Show-and-Tell 21st century style

A small technology revolution has just landed at our school. What initially appeared a minor upgrade to our kit has proven to be a powerful tool to increase learning and engagement in my classroom. Two weeks later,  I'm still amazed at the impact for such a small investment in time, money and training.  As one of my students observed  "You love your new toy don't you Sir?"  Yes I do - the HoverCam is an amazing tool for teaching!

Make your student work the star attraction!
HoverCam image from http://www.thehovercam.com/ 

What is a HoverCam? It's a document viewer which allows you to place documents under the camera and instantly display them on a data projector or interactive whiteboard. You can write on the document and see changes in real time on your display, and with one click you can capture the image for posterity.  

Why has this made such an impact? My students can now see - in real time - the work their peers are doing in class. The real secret to getting the most out of the HoverCam is to display student work, not teacher work. The first time I showed some student work, the rest of the class was stunned - and immediately hands went up everywhere "Show mine, show mine!".  Yep - it's really just the old "Show-and-Tell" using modern technology.  As I continued using the HoverCam over the next few weeks, I kept being surprised at the student reaction. Students eagerly completed work, keen to show it to the class. The more I experimented and thought about what our class was "showing" and how we were "telling", the more powerful I realised the tool could be.

The biggest surprise for me has been the effect in more challenging classes. Students in these classes are often disengaged, no surprise after years of being told they are incapable of doing the work, and their written work is often very poor quality. But even in these classes, there are usually one or two students with beautiful written work - although - and here is the catch - most other students in these classes have no idea quality work is being done by their peers. They have never seen it. Until the HoverCam! The effect on the class when they do see this quality work is really amazing - suddenly pens start moving across the page - keen to show they to can do good work.

Here's what I have learnt so far using the HoverCam:

• Make student work the star attraction. Are you showing your work or your students' work? It's the student work the class is really interested in!
• Find a (genuine) reason to show work of struggling or disengaged students: Watch how a distracted or disruptive student suddenly gets focused after receiving honest public praise for some aspect their work.
• Select student work carefully. Search for student work that allows you to develop your lesson goals. Ask students "Does anyone have something like (or not like) this, and would like to show it?". 
• Give very specific feedback on displayed work: "This is good because ..... ",  "I like how (student) did (this) .... and they could add (this) to make it even better". 
• Use student work as the source material for your explanations and examples. So much more interesting that your own examples - especially if the lesson activity is generating creative, interesting or fun student output. 
• Always ask student permission to show their work to the class. Sometimes even though a student has done interesting high quality work, they may not want the attention - be sensitive. If you tell them in advance what you will be highlighting about their work, they may be more willing. And thank students for allowing you show their work.
• Teacher "Show-Don't-Tell":  Demonstrate how to do lesson activities using the camera - especially for paper based activities. For example, I will show how to grade work during pair marking, how to record dice throwing results, how to mark up diagrams. So much more efficient than trying to explain in words how to do it!
• Use the HoverCam to record student output. I was able to scan twenty samples of student work in just a few minutes - a fantastic record of student work for the lesson. Something to capture the work for later analysis or to show to parents.
• Silliness: Students will want to put their hands and face under the HoverCam - you will need to keep an eye on it if you leave it running while you are working with other students.

Will the effect wear out as students get used to seeing their work displayed to the class? I haven't seen it yet - I think so long as lesson activities are interesting and give students opportunities to produce work they want to share, the class will remain interested in this digital take on "Show-and-Tell".

A special thank you to my head teacher for his vision and energy getting HoverCams installed across our school, and to a colleague who encouraged me to try the HoverCam with a challenging class despite my initial skepticism. 

Disclaimer: This post is written as a personal response to my classroom experience using this product. Neither my school nor myself have received any consideration from HoverCam.

No geometry, no banana!

One of my math heroes, Michael Atiyah describes how mathematics is often partitioned between algebra and geometry, but warns about the seductive power of relying exclusively on algebra. In that colourful tone of his, he writes:

With apologies to South Park.

"Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.’”
Atiyah (2001)

When I was a young student, while I found Euclidean geometric proofs fascinating (yes - I was a math nerd), the glory of Algebra - the revelation of being able to work with symbols and expressions blinded me to anything else. Perhaps it was because I'm not that spatially aware (as my long suffering World of Warcraft guild mates found out this weekend when I kept running into the fire breathing dragons despite repeated deaths...) - but I just took to algebra like a duck to water. And then I discovered coordinate geometry - who needed shapes anymore? I could turn every geometry problem I met into an algebra problem .. even it if was a bit messy sometimes!

