I always start out with good intentions, but laziness often gets in the way and they become left by the wayside. I am aware this is the case, so I try to develop ways of preventing this from happening. This is typically achieved by attempting to implement improved, more efficient methods but, occasionally, things slip through the net.

This is either because the new method cuts a corner, thus omitting the good intention altogether (albeit unintentionally), or just because my laziness kicked in and the good intention was omitted over a period of time.

The purpose of this post is to remind me of some of those things which are a good idea, were briefly forgotten about, and are about to slip through the net. This post is therefore acting as that net.

Reflection in teaching is one of the most important tools to help improve practice, but other pressures of the job often get in the way. This blog, as a whole, together with social media and online forums go a long way to helping that (if used correctly), but nothing beats a colleague entering your classroom, observing and feeding back. Thanks SB.

SB was in the classroom for no more than two minutes.
“[Student ‘X’] looked bored while you were taking the register” was the feedback.

Ouch. That hit me right where it hurt. Any pride that I may have had felt like it had disappeared instantly. But it has played on my mind since. I have thought about it long and hard. Was he bored? Why was he bored? What would stop him from being bored? What could I do differently to stop him from being bored? What am I going to do to stop him from being bored? And, to help my pride, how can I demonstrate to SB that he has never been bored since and will never be bored again? Ok, so that last question may seem far-fetched, but it is what I strive to achieve [right?].

While this feedback does not necessarily follow ‘the rules’, or guidelines, of giving feedback, it certainly made me stop and think – and reflect when I may not have done so otherwise. I need to somehow not forget that feeling I felt when SB pointed out to me the observation. The clearer I can remember that feeling, the more likely I am not to want a repeat – and the more effective the ‘net’ will be at saving those good ideas – and defeating laziness.

I continue in my quest to improve.

Just be careful

Teachers, like many other professionals, are busy people. They care about what they do and they want to improve, but their network consists of a limited number of people (typically friends and colleagues) who they turn to for ideas and suggestions of how to do so*. Consequently the smaller the network, the smaller the potential scope for improvement.

Until Tim Berners-Lee created the internet.

The internet breaks down barriers and can provide a fantastic way for teachers (and other professionals) to communicate with each other and increase the size of their network. The internet as a whole, together with blogs and social media, provide a platform for reflection, discussion and the sharing of good practice. Questions can be asked to, and answered by, thousands of people. Instantly, the size of the network is increased and the potential for improvement is magnified.

However, the internet is a public domain and, as such, must be treated with respect. Anyone has access to the information that is published and it is very difficult to permanently remove comments that have been made in error or otherwise. It is also easy for information to be misinterpreted if not expressed clearly enough, which can be further hindered by the lack of feedback from the ‘audience’.

People are becoming increasingly aware of this, so the following recent warning from the Scottish Secondary Teachers Association (SSTA) would appear to restrict the improvement of good practice (see article here):

First thing is don’t bother telling anybody else about your social life. Nobody is interested about your social life and it doesn’t help.
Secondly, never make any comment about your work, about your employer, about teaching issues in general.
There is always a possibility it will be misinterpreted.

The article here dissects the specifics of the comments, but some common sense and careful thought should ensure that nothing inappropriate is published. For all intents and purposes, Twitter and blogging are public. Similarly, Facebook changes so often that it is very difficult to stay on top of privacy settings so, to avoid complications and potential difficulties, may as well be treated as if its information were public.

If everyone took this stance, maybe the SSTA and other professions (see similar discussions regarding footballers here and here) would realise the potential benefits that these improvements in communication can offer. People should be encouraged to use the internet responsibly (and educated accordingly, if necessary), rather than prevented from using it to its potential.

As a form of insurance, my pages state that views and opinions expressed are my own and do not reflect the organisations that I belong to, but I am not naïve enough to assume this absolves me of any responsibility. I do, however, hope that the organisations that I belong to would not disapprove of anything that I publish**.

*other limiting factors exist such as time, money and imagination; to a degree, the internet generally overcomes all of these.

** I would like to apologise to anyone who may be offended by anything I have written or whose future I may have jeopardised by writing this article. Just in case.

Factorising quadratics

Many students who are comfortable factorising quadratics of the form x2 + bx + c struggle when there is a coefficient in front of the x term, as in ax2 + bx + c. They usually understand what is required, but become frustrated through the apparent ‘trial and error’ process that they apply until they identify the correct solution.

When I was a student at school, I remember being taught a structured trial and error process that worked methodically through all the possible combinations. It worked every time, but it was laborious and the greater the number of factors that the ‘a’ and ‘c’ terms had, the more tiresome the process became.

