Sunday, March 3, 2019

The ultimate formative feedback? Tales from the whiteboard classroom.

The idea is extremely (and deceptively) simple: Cover your classroom with whiteboards, then think about how to get students to spend most of their time standing at the whiteboards, rather than sitting at a desk listening to you talk from the front.

A "fish eye" view of my whiteboard classroom - Dec 2017. 
There are 7 whiteboards (not all are shown here) and the room comfortably fits 16 students, and 24 at a pinch. In mid 2018 I removed half the desks to make more space.

This was the challenge posed to me by Tricia Forester (University of Wollongong) at the 2017 MANSW Conference, referencing the "Building Thinking Classrooms" work of Peter Liljedahl.  Well actually, Tricia was more clever than that - she posed interesting maths problems to groups of teachers, providing a "vertical non-permanent surface" space to try it out.  It didn't take long to see the power. When I got back to school, my principal gave me permission to try it out for real, transforming a smallish classroom no-one really liked into a fully fledged whiteboard room with 7 large boards.

It's now been nearly one and half years that I've been working in a whiteboard room for my senior classes (unfortunately I can't fit 30 junior class students in the room) and there's no looking back! There are many things to share about the experience and why it supports such powerful learning, but for now I'll confine myself to a few big ideas.

The power of the whiteboard room is the way it provides continuous formative feedback - to both the teacher and the student. In fact, for the teacher, it's overwhelming - you will be bombarded by feedback. You instantly see what students can and cannot do, you see the approaches they take and you hear many simultaneous conversations discussing the problem. For the whole lesson. It's almost too much feedback. In the whiteboard room, we have a "live stream" answer to the "How do I know what my students learned?" question. I don't have to give them a diagnostic test every day - I can see it every minute. This continuous formative feedback allows me to dynamically alter the lesson according to needs of the students. It's a bit of a high-wire act some days, and often full of unexpected surprises, but it makes for exciting and rewarding lessons that students really enjoy and demonstrably produces learning. I'll write some more soon about the strategies I've developed to dynamically respond to so much continuous feedback.

Students working in "interleaved" mode on a problem.  The students worked out on their own that harder integration questions are easier done in "parallel" by two students - but they still end up correcting each others work! Different colour markers let me quickly see how (or if)  students are working together.

The student engagement factors at play in the whiteboard room, especially when combined with the Visibly Random Groups technique, are truly astounding. I have seen students who previously had difficulty making friends,  difficulty communicating, who were disengaged, lacked confidence, transformed by the whiteboard room experience. The combination of the social interaction, the risk free "non-permanent" writing surfaces and physical movement in the whiteboard room seems to work magic on students. One day a learning support teacher came into the room and asked a notoriously lazy student why he liked the room so much, he replied "Because I can't sleep in the back of the classroom any more!".  Everyone, without exception, is engaged in the whiteboard room - there's no option to not be involved *.

We also use the whiteboard room for our before school "Year 8 Algebra Workshop" - a weekly session for students who need a little more support with algebra. The whiteboards allow them to work together while developing fluency. The social interaction and support created by the whiteboard room is an essential component of this program which seeks to build confidence and a growth mindset attitude.

Is this "student-led" or "teacher-led" teaching? I think it's a blended approach, and a good example of why we need to get beyond using labels.  However I do think I need to be clear: while students are doing most of the work, the lessons for my senior classes are very strongly guided by me. The student feedback steers the lesson according to what they need or where they are ready to go, but I'm still the "guide", it's just I don't "teach from the front". Several times during the lesson I will regroup the class in front of a student whiteboard I selected because it has a "teachable moment" and then "microteach" to summarise or clarify, and then teach a small amount of new content as required so students can continue with the next set of problems.  Other times, a carefully selected sequence of problems means I don't need to do that - the learning follows naturally, and I can "spot teach" to the groups who need more scaffolding. My lesson designs have changed significantly. Because the students are doing so much work, the whiteboard room forces me to seriously consider the "What will my students be doing?" question. I spend much more time on thinking about examples and problems than on what I will say. 

It's scary to change how you teach, and even after all this time, I sometimes worry I'm doing something too different from the other classes at my school. This year, the whiteboard room is less available to me (because I'm encouraging other teachers to use it!), and due to timetabling, while I still have the whiteboard room for my Year 12 class, I can only use the room with my Year 11 class for four out of every thirteen periods - so it's been interesting to see what happens when you mix things up.  Sometimes old habits resurface, and I'm tempted not to use the whiteboard room when it's available - because it is easier to stand at the front, or you just haven't had the time to plan a whiteboard room lesson sequence. But much to my (pleasant) surprise, my Year 11 students complain and insist we go to the whiteboard room, so we just do it. And you know, when we finish the period, it's so clear they were right - it's a better lesson, even if I hadn't planned it as a whiteboard room lesson.

