The disasters of well taught maths classes

The article on the right was one of the first that began to change my thinking about education and maths in particular. This article asked this question: What if traditional math education, that was taught well, was harmful to students?

The answers would likely surprise you.  You may think that Schoenfeld was talking about damage to emotion or self-worth.  He wasn't.  He was talking about how attending well taught maths classes turns students into poor mathematicians.    

Schoenfeld makes a convincing point that a 'well-taught' class teaches two things; content, and beliefs (he calls these 'orientations').  The content is the stuff people are aware of.  The beliefs however, are built into the architecture of maths lessons, curriculum structure and content approach.  They are woven into the classrooms' historical, cultural and social dynamics and they shape the belief systems of the individuals within them.  They are obtained by recognising and internalising the sociocultural norms of maths classes - in short, the things that are valued in maths classes and the accepted way that things are done.

The troubling part of this is that these beliefs run counter to the behaviours and dispositions that good mathematicians possess.  That's right - maths classes not only do not teach your child to be a mathematician (in the broad sense of the word) but develops within them a belief system that sabotages your child's future maths ability.

While students learned how to solve sums and canned mathematical problems provided by text books and teachers, they never learned how to use mathematics to inform real life decision making.  I mean, think about this, we all spent at least ten years in classes with trained maths teachers and almost none of us USE the very tool we were supposedly gifted from the education system.   Other than the maths we have to use, we generally do not apply the skills we were meant to possess.  What other subject has been so monumentally wasted?

So, here is an interesting maths problem - work out how many hours of your life you spent in maths classes over the course of your schooling.

Then imagine that you could have those hours back and invest them into any educational subject you wanted. What would you choose and would you expect a better outcome from those hours?

Being VERY conservative my number is about 1300 hours (upper range about 1800).  1300 hours of straight math instruction, minus homework.  If only I could re-invest this time.  Because by any cost/benefit analysis I can think of, what I got for those 1300 irredeemable hours of my life simply wasn't worth it.

So, if this is the situation for the central bell curve of learners, what about the poor students at the lower end of the curve.   What do they learn, retain and actually use?

In 1984 Schoenfeld sounded the alarms, in 2014 nothing has changed.

Something has to change.

You can’t spect a kid to change if all ya do is tell em…

It may seem strange but I have watched this clip about five times.  Something in it really gets to me and I think it is this:  injustice.

Perhaps it is because it touches on my own background.  I have been this student, and I have met many many other learners who are intensely interested in having great futures, achieving things, changing the world and making a difference.  But... are unable to do so because they lack the skills to do so - and they know it. Worse, the services offered to them are of such low quality that gaining those skills becomes impossible.  They would work hard if they just knew what to do.  

This young man happened to possess the vocabulary and intelligence to articulate what many young people don't figure out until years later.   

Jeff is a pain in the budunsky, but for all the right reasons.  We need more like him.

Desirable learning behaviours

There is some fantastic research about how we should go about learning mathematics or numeracy.  In a nutshell: Everything you think you know – IS WRONG!

Comforting for tutors huh.

Here’s the skinny.

The strategies learners use to learn mathematics can be categorised as either ‘desirable’ or ‘non-desirable’.

Non-desirable strategies include:

•       Memorisation
•      Single-strategy problem solving
•      Repetition
•      And my personal favourite: listening

Now there are a lot of reasons these approaches suck – the main being that these strategies are effectively useless in regard to understanding mathematics – despite their persistence in the education zeitgeist.

The reason these are not effective can be summed up in two words – passive learning.  Learning is an active process of making meaning and connections about and between ideas and concepts.  As my old lecturer used to say ‘the brain is not a data collecting machine but a meaning making machine’.  The strategies above are generally considered ‘passive’ and facilitate only ‘shallow level processing’. 

Mathematical success requires understanding.  If a learner believes and then actually tries to use these four strategies they are doomed to poor academic performance.  At best they may experience some short term success, but they are destined for failure.  It ain’t good.

Desirable strategies include:

•      Elaboration
•      Strategic problem solving
•      Collaboration
•      Connection making

These strategies are considered to engage learners in ‘deep level processing’ and have consistently been shown to produce incredible results.  In a latter post I am going to go into detail about what ‘elaboration’, ‘collaboration’ and ‘connection making’ mean and look like in practice.  There is some description of strategic problem solving here.  But the basic idea is that they all require ‘mental action’ by the learner. 

