## Sunday, July 31, 2011

### Using laptops in my classroom - is it real maths?

The following is the first of two short reflections on some recent experiences using (or not using) laptops in my maths teaching. I used these as openers to a recent presentation about one-to-one laptops in the mathematics classroom to show some of the upcoming themes.

I am big fan of using software and the internet in the classroom to enrich mathematics teaching and learning, so I was a little surprised when I caught myself being skeptical about getting my class to try out a new interactive application from the Australian Bureau of Statistics.  Just a few days ago, a colleague eagerly showed me Spotlight, an interesting and engaging site that allows the user to explore past census data presented in the context of their own demographics. My colleague suggested we should do this as an activity with our two top year 9 classes.
 A colleague tells me about the new ABS Spotlight application ... but it is "good enough" for my top year 9 class?
I'm slightly embarrassed to say my initial instinct was that while it was a fun and interesting application, it wasn't really 'serious enough math' for my top year 9 class. Some sort of pride kicked in - these students needed to do 'real maths'. That evening at home I logged onto the social network my school system uses, and into the maths group where around a hundred and thirty maths teachers hang out (sad isn't it? :-) ). And what do I find but other maths teachers raving about Spotlight? One  teacher pointed out there were two versions of the app - one without sound - which might make it better for in-class use. They also referenced some of their other favourite statistics resources, including Hans Rosling's amazing 200 Countries, 200 Years in 4 minutes - which happens to also be one of my favourites, so I thought perhaps I ought to take this idea seriously! And I figured it was time to go apologise to my colleague...

Thinking about my top year 9 maths class, we actually haven't been doing that much with the laptops - the students all have digital versions of their textbook on it, and we sometimes do activities with Excel or using edmodo - but so far we haven't actually done that much in class with the laptops.. Which is kind of surprising given how much I like using technology. Partly I put it down to the fact we haven't done topics yet suitable for my favourite tool GeoGebra - but it's still surprising.  So our class did spend a period on the Spotlight site, and then exploring the ABS Census information. Before I realised it, we were having some quality discussion about the political and social reasons for the census, political controversies in the United States about counting homeless people and the use of sampling to fill in the gaps, and how in Australia we didn't even count our indigenous people until recently. I realised what a rich topic this was going to be - and the potential for students to do a quality mathematical and social investigation using the ABS website.

So what does this little story highlight? Quite a few things that are important in the one-to-one laptop discussion:
• Even when the teacher is keen on technology, there still isn't necessarily much use of the laptops in many mathematics classrooms,
• The idea that 'real maths' is done in the traditional way is buried very deep inside us - no matter how differently we think we feel about learning maths. It's the way we were brought up and sometimes anything that's not pure abstract symbolic manipulation, expressed on paper in exercise books, or presented by the teacher .. well it just doesn't feel like top level maths.
• The important role of a professional learning network (PLN) to encourage quality use of technology. Hearing other teachers give positive feedback about their own very specific use of an idea or application, in the specific syllabus area you are working in makes a huge difference
• A wonderful thing about having laptops in the classroom is I was able to make a very quick decision to do the Spotlight activity. If the students weren't bringing in their laptops, I would have had to try to book a computer lab - and most likely had a two-week waiting time, the hassle of moving the class there,  hoping all the computers were working ....
• Why did the students have their laptops ready to go? Because they use them in place of lugging their textbooks around. As mundane as it may seem, offering students the ability to use digital copies of their books greatly increases the chance they will come equipped with their laptops, and fully charged. Certainly for those classes where the students want to do mathematics. (And that's another story for later).
The next post in this sequence will consider our willingness to hand over control of the technology to students, and then I will look at the current situation of laptops in the mathematics classroom as reported by several major research studies.

## Saturday, July 30, 2011

### Conference: Using one-to-one laptops in the mathematics classroom

Using one-to-one laptops in mathematics lessons
July 30, 2010
Engaging with the Australian Curriculum: Mathematics for 7 - 10
University of Sydney

Zuber Presentation v03 Public

The most important resource links for this presentation:

Using Laptops in the Mathematics Classroom
Here is a list of posts which tell the story in more detail (more coming over the next few months):

Is it real maths?
Who is holding the mouse?

