Lesson four - Problem solving

One of my primary goals is to change learners’ behaviour with mathematics and numeracy.  A big part of this is accomplished (I hope) by using problem solving activities.  My main challenge has been overcoming the learners’ absolute resistance to working together to solve problems.

Essentially, the learners in the tertiary sector treat all activities as individual tests rather than learning tools.  While they may talk to each other, they do not use each other as resources for solving problems and hence miss the learning opportunities that good problems provide.  Yesterday I included the fish tank problem in my lesson, below is a quick account of how it went.    

The fish tank activity (Hat tip Dan Meyer)

Objectives: 

• Develop group problem solving discourse patterns such as conjecture, clarification and justification;
• Develop trial, error and evaluation skills;
• Appreciate the need for dimensions of measurement such as milimeters, centimeters, area, volume and time.
• That answers can be expressed as tolerances (not just right or wrong)

I video recorded a fish tank being filled slowly and consistently from a hose.  I showed them the first quarter (roughly) being filled up and timed it.  I then stopped the video and the timer and asked them to work out how long it would take for the tank to fill.  I had the fish tank (empty) in the room with us and each group had measuring tapes.   The partly filled tank was up on the screen where all could see it.

My first question to them was to think about what information you needed to know to help solve the task.  The groups did not engage as I would have hoped.  All groups guessed based on the water taking 48.31 seconds to reach roughly a quarter to a third and used additive thinking to identify a number a little over 3 minutes.  Each group estimated between 3:10 minutes and 3:30 minutes. 

Unfortunately these guesses came from individuals within the group working on their own.  There was very little collaboration and hence objective one was completely missed.  Yes, I did discuss the need for this before the activity began and have been trying to cultivate this in the class.  Those who were not sure simply let the others do the working.  I’m struggling to overcome this issue.

We wrote the estimates on the board and then watched the remainder of the video.  As the tank filled groups began to change their guess.  I had total engagement for a moment or two as they all watched the tank fill (a painfully long 3:48 seconds) and all wanted to know the final time.  So even the students who did not contribute to the estimates were still engaged.  That’s one good thing! 

What I wanted was groups to use the measuring tape to at least measure the height of the water tank and then determine from the video the exact water level at the time given.  Then I wanted them to extrapolate this out.  Ultimately what they could have done was to find the volume of the tank and then use the video to determine the remaining volume and the exact time to fill it.

The positive of the activity was that all the learners cared!  Even that one student who is always sleepy.  They looked carefully at the video, they asked about how to add seconds given that every 60 is one minute.  One learner had one minute and 92 seconds which started a very good conversation. They stayed five minutes later than they had to.  I'm taking this one as a win!

In a few weeks I think I will redo the activity and actually model how to solve it.  We are going to be working on measurement from here on in so it will facilitate content around time, length, area and volume.  Perhaps even Pascal’s law?

Have you ever tried this activity?  Check out this TED Talk for an overview – the tank activity is toward the end.

A very good clip for tutors teaching numeracy.

https://www.youtube.com/watch?v=BlvKWEvKSi8

The life of a tutor...   NOT!

Happy Monday to the NZ tutors.  Well done - you may be tired, but you are also having a positive impact on the lives of those who need it the most.

Changing lives.

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 learned anything.  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?

Cheap and nasty?  No way!

Engagement!

Some people may wonder why I do not use better materials than paper and dice to develop numeracy skills.  Well, I do.  But only when they improve learner thinking. 

My philosophy to teaching mathematics and numeracy is to focus entirely on student thinking.  Student thinking (which I call 'engagement') is a moment in time when the student stops thinking about everything except the task at hand (some call it 'flow')- they are absorbed fully in the moment as they struggle to make sense or meaning out of the information at hand.  It is in these rare moments that the learner 'constructs understanding'. And hence these are the most precious moments in a teacher or tutors day.  

Everything else is superfluous.  Everything else is killing time.  Everything else distracts.  

So, because I believe that it is only 'what happens in the head' that results in learning I am very utilitarian about my gear.  

