21st Century Learning

I get nervous when I hear people talk about ‘21^st century education’.  In my business I hear it a lot, from the current Government, from colleges and from friends.  It’s a buzz word that pushes all my buttons.  So why do I find it so threatening? Because it is usually used with a vague reference to technology and distance learning, with the implication that by using technology we improve learning.  It is promoted with the idea of greater access (not quality) and the specifics are very rarely defined.  Press the speaker harder on the specifics such as 'how' it will improve learning outcomes and the arguments are not as compelling.

There are advantages.  In my opinion 21^st century technology, such as computers and tablets, enables three opportunities to improve upon 20^th century mathematics education.    

1. Clarity of expression
2. Clarity of representation
3. Playfulness/experimentation with feedback

By clarity of expression I mean the ability to provide the learner a coherent explanation of the concepts being taught.  The learner can then repeat the viewing as necessary.  In the classroom, even experienced teachers speak in half sentences, um and ah, and generally convolute the message.  Believe me, I have transcribed hours of recorded real class interactions – people don’t speak good (that’s a joke by the way).  When using technology you can plan the message and get it right.

Clarity of representation.  To understand mathematics you need to have mental models of the concepts under operation.  No mental model and you will suck at maths.  It’s that simple.  Using technology we can create great visual models of the concepts being taught.  We can also create a range of models so if one doesn't quite work we have others that might suit the learner better.  These should be massively powerful in visually demonstrating the concepts the learner is seeking.

Lastly, the representations should be provided in an environment in which learners can experiment and get immediate feedback.  They should be able to stack things up, pull things down, twist, bend, and stretch stuff.  They should be able to see what happens when you do or do not do something.  The more the better.

So, next time you hear ‘21^st century education' or 21st century learning’ ask some questions about what they mean.  Those are my three criteria – what are yours? 

Mathematical discourse:  Questioning

How learners ask and respond to questions reveals much about the type of information they deem important within a numeracy class.  And what learners deem as important reveals much about the goals they set for themselves, and goals, be they implicitly or explicitly defined, reveal much about the beliefs learners hold.     

And that is why this particular finding about adult learners' questioning practice is so troubling.

First let me explain the differences between good questioning and poor questioning using the model provided by Huffard-Ackels, Fuson & Sherin (2004) called the ‘math-talk community’.  To my mind this model is the most appropriate for the adult sector and has been used everywhere from young school students to University mathematical classes.

Good questioning in a numeracy class:

Student to student talk is student initiated, not dependent on the teacher.  Students ask questions and listen to responses .  Many questions are ‘why?’ questions that require justification from the person answering.  Students repeat their own or others questions until satisfied with the answers.

Poor questioning in a numeracy class: 

Teacher is the only questioner.  Short frequent questions function to keep students listening and paying attention to the teacher.  Students give short answers and respond to the teacher only.  No student to student math talk.

These two descriptions act as book-ends on a continuum.  There are two intermediate descriptions also that demonstrate how learners may transition as they move from the poor description to the good description. 

The findings

The findings of my classroom observations reveals that the greater part of the questioning discourse resembles the ‘poor’ description – disturbingly so.  This is not to say there were no exceptions.  There were some really good examples of questioning – but not many.  Overwhelmingly this is what I found:

• Tutors asked almost exclusively polar questions while teaching
• Learners responded to these questions with single word answers
• Learners only responded when certain of their answer (no one risks being wrong in public)
• On occasions when tutors asked learner to explain their thinking the learners still responded with single word answers (even when prompted).
• Learners rarely asked questions and when they did the questions were requests for answers, not for explanations, clarifications or justifications.
•  Learners very rarely initiated their own ‘why?’questions and when they did it was to the tutor not  each other.

 

The pattern is simple:  Tutor asks a closed question to the class, a learner who knows the answer answers, and the tutor carries on - over and over and over again.  This is replicated in group work environments with no tutor present.  This pattern is known as – Initiate, respond, feedback (IRF).

