Saturday 27 September 2014

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 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,

Monday 22 September 2014


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:

 and I'll send the tickets.

Thursday 18 September 2014

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.

Sunday 14 September 2014

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!"


"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?

Monday 8 September 2014

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.
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?

Wednesday 3 September 2014

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.

Tuesday 2 September 2014

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


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?