D1. What interesting things you can say about the people who contributed to the discovery and/or the development of this topic?

According to Shosky (1997), the truth table matrices was claimed to be invented by Bertrand Russell and Ludwig Wittgenstein around 1912. However, there was evidence shown that the logician Charles Peirce (1839-1914) had worked on the truth table logic (1883-84) even before the other two mathematicians worked on the same logic. However, Peirce’s unpublished manuscript did not directly show as a “table”, but the “truth functional analysis”, and was in matrix form. Peirce used abbreviations v (for true) and f (for false) and a special symbol ―< to connect the relationship between statements, say a and b. Later, Russell and Wittgenstein (1912) claimed the first appearance of the truth table device, causing doubts if they worked together or separated and evidences needed to make the claim. In short, the invention of the truth table was credited to Charles Peirce in “The Algebra of Logic” (around 1880) and the “table” form was developed to be clearer and easier for understanding, along with many important contributions of Russell, Wittgenstein based on their knowledge of matrix, number theory, and algebra.

E1. How can technology be used to effectively engage students with this topic?

The truth table topic doesn’t have many engaging activities for students to learn even though it has many applications, especially in digital designing, electrical systems. However, we can include some use of technology so that students who finished group activities early or students who needed more practice can find. This website is an interactive activity for students to do so:

http://webspace.ship.edu/deensley/discretemath/flash/ch1/sec1_3/truthtables/tt_control.html

There are different conditions represented by p, q, and r on the first three columns. The next columns, students are asked to fill out the answer (True or False) to each corresponding condition. When they are done with one column, just click on the statement “I’m done with this column”, and then the students will be directed to another one to try. In addition, they can always click on the pink rectangular box in the bottom to change to a different truth table.

Source:

http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1119&context=russelljournal&sei-redir=1#search=%22truth+tables+history%22

http://www.math.fsu.edu/~wooland/argumentor/TruthTablesandArgs.html

http://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf

Engaging students: Distinguishing between axioms, postulates, theorems, and corollaries

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 Michael Dixon. His topic, from Geometry: distinguishing between axioms, postulates, theorems, and corollaries.

A2. How can you create a project for your students?

A project that I would have my students do to show that they know what the differences between these four logical terms are to ask them to write a story to model each one. There are several subtleties between these terms that require defining. Axioms and postulates are very similar, both are terms to describe something that is held to be true, and neither require any proof. The general idea is that these are supposed to be “obvious”statement that require no argument. Theorems are ideas that are heavily proven to be true, following the axiomatic method. Corollaries, however, generally follow directly as a result of a theorem, usually requiring only very short proofs.

As an example of what the students could come up with, they could write about two different doctors, who happen to be brothers. The first is a successful general physician in a remote village. He studied for many years to become the man in his village that takes care of all the illness and injuries that the villagers suffer from time to time. He is able to take care of almost anything that requires medicine or general care. But occasionally, the physician decides that a villager needs extra care or surgery that he cannot provide, so he sends them to his brother. His brother is just as successful a doctor, but instead of studying general medicine, this brother focused only on learning how to perform any kind of surgery. When the physician sends a villager to the surgeon, the surgeon figures out what needs to be done and then operates on the villager. Between the two of them, the village hasn’t suffered a death due to sickness or injury in several years.

In this example, the physician would model an axiom, and the surgeon would represent a postulate. Both of them are known by everyone to be excellent in their functions, modeling that they are known to be true. But axioms are held to be true in general, across many categories and sciences. A postulate, however, is known to be true, but is specific to one particular field.

C3. How has this appeared in the news?

If I ask you, “who is the most famous mathematician?”what would you say? Its probably not a question that can safely be answered without causing an argument among mathematicians. But to the layman, the best answer would most likely be Albert Einstein. He is famously known for his General Theory of Relativity. After publishing this work in 1905, Einstein steadily rose to fame, for this work and later for his work on the Manhattan Project and his work in quantum mechanics. And even still today, Einstein’s work still influences the scientific community. Recently it has been reported on PBS that a previously unknown theory that Einstein was working on has surfaced that leads to the idea that he might have supported the idea of a steady-state universe. Pioneered by Fred Hoyle, steady-state theory states that the universe is constantly expanding, but not becoming less dense, hence it remains steady throughout time. Einstein even used equations from general relativity to support his theorem. The article states that Hoyle did not know of Einstein’s support, and though Hoyle’s theorem was mathematically sound, it did not become universally accepted. With Einstein’s support, that result could have turned out differently.

D2. How was this adopted by the mathematical community?

