Thursday 29 May 2014

Learned helplessness

Learned helplessness is one of the biggest problems in adult literacy and numeracy.  It is the creeping, silent, invisible enemy of educators (and parents).  Unfortunately  educators or parents rarely recognize its' impact and they rarely know what they are up against.  They just know that their children or their learners are not learning.

Let me begin be describing what learned helplessness is and how it began.

Story of two dogs

Once upon a time there were two dogs.  The first was hung up in a harness and subjected to a series of mild/medium electric shocks.  However, the harness was configured in such a way that if the dog touched a plate in front of its nose, the electrical current stopped.  The dog figured this out and when it was shocked it touched the plate.  It learned that it could stop the shocks.

The second dog was hung in the harness and also subjected to the same treatment.  However, this time there was no plate and no way for the dog to stop the electric shocks. This dog learned that it couldn't stop the shocks.  They continued this for some time with both dogs.

Now they took these two dogs and ran an experiment to see if they behaved differently under certain conditions.  They made a container with a small wall dividing it into two parts.  The dogs could easily step back and forth over the wall, but could not escape the container.  The floor on one side of the container was a steel plate allowing electricity run through it. Any dog standing on this side would receive electric shocks.  However, before the shock was administered a light would begin to dim warning the dog of the shock.

When the first dog entered the container it stood on the steel plate and received a shock.  It immediately jumped to the other side of the container and stayed there.  It quickly recognised that the shocks were preceded by the dimming of the light and moved as the light began to dim, therefore avoiding the shocks.  It repeated this behaviour in a variety of different containers - always responding to the dimming light and moving to avoid it.

The second dog however, was very different.  When this dog received a shock, it did not move.  It just hunkered down and continued to receive the shock - always, even when safety was one step away.  It did not respond to the warning (the dimming light).  It did not believe it could make a difference.

You see, the point is, they trained the second dog to be helpless.  They trained it to believe that it would get shocked no matter what it did - so it did nothing.

The question then turned to this: - Can the same thing happen to people?  Yes

Learned helplessness and maths 

Mathematics classes actually recreate the conditions necessary to develop learned helplessness.  The way they do this is complex.  However here is a simplified example, a learner begins to fall behind in a class, he stops understanding most content but does answer some questions correctly here and there.  The learner takes some action to improve.  They may ask a friend for help, ask the teacher to repeat the problem, go home and repeat the classroom problems and so on.  However, if the learner still continues to struggle they begin to get the sense that nothing they do actually makes any difference.  They sit in class and become bamboozled by the content, occasionally getting some things correct but generally failing repeatedly.   When they do get questions correct they attribute it to the work being easy, and when they struggle they attribute it to being too hard.

Sustained failure cultivates learned helplessness until it becomes self-perpetuating - at which point they are lost.  

The end result is that these learners stop trying the moment things get a bit tough.  They will give up trying in the face of any difficulty and relegate the content as too hard.

Key characteristics are expressed by this advertisement.

Anyway, I didn't want so much to describe LH but rather to give you a checklist.  The list below will describe it better than I can.  I have stolen it from a wonderful researcher named Shirley Yates (2009), who tested it to see how reliable and valid it is.  Use it to think about your learners, children, or self.  I am going to write this in the masculine to save time (no sexism intended!)  Think about your learner while they are doing maths.

  1. The student gives up when you correct or find a mistake in his work.
  2. The student takes little independent initiative; you must help him to get started and keep going on an assignment.
  3. When he fails one part of a task he looks discouraged - says he is certain to fail entire task.
  4. Tries to finish assignments, even when they are difficult (reversed).
  5. Does not respond with enthusiasm and pride when asked how he is doing on an academic task.
  6. When experiencing difficulty he persists for a while before asking for help.
  7. Says things like "I can't do it" when he has trouble with his work.
  8. When he receives a poor grade, says he will try harder next time (reversed).
  9. Prefers new and challenging problems over easy ones.
  10. Prefers to do easy problems rather than hard ones. 

The idea of these items is that you rate each one on a five point scale (1 = not true - 5 = very true).  The higher the score the more indicative the learner has LH.   However, I think that if any of these ring slightly true for you (regarding your learners) you need to act immediately.  Once LH is established the consequences are serious and persistent.

