The power of nerves!

Okay, so here is a quick talk I gave a few years back.  I had never been more nervous in my life and in many places my words came out backwards.  Please know - I have gotten better!

But, for you viewing pleasure.  (I come on about one minute in).

Long division!!!

Oh boy - If I had a dollar for every person who has complained about not understanding long division  in school - Well, I would have at least $23 by now.

Long division seems to be that thing that defines our school maths experience.  For those of you who 'got it', you felt good, gained confidence and progressed.  For those who didn't quite 'get it', you felt slightly worried, unsure whether you would be able to understand the next bit (and hoping long division was not pivotal to future learning).  For those who never understood a bit of it - we knew we were screwed.

Let's revisit it, and see what all the fuss was about.

But first, have a go at the problem below and try to use long division.  Don't worry if you have forgotten, just have a go - all that knowledge is sitting in your brain ready for action.  Just begin, write down the numbers, put it in the format you remember.  It'll come back.

The problem is a famous research question.  

An army bus holds 36 soldiers.  If 1128 soldiers are being bused to their training site, how many buses are needed?

Mythologysing the 'Aha' moment

I've spent the last few weeks working through some data I collected on learners' beliefs about quick learning.  Here is a brief summary:

Learners tend to believe that 'understanding' occurs in a single moment, usually in response to listening or watching a tutor demonstrate or explain some aspect of numeracy.  They say things like, "Some people just 'get it', but I usually don't", or "I love it when I get it".  This notion of 'getting it' permeates their thinking around mathematics and numeracy.  And it's all bad.

If a learner believes that understanding happens in a 'moment', then when they do not 'get it', they may begin to doubt their ability to understand it at all.  Often, they will ask the tutor to repeat the content, example or demonstration, or they may just ask the tutor to 'show us again'.  They hope, and expect, to 'just get it' and believe that they ought to be able to do so.  When they cannot (and see others getting it) they often use this as evidence of inability.

The truth is that understanding can on occasion happen quickly, but only when the groundwork for understanding has been laid.   Understanding is a 'process', not a 'state'.  It takes time and effort. Often it emerges from an extended period of confusion.  True mathematical understanding develops over time, not in response to someone telling you something.  Yes, there are moments of insight, but these are the conscious outcome of temporal subconscious processes.

If learners believe that understanding ought to happen quickly, then they are set up for failure and negative affective responses.  For example, if Kelly believes that she should understand the concept of ratios as the tutor tells her (in that moment), and she doesn't, then she may believe that she has a mathematics problem.  She may then give up, and reaffirm her belief that she is no good at mathematics.

Add to this, the learners lack of strategic learning repertoires, and their subsequent reliance on tutors, and it becomes clear that for these learners, listening to the tutor and hoping they 'get it' is the only option.  This means learners who do not 'get it' have to ask the teacher effectively limiting their learning opportunities to periods of tutor instruction.  In an adult numeracy classroom in NZ, tutor responses to learners questions are abysmally small.  Learners don't ask for the tutor to repeat information because it reveals to the rest that they don't understand.  And that noisy learner?  You know the one that asks all the questions, all the time?  Well that learner isn't asking the right questions, and yet tends to dominate the tutor.  We have the conditions for a perfect storm.


Finally, and to the point.  How many tutors talk about the 'aha' moment. In particular, how we feel good about our roles when learners suddenly 'get it'.  We may in fact be mythologizing a negative meme. The 'aha' moment is a passive response to a usually accidental delivery of content.  Instead, we should be talking about the learner we motivated to go home, and spend hour after difficult hour, working on that confusing concept until they finally began to make sense of it.  That would be a real inspiration.

Mathematical understanding is the result of hard work, time, effort and often confusion.  Learners who think that this experience means they are dumb, are not going to persist for long.  It's a bit like a hopeful marathon runner who interprets discomfort as a signal that they are no good, - because all the good runners do it so easily.

Questions:
Do you think understanding happens quickly?  Or gradually over time?
Does it make a difference?

For parents of high-achieving girls

Girls are consistently out performing boys across all areas of education.  The advantage that boys had in mathematics has largely vanished, and girls have always enjoyed an advantage in the areas of language.

However, one area that girls do have difficulty with is at the higher levels of education, particularly high-achieving girls.  What seems to be happening is that girls achieve well, and expect to achieve well.  However, the constant success may not develop the tenacity, or grit, that is required at higher levels of education, particularly in maths.  Hence, when the learner fails, and they are not used to it, it hurts.  Often it damages, or collapses, the learners self-belief.  Some of these learners have not experienced 'fighting out of the hole', and instead feel defeated and as thought they have reached their limit.  Many brilliant people quit.

Here is a sentence from Carroll Dweck that should get your attention, "What we found was that bright girls didn’t cope at all well with this confusion. In fact, the higher the girl’s IQ, the worse she did."

Contrast this with the average achieving learner who has scratched and clawed their way through, constantly fighting to stay in the game.  They may have developed resilience, persistence and grit -which count for so much at higher levels of education.

Of particular note, is the impact of beliefs on how learners interpret and respond to failure.  Your beliefs as a parent will strongly influence your children, positively or negatively.

Anyway, the link below is to a great article by Carol Dweck.  If you are a parent of daughters, it is well worth a look.

Click here for - Is math a gift? by Carol Dweck

If you are too lazy to read, here is a TED talk by Duckworth about grit.  A great message.

P.s the Duckworth set up at the TED talk is a bit weird right?  She has people in a circle around her, and no one in the audience is actually moving or making eye-contact.  Looks a tad awkward!