When speaking of the axiomatic method and the history of proofs of this nature, naturally the conversation takes a turn towards the ancient Greeks. Most famously, Euclid developed his geometry using postulates, axioms, theorems, and corollaries. No history would be complete without mentioning these facts. In fact, it was Euclid’s Elements and the parallel postulate that led to a focusing on deductive reasoning and a general application of the axiomatic method in the early 19th century, after the discovery of non-Euclidean geometry. When it is assumed that the negation of parallel postulate is true, an entirely different geometry than we are used to comes into being. Logically it can be reasoned and soundly proven using exactly the same method of logic as Euclidean geometry. This led to a mathematical revolution of sorts, where mathematicians began trying to formalize axiomatically all of mathematics into a system. This led to all kinds of interesting paradoxes, including the incompleteness theorem, among others.

http://www.differencebetween.com/difference-between-axioms-and-vs-postulates/

http://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/

http://www.pbs.org/wgbh/nova/next/physics/einsteins-lost-theorem-revealed/

http://www.encyclopediaofmath.org/index.php/Axiomatic_method

Engaging students: Fractions, percents, and decimals

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 Billy Harrington. His topic, from Pre-Algebra: fractions, percents, and decimals.

Application:

1) Problems that arise with integrating fractions, percents, and decimals include instances such as shopping during a sale at a certain store or shop. The type of shop does not matter whether it is a flea market, or a high-end clothing store. A sale affects all types of stores in the same way. When an item is (1/3) of its original price, people must convert this into a fraction and then convert to a decimal to find out the whole dollar value which will most likely involve decimals as well as the fractions/percentages indicating the amount of money off the original price.

I used this website as an example of problems:

http://www.bbc.co.uk/skillswise/worksheet/ma18comp-l1-w-problem-solving-with-fractions-decimals-and-pct

Another really good exercise in percentages, fractions, and decimals is budgeting a certain income over a year. Students should calculate the percent of their budget that they spend on a home, food, necessities, and their leisure activities. Some students can be told to start budgeting using fractions, while another group of students is told to budget based on percentages. When the class is done, students can come together for a class discussion, and share the benefits, and obstacles of budgeting using the method they performed.

2) For a full activity, each student will get one full sheet of printer paper, and a pair of scissors (or be split into small groups of 2 to 4 four people in each group to save paper). Each student/group will start by acknowledging that their full page represents 1 part of 1 whole and represent this as a fraction and a decimal. Students will then continue by cutting their paper in half and notice that there are now two pieces in front of them. They will continue to cut their paper in half another five to six times and then represent each stage by a fraction.

Stage 1	1 part of 1	Represented (1/1)
Stage 2	1 part of 2	Represented (1/2)
Stage 3	1 part of 4	Represented (1/4)
Stage 4	1 part of 8	Represented (1/8)
Stage 5	1 part of 16	Represented (1/16)

Curriculum

1) When students get to their upper level math classes or even when they get to college, they must calculate their own grade/GPA. Not all classes or grades are going to be graded equally and on the same scale. Some classes are graded on a 1000 point scale where as some classes are weighted on a 75 point scale. To convert their weighted total number of points to calculate their letter grade, students must either set their percentage total in a proportion and weigh out the actual score on a 100 point scale to calculate their grade based on the letter grade scale. A student may say, “I have a 130 in this class, this must be an A!” This may be great, or it could be terrible depending on the grading scale, that’s why students must weigh it against the total point value, then convert it to a percent to find out their true letter grade and see in fact if their 130 is truly a good grade worthy of passing.

2) Students will always need basic math in their lives, even throughout adulthood. Percentages, fractions and decimals should be part of that foundation of mathematics that they know. A big part of this topic that students should learn is budgeting, even if it is a small allowance they receive on a weekly, or bi-weekly basis. If they’re given $20 every other week, how are they going to spend or save that money over the 2-week period they have? Students could spend it all, save it all, or spend some, and save some. Students could calculate the percent of money they did spend if they decided to spend money and see what fraction, percent or decimal value best represents what money they spent, and/or saved.

Culture

1 & 2) Percentages, fractions, and decimals is actually really important in the media world such as music and film industries. Take ITunes for example as the sole business that sells music, and also a different assortment of films. The consumers are drastically affected by other media sources, such as a television, or even a newspaper. If a “huge hit” is coming from this new movie coming out next Friday, chances are that a huge percentage of people are going to partake in the new film and go watch it at the local theater. If the movie is a success, then chances are that the movie will reach the top of the box office. The box office is determined by profits over a short amount of time when a movie/film is released into theaters. Movies such as Harry Potter and the Hunger Games were big sell-outs in the box office because there was such a huge profit made off of the films. Profits based on ticket sales are depicted by a percentage of average sales, which means the higher the percent of people that went and watched the new movie, means that the profits are going to be higher. Based on these statistics, movies are then ranked in the box office to see which movie was the most successful at the end of the year.