Some starting resources and ideas:
Laura Wheeler's @wheeler_laura blog post Building thinking classrooms is an excellent starting point.  Here's a handy link to all her posts with the "thinking classrooms" tag.

Can't set up whiteboards in your room? Try out Magic Whiteboard to transform any wall into a vertical non-permanent surface.

Peter Liljedahl's @pgliljedahl papers "Building Thinking Classrooms (pdf)" and Visibly Random Groups (pdf) are highly readable and motivating!  Liljedhal's "Thinking Classrooms" framework is much more than just vertical non-permanent surfaces (VNPS) and visibly random groups (VRG). For me this is start of a journey.


* If a student really wants "to be alone", I'll give them a break and let them work solo for a while, hoping they will change their mind later, or next period. They always do.  Also at the end of a really hot day, we might decide to sit on chairs for a change - sadly the whiteboard room doesn't have air conditioning.

Some lessons work best when you do (almost) nothing

Returning to my blog after many years absence, I found this Feb 2017 post in my "drafts" folder, waiting for posting.  I think it's still worth posting now in March 2019 (!).

It's a new school year, for me the start of the seventh year teaching. And this week I think I taught my best lesson ever. The surprising thing is that I barely taught anything. In the words of one of my mentors, a teacher with 35 years experience, "you know it's a great lesson when you do almost nothing. You sit back*, close your eyes, and you hear the student conversation taking place - they are talking about mathematics - and you'll hear how they think - and then you know what you need to teach, specific to each student."

For this lesson, I handed out "A/B quizzes" to the students which looked liked this:



They are designed to test skills in way that encourages students to help each other. In the A/B quiz lesson design, different students get different papers (see the letter in the bottom right hand corner of the paper). They can help each other if they like, but since the questions are slightly different, they have to actually teach each other rather than just share answers. Half way through the period, the students form into groups with those who did the same paper and compare their work, teaching each other anything they need to. Then they repeat the process for a second round, using a different version of the "A/B" quiz.

It was amazing to watch this group of students work together. I saw them struggle through the harder integrals with negative fractional powers, debating with each other what the correct answers were. Every single student was involved, no-one was left out.  I mistakenly had some indefinite integrals on the quiz, which I hadn't taught yet, but the students who had worked ahead taught the others how to do them. And they loved the lesson, "Integration is so much fun, Sir!" I think they packed a week of learning into forty minutes. I only taught for the last ten minutes, reinforcing some of their findings and explaining a few finer points of the setting out and reasoning required to provide the highest quality solutions.  It felt like one of the best lessons I had ever taught - but strangely I had barely done any teaching.

OK - I'm exaggerating when I said I did nothing. The lesson happened as a result of many months working with this group of students, building trust, confidence and openness.  It does take a fair bit of training to get students working effectively in groups for this process - to ensure they actually help each other and look carefully at each other's work, rather than just "looking for an answer". I did design a very specific learning sequence for the topic, selecting items in the diagnostic quiz designed to elicit discussion and to expose any difficulty students had executing the required skills. And it does take many years of experience teaching a topic such as Integration to anticipate what difficulties students will have, how to diagnose them and how to provide the necessary support.

And that's the joy of the Art and Science of Teaching - sometimes the very best lessons have very little 'visible' teaching. But if you look closely, you'll see a lot of visible learning - and it will take many years of experience to feel like you got the best out of a particular lesson plan.

Would this work for every lesson? Absolutely not! It's my view, after trying many different approaches, that when teaching mathematics, four out of every five lessons should follow more traditional, explicit instruction supported with ongoing and regular formative assessment. And for the fourth or fifth lesson - try lots of different things - the more student-centred the better. "A/B" quizzes are a lesson design that works really well - the gift that keeps on giving, year after year.

* Update for 2019: I've found an even better way to run a lesson like this! Coming soon...

Sunday, August 2, 2015

"Why are we learning algebra?"

It had been several weeks since my Year 7 class had the discussion of why we were learning algebra, so I was taken off guard when the perennial question came up again: "Mr Zuber, why are we learning algebra (again)?" 