Look at the reversed statements (numbers 4, 6, 8).  These are what you want to cultivate in your learners/children.  In fact if you do - it is the greatest gift you can give your child.

If you would like to know more about learned helplessness - make a comment below.

Yates, S. (2009). Teacher identification of student learned helplessness.  Mathematics Education Research Journal, 21 (3), 86-106

Tuesday 27 May 2014

New findings regarding the relationship between beliefs and adult engagement in mathematics/numeracy

One primary aim of my research has been to identify how beliefs may act as constraints or affordances to learning. Affordances are things that help us to achieve things.  For example, door handles help us to open doors and therefore are affordances to opening doors. It's a word that doesn't see much sunlight outside of acadamia.

The term 'constraints' on the other hand is pretty easy to grasp. Constraints can also mean items designed to reduce movement and behaviour, and my use of the word is much the same but related to mental and physical behaviour.

My research now clearly demonstrates that a wide range of beliefs operate to constrain adult engagement in mathematics. These include epistemic beliefs and beliefs about mathematics, oneself, and one's relationship with mathematics.

New findings
My most recent finding identify that many (almost all) lower-achieving NZ learners dichotimise 'understanding'. That is they view understanding as an either/or state, a bit like a switch.  Either something is understood, or it is not.  Secondly, they believe that understanding is something that will happen in a single instant.  This is often expressed by them as 'getting it'.

"Sometimes I just get it, but when I don't get it I stop trying".

By 'getting' it, they mean sudden understanding.  They don't necessarily mean they will understand something on the first exposure but they do expect the content to just 'click' at some point.

Thirdly,  lower-achieving learners believe understanding is transferred to them by an expert, in a 'content packet'.  If they don't understand the first time, then repeated explanations of the content are needed. My participants only had one responses to this question -"What should someone do if they haven't understood what is being taught?"

Answer:  Ask the teacher to repeat it.

Classic non-agentic responses. Contrast it with higher achieving learners answers, "Oh, go and study it at home, go on Youtube, hit the teachers up, work through the problems till you start to see patterns".

The harm...
How might these beliefs hurt low achieving learners?  First let's quickly discuss how understanding really develops.  There are multiple theories, but they all basically say that understanding is created by the individual as they struggle to make sense of and incorporate new information into existing understandings.  New understandings are also thought to result from receiving information you do not understandthat challenges your existing understanding, causing you to have to reconfigure your existing ideas to accommodate the new ones (long sentence!).

In other words, new understandings require you to 'make sense' of information that you do not understand. It requires that you struggle and ultimately reconfigure your existing understandings.  This takes time and effort.  It is an entirely different process to hearing something and understanding it.

Do you see the problem?
This set of beliefs leads students to expect to understand information when they hear it.  If they don't understand they think there is something wrong.  Then they think they are dumb, and that others think they are dumb, so they disengage.  If they are persistent and motivated they ask the teacher for help.  By 'help', they mean repeating the information, again and again.  If they still don't understand then they are out of options (often blaming the teacher or themselves for not being smart enough).

By expecting understanding to occur in response to an explanation learners take a very passive role in their education, expecting the packet of content to match their existing understandings.

The truth is that understanding is a process, not a state.  It takes time, and it is constantly evolving.  There is no 'either I understand' or 'I don't understand', there is only coming to a further understanding.

You are in trouble if you believe any of the following:
  • Learning is about understanding what you hear at the time, and then applying it.
  • Not understanding what you hear, might mean you are not as smart as others who can.
  • If you don't understand the content you must get the teacher to explain it to you again so that you can understand. 

If you answered yes to more than one, then you have either been very lucky, or have some issues around education and learning.  Yes, the first one is what our 'training' sector is based on but has quite different goals than developing understanding. 

All of my low-achieving participants believe these three, and all my high-achieving participants don't.  That should get people thinking.

Beliefs are the foundation of the mind.  They determine how you interpret the world and what you deem important, and as such are the most essential factor in all education.

