Solving triangles decision tree

Before entering declination and inclination in year 10 I wanted my students to really consolidate how to solve a variety of triangles.

I set up a decision tree on the whiteboard splitting the various methods for solving triangles.
My students are heavily reliant on notes to solve problems but now can see which parts of their notes to use for a variety of problems. They are even labelling triangles correctly!

Obviously the tree has limitations - finding third angles when two angles are given, finding the unknown angle with the cosine rule (when not the central angle) and the "ambiguous angle" with the sine rule.

Another area I have focussed on is providing differentiation for students of varying levels of algebraic skill. For some I have written all variations of the various rules on their notes page and ensured they can find the correct rule and use correct mathematical notation when recording their logic for solving a problem. For more capable students I have suggested only the bare minimum in their notes and encouraged them to identify the subject and manipuate equations to suit (as it is great practice). Some students have worked more on mastering sine & cosine rules, others have experimented with 3D trigonometry.

After discussing the limitations and completing a number of examples, suddenly the lights turned on for many of my students.. for the first time in a while I felt they were ready to move on. This is my cue to run a revision session of mixed and composite examples and check for further issues.

3D trigonometry

3D Trigonometry in year 10 is one of those subjects that lots of kids have issues with. Students traditionally learn just one topic at a time, are tested and then move to the next topic. With 3D trigonometry are all of a sudden faced with using lots of maths all at once.

Topics that all come together:

• Parallel lines (in 3D)
• Alternate/Complementary/Supplementary etc angles
• Polygon properties
• Trigonometric ratios
• Pythagoras' theorem
• Sine Rule
• Cosine Rule
• Oblique and isometric drawing
• 3d visualisation
• Volume, Surface area
• Algebra (in all its guises)

My students have seen some success in this topic through the use of "wireframes" (made of skewers and connected with a hot glue gun) and templates placed in the wireframes to indicate various right angled and non right angled triangles. It also helps them see parallel lines and equivalent lengths/angles. This seems to be the concrete element they need to put all the other stuff in place.

I also try to introduce the topic multiple times throughout the year as new concepts are introduced (eg. parallel lines - > volume -> pythagoras -> trig ratios -> sine rule -> cosine rule)

Also useful is the discussion on solving triangles here.

Burnout Week 5 term 3

As a new teacher we have to cope with a lot of things.. The kid with the difficult home life, the aggressive learner, physically agressive students, bullying and social issues between students, unachievable aspirations of students, students with low self esteem, politics between teachers, goals of the school, TEE, the list goes on.

And sometimes you feel it is all a bit too much. You grit your teeth and look forward to the next break. This is especially prevelent in term 3. You can see it even in the senior teachers as they confide they are not coping. I think to myself if they're not coping, how can I do it?

Then you stop sleeping and eating.. You get a little grumpy.. classes stop responding to your teaching methods. You're not as quick at coming up with alternate responses to poor behaviour and end up being the teacher you said you would never be.

At this point, talk to someone you trust - hopefully a team member that has gone through it all before. Don't pick someone struggling that will reinforce your issues, but someone that is doing well. Listen to what they do and why they are having success. Ask for some time out. Don't talk to your line manager - my past experience is that their responsibility for performance causes an imbalance in their judgement as they look at the now and only have time for a quick fix rather than investing in the whole picture.

I suppose the main thing is not to burn out. Your classes can suffer for a little while whilst you slow down a little. The profession needs you - three years from now you can laugh at what caused you such grief before, even if sometimes you can think it doesn't care. At the end of term four you can reap the rewards and next year be recharged. This is the hump - know this and just get over it the best way you can.

At least I hope so.

Setting difficulty level of classes

I've copped a bit of criticism about the level of difficulty of my classes. Some students have found it too hard, others too easy. This to my mind is an indicator that I'm probably doing ok.

After all it's better to push kids too far and get the difficulty level wrong occassionally than not push students and get the difficulty level wrong all of the time.

I try and manage the difficulty level through a number of factors:
1) Are the majority of kids engagaed in learning?
2) Are mean scores in tests increasing?
3) Can those not engaged be moved next to engaged students?
4) Can I make the course more difficult?
5) Is the top student being extended?
6) Are students endeavouring to pass the top student?
7) How long have we been on the current topic?
8) Where are similar classes at?
9) Have I covered the intended content?