## Saturday, July 16, 2011

### Becoming a teacher: it's a marathon not a sprint!

Finally you've graduated teacher training - after so many lectures, so many lesson observations, practicums, and now you have your own classes - shiny new teacher, keen and eager to launch into the new career. BAM! You're off and racing!
﻿
 Jeremy Wariner shows how to power off the starting block for a winning sprint. This is not recommended for a new teacher.
Small problem though - you're sprinting ... but this is a marathon!

Feeling absolutely exhausted during most of my "holiday" break, I've belatedly realised I've been working at completely the wrong pace: that it will take three to five years to build the foundation - and I better start pacing myself accordingly. Sprinting isn't going to make it happen any faster.

I'm realising now that for each topic you teach, you actually need to teach it three times, over three years to have those basics covered.  The first year you engage deeply with the topic as you encounter it in the teaching program. You may think you fully understand a topic, but when you go to teach it, you'll realise you were only touching the sides. Thirty pairs of eyes and active minds will see, hear and do completely unexpected things, have completely unexpected questions and reactions to the topic and to the way you teach it. As you strip the topic down, and then build it back, weaving a sequence, a narrative and ornamentations around the topic, you will find your own understanding of the topic deepens and changes. I've been amazed how even the most supposedly basic concept (the area of the triangle) could require so much thinking.

Then a second year to repeat the topic, this time knowing what hurdles you will face and designing your teaching sequence and activities to match. But you're not done yet. You need a third go - because chances are you are now seeing a different type of class dealing with this topic - and the ideas you thought were good ideas for the second year needed reworking - or even ditching. So hopefully at the end of the third year, you have the topic well understood, you know some ways to successfully teach it, and you have a resource kit that matches your teaching approach, at your fingertips which can form the basis for future development.

Now - consider you are most likely concurrently teaching five different topics to five different classes, and moving to new topics every few weeks, usually without even a moment to collect your thoughts. Learning how to teach, making the topic connections and building your resources (even if people give you resources) really is going to take three years - just to lay the foundation of a long term teaching practice. And that's just thinking about content - we haven't even considered learning about classroom management, school procedures, working with parents and with other teachers. No wonder the new teacher sometimes feels like they are adrift at sea, trying to build a boat with a few planks of wood.

What to do? Adjust the mindset for a marathon. This is going to be slow steady pacing - with some sprint training sessions for sure, but mostly about building endurance for the distance. It will take time. The so-called 'holiday breaks' aren't enough to catch up work, sleep, personal life and health if you have been sprinting for ten to thirteen weeks - so you need to be working efficiently and don't overdo it, allowing time for healing and recovery.  The pace you work at during school term has to be sustainable over the long term - there really will be no breaks, and it will be like this forever. Adapt - and pace for the marathon.

 Nearing the finish line at the 2005 Gold Coast Marathon
Writing this post has reminded me of an an old dream I never finished: to run a marathon in under four hours. I finished my first marathon in 4 hours 32 minutes - was very painful, but very happy to have completed it (everyone is a winner in the marathon!). Second attempt was better, but a heartbreaking 4 hours and 2 minutes. Unfortunately I never went back for a third attempt - and now I'm old and fatter.  Maybe it's time to take up that challenge again (8 years later ...) and see if I can fit marathon training into a teaching schedule - now that's a challenge!

## Tuesday, July 12, 2011

### Nothing like chocolate to focus the mind ...

There was a certain anticipation in the last SBG Outcomes Grid I gave out ....

and what an amazing effect it had on concentrating student attention! "When is the chocolate lesson, sir?" ... "Umm.. I don't think we're quite ready for it yet .. we need to learn more before we can that do activity ...".

So ... WCYDWT?

 WCYDWT : What Can You Do With This?  It's Dan Meyer's catch cry for starting a quality maths investigation.