And on a lighter note...

Mass enrollments, qualifications and tax-payer funding.

Well, the Herald reveals the investigation will roll on.  I suspect that it may go for some time.

It would be a good time for organisations to run a few 'mini' investigations themselves.  So far, no organisation has been found to be systemically at fault.  What has been revealed are pockets of poor practice, or 'very dubious practice', that are allowed to develop due to poor oversight by the organisation.

An old friend of mine used to say 'if you expect, inspect'.  I always hated that saying, but perhaps he had a point.

Financial literacy

49% of Meridian shares (MELCA NZX) were released to the public by the National Government about a year ago.

The shares were valued at $1.50 each.  However, in an effort to sweeten the deal, the buyer only had to pay $1 per share initially.  In eighteen months (I think) the owners of the shares are obligated to pay the remaining 50 cents.  In the meantime the owners of the shares still receive the full dividends, so these can help offset the remaining 50 cents.

It was a good deal

Many New Zealanders who had never bought shares bought these.  They are now faced with an interesting mathematical question.  Next year they will have to pay the remaining 50 cents on each share they purchased.

There are two choices as far as I can see.

One: Pay the remaining amount out of cash reserves (savings, or whatever)
Two: Sell off existing MELCA shares to the same value as that owing.

But which is the better option? 

I have no idea...  Trying to think this through...

Lets say you own 10,000 shares.  This means you have paid $10,000 so far and you owe $5000.  The share price is currently 1.73.

Option one: If you pay the $5000 out of cash you are getting a 23 cent discount.  Right? (tell me if I'm wrong please!).  You are still only paying $1.50 for shares currently worth $1.73.  As soon as you buy them presumably the price shifts to $2.23 ($1.73 + .50).  In total you have paid $15,000 and the investment is worth $22,300.  So $22300 - $15000 = $7300 Cap gain.  Or a 48.6% increase.

Option two: You sell 2890 shares for $5000 (at the current price).  This leaves you with 7110 shares that are now worth $15855 (at 2.23).  In total you have paid $10,000 for 7110 shares and the investment is worth $15855.  So $15855 - $10000 = $5855 cap gain.  Or a 59% increase.

Option two looks better right.  But, you are actually giving up valuable shares, worth $5000 in real terms.  And, you are giving up the dividend.  And any capital gain.

Two questions: 

1.  I am making a mistake.  Can you see it? (something to do with 49% or 59% gain of what?)

2.  Where is the financial resource that New Zealanders can go to to get the answer?  (Website?) Or is this a case of 'Mum and Dad' investors not fully benefiting from the sale of Meridian because they lack financial literacy?  The political left told us not to buy (a mistake), and the right said to buy,  but then have not provided information (should they?).

Please, someone in the know feel free to enlighten us. I would really appreciate some feedback on what they think.

Financial literacy - a quick critique

Current financial literacy initiatives tend to focus on issues that were a concern of the 90's, like basic consumer skills. I find these very condescending and short sighted.  2014 presents a range of more complex financial thinking that requires a new set of skills.  Someone will say that the poor don't buy shares.  Granted, but neither does the financial literacy of yesteryear attempt to develop wealth - its focus was primarily about meeting immediate needs - not wealth creation.  Basically how to live poor but still pay all the bills.  Not good enough.  

Time to think a bit more strategically

Ninja math

Talk at our house lately has been about inspiring young people to persist with challenging maths problems ALONE, only asking for help once they have exhausted all other possibilities.  In other words developing 'agency' - the ability to act.

Non-agentic behaviour is getting stuck and immediately asking for help.  Research shows that many learners lack the emotional skills to solve a task once they get stuck, or hit a wall.  All they can do, is ask for help, and hope they get it.  Hence they become helpless.  (see here and here for learned helplessness)

Not so the true mathematician - the true mathematician loves hitting the wall, because he/she can dig into their agency, their ability to ACT, to think, experiment, play, to solve and conquer that problem in a unique way.

The trick is to cultivate these dispositions.  Young people however, often have trouble knowing what to do when they get stuck.  I've been thinking about how to help them develop the skills.

Anyway, below was a poster I designed to be used with 10-13 year olds.  It's a draft, unfinished, but if anyone wants to take the idea and make it rock, feel free.

Tips for home educators teaching maths

Three tips for developing maths skills in a home-education environment

No 1. Before any math work begins ask you child what they did yesterday (with maths). Have them explain as much as possible, in as much detail as possible. This helps your child develop a sense of continuity between math lessons and work. It also has been linked to significantly better learning outcomes. There is some very cool research around this (another time). Suffice to say this one difference is linked to significant increases in learning outcomes.

No 2. The pokerface. Whenever your child gives you an answer to a problem NEVER let on whether it is correct or not. Hear their answer and then say “tell me how you worked it out”. As they talk through their thinking process they do two things. Firstly, if they were wrong in the first case they may self-correct. My Masters research found that learners often self-corrected as they explained their thinking even when they didn't realise it. Secondly, the process of articulating their process will help clarify and consolidate their thinking. Finally, be sure to reward thinking and effort, not correct answers.

No 3. Use equipment as much as possible and where not possible have your children draw pictures. The nature of maths is that it becomes more abstract as it progresses but the human mind passes through stages before this is possible. Abstract thinking MUST be built on a foundation of empirical knowledge. For example if you teach a half plus one third equals five-sixths without your child being able to get a sense of what this really means or looks like in time your child will struggle to make sense of new concepts. Getting children to draw pictures to explain their thinking rocks. You can then use those pictures with your children – asking them to explain what they have drawn and why. Also, save those pictures and pull them out six months later and ask you children to explain what they think was going on.