However while relearning mathematics 20(ish) years later as part of my post-graduate teaching degree, I discovered the joy and elegance of the geometric view. So many ideas, including algebraic ideas, are clearer, easier to learn and remember when seen geometrically. As Lockhart points out - how much more interesting is the area of a triangle when seen geometrically as a half rectangle, as opposed to a formula learnt by rote? 

Now that I'm teaching maths, I'm exploring what happens when we use geometry to help understand algebra. It was interesting to observe my top Year 9 class when set the task to explain the "difference of squares" using a geometric argument:

I handed out sheets of coloured paper and scissors, but how they resisted! Some drew small sketches of the idea, but just refused to cut out accurate representations and manipulate the shapes. I chuckled at their resistance, because I knew that's what I would have done in my school days - the algebra is 'obvious' - what's the point? We got there in the end - no doubt that will be recorded as one of those 'weird activities' I made them do :-)

Does using a geometric demonstration help learn ideas? Absolutely! I have no difficulty remembering the formula for the sum of angles in a polygon now, because I can see the triangles arranged inside it; I feel the triangle area formula - and now even the odd looking rhombus area formula is in my bones - because I can see the triangles; and the difference of squares - well it's just so clear; and you absolutely know that  because you can see it in the diagram - those two extra bits are in there!

So many ways understanding the geometry helps understanding the forumulae!
Click for a larger view

Michael Atiyah was right when he said that having to choose between algebra and geometry is like asking someone if they would rather be blind or deaf : "On the whole we prefer to have both faculties" (Atiyah, 2001).

Atiyah, M. (2001). Mathematics in the 20th Century: geometry versus algebra, Mathematics Today, 37(2), 46- 53.

Illustrations made in PowerPoint, GeoGebra and Paint. GeoGebra is an important part of the solution to breaking down the false dichotomy between algebra and geometry - a very good reason to use it in class!

Answer to difference of squares problem: Cut the top green rectangle into the two obvious pieces, rotate the narrow green piece to line it up below the larger green piece. The area covered is very clearly

Would you like 2 yaks or 3 yaks with that test?

I took a (small) risk last week and tried something different for our Year 8 Algebra test. I'm calling it the "2-Yak/3-Yak test". The idea in a nutshell: students choose the level of difficulty of the test.

Each section of the test provides questions grouped into level of difficulty indicated by the number of yaks. Students were required to do the 2-Yak column, and then for each section choose between the 1-Yak or the 3-Yak column. If students selected the 2-Yak/3-Yak combo, they would automatically get the marks for the 1-Yak questions. I suggested to students that if they wanted to do the 3-Yak questions but thought they might be too hard, to just do the 1-Yak/2-Yak, move on to the next section - and then at the end, if they had more time, go back and try some 3-Yak questions.

So why do this? And what did the students think?

Mandarins in maths class

Think you're perhaps giving too many chocolate rewards to your class? I decided to switch to a healthier treat : mandarins. They are on special right now in Sydney - so even better for the underpaid teacher. Much to my surprise, students really take to them. I put a big pile on my desk to build interest and then at the right time, it's Mandarins for Maths!

A special mathematical treat. Read more about mandarins at
http://www.sevenfields.com.au/products/citrus/mandarins.aspx

My Year 8 class is studying probability, so I posed the question: "What is the probability a segment in a mandarin has seeds in it?"  Someone asks: "Do you mean we get to eat them?" I smile : "Of course!" I asked the students to draw a table in their exercise book to record the total number of segments, and the number of segments with seeds. We agreed how to count them, and then I handed out the mandarins - one for every two students, and we are off doing mandarin maths. Five minutes later I asked a student to each visit each pair, collect their data and then aggregate it. We worked out the probability together.

So what's so great about mandarins for maths?

• Mandarins are healthy - maths is good for you
• Mandarins are yummy - maths should be yummy
• We get to share - just like a good maths activity - it's better in pairs! (Feel the love)
• Mandarins are clean and easy to handle
• Low risk of allergies (still worth checking though)
• There are so many maths questions you can ask about mandarins. 

The disadvantages? Mandarin peel and seeds everywhere. Easily solved - ask a student to walk around the class with a bin to collect the scraps.

There is of course a place for chocolate in every maths class ( I have been building a stash of Toblerone bars ready for volume of prisms work), but definitely think about the healthier options - mandarins were an absolute winner with this class.

And the answer to the question? We ate 194 segments, and 46 of them had seeds - so we worked out a probability of 0.24.