With apologies to Dan, the truth is we just wanted an excuse for a treat at the end of the term! What better way to explore volume than with a quality chocolate exploration? And having those (now empty) Toblerone boxes around sure proved handy to demonstrate the idea of the "net" of a shape:

We also ate explored some rectangular prisms (Maltesers appropriately boxed) and an amazing rhombus prism package that held three Lindt balls, the contents of which went to the first three students who could draw the net of that prism. And for those who think that's too much chocolate - try a lesson using mandarins instead.

Two 200g Toblerone bars are more than enough for one class of 30. Although better to get smaller bars so you have more boxes to hand out. Nice thing about Toblerone is the bar is wrapped in foil, and students can break of a triangle and pass to the next person without getting messy.
Warning: Toblerone contains nuts (almonds) - so provide an alternative.  My nut-allergic students were most unimpressed with my suggestion we substitute with a box of cereal!

## Monday, July 11, 2011

### SBG: Taking the blinders off your horse!

Got too much content to get through in your course? It's a race you know! A race to complete the prescribed material. And how do you keep a horse running the race? How do you prevent it from being distracted by unpleasant things happening to other horses? Easy: put on some blinkers - or better yet - some real full scale blinders:

A teacher using the traditional "end of the topic test" assessment method can, if they choose, run as fast as the program says without too much distraction. And the students won't be too distracted either - they get their test results maybe once a month or two - perhaps an unpleasant day that reaffirms what they can't do - but no fear - we keep on racing to the end.

Enter Standards Based Grading - or indeed any form of continuous assessment - now both the teacher and the students are running that race without blinders. It impossible for the teacher to avoid seeing if the race is falling apart - if a significant number of students are falling further and further behind. And for those students struggling - if you choose to maintain a pace too fast for them - they are getting constant feedback that they are not keeping up the pace.  The blinders are off - everyone can see what is happening, all the time. To make it more interesting, the SBG version of continuous assessment encourages turning your horse back to rerun the part of the track you couldn't handleSo with SBG, you just don't have a choice to keep running ahead - certainly not with junior classes where students have yet to fully develop learning skills and confidence to take full control of their academic progress in their own time outside class.

And that's the problem and the joy of SBG. It will disrupt your teaching program. If the race is going too fast (and it seems it always is - just too much content in our programs), SBG will stop you in your tracks - forcing you and your students to stay with the standards being worked on until you are happy a satisfactory level of mastery has been reached by enough students.  It's going to get even messier when you find some in the class have mastered the current set of standards and are ready to keep moving, but another part of the class has only just left the starting block.  Or some haven't even entered this race - because they never mastered the material from last year ... or even the year before that! So SBG will not only delay your program, it will also force you to work out how you are going to cope with the spread: how can you differentiate so your strongest students are able to keep running, while providing support so others don't give up the race?

Call me naive, but I'm of the view it is better to get through half or three-quarters of the program with students fully mastering the content they did cover, rather than ticking a box to say the program was completed on time, and ... oh .. too bad the class average test result was 60% (we won't ask about the spread!) and that many of them reinforced their negative views on mathematics and low self esteem in the process. Of course this approach is not possible with some courses. In senior courses for example, with a sequence and pacing strictly prescribed by state education authorities, you just have to stay on the schedule - the race will go on regardless. For courses where the teacher has more flexibility to adapt to the class needs, the question of how many students in difficulty constitutes a significant enough number to justify changing pace, and what they should be expected to do in their own time is a professional judgement. And that's a hard one for a new teacher!

 Blinders help with compliance!

While searching for images of horse blinders, I was amazed to discover an article about a horse called "In Compliance" and how putting blinders on the horse help it win races ! No kidding. Mind you - even this horse met its limits - blinders or no blinders - it just couldn't jump the 3m hurdle.

## Wednesday, July 6, 2011

### I see parallelograms everywhere

Having come to terms with the area of the triangle - and realising how parallelograms unlock the secrets of so many of our basic regular shapes ....

... it's time to look at our friend the trapezium (or trapezoid if you insist).

