Saturday, December 10, 2011

The gravest sin : favouritism

"You're a great teacher, but you have one problem - you have favourites"

When I read this feedback in the anonymous survey forms from five students I was shocked and disturbed - favouritism is one of the worst things a teacher can do - it distorts and poisons the classroom. What surprised me was that a teacher could have favourites without really being conscious of it. But I knew it must be true when so many students had written this in their feedback. 


It didn't take me long to realise the root cause : as a new teacher with a great class, I've made the mistake of not imposing a strict 'hands-up' rule during whole class discussion - with the consequences that a group of (well intentioned) students have dominated these sessions by calling out. And so week after week, the quieter students at the back of the class have been watching silently, building up the idea I preferred the ones in the front. Did I have favourites? I may not have thought so, but it surely worked out that way.

There's nothing for it when you get feedback like this - an immediate apology is required. I shared with the class my distress at having made this teaching mistake and apologised.  I thanked them for their trust and openness with me - and how grateful I was that they helped me be aware of this problem with my teaching. I was careful to put the blame where it rested - on me - and not on those students thought to be the favourites. I made a committment on their behalf to do better with my classes next year.  By small way of recompense - and as a test for myself to see if I could do it -  I made up a set of thirty "Thank You" cards and wrote a personalised comment to each student, taking care to demonstrate through the comments that I had indeed noticed what they had contributed to the class during the year. 

I'm so grateful my students gave me this feedback, and I really do believe the only reason they told me was because I collected anonymous feedback from them several times throughout the year. Without this framework and trust, students would never have dared tell a teacher about this until it was too late. Without the feedback, I could have blissfully continued unaware making the same mistake for years - building up a reservoir of hurt and misunderstanding along the way.

Was it all bad feedback from this class? Not at all :-)  Overall, the students really enjoyed the year and gave lots of positive feedback and encouragement. But as always - the most valuable feedback is the one that helps you see what didn't see before - appreciate the positive feedback, but it's the uncomfortable feedback that pays the biggest dividend.

Reaching the end of term for 2011? Try some anonymous student feedback - you will be amazed what you discover! Take the plunge - you will never regret it. Every time something special  happens - for the teacher and for the students. See my post on getting student feedback for templates and some suggestions. 

So what can I do better next time to avoid favouritism?
  • Use the "hands-down/paddle-pop stick" system for asking questions. I had actually done this for the first month with this class but as we got more comfortable dropped it. Bad decision.
  • Enforce "hands-up" for when students want to ask questions. How silly I feel writing this down - it's Teaching 101. I thought I didn't need to do this for this class - they were such enthusiastic students and it felt too authoritarian. Wow did that idealism bite me back!
  • Split up or move dominant groups to a less prominent position in the classroom.
  • Ask myself : have I visited each group table during this period? Have I asked them questions? Have I given them some of my time today?

Saturday, December 3, 2011

End of year maths classes: a precious opportunity

If you live in the Southern Hempisphere and your school is anything like mine, everyone is winding down as the holidays draw near. Final tests and reports are done, and the students have decided there is no more formal learning for the year. I'm sure the same effect happens in the Northern Hemisphere around July. For some teachers the response is to provide 'find-a-word' worksheets or watch an entertaining video. And while it is a struggle to get students in this frame of mind to do any work, I've come to realise this "winding down" period is a fantastic and precious opportunity to do some maths beyond the confines of the standard syllabus. Now is the perfect time to bring out your big gun 'fun' mathematics activities.  




Some ideas for end of year activities

Bring out your concrete maths objects and just let students play: this week my Year 7 students helped me unpack a recently arrived box of 250 GeoShapes.  Once they saw someone construct a dodecahedron, there was pandemonium in the class for the fifty minutes while groups traded, cajoled and bargained to obtain the required 12 pentagons from the surrounding tables: "I need more pentagons!" demanded a student who had never used that word before. Students who didn't have the required shapes tried to build them using other shapes. One group wanted to know why they couldn't make a 3D regular solid with hexagons. I think this class have a better understanding of prisms and polyhedra than when I was actually 'teaching' them the topic.  From what I observed, just letting students 'play' with my concrete object kits, without any overall objective, produced very interesting results. If you feel the students aren't challenging themselves enough mathematically, join a group and ask a few questions, or just quietly construct  something interesting and then walk away. I really think we don't give students enough time to 'play' and get a feel for these mathematical objects.  So this end-of-year 'winding down' time is the perfect opportunity to have some quality play time.  Play helps older students too - I found even my Year 11 students gained benefits from more hands-on time with the concrete objects. 