So, the discourse within adult numeracy classrooms resembles that of a traditional transmission-based classroom.  Given the picture above it kind of makes sense how this idea may have sneaked into our psyche. A description of this pattern has been attributed to Friere: ‘The pedagogy of the question’.  In other words a classroom culture in which questions and answers are used to control the structure, discourse and content of the class.  But, Friere is suggesting it is the teachers ‘fault’ for implementing (or conforming) to the structure.  But things are very different in this sector because the adult learners are now apply pressure to the tutor to conform to this model.  And my thinking is that this is because adult learners believe that this is how maths classes should operate and therefore both exert influence for the pattern and conform easily to it.

Are adult learners engaging in the types of discourses that promote the development of the conceptual understanding they need.

My answer: No.

Do the adult learners themselves contribute to this pattern?

My answer: Yes.

Can we as tutors change this?

My answer:  I hope so but I'm not sure how.

Thoughts?

Mathematical discourse

We were made to talk

When thinking about education I can't avoid working from this basic premise:  We were made to talk and laugh and have fun.  And learning was meant to be joyful and easy.  Unfortunately things in the educational world don't always turn out this way.                 

I have spent the last few months analysing discourse patterns that occur between adults (learners and tutor) within numeracy lessons.  The findings shed light on whether learners are "talking and having fun" in mathematically productive ways.  In my mind, the findings have huge implications for the tertiary sector.    

The word ‘discourse’ has several meanings relating to slightly different contexts and purposes.  My use of the word really just means the type of talking/communication that occurs between students and the tutor in classrooms.  I have analysed particular features that have emerged as important. 

Setting the scene

There is strong evidence that the quality of learner discussions has huge implications for how learners engage in, and consequently develop, mathematical knowledge.

In other words:  The type of discourse that occurs in numeracy classrooms COUNTS.

So, in adult numeracy classes do the discourse patterns enhance learning, or constrain and inhibit learning?  Well, unfortunately the discourse patterns as they relate to mathematics are particularly poor.  There are four areas of specific concern.  I’ll post the first of the four below with the other three to follow in a further post.

The source of mathematical ideas
The learners in my study believe that the tutor, and associated resources, is the only valid source of mathematical ideas.  They view the tutor as the expert who possesses the right type of mathematical knowledge.  That is, the math knowledge that is generated, owned and passed down by established academic institutions (and official mathematicians).  Because of this belief, learners see little value in generating their own ideas, sharing their ideas with others, or listening to other learners ideas. Learners do not see group problem solving as a learning opportunity and require the tutor to verify any and all answers before they celebrate success (note that whether an answer is correct or incorrect should be self-evident if learners understand the problem and answer).  Keep in mind that learners may be motivated to group problem solve due to the social aspect and a sense of fun.  They will still engage but not really believe that they will learn what they need from the experience.

The discourse patterns during group work were dominated by learners who ‘knew’ the correct method (taught to them by previous teachers and therefore authentic) and simply told the group how to solve the problem.  The group did not discuss ideas such as solution options, alternate interpretations or alternate solution strategies.  Also, once the tutor began to work through the problems the learners simply used the tutor’s ideas.  Learners very rarely spoke up and gave a new or unique method of solving or thinking about a task.  The one exception was ‘numeracy experts’, that is, those learners who had high numeracy skills and wished to show their proficiency to the class.

In essence, learners do not believe that their own mathematical ideas have any value and hence do exert energy in producing or evaluating them, nor do they exert energy in listening to other learners’ ideas.  However, research strongly suggests this is an essential process in order to develop conceptual knowledge of mathematics. Learners must generate their own mathematical ideas. However, getting them to do so requires changes to occur at a multitude of levels.  The first and not least, is to change the belief that mathematical knowledge is only produced by experts and cannot be self-discovered, a belief held by many tutors.