My only question when using material is:  does this facilitate learner thinking?

 

Lesson three! 

Lesson three is done and dusted.  Here are my objectives and feedback.

Outcome:  Recap the place value system

We began with a recap of the washer scenario in which learners had to work out how to send various quantities of washers.  I wrote order quantities on the board and learners worked in groups to organise how to send the correct amount.  The task was two-fold.  First, I asked learners to develop a system for easily sending orders out.  They were meant to write out a place value chart and write in the amounts.  We did this in the last lesson and it was meant to be a refresh and act as a formative assessment.  But… no one did it. 

The second task was just to describe the method of packing the washers.  They all did this.  For example an order for 3240 washers would require three pellets, two boxes and four packets.  See the ‘lesson two’ post for a full explanation. 

Outcome:  Know how many tens are in numbers up 1000

As with last week I asked learners how many tens were in:

• One box
• Three boxes
• One pellet and two boxes
• Three pellets, four boxes and one packet.

This went well!  My plan was to then repeat the activity but with the scenario of money.  If I have only $100, $10, and $1 notes and coins, how do I make the following amounts?

• $45
• $125
• $368
• $1230
• $1083

I forgot to do this! 

Outcome: Understanding what decimals represent 

We repeated the decimal game used in the last class but this time had groups construct the numbers using paper on their desks.  I handed out ones, tenths and hundredths to each group (5 in total).  I then wrote two numbers on the board and asked them first to discuss which was bigger and to make each numbers.  I began with ones and tenths:

• 1.30
• 1.5

I walked around and observed groups.  There were some really good conversations taking place.  I then asked which number was bigger and then drew a picture of what the number would look like with paper.  Everyone was winning.

I then repeated this with ones, tenths and hundreths.

• 1.23
• 1.32

Learners constructed both and were able to see which was the biggest. 

I stressed to them that they needed to visualise how the numbers would look before they made them with paper.  Numbers represent things! We did five rounds and then moved to the next activity just as boredom was setting in.

Outcome:  Add numbers with decimals to hundreths

I then wrote on the board the following:

• 1.2 + 1.4

 and asked them to put the two together.  This worked well.  Most groups were able to work through this systematically. I then used the classic Vygotskian method and modelled it verbally on the board.

“How many ones are there?  Two.  How many tenths? Six.  Answer = 2.6”.

Next round included hundreths.

• 1.23 + 2.44

Learners were able to solve this easily and constructed the answer with paper.  I then asked groups to read the answer out loud e.g “three point six seven”.

By round three a few people were beginning to yell out the answers before using the paper.   The groups began to predict the quantity by simply adding the ones, tenths and hundredths.  Eventually when I wrote 2.34 + 1. 25 they learners were able to get the answer without using the paper. 

I asked various groups questions like “how many tenths are there?” or “how many hundredths are there?” to make sure everything was being understood. 

 

So… the learners now at least understand what decimals represent and can add them together.  Confusion regarding which is bigger 0.24 or 0.3 are a thing of the past.  The next step is to work on crossing the place value barrier i.e. 2.3 + 0.8.  Well nail this next week.

Feedback

Learners left this class saying “Oh, now I get decimals”, and “Shot bro, that’s pretty easy”.  I even got a pat on the back as one student left. 

Some initial findings regarding the theme – Emotion and attitude

Having spent the last two months analysing and gradually synthesising the findings from the observation data I’m beginning to make some progress.  Below is an very early overview of findings relating to the theme ‘emotion and attitude’.  

Method

I coded and analysed ‘expressions of affect’ which include external expressions such as emotionally expressive words, interjections and exclamations, eye contact or lack thereof, facial expressions and multiple modes of body-language.  Interestingly, the majority of emotive speech happened privately between two learners or as self-talk.  Without the use of multiple microphones the entire range of emotional expressions would have been 'under-cover' of the layer of audible classroom dialogue.     

The expressions can be broadly organised by the responses at consistent routines of the each lesson.  Most lessons I observed were structured in three parts.  These were: the initial introduction and demonstration of the skill/method/concept to be learned; learner application/practice - usually in groups; review, discussion and evaluation and marking of learner work.

