Blinded and silenced by a vision of working mathematically

At the risk of being overly dramatic, I can only say that it's been a "road to Damascus" experience. A realisation that everything I did in the first three terms of my teaching career may have missed the point. That shattering moment when someone shows you something so different, you have to rethink everything. 



You want drama?
Nothing beats a Caravaggio.

The strange thing is the vision was always right there in front of me. My teachers regularly presented the idea, I've read the articles,  I've even written essays about it, but I don't think I truly understood the central truth and importance of the idea. Maybe I had to experience the reality of teaching mathematics long enough before I was ready to see clearly. Fortunately I had a chance to hear the message again, this time from Charles Lovitt at the MANSW 2011 conference earlier this month.

The pivotal moment of clarity came when, after we participated in one of his lessons, Charles Lovitt asked us to consider the question: What does a mathematician actually do?   When you unpack the answer, when you look at what "working mathematically" really is about, it raises so many challenges about our classroom practice. About our emphasis on skills and fluency at the expense of understanding, problem solving and reasoning. It offers us a roadmap to a richer and more rewarding experience for all our students. And the part I like the most: it gives a substantial answer to that student who asks "So how is this going to help me in the future?".  The amazing thing is that the answer was there all along, right at the core of our subject. We just had to see it.  

And so what is the answer? And how does it give us this roadmap to richer and more balanced mathematics lessons? I'm not quite ready to put it into my own words. I leave that to Lovitt and Clarke who gave a recent explanation in "A Designer Speaks". 

These snapshots from Lovitt and Clarke's recent article
on  designing rich and balanced mathematics lessons.

Three weeks later, I'm still reeling from the impact of this presentation - and feeling a little blinded and silenced by the vision. It may take me many years of practice before I can speak in detail about it because I think you have to do it before you can share it.  This blog might be a little quieter for the rest of the year while I try to work it out.  What I do know is that all the things I've been working with and writing about this year - student engagement and motivation, standards based grading, using technology in the classroom, and student voice - are not the most important place for me to focus. They are important, but ultimately it is the degree to which they support working mathematically that matters - and this is what will contribute to the bigger picture, to better life long learning outcomes for my students.

And just to ram the message home, there was that final kick from my Year 9 class, who helped me see that my deeds were not living up to my intentions.  I think I'm ready now to start again.

On the Road to Damascus

Here is a set of resources, in the order I encountered them, which led me to this place on the road when I was struck down:

• Paul Lockhart's famous lament.   Make sure you also read the sequel - it's not widely quoted but important for balance.
  Warning: this article can be fatal to some aspiring mathematics education students. I think it was responsible for at least two colleagues leaving my teaching course.
• I wish I could point at a link to a digital scan or even a place you could purchase Lovitt and Clarke's amazing 1980's MCTP resource kit. The closest you can come to it is in its modern incarnation at Maths300.
• Dan Meyer's TED presentation "Maths class needs a makeover" throws down the gauntlet.
• Malcolm Swan's  Improving learning in mathematics: challenges and strategies offers a highly readable exposition of a better way to learn mathematics.
• Peter Sullivan's 2011 ACER report Teaching mathematics: using research informed strategies is well worth the 80 page read - an up to date and highly topical document in the context of the new Australian Curriculum work,
• The Australian Curriculum Mathematics K-10 content page provides an excellent description of the proficiency strands (called Working Mathematically in the NSW BOS update).
• Lovitt and Clarke summarise their work in this "A Designer Speaks article.

Moving out of the way

Reflecting on my dangerous habit of talking too much in class, I found myself remembering a doodle I made two (!) years ago after a great lecture while on my Master of Teaching course:

As we work to become better teachers, inspired perhaps by dramatic "teacher as hero" stories, we can fall into the trap of thinking a better teacher is one who does more, who is more prominent and more active in the classroom. Now I'm beginning to realise a wiser teacher is much less obtrusive. You're still there, you're still doing a lot of work - but you also need to get out of the way and let the student interact with the subject.

Blah blah blah blah .... teacher's voice, student voice

I sort of knew something was coming my way as I handed out the end of term student feedback forms to my Year 9 students:

There was a gentle warning a week earlier when a student handed me this drawing of her impression of my teaching:

A bit hard to explain all the references in this picture.  We have been using a 'save the unicorn' motif (that's a future post) and Justin B. makes regular appearances in topic tests.  "Slow down Mr Zuber" is a sign I made for students they can wave at me any time as a safe way to show they don't understand what I'm explaining.  Thanks to L. for allowing me to share this - and extra thanks for making me look thinner, younger and sort of cool!