Anatomy of a pretty stink lesson on the Pythagorean theorem

Despite planning well, having some reasonable materials and a good story (the history of Pythagoras) today's lesson was less than inspiring. 

So what went wrong?

Well, the older students were unable to attend because they were sitting an assessment.  So I had a class of younger learners who I felt would rather be somewhere else.  It was after lunch (the dead zone) and the other students were going to go home early.  Many were thinking about their imminent escape to freedom – Not about a potentially life changing lesson on the Pythagorean theorem!

However, my job is to get their attention by being interesting and relevant.  Despite having a pretty funky little lesson planned I didn't really manage it.  Here is what I did: 

1.  My first activity was to fold an A4 piece of paper into a square.  This would set the stage for a discussion of right angles and the hypotenuse.  About two-thirds did it – the rest didn't care.

2.  I then asked who knew about Pythagoras and how he related to the activity.  A few had heard of him but no one really knew.  I told the heck out of that story – cults, murder, violence, numbers, meta-physics and the nature of reality.  It was good people, it was real good.

3.  I handed out a grid.  The idea was to find the smallest square (1 by 1), then the next, then the next until we had the square numbers.  This was to clarify ‘squares’ and exponents which I know most of the class is confused by.  Unfortunately, what happened was various people in the group became distracted and talked through that little bit.  I pressed on.

4.  Question to class: Is the Pythagorean theorem just some maths thing that you never use in real life, or does it have a real usefulness in your life?  Excellent – we came up with lots of practical examples – including one to do with a support beam in the classroom.  This rocked.

5.  A few worked examples on the board.  Bit boring.  

6.  Find the hypotenuse of the classroom window.  This was a good excuse to draw on the window (hey, all the greats did it!).  I measured the top as 8 and the side as 5 and let them find the hypotenuse in groups.  They did it!  With support.  Then another, then another.

Sounds okay right?  But about four students are at the end of the room just cruising and mucking about.  I felt that I didn't make my points and they didn't make sense out of it as they should.  I need to rethink this lesson.  I’ll have a second shot next week.

So, any ideas for teaching Pythagoras?  Please let me know in the comments.  And, have you ever taught this without just ‘telling’ and ‘showing’ students?  If so, please give me some hints!

Pythagoras theorem

 Ah yes, it has been mentioned – the Pythagoras theorem.  I had planned to introduce this to my class after we had carefully covered exponents, squares, area and basic geometry.  But it arrived yesterday with a bang!  Here is a transcript that followed a brief discussion regarding the words ‘perimeter’, ‘circumference’ and ‘diameter’.

Damon: Any other weird words that have been popping up?

Joe:  Pythagosi.

Damon: Oh yeah, Pythagoras.

Joe: I hate that bastard.

Hmmm, a bit of bad blood toward old Pythagoras in the room.  We did an impromptu lesson on the concept and now I’m putting something together for tomorrow. I've searched the Learning Progressions material – not too helpful so far (but very good for measurement).  Now moving to NZMATHS which has some great ideas and resources.

It’ll be interesting to see how the diagrams work tomorrow.  See if we can’t make Joe love the bastard!  

Book review:  Mindset! 

Mindset pulls together much of the research by Carol Dweck, one of my personal favorite researchers. Her theory: essentially, everyone one of us holds one of two beliefs (or implicit theories) about intelligence. One of these beliefs leads to resilience, growth, and success. The other does not...

Over the next few weeks I'll post some of the ramifications of both beliefs as they are broad and far reaching. But now for a brief overview. 

According to Dweck, some people believe intellect is inherent and fixed.  This is called the 'fixed' mindset. This suggests that some people are just 'smarter' than others. Words like 'talent' or 'gifted' are often used to suggest that some people are just born 'smart'!

The second is the 'growth' mindset. This view of intelligence states that people get 'smarter' (or more intelligent) the more that they engage in new experiences and or learning. Thus, we all start on a similar playing field (it doesn't suggest there is no genetic influence) but become more intelligent the more we engage in various and diverse activities.