Second best question you can ask learners when teaching mathematics

The number one greatest question in  your math/numeracy educational arsenal is here.

The second is the following question:

If somebody was going to make a mistake with this, which part would they most likely make the mistake?

Get them to verbalize it, and be very clear.

There are two good reasons for this.

One, the part they identify will most likely be the very part they struggled with.

Two, speech is the key to learning,  By getting learners to talk thorough the problem they are actually processing the information.

Easy or hard?

A x A = BC

BC x BC = DEC

Which digit does each letter represent?


Level two GCSE problem.

Technology, Environment, Mathematics & Science (TEMS) Symposium Abstract

Done, and dusted.  Sorry about the quality - but if you are interested here is how the PhD is shaping up two years in.

How to set up a system of observing tutors that is safe and effective

A previous post raised a range of issues that relate to effectively implementing and evaluating L&N provision.  This post is designed for the person in a professional development role tasked with overseeing the implementation of ELN within a medium to large organisation.  A position I have held.

This position is tough as the role requires bridging two different worlds.  On the one hand, management, and on the other, tutors.  As such the role requires that you take on-board the unique stresses and demands of each domain without ever being a complete part of one or the other. This is partly what makes this role essential in an organisation as it enables a unique perspective on the demands of the roles while avoiding the extremes of both.

One of the key tasks for the person in this role is to observe and provide professional feedback to tutors and work toward meeting performance outcomes set by the organisation.

Beginning tutor observations

Tutor observations are essential to improving organisational performance.  They are also a key tool for cultivating an internal culture of constant improvement.  Yet, as a tutor there is simply NOTHING more intrusive and threatening than being observed.  The solution is to set things right at the beginning and stick to it.  I will describe some key ways to do this below.

Selection of the observer

Managers take note:  The person you select for this role (and observations) must be an experienced, trusted and respected staff member.  If this is not the case, abandon the whole idea now.  If you are not 100% certain you have the right person, have the tutors select their own person from the staff.  Or if it looks like the whole thing is too difficult, distribute the task to all the staff and work through every staff member.  That is, every tutor will have a go at observing other tutors and giving feedback. This has proven to be effective in the past.

Change the notion of 'observing' 

 Assuming you are this person, here are some ideas to help reduce tutor concerns regarding the observations.  First, change the connotation many tutors have with observation.  That is, the top down quality control view.  Observations are done to provide another perspective on tutoring practice in order to develop skill, not to check up on how tutors teach.  What helps is to create a divide between the findings and management.  There is an easy way to do this.

1.  Have a meeting with the tutors and tell them what is happening.  Make sure you tell them that this is to improve performance and provide positive feedback.  Note, that they won't care what it is for, they will likely simply view it as threatening.  Don't say anything else -  just ask for their concerns. Then just listen.  

2.  Address those concerns.  Most will relate to security and vulnerability, and the possibility of bad performance reviews.  You are adding to their workload and stress.  You MUST address these concerns. Here is how:  

Confidentiality

1. All the observation feedback must be confidential and anonymous.  That is, the feedback will be only between the tutor and observer.  

2. The feedback from all observations will be thematised and summarised into a report that management will see.  This report will present themes that CANNOT be traced back to any one tutor.  It is broad and not specific to single tutor performance.

3. This report will be sent to all the tutors first and if they have any concerns they can talk to you privately before you send the report to management.  Give them all time to read it.  Make any changes the tutors want.    

4.  Positive feedback.  In my experience most tutors expect this to be an unpleasent experience- it should be incredibly encouraging. Make it encouraging and rewarding for the tutor.  Give great feedback that specifically identifies the things that are really good at.  How they handled difficult learners, how they introduced certain content, their rapport with learners.

As for constructive feedback, simply ask them.  "What do you think you could have done to improve the lesson?".  Let them talk.  Give no constructive feedback.  None. Nadda.  Thank them and reiterate the positive feedback.  Promise to send them their feedback form and make sure you do within one day.

At the next observation, however (one, two,  months  later) during the feedback, read back to them what they said last time about room for improvement, and ask them how they addressed what they had mentioned.  Let them talk.  People will quickly realise what's happening.  Ask them again what areas they would like to improve and then repeat again in a month or two.

The simple fact that you have put in a system that gets tutors  to state where they may need to improve, and that it will be regularly talked about, will get tutors taking action. And... tutors will begin to enjoy this process - once they realise it is safe, and not a system to beat them with.  It is a time of validation and reflection.  There is no stick.

What you are trying to develop is a culture of constant improvement in which tutors do not fall into ruts and can make change.

Use the second themed report to show that tutors had identified and acted on key areas.  Ask them for feedback and then send it to management.  The first three reports will not really any use for real measurement, but they are a way of implementing a process that serves the tutors and the organisation.

In time the reports will indicate key areas that might require training or PD.  Themes will emerge (say classroom management). Once you have a pattern you might decide to bring in a classroom management specialist.    It is also a great way to remind management of what is happening in classrooms.  They will like this because once the tutors trust the process and don't dress up the results then management have an extra mode of communication to inform their thinking.

In my experience, observing tutors is wonderful for boosting morale and getting a buzz going.  Once you have told tutors it is happening do it quickly, with a day or two.  Otherwise they worry and they shouldn't.  Make the experience positive.  They will tell the others and word will spread that it is good.