And so on..

2009 subject selection

Many graduate teachers and even experienced teachers feel some anxiety when it comes to subject selections. It's one of those times where your judgement can make or break the aspirations of a student.

Ultimately school should be a safe place where students feel comfortable to attempt what they previously thought of as impossible. Too safe subject selections will put students into classes that they will not be challenged in and surrounded by students with weak work ethic / too hard subjects will place students at risk as their self esteem takes a pounding.

Teachers feel this pressure. It is important that any conflict over what students should and shouldn't be doing be overcome before students are counselled into their subjects. The issue over 3A MAS/MAT (old Calculus/G&T stream) subject selection has caused much angst as this is a class in small schools typically with less than 14 students (the DET magic number for whether a class will run in 2009) yet also a draw card for top students entering year 8.

The plus side is that theoretically the new level 3 courses have a reduced difficulty level / the downside is that nobody really knows what the exams will look like and how the CAS calculators will alter treatment of certain topics. With very few true mathematics graduates entering teaching, it has been a concern for some time as to who will teach calculus courses in the mid term.

Cases where counsellors and administrators are counselling 'A' students out of level 3 courses due to misconceptions of course difficulty, available staffing levels, questions over quality of recommendations and issues with grades given to students need to be resolved well before the counselling process. Students being told to pursue higher maths by teachers and then convinced otherwise by counsellors is an avoidable situation that requires clear communication from administration what subjects can run and clear guidance on how to guide students forward.

I think schools need to consider whether the prime role of schools is to guide students into their first job (eg. narrow but job focused education) or into lifetime learning (eg. broad pursuit of highest possible education and widest vocational choice). Where schools assume a split role (eg schools with strong competing VET & TEE courses) close attention needs to be made over who is part of the counselling process and their mandate in giving subject advice and applying/overriding teaching or vocational recommendations.

You just don't get it sir!

Yes, the kids do call us 'Sir' still, it's a mark of endearment for some kids - but before anyone gets too excited, there's usually a 'f*ck off' in a nearby sentence.

Occasionally you get a disruptive student that says "I just don't get maths" or "I don't need maths in "... and I think to myself here we go again. Do I have an honest student that has no faith in their ability or a student looking for excuses to maintain poor behaviour?

I have a great test for this but it requires being done in a period before lunch. I suggest that if they don't get it, it's ok, wait behind after class and I'll give them as much time as they need. In fact I insist, it's very important that they master the concept and not fall behind. It's amazing how many students get instant inspiration and get the task done.

Some students think I came down in the last shower and copy another student's work. A quick question usually indicates that I know when they are 'borrowing answers' and then I insist that they remain to get additional assistance to avoid such dependence on others. At this point they can be quite stroppy.

I start them off and they usually take about half of lunch procrastinating over the first question, arguing, complaining and whinging. They eventually realise I'm happy to wait until I see some success and start work..

.. and that's where my real work begins. I ask questions about how relevant they find maths, what do they find hard, I watch their every move and slowly, subtly start helping them with each answer. How can I make learning easier? Where in the class would they be less disturbed? How do they think I feel when teaching becomes impossible due to the number of interruptions? How can I connect maths with their aspirations? Other students come in that I have great rapport with (in senior years) and I give them a hand whilst the student is working. By the end of the period they see (in most cases) that a) they can do maths and b) if they just read their notes they could figure out what to do c) despite being noisy others students get work done and d) most students in my class have had this talk. I then let them in on the real secret - if teachers really didn't care, they would just let students fail and not try to get them to succeed. I suppose it also helps that I grew up in the area, know the type of kids to some degree and was reasonably successful prior to teaching.

If this fails, the next level is discussion with other teachers and the parent call/three way conference with pre-prepared wads of homework and suggestions of reduced TV/sport/PS3 time. Word quickly gets around it's a bad idea to get me 'organised'.

Maths is a confidence game and building the confidence of key students in a class can help make it all work. It's amazing what a class can do when disruptive influences become positive and the right opportunities are allowed to flourish.

