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.

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!