Last thing:  we all know that we prepare better for the lesson we are being observed on and hence it does not reflect a 'real' session.  That's okay.  You see, hopefully, the extra time the tutors put into that lesson will ensure a good class, and generate new ideas and behaviours.  It will pay off.

The inquiry 

McDonell:  I'm going to be asking you questions pertaining to two different but related areas.  The first is to gain clarifications regarding your organisations' definition of embedded literacy and numeracy and the process you took to verify that this aligned with your funders' definition.  Second, what actions you took regarding your quality control measures for your organisations' embedding of literacy and numeracy into your level one and two programmes.

CEO: Understood.

McDonell:  Please explain what your Organisations' understanding of embedded literacy and numeracy is?

CEO:  We have experts who are well grounded in the details.  My role does not require in-depth knowledge of ELN, only that systems are in place to ensure it is.  My understanding of ELN is that literacy and numeracy delivery is integrated into programme delivery.

McDonell: Are you aware that the TEC had released (some time ago) a document that defines its' high level expectations for embedded literacy and numeracy?

CEO:  I was not.

McDonell:  What efforts did your organisation take to ensure it's understanding of ELN was correct to ensure compliance with funding criteria that you were receiving?  

CEO:  We have specialist staff members who's role it is to stay updated.

McDonell:  And it was these, or this, staff members role to ensure the entire organisations' level one and two programmes were aware of the definition and were compliant with it?

CEO:  Yes.

McDonell:  And you assume your staff are aware of the definition, and subsequent expectation for their provision of ELN?

CEO:  Yes.

McDonell:  Well, we are compiling their responses to a questionnaire, and interviews,  as we speak.  We can review the findings tomorrow.  For now, let's continue.

McDonell:  Can you please explain your organisations' process for determining the quality of your embedded literacy and numeracy provision, or even if it was occurring?

CEO: We acquisitioned a staff member to implement ELN across the organisation and ensure tutors were embedding literacy and numeracy.

McDonell:  Can you explain to the house the criteria for selection of the staff member for the position?

CEO: The individual had experience with ELN and was an experienced staff member

McDonell: In what way?

CEO: Sorry?

McDonell: In what way were they experienced with ELN?

CEO:  Ah, they had delivered literacy and numeracy.  And they had been to a range of workshops.

McDonell: In what capacity had they delivered ELN?

CEO: I'm not sure I understand the question?

McDonell:  How do you know they were qualif... I'll rephrase.  What were your organisations' appointment criteria for this position?  For example what qualifications did the position require?

CEO:  I'm not sure.

McDonell:  Can you at least tell us what the position description defined as qualified?

CEO:  I can't.

McDonell:  So your appointment of the position went to an individual that did not meet any criteria other than having experience with literacy and numeracy that cannot be defined?  We don't really know what they did do we?  Or their quality? You see the problem don't you?  How do we know they didn't get their previous job based on having experience somewhere else?  We really need some criteria of their expertise that is not simply holding a role.

CEO:  They are a trusted staff member.

McDonell:  Now this individual was tasked with determining whether literacy and numeracy was being embedded and whether it was of  a sufficient quality to warrant accepting funding for it?

CEO:  They were tasked with making sure it was occurring.

McDonell: Not quality?

CEO:  Well yes, PD was provided to tutors.  Tutors were trained.

McDonell:  And the outcome of that training was?

CEO: To embed literacy!

McDonell:  Yes, and I am asking you what internal systems were in place to ensure this was in fact happening and that the quality was of a sufficient level to warrant receiving funding.  Did your staff member report to the organisation on the quality of provision?

CEO:  We use the TECs' Assessment Tool to determine the effectiveness of ELN provision.

McDonell: So the staff member had no mechanism to report back on quality?  Did the staff member actually review quality?

CEO:  As I said we have an extensive assessment process.

McDonell:  So I will assume that despite vying for and receiving funding that required quality ELN provision you had no internal mechanism for determining whether or not literacy and numeracy was being embedded in programmes, or that the quality was sufficient, other than your Assessment Tool results?

CEO: Correct:

McDonell: So there were no tutor observations, no interviews, no student interviews, nor internal reviews?  How do you know your tutors' were making any change to their practice, let alone doing a good job?

CEO:  I trusted my staff member to tell management if it required attention.

McDonell: But you will admit no formal mechanism existed?  Despite an extensive shift by the NZQA toward internal evaluation systems?  Out of interest did the staff member ever give an indication that tutors were not meeting the requirements of the funding?

CEO:  No.

McDonell:  That seems extraordinary given your present situation.  Surely you see that you would not be here today if things were operating as you presumed?  I'll move on.  The data from your Assessment Tool results have been largely static over the last four years and suggests that you have failed to improve the quality of your provision each year. What does this suggest to you?

CEO: Actually our results show that many learners do improve their literacy and numeracy.

McDonell:  I'm not talking about improvement within a given year.  I'm talking about a trend of improvement.  One year compared to the next.  Yearly improvements suggest your organisation is improving.  Your results do not.

CEO:  That is due to a range of issues in the sector such as the nature of the learners' skills, attrition rates, transience, social issues and a raft of ever changing TEC funding criteria.

McDonell:  Do you think that it is reasonable that a tutors' performance will improve the longer they are in the job?

CEO:  For the reasons I mentioned earlier, not necessarily.

McDonell:  In every other educational domain learner outcomes improve as tutors gain experience. Please explain to the house why this is not true in your organisation?