Monday 19 May 2014

The focusing of emotion into rage

Learned helplessness versus learned agency

I must confess, I was once deeply into power-lifting.  Power-lifting is the art of combining massive amounts of discipline with massive amounts of manufactured focused rage.  Again and again.

How does this relate to maths?  Well actually it was through power-lifting that I learned maths - both the concepts and the attitude.  Today is about - ATTITUDE!

Much can be learned and applied to the learning of mathematics and in particular remediating learned-helplessness (LH) from power-lifting.  This is because while students with LH are slaves to emotion, power-lifters are masters of emotion.

Learned helplessness is the destructive belief that nothing you do will have any effect on the outcome. You are completely helpless and dependent on others. Unfortunately learned helplessness is found within the maths domain - usually at the high school level.  Many young people simply struggle so much with maths that they begin to feel that one is either good at maths or not, that solving maths questions is due to luck or because the questions are easy, and that teachers are either good or bad.  They do not see where their own effort fits into the picture.  I'll do a full post on learned helplessness later.

But briefly:  Students with LH will attribute success or failure to anything but their own effort.  They will attribute math success or failure to:

  • the teacher
  • the content
  • the behaviour of other students
  • innate ability/disability
  • luck/bad luck
They will not attribute it to factors over which they have control.  They are perpetual victims of the mind.

Power-lifters need to manufacturer the complete opposite mindset.  They have to believe they can do the impossible.  They have to believe with all their mind, soul, heart and body that the power to move an immovable object exists within them.

Now keep in mind that while power-lifting is highly social, each person is competing directly against themselves.  That is, they are competing against their limiting beliefs, genetics, environment, access to training, work schedules, other commitments and so on.  They compete and challenge the very things that those with learned helplesness rely on - uncontrollable factors.  And you know what?  - they win.  I am amazed at what they end up doing.  Let me tell you, it is truly amazing what these guys and gals move.

So what can we learn from power-lifters and apply to students learning mathematics.  Here it is:  In the moments before a record lift most lifters engage in a psyche up session.  This is usually solitary, and during this time the lifter searches their mind and confronts the challenge before them.  They think about how hard this will be, how they will have to dig deeper  than ever before, find and exert more effort than ever before. They make that immovable weight before them a representative of every uncontrollable hurdle, hardship and struggle they have encountered.  They begin to manufacture rage out of emotion... not wild rage but very focused rage, positive rage.  And finally in those last moments they let it loose.

Here is what I see with adults who struggle with maths.  They have huge emotion, anxiety, frustration, and anger and it builds up quickly.  But instead of being focused back into positive determination it gets blown on outbursts of many kinds.  One of my learners got so upset he said he was going home to hit the bong.  All that emotion, that could have been focused got dissipated through drugs.  I imagine he felt much better the next morning with all that stress and frustration gone.  But that frustration was ENERGY and it should have gone into mastering the material that was making him angry.  He could have focused it into determination.

Power-lifters have a special mindset - It could be called 'learned potency' but really it is 'learned-agency' and it is the ability to harness emotion into energy and then into action.  It is the challenge between all the reasons this cannot be done, with the will to make it happen anyway.

How do we develop this skill in adult learners who have histories of learned helplessness?  And how do we cultivate it in the minds of our children?


I've been accused of being a little corny at times.  Well, here I go again.

Learning is joyful, it energizes you, it FEELS good.  Or - it should.

It would be strange if a process so vital to our species survival, our family's well being and our individual development was unpleasant, wouldn't it?  Let's compare it to eating, making children, and spending money! And yet among the people I talk to, the experience of learning is anything but.  It is generally described as hard, as work, as something from which someone needs a rest.

In my research I asked 150 adults to describe what maths is and how it is learned.  The funny thing is that if you substitute the word 'maths' with 'dentist' you pretty much can't tell the difference between the two!    Nobody likes it much and nobody really expects to.

I remember figuring out one night that what I had learned about linear graphs could be applied to the handshake question (see previous post).  That's right, I was playing with maths one night instead of watching TV and realised that one area of maths could be used to solve another.  I had transcended the text book (much like Neo in the Matrix). The feeling was - joy.  Not happiness, but a deeply fulfilled sense of contentment mixed with achievement.  Now the maths was not hard, anyone can make the connection (and do everyday), but making the connection is what feels good. What if we could spread this feeling?  What if this was the feeling everyone experienced on educational programmes?  The world would be very different.