Don't watch "The Lion King" - share your favorites maths videos and digital interactives: choose the right material and the response can be surprising. I showed my Year 8 a section of Marcus du Sautoy's "The Code" on the mysterious places π turns up. Once the class got over groaning that were going to watch a maths video ("Can't we watch Harry Potter?"), they were quickly drawn into du Sautoy's 'spooky' presentation. "Is this going to be scary, sir?". And they were hooked! The students were riveted by the exploration of how strange and interesting the number π was and demanded to watch more of the video: looking at the mystery of negative numbers and I even let the video keep going into imaginary numbers ("This is Year 12 maths", I said,  but they insisted on watching it). We then zoomed in and out of an amazing digital π poster (π to around 350,000 decimal places) and looked at a Buffon's Needles simulation for generating π.  The questions and conversations this material produced was amazing. Students who previously were bored or disengaged were asking very deep questions about numbers: "How do you know the decimals go on forever without repeating?", "Why is the ratio always π?"  du Sautoy's presentation and the follow up material really had stimulated thinking and wonder about mathematics.

Bring out your maths games :  I'm a huge fan of SET. I never cease to be amazed how students who don't like maths, or say they can't do math problems, can get hooked on SET. The secret is how you introduce the game and ramping up the complexity carefully. (I will write some more about this in a later post).

So - use this precious time - bring out your favorite maths activities, 'toys', games and videos - and very soon you will be wondering why you don't do this through out the whole year!

Still have syllabus content to get through these last weeks? I do! So I'm blending these activities into the new content I still need to teach. I'm doing circumference and area of a circle with my Year 8, hence the selection of π materials.  I have a sense they are going to learn this topic better than many others I did this year because I'm using so much concrete material and interesting, challenging digital material.

Saturday, November 26, 2011

At play in mathland

I recently started a separate blog At Play in Mathland to store and share maths problems I like.

Mathland is like Homer Simpson's Land of Chocolate - except with maths.

In words of Seymour Papert, "If we all learned mathematics in math land, we would all learn mathematics perfectly well".

With apologies to Matt Groening.

Wednesday, November 16, 2011

SBAR with light bulbs and spanners

So how has my experiment with Standards Based Assessment and Reporting been going? Here is the first in a series of reflections.

Half way into my first year using outcomes sheets with my students as the basis for Standards Based Assessment and Reporting, I realised the outcome lists I was making every few weeks were really just a list of skills I expected students to master. When the penny finally dropped that skills were only one part of Working Mathematically, I realised my outcomes sheets had to change.

Here is my first attempt to be different - which I've been doing now with my outcomes sheets for a few months:

Click on the image for a full sized image.

The key idea is to separate outcomes into the categories of the Working Mathematically proficiencies:

A light bulb icon indicates an outcome that requires some new understanding of an important idea.


A spanner icon indicates a skill to be acquired - 'fluency' in the Australian Curriculum description.

A balance scale indicates that a reasoning process is being used.



How has using light bulbs and spanners changed teaching and learning in my classroom?
  • Clearly showing the understanding and reasoning outcomes forces me to focus on these important elements. If I find my outcomes sheet for a new topic is full of spanners and no light bulbs or balance scales, I know I've made an unbalanced teaching sequence.
  • It sends a clear message to the students that understanding and reasoning are important - it's not enough to just be able to mechanically follow a process to get an answer to an exercise. I will be expecting them to be able to explain and reason.
  • Any time during a lesson when I'm about to introduce or consolidate the development of an understanding outcome, I stop for a moment, and point to it on the outcomes sheet, making it very clear to students were are working on a "light bulb" outcome. I emphasise this means it's a time for quality intellectual engagement: thinking, listening and asking questions. While we can get the idea behind a skill outcome by reading a text book, or perhaps watching Salman Khan do it on YouTube, the understanding outcomes are much better learnt interacting with peers and the teacher.
Where is Problem Solving?
The big challenge I'm facing now is how to integrate the problem solving proficiency into a set of outcomes related to a content heavy topic.  What this really reflects is the fact that real problem solving (beyond just "harder skills questions") aren't yet integrated into my content heavy program. For now, I'm experimenting with specific Problem Solving lessons which stand outside the regular content sequence - and that's something I'm going to work on in 2012.

