The Magic of Maths


The phrase “The Magic of Maths” is a way of introducing a personal reflection on my relationship with maths and how I see its manifestation in society.

“Love at First Sight”

My fascination with maths started in my very first year at school when I learned about inches. It continued throughout my school years, and continues to this day. I am currently learning about number theory.

For some people, the motive to become numerate lies in being able to use that skill for personal advantage in everyday life. Budgeting for a holiday is one example. My own motive comes from a fascination with patterns – the consequent advantage of being numerate is in a sense a by-product of that fascination.

Hidden in Plain Sight

Some people see the world through rose-tinted glass. I see the world through maths-tinted glasses. This includes things such as:

  • When will the next train leave Midland Station?
  • If that child leaps out into the road, will I be able to stop this car in time?
  • Judging a safe distance to overtake a cyclist when I am driving.
  • Ensuring that all bills are paid when they are due.
  • Using a PIN with a credit or debit card.
  • Checking that the amount shown on an EFTPOS machine is correct. (I started doing that after I was inadvertently overcharged.)
  • Avoiding speeding.
  • Checking wind speed and direction for nearby bush fires.

While these may appear to be ordinary everyday questions and activities, they all have an underlying maths component that can be used when delivering learning in numeracy. I have wondered if there is anything that cannot be seen in this light. Human interaction, the stuff of sociology, is also susceptible to numeracy through its classification of types of behaviour, durations and interactions with others.

The Floating Iceberg

There is a lot of maths that is perhaps not quite as obvious as the bullet points above. They tend to be “behind the scenes” stuff, but there are great benefits to the wider community from the people who use their numeracy skills in these contexts. Examples include:

  • Ensuring the security and reliability of money transactions across the Internet, and that includes your cash withdrawals from ATMs.
  • Keeping petrol stations stocked with fuel. (Doesn’t it drive you crazy when your local petrol station runs out?)
  • Planning bus routes and timetables to meet social needs.
  • Calculating mortgage repayment rates.
  • Calculating insurance premiums.
  • Designing computer hardware.
  • Writing computer software, including the browser that you are now using to read this post.

The numeracy skills used here tend to be a bit more advanced than what might be termed “everyday mathematics”.

Sharing Maths

It has been my pleasure to help others learn numeracy. Two outstanding examples come to mind. The first was a woman who had been awarded the lowest grade possible for mathematics at school. My efforts helped her to gain a thorough grasp of numeracy which was a crucial factor in her gaining a degree in environmental biology as a mature student. The second was a youth at risk for whom numeracy was initially a waste of time and effort. He left college with a Certificate III in maths, and went on to become a productive member of society by harnessing his passion for cars at a local automotive workshop. (As an aside, the motor car turns out to be a very rich source of material for delivering numeracy.)

Where to from Here?

The answer to the question “Where to from here?” perhaps depends on what you think is the purpose of being an educator. For me, being an educator is about the empowerment of others. Different learners have the potential to excel in different areas. I hope that my passion for numeracy gives them a chance to make an informed decision about which area or areas they wish to go into in their own futures.

9 hexagons

9 hexagons

The Curse of the Calculator?

An Unapologetic Rant

I have watched the debate about calculators in the classroom for over 20 years, and I have always had some reservations about the promotion of calculators as being a harmless or even useful addition at the lower educational levels. As somebody who has used mental arithmetic from the age of nine, I have wondered what impact calculators would have on today’s young adults. Yesterday I had my answer, and it was not pretty.

I happened to be present at an impromptu meeting between a trainee and her line manager. The business context was hospitality (I am being deliberately obfuscating here, this for the avoidance of embarrassing anybody), and in that context the ability to give the correct change to cash offered in the shortest possible time is a necessary skill. The trainee also worked in a supermarket, and she was skilled at using the electronic till to know what change to give. Yesterday her inexperience at being able correct change without the use of a till was a cause for concern for both her and her line manager. The trainee mentioned that she been using a calculator since the age of six, and had relied exclusively on calculators, even if in the form of electronic tills, to “do her sums for her”. The line manager suggested that the trainee acquire the skill of counting out the change, and the trainee agreed with alacrity.