The reason I bring this up is because I have made another great finding hidden away in my data.  It's a gem and it makes me feel good to have discovered it.  I'm currently basking in the afterglow.

What I have found is another way adult learners judge themselves as failures - heaping up the opportunities for a loss of confidence, anxiety and feelings of low self-worth.  Where is the joy?

How do adults beliefs' influence their cognitive and affective engagement with mathematics?  Well, another piece of the puzzle has dropped into place.  Perhaps by identifying it, we can begin to deal with it.

Thursday 8 May 2014

A classic

If you are in need of a good numeracy problem that really draws out learner thinking and sense making - this one has a pretty good hit rate.

I frame this one as a group of mathematicians attempting to track a deadly zombie contagion.

If eight people meet and each shakes hands with the other, how many handshakes are there?

Each handshake represents a point of contact.

The mathematical journey you take as you seek to solve this is a similar journey you would take if you tried to track the movements of a contagion through a population, a computer network, or sports betting formulas.

You will deal with triangular numbers and have to navigate through additive thinking and into multiplicative thinking.

Once you have the solution, take your system and use it to solve for 100 people.

The end result should be a nice formula that elegantly solves the problem.

I beef this problem up by introducing statistics (what if one, two or three people are contagious) and the contagion has a 100, 50 or 25 percent infection rate?  You can go on developing the thinking indefinitely.  Mortality rate, contagious period, time range within which the infected succumb.    

Good luck.  I'll reveal the answer and formula in a later post.

Friday 2 May 2014

The end.

Today I am cleaning out my office.  Out go all the sketched diagrams, the ideas jotted down on countless scraps of paper, the rulers, grids, empty coffee cups, receipts, USB’s and flipshares.  It marks a new beginning.

As of Monday I stop collecting data for the PhD, I stop talking to the very people I’m hoping to help and I stop thinking about teaching and learning and move entirely to analysis.  The data collection phase is over.  

To be honest, it was hard to see this day coming.

Over the last two years I’ve concentrated on seeing through a fairly ambitious project plan.   The plan required that I build relationships with organisations, tutors and learners and get inside the mind of each of them (primarily the learner).  As a tutor I know how uncomfortable it can be to allow a researcher into your private domain of the classroom.   A lot of energy was expended early in the project developing access to organisations.  In fact this was one of my high risk areas.  What if no one let me in?

Some more challenging aspects included talking to managers who don’t know me, and quickly articulating my values, purpose and needs within a five or ten minute conversation.  Their question is ‘why would I let you look around in my organisation?  I don't know you and you could be potentially destructive”.  I have to answer that question and demonstrate what I might offer them.  However, generally it has been down to managers recognising that both them and I share the same values and outcomes – to improve the lives of disadvantaged learners.  I never had a single manager say no and that is because I have never met a manager in this sector who didn’t share those values.

One of the most demanding parts of the research was walking into a classroom of suspicious adults ‘cold’, building rapport instantly, and then getting their consent to participate.  This was really tough and required clear messaging and frank honesty.  It also cost a lot of time and money.  I have driven multiple times to Auckland only to find three or four learners in attendance.  Or I have spent hours upon hours gaining access through managers, then tutors (each requiring a separate visit to Auckland), only to have a small number of learners available due to attrition.  However, the process has been wonderfully enlightening and I wouldn't change anything if I could.          

The project has included:
  •       Visiting 12 different educational organisations
  •       Speaking to the learners and tutors of 23 different education programmes
  •       Talking with over 200 adult learners
  •       Teaching a numeracy/math class to high need learners

I have also had the opportunity to interview many adult learners and hear their stories.  Some are simply tragic, and I have come away with a new dedication to my original mission statement.

Re-engaging adults with their dreams, passions and potentials through maths education.