CEO: Tutors are busy, they have demanding students, the schools have failed and we have to meet unreasonable outcomes to continue in business.

McDonell:  Then it would seem reasonable that your organisation should have exceptional evaluation and training systems in place - particularly in funding sensitive areas such as embedded literacy and numeracy?

CEO:  We do have evaluation systems.

McDonell:  You made the statement that the assessment tool is your primary measure of ELN?  Two questions.  What other measures are you using to determine the quality of ELN?  And second, did the results of the assessment tool suggest in any way  that the PD you speak of improved learner outcomes?

CEO: In answer to the first, as I just said, no, the Assessment Tool is our measure.  As to the second improving tutor performance takes time.  We need time to bring the staff up to speed, and we are doing this.

McDonell:  The truth is though, isn't it, that because you haven't implemented systems to inform you of the internal provision of ELN that you don't actually know the answer to either of those two questions?

CEO: Not at this moment, no.

McDonell:  Tell me if you disagree with my summary.  You selected a staff member based on a trivial notion of experience.  No criteria existed for this role despite this role overseeing significant funding streams deemed priority areas by the Government.  Second, no formal mechanism existed for this individual to feed back concerns to management regarding quality.  I suspect the individual was left without support or direction from management.  Third, this role did not include a quality control aspect in any 'real' sense. For example, the person did not observe tutors and evaluate performance in any way.  There were no tutor observations, no interviews, no learner interviews, no learner feedback mechanisms regarding literacy and numeracy provision.  If this did occur, it happened in a laissier-fare way.  Your organisations' sole method of evaluating whether tutors are embedding L&N, and the quality of it, is the Assessment Tool results.  Yet these results have not been interpreted in any meaningful way.

I fail to see how you would become aware if no real change occurred in any class.  Would you say you have been negligent in your approach to embedding literacy and numeracy?

CEO: No

McDonell: Oh boy.  Okay, the next questions will relate to your knowledge of your tutors' practice, in regard to ELN.

Invisible numbers, ratios and the eternal cement/concrete conundrum: Musings (part one)

Travelling across the country the last few weeks as a part of the workshop series that I have helped deliver, I have encountered many interesting, strange, and useful things.  The next few posts are a few musings about various things I have learned/observed.

I'll start with the least interesting and move to the weird and wonderful.

First, I learned quite a bit about how adults approach ratios.  A key finding is that adults are largely unaware of the need to identify the 'relationship' that exists between two or more quantities.  I have begun using the term 'invisible number' to create some interest about it and basically 'pizzazz' it up a bit.   

Did you know that being able to recognize the invisible number is a key part of being a 'proportional reasoner'?  Being a proportional reasoner is highly desired as it allows you to do a whole lot of things you otherwise would struggle with.  

Lamon (2006) has estimated that 90% of adults are NOT proportional reasoners despite this being a year seven requirement.  That's right - you should have learned this when you were about 12!  Not being so essentially means that you will be forced to learn algebra by memory, not as a meaningful system, and subsequently struggle and perhaps decide that you don't much like it and would rather play with a ball.  

Below is a power point I used to help make the point.  The participants were asked to make various ratios using Postit notes.  I showed them the following and asked them to model it with the Postits and discuss where else these ratio may apply.

Then I increase the quantities on one side, while removing the other.  They had to determine the amount necessary to add or subtract to ensure the correct ratio.  For example in the Power point below you need to work out how many  Postits would be needed on the right side to complete the ratio.

If the ratio of tan to grey Positis is 2:3 and you have 4 tan Postits, how many grey ones will you have?

Or, imagine you are a caterer.  You are providing one savory and three sandwiches for each person.  The ratio is 1:3.  If you have two savories, how many sandwiches will you need?

See below for the answer.  The tutors would have discussed this and modeled it with real Postits on their desks.

After a while we discussed the 'invisible number'.  The invisible number is the relationship between the two quantities.  However it became just as useful to talk about the invisible number as 'any' relationship (the mathy's out there will know this as invariant relationships).  In the case of the ratio above (2:3) the invisible number is the relationship between the 2 and the 4 tan Postits.  This relationship is double or multiplied by two.  Once this number is identified we simply applied it to the other side.  3 doubled is six.

For the 1:3, we simply doubled our amount, hence double the other side also.

Eventually we ended up here.

Proportional reason according to Lamon (2005, p.4) includes "detecting, expressing, analyzing, explaining and providing evidence in support of assertions".  The tutoring workforce need to work on this to be sure learners are able to develop these skills.

I'm not explaining it too well but the 'invisible number' idea has legs.  I'm going to develop the idea and phrase some more.  Lets see where this takes us.

My next musing will be about the concrete/cement paradox that you never knew about or cared about.  Until now...

Self-regulated learning

The single biggest factor in learner success is a learners ability to self-regulate their learning.  It is the exact opposite to 'learned helplessness' that you should read about here.  Learn about it, understand it, and you will become a better educator.

Of course, self-regulated learning (SRL) is not easy to develop in learners particularly learners who have been beaten around by the educational sector.  Also SRL can be broken down into many sub-skills and dispositions that are generally developed separately but used cohesively.

I used to think that SRL was the secret to developing the potential of adult learners with problematic learning histories.  I used to think it was the answer to the literacy and numeracy problem.  No more embedded ELN, no more literacy or numeracy provision - just SRL.

I now realise that sitting below SRL are belief systems that dictate SRL - beliefs are the mainspring of downstream effects.

As such the form below is no good on its own, even with an extensive PD package that was almost developed and ready to go.

