## The answer to the chain problem

The other day I posited a problem regarding linking a chain.

A woman has four pieces of chain.  Each piece is made up of three links.  She wants to join the pieces into a single closed ring of chain.  To open a link costs 2 cents and to close a link costs 3 cents.  She has only 15 cents.  How does she do it?

Most likely you drew it like this?

If so, you are totally normal.  Normal but wrong.

Drawing it like this means you start thinking about how to join the links at FOUR points.  Your brain gets stuck on four... you do the math and you are over the 15 cent limit.  Now perhaps you suspect a trick, so you begin to think about that perhaps you don't have to join them all.  Wrong.  These are all avenues your brain runs down but essentially lead you astray.

Rule number three for getting smarter is to practice and learn to change the representation.  That is, look at things differently.  The easiest way to do this is to draw or model it differently.

So here you go - I've represented it differently.  Think about opening the three bottom links and using them to join the other pieces.

Hope you got this.

To recap:  The ways to get smarter:

• Increase your world knowledge
• Automatize your thinking as much as possible
• Change the way you represent things
If you follow the link above, I highly recommend you click through to the Hanoi tower.  Like steroids for the brain.

## School

I have spent the last few months 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 judgements 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.  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, or perhaps in his now private book - 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 private 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.

## Catching the Predator

What's the point of improving your maths and numeracy skills if you don't use them to do something sensible?

Grew up thinking about how I would deal with the Predator alien. Let's face it - ya gotta be smart. Here is how.

As Sun Tzu noted you need to take advantage of the differences between yourself and your enemy.

Key differences:

1.  Height, body weight, strength.
2.  Superior technology
3. Sensory design  (Eyes too close)
4. Body mass distribution (Big freaky head).

Thinking about how to equalize number one is pretty simple. You must battle the Predator in muddy, swampy terrain.  Predators are big and heavy - this will work against him in deep mud.  If you could possibly drop him into a pit of mud you could trap him.  If not, make sure the general territory is as muddy as possible.  The more he places his foot down to push out the more he sinks and all that muscle tires him out.

Second - Use bridges with breaking points between your body weight and his.  This means that although you can run over a bridge, he will collapse it.  Great way to evade and trap him.  Punji stakes anyone?

Number four is a huge advantage to us - balance.  The Predator has a big fricken head.  His center of gravity is way too high.  Should be able to get him off balance quite a bit.  A log bridge tied by ropes that you have to walk along as it moves (like at the playgrounds) will cause this guy problems.  If his feet move out from directly below him he'll fall over.

Number three - Blind spots.  No wonder he is so mad.  Look at those little deep set eyes.  No wonder most of the technology has gone into the battle helmet - the dude has a field of vision of about 60 degrees.  That gives you 300 degrees to bust him a new one.  The dude is constantly wondering who is sneaking up behind him - probably explains their whole hunting culture.  They got so annoyed at being snuck up on they decided to dedicate their lives to the hunt.  Really just bitter little people.

### My point - and challenge

Moving along... The way to beat the Predator is to plan and set a trap that destroys him.  Now there are lots of complicated traps but I think Arnie had it right.  Drop a great big tree on his head.  Pummel him into the ground.  Forget spikes, and swings and trip wires.  It's got to be massive blunt force trauma.

### Scenario:

You are in the forest and have one night to prepare.  You find a sweet bottleneck area flanked by two tall trees and there just happens to be a massive log lying there on the ground.  The log weighs 200kg.  If you drop that on the Predator - it is history.  There is also a huge tree with a branch that will hold the weight 20 meters up.

Equipment:  You have in your bag, two pulleys and a fifty meter rope.   You have a knife and you are able to climb the tree.

### Question:

How do you get the 200kg log 20 meters up?  Let's just say you are a normal guy.  How are you going to lift that sucker so it hangs just below the branch?

If you work this out - you may live.  If not, it'll be you hanging from the tree.

### Options

Okay, below are three possible configurations for hoisting the log.  Have a good look.  Which one would you have done without the pictures, and which one would you say offers the best possibility of lifting the log?

A rookie mistake is to tie pulleys to immovable objects.  The key point to maximising the power of pulleys is to make sure you change the ratio of distanced pulled to object moved.  In other words, the pulleys should move toward each other.  The difference is power.