I have occasionally chatted with shop assistants about about this topic, and I particularly recall a woman in her mid-50’s who took delight in counting back the change to me, while other people (mostly younger) confess to relying on the till. This reinforced my view that today’s younger people were probably missing out. Yesterday’s experience confirmed that view.

At the risk of sounding like a frightful reactionary, I think the time has come to re-assess the role that calculators have in the classroom, and give younger people an opportunity to develop a much greater command of numeracy.

Good Font, Bad Font


I have been an occasional student of fonts ever since 2010 when I designed a font for an organisation. I have also watched with interest fonts aimed at dyslexic people. All these fonts are aimed at improving the visual readability of “printed” text, and we have come to take for granted the efforts of font designers. To put their efforts into some relief, I thought to make a font that sets out to be very difficult to read.

Some Design Ideas

We have grown used to having a number of features common to any particular family of fonts. For example,

  • curves are common
  • the visual bulk of lower case letters tends to avoid the top edge
  • upper case letter usually have a stronger presence on the top edge
  • each occurrence of a letter looks exactly the same every time

The task was to design a font that turned as many of these features as possible on their head.

The Result

The result was built around using straight lines that went from point to point on a rectangular matrix. The lines themselves came in two different thicknesses. The thickness of each line was chosen at random. The result looked like this:ugly_font_1People of a masochistic nature will notice that the text in the image above are the words at the start of this post.

Enough of this foolishness. :)

Should We Let Students Struggle?


My thanks goes to Aviva Dunsiger (web site, twitter @avivaloca) for prompting me to write this post. She made the following tweet:avivalocato which I repliedavivaloca_1
Aviva then expressed interest in how I had done so, hence this post.


The nature of the struggles that I set are governed by the subject that I happen to be teaching. The struggles needed for mathematics are quite different to those for English, for example. The subject of computer programming lends itself to making the nature its struggles easily explained.

It is my practice to create learning activities where the goal is made explicit at the start, and the learners then have to solve a series of problems to be able to reproduce that result. For example, the goalrgb_resultcan be achieved with the following code:
private void show_colours() {
show_red.setBackground(new Color(red_slider.getValue(), 0, 0));
show_green.setBackground(new Color(0, green_slider.getValue(), 0));
show_blue.setBackground(new Color(0, 0, blue_slider.getValue()));
show_rgb.setBackground(new Color(red_slider.getValue(), green_slider.getValue(), blue_slider.getValue()));

To present the above text to learners would simply be to over-face them. The learning activity consists of describing what each word means (for example, “setBackground” sets the background colour of something on the screen), and offering a collection of “code fragments” which they can use to build a working program. A “code fragment” can look like this:

show_blue.setBackground(new Color(0, 0, blue_slider.getValue()));

The struggle then comes down to three components:

  1. Understanding how each code fragment works internally;
  2. Working out how each code fragment contributes to the goal;
  3. Figuring out how to assemble all the code fragments into a working whole.

The learners are usually very quiet during this activity: it is rare for any learner to ask anything. The learning activity allows self-evaluation: when the learner has completed it successfully, they know immediately. The most common issue that I find in such learning activities that a learner may have mistyped something, which then affords an opportunity for a whole-class discussion on debugging. In my experience, learners always feel happy when they complete such learning activities, and one learner went to far as to shout “Hooray!” to the whole of the class when he completed it.


I suggest that this can be related to similar problem-solving issues in other subjects. Using English composition, for example, how do you want to break down and organise your ideas in a blog post so that they are presented logically and can be easily understood by your audience. And at the risk of turning this blog post in on itself, you may care to critique it for those qualities.

Trouble with (information) technology (or: A Study in English Composition)

A Good Start

I was recently in Adelaide, and I had cause to use a car park courtesy (if that is the right word) of Wilson Parking. The entry to the car park offered the use of a credit card by way of payment, with its obvious means of identifying both the duration of the stay in the car park and a means of payment upon exit. The entry was quite uneventful, with me inserting and retrieving my credit card in a wholly natural way.