And yes - I got some pretty harsh feedback from my students.  While I'm getting good scores on the understanding and the difficulty questions, the percentage of students who are enjoying the class has dropped from around 75% in Term 2 to 50% in Term 3.  No-one is 'hating' yet, but nearly 40% said 'it was OK' - which isn't OK by me. There were also some pretty rough comments in the free text responses. I am indeed talking too much, and not giving them enough quality time to work on their own or with each other, but I'm also getting push back for not using the textbook enough, or doing enough exercises from the book - my 'weird activities' just don't feel like 'real maths' to many in this class. Beyond my own limitations as a new teacher talking too much, I hadn't effectively communicated to the class the reasons why I was doing problem solving and reason activities at the cost of doing less skills based lessons.

Fortunately I had two days to reflect on the feedback before seeing the class again, which gave me time to think more deeply about it - and get over the ego hit :-) I showed the feedback to my head teacher, who also gave me support and encouragement.

So after sharing  feedback with the class, here's the commitment I made to them today:

I realised in my eagerness to help everyone understand the content, I was doing way too much whole-class discussion (to be honest - that's mostly them asking questions and me talking) and this was getting in the way of learning for many students.  So I've resolved to do something about that. Less teaching, more learning. I also started the process today of being more explicit about why we are doing problem solving and reasoning activities, helping students understand why this is just as much 'real maths' as is doing skills exercises from the text book.

The real story I want to share is the value of asking for anonymous student feedback and then responding to it. Don't miss the opportunity - it can be scary sometimes - but it can be very rewarding for you and your class. So many teachable moments - demonstrating to your students your trust in them and the fact that you too are a learner.  It will be challenging at times, and you may well discover that a class you thought was going just fine is actually hiding some discontent, but you will be so glad you took the risk to hear the student voice.

Practicalities

Here's some key tips for getting student feedback:

• Make it very clear the feedback is anonymous. Repeat many times to students they must not write their names on the form.  You don't even want to know who is giving you 'nice' comments. Stay away from the students as they fill it in, and ask a student to collect up the folded forms.  Treat the responses confidentially. A recent addition I made to my form is to have an opt-in tick box in the comments area to ask students permission to share their comments - sometimes they may not want to.
• Share the results with your class as soon as possible - preferably the next time you see them. This shows you take their feedback seriously. Show you can accept - or at least are prepared to  consider negative feedback - and that you are not embarrassed to share this with the class.  Don't allow students to attack negative feedback given by other students - reinforce you accept the negative feedback - even if you don't necessarily agree with - the feedback is valid for the students who gave it - it is what they think and feel.
• Try not to be defensive. If you remain open, there is a good chance you will hear more detailed explanations of the feedback and prompt further discussion. So example today I found out the comment 'GeoGebra is boring' really meant 'You haven't really showed us how to use GeoGebra'.

Want to read more? See my earlier post Putting student voice into practice, which includes links to some resources to make doing student feedback quick and painless.

A 'new' approach to geometric proofs

A brief follow on from the previous post on rediscovering Euclid.

Check this out for a 'new' teaching idea for presenting geometric proofs:

Euclid's proof of the equal angles in an isosceles triangle
(the famous Pons Asinorum),  as presented by Oliver Byrne in 1847.
Image from the Oliver Byrne image project at the University of British Columbia

This comes from the amazing 1847 Oliver Byrne version of Euclid's Elements. I'm thinking a page or two from this image library will make for a great exploration activity with my Year 9 class currently learning about geometric proofs and congruent triangles.

Of particular interest to modern educators is Oliver Byrne's introduction where he argues:

"Illustration, if it does not shorten the time of the study, will at least make it more agreeable. This work has a greater aim than mere illustration ; we do not introduce colours for the purpose of entertainment, or to amuse by certain combinations of tint and form, but to assist the mind in its researches after truth, to increase the facilities of instruction, and to diffuse permanent knowledge." (Byrne, 1847, p vii)

and continues with a decidedly modern take on how using visual imagery aids memory retention and understanding. I love how he denies this is merely a form of entertainment - anticipating the charge of  "mathotainment" sometimes cast on alternative teaching approaches today.

Read the full story at http://www.math.ubc.ca/~cass/Euclid/byrne.html. The German publisher Taschen has recently published a facsimile copy of the work - mine is on order from Amazon!

Euclid who?

Raphael "School of Athens" - detail showing Euclid.