You 'hold' one of these views subconsciously and the clues slip out in your speech. For example, when your child scores 100% on their maths test do you say:

'Gosh you are really talented!'
Or, 'Your hard work really paid off, well done'?



One of these views will limit your life potential and that of your students/children, and the other will open opportunities for exponential growth.  

Comment below, which one do you think is better - a fixed mindset or a growth mindset; and which to you feel is true?

Book link here

Problem solving revisited

Today’s problem was designed to get people talking about mathematical ideas, measurement, using measuring tapes and exploring volume.

Here is how I did it.

The preparation

This morning I videoed myself and my trusty fish tank out in the backyard.  I filmed the fish tank and an empty one litre milk container and asked:

How many of these one litre containers will it take to fill this fish tank? 

I then began to fill the litre container and tipped it into the fish tank. Fourteen litres later and it was full.

The execution

In class we discussed area and then volume.  We had already spent time working on using measuring tapes, and all their associated features.  We had measured desks, walls, ourselves etc and converted the measurements from metres to centimetres to millimetres.  We were pumped.

I brought out the fish tank.

“Oh here we go again” said someone.

“That’s right, and this time I bought lollies to bet with”, (External motivators are always a mistake but I couldn’t resist).

I showed them the litre milk container and the fish tank.  “Your challenge” I said, “is to work out how many litres of water it will take to fill the fish tank”.  I then showed my beautiful videoing from the morning but paused it as the first litre hit the tank.

    

I had almost instant engagement.  People guessed at first and then some actually began to measure!  Height x length x width.  Students had difficulties around writing down the length, but asked each other and came up with some great solutions.  We also had difficulties around the thickness of the glass.  Do we measure from the inside or the outside of the tank?

In the end we wrote up every groups’ measurements on the board under length, height, width.  They were all pretty much the same (better than I had thought) and some were a clean centimetre apart (the glass was 5ml).  I asked why this might be and got some good answers.  Most had thought about the thickness but were not too sure what to do about it.  I suggested multiplying the thickness of the glass by two and subtracting that amount from each measurement.  Writing this algebraically will be in next weeks lesson. 

We then had to work out how to enter the three numbers into the calculator.  This was another great activity.  We had something like:

  

22.6 x 29.5 x 21.6

Finally the answers started coming out.  Big fat ugly answers like…

14400.72

The next bit was awesome – learners wanted to know what the number meant!  Ha ha – victory.

We discussed that it meant 14400 little cubes of square centimetres could fit into that fish tank.  We did a visualisation trick – Imagine we had little frozen cubes of water one centimetre by one centmetre and began to stack them in there…

The next question was how this relates to the litre.  I explained that one millilitre occupies the same space (at sea level you numeracy geeks) as one cubed centimetre.  So we need simply work out how many little cubed centimetres fit in the litre bottle.  We got one thousand and then figured out we needed to see how many thousands fit into our big number.

 

The students simply read the first two numbers of their answer (some thirteen some fourteen) which was pretty clever I thought.

 

We locked in our answers, placed bets with packets of Smarties and let my video roll.  The class stayed five minutes late just to see the answer.  Cool.

The bad

One learner found it pretty tough and left feeling discouraged by the lesson.  It was too much and I pushed him too far.  Stink.  He is really smart and making great progress.  I may have introduced this before really building the necessary skills.  My idea was that it would help them think about the skills they need to learn.  

The unmotivated, dispassionate New Zealand adult learner... Think again!

A wonderful finding from my research – New Zealand adult learners are wonderfully passionate about mathematics and numeracy!  They care…  They complain bitterly when they have an answer wrong, they shout and cheer when they get answers correct and they chatter away to themselves when working on problems.  All this wonderful energy and passion is at our disposal.  Don’t ever think again that low achieving adult learners are not motivated.  They are motivated.

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.