Findings

Anticipation of failure

Learners reacted strongly to being compelled to participate in numeracy lessons or activities.  I’m struggling with the word ‘compelled’ but given the negative emotion expressed most adults would simply leave if they could.  There are reasons why these learners don’t which have to do with PTE attendance procedures/rules and sociocultural pressures.  When tutors introduced the topic learners become visibly anxious, exclaimed their hatred of maths or their poor historical performance with maths and often begin to withdraw their participation.  I’ve linked these responses to the concept ‘anticipation of failure’.  The anticipation of failure has links to learners’ prior experiences.  Learners anticipate re-experiencing the worst aspects of their mathematics experiences.  This answers one question: do adult numeracy environments activate mathematical beliefs?  Without a doubt.   I have been playing with the term ‘emotional inertia’ to describe how difficult it is for a tutor to positively engage learners in numeracy.

Emotional investment

The second part of the lessons (Learner application – group problem solving) evokes emotional responses also.  But here they are a mix of positive and negative.  On the positive side many learners emotionally invest into solving tasks, they work hard, persist and care deeply about the results.  They appear motivated by the desire to demonstrate to others and themselves how good they are.  They want to be the best and as such are performance oriented.  These are the learners that challenge tutors, they are usually the best in the class and interact the most with the tutor during numeracy sessions.  They care deeply about being right and will work hard to be so.

Learners also express negative emotions during problem solving phases of lessons.  Often they become overwhelmed by the complexity of the task and disengage.  These learners often are the hardest to engage as they withdraw at the first sign of difficulty.  These learners have deep doubts about their ability and are motivated to avoid losing status.  If the task looks like it could possibly damage their image they choose not engage.  Other learners do engage and become frustrated at the lack of progress (they often resemble the positive emotion described above) but these learners are motivated by the desire to get the job done.  They want to finish the task as quickly as possible.  When it proves to be a challenge and they feel they are no longer moving toward completing the task they begin to get frustrated in a negative way.

Loading meaning onto correct/incorrect answers

At the third phase of the lesson (during which the tutor works through the answers) all learners who have completed the tasks are fixated on the tutor and answers.  Many learners describe themselves as either ‘math people’ or 'non-math people’.  Many adult learners use numeracy classes to either verify their existing identity or to reconstruct it.  Hence many learners ‘load meaning’ onto their answers.  When they have an answer it is not just a quantitative answer to a quantitative question rather it represents whether one is ‘smart’, ‘intelligent’, ‘good at maths’ or ‘bad at maths’.  There were occasions in the observations where learners who got answers wrong had huge emotional responses and withdrew participation from that point on.  For these learner’s getting the answer wrong confirmed their historical recollections and beliefs that they are not intelligent. 

In summary, participating in numeracy is highly emotional for almost all learners whether they are ‘maths people’ or 'non-maths people’.  Learners load meaning onto what happens in the class, how they learn, the answers they get correct, and the answers they get incorrect.  Adult learners care about their performance with maths and numeracy.  This is a huge opportunity for educationalists.   

 The question next is:  Why do learners care?  What do they believe about numeracy and mathematics that makes it so meaningful for them?       

Lesson Two! 

  Or “You want to try something stupid?”

The plan

After reviewing my assessment data it is evident that there is a general lack of knowledge around place value.  Most learners are best represented by Step two/three. The target vocation is engineering.  Let's get busy.

In the last lesson we introduced the place value system and the outcome related to understanding that the quantity increases by a factor of ten as progress up the place value chart. The activity use to generate student thinking was the washer activity.  In a nut shell:

A company makes washers and has large orders to fill.  They decide to package and send the washers in tens.  That is ten washers are wrapped in paper called a ‘packet’, ten packets fit within a ‘box’ and ten boxes fit on a ‘pellet’.  The activity was to work out how to send various amounts.  Learners were to make links between the place value chart and the packaging technique.

The activity was a success.

