## Saturday, June 23, 2012

### The wisdom of Year 7 : thinking about groups

It's been nearly a year now since I changed my classroom configuration from rows to groups:

Overall I'm very pleased with the results - it's working for almost every class. The one class where I have wondered if I should revert to rows have begged me not to - challenging me to think more deeply about my classroom management for this class - I'm working on it! But there is no looking back now. These insightful comments from my Year 7 students, given as anonymous student feedback, reveal the benefits and challenges of setting up group tables.

Group Table Configuration : The Good
"Helps with learning because always someone u can ask 4 help"
"You can ask for help when the teacher is busy"
The group configuration helps deal with the challenge of answering questions from thirty students at once. Sometimes the group may come up with the wrong answer - but I rarely see this happen and am much more likely to detect any misconceptions if four students share them. What I do see on many occasions is students debating the answer, and they will ask for help if they aren't sure of their answer.

"Interesting learning from different perspectives"
Students discover how their peers view and understand the content - enriching their own understanding, and providing opportunities to develop metacognition : becoming aware how knowledge is obtained and processed.

"Let's me compare my answers and help people"
A powerful gift to offer to students : creating an environment for helping each other - developing generosity. One of my four pillars from the Circle of Courage.

"It helps cos if u don't understand something and 2 shy 2 ask u can ask ur friends"
Wow! How many students are held back because of this? A powerful insight on how a group table structure can help overcome emotional, personal and social barriers to learning.

"No need to be a loner - there are people around to help and support"
"Being alone is lonely"
How can we know the emotional needs of all students - let alone be able to help thirty students? Sitting students together, supporting them during class time to be together may just turn out be very important to some students who might be alone at other times. I was moved to read these comments.

"I would like to sit with different people"
I'm still uncertain if it's best to organise students or let them self-select groups. I worry about bullying and social exclusion, allowing students to set up hierarchies "you are in my group, you aren't". For now my answer is I assign the groups for Year 7 and Year 8 - and consider any problems on a case by case basis. A part of me also thinks it's important I maintain control of the seating.  Any ideas welcome!

"I can never see the board properly"
Ouch. This is the biggest issue  - and I think it's serious - especially since perhaps 30% of my lesson time is whole class instruction. Research that argues for sitting in rows claims this is the major problem with the group configuration. There are four seats in my configuration where this is a problem. I wish my classroom was wider to optimise the layout, but some tables don't get a good view of the board. I'm now going to establish it as a norm that those four students to turn their chairs to face the board during whole class instruction.

"Sometimes people just give you the answer"
An insightful comment from one student! Group configuration discourages solitary work - which is essential at times - and allows for students to just give each other answers. I often use an A/B/C/D paper approach to ensure each student at the group table has a different set of problems to work on - getting help has to be real help, not answers. But time doesn't always permit this, and if we are using the textbook, they are working on the same questions. I think I will need to be more explicit with students about ways of helping to maximise the learning.

Something not mentioned in the feedback is groups can encourage off-task behaviour and conversation. Fortunately with this class, that's not a problem - when they do go off-task (they are students!) they respond quickly  to my request to get back to mathematics. This isn't the case with all my classes - more on that in another post.

The Verdict?
Unquestionably (for me) : Yes. The learning and social benefits are so high, it's worth persisting to deal with, or minimise the down sides.  I'm looking forward to seeing how the comments next week from the class where I am having some class management issues will compare to those quoted above.

Note: My student feedback forms have an Opt-In indicator "Tick if you are OK for these comments to be shared with others". The forms themselves are completely anonymous, allowing for students to give me frank feedback without concern for any consequences.

## Monday, June 11, 2012

### It's trigonometry Jim, but not as we know it!

Do you remember high school trigonometry? Was it a blurred sequence of formulae with a recipe book of incantations for solving standard exam questions? I'm afraid to say that was my experience and it has taken many years (decades!) for me to see the full beauty and unity of the subject. Teaching it for the first time this year I was determined not to inflict the same on my students. My attempt at something a little different was based on a few key metaphors and heavy use of diagrams and graphs, supported by stories to help students see why trigonometry is indeed an interesting study for scientists, mathematicians and historians.

Forget triangles - let's cast a horoscope!
In senior school trigonometry, we're not just playing with triangles any more. Welcome to the weird and wonderful world of circles and periodic functions. I think a good way to introduce the unit circle is to place it in the historical context of our ancient star gazers watching the heavens. That's our unit circle: the night sky. I suspect our ideas of angles, triangles and projections onto a circle owe as much to calculating the pharaoh's horoscope as they do to measuring irregular sizes strips of land to work out the taxes owed him.