Things Start Getting Exciting

Having spent a pleasant time in Rundle Mall, I returned to the car park to pay for the privilege using it, and my subsequent departure. It is at that point that things became exciting. The pay station that I used had a credit card reader, much the same as the credit card reader that I had used upon my entry to the car park. I put my credit card in, and nothing happened. I tried to extract my credit card, but without success. I then used a second credit card to force the first credit card in so far into the credit card reader to try to cause it to eject the first credit card, but still nothing happened. The credit card reader was plainly malfunctioning: it had failed to accept the first credit card, and it had failed to reset itself under a fault condition.

There was a communications station next to the pay station, complete with push-button, speaker and microphone. I pushed the button. Nothing happened. I pressed the button for longer, and still nothing happened. After about six attempts, all unsuccessful, I called the help telephone number displayed beside the communications station. The first call was dropped. The second call was dropped. I got through on the third attempt. The first question that I was asked was “which car park?”. I had no idea. There were no identifiers showing the car park in which I found myself. It was only the kindness of a passing stranger that help me to provide an answer. It took a very long time for the person at the other end to finally comprehend what had gone wrong, at which point me she informed me that the manager of the car park would arrive in 15 to 20 minutes to retrieve my credit card. He arrived late.

In the meantime, my partner had gone to a second communications station, and had got through. She informed the person at the other end that the communications station that I was using was non-functional, and the person at the other end denied that this was the case. She was asked by the person at the other end which pay station was causing the issue. There were no identifiers on either of the pay stations.

While we were waiting for the manager, we were informed by other car park users that these pay stations had a reputation for swallowing credit cards.

The manager turned up, and was unable to retrieve my credit card. He felt the need to ask me if I actually had a credit card in the reader. He then stated that he would need to get a technician in to retrieve my card, and that this would take some time. We agreed that we should make contact with him in an hour or so. As we were planning on going elsewhere at that time, this then raised security issues, and the manager stated that me quoting the first four digits of the credit card would be an adequate identifier. Anybody with even a rudimentary knowledge of credit card security knows that this is inadequate.

I work with information technology as a matter of course, and I am normally more tolerant that most other people of its foibles. When we left the car park, I was livid and my partner was distraught.

An Analysis

I formed the following opinions, and here I must choose my words with extreme care, this for obvious reasons.

  • The pay stations also had a ticket entry point, and appear to require the insertion of a ticket (not provided to me at the point of entry), before the insertion of a credit card. This might indicate a failure to fully think through how the whole system worked.
  • The software within the pay station was inadequate. Whether this was due to an inadequacy of the Use Case diagram at the system design stage, or a coding failure, or an omission from the test procedures, I cannot determine.
  • The technology exists to prevent a credit card being inserted into a reader at inappropriate times. This had not been selected at the design stage.
  • The failure to eject any contents under either a fault condition or a reset condition could be considered a design failure.
  • The reputation for swallowing credit cards could indicate a design weakness, or quality control issues during manufacturing, or some combination of both.


On returning to the car park about an hour and a half later, the technician had just retrieved my credit card, and gave it back to me.

Hyperlinks opening the wrong application?

My Thanks Go To …

My thanks goes to Gail Poulin (Blog:, Twitter: @poulingail) for asking a question about something that seemed to be going wrong on a computer.


Gail, others and I were in a Blackboard Collaborate room when she reported that all the hyperlinks were opening OpenOffice for her, rather than taking her web browser there. I was not able to solve this in the time available in the webinar, so we agreed that I would find an answer, and get back to her, and I am using this post to do just that.

The Cause?

It is not possible to be completely confident that I have found the cause until the proposed solution has been tested and been found to be effective. With that disclaimer out of the way, it seems that there is something wrong with something known as “file associations” in her Windows 8 computer. The “http” protocol (don’t worry if that is meaningless to you) was linked to OpenOffice instead of her default web browser.

A Solution?

Gail is running Windows 8, so the steps below are tailored for that environment.