Activity one

This week we repeated this activity and then shifted to what is described as Step three knowledge.  Learners need to know how many tens and hundreds in numbers to 1000 and the place values of digits in whole numbers up to 1000.

Building on the washer activity we added a new scenario.  Customers are ringing the washer company asking questions about their orders.  The questions include the following:

• How many tens (packets) are in a box?
• How many tens are in three boxes?
• How many tens are in a pellet?
• How many tens are in a pellet and two boxes?

These questions really got the learners thinking.  They started to really discuss these problems for the first time.  This requires Step three knowledge but really can be quite challenging.  I had large pictures of the quantities on the board – a washer next to a packet, next to a box, next to a pellet.  It was a nice visual representation of the place value chart. 

Following this the learners linked the whole activity to the place value chart. Very successful.

Activity two

Ah, this one was risky!  This activity is designed for kids but ROCKS for teaching the place value for digits.  I talked it up – “Hey, you want to try something stupid?”  Everyone was in!

I took five volunteers (the youth of course!).  Each stands in front of the class and holds an A4 sheet with a single digit on it.  As they are lined up it makes a number.  For example, 24,578.  A learner yells out a number using those digits (87,542) and the volunteers have five seconds to rearrange themselves to make the number.  If they can - they survive the round.  

So the learners sitting in class have to take those digits and rearrange them into a number (and say it properly) and the volunteers have to move fast to make it.  Three groups are seated and one in front of the class.  Swap out after three or four times. Great game – really worked well!

Next week I’ll add a decimal (the dot) and the game will become more complex.  You can easily add people (the numbers get bigger) or have groups make the biggest number and so on.

Activity three – Knock down

As the learners are standing at the front of the class, it’s a good excuse to play 'knock down'.  Each number must be reduce to zero by subtracting the correct amount.  I say to the class ‘knock down Jake’ and the class must look at the ‘place’ Jake is occupying and subtract that amount.  For example, in the picture Jake is holding a 5 that represents five-thousands.  So the learners must write down ‘minus five thousand’.  When they get this correct – I hand Jake a sheet with zero on it.  Then I say knock down Phil, and so on.  This gets fun if you decide to have a very large number. 

I am not an artist - forgive me.

Anyway – I kept this activity short – five minutes at the max and then we moved on to the final activity.  

Activity four - Bet on it

Resources: 8 sheets of A4.  A4’s cut into tenths (you need at least 17 tenths), A4’s cut into one-hundreths (hard work!  But you need about 17).  See the picture below.

This activity was preceded by group discussions about what a decimal is and where you see them (more on this later).  We then discuss what the tenths place represents.  I show them the A4 and say “this is one whole piece of paper”.  Then the tenth and say "this is one-tenth of one whole piece of paper” and then show them. 

I wrote the following numbers on the board:

• 1.7
• 1.6

And asked which is the biggest number?

Groups then bet on which is bigger.  I take their bets and then make the two numbers on the ground.  First I make the 1.7 by laying down one A4 and then 7 tenths.  Then right next to it one A4 and six tenths.  At this point I get their interest and hear things like “oh, that’s how that works”.

We repeat – which is bigger?

• 1.35
• 1.29

Place your bets!

You get where I am going with this.  I introduce the hundredths.  They guess and then they see.  By round three everyone is winning – every time. 

I then ask them to mentally visualise what each number will look like.  They are able to look at the number and visualise the quantity of paper.  It's the ability to visualise the decimal numbers that makes this such a powerful tool. 

Tomorrow I will write the numbers on the board and they will make them on their tables.  My master plan is to increase the size of the paper so a ‘whole’ is one metre, tenths 10 centimetres and hundredths one centimetre.  When this is easy for them I’ll over lay it with a measuring tape and we will transfer our place value knowledge to measurement.

Four low tech but effective activities – each building on the other and each developing the right skills at the right time.  Someone will ask why I bother cutting out paper when I could purchase some nice plastic place value blocks.  Well, the truth is I've used paper and blocks many times.  The paper works better!

The goal is to get learners thinking and making meaning - it doesn't need to be pretty.