You may notice the outcome sheet above doesn't make provision for recording quiz results - which you would normally see on my sheets. That's because for this course I'm actually not permitted to use SBG, but have to follow a statewide assessment method and schedule. But this doesn't stop me using the idea of standards, or using them for formative assessment. More on this in the next few posts.

Saturday, November 12, 2011

In my toolkit: JFileSync

Been a while between posts - school year is getting very busy and as usual I'm trying to do too many things. 

The problem: I have three Windows computers I use all the time, in conjunction with an array of USB portable disk drives. I want to be able to use which ever computer or disk drive I have at the time, while keeping everything in sync. To make thing harder, I don't have admin rights to some of those computers (read: school issued laptop *).

The solution. A free open-source Java application called JFileSync.


Some tips to make the data management painless:
  • Nominate a portable USB disk drive as your master data repository. Give it a name - this gives it a distinct identity, helping you form a mental image of the data store as a distinct entity. I call mine 'wombat' - and I have stuck a label on it.
  • Carry your portable USB disk drive in a protective case. They are robust, but still vulnerable.
  • Don't use a USB memory stick as your master data store - while they are convenient, they are also fragile. I have seen several teachers lose their entire collection of vital documents when their USB memory stick died. Just too fragile - a zap and they are dead.
  • Any time I sit in front of a computer, I plug in that drive and immediately run JFileSynch to resynch that computer to the current state of my USB disk drive.
  • I work on the computer's local drive.
  • When I'm about to leave that computer, I run JFileSyn again to get my USB disk drive up to date.
If I don't have my USB disk drive on me, or I forget the ocassional JFileSync, it's usually fine to just resynch the next time - JFileSync will 'do the right thing'. The only challenge is if I edit the same file on another device - because JFileSynch will not merge files - and then you have to think. Best not to do that.

I've been using JFileSync now for three years and couldn't recommend it highly enough. And as a side benefit, I have four copies of my data in several locations. If worst comes to worst, there should always be a place I can recover my data (barring an asteroid strike on Sydney - in which case I'm dead anyway).

* If you have a NSW DEC laptop: JFileSync is a java app - so you can just put the Java file somewhere on your laptop and run it any time - no install required.

Sunday, October 23, 2011

Getting personal with rates of change

The key to mastering calculus seems to be gaining a good understanding of rates of change, how this relates to the idea of a function and then seeing how we can use the tools of algebra and geometry to develop the gradient function. Now as exciting as it is for some of us to play with a quadratic or a cubic function, I recently discovered, quite by chance, a very personal and highly engaging way to explore rates of change of a function.

This data is all about ME!

I was exploring the introductory concepts with a Paul*, a teenager just starting on the calculus road, when he made the connection that he was experiencing a very dynamic change process: his height had started shooting up in the last few years and very soon he expected to be nearly 2m tall.  He knew he was experiencing a "growth spurt", growing at a faster rate than when he was younger. Just as I was wondering how we could use this connection in our exploration, Paul told me his parents had been marking his height on the kitchen wall for the last 10 years. Wow! This was exciting - some real data we could plot and explore.  Paul measured the markings off the kitchen wall and made a table showing his height at different ages. And here's what we were able to do with that data in GeoGebra:




In the process of this exploration we uncovered many ideas about slope, functions and the use of modelling, each time applying them in a context Paul had a profound personal interest in - we even named our  polynomial the "Paul function" in his honour. I'm certain he will never forget the idea of rates of change, the gradient function or the power and fun of modelling a function based on data points.