Can you turn this into a parallelogram? And how does it help work out the area?

Try it out, before clicking to see more. A teacher resource is also provided at the end of the post.

## Tuesday, July 5, 2011

### Really, really understanding the area of a triangle (Part 2)

Continuing from Part 1, we look for an elegant and convincing way to demonstrate to students that our triangle ideas also apply to the infamous (and well named) obtuse triangle.

How can we show that the area of this triangle is the same as half the area of the rectangle made using the base and height of that triangle? I struggled with this - and it turns out drawing a rectangle was not the best way - the secret is to turn this triangle into a parallelogram.

Here is a neat trick:

 By rotating our obtuse triangle, and putting it on top of our original obtuse triangle, we now have a parallelogram. The parallelogram area is base times height, so the area of our obtuse triangle is just half that! Woot!

So long as we are solid in the knowledge that the area of a parallelogram is the base times the height - we are there : clearly the area of even the obstuse triangle is half the base times the height!

Just one small problem ... have we taught parallelograms yet?

Should we change the order we teach area?
Realising how central the parallelogram is to gaining a deep understanding of quadrilaterals and now even triangles, has led me to wonder about the classic teaching sequence for this topic.

The classic sequence for area is:

We start with a very specific rectangle 'the square', then extend to the rectangle, switch to a completely different shape, then are back to another quadrilateral. No wonder we have these puzzled looks when we say a square really is a rectangle, especially when in some textbooks the formulas are even written using different letters, and students are puzzled why some area formulas have that half factor and others don't. And we haven't even looked at the trapezium yet - which uses the half in yet another way!

So - with the goal of students better seeing the relationship between all these shapes and their areas, I'm going to try this sequence next time:

Why?
• Rectangles make sense to most students, the math is very simple - so we can focus on the real meaning of area as a measure of 2 dimensional space. Understanding what area is, and how it is different from length and volume is non-trivial - it's quite a deep teaching - and well worth quality time.
• We look at squares as an aside - so it's clear the square is 'just' a special rectangle.  This is time for a discussion of 'all squares are rectangles, but not all rectangles are squares". My students liked this version: "All crows are birds, but not all birds are crows".
• We then move to parallelograms.
• What makes it a parallelogram?
• Now we focus on what happens to the side lengths as we change the angles while the height remains constant. What happens to the side length?  How does it compare to the height?
• We draw a bounding rectangle and show that so long as the height and base stay unchanged, the area is also unchanged. We can show this with guide lines, and through demonstration or student exploration with a dynamic geometry tool such as GeoGebra.  Keep the focus on the height - this will pay off soon.
• We then observe that when the angle is 90 degrees we have a rectangle - so a rectangle is just a special parallelogram - which explains why the area formula is the same!
• ... and when the side lengths are just right, we have a square - wow- that's a parallelogram too! Same formula.
• And now we are ready for triangles.
• Compare triangles to quadilaterals - so it's clear how they are different.
• Explore the area of an acute triangle using the rectangle as the base to develop the 'half' part of the formula, and emphasise the triangle height.  Don't start with a right triangle - it allows students to think the height is the same as the side length!
• Extend the idea to show how triangle area can be explained in terms of a parallelogram made by reflecting the triangle on a side. Now we can show the area calculation is also true for obtuse triangles.
• Now hammer home the calculation methods - with clear focus on the height versus the length of the side. Make sure students do the right thing, even when given a triangle with all three side lengths and a height provided. And now is the time to look at algebra issues students may be having
• And finally, go back and look at our rectangle again:
• Time for a deeper understanding of what happens when you change the units of measurement,
• And time to get a good physical sense for how big a square metre actually is - and why you can fit 10,000 square centimetres into just 1 square metre.
So in this sequence, we use the parallelogram as our central organising concept.

The rectangle and the square are special parallelograms - all using the same area formula : base times height. We can see triangles in terms of being two halves of a rectangle (this is the simplest), with the fallback of knowing the parallelogram is the secret for understanding why this also works for the obtuse triangle. And with the parallelogram well understood, and our technique of seeing areas in terms of rectangles, we are ready for our next set of friends: the trapezium, the rhombus and the kite.

