Occupy Math starts this week by revisiting the issue raised in an earlier blog: Believing things are impossible. Another blogger, Junaid Mubeen is writing some excellent articles about math education. These include wonderful examples of problems without solutions and the incredibly poor follow-through by math teachers when they pose these problems. It is critical for citizens to be able to notice when a politician makes a completely impossible claim and it is important for everyone to be able to tell if they are committing to an impossible project. This speaks to the title of the blog and here’s the problem.

**The way we teach math make people certain problems have solutions.**

Occupy Math has decried the way we teach math before. The blog post Math is not a Form of Ritual Magic explores the problem that many high school math classes no longer teach math. Instead, they teach how to get a really good score on the math part of a standardized test. The students are taught to recognize key features of standard problems and, having classified the problem, they apply — as a conditioned response, not a thoughtful act — a procedure that they have been drilled in but probably do not understand. They turn a crank and produce an answer. These students are not able to notice when a problem does not fit their classification system for problems. They do not notice when they have incorrectly classified a problem. They cannot solve problems of a type they have never encountered before. Worst of all, they gain no problem-solving skills *at all* from classes in one of the most useful problem-solving disciplines known to our species.

This behavior verges on malfeasance in the execution of our duty to educate our children. It is also poor preparation for life. This sort of teaching supports the idea that math is useless to real people. We are burning down one of the pillars of our civilization to get good scores on standardized tests. Occupy Math has as a goal to avoid pure negativity, so, yes we are dropping the ball on math education is a really big way, but what should we do instead?

**Instead of bite-sized math problems that all have simple answers, use open-ended problems, discussion, and exploration.**

One of Junaid Mubeen’s blogs is a good starting point. Consider the following problem:

There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

A person who has not been conditioned to ignore the problem and just start crunching the numbers through some procedure will think for a few moments and say “the number of sheep and dogs in a flock does not contain any information about the age of the shepherd. This is a dumb problem.” Dr. Mubeen reports that when this problem was posed in a math class the students went to work multiplying and dividing 125 by 5. This means that they completely ignored the real world context supplied by the words and just waded in calculating. There are two hypotheses that explain this behavior. The first is that these students were taught math as if it were ritual magic. The second is that these students are not too bright. It is extremely unlikely that a whole class is made, uniformly, of students that are not too bright. Here is another example of a problem of this type from Dr. Mubeen’s blog.

Yesterday 33 boats sailed into the port and 54 boats left it. Yesterday at noon there were 40 boats in the port. How many boats were still in the port yesterday evening?

Since boats are presumably arriving and leaving all the time, knowing the number of boats present yesterday at noon is not relevant information and, in fact, there is no reference or anchor anywhere in the problem for the number of boats, only information about how that number is changing. The score on this one is, of 101 students, 1 noticed it could not be solved and 100 waded in crunching numbers. When asked to reflect and explain their methods, only 5 of the 101 students realized something was amiss. The others had absolute faith that their teacher would only pose problems they could solve.

**Is it useful to pose problems that cannot be solved, intentionally?**

In the post linked above, *Believing Things are Impossible*, Occupy Math recounts the experience of a former student, Carly Rozins, who intentionally posed an impossible problem to students at a math camp. One of the startling outcomes was that one of the students, after seeing and accepting a mathematical proof that the problem *could not be solved*, worked for hours to find a solution. Occupy Math’s blog contains the proof — with lots of pictures — if you are interested. Another of Dr. Mubeen’s blogs is about an impossible problem posed to his nephew by a math teacher. Its a good problem: read the blog! The blog contains an accessible proof that the problem has no solution — and is a nice example of several problem solving techniques.

There are two important take-home messages in that blog.

- Having posed a problem he knew to be impossible, the teacher did not follow up with a discussion, a demonstration of impossibility, or anything else that would alleviate the frustration of students who failed to solve that problem and did not have a mathematician as an uncle.
- Dr. Mubeen’s nephew worked for hours trying to find a solution
*after having seen and accepted a proof that no solution exists*!

Carly Rozin’s persistent student and Dr. Mubeen’s nephew both speak to something that strongly informs how we should teach math.