Do your students have a wall somewhere with their heights measured over the last 15 years? If so, I highly recommend working this into your calculus activities.

I'm a deeply indebted to the work of Mary Barnes, in particular her 1999 series "Investigating Change", on teaching and learning calculus, still available at Curriculum Press. Google Books previews are available. Thanks also to a new Google+ friend and teacher Steve Phelps for showing me how to use the GeoGebra FitPoly function. Using GeoGebra to build a gradient trace function comes from the original GeoGebra documentation by Marcus Hohernwater, however I have found in practice it can take students some time to understand what is going on. Priscilla Allan's YouTube demo shows a good way to use colour to make it clearer and I have extended her idea to actually show the trace point moving along the x-axis, prior to adding in the gradient value. 


And special thanks to Paul (*not his real name of course) and his parents for allowing me to share this story of our ongoing exploration of mathematics.

Saturday, October 15, 2011

IWB Raw: Demonstrating similar triangles

Here's my first ever (be kind!) screencast showing how I work with my Interactive Whiteboard + SMARTNotebook software to turn static diagrams into something that hopefully shows ideas more clearly:


Friday, September 30, 2011

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:

Saturday, September 24, 2011

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.

Thursday, September 22, 2011

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 'hatingyet, 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.

Wednesday, September 14, 2011

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!

Tuesday, September 13, 2011

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.

Sunday, September 11, 2011

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.

Wednesday, September 7, 2011

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.

Sunday, September 4, 2011

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

Saturday, September 3, 2011

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.

Tuesday, August 23, 2011

Magic Expanding Chelonian - Day 3

Here is a compilation of images over the 3 days. The scale was preserved by using the Day 0 chelonian as a reference.  So the question: How valid was the claim it would grow 600%?  Do I need to go back to the $2 shop and demand my money back? Keen to see what my Year 9 maths class makes of this one....

Click for a full size image

Actually - they won't be getting this ready-to-go picture - that's way too easy. Instead they will be offered 4 raw photos of each day, each picture with a Day 0 turtle for scale reference, as well as access to some real-life expanded chelonians if they want to try some hands-on physical measurements.  If this activity works, I might go out and buy another 15 chelonians.





Saturday, August 20, 2011

Magic Expanding Chelonian - Day 1

This magic expanding chelonian is turning out to be even more fun than I expected.... although it is kind of sad what a math teacher does for kicks on Saturday night...

Download GeoGebra file 
I'm guessing the promised 600% growth in 72 hours will be by area - or maybe even by volume. I'm getting scared ....

Am I being too negative (about you-know-what)?

Some days will be rainy days.
So am I being too negative in my recent posts about what I believe are some of the barriers to using laptops in mathematics classrooms?  Some responses to recent feedback. Please feel encouraged to contradict (or support) by posting a comment!

I'm an unashamed enthusiast about laptops. I'm thrilled to have students with laptops in my classroom - along with my Interactive Whiteboard and my HoverCam we can go many places which would be much harder to do without them. I can't imagine not having the technology - and I'm frustrated on a daily basis that my Year 7 and Year 8 students don't have laptops. But what about teachers who aren't so keen on the laptops?

An honest appraisal of what is happening is a prerequisite to understanding and to encouraging change. I really don't think this is being negative. Mathematics teachers have reasons for not using the laptops - they are motivated by concern for the best learning outcomes for their students. I'm not saying I agree with those reasons, but I respect and understand these views. In many of my interviews with teachers not using the laptops, they shared their strong sense of hurt and feelings of being ridiculed or ignored by their technology-enthusiast colleagues.  I suggest we need to work with those concerns - and not avoid discussing them.