Moving on.

One of the problems with mathematics teachers and resources

I'm playing the devils advocate here - so bear with me.


I've been looking at the age problems used in all algebra courses.

For example: Mary is three times older that Dave.  In 12 years Dave will be one year younger than twice Mary's age.  How old are they both?

I know someone is going to ask, 'why is this useful?  Couldn't we just ask them how old they are?'  And they will have made a reasonable point.

So I Googled, "Why are algebraic age problems useful?" and got:

The purpose of age problems is to determine the age of the people in the equation.

Really?  So the purpose of driving is to drive?  The purpose of eating is to eat?  The purpose of learning to count is so you can count?  Crikey.

Mathematics educators need to provide FIRST and foremost, the reason that this is useful and give practical examples.  The point of learning is not to help facilitate more learning.  Okay, in some cases it may be (developing automaticity for example) but not in regard to reasoning.

But what's the big picture baby?  What's the end game?  Why should I learn algebra?  When will I ever (!!!!) need to calculate someones age based on the crazy data provided?

Now, I can answer the 'why do I need algebra?' question as there are solid reasons (another post). But I'm still struggling with the application of the age problems.

Someone help me.  Give me a real example of the usefulness of the type of mathematics used in age problems in the real world.  Please.

Thoughts on numeracy 'in the real world'

Several weeks ago I asked this question: How do you write one-half as a ratio?  It started a few people journeying into the world of fractions and ratios.

Although it appears easy, it isn't.  Have a think about it, and most of all, 'prove' your answer, no matter how sure you are or how simple you think it is (no, it is not 1:2).

I have been delivering workshops and I have asked all the participants across the country  the same question.

They all all struggle initially, and finally crack it.  But here is where its interesting.  They all think its easy and yell out the 'answer' 1:2.  It isn't 1:2 and its isn't 2:1.

Here is the second interesting thing.  Before I give the question the groups tell me how important ratios are and how they use them almost every day in real world tasks.

They must be wrong, because they all hold the wrong beliefs around ratios, regarding how they work and what they represent. They do not have a conceptual understanding of ratios and therefore have not been directly using them in any of the examples given.

Are we as tutors in danger of mythologizing some aspects of numeracy?  I believe we are.

We definitely need strong numeracy skills, and more than ever, but do we really recognize where it is hiding in our lives?

Food for thought.

Work!!!

The posts have been a little slow as of late because I am currently co-delivering a series of workshops across the country.

So far we have been to Dunedin, Invercargill, Christchurch and Auckland.  What a great group of tutors in each of those places.

There are some fantastic people doing some fantastic things out there.

The challenge has been won!

Well, I had a huge response from the math challenge last week.  Well done to all of you who had a go at the problems.  Lots of people emailed through solutions to the three problems and everyone was in the ballpark.

The problems were tough, but we have a winner.  One young chap worked really hard and managed to solve all three problems.  Tickets will be sent this week.  Again, well done.

There were people all over the country working away on them.  In particular, problem one, the car and the truck problem.  I will be posting in-depth solutions to each of the problems and showing how others tried to solve them.  You will be amazed at the different approaches.  This will be later this week unfortunately.
 
However, three quick points.

GRIT

Grit is your ability to work through frustration and stay on task.  The car problem requires grit.  It simply cannot be solved just by throwing a few minutes at it.  You MUST engage.  For those of you who did.  Well done!

Working with the unknown

Here is a secret...  'Learning' requires working in the 'unknown', the 'confusing', and the painful.  This is why so few people are great learners.  You must become comfortable with these three.  To learn maths you have to be comfortable in this zone.  This can and must be done.  For example, any exercise or sport is painful, but people love it!  Making a new PB in the gym, running faster or further than you ever have or pushing to win that game of badminton or squash - all painful but fun.  Some people never realise how awesome their bodies are and so think these things are painful.  But for those who have experienced it - they LOVE IT.

Maths is the same.  Learn to love the unknown, the confusing and the painful.  

P.s - any maths in which you know what you are doing, are not confused and is not painful is not maths.  It's more like playing Trivial Pursuits once you have memorised all the answers.

Those of you who struggled, or are still struggling, with the car question are in the zone.

 Picture and 'Stuff'

Maths is all about pictures (representations).  Numbers are just an advanced version of this.  How many of you drew little pictures of cars, and graphs etc.  That is maths.

I'll post the solutions soon,

Challenges

There are some amazing young problem-solvers out there.  Yet many of these learners are slogging away on algebraic formulas, percentage problems, fractions and ratios with little immediate reward. You need to be rewarded for your efforts.  Below are the two final problems that need solving.

The rules are:

• No help from your parents,
• You CAN work together
• Must be 15 or under

Reward for the first person who puts the answers in the comments (or email me) - two movie tickets to Hoyts cinema.

Scenario number one

See previous post

Scenario number two

The zombie apocalypse has happened.  You and a rag-tag group of survivors find yourself on one side of a broken bridge that used to cross a large lake.  You cannot swim the river (you are not sure what's in there).  Worse, a horde of zombies  is slowly making its way toward you (millions of slow walkers).  You have about a day before they arrive.  Crossing the river is an imperative.

The gap between your side of the bridge and the other is about fifty-two foot.  All you need is a long board or pole and you could lay it between the platforms and walk across one by one.  However, you cannot find a pole long enough.