I caught myself out again today - (see future post : "Assume : making an ass out of you and me") : I assumed my students, and indeed my colleagues, knew the wonderful story about Euclid and his five postulates. Assumed they knew how our high school geometry is built on the foundation of these five unprovable axioms. But I'm of course being unreasonable and unfair - Euclid isn't in our school syllabus any more.

While we still ask our top students to replicate and develop geometric proofs, Euclid and the idea of axioms is effectively removed from the content.  There is just a single reference to the word 'axiom' in our syllabus - a background note tucked away in the NSW Board of Studies 7-10 Mathematics syllabus (p161): "The Elements of Euclid (c 325-265 BCE) gives an account of geometry written almost entirely as a sequence of axioms, definitions, theorems and proofs. Its methods have had an enormous influence on mathematics. Students could read some of Book 1 for a far more systematic account of the geometry of triangles and quadrilaterals."   Fortunately these precious words survive on in the new 'Australian Curriculum' version of our syllabus.

But  I wonder. What have we done? Why did we keep the formalism of doing proofs, keep our students busy with it for months of syllabus time while letting go of one of the most powerful ideas in mathematics: the idea of proof built on axioms?  How will our students relish those mind-blowing moments in their future when they encounter parallel lines that do meet, or better yet, encounter Godel or sit in a philosophy class wondering how we know something*, if they don't first meet Euclid? 

And sad to say, I had to confess to myself I couldn't actually remember those five postulates. So it was time for a visit to Wikipedia.  

And then I discovered something I hadn't seen before: 

The Pons Asinorum 
aka "the isosceles triangle theorem"

I smiled.  It was if Euclid himself was winking at me across the millennia. So tomorrow my class is exploring the idea of axioms and getting a history lesson! And like students of many generations past, I think they will appreciate the humour of the Bridge of Assess.


Two interesting resources for The Elements:

A full digital copy of  Oliver Bryne's 1847 famous pictorial version of the Elements (most of the proofs are done without words) is at http://www.math.ubc.ca/~cass/Euclid/byrne.html.  This could make an interesting source document for students.

A good explanation and commentary from David E Joyce http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html - who has an amazing website exploring the whole contents of the Elements.

* Say it quietly : epistemology.

Embracing Rebecca Black: YouTube for statistics lessons

My classroom this year was invaded by Rebecca Black:

Yes - for those of you not living on Planet Earth - that's the latest hit from the "Friday, Friday" girl. Rather than fight it, I decided to embrace it - and you have to hand it to her - whatever you think of the lyrics and the singing, she's got a positive "go get it" energy which is catching.  To give me add some interest to my statistics lessons, I've been collecting Rebecca Black statistical data by hand from YouTube - visiting the site each day recording each day the number of hits, likes and dislikes.  It was getting a little tedious and then I noticed something wonderful - just quietly sitting there in the bottom right corner of the YouTube screen:

YouTube - ready for your statistics lessons!

Click the icon and you get something like this:

Yes - that's right - YouTube has real world data about things your students are interested in - just ready to put into your statistics lessons. Lots of possibilities for rich questions or skills practice. Here are some thoughts on how you could use this:

• Compare the statistics between two similar songs.
• Compare the statistics between two successive songs by the same artist.
• Discuss the reliability of the data.
• Discuss 'likes' and 'dislikes' : what is this data actually telling you? Consider the percentage of like/dislike comments compared to the number of views? This could lead into a discussion about online participation.
• How is the gender and age profile different to that of another artist?
• How does YouTube even know the gender and age profile of visitors? How accurate is it?
• How does YouTube know what country viewers live in?
• What quantitative or qualitative questions do you have about the viewers of an artist? How would you answer them?
• A homework activity: look up your favorite artist - make a PowerPoint or paper poster about your statistical analysis of their YouTube hits. This could be extended as desired.

The secret to coping with Rebecca Black is to enjoy it even more than your students - start singing "Friday Friday" yourself - and take pleasure annoying your faculty colleagues - just don't overdo it.

Warning: There are many many hazards using YouTube live in your classrooms. However YouTube is just too valuable not to use it - so you need to be aware of those hazards and ways to work around them. I'll be discussing strategies I've learned in my classroom in a later post.

IWB Tips: Making invisible algebra visible

Second in a series on quick easy tips to get more from your Interactive Whiteboard + SMART Notebook software - mostly for maths teachers but might apply to other subjects too.

A perennial challenge when teaching algebra is getting students used to the conventions - and in particular, the conventions of what we don't show.  When I look at the algebraic expression "3x", I don't just see a '3' and an 'x' - I see, or at least I know, this is 3 times x.  And when I see x on its own, I know it's the same as 1x.  The challenge is to help our students see and understand the presence of these implied algebraic ideas amongst the more visible characters.