## Monday, July 4, 2011

### Really, really understanding the area of a triangle (Part 1)

We all learnt this one at school - recite with me : "The area of a triangle is half base times the height", and then the teacher wrote on the board: $A = \tfrac{1}{2} b h$ and then the students practiced endless boring substitutions, triangle after triangle after triangle.

But how many of our students really understand what is happening? What is the height? Where does the half come from? Why are we multiplying by a half - aren't we dividing by two? And what's with that strange  obtuse triangle?

 Observe how these diagrams are almost monochrome making it all that harder to explain the ideas.

After spending several weeks with Year 8 and Year 11 classes exploring the challenges of teaching and understanding area of the triangles, it's become very clear to me that many students are going all the way through high school without ever really 'getting it' - which is a pity because the idea is so wonderfully powerful and actually quite simple when you strip away the ornamentation.

Here are the lessons my students taught me about being a better teacher of triangle geometry:

Understanding the key idea is easy - for right triangles and acute triangles
So long as your students are happy with the formula for the area of a rectangle, the logic of the "half the base times the height" formula is stunning clear:

The 'wow' noises that came from the class showed the power of these diagrams! Even better if you have a dynamic geometry tool like GeoGebra. It's just so clear the area of the triangle is half the area of the bounding rectangle.

 That's so obtuse!?!
The hard part came when we looked at obtuse triangles. I am deeply ashamed to confess my bag of tricks was empty at this stage and I had to resort to saying: "Well, the formula works the same for obtuse triangles - trust me!" (blush).  Demonstrating this result using a dynamic geometry drawing tool, comparing the bounding rectangle area to the triangle area didn't quite cut it.

Extra work is needed to understand the diagrams
Many of my students really struggled with the height markings on the obtuse triangle. I think in part because I was initially using monochrome diagrams, and in part because the idea of the 'height' of triangle (as opposed to the length of the sides) wasn't clear.  Here is what I'm going to do next time I first show this diagram ...

 Many students don't understand what that dotted line is all about. Replacing it with a ruler, and explaining the idea of triangle height  better helped understanding.

Extra work is needed to explore the idea of the 'height' of a triangle

... and at the same time as showing the diagram, we need to reinforce thinking about the difference between the length of something and its height above the ground. Here's what seemed to work: I started by standing tall,  asking how high my head was, and then I leaned over - was my head still as high? No. Then I made a triangle using a school desk as my base, and a stick (a long ruler) as one side of my triangle. I put the ruler vertically on the desk, so it was clear the height = length of the ruler. Then I leaned the ruler over at an angle and asked what the height was now? Some wonderful students jumped up, grabbed another ruler to measure the height. Another look at the diagram of the obtuse triangle with the height marked, and more 'Aha' light bulbs going off!  Next time I'm going to use a loop of string wrapped around the desk to make my dynamic triangle.

Don't assume students are comfortable with the standard formula
I made the mistake of assuming students would be happy with 'half base times height' expressed as $A = \tfrac{1}{2} b h$ . I was wrong! I heard several students asking - "Why are you multiplying by a half? Why not multiply base times height and divide by two?". And another "Huh? Doesn't multiplying make something bigger?".  For the time being, I'm letting students write $\small A = \frac {b h}{2}$ or even $A = b h \div 2$ . We'll save that story for another day!

And the obtuse triangle?
So is there are easy way to show how an obtuse triangle fits into half the base times height? It turns out there is!  Here's a hint:

 Another way to explore triangle area - turn it into a parallelogram. So powerful it works for obtuse triangles too! Image: Wikipedia Triangles article, by Herbee
The idea is so powerful, I'm now seriously thinking we are teaching area of shapes in the wrong order. But that's for Part 2.

With many thanks to my Year 8 students who felt safe and secure enough to ask every possible question - and more - they had about area, triangles, multiplication and algebraic expressions.