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.

Planning for Christmas

With Christmas approaching, lessons in class are often littered with appropriately themed resources. However, I have a somewhat scrooge-like reputation to protect, so am not too frivolous in my giving.

I don’t spend time or money decorating the classroom, nor do I introduce Christmas into the theme of lessons, unless it has a real purpose; there’s no room for ‘painting Father Christmas by numbers’, even if maths problem have to be solved in order to work out the number first.

But there are a few places where Christmas and maths naturally coalesce. The first is applying mathematics to the journey that Father Christmas must travel on Christmas Eve. There are plenty of opportunities to apply maths in this instance; to the speed of travel, the size of the sleigh, the minimum number of reindeer required and so on. But, unfortunately, Father Christmas and physics are not the best of friends so, in order to avoid an unhappy ending, I steer clear of this one.

Some people may be familiar with the desk calendars that are printed onto 3-D shapes, similar to that shown in the image below. This page [navigate from here if the link has expired] has a number of different options that are more exciting than what is essentially a cuboid – although I would encourage students to generate their own nets.

Moving away from ‘making things’, the American bank, PNC, has developed a Christmas Price Index – based on the amount it would cost to buy items from the song ‘The Twelve Days of Christmas’ (see here). I tend to start with a calculation of the number of presents, which is a source of surprise for many, before a calculation of the cost of buying all of those items – more than $100 000. The list of possible directions from here is endless, aided by the PNC supplying data as far back as 1984.

Note: The trend and the pie chart, below, refer to the PNC Christmas Price Index, which are both based on the cost of Day 12 alone, not the total cost of all days.

Word problems

I have recently started playing with Google Forms and am currently investigating how useful this software may be in a classroom environment. It’s a very powerful method of questioning many people and collating their results. This is great for me, the teacher, a data junkie, but I’m not convinced that there is much benefit to be had for the individual filling in the form, i.e. the student.

Earlier this week, I created a small form (see here) which collates opinions on the forthcoming strike; this worked well. I have now created a second form which could (theoretically, at least) be used in the classroom. I have embedded the questions here, and subsequently comment on the form below.

Here is the most recent score (a manual refresh may be required):

The form can be seen externally here (so students wouldn’t have to access this page) and the analysis (including people’s results) can be viewed here (the intention is not for students to access the analysis).

The questions are on the topic of ratio and proportion, with a focus only on word problems. Despite being maths questions, they have to be read very carefully in order to be understood fully. This is a difficulty with many maths questions and is where many students struggle in an exam (although not necessarily in ‘real life’). Some questions are direct proportion, some are inverse proportion, others are a mixture of the two, while there are also some trick questions.

As well as the ‘front end’, I have also done some work with the ‘back end’ of the form – the spreadsheet part of the document. This analyses overall performance as well as performance of the different types of question. Furthermore, I have generated a formula that attempts to assign a percentage to the level of concentration the student has maintained throughout the task. It does this by computing how many consecutive wrong and right answers there are – therefore assuming that a student who has become bored and skips to the end is likely to have a long string of zeros [I doubt that this is (a) necessary, or (b) accurate, but I was also interested to know what formulas could be achieved through Google Spreadsheets].

I like:

The ‘drop-down’ list on page 1 so students can select their class – which could subsequently be filtered in the analysis.
The navigation feature after each question (although I think it could be improved*).
The box that displays the most recent result on this page (which could be adapted as necessary).
The fact that students rate their own performance at the end, and could add any comments if they choose.
I have tried to get the balance right between compulsory questions and those that are non-compulsory.

I’m not sure that I like:

In this instance, I am not sure that multiple pages are appropriate; they prevent students from copying each other, but the process may be more user friendly if they were on the same page.
Twenty questions are probably too many for this task.
The questions themselves could be better. These particular questions allow for single number answers, and easy marking, but I’m not sure this would be the case for algebraic answers or sentences.
In its current format, I think this may be best suited to assessments and maybe some homeworks, but I hope to be proved wrong.

I don’t like:

I wanted to be able to provide students with a mark instantly, but I think this is only possible by giving them access to the spreadsheet. I think it is possible to set up an automated reply (to an email address), but I do not like this idea.

*a (possibly hidden?) contents page might help, but I still think there is a better solution

My lesson in classroom management

After five years of teaching, I still find classroom management more difficult than I would like. The number of strategies I have continues to increase, while I better understand what works well, why it works, and what I like.

Picture the scene:

It is Friday afternoon, which is typically the single hardest lesson of the week. I was tired and had been at work until past 8pm the previous night. It was Children in Need and would previously have been a non-uniform day but, at relatively short notice, it was revealed that this would not be the case – which caused some unrest. The Year 8 class I was to teach is set 3 out of 4 and has 30 students (this would be a lot for any class). The previous lesson, the students had not progressed far enough with their task, so needed to use communal resources (laptops) to finish off. I discovered at 11.15am that another class would be using the laptops.

Issues like this are not uncommon throughout schools, and teachers have to adapt to whatever hurdles may arise, but that was the hand I had been dealt for this particular lesson.

The students didn’t take the news about the laptops well, and things started to deteriorate. As my stress levels started to rise, I sensed I may have a battle on my hands.

I have a few extreme strategies in my repertoire that I turn to only as a last resort. One, which I have used on just one previous occasion in five years, is the “you’re wasting time, so I’m going to get on with my own work until you’re ready to learn”. I said it. Sooner than I would have intended. But it was too late to take those words back. In the five seconds it took me to sit at my desk, I immediately thought to myself “what have you done? You’ve given yourself no room for manoeuvre if things become any worse, and those students who actually do want to learn do not have sufficient information to get on”

Those precious seconds saved me.

I started typing instructions on the board. The ‘good’ students followed the instructions. The ‘middling’ students followed the ‘good’ students, and soon, the ‘bad’ students followed the ‘middling’ students. I still had to think on my feet; there weren’t enough resources to go around and some students (a small, but manageable, minority) were causing disruption. But these issues were shrinking in their gravity.

Problems with resources were soon resolved, and I called key students up to my desk one at a time. Some to praise, and others to advise how they needed to change their actions. Gradually the class moved in the right direction. I didn’t leave my seat for the remainder of the lesson. I continued to call up different students to offer differing words of advice. The result was that the entire class produced more work than they might have done otherwise. Furthermore, my stress levels were significantly less than they could have been.

This was not, by definition, a ‘good’ lesson, but it was a ‘satisfactory’ lesson – and no amount of planning could have accounted for the sequence of events that unfolded. I will mark that one down as a win.

Keeping me on my toes

Today’s lesson on real-life graphs was based partly around Dan Meyer’s Graphing Stories (see here for the original). It was an introductory lesson for a set 2 class (out of 5) who I will only see 3 (yes, three) times as part of a rotation. I had already used it successfully in a similar manner with sets 3 and 4, so it formed a basis to quickly assess current understanding.

I was pleased with how the lesson transpired but, what stood out most was the repeated comment from one student: “that man must have been paid loads of money to make that video”.

First off, congratulations Dan – that student believes there’s an acting career waiting if all else were to fail. But second, this also demonstrates that these students are pretty sharp – I’m gonna have to keep on my toes for the remaining two lessons; I’m thinking something along the lines of an nrich task (see here or here).