Take picture number 2.  If you pull the rope 12cm, the log moves 12cm.
Take picture number 1.  If you pull the rope 12cm the log moves about 6cm.

But what about picture number three?  Better or worse?

I have run some maths classes using pulleys.  It's great fun and allows learners to experiment and also creates lots of possibilities for maths exploration.  Ratios of pull to lift.  Ratios of distance pulled to distance lifted etc.

Final thing, Arnold lifted the log using vines and no pulleys!  Although he did use a fixed branch as a pulley.  Of course this only increased the friction!  Come on Arnie, Think!  Lucky he has those massive arms.

## Presentation

I gave a brief presentation today about adult numeracy.  I decided to give an authentic task to the class to 'put them in the zone'.  The idea was that the task would help them experience the discomfort many adult learners feel in numeracy classes.  We were then going to talk about how adults have affective responses to numeracy tasks that impact on our engagement.  Then we were to discuss our actual approach to solving the task.

It worked.  Horrible task - but here it is.  (oh - yes I know that it should be an 80 litre barrel...)

As you can imagine we had some affective responses.  Some good, some not so good.

The final slide was meant to be that below - but we never quite made it.

But - it is still a very good question.

## On tutor turnover

We have some great things happening in the entry level layer of the tertiary sector.  We work with learners who have the high needs and learners who are very demanding.  We work with learners who have experienced failure and without our help will repeat those patterns of failure.  We work in a tough environment and do it well.

We also have some systemic problems that continue to limit the kinds of success we all want.  One of our biggest problems is high tutor turnover.  This is compounded by poor organisational knowledge management.  In other words, tutors leave often, and when they do, they take large chunks of knowledge and experience with them.

### Why is high turnover a problem?  Isn't it feeding us new blood?

Schoenfeld's recent work with teachers provides some troubling findings.  He found that teachers up until the fourth year of teaching were primarily focused on mastering classroom management.  The actual educational gains of students were low, because the teacher was attempting to master the demands of running a class and learning the system.

The next four years of a teachers' career was focused on learning content.  They needed to master what they were teaching.  Let's call this subject knowledge.  Again, the learner gains were very limited.  The teacher's attention is split.

The next four years were spent mastering pedagogical knowledge - actual teaching to students.

Consequently, it takes 12 years for a fully trained teacher to begin to teach students really well.  And this is with a teaching degree, support and constant PD.  Not to mention regular pay rises and a supportive organisation (we assume).

Our sector has about a three year turnover period for tutors.  Several years ago it appeared to be less. There is little hard data so this is based on my observations.  At the time I asked every group (at workshops) that year how long they had been in their jobs and took note of the results.  One quarter had been in their job less than six months (sample was about 1000 tutors) in 2012.

It is not hard to see that tutors are occupied with mastering classroom management, developing content knowledge, and administration knowledge (NZQA, Assessment Tool etc) in their first few years, rather than developing great pedagogical knowledge.  To be blunt - Tutors are struggling to get a handle on the job and leave before developing high quality skills.  Consequently learners (the most demanding across all spheres of education) are not receiving the quality they require.  I don't say this in a judgmental way, I tutored for eight years straight (25 hours face to face contact 49 weeks every year) And STILL struggled.

The turnover erodes learner progress,  tutor development and institutional progress.

### Why do tutors leave?

I believe the reason tutors leave is two fold:  First is the growing discrepancy between their actual work outcomes and the demanded work outcomes.  Most tutors enter the sector because they want to help people get more out of life.  This is the real reason they are in the job, or at least a large part.  This can be called the 'actual' outcome.

Actual tutor outcomes:
• Positively impact people who need support
• Build confidence
• Work with and encourage young people
• Developing real skills that will pay off in the real world
• Minimising negative behaviours and maximising positive ones
• Demonstrate to people what can be achieved

Demanded Organisational outcomes
• NZQA Unit & Credit achievements
• Movement on L&N assessment tool
• Attendance
• Completion rates
• Learner employment

Now I know it is not as easy as all that.  But my suspicion is that the tutor role no longer reflects the values that attracted people to the role in the first place.  Hence, the downside of the role is no longer equally balanced by the positive side - and people simply leave (actually, the people with options leave).  All jobs have aspects that we struggle with, but those tensions are balanced by other positive factors.  The tutoring role is out of kilter.