As a teacher, I continually look for improved methods of carrying out this process. However, many of them lose the understanding of what is going on – and students can end up struggling to remember a method, rather than remember what they are trying to achieve. To this end, I prefer for students to begin with a trial and error method, so that they appreciate the nature of the problem, before moving onto an appropriate method.

Consider the example 2x2 – x – 6 in order to study some of those methods.

Most methods start by multiplying the ‘a’ and ‘c’ terms…

…then finding factor pairs of their product, i.e. of -12…

…and identifying the pair that sums to give the ‘b’ term (i.e. -1). In the chosen example, this gives 3 and -4.

The ‘Lizzie method’ separates the pair, as shown, divides by the ‘a’ term, then multiplies the ‘x’ and the fraction (simplifying if necessary) until the denominator has been multiplied out.

A variation (see here for more detail) on the Lizzie method, separates the pair and multiplies the ‘x’ by the ‘a’ term, dividing by any common factors.

However, in my mind, these are rather formulaic and I prefer a method which is more intuitive. The method that I like to use (see here for more detail) involves rewriting the ‘b’ term as the sum of the factor pairs. The expression is then factorised in two parts; there is no need to have to ‘remember’ which number to write where, and which terms to multiply or divide by other terms.

OrHere is another example, 6x2 – 5x – 6.

Multiply ‘a’ and ‘c’ terms  Find factor pairs of -36  First method
 Second method  Third method  Third method, again

However, as a teacher, judgement is required as to which method, if any, may be more appropriate for the class, or just for individuals within it, to use. Experience has shown that time spent explaining lots of methods to lots of students can be detrimental. Many students don’t want to see more than one method, so finding time (*sigh*) to individualise explanations would be beneficial – even if this is done ‘out of class’, or otherwise.

New beginnings

As the year ends and a new one begins, thoughts turn to new beginnings and a chance for a fresh start. I’ve also been enjoying the Christmas holidays, and feel it’s about time I think about some school stuff. However, there isn’t too much that I think is ‘blog-worthy’ at present, so I shall use this space to remind me of a few things I have seen recently (while reading through the backlog of unread blogs) which have caught my attention (and that I may wish to refer back to at some point).

This quote from the German writer, Johann Wolfgang von Goethe:

I have come to the frightening conclusion that I am the decisive element. It is my personal approach that creates the climate. It is my daily mood that makes the weather. I possess tremendous power to make life miserable or joyous. I can be a tool of torture or an instrument of inspiration, I can humiliate or humour, hurt or heal. In all situations, it is my response that decides whether a crisis is escalated or de-escalated, and a person is humanized or de-humanized. If we treat people as they are, we make them worse. If we treat people as they ought to be, we help them become what they are capable of becoming.

This video from Teaching Channel:

And the similarities and differences between various methods for factorising quadratic equations:

See this post.

I hope that somehow these will help me to improve my practice in the future.

Fish out of water

The final day of the winter term consists of a trip to the village church for carols and a pause for religious thought, before a whole school assembly and the subsequent end of term. Staff are encouraged to position themselves strategically among their tutor group to ensure the service proceeds as smoothly as possible*.

There are a small handful of students in my tutor group who could pose problems in such circumstances. Last year, my tutor group were in Year 7, so the biggest threats were at my side. This year, 80% of those potential threats were absent from school. At this stage of a term, I question the motives of such absences. It is therefore a pity that one of the absentees will not receive the voucher they had won (and arguably earned) because they were absent today; one requirement of claiming the prize is being present on the day it is given to claim it.

But isn’t Christmas and New Year an opportunity for forgiveness and new beginnings, or something like that anyway. Either way, have a good one.

*Generally in this situation, I almost feel hypocritical – by demonstrating how to behave in an environment where I feel quite uncomfortable myself. Fortunately, there were no issues.

No room at the inn*

The last week of term is here, and I am determined not to fall into the mould of showing films that students could equally watch during the holidays.

Instead, we will discuss infinity through the Hotel Hilbert (see here for a description), Zeno and other paradoxes (see here for others) and propose that it will never be Christmas.

Yours, Scrooge.

*Perhaps Mary and Joseph should have tried Hilbert’s Hotel

Assessment pages

A page dedicated to assessments (see here) has been added to the teaching pages of this site.

It is a work in progress and more levelled assessment materials will be added in the future. It is not necessarily the intention that all assessments will be of a similar format to those which are currently available.

Assessment is a topic of much debate so, at present, this page will act only as a central place to collate some resources.

Any thoughts and comments would be appreciated.