Polynomial stories

What's not to like to about polynomials? They look amazing - and they are just great fun to play with - especially if you have dynamic graphing software to explore their shapes. Here are a few teaching ideas I developed over the last few weeks.

First and foremost we need a character : meet Polly the Amazonian parrot. There's a reason she is from the Amazon... you'll see soon.

Amazona agilis by Jacques Barraband(1767-1809)
http://commons.wikimedia.org/wiki/File:Amazona_agilis_-_Barraband.jpg

Polly featured throughout my lessons - my favourite was the zero polynomial $P(x) = 0$

Reminds me of a Monty Python sketch ....

and later when we looked at taking the second derivative, then the third, fourth, and fifth derivatives - the disappearing Polly:

The Disappearing Polynomial

I'm a big believer in having 'characters' to help teach mathematics - I think they act as 'mental anchor points' to help link related concepts, and then make high level linkages across topics more visible - as in the example above of the disappearing polynomial.

Then we need to explore the properties of polynomials. If we have a quartic polynomial, how does its graph change if we change the roots? I got a good response to my students with this homework exploration tool - I used it for a 'flipped' lesson:

Locus and Parabola Make a Parabola GHT0301

Lesson 02 FLIPPED Graphs of Polynomials

Early on in the presentation of polynomials, I think it's a good idea to show some of the interesting applications:

Using polynomials for modelling. I used this example of a photograph of the Amazon river, loaded into GeoGebra, fitted to a polynomial using the FitPoly function.  Why would we want to do this? I suggested in this case, having an equation for the Amazon River could help us model water flow - perhaps helped by working out the gradient function:

Amazon River – photo from NASA. 
Curve fitting using GeoGebra FitPoly[] function.   
http://commons.wikimedia.org/wiki/File:Amazon_57.53278W_2.71207S.jpg

Using polynomials to approximate other functions: A good time I think to introduce the Taylor Series:

No need to go into deep explanations - just show what is possible with polynomials. I returned to this idea in the next topic when showing higher derivatives, and have another visit planned when we do complex numbers to help demonstrate the famous Euler Formula.

And finally, after demonstrating the closure of polynomial operations for addition, subtraction and multiplication, students may find it interesting to learn about the role polynomials play in many encryption systems.

I looked hard for an online paper suitable for advanced high school students - the best I have found so far is Christopher Cooper's notes for his "Languages and Machines" course at Macquarie University.

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More GeoGebra HowTo Sheets

What's in a word: low ability or low achieving?

While working on a paper I'm writing, one of my teachers suggested I change the words 'low ability' - as in 'low ability students' -  to 'low achieving'. The thought 'need to be politically correct' popped up immediately - but then I did a double take ... is it really just about using socially acceptable labels? Or does changing one word actually make a difference?  Reflecting further, I've come to be conclusion it makes a huge difference - especially in the context of mathematics education. 

"low ability students", "low ability classrooms" : says there are limits to what can be achieved with these students, says there is a limit beyond which further effort from the teacher is wasted.  "Low ability" says there is a limit to the learning that is possible for this student.

"low achieving students", "low achieving classrooms" : says the students are not meeting the outcomes we would expect students of this age group to achieve. "Low achieving" forces us to consider why they are low achieving. Are there problems with engagement? with effort? with attitude? with learning strategies? with the teaching? Are the outcome expectations reasonable? We no longer attribute low achievement to limited student ability, or at minimum, we are prepared to consider other factors are at play. 

Almost all secondary school mathematics faculties sort students into streamed classes based on previous mathematics achievement. Although the sorting is based on achievement, it is all too easy to accept this a proxy for mathematics ability - and it doesn't take long before we talk (discretely) among ourselves about our "low ability classes" and our "low ability students".

By focusing on the 'achieving' word, rather than the 'ability' word, we can better access other important teaching and learning ideas in our mental framework:

• Andrew Martin's work on student motivation and engagement, which encourages students (and teachers!) to see performance as a result of effort, strategy and attitude;
• Anders Ericsson's important work on expertise and ability - which shows how even the people we think of as having 'natural ability' require serious effort in deliberate practice to reach their potential;

and more fundamentally,

• John Mighton's questioning of the very idea of mathematical ability - as presented in his books The Myth of Ability (2004) and The End of Ignorance (2011).

It's more than changing one word - it's changing your mindset. You won't hear me saying 'low ability' ever again.

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 handle. So 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.

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?

Year 7 Maths : Extreme Differentiation Needed?

This post is the second in a series of reflections on my first experience of teaching a Year 7 mathematics class.

I'm almost embarrassed to say it, but it's taken me almost ten weeks to realise the extreme variation of mathematical ability in my Year 7 class. I knew there were differences across students, but underestimated just how wide that range is. What made the scales drop from my eyes? A second round of summative assessment (topic tests) and feedback from the end of term anonymous class survey. And I don't think there's anything different in my Year 7 class to any other Year 7 class in a comprehensive school.

Staggering mathematical bio-diversity in Year 7!
© David Hall seaphotos.com Used with permission.

Here's an extract from the survey that demonstrates the challenge:

I've never seen such variation in any class I've surveyed in the past - and the student self-reported feedback matches the most recent round of test results - which ranged from 3/30 to 30/30. How devastating to self-confidence must it be to receive 3/30? I don't buy any argument this will encourage them to "try harder" - especially since "try harder" just won't help when the content and skills are so far ahead of where the student is now.

In a follow up session I explored this with the class and confirmed that the more mathematically advanced students are getting frustrated and feel like they are being treated like babies (the math is too easy), and that other students are struggling - which may explain some of the work avoidance patterns beginning emerge. Looking at primary school records (something I recommend all high school teachers do for their Year 7 students - and wish I had done earlier) revealed a vast range of difference in mathematics learning outcomes - some students have already mastered Year 7 outcomes (through tutoring?) and others are still mastering early Primary School mathematics.

Looking ahead, I'm thinking it's time to implement an SBG approach to assessment (as with my Year 8 class), but even that isn't enough: extreme differentiation is called for!  I've asked many experienced teaches for advice and here's what I'm going to try out next term:

Extreme Differentiation: Ideas for Term 2

• Design a pre-topic diagnostic that indicates student readiness for this topic. I've come to realise I need to design my own diagnostic test - the standard "Are you ready?" diagnostics in the textbooks aren't always up to scratch!
• Communicate the result of the pre-topic diagnostic for this topic to the student. I'm keeping in mind that a student who is not ready for, say, algebra, may well be ready for geometry.
• Keep a copy of the pre-topic diagnostic on file. I need to be able to justify why I offered this student the option to work on different, easier material.
• Differentiate the topic into three levels: Essentials, Development and Challenge. The Essentials level will also include material from earlier 'stages' of mathematics (ie: some primary school material).  I'm considering using an easier text book for the Essentials material.
• Course material will contain a level indicator: one star for Essentials, two stars for Development, three stars for Challenge. Each lesson and each assessment tool will offer students material at each level.
• Offer students the option to select the level they want to work at in the topic. I believe most students will make the appropriate choice. More advanced students will be able to skip the Essentials and go straight to Development and Challenge. Of course I'll be watching for what happens, and encourage students making the wrong choice to consider the alternatives.
• Class summative assessments will report marks for each level the student attempted.

In the final post of this series, I'll consider the question  "Year 7: Children or Young Adults?"