But Sir, the person next to me has a different quiz!

There's a special moment that happens just once each year with each of my classes. It's the first time we are doing a written quiz on the first week's work. I settle the class down, and innocently hand out the quiz paper.  After a few minutes of working quietly, when students are in theory working on their own, someone always calls out: "But Sir, the person next to me has a different quiz!". One or two students gasp - oops! - as they frantically check their neighbour's paper just that little more closely.  I chuckle, "Yes - they are different, because I'm keen to see what you know."  Usually someone else says "But don't you trust us?", to which I chuckle a little more, and say "of course I do" - and leave it at that. On more than one occasion, the students have cheered and clapped - I think they are impressed that I took the effort to make different quizzes, and that I'm not an idiot.  

What the students don't realise (the first time) is that there are actually only two versions - I stacked the papers and handed them out so alternate students received alternate papers. The differences are minor - I'm careful to ensure they have the same level of cognitive and mathematical difficulty. If this sounds like too much hard work for the teacher, see the Practicalities section below on how to do this without too much extra pain!

Here is what my quizzes look like:

Click on the image for a larger view

and this is what the (B) version looked like:

Once the dust has settled down, and the class gets used to the idea, we can then do some great things with having multiple versions of the quiz:

• Pair Marking: Students swap their papers, and then do a whole class activity working through the answers. Each student marks their partner's paper, assigning a grade for each section. They have to analyse and discuss with their partner if any errors were due to minor mistakes, carelessness, or if the student really had no idea. Because there are two versions of the quiz, each student ends up doing the quiz twice and they end up discussing the answers with each other.  They love "playing the teacher", giving each other grades and writing "teacher style" comments.  Can I trust the students to do the right thing? So far I haven't seen anyone do the wrong thing - they are so keen to mark. And if they cheat - well - it means they had to explain it to each other, and it will show up in the next quiz when I don't do pair marking.  After they have swapped the papers back, I give them five minutes and then collect the papers for myself to check the marking was done correctly.
  Note: This Pair Marking activity can take a whole 50 minute lesson - so I don't do it for every quiz.
• Take home version: You can give the alternate quiz to each student as a practice paper or for homework.
• Second attempts: If you use a system liked Standards Based Grading where students are encouraged to re-attempt assessments, you have a spare unused quiz ready to use! In fact, I usually make four versions of each quiz - I hand out the (A) and (B) in class, and have (C) and (D) in reserve for reattempts.
• Topic Revision: I make the full set of the quiz (all the versions) with solutions for one version available on the class online edmodo group. This makes it available for topic revision prior to the end of topic test.

Some more benefits of having multiple versions of the quizzes:

• I don't have to impose super-strict test conditions on the class while doing the quiz as the opportunity for cheating is greatly reduced. Since we are doing quizzes regularly, I don't want the class to feel like they are constantly being tested. 
• If I sense students are experiencing difficulty with the quiz, I actually let some of the pairs help each other - especially if one pair member is much stronger than the other. I know they will be learning and explaining - they won't be just copying because that's not possible.
• I encourage students to write "I don't know" if they can't do a question, or to write "I'm not sure", "I think this is the answer".

For those of you interested in Standards Based Grading:

• The use of a simple grading rubric, applies to each section - as opposed to the whole quiz. The quiz above tested three outcomes, hence the three circles for recording each grade.
• A student experiencing difficulty, can still end up with a quiz result that has a "B+" or an "A" for one section, and a "C" for another. Usually I design the quiz so that the first outcome tested is one tested in the last quiz - increasing the chance for success.
• This system immediately gives me a set of quizzes for reattempts. I allow students to re-test using the online edmodo version and bring me in their completed paper.
• Pair Marking increases student understanding of the grading method. Unfortunately this can take a whole lesson, so I don't do Pair Marking all the time.
• When I hand back the quizzes, I ask students to record their marks in their outcomes sheet (this sheet is glued in their exercise book and tracks all their attempts and outcomes).
• I encourage students to retake the quiz until they get an "A" for all outcomes. If they end up with the odd "B+" on their second or third attempt, I reassure them that's fine for now (else they burn out). If they get a "A" on the topic test for that outcome, I bump up the corresponding quiz grade.
• If an outcome needs more work, I will re-test it in the next quiz. Students usually greatly improve on this whole-class second attempt.

Practicalities

Isn't this all extra work? Surprisingly, it turns out to be much much easier than you might think. 