### Solutions:

Many years ago during my degree I did a case-study of an organisation that was having difficulty with staff. Upon investigation it was clear that the staff had gotten themselves into a cycle of complaining and this cycle was now self-perpetuating.  The staff worked hard and well.  But when they got together the conversation always gravitated toward how bad things were getting.  Any negative thing that did happen tended to be taken as evidence that the whole organisation was negative. Management was largely absent but because the staff were highly committed they continued to perform at their very best.  The management were unaware but the staff as a group were on a downward trajectory.

Long story short, I interviewed staff and came up with some solutions based on some theories I was using (mostly motivational theory). The main solution had to do with the personality, traits and motivational make-up of the tutors.  The tutors were short term task oriented, yet highly relational.  They were motivated to make a difference for their students - not motivated by money and  not by achievement.  What did they need most?

Affirmation

I designed a plan based on the make-up of the staff and also drawing on Herzberg's two factor motivation-hygiene theory.  To my mind this is one of the most valuable ways of looking at the PTO sector. Look it up if you don't know it.

As a part of the plan I designed a meeting every two weeks with each staff member.  This would I believe have interrupted the cycle.  But it had to be personal and face to face.  Group rewards were ineffective with this group.  For example, a group trip away as a thank you to everyone would not be effective (this turned out to be a hygiene factor rather than a motivational factor).  They needed individual affirmation.  It had to be real and specific.  Any hint of in-authenticity would blow it. It didn't require money, it required genuine appreciation of a job well done from a senior.

The managers did not do this.  They had their reasons.  However key staff members dominoed out the door. Once one person left, they all went.  I have seen this happen about five times now in the same situation and sector.  The organisation then has to recover and this takes time and hurts learners.

Career pathways

The second way to begin to slow down turnover is to do some thinking around progression plans.  Tutoring has become a terminal job.  That means no pathway of progression.  It means anyone with aspirations must leave to get ahead.  Anyone who needs to save money must leave to get ahead.  As part of an annual review the organisation MUST build in a progression for the tutor.

You can also choose, as many do, to simply let tutors leave and train new ones. But in truth we are kidding ourselves if we think this is a sustainable strategy.  This simply places huge strain on the new tutor (see issue one) and on supporting staff.  But here is the kicker:  The learners will not learn.  You may get them through units, but these skills will be gone within 2 months.  You may get them jobs (great) but you will not have improved their skills.  You may have inducted them into a way of behaving, but you have not improved their skills.

Your organisation needs to be improving every year, year on year on year.  You can't do this if organisational knowledge is rolled back to zero every few years.

### Tough talk

The pressure will be coming on soon for PTEs to produce 'real' learning outcomes.  There is a cull coming (we have already seen the start of it) and those organisations that don't begin the transition to become real education organisations will not make the grade.  Mastering the art of moving learners through Unit standards will not be preparation for the next wave.  Those credits will have to reflect authentic skills development. That means if you re-assess any learner in 6 months they should easily pass the assessment.  Ask yourself, what if you reassessed all your learners right now on the units they have passed, with no teaching or support, would they easily pass the units?

We need professional tutors, who want to do this for a career.  This may require an entirely different business model in order for current funding streams to make this possible.  I have some ideas - would love to hear others.

Updated:  Graeme Smith has some great thoughts here.

## Unwarranted inferences - part two

A few weeks ago I began a series about how to get smarter.  One of the ways to get smarter is to understand the ways in which your brain limits your ability to solve problems.

One of the main ways your brain limits you from solving problems is due to unwarranted inferences.  In the reading comprehension research an inference is the information you bring to a context that is not explicitly stated.  For example:

Aaron spotted a coffee shop.  Right now, the warmth would be great.   He stopped, shook out his umbrella, closed it and then entered.