In my class, we have names for these algebra conventions:

If you're wondering about that hat ... see my post
on how the royal wedding helps teach algebra.

The joy with an IWB + SMART Notebook is you can show the invisible symbols. Here's how....

Mr Zuber:  " ...and don't forget the invisible one. Can you see it? I can see it". 

Step 1: Can you see the "invisible 1" ? It's there! Really!

Student#1: "I can't see it! Where is it?"

Student#2: "Show it! Show it!"  (they all know what is coming now!)

I switch my SMART Notebook pen to the Magic Pen, 

and write in the 'invisible' part of the expression.

Step 2: Use the Magic Pen to write the 'invisible 1'. 

The whole class holds their breath in anticipation .... waiting, waiting ....  and sure enough, five seconds later, the Magic Pen marking fades - and my invisible 1 is gone.

Step 3: Five seconds later - the 'invisible 1' has disappeared.
Back to where we were - but with the 'invisible 1' in place!

But I can now quite reasonably continue talking as if it's really there - it was there wasn't it? Did you miss it? Maybe you weren't watching? :-)

Any time I want to show 'invisible' elements, or hidden, assumed conventions, I use the Magic Pen to temporarily write them in. It really grabs the class attention and drives the point home.  OK - it's only a little gimmick - but it seems to have a real impact. Something about the anticipation of waiting for the fade, and seeing it fade automatically, combined with just the sheer fun of the trick really does seem to drive the point home.  I've been using the Magic Pen this way for six months now and my Year 8 class still  hasn't tired of it - they still watch, wait, and then ooh and ah and laugh when the text disappears. Indeed, whenever I mention the invisible one or the invisible multiple sign, they usually insist I demonstrate it. Funny thing is, even Year 11 students, who are "way beyond childish things" still get a chuckle from the Magic Pen.

My thanks to my wonderful colleague Ms Tran who gave me some early lessons on using SMART Notebook and showed me the power of the Magic Pen. The Magic Pen changes function depending how you use it: if you write with it, you get disappearing ink. Try drawing a circle or a rectangle to see some other fun tricks.

What's in a word: low ability or low achieving?

While working on a paper I'm writing, one of my teachers suggested I change the words 'low ability' - as in 'low ability students' -  to 'low achieving'. The thought 'need to be politically correct' popped up immediately - but then I did a double take ... is it really just about using socially acceptable labels? Or does changing one word actually make a difference?  Reflecting further, I've come to be conclusion it makes a huge difference - especially in the context of mathematics education. 

"low ability students", "low ability classrooms" : says there are limits to what can be achieved with these students, says there is a limit beyond which further effort from the teacher is wasted.  "Low ability" says there is a limit to the learning that is possible for this student.

"low achieving students", "low achieving classrooms" : says the students are not meeting the outcomes we would expect students of this age group to achieve. "Low achieving" forces us to consider why they are low achieving. Are there problems with engagement? with effort? with attitude? with learning strategies? with the teaching? Are the outcome expectations reasonable? We no longer attribute low achievement to limited student ability, or at minimum, we are prepared to consider other factors are at play. 

Almost all secondary school mathematics faculties sort students into streamed classes based on previous mathematics achievement. Although the sorting is based on achievement, it is all too easy to accept this a proxy for mathematics ability - and it doesn't take long before we talk (discretely) among ourselves about our "low ability classes" and our "low ability students".

By focusing on the 'achieving' word, rather than the 'ability' word, we can better access other important teaching and learning ideas in our mental framework:

• Andrew Martin's work on student motivation and engagement, which encourages students (and teachers!) to see performance as a result of effort, strategy and attitude;
• Anders Ericsson's important work on expertise and ability - which shows how even the people we think of as having 'natural ability' require serious effort in deliberate practice to reach their potential;

and more fundamentally,

• John Mighton's questioning of the very idea of mathematical ability - as presented in his books The Myth of Ability (2004) and The End of Ignorance (2011).

It's more than changing one word - it's changing your mindset. You won't hear me saying 'low ability' ever again.

How to hypnotise your class

This amazing Pendulum Wave video clip can totally transfix even the most hyper or disinterested class.

You can discuss what is happening on so many different levels. For students who don't believe it's real (some are absolutely certain it's special effects or computer generated), suggest they focus on just one pendulum.

Yesterday one of my students cottoned on to my game : "Are you trying to hypnotise us sir?" I smiled, and suggested she return to focusing on one of the pendulums.