**People are wired to prefer solutions to dead ends. This means that believing in the impossible is difficult and must be learned.**

How then, do we teach the impossible? The robotic process of classifying a problem, picking the correct gun, and shooting the problem dead in a few minutes is exactly not the way to go. Mathematics is far more than brute calculation and what is needed something like this.

- Pose a problem and discuss until the students actually understand what the problem is. This models the behavior of thinking things through and does not teach the students to flash identify problems and shoot from the hip.
- Explore, discuss, and critique possible approaches to the problem. In the first-year class Occupy Math is teaching at present he often works problems multiple ways because, in a given category of problems, different techniques are harder or easier for different instances of the type of problem.
- Ask questions and
*wait*for the students to answer. Questions like “why do you think that will help?”, “does that buy us anything?”, and “Can you try that on an example?” - Be
**gentle**when getting someone to realize they are heading in a bad direction and be**open**to the possibility that the direction might not be as bad as you thought.

Listening to your students pays dividends. When Occupy Math was first starting out, a student named Jennette Tilliotson came by all enthusiastic about a cracking sequence she had devised for a lock in a video game. To open the lock you had to enter the digits 1, 2, 3, 4, and 5 in some order and when you entered a new digit it appeared on the right, everything slid over one, and the leftmost digit fell off. What Ms. Tilliotson had done was find the smallest string of digits that included every possible combination under the slide-one-left entry system. Four other faculty blew her off before she walked into Occupy Math’s office. The result of listening was a scientific publication: *Daniel Ashlock and Jennette Tilliotson*, **Minimal Superpermutations**, Congressus Numerantium number 93: pages 91-98, 1993. Occupy Math managed to generalize Ms. Tilliotson’s solution to any number of digits and prove it was correct and optimal. If Occupy Math had it to do over, he would make her first author. Occupy Math did most of the work, but the precious commodity is the *leap of insight* which Ms. Tilliotson supplied.

Occupy Math will conclude with examples of open-ended problems that could be used in the classroom, using the techniques above. Carly Rozin’s problem posed in the linked blog is a good open-ended discussion problem, and it is impossible. Here are some others.

**Five component cable problem.**

Suppose that we need to wire together five electronic components so that there is a cable from every component to every other component. If two connecting cables cross, we must use an expensive shield where they cross. Show the objects and the cables in a configuration that minimizes the number of cable crossings.

(Teacher’s Gloss) Below are the worst possible and best possible solution. To make the problem open ended, ask “what if there are more components?” Cables do not have to be line segments, they can curve — but all optimal solutions can be constructed using line segments.

**The glow worm problem.**

If two glow worms are beside one another, one will eat the other. What is the smallest number of glow worms that can be placed in a 6×6 grid so that no more glow worms can be placed? Assume that it is okay to put two worms diagonal to one another.

(Teacher’s Gloss) Read the Occupy Math blog The Bug is in the Nature of Reality which is about this problem. The smallest configuration that will not permit another glow worm to be placed has eight glow worms in it.

**The Friends at a party problem.**

Suppose that, for two people at a party, these people are either friends or strangers. Is it possible for each person to have a different number of friends at the party?

(Teacher’s Gloss). It is not possible. If you consider the number of friends a person can have, the largest possible is all but one. The smallest is none. The numbers {none, one, …, all but one} are a set of numbers the same size as the number of people at the party. For everyone to have a different number of friends, we must use all these numbers. That means that one person is friends with everyone else while another is friends with no-one. These two conditions contradict one another, showing the answer is no, this is not possible.

Occupy Math hopes that teachers that read this will take some of it on board. There are some obstacles. This method of teaching is completely not in line with the current thinking about how to prepare students for standardized tests. It is also time consuming and a lot of work for the teacher. Let’s end on a positive, reasons to adopt these techniques. Occupy Math is sure, and his editor concurs, that math taught this way is far more useful to student when they gain employment. Also, based on years of trying to motivate unenthusiastic students, these techniques that engage the students conversationally can make a math class a much livelier place. Would you like more problems like this? Do you have some to share? If so, please comment or tweet!

I hope to see you here again,

Daniel Ashlock,

University of Guelph,

Department of Mathematics and Statistics