Rank 1 in Box Office for 2013 –

Hunger Games Catching Fire at over $420 million dollars

This concept applies in Theater as well such as Broadway plays they make huge profits on ticket sales

3) A huge way fractions, percents, and decimals has influenced the world and our culture is by our economy and our market system. Our current economic system is currently in shambles and is desperately trying to fix itself through many irregular and unorthodox ways that sometimes turn out for the worse. The economy is not easy to understand and explaining how the market works to an average citizen probably will not go well, so the market and its different branches are represented in simple, yet intricate graphs, percentages, and decimals to represent how the current day has progressed. There are some days where the DOWJONES may be below 13% where as some days the NASDAQ may be up 10%. Different branches of the economy are each shown in simple percentages, if people don’t understand the values of percents, fractions, and decimals; there is almost no hope for that person to understand the current economic situation.

Formal logic bumper sticker

Source: http://www.xkcd.com/1033/

Can You Solve This?

A friend forwarded this very interesting video to me. It’s not so much an exercise in mathematics but an exercise in problem-solving and logic and especially confirmation bias. I won’t ruin the video but I’ll give the punch line at the end:

If you think that something is true, you should try as hard as you can to disprove it. Only then can you really get at the truth and not fool yourself.

Two-Column Proofs that Two-Column Proofs are Terrible

I’m not entirely sure that I completely agree with the author of this post (http://mathwithbaddrawings.com/2013/10/16/two-column-proofs-that-two-column-proofs-are-terrible/), but he certainly provides food for thought and so I’m happy to link to it. Among the most provocative quotes from this post:

In a good proof, each individual step is obvious, but the conclusion is surprising. In many two-column proofs—especially those taught earliest in a geometry course—each individual step is mystifying, while the conclusion is obvious.

Engaging students: Truth tables

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

This student submission comes from my former student Elizabeth (Markham) Atkins. Her topic, from Geometry: truth tables.

D. History: Who were some of the people who contributed to the development of this topic?

In “Peirce’s Truth-Functional Analysis and the Origin of Truth Tables” it is said that Charles Peirce was the first to start studying truth tables or rather developing the idea. He created the truth table in 1893. Peirce stated “the purpose of reasoning is to establish the truth or falsity of our beliefs, and the relationship between truth and falsity”. Nineteen years later, two mathematicians developed the truth table as we know it today. Ludwig Wittgenstein and Bertrand Russell both knew of truth tables but formalized them into the form we know today. In “The Genesis of the Truth-Table Device” it is said that George Berry stated “Peirce developed the technique, but not the device”. Wittgenstein developed the terminology that we today associate with truth tables. All in all it is the work of many people that finally developed the truth tables that we know today.

APPLICATIONS: What interesting word problems using this topic can your students do now?

Truth tables state that if P is true and Q is true then both P and Q are true. If either P or Q or both are false then P and Q are false. So I could have the students construct many truth tables to demonstrate their knowledge of the subject or I could come up with some interesting word problems. Word problems such as “True or false: If Billy Joe graduated and Shawn graduated then both Billy Joe and Shawn graduated.” There are not many word problems you could create that would deal with truth tables. You can have the students begin to think logically. You could give them a statement to complete such as, “Good apples are red. Granny Smith apples are green. Thus ____” This enables the teacher to get the students in the logical process of thinking in order for them to correctly understand truth tables.

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

By teaching my students truth tables and how to use them correctly it prepares them for future classes and for everyday life. In high schools now the students are learning twenty first century skills. To learn truth tables it will help with the twenty first century skills. When you learn truth tables you learn to think logically. The students need to learn logical thinking for science and economics. In Science, they need to learn logical thinking for when they do experiments. It will allow them to process, “well if I do this then this might happen.” In economics students need logical thinking so that when they learn to invest money they can weigh their options. In everyday life students make decisions that they need to think about. Teenagers in the modern day are moving so fast that they often do and say things without thinking. If they learn to think logically then they might be able to think, “If I say or do this then this might happen.”

Irving H. Anelli’s

PEIRCE’S TRUTH-FUNCTIONAL ANALYSIS AND THE ORIGIN OF TRUTH TABLES http://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf

THE GENESIS OF THE TRUTH-TABLE DEVICE http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1119&context=russelljournal