About the evidence:
  • Where is the evidence for lower use in mathematics classrooms? See the references in the earlier post - they are the major longitudinal studies done in the US. They are conducted by researchers with an active interest and long term commitment to studying 1:1 programs, some of whom have been involved in technology and education study for decades - and typically with a positive disposition to the programs. Their evidence is credible.
  • Isn't that old evidence? Yes and no. These reports were published between 2005 and 2010 - and most likely this data was already a year old when published, however, they do reflect findings after many years of 1:1 laptop programs in the USA. In my personal experience, I do see laptop usage increasing over time - I see it in my own faculty - so we can hope there is greater use being made of these new resources.
  • Aren't we different in Australia? Yes and no. We do have a lot more support in our 1:1 programs, with major funding for professional development, great technical support and a vast array of teacher support resources. The challenge is not the funding - the challenge is engaging a time-poor and typically conservative profession to participate in the change.
  • "But you only looked at a few schools". Absolutely - I was an honours student with a research budget of $0. I make no claim to a representative sample or exhaustive research. All I can say is: "Do  my findings suggest credible reasons why we have lower levels of use of the laptops in mathematics classrooms? Are they consistent with existing research on 1:1 laptop, with mathematics teacher responses to curriculum and technology change?"  There is a wide body of evidence, beyond the studies of 1:1 programs, about mathematics teacher beliefs in their response to technology and curriculum change. In considering our response to mathematics curriculum change, the work of Michael Cavanaugh (2006) is particularly interesting. For recent comprehensive longitudinal research on mathematics teacher beliefs in relation to technology, see the work by Australian researchers Goos and Bennison (2007, 2008) and Pierce and Ball (2009).
  • What I'm not saying in my research: I make no claims my work can be generalised to all schools:  my sample is too small, the duration of the work too short, the methodology too limited. I make no claims on laptop usage or teacher beliefs for other school subjects - I haven't read or studied any of this material. And I make no claim about outcomes or the value of 1:1 laptop programs.
  • Why pay attention to this exploratory research? It may offer some insights into the reasons behind the previously established lower levels of laptop use by mathematics teachers in their classrooms - and may help us consider what to do about it. There is something different about mathematics when it comes to 1:1 laptops - and the question needs answering. I argue some key beliefs unique to mathematics education in combination with some specific mathematics teaching practices create a strong barrier to widespread adoption of the laptops.
  • "In my faculty we use the laptops all the time with all our classes". Yes - there are mathematics faculties where full use of the laptops is being made, where teachers are actively collaborating and supporting each other. We need to get their stories out into the wider mathematics teaching community - told by maths teachers to other maths teachers. And remember : many of the teachers not using the laptops with their students aren't online. Talk to some maths teachers who are not in your online network, who are not in your faculty (if your faculty is one of those actively collaborating on using the laptops) - ask them the questions: What does a student need to do to learn maths? Which  classes will most benefit from exploration activities? Which classes will get most benefit from using the laptops? I think you may be surprised - as I was.
  • Won't discussing barriers to laptop usage pander to those who don't want the laptops? People hostile to laptops don't need me to justify their reasons - they have more than enough arguments already. And they probably aren't reading my blog :-) Indeed, it's unlikely you are reading this blog if you are strongly skeptical about using technology for mathematics learning.
So - what can we do? Talk to our mathematics teacher colleagues about the things that concern them - not us.  Better yet - don't talk - show.  Show ways in which laptops can and are successfully used with lower achieving students. Show how laptop use can be integrated into current pen-and-paper practices. Just today I read a great post about getting students to keep a learning journal while doing online learning. Small changes and practical ideas like this can make a big difference in encouraging teachers to try out the technology.

Last word goes to a more experienced and no doubt wiser colleague @Deborah_morton. How do we present the message?  Deborah Morton suggests there are three key elements needed in a presenter of ideas trying to encourage change:  inspiration, trust in the presenter and evidence it works. We can do this!




Do you have a story about using the laptops in a mathematics classroomm you would like to share but don't have time to get a blog going to share the it? Happy to give you a guest slot here!

