What to do when you don't know what to do!

Back in the video game: Maths Educator at Large

Whiteboard Problem no.7

Below is one of the best puzzles I have ever come across.  The clip below embeds it within a communication context, but you can just copy it straight to the whiteboard.  My advice is to try and solve the puzzle before watching the entire clip.

It's a little long winded I know. Endure.

Good luck.

On school and learning identity

I have spent the last years analyzing the data from a series of interviews I did with adult learners.
The one overwhelming finding is: Self-worth is tied up with academic performance - especially mathematics. Moreover, some environments are far more prone than others to enable judgments to be made about your worth (both by you and others).

Societal beliefs about what mathematics is and what it means to be good or bad at mathematics has damaged some of us.  And may still be doing so.

I'm trying to put into words how maths classes were described.  This is a rough start but perhaps this tale will help generate some thought about the impact social-pressure can have on us.

The tale

John has just turned 13.  Today is his first day at High School.  He is excited to meet new people and take part in what others are doing.  He is really looking forward to taking part in classes.

John has a positive view of himself, he is capable, inventive, gets on well with others and loves to play and have fun.  John is excited about going to maths class, he loves solving problems, talking about maths and enjoys reading about famous mathematicians like Aristotle.  His mum bought him a shiny, tidy new work book and he received an exciting text book.  The new pencils, pens, rulers and calculator made it even more exciting. 

On the first day of maths class he noticed the teacher would ask the class very direct questions.  He quickly realised that they weren't real questions because the teacher already knew the answers - they were more like mini-tests, however, he had a go at them  - his hand usually went up first.  When he answered, the teacher just carried on, as though the answer had come from his own mouth -but it felt good to answer, and the teacher seemed to approve.

One time, he answered wrong, and the class laughed.  He noticed that they laughed 'at' him, not with him. They didn't laugh because he was funny or had made a joke.  In fact, he wasn't sure why they laughed, but it didn't feel good.  It made him feel 'alone', for a moment, different, a sense of 'outsidedness'.  This didn't happen at his old school. 

Another time he answered a question in what he thought was a conversation with the teacher.  He realised too late that the teachers' question had been rhetorical.  He also answered with the wrong numbers. Other learners laughed at him again. One of them called him a 'dummy', but in a funny sort of way, not mean.  No one else had tried to answer.  Maybe he was breaking the rules about when to speak?  Why did they laugh?  Perhaps because he was so wrong, so surprisingly wrong, that it was funny?

He didn't want to be laughed at anymore.  He wanted to laughed 'with'.  So when a different student answered incorrectly - he laugh 'with' the others.  He stopped answering questions himself - rather he waited until someone else answered first and then he checked if he was right.  Often he was wrong, but so long as he didn't share it, it didn't matter.

John found that two worlds developed -his own world of math, conducted in his head where he answered and asked questions, and the public math, the math that occurred around him as he spoke to other learners and engaged with the teacher.  Two domains to think about math, one in the privacy of his head, and one public. 

Often he got things wrong in his private world, and hid it.  Often if he was wrong, and no one knew it, he would subtly rewrite the answer in his book.  If he wrote it lightly in pencil he found he could erase it and leave no smear marks.  But other times, as much as he tried to hide it he couldn't.  The teacher would ask him for answers directly and he had to give his answer to the whole class.  Sometimes they had tests and everyone found out how everyone did.  He would try and hide his score but someone would always ask.  To not tell would show you cared.  But that 'dummy' word stuck with him.  He wasn't sure why, but it came up again. 

He stopped enjoying maths, the class felt like a test and at stake was his reputation, his very self.  But outside the class he was great, especially away from school.  It gnawed on him though.  He was good at sports, and enjoyed some other subjects.  English wasn't too bad. 

What he didn't understand was how he could be confident outside class but be a dummy inside the maths class.  Which was he? Confident or dumb.   

He concluded that he was only dumb in the maths class.  That when it came to maths, he just didn't have it. But being good at maths means you are smart, so being bad at it means..?  This was too painful, so he told himself that maths wasn't his 'thing'.  He told his mum, "I'm good at sport but not maths.  Maths is for geeks anyway'.