You do notice a pole sticking out of the water right beside you.  It is thick, and looks like it is strong enough.  If only you knew how long it was? Andy an engineer, suggests that you could pull it out of the ground using ropes and pulleys.  He assures you that given the equipment on hand it can be done.  BUT, it will take a day at least.  In other words, if it is not long enough to cross the gap (52 foot) you will not have time for a second option.


You lean over to inspect the pole and notice some specs written on it.

It says: Half of the pole must be in the ground, another one-third of the pole to be underwater. Nine feet of the pole remains above the water.

Question: Is the pole long enough?  Is it worth spending all day pulling out?

Scenario number three:  Age problems 

Zombie Mary is three times as old as her son.  In twelve years Mary's age will be 1 year less than than twice her son's age .  How old is each now?


Once the result is in, send me your address via my email:

whittendamon@gmail.com

 and I'll send the tickets.

Teenagers versus adults... (Updated)

In the next week or so, I will be posting three problems for anyone under the age of sixteen to have a go at.  The prize is epic - two whole movie tickets.

The reason is, is that many folks are busy working away on number problems that have little extrinsic reward.  To those of you actively learning mathematics - well done.  It will pay off, plus it's just good fun.

This weeks challenge

There is a problem below that will test you.  What I would like to do is have a little competition between teenagers and adults.  I am wondering if teenagers can solve this quicker than their parents!  I suspect they might.

Game on people - post your answer in the comments.  Good luck.

Zombie apocalypse

The apocalypse happened.  It was bad, and only a few survived.  However, you have become the leader of a rag-tag group of survivors.  You are doing well, but MUST get out of the city.

As you walk along you come across a truck that will be able to carry almost everyone.  You load everyone on board except you and three others.

As the truck leaves you tell the driver to travel at fifty kilometers an hour only.  They will be on a highway so will be able to keep this speed steady.  You do this so that you will know where they are at all times.  All phones and electricity are out so you don't want to get lost.  You want to be able to catch up to them.

Because the four of you had to find a car, you leave one hour later.  If you travel at 80 kilometers an hour, and they are traveling at 50 kilometers an hour, how long will it take to catch the rest of your group?

Good luck.

Update:  Seems the good people of Hastings are the quickest!

Update two:  Maths is really only maths if you have no method that can be used to solve the problem.  Instead you have to draw on your wits, cunning and resources.  Some people draw pictures of cars, some work out where each vehicle will be at certain time zones, some work backwards, some draw graphs and so on.  It's the HUNT, that makes maths cool, not the answers.  Enjoy the process... that's where the fun is.

Fixed versus growth mindsets...

If you want to discover one of the reasons some people succeed and others do not, then explore the difference between fixed and growth mindsets. But prepare for a journey into your own psyche.

I have talked about them here.  But Graeme Smith has found a great video clip that you must watch and show your kids!  It is very watchable and sums it up very well.  I recommend all parents read it and investigate it closer.  Click here for the article.

One of the most interesting things for me personally, is what the research has to say about how we encourage our children.  In particular whether we should say:

"Well done, you are really talented!"

or

"Well done, all that hard work you put in has really paid off!"

One of these is good and one is bad, see if you can work out which is which.

Question of the day...

Try this, and then try it with your learners.


How do you write this

                                                                                                              as a ratio?

How do adult learners approach numeracy problems? Updated...

How do adult learners approach numeracy problems?  This is a pivotal question relating to exactly how much we expect adult learners to learn in their respective courses.  Why?

Well, in a nutshell numeracy or mathematical problems are only useful because they help the learner LEARN something.  They have no other purpose.  If a learner can solve the problem using an existing method like an algorithm then you and the learner may be wasting your time - if your objective is to develop numeracy skills.

Do adult learners ‘use’ numeracy problems to learn, or do they treat them as micro-tests? 

The research is reasonably clear that good problem solving behaviours should include the following phases: 

  

1. Discuss the problem and generate several strategies for solving it.  (What exactly is the problem? What are the variables?  How might we best solve it?) 
2. Select the best approach based on its merits.  (Which of our ideas is the best and why?)
3. Enact the strategy (Let’s get busy)
4. Monitor the strategy (Is this working?)
5. Evaluate the answer and review.  (Has the strategy worked?  How do we know?  How can we prove it?)

The above is more in line with how mathematicians approach problems.  The trick is to separate ‘calculation’ from other aspects of mathematical problem solving.  For example, have a look at the following example:

In this example if you begin calculating before examining the problem you will miss the point of the problem.  Clearly the answer is 720 because there are three of them.  We know that the answer to 720 + 720 + 720 is meaningless in this scenario.  But to 'see' this requires taking the time to work through the problem.  Engaging in ‘calculation’ is the worst thing you could do to solve this problem. The answer is evident if you take the time to look at the whole. However, imagine this problem is written as a word problem.  Suddenly it becomes harder to see the big picture.

The question then is this:  Do adult learners actually engage in the first two parts of problem solving?  Do they think about the problem situation?  Or do they simply look for the important numbers, adopt a calculation they think fits, and begin the calculations?  

The findings: 

Well, my research has found that adult learners are primarily oriented toward solving numeracy problems by applying pre-learned procedures to the quantities provided.  They skip the first two steps, attempt to identify numbers to preform calculations on and then immediately begin calculating.  Finally, they engage in very little evaluation or review.  Rather they move quickly to repeat this process with the next problem.  Below is a quick review of the behaviours used during problem solving sessions.

Lack of discussion about the problem situation or potential strategies

When a group of learners begin to analyse the numeracy problem they do not discuss the problem situation.  Rather they identify the key numbers and begin to engage in calculations immediately.  