Further reading
Cavanagh, M. (2006). Mathematics teachers and working mathematically: Responses to curriculum change, in  G. Grootenboer, R. Zevenbergen & M. Chinnappan (Eds.), Identities, Cultures and Learning Spaces (Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia) Vol 1, pp. 115-122. Melbourne: MERGA.  http://www.merga.net.au/documents/RP102006.pdf

Goos, M., & Bennison, A. (2007). Technology-enriched teaching of secondary mathematics: Factors influencing innovative practice. In J. Watson & K. Beswick (Eds.), Mathematics: Essential Research, Essential Practice — Volume 1, Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia (pp.315-324).  Wahroonga: MERGA. http://www.merga.net.au/publications/counter.php?pub=pub_conf&id=398

Goos, M., & Bennison, A. (2008). Surveying the technology landscape: Teachers’ use of technology in secondary mathematics Classrooms. Mathematics Education Research Journal, 20,(3), 102-130.

Pierce, R., & Ball, L. (2009). Perceptions that may affect teachers’ intention to use technology in secondary mathematics classes. Educational Studies in Mathematics, (71)3, 299-317

Weston, M.E., & Bain, A. (2010). The end of techno-critique: The naked truth about 1:1 laptop initiatives and educational change. Journal of Technology, Learning and Assessment, 9(6), 5-25. Retrieved Aug 14, 2011 from http://www.jtla.org.

WCYDWT? Magic Expandable Chelonian

What can you do with this?

The Magic Expandable Chelonian : Grows 600%!
(Do not swallow!)
Yes - it's fun AND educational too!

My head is spinning with possibilities!  I'm very curious to see the questions my different maths classes ask. I bought three of these today at the $2 shop - the plan is to show the class the original packet then see what happens. I'm hoping to try out some version of Dan Meyer's WCYDWT rubric and half way through the period bring out some turtles "I expanded earlier". The packaging suggests dramatic growth in 48 and 72 hours - so I'll get some going a few days earlier.

It's fun and educational - or so the packaging claims. To be honest, I don't even mind if it's not educational - I just want to see what happens to the turtle! And what would happen if I did swallow the turtle and then drank a litre of water?

Next in this series: The Magic Expanding Chelonian - Day 1

Thursday, August 18, 2011

Why is maths different when it comes to laptops?

Continuing the series on 1:1 laptops in the mathematics classroom. This post may be a little uncomfortable for all of us, but the factors considered come up again and again for all mathematics teachers - even those of us (like myself) who have drunk the Cool-Aid and are eagerly looking for ways to enrich our teaching through use of technology. In later posts we shall consider the wonderful and amazing things mathematics teachers can  do with laptops - but first we need consider some of the barriers.

In the previous post, we looked at what for many people is an unexpected finding: mathematics teachers have their students use laptops much less than teachers in other subjects. Some reports put this figure at 50% less than other subjects.  When all other factors are taken into account - access to technology, training, confidence, skills - we still find a reluctance to use the 1:1 laptops in the mathematics classroom. And so we ask: Why do maths teachers make these decisions? Is there something different about mathematics?

My research has led me to conclude there is indeed something different when it comes to mathematics.

Mathematics teacher beliefs + mathematics teacher practices :
a powerful combination which often acts as a barrier to using technology.

A core set of beliefs about mathematics and mathematics teaching in conjunction with some strongly entrenched mathematics teaching practices act together as a powerful barrier to widespread use of the laptops in mathematics classrooms.  I see the three key themes at work:

  • "Maths is something you do on paper"
  • "Laptops aren't suitable for low achieving students"
  • "The teacher leads, the student follows"

I make no explicit comment on the validity or otherwise of these commonly held mathematics teacher beliefs and teaching practices - but there is no getting around their effect on 1:1 laptop programs.

"Maths is something you do on paper"


When you ask mathematics teachers what 'real' maths is, and how you 'really' learn it - pen and paper, and I really mean pen and paper the physical media - eventually emerge as a key requirement. Software may be fine to demonstrate and maybe explore mathematics (for some mathematics teachers)- but it's not properly learnt until it's done on paper.

Now consider the very strongly established practice of managing student learning by working in and monitoring output in the student exercise book. Learning outcomes aren't tangible - can't be verified until they are seen in the exercise book.  Entire sequences of classroom practice, homework, outcomes tracking are based on physical movement in and around the student exercise book. You won't find this combination of beliefs and practice in most other subjects. No-one would say you don't understand science, or history, unless you do it on paper. And other subjects are much more receptive to accepting digital learning artifacts as evidence of learning.