 Trigonometry : was it really about casting the daily horoscope? Celestial dome cartoon (inset) from http://www.herongyang.com/astrology_horoscope/Astronomy_The_Celestial_Meridian_and_Zenith.html

Now that we have the idea of the rotating angle, and students see (or better yet explore) the sine curve, I think it's well worth showing why this is such a relevant and powerful idea. Share with students some of the interesting physics - show wave motion, show how different light frequencies relate to different colours. Hint at the mathematical treasures that await them: I showed some pictorial sequences introducing Fourier analysis. Trigonometry is about so much more than triangles - and it's relevant, interesting and surprising.

A powerful metaphor for explaining the rotating angle and the periodicity of trigonometric function is the carousel:

 Think of the rotations around the unit circle like riding a carnival carousel.Carousel photo CC-BY-NC- SA http://www.flickr.com/photos/jaremfan/3478916095/Carousel horse photo CC-BY-NC-SA http://www.flickr.com/photos/birminghammag/6045458462/
This emphasises the periodicity of the rotation and allows extension to the idea of spinning faster, going from $\sin(x)$ to $\sin(3x)$, or slower, $\sin(\frac{x}{2})$, and then spinning wider to $5\sin(x)$.

Draw a diagram!
At this stage I believe it's worth spending quality time looking at the different graphs of $\sin(x), \sin(3x), \sin(\frac{x}{2}), 5\sin(x), 5\sin(3x)$. This helps visualise the functions and helps avoid the problems we seen when students start working with $\frac{\sin(3x)}{3}$.  Time also to bring out your function machine analogies and emphasise that $\sin()$ is function operator - not a multiplication of $\sin \times x$.

When it comes to solving trig equations, so many text books are filled with pages of algebra and barely a single diagram. Want to really understand the equation $4\sin(3x) = 1$ in the range $0^\circ \leq x \leq 180^\circ$ ? Draw the graph (sketch or use your favourite graphing tool):

Now we have many stories to tell! Why are there so many solutions? Because your students understand we have periodic functions (riding the carousel) and have seen the difference between the graphs of  $\sin{x}$ and $\sin{3x}$ the reason for all those solutions becomes less mysterious. And a quick sketch can help check the solutions. Drawing the graphs of trig equations also reinforces the difference between equations (sometimes true) and identities (always true) - another source of confusion when students start trigonometry at this level.

A helpful place to use graphs is when teaching the Auxiliary Angle transformation of $a\sin(x) + b\cos(x)$. Remarkably, very few textbooks show the graphs. I started with the graphs: looking at the unexpected result that when we add a $sin()$ curve to a $cos()$ curve, we get another periodic function - just with a different amplitude and a slight phase shift.

Yes - that's physics coming in - I share this language with the students - many of them are studying physics. Once you show this remarkable graph, actually working out the equations for the transform is straight forward and it makes sense - it's not just abstract symbolic manipulation.

Trigonometric Identities : Same person, different clothes
A helpful metaphor to distinguish identities from equations (why oh why do we do regularly use the same equals sign to mean different things?):

 Same equation, different clothes.Superman/Clark Kent graphic by Ian MsQueehttp://ianmsquee.deviantart.com/gallery/3370060?offset=24#/d1onlf2

Later, when it comes to teaching the transformations, I add in the telephone box:

 The half-angle t-transform helps Superman transform back into Clark Kent.And Clark Kent is easier to pin down (solve) than Superman!

Trigonometry at this level really is a lot of fun - it brings together many different ideas and skills, producing some beautiful and unexpected results. And so many more wonderful surprises to come for those students who will later visit the world of complex numbers! Hopefully we can share that wonder with our students, so they don't just view trigonometry as a set of definitions, formulae and algebraic manipulations, but instead develop a strong intuitive feel for working with the circular functions.

And don't forget the graphs - or as a colleague repeatedly tells her class "DRAW IT!"

Some Teaching Resources

• The Maths 300 "Trigonometry Walk" lesson (subscription required - but an internet search will find you some worksheets) is an outdoor exercise that helps students get a strong sense for the idea of projecting onto the unit circle.
• James Tanton's whimsical Squine and Cosquine presentation explores what would happen if we used a unit square instead of a unit circle. Perhaps save after students have consolidated the knowledge?
• Vi Hart's 13 minute video "What is up with noises?" is a wonderful exploration of the physics of sound, music and hearing. A little long to fit into a busy schedule, but good for a rainy end-of-term day.

So now I can write $ax^2 + bx + c = 0$ and $\frac{ \sin{x}}{\cos{x}} = \tan{x}$ without pain!