Summary

John is struggling to construct a single coherent identity.  He wants to be strong, smart, capable of exerting his strength and getting things done.  But in the maths class he is positioned as the opposite, he can't be the guy 'who gets things done' and be the guy who is a dummy.  He can't be the dummy and the doer.  And hence there is a conflict between two identities.

John emotionally disconnects from maths.  He stops caring - because he has to. 

Statistics and critical thinking

Some quick thoughts on statistics, critical thinking, and foundation-level learners. 

Type one statistical thinking – Understanding how data is collected

Mark Twain noted ‘There are three kinds of lies: Lies, damned lies, and statistics’. He seemed to be implying that statistics were a tool used to manipulate the truth. Perhaps he was suggesting that politicians, advertisers, and others use statistics to mislead the public? What a concept.     

When teaching statistics to foundation-level learners it is useful to think of statistical critical thinking skills as two skills. The first is to understand ‘how data was collected’ and the second ‘how the statistics are represented’ to the reader. The first is closely related to typical critical skills, and the second more mathematically oriented. Foundation-level learners often enjoy the first and are not so familiar with the second.   

Understanding ‘how the data was collected’ is useful when presented with information such as this old advertisement for smoking.

Critical questions here might include:

Did they ask every single Doctor whether they smoked? How many did they ask, where were they from, how many answered, and how didn’t smoke and how many did? Of those who did smoke how many smoked Camel? Why did they smoke Camel? Were the Doctors given any incentives to promote Camel?

Question: Based on the evidence in the advertisement above is this scenario possible.

The researchers asked 50 Doctors whether they smoked. Five said they smoked. Three of these smoked Camels.

Could this be true based on the advertisements claims?

An old ‘good boy’ dog food scam.

A large company sold a popular brand of dogfood called ‘good boy’. The company was putting together a new advertisement and wanted some statistics to use in the advertising.

Here is how they produced the statistics.

An employee rang ten people and asked them what dog food they used to feed their dog. They recorded the amount of people out of ten who named the company.

They then rang another group of ten people. Then another. Then another. After doing this they simply selected the group of ten in which the highest number had selected their dog food. 

The statistic used was ‘six out of ten dog owners use good boy’. It was true, one of the groups of ten people had six members who used Good Boy. What wasn’t mentioned was that this was the highest of the ten groups. Some of the groups included only one or two people who used the dog food, others three, four or five. If the groups were combined the statistics would have read ’36 out of 100 dog owners use Good Boy’, or ’66 out of 100 dog owners do not use Good Boy’. Quite different. Yet the statistics are accurate – it’s the method needs to be critiqued.

However, be careful...

Despite these dubious methods of generating data, don’t allow learners to become cynical of all data. Most data collection methods are very good; understanding the strengths and limitations of the methods is the goal. Critical thinking when taken to the extreme means you can never know anything.  Don’t teach learners to be cynics, teach them to be critical evaluators.

Type two statistical thinking: Understand how statistics are represented to support certain ideas.     

Adults also need to know what statistical data analysis methods have been used and how the approach produces and represent findings. For example, if the term ‘average’ is used, does it refer to the mean, median or mode? Is the average even relevant? The ‘average NZ house price’ is frequently used in the NZ Herald. Would it not be better to know how many houses were sold in various price brackets? If two houses are for sale in a neighbourhood, one for 100,000 and one for 1,000,000, does the average of $550,000 fairly represent the situation? I don’t think so because an average of $550,000 suggests that people with one fifth of this amount cannot buy a house – and it isn’t true based on the situation.    

For example, the classic statistical scenario: There are two buckets, one full of boiling water and one full of freezing water. Stand with a foot in each. Stop screaming, because on average you feel warm.

If you only know the average, you don’t know enough to form an opinion.  

In sum, adults are bombarded with statistical information through politics, advertising, workplace statistics, dietary recommendations, safety courses, sports and health advice. In order to make sense of the information, adults require some knowledge of how data can be collected and how it is represented.