Unfortunately there is also a technical hurdle : unlike other subjects, writing in the language of our subject with a computer is hard. It's actually very awkward to write a continual flow of mathematical ideas with standard or even specialised software. 
Try writing this   without taking a software detour. Now do twenty lines of it. For now at least, the technology gets in the way of expressing the ideas. While there is powerful and non-intrusive software such as GeoGebra for exploring and demonstrating some parts of mathematics, actually writing long sequences of mathematical language is hard work on the computer.

So: combine the belief that real mathematics is done on paper with a key teaching practice based on writing in exercise books and there isn't much space left for using laptops beyond peripheral extension activities.

"Laptops are not suitable for low achieving students"


We have a real problem in secondary school mathematics: many students are not achieving the learning outcomes. It's no surprise these students don't enjoy maths and are looking for other ways to occupy their time and energy during maths class. Now ask mathematics teachers if using laptops might help make the classroom more engaging, or possibly even provide new ways to help these students with learning mathematics. The answer is a pretty resounding 'no' - there is a widely held belief that laptops are not suitable for low achieving students. Two lines of reasoning are offered: the low achieving students are actually incapable of using the software; and the low achieving students are using the laptops to escape from mathematics and instead engage in off-task behaviour - watching videos, listening to music, playing games. Not like the high achieving students who want to use their laptops for maths.

Some mathematics teachers strongly believe it is in their lower achieving students' best interests to turn off the laptops. These students need to do more maths, and allowing them to use the laptops, which provides more distraction, is actually harming them - teachers motivated by care and compassion for their students make the decision to block use of the laptops. Personally - I don't agree with this approach - indeed I believe the laptops offer us possibilities to re-engage students with mathematics - but this reaction is understandable and consistent with those teachers' beliefs.

Now consider the strongly entrenched teaching practice of  ability streaming, used in mathematics faculties across the country almost without exception, and to a degree not seen in any other school subject. We put the highest achieving students in one class, and then progressively lower achieving students into progressively "lower" class groups, creating entire classes of disengaged, low achieving students.

Combine the belief that low achieving students can't or won't use the laptops for learning with the practice of ability streaming, and we have effectively created entire classrooms where the laptops just will not be used. And indeed this seems to be the case.  Chances are when it comes to secondary mathematics, you will see the laptops being used almost exclusively in the top achieving classes.

"Teacher leads, student follows"


And finally, we consider the strong prevalence in secondary mathematics education of the idea that the teacher should show-and-tell, and that students should follow-and-practice. While it would be an overstatement to say this is always the case, it is the prevalent belief among maths teachers.  A student armed with a laptop can be disruptive to 'teacher leads, students follows' - and although the presence of the laptops doesn't automatically guarantee a change in pedagogy, the benefits of the laptops seem to me to be diminished if they are merely used to automate lead-and-follow practices.  This combo of belief+practice isn't unique to mathematics teaching by any means, but I do think we are more likely to follow traditional teaching and learning approaches than other subjects.

In conclusion ...
So by considering these three powerful belief+practice combinations, which are to a large degree unique to secondary mathematics education, we can begin to see just why laptops are used at up to 50% less than in other subjects. I find that even in my own practice, keen as I am on using technology with my students, I'm often falling into these memes: I do worry about not doing the maths on paper - "is it real maths?", I do worry about exercise books, and I do find myself dismissing using the laptops with my lower achieving students. And I catch myself 'holding the mouse' often.

Am I optimistic about using technology in the mathematics classroom? Absolutely. But I also recognise there are powerful beliefs and practices in our subject domain - and these contribute to making using the laptops harder in mathematics teaching and learning.

This post is high level summary of research I conducted during 2010. The study examined the use of the laptops in mathematics classrooms at five schools, looking at usage levels, how the laptops are used and the role of teacher skill, confidence, knowledge and beliefs factors. The work builds on a body of knowledge as found in nearly 100 published papers on technology in education, mathematics teaching using technology, and mathematics teaching beliefs and practices. An academic paper is currently in preparation.