The inference we can reasonably make is that it is raining and cold.  It wasn't explicitly stated but it was implicitly inferred.  Now when you read you do not assemble clues and make conclusions consciously - rather your brain does this for you subconsciously.  Usually it does this very well. However, an unwarranted inference is when your brain draws a conclusion based on incorrect assumptions.  In many cases you are unaware that these inferences have become part of the context you are now operating on.  They are now limiting your thinking.

This is why a new member of a team can often 'see' the causes and solutions for problems better than those fully immersed.

Question:  Did any of you attempt the chain links problem?  If not (and you want to get smarter) have a quick go with the problem below.

### Chains and links

A woman has four pieces of chain.  Each piece is made up of three links.  She wants to join the pieces into a single closed ring of chain.  To open a link costs 2 cents and to close a link costs 3 cents.  She has only 15 cents.  How does she do it?
(I love this one... very satisfying)

Some helpful solution methods:

Draw pictures.  If this doesn't help draw it in a different way (change the representation).

If you are struggling with this it is because your brain is representing it to you in a certain way.
Probably like this:

If this is how you have drawn it then you may be suffering from an 'unwarranted inference'.  What are you inferring?

Looking at the picture you may be thinking that you have four points that need connections.  That's four 'opens' (8 cents) and four 'closes' (12 cents).

Look at the picture again.  What are you NOT seeing?

## Home Educator Options

Graeme Smith has been posting material lately that would be hugely beneficial to home educators with children in the high school years.  In my conversations with parents, they are often unsure of how to begin navigating through the accreditation process.  The world of NCEA and NZQA can be challenging to say the least.

Graeme is developing some templates that can be used to collect evidence for some of the literacy units.  I think they look great and we will probably use them.  Definitely have a look.

You can find Graeme's material here and here.  But I would suggest hunting around the posts for other ideas as well.

Graeme's most recent post discusses the minimum criteria necessary to enter University.  Well worth a look, and a good stimulant to thinking about ways forward for home educated kids.  Click here.

## How to win the 3MT!

Well, I don't really know how to win as I didn't quite manage it.  But I do have some tips.

But first - just entering the 3MT is awesome.  It is a great experience and one of the coolest things I've ever done.  I had fun, met a group of talented people.  And did I say - I had fun.  Do it, you will too.

The following post is purely my opinion on how to win.  I'll cover the basic mistakes people made and the things you have to nail to make it.

### Mistakes

The problem with PhD students is that they are all so smart and passionate they forget all about the audience's experience of the presentation. They tend to think this competition is about their research.  So let me say it clearly.  The 3MT is not about your research.  It is about connecting with the audience and bringing them into the world of your research.  Remember, the people putting on the 3MT want it to be a great show.  They want the audience to love it and come back.  So be entertaining.

First up are the common mistakes that I and others made.

## Number one: Not looking at the audience.

Lots of participants looked at their notes, at their power point, at their shoes, at the desk in front of them - anywhere except for the audience!

## Number two: Reading from notes

It doesn't matter how well your pronounce words, or how much expression you use-if you read you lose connection.  Do NOT take notes.  Ever...

## Number three: Not mastering the time limit

You somehow need to memorise the entire presentation while making it look and sound as though you haven't. This needs an explanation.

The time limit changes things.  It makes both you and the audience nervous.  The natural reaction was to speak too fast, following a script from memory.  This simply reminds the audience that a clock is ticking - they concentrate on the time and not on your message.  Also most speeches became a monotone humming noise that lulled the audience to sleep (much the same as reading from notes).  Do not talk fast.  The way to win is to make the audience forget about the time.  Most people I saw rushed through their talk (including me).  They lost all expression, timing, beat, pace rhythm and tone.  You are not trying to download information to the audience - you are taking them on a journey.

## Number four: 'Telling' us about your research

Don't tell us about your research- tell us a story.  Most 3MT contestants I saw told the audience about their research.  Here is the hard truth - after sitting through 2 or more 3 minute speeches the audience is bored senseless.  At this point ANYTHING different will be a welcome change.  Take us on a journey damn it.
It shouldn't be called the 3 minute thesis, it should be called 'the book you write after doing your thesis'.

## Number five: Butchering the presentation due to nerves

Being tripped up by nerves and subsequently missing your place, forgetting your next sentence or getting out of order wrecked many presentations.  The nerves are exacerbated by the time limit.  This whole competition is about managing the nerves.  If you screw up, recover quickly and press on.  A good ending will more than make up for a few errors here and there.