Avoidant behaviours on the nose

I thought I'd start with a lighthearted post as today this happened in two different classes. I've noticed an increase of stinky kids - I don't mean kids with poor hygiene - I mean the kids (usually boys) that let one rip and fill the room with godawful pong..

There are times when I say don't exaggerate, yes he made a noise, it's a natural function now get on with your work.. but today there was a dark cloud surrounding the little darling. It was capital S, STINKY!

Now, flatulence happens to the best of us I'm sure, but no kid wants to be known as the stinky kid. This is one problem that can be faced and dealt with, although I'm not sure by whom. Do you refer them to the school psychologist, call the parent, have a talk with their health teacher, direct the kid to the loo? I don't know..

From a teacher point of view, you need to hose it down or you get the 'wanna be' funnies making stupid noises at decreasing intervals. In my classes repeat offenders that call attention to themselves get called stinky and soon get the message - especially if it continues past the class by students in the playground. This is a reasonably effective strategy for a popular student but can have long lasting repercussions if it goes too far or with an already unpopular student. Perhaps a quick warning and anecdotal story to the student about how 'another student' was labelled 'stinky' for life might do the trick.

Another common class disruption is the student that hasn't eaten and claims that hunger is preventing them from working. We are lucky that we have good in school process for feeding these sorts of kids albeit they still miss most of class whilst being fed.

I'm not even going to touch on the 'I'm too sick to work/I need a drink/I need to go to the toilet/I need to see the counsellor or nurse' plague.

The armoury of student behaviours to avoid work and teachers strategies to sidestep avoidant behaviours is like walking a tightrope - especially in lower classes. One push and a student/parent complaint, a lack of push and a class of low performing students. It's a tough one experienced teachers navigate naturally. I look forward to the day when I can do the same.

Preparing students for exams

We had a great lecturer (now retired) Kevin Casey at uni. He was of the view that learning was basically a factor of time. Given extra time most students could learn the same concepts.

Confidence in exams is a similar concept. Take one student that had extreme exam anxiety - she would frequently throw up before exams, have shakes and rarely display her progress. Some would say that she is a prime candidate for alternate assessment. I would suggest that in many cases this is exactly what not to do. (In life she has proven that she has great coping skills in highly stressful (eg exam like) situations).

In considering her exam phobia I sat her down and looked into what her core issue was. She was a student that had had repetitively reinforced 'I can't do maths because I'm a girl syndrome,' from both parents and past teachers. She also had an almost photographic memory - anything placed in front of her could be instantly recited back.

So I thought to myself let's fix the confidence issue then fix the maths. I considered learning methods I was familiar with and selected mastery based learning as the key tool. Mastery based learning is a great way to attack confidence issues as students repeat activities to the point of automaticity - the point where they don't have to think. This was important as she was brain freezing in exams, thrashing and stressing to the point of uselessness. Mastery based learning is not the most fun way to learn all the time - but it is a great way of showing to a student that they have ability.

Next we did a lot of boring repetitious stuff until she hated me, maths and statistics. But.. she (for the first time since year 7) not only passed but managed to get 80% on her test... and this was the pattern thereafter in not only maths but in other subjects.

We also went through the main steps when attacking an exam:

• Don't panic
• Determine the time per question/mark and monitor regularly
• Read the whole paper before answering anything
• Order questions from easiest to hardest - do the easiest first and build confidence
• Don't get stuck - move on..
• Don't write comments about how hard it is to the marker or doodle on the page
• Don't rub out working / show all working
• Go back over the exam at the end - ensure that each question is attempted
• Take note of when the teacher/lecturer says 'this is in the exam' - put a star next to it.
• Prepare your notes 1 week before the exam and add to them throughout the remaining week
• Don't panic
• Pick your study buddy carefully
• If revision sheets or trial/past exam papers are available... do them
• Complete assignments to the best of your ability - this will lower exam pressure later on.
• Avoid friendship based group assignments
• Complete as much as you can in class - keep an eye on what the 'top' students are doing
• Try and guess the types of questions in the exam - make sure you have attempted the questions at the end of exercises and chapters
• Ask your teacher or lecturer for direction (they can only say no)
• Don't panic

Students that do well in mathematics have well practiced skills before they attempt to apply them (in assessment or practical applications) and before they are able to choose between multiple skills to determine an optimal solution.

Practice makes perfect.