• Making extra quizzes: If you have time to make one quiz, making the extra versions isn't much extra effort. So long as your quiz is made in a digital tool, it's as simple as copy-and-paste, and change a few numbers and letters. Well - it is for middle level school mathematics :-) It is a little more work for senior school mathematics, and I imagine harder for other subjects.
• Stacking the papers: Print off your (A) and (B) paper. Put the two sheets into the photocopier at the same time, set the copy count to half the number of students in your class plus some extras. You now have a pile of alternating (A) and (B) papers ready to hand out.  Just make sure you hand it out pair-by-pair.
• Marking: When you collect the quizzes back, sort them into (A) and (B) piles. Find the best paper in each pile and use it as your marking guide. Because the quizzes are short and sweet, the marking is fast.
• Yikes - so many more resources to make: Yep - that part is tough. As a new teacher, I'm gradually developing quiz sets for topics and classes. It is too much to do for all classes for all topics at once. But I know that in year or two I will have a full set of powerful tools!

Important update June 10, 2011:  Some recent anonymous class feedback on the quizzes, shows very strong support among the class for the use of regular quizzes and pair-marking. However one student did express concern about other students seeing their work - so some sensitivity is required, and options for these students may be necessary. It is probably also good to explain to students why you are doing pair-marking and set some guidelines on respecting each other's learning and privacy.

Do you edmodo?

Looking for a fantastic way to extend your physical classroom into an online space? You can't go past edmodo.  If you haven't heard of it yet, you could say edmodo is like Facebook for the classroom (that's how the students see it), but that would be scratching the surface of what you can do with edmodo.  Like Facebook, edmodo is a free online social networking application that young people love to use, however unlike Facebook, it has a strong privacy and security model, designed specifically for the classroom.

I've been using edmodo for nearly half a year now and can't say enough good things about it. So much so, a few colleagues and I felt it was time to set up the Edmodo Teacher Hub wikispace to share our experiences and teaching ideas. It's still early days, but already some teachers have shared amazing stories - my favorite is this story about using edmodo to support a student voice program. Coming from a computer systems background, I'm particularly impressed how the edmodo designers extended physical classroom management practices to implement secure virtual classrooms - something I wrote about in the article Is edmodo safe for schools?

Do you have an edmodo story to share or an idea to share? All contributions welcome!

Disclaimer: The edmodo teacher hub wiki is not officially related to edmodo in any way. Views expressed are those of individual teacher contributors and not necessarily those of their employers.


* A special thank you to Bianca Hewes, our resident Australian edmodo evangelist for encouraging and helping set up the wikispace (and designing the logo!)

Algebra with style, or, how to get the royal wedding into algebra class

This week we've been laughing way too much in Year 8 algebra class - and I put part of the blame on the royal wedding.  But let's backtrack a bit ....

While preparing to teach my first ever full unit of Algebra to junior high school students, I was amazed at the number of ideas and skills a student has to master in the first few weeks of algebra. For those of us who "just get it", it really is an eye opener to list each individual idea and skill. I counted twelve logical steps, seven starting definitions and six little 'gotchas'. Would you believe some textbooks combine most of these ideas in the first one or two sections of their algebra chapter? No wonder so many students struggle with algebra! 

So what do I mean by the 'gotchas'? Take for example whether we should write 1x or just x.  Is it wrong to write 1x? Of course not - it's just not the way we usually write it.  Maths teachers often use the word convention at this stage - and discuss why we have conventions. But as I was writing up the list of the algebra conventions, I found myself thinking that really it's about style. If you write 1x or 0x, you are sort of lacking a certain style - but you're not wrong. I teach at a girls school, so style really is something we all want to have! I found that by talking about style, my students understood the idea immediately: it's a matter of taste and "class" - not that you are wrong. My students, like most I suspect, are terrified of being "wrong" in maths class - so talking about "style" helps work around that. We have similar style discussions as to whether we should write 5xy or 5yx, and if it really matters if we write 5x +2y  or 2y + 5x.

And now, courtesy of Princess Beatrice, I have a wonderful image to use with my class about what it means to have good style.

For the record - here is the full set of "Algebra Style" elements we cover in our Year 8 class. The numbers in the circles relate to the step number in the teaching sequence. As a class we build up a table of "Algebra Style" elements and then take a moment to consider why we have these conventions.

Click on the image for a larger view.

Speaking of the royal wedding - don't miss this wonderful post at MathsPig about the forces at work in the wedding dress train. I'm intending on building a short sequence in my algebra class next week using this idea - see if the students can model the problem.