## Number six:  Busy power points.

If the audience is reading your power point they are not listening to you.  Five or more words is too many.  If you plan on using words - rethink it.  The one in the picture is pretty darn good (but risky with 8 words).  It sets the scene, pulls people in and doesn't demand that they exert mental energy working it out. I used three words - Aaron hates maths.

### To win

Make the audience forget about the time.  Be the most relaxed person in the room. This is your show, take them into your world and amaze them.

## Use discourse markers:

To help you keep your place in the talk use discourse markers.  These will help you know what is next.  In my speech I had a 'hook' halfway through.

"And that is why Aaron has avoided it [maths] like the plague.
[2 second pause, quiet voice, lean into the audience] "Until this year."

This phrase (until this year) was a signal to me that we were moving into the next paragraph.  The 'findings' paragraph.  I had increased the tempo up to this point and now had s l o w e d it right down and brought the audience in.  They want to know the next bit of the story!  Build and release, build and release - each paragraph should do this.  I had five paragraphs that made up the 3 minutes.

## Polish.

I think this is why I lost.  The winner was relaxed and polished.  Very nicely done.

## Humour.

The crowd is nervous for you.  Release the pressure with a joke.  Not one person had a single funny thing in their speeches except for one contestant.  She got a massive laugh.  All that nervous energy packed into one small moment of humour.

### Frequency and location of practice

We got 10 minutes on the actual stage to practice.  I had two runs - the first was horrible!  The second slightly better.  Two hours later the live version was better again.

Take every opportunity to practice.  Bribe your friends to stare at you while you do it.

## End

Win their hearts, take them on a journey, look smart, and be polished.  That's the winner baby.

## Are you a numeracy expert?  Updated...

In 2006 a numeracy survey was conducted in NZ (and around the world) to determine the distribution of numeracy skills in the adult population.

We didn't do so well...

Anyway, below is the toughest question in the survey.  If you got this (and a few others) you were likely to be placed in the 'expert' level (level five to the L&N folks).

Good luck!

### Problem

You deposit \$1000 into a term deposit that gives a rate of 7% interest per year, and you leave it for ten years.  Assuming nothing else changes, will your money double?

Well, guess what - only about 5% of the population actually managed to get this one.

### Solution

Here is what you need.

Where 'x' is the total and 'p' is the principle, 'i' is the interest rate and 'n' is the years.

x = p (i + 1)^n

Here is a little project for you.  Write this as an excel formula into excel and make it work.  Plug in the numbers above and see if it doubles (right now someones is saying, 'just give me the ----- answer').

### UPDATE:

Okay, here is how you solve it.

Remember Bodmas/bedmas?  They (depending on what year you went to school) define the order of operations. Brackets and exponents first, then multiplication and division, and finally, addition and subtraction.

Knowing this we can just plug the numbers into the formula and use a calculator.

p = \$1000
i = 7% = .07
n = 10

1.  So, brackets first = (i + 1) = 1.07
2.  Then exponents = 1.07^n = 1.07^10 = 1.97
3.  Then p x 1.97 = 1000 x 1.97 = \$1970

Now most folk get stuck on step 2.  Remember that with exponents you are multiplying the number by itself.  You do this ten times (once for each year).

1.07 x 1.07 x 1.07 x 1.07 x 1.07 x 1.07 x 1.07 x 1.07 x 1.07 x 1.07.  DO NOT do this on paper or in your head.  Grab a scientific calculator and use the  'x to the power of y' button.  All your phones have this.

Just enter 1.07, then push the button, then enter 10, and hey presto!  You have your answer.

Does \$1000 double in ten years at 7% interest? No.

Now, while this is a type of skill that we each need to thrive in the information age, it's your attitude that counts.  It's not just about being able to solve this one task, but rather possessing the courage, AND the skills to actually have a go at it.  Numeracy = skills and agency (your power and proclivity to act).

If you are happy to just let others do it for you, or to use an existing formula calculator - haven't you given up some of your independence as an adult?  Are you vulnerable in some sense?  That's what the architects of the question thought.  Do you agree?