I have a whole range of answers I like to offer to this favourite question but this time something unexpected came out of my mind.  "Have you seen those amazing new pictures of Pluto that came in this week from the Horizon spacecraft?"  I was pleased to see many students in the class start to get excited - they certainly were inspired by those photos.  

Global mosaic of Pluto in true color (NASA) July 2015

"Well", I said, "that was algebra. Algebra brought us those pictures. Very complicated algebra, and physics and engineering worked out by smart people helped get that spacecraft just at the right place, at the right time above Pluto, millions of miles away from Earth, to get that photo and send it back to us. That's why we're doing algebra."

I think that was the best answer I gave in class all week - and the students seemed to like it. Thank you NASA!


Here are four reasons for learning algebra that I like to offer students when I start the introductory algebra topic.  

Firstly we have some utilitarian reasons:

Algebra is a tool to help solve problems.
We use it to find values of something we don't know.

Algebra allows us to record information about relationships between numbers in a formula.
We can then put values into those formulas to find related numbers. This could be the area of a triangle, or the dosage of medicine to give a child based on their weight.

At a deeper level, algebra has an important place in our discovery of the world:

Algebra allows us to describe how the world works.
Students like this image. The picture in the centre is a matter-antimatter collision and the formula is Heisenberg's Uncertainty Principle.

and in supporting our exploration and representation of mathematical ideas.

Algebra allows us to represent and explore mathematical ideas and mathematical objects.
At least some students in your class will have seen the Mandelbrot Set and know how complicated it is - they will be very surprised how 'simple' the algebra looks.

Putting all these ideas together, I like to summarise with the one big idea: algebra is a language.


So - for those people who say "but I will never use the quadratic formula in my future work", I would respond: "Wouldn't you like to learn this amazing language? It will open up so many career possibilities to you (a utilitarian argument) and it's also a fascinating and rich language that will let you access a whole new level of knowledge and ideas (a sheer pleasure argument)"

What's even more amazing about this language is that it's an international language. I can speak algebra with a Russian or a Chinese mathematician. Somewhere out there in space, a class of Year 7 students with green skin and three eyes is also learning algebra.  Can you think of another subject you're learning at school which is also being taught in Alpha Centuri?  (OK - science... but let's pretend that's the same as maths :-)




Postscript: Should I have mentioned that algebra helps us develop reasoning skills? Possibly.... but I'm not sure most students buy the "it's good for your thinking" argument. So I take the "what algebra will offer you" line, and make sure I give emphasis to its role in abstract thinking as well as in 'practical' applications.

Thursday, July 9, 2015

Wormholes and Tesseracts in the Classroom - Part 2


In Part A I looked at some inspiring ideas about teaching in the movie Interstellar. Here are some ideas for using some Interstellar content in the classroom.

Exploring Dimensions: whether you're doing lessons on 2D and 3D solids, or just having a discussion why we say "x-squared" and "x-cubed" but "x-to-the-fourth", it's time to bring out The Tesseract.  In the Interstellar version, it's an object that has spatial and time dimensions: as the main character Cooper moves through space, he's moving into different time "rooms".

"Time is represented here as a physical dimension"
Warning: SPOILER for the film!

My Year 7 students liked this, and were very fast on their internet devices to find more traditional mathematical representations of the tesseract (the hypercube) which made for a good discussion.

Permutations and Combinations: I haven't worked it out yet, but there's definitely a perms and combs activity to do with the CASE and TARS robots! Might link in well with a Quadrilaterals exploration too.



Watch CASE at work on Miller's Planet: 



Some good resources:


An exciting way to introduce circular motion:


A nice adjunct to the more classic and sedate circular motion sequences in my other all time favourite movie, "2001: A Space Odyssey".

And for something different, you may like to point your music teacher friends at this mini documentary about the making of the soundtrack:



Wormholes and Tesseracts in the Classroom - Part 1

"You have to go see the film Interstellar", I told my Year 12 class recently, "Maths saves the human race!"  They corrected me immediately:  "No Sir - that's not true, LOVE saved the human race".  Yeah - we have a lot of science fiction movie buffs in our school...

But quibbling aside, what a wonderful resource Christopher Nolan has given with this film. Here are a few highlights that inspired both my teaching and enriched the conversations in class.

Interstellar: Inspiration for Teachers

Need to describe the projection of 4 dimensions into 3 dimensions in under 30 seconds? This explanation of why a wormhole should be a sphere is astoundingly concise (start at 03:00)



Wow! Imagine if we could teach complex content this easily.