Now for a brief thought.  Is developing mathematical skills synonymous with developing learners strategic learning repertoires?   

What message do we as educators send to learners about preferable behaviours?  Do we implicitly send the message that if learners will listen, concentrate and practice they will learn? 

If so, we are setting learners up for failure - again.  If I could sum up all this research in five words it’s these. 

Nobody learns maths by listening.

Provocative?  More soon…

Freaking out over maths.

Math anxiety has been an area I have been highly interested in over the years. I have seen the fear in learners eyes (and tutors) when they feel they will be publicly 'outed' as 'bad at math' or, as in the elegant words of one of my participants a "dumb so and so".

Math anxiety  has many tendrils and it's existence manifests in a variety of ways.  It is debilitating, misunderstood and one of the most relevant issues in adult education today.

Graeme Smith from ALEC has a great article here, in which he covers a module he is designing within The Diploma in Adult Literacy and Numeracy (NZDipALN).  He does a great job in covering the main issues and looks at how the new qualification may approach the topic.  Graeme is always up for input so please make comments via his blog.

ALEC is currently designing a fantastic programme and qualification for tutors seeking to move forward from the National Certificate in Adult Literacy and Numeracy Education.  It is a great step and I highly recommend it for any tutor who would like to have a greater impact in the sector.

Again ALEC leads the way in adult literacy and numeracy.  Really worth checking out the entire site - just click here.

Circles

The class and I have worked hard on gaining a conceptual understanding of 'area'.  We have made great gains toward understanding and using the Pythagoras theorem, and now we are moving into finding the area of circles.  The attached video is a brief overview of the general approach.  It's raw.  And by 'raw' I mean budget.  Read the rest first, then watch the video.

Activity one

I handed out graph paper and gave the class this scenario: 

The company you work for has built a new workshop.  Because you are so good at your job the boss has let you go out and choose your own work area.  He tells you to go into the empty workshop and tape out your work area.  But… you are only allowed 24 metres of tape.  How would you do it?

The learners said things like “I’d do it up against a wall”.  That’s great thinking, but the wall will count in the 24 meters.  “So it’s 24 meters around the outside?” – “Yes and the word for that is… perimeter.” 

So with a perimeter of 24m what is the largest area you could make?

  

Most of the learners got engaged and began to draw rectangles on the graph paper.  Others didn't quite get the idea so we drew some rectangles on the board and began to compare the area.  A major moment of learning was when learners realised that the area can change even when the perimeter remains the same. 

We finally came to the conclusion that a square has the best perimeter to area ratio.

The argument 

At this point an argument started in the class.  A learner said “See, a square is always the best”.  The other learner looked at me and said “Damon, can you beat a square?” 

“Yes, with a circle”.

“How would that work?”

“Good question”  I drew a circle on the grid on the whiteboard.  "Seeing as we always measure area in squares, how do you think we deal with shapes like circles?”  

The lesson
Objective:

• Demonstrate the shapes can be reconfigured to calculate area easier.
• Gain an understanding of the formula a = πr²

I bought paper plates and handed them out to the class.  We folded the plate in half and drew a line down the middle.  Again in half, then again, then again.  We then cut out the eighths. 

Now arrange the eighths into a rectangular shape.  A learner Pipes up “that’s not a rectangle!”

True!

"Cut each piece in half and do it again.  Now it starts to look like a rectangle.  Cut each piece in half one more time".  Finally it resembles a rectangle. 

“If we kept cutting these pieces in half would it continue to approach a rectangle?”

They agree that it would.

“And how do you find the area of a rectangle?”

Ahhh, now they start to see it and someone says “length times height”.

“Is the height the same as the radius?”

“Um, yes”.

I ended up talking through the end and explained that the sides were half the circumference and the height was the radius.  A few students seemed to ‘get’ it.  Unfortunately I was running out of time and learner concentration and made a fairly lame link between this approach and the formula.  I’ll hit this hard again on Monday.  So much for discovery learning! 

So, in summary, a great idea, a reasonable execution, and a fairly lowly lesson (story of my life). 

Pythagorean theorem - Success!  

I’m not sure if you remember but I had been slugging away at the pythagoraen theorem with the class.  To give you a feeling for the mood of the room when he was mentioned, the first thing one student said was:  “Pythgoras.  I hate the bastard”.

Well guess what – the last two lessons have been pretty damn awesome.  Here is what I have learned.