Final question:  At what percentage interest point does your money double in ten years?

## How do maths/numeracy teachers contribute to YG learners lack of progress?

Heyd-Metzuymin is a maths teacher working with a class and conducting research.  One of her students particularly struggles with mathematics and is fairly typical of many YG learners (in my opinion).  This learner gets up frequently, fidgets regularly around in her bag, goes in and out of the classroom and talks 'at' other students loudly.  She also attempts to answer questions, (in a drama-queen type way), and generally takes over the class with her explanations.  She also has the wrong idea about almost everything all the time.

Now the article is exploring the notion of 'disabilities' and whether they are neurologically based or socially constructed.  I tend to sway more toward the social construction end of the continuum but do accept the role of neurological factors.

The teacher/researcher makes a good point about the way she herself treats her student.  Because the student (Dana) is such hard work the teacher limits the types of interaction she engages in with her.  She rarely asks her to explain her thinking, she never gives her harder work, she structures things for her and early on decided that lots of work on the basics was pretty much all they were going to cover.

Here is the problem:  The learner tacitly agrees to this.  It suits Dana fine.  From Dana's perspective, everytime she is asked to explain her thinking or describe something, she gets it wrong and ends up looking foolish in front of the teacher and fellow students.  So she is happy to just stick to the teachers plan also.

The interaction patterns between the two of them conforms to the 'instruction, question, answer' pattern.  In other words the teacher would explain a rule or process, Dana would ask a question and the teacher would evaluate Dana's contribution as right or wrong.  Then repeat, repeat, repeat...

This type of interaction never allows Dana to engage in the kind of discourse that will develop her mathematical thinking.  It is dull, boring and ultimately led to Dana making zero progress in the class. In other words it may not be Dana's disability that stops her learning but sociocultural factors that reduce her opportunities to 'act as a mathematician' and therefore forever be the passive learner.  (And to reinforce her identity as a person who is bad at maths)

Some learners are very difficult and you find that you position them in certain ways.  Our interactions with learners become 'routinized' and often they are not in the best interests of their education, but rather are an outcome of difficult personalities and classroom management styles.

Have you fallen into routinized patterns of interaction with any of your learners?  Is this in their best interests or simply a result of trying to make life easy?  And... does it make all the embedded numeracy in the world effectively useless?

Ref:
Heyd-Metzuymin, E.  (2013). The co-construction of learning difficulties in mathematics -teacher- student interactions and their role in the development  of a disabled mathematical identity. Educational Studies in Mathematics, 83, 341-368.

## Unwarranted Inferences

Back in the day, I used to know every one of the 'problems' that you get in problem solving books.  The reason is that I used to collect them and use them to teach numeracy.  I realised after a while that the problems were always of a certain type and easily solved.  In fact I used to love finding a new type because it would spark off new ideas etc.

My students would of course, try to find new problems and give them to me with the hope of stumping me. The thing is, I would always get them.  Now this is not because I am smart (believe me - that ain't it), it is because I have worked out the types and subsequently reduced the types of solutions possible.  Once you have twenty under your belt, you really need something a bit tougher.

### Why people fail to solve problems

Anyway - many of the problems that you see in workshops or in books are simply based on your ability to make 'unwarranted inferences'.  Unwarranted inferences are inferences you make about the rules of a problem.  No one states them, your brain just puts them in.  The designer of the problem has placed information into the problem structure or description to evoke certain inferences.  Your job is to overcome this.

Below are some well known examples.  Try and solve them and figure out what the unwarranted inferences are that the problem is trying to get you to make.

### 1.  The sixteen dot problem

Without lifting your pencil, join all 16 dots with six straight lines.

### 2.  Chains and links

A woman has four pieces of chain.  Each piece is made up of three links.  She wants to join the pieces into a single closed ring of chain.  To open a link costs 2 cents and to close a link costs 3 cents.  She has only 15 cents.  How does she do it?
(I love this one... very satisfying)

### Summary

If you want to master these things then think about 'how' the problem is making you think.  Then re-think the boundaries you have placed on your interpretation.

Remember - avoid making unwarranted inferences, draw pictures, keep cool.