Adoption of the first strategy proposed

Generally, the first calculation strategy proposed by a group member is adopted.  The acceptance of the strategy appears to be on the basis of ‘who’ proposed it. It may be, that being the first to generate the strategy indicates speed of comprehension, knowledge of the procedure and therefore high proficiency.   In other words, the first learner to speak must know what they are doing.  Let's go with their idea!

Focus on step-by-step procedures

Learners relied on procedural step-by-step (algorithmic) approaches to solve problems rather than engage in reasoning.  That is, answers were assumed to ‘appear’ once the correct formula had been applied.  Moreover, learners often acted as though problems could not be solved without a procedure. If a learner asks "how do you do that again?" it means they do not understand how it works.  This small sentence is an indication to you that the learner has yet to develop understanding (even if they can repeat the procedure). 

Disconnect between problems and answers

In many cases there was a disconnect in the relationship between the solution strategy used and the answers.  This was evident in several ways.  Firstly, the way answers were described as a product of a procedure and answers lacked any contextual aspect.  For example answers were described as ‘popping out’ after performing the correct procedure on the calculator and answers were often spoken as digits rather than meaningful quantities. 

 “The answer is four, two, three, eight”.

The answer was “Four-thousand, two-hundred and thirty-eight dollars” but the learner had stripped it of all meaning. 

This is a serious finding as it undermines some of the key assumptions made when teaching numeracy skills. The first being that numeracy is the bridge between mathematics and the real world.  The second that placing numeracy in a context meaningful to the learner improves outcomes.  Almost all of the learners in this study simply ignored context and engaged in the calculation of numbers.  Just like in maths class. 

Goal orientation
The primary goal of the learners was to identify the correct answer and finish the numeracy problems quickly. While this sounds reasonable to most people, it conflicts with the goal to learn from numeracy problems.

For example:  Groups would often unevenly distribute responsibility for task completion.  That means that if two or three learners were asked to solve a series of problems, they would allocate the task of calculating to the most proficient learner, the second would take responsibility for putting numbers into the calculator, and the third would write down the answers. 

Yes, they would achieve their desired goal – to finish on time and have all answers complete.  But no one necessarily learned anything new.  The learner who did the calculation was already able to do the task, and the other two learners never actually engaged in the thinking necessary to develop new understandings.

Beliefs
The learners in this study have developed beliefs about what mathematics is, how it is learned, and what goals and behaviours are appropriate in a maths class.  These beliefs clearly transfer to the numeracy classroom.

This is a problem.  As tutors we need to change these beliefs and the associated goals that these learners hold.  Otherwise we risk diluting the educational experience the learners are engaged in.  Do they really need another classroom experience in which they make very little progress? 

The next question is:  How do we as tutors begin to change these beliefs so that learners adopt behaviours better suited to learning?

The hardest work there is...

Henry Ford is purported to have said "Thinking is the hardest work there is.  Which is probably the reason why so few engage in it".

Well, I equate thinking with writing, they are one and the same.  Writing well is absolutely the hardest work I have ever done.  Challenges include, little sense of progress, frustrating amounts of editing, and absolute concentration needed for long periods at a time.

For those of you who notice the odd, grammar or spelling error on this blog, please understand that I permit myself this one small dollop of relaxation once a day.

Master writing, and you master the world.

By the way:

World Literacy Day - 8th September

A day in honor of the gift of literacy to humanity.  It raises us from the limitations of our own minds by introducing us to the minds of others.

Will post something more substantial in coming days.

Adult learners' perspectives on cheating.

I had the pleasure of spending some time with Professor Diana Coben and Associate Professor Jenny Young-Loveridge on Monday.  I could spend hours with both of them as they each have so much knowledge and experience to share.  They also both have a wealth of interesting stories that I could sit and listen to all day.  If you ever get the opportunity to sit down with either of them over a coffee or cup of tea do so, it'll be one of the best insights into mathematics and numeracy education you can get.

In regards to adult numeracy Diana has been at the cutting edge for a long time and her involvement in adult numeracy across the globe is simply astounding.  If you are conducting research in adult numeracy you cannot but be amazed at the quantity and quality of the publications, projects and networks Diana has been involved in.  She really is inspiring.

We were talking about the attitude of many adult learners toward calculator use when Diana remembered having developed the poster below.  I asked her if she would mind if I could post it.  I thought it could be a great conversation starter with learners.  She wrote the following:

Here is the Cheating poster. Happy for you to include it in your blog.

It came about through a discussion with adult numeracy students in the East End of London, UK (Tower Hamlets Adult Education Institute) in the early 1980s. I was the tutor.

The poster came about entirely spontaneously. The words are those of the students themselves - spoken in that order by different students - one sentence each.

I said "Wow - write that down" and somebody did. Then one of the students who was also doing silkscreen printing in another class in the same building made it into a poster and we sold the posters for a while.

When the Numeracy Pack was published we used it on the back cover because it summed up so well how many adult numeracy students feel about cheating.

Best wishes

Diana

It would be very interesting to hear what opinions learners hold about calculators in your classes.  I've noticed that older learners have a bit of a bias against their use while younger learners have no qualms about it at all.

I really liked the "Cheating is pretending you understand when you don't".  Good to get that reaffirmed.

 Also, almost everything I know has been gleaned from the back of the book, with a process of working backwards to understand it.  Answer pages are not the enemy if you genuinely want to learn?  Thoughts?

Cartoon pretty much sums up the difference between academic success and failure

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.