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.

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.

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.

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.

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:

Working Mathematically - a picture essay

Click on an image to see a larger version.

See  Australian Curriculum Mathematics K-10 content for definitions, and Peter Sullivan's 2011 ACER report Teaching mathematics: using research informed strategies for a deeper discussion.

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.