Probably the scariest conversation a maths/science teacher could ever hear takes place early in the film - a wake up call to all us! In a four minute sequence, Nolan describes the futility of assigning a single number to measure student success, text book wars and the battle we have on our hands to defend the scientific world view:


So why do I teach? In an astounding sequence, Cooper mumbles muses:
"We used to look up in the sky and wonder at our place in the stars. Now we just look down and worry about our place in the dirt."  
(sequence starts at 3:00)



We need more star gazing. If I can get even one student a year to decide to look at the stars, it's all been worth it.


Part 2: Curriculum links : ways to use Interstellar in your classroom.

(The videos are probably not going to be around for long ... watch them while you can!).

Sunday, April 26, 2015

Just change one word

Ever found yourself describing a student or a class to another colleague as "low ability"? It's a shortcut we use more frequently than we may realise, even if said in the most caring, well-intentioned way. Early in my teacher education at Sydney University, I was very fortunate to be given a very simple and powerful idea: change one word.  Replace the word "ability" with "achievement". 

The result may surprise you. Here's my favorite example: next time you hear yourself saying: "Let's save that (interesting, challenging) activity for the high ability class", change it to: "Let's save that (interesting, challenging) activity for the high achieving class". Wow! Would you really want to do that? Changing one word isn't about being politically correct - it's about altering our mindset from a fixed mindset to a growth mindset. When done with awareness, changing one word can make a real impact in your classroom and your school.

It's approaching five years now since I graduated from Sydney University, and this simple idea continues to pay dividends in my teaching. So it seemed only fitting to make it the topic for my presentation at our alumni conference, SUSMAC 2015. 



With thanks to Judy Anderson and Maria Quigley for organising the conference, and to Eddy Woo for his work to make the conference available on the internet.

For a full set of videos and notes from SUSMAC 2105, featuring a keynote from Andrew Martin, and over 20 short presentations from teachers and preservice teachers, see:

SUSMAC 2015

Tuesday, November 11, 2014

The Pi Collection : 50 maths enrichment books for your school libary

A little project I've been working on for the last year with my school librarians: we call it "The Pi Collection".  


We've built a carefully selected collection of around 50 books with maths themes - fiction and non-fiction, in an effort to entice our students to engage with mathematics beyond their regular classroom work.

Officially our idea is to encourage students to read more widely and deepen connections between maths and other subjects, but in truth, we just love having these great books available to share with our students. So, when, for example, we're exploring extra dimensions, we can ask "What if someone built a house in four dimensions?" - and point at the classic Robert Heinlein short story "And he built a crooked house" which just happens to be the library waiting for you to read.  Or when we're talking about equations, and wondering if we could use them to describe everything - point students at the wonderful Isaac Asimov "Foundation" series - every thirteen-year old nerdy boy's dream of running the universe through maths.   Or perhaps someone thinks girls don't math? Well have we got several books in the Pi Collection to show you otherwise!

Along the journey of building the collection, we've discovered books about people who think differently ("Born on a Blue Day"), a terrific manga-style book questioning the inner truth of logic in mathematics ("Logicomix") and short stories about what could happen if you were allowed to divide by zero ("Stories of your life"). And who could forget that the answer to the meaning of life is 42 ("Hitchhikers Guide to the Galaxy") - a book which also has great fun exploring probability - remember the "Improbability Drive"?  Doing permutations and combinations? Well that's just begging for a reference to Arthur C. Clarke's short story "The Nine Billion Names of God".

Nothing however quite had the impact of "The Cold Equations" - this flew off the shelf as soon as we posted it up:

Maths tells our hero he needs to eject his stowaway into space.
His heart tells him otherwise. What decision will he make? 

I was a little worried about the unsubtle appeal  to baser instincts - but hey - anything to get the students reading! And it turns out our English faculty teaches Science Fiction in Year 7 and The Detective Novel in Year 8 - so we made sure to select books in these genres that also had a maths element.  As we expand the collection, we're finding more connections to Geography, History, Science and Art.

We've also included maths extension and enrichment books for curious students who want to go beyond the official high school curriculum. Collections of classic puzzle books, short articles on maths topics as well as some more challenging books. We even snuck in a few that might encourage some students to consider a teaching career (thank your Mr Lockhart!).

The full list is available here. Do you know of any books we should add?

Looking for more maths themed books?  An invaluable resource is Alex Kasman's collation of titles. Some care is required though, because not every book here is suitable for high school students.

Updated: 2015 Semester 1 Pi Collection