Use tree houses

Today I showed lots of pictures of tree houses.  You should Google image it because they rock.  They also almost all have supporting beams that we can lock into triangles and solve for the hypotenuse.  Got the learners interested without having to speak – simply put the picture up and then outlined the triangle with the marker.  Perfect.

Use the data projector to shine a grid onto the whiteboard. 

With a grid on the whiteboard the whole class can see exactly what is happening.  We can draw the shapes so everyone can see them and the grid functions to provide the measures.  So we can get down to conceptualising immediately. 

Zombies. 

Okay – the class was distracted (they were ignoring me) so I needed something special.  Here it is:

Zombie Apocalypse!  You have survived a zombie apocalypse (for now) and all the other engineers have been eaten. You have to measure the length of a support beam on a bridge.  The bridge leads to safety.  I drew this scene on the board using the grid zombies and all.  The bridge was four meters long, and three metres above the water.  I had their attention immediately.

 

“Well, how long will the support beam need to be?  And hurry up because THE ZOMBIES ARE COMING!”

If that ain’t motivating what is?  It worked – I had instant engagement.

Continuity

This is the unsexy secret of education.  Learning happens because of continuity.  You need to keep chipping away, using new ideas to keep it interesting.    Also – I have the privilege of working with some fantastic tutors who are working full time.  These guys are great and deal with all the flack, drama and behavioural issues.  They are also teaching and supporting the work that I’m doing so it is all adding up.  They deserve a medal- but will continue to do great work with little recognition.  

Context

This happens to be the week that the students are sitting their assessments so they are motivated. They have questions and they want answers.  Perfect.

The result

Today – everyone got it.  In fact they were correcting me and hassling me for being slow.  Even the student who hates Pythagoras was finding the hypotenuse.  I handed out some problems in which learners had to find the length of the legs (the legs are the sides not the hypotenuse).  Victory.

The problem

None of this was taught via a problem solving approach.  It was very much a watered down version of chalk and talk.  So, chalk and talk works… but does it result in the kind of learning we want or have the learners simply memorised a method?  I’m not sure.

What's really happening in adult numeracy classrooms

I have been asked to present at the National Centre for Adult Literacy and Numeracy symposium in Wellington this year.  I have almost always been asked to speak on a specific area but this year I plan on presenting some findings from my research.

The presentation is called:

                        What’s really happening in adult numeracy classrooms.  

The background to the research is this:  While we were learning maths in school, we were also learning lessons about what maths is, how maths is learned, and what it means to be good or bad at maths.  For many learners who struggled with mathematics these ‘lessons’ have resulted in beliefs that now constrain or inhibit positive engagement with numeracy.   Now in the tertiary sector, these learners are required to re-engage with numeracy.  The question is:  How do adults' beliefs about mathematics relate to cognitive and affective engagement in numeracy contexts?

To answer this question I have entered multiple adult numeracy classrooms and placed microphones around the room.  I then observed the classes, videoed the classes and listened to ALL of the conversations that took place simultaneously during the class.  The recordings reveal what learners really think and do while ‘participating’ in numeracy lessons.  I have also triangulated these findings with surveys and interviews. 

The symposium presentation will cover the findings – which are fascinating.

 

A brief overview

Four themes emerged (some of which I've discussed on this blog).

Pedagogical preferences and expectations
This theme describes how learners approach the learning of numeracy and their expectations for numeracy classes.  It reflects learners’ beliefs about ‘how’ a ‘good’ numeracy class should operate.  Divergences from learners’ expectations are viewed as inefficient or superfluous and therefore prevent full engagement in the very activities that support development.

Reasoning and engagement
This theme describes the discourse patterns that dominated each of the classrooms observed.  There is a quick review here and here.  This reflects the learners’ beliefs about how numeracy is learned and the goals that support these.

Emotion and attitude
This theme emerged in response to the highly emotive nature of numeracy lessons.  You can read more here.  I haven’t released the attitude component yet but it will stir the hearts of tutors (yes, we do make a difference).

Image-management 
This theme describes learners' strategies to either protect, raise or modify their status within the class.  The findings are nuanced yet reveal almost everything about the beliefs learners hold about what being good at math means.  The findings are not predictable and our New Zealand context is different to the States, the UK and Aussie.  We really are wired up a bit differently.

Then I’ll move into a ‘where to from here’ moment, have an open discussion and ‘hey presto’! Done!

Anyway, the presentation should be great.  Bit of drama, bit of controversy, and some very interesting findings.

Love to see you there!    Come and say hi.