There are large numbers of people who are emotionally wounded and hamstrung by their life experiences.  Education has a bigger role in this than you might think.  Our early experiences with education shapes our self-identity and the shadow stretches long.  I had a 57 year old lady tell me that the feeling she got sitting on the mat in primary school when the other children laughed at her, is the same feeling she fears in her current class.  50 years later and the scars are still having an impact.

My opinions have hardened in a few places also as a result of the research and I have to be sensitive about how I articulate them to the sector.  The education system is dysfunctional at a macro and a micro level.  The dysfunction runs so deep that no single change would correct it.  But the stakes are high.  

This week I interviewed a man whose early life was terrible through no fault of his own.  It stripped him of confidence and knowledge and has set his life on a path that simply isn’t fair.  It has been hard.  He is a causality of the system.  It’s not fair and I’m not sure how to change that.  I do know that my current influence in the sector is limited and I need to do something about that.

There is the upside.  People are strong, they have big hearts, they care, they have unique ideas and gifts and styles and talents.  And God loves the little guy. 

Thank you everyone who helped me with this project – there are some of you that I couldn't have done this without.  Thanks for trusting me, letting me in, and being part of the project.

From here on in I write.  The pressure is on to now to actually produce something.  The clock is ticking.

Thursday 1 May 2014

Profound Understanding of Fundamental Mathematics (PUFM).

This post probably contains the most powerful key to teaching and learning mathematics.  So if you are home educating, tutoring or a teacher who has control over what you teach then this post is for you.  The key to powerful teaching and learning is PUFM.

PUFM is the thing that separates the people who are good at math from the people who struggle.  It separates the kids who never quite 'crack' math, from those who always appear to 'just get it.'  It separates the tutors/teachers/parents who are great, from those who are average. It is the "difference that make the difference".

Why?  Well, first a little story... Liping Ma is a researcher who wanted to discover why Chinese maths teachers, despite being less qualified and resourced, get far better results than their American counter-parts. That's right, highly qualified American maths teachers (with a significantly larger budgets) get worse results than the Chinese teachers.

Have a think about how and why the Chinese maths teachers get better results?

Perhaps it's culture, family pressure, better behaved children, a natural affinity for mathematics?  It isn't.  While these account for some small variance none of these explains the quite significant difference.

The answer lies in PUFM.  It stands for a 'Profound Understanding of Fundamental Mathematics'.  It refers to not only knowing how to 'do' maths, but profoundly understanding the workings of mathematics.  Not necessarily understanding higher level mathematics, but how the number system works and how methods work and why.

The Chinese teachers, although less qualified, had a PUFM and they used this to develop a parallel understanding in their students.  In one example Liping Ma examined the difference in how the Chinese and Americans taught their students to 'carry' when using an additive algorithm.  The American teachers showed the students how it was done using a blackboard and then had their students practice the method by solving additive problems in their workbooks.  They did about 30 additive problems every lesson.  That's about one every minute and a half.

In contrast the Chinese teachers taught their students why carrying worked.  They had the students use equipment to recognise that you could not have more than nine units in a single place, and therefore could carry a ten to the following place.  The Chinese students did an average of THREE problems a lesson but explored those three problems in depth.  So they spent an average of fifteen minutes per problem.  

You see, the Chinese teachers were experts at getting their students to understand why carrying worked.  The Americans were experts at 'getting' their student to carry. Understanding fundamental mathematics is necessary to continue to understand mathematics.  Unless you understand mathematics you will not be able to use your knowledge to learn more.  You may be very good at solving 'canned' problems but you will struggle to learn new maths and apply your skills to unique situations.   

So, the trick is to begin to incorporate more of the Chinese philosophy into your current practice.  Less of asking students to practice solving similar problems and more digging down into the workings of mathematics.  Make the hidden workings of math methods explicit to your learners and you will begin to see drastic changes.

A project for you - if you wish to accept the mission!  Why and how does the formula πr2 actually work (that's 'Pi R squared')?

Once you know - then you will begin to have an understanding of its underlying mathematical workings.  A little hint is found here.

PUFM is a journey - I began it a while back and have loved every minute of it.  The joy of maths is not only found in solving maths problems that no one else can solve - it is found in understanding how simple things actually work.

Long live the joy of maths.