  1. Start Windows 8.
  2. If required, log on.
  3. Press the Windows+W key. The search panel should appear.
  4. Type in the word “default” (without the double-quotes). A list of items should appear.
  5. Click on the”Default Programs” item. A “Default Programs” window should appear.
  6. Click on “Associate a file type or protocol with a program”. A list of file extensions should appear.
  7. Scroll down to the bottom, and look for “URL:HyperText Transfer Protocol” in the description column.
  8. Verify that the “Name” field looks like “OpenOffice” (or possibly “OpenOffice Writer”, or just “Writer”)
  9. Click on that line. It should become selected (i.e. be displayed in blue).
  10. Click on the “Change program” button. A dialogue box should appear.
  11. Click on “Internet Explorer” (or, if your preferred web browser is different, click on that one instead). Wait a few moments, and check that your change has been effected.
  12. If necessary, repeat the above few steps for “URL:HyperText Transfer Protocol with Privacy”.
  13. Close the “Default Programs” window.
  14. Verify correctness of operations.
  15. This procedure is complete.

Firehosed by Professional Development


My thanks goes to Rodney Turner (web site, twitter @techyturner) for prompting me to write this post. He mentioned the word “firehosing” in connection with professional development sessions, and brought into sharp focus something that had been at the back of my mind for some years. I then tweeted firehosingPDwhich provoked two retweets and some jaundiced exchanges about professional development.


Rodney’s comment made me re-evaluate both the professional development (PD) sessions that have been to, as well as those where I had been an observer. What, I wondered, would people remember of being talked at for eight hours in a PD session? A little bit of Internet research came up with a study which can be summarised as learners remember

  • 10 percent of what they read;
  • 20 percent of what they hear;
  • 30 percent of what they see;
  • 50 percent of what they see and hear;
  • 70 percent of what they say; and
  • 90 percent of what they do and say

(Metcalf, T. (1997) Listening to your clients, Life Association News, 92(7) p16 – 18). Adult learning principles indicate that activities in a PD session should as varied as possible. See, for example this article by the Journal of Extension. There is one sentence in that article which is particularly telling: “Their motivation can be blocked by training and education that ignores adult learning principles (Knowles et al., 2005)”.

There is also the question of the amount of information that is presented in any one day of PD. If the PD is properly planned and executed, this would be the same amount of information that you would expect your own adult learners to learn in the same amount of time. From what I have seen for myself, and what the jaundiced exchanges would also indicate, is that this does not happen!

Why do we, who are educators and who should know better, allow this sort of disjunction between when we know to be good practice and what we allow to be done to ourselves? Why are we wasting our time, our presenters’ time, and our administrators’ budgets on this sort of thing? How many people can remember all the relevant details of the first session in a day’s PD by the end of the day?

My gut feeling is that less than 20% of all the relevant information is retained at the end of such a day. To put it into the perspective of personal finances, would you be happy for your garage to charge you five times as much as normal for an ordinary service for your car? No, I though not.


Enough of this bellyaching on my part.

Rodney described a situation where PD is built into the method of work. There is no “once a year formal PD session”. There is, however, a culture of continuous learning. (As I write, it occurs to me that this mirrors a culture of continuous [self-]improvement.) Just-in-time learning comes to mind here. As an aside, when I have seen Information Technology (IT) specialist come across a novel problem (and given the nature of information technology today, there are plenty of those), the first thing they do is use the Internet to see if somebody has already found a solution. Rodney’s comments made it plain that this just-in-time IT solution can be morphed effectively to suit educators.


What would it take to sell this idea to the relevant decision makers?

Don’t Be Scared, It’s Only Maths!


I was co-moderating a webinar a few days ago when the subject of numeracy came up, and I had my usual rant about the teaching thereof. Now, I would like to share my thoughts on some of the challenges that teachers might face when it comes to maths.


One of my teachers in kindergarten had a very good understanding of how to teach place value. She made it simple for this six year-old child to see the pattern. She praised correct answers to questions, and fostered my own fascination with patterns. Later teachers also taught the patterns of numeracy, and I was hooked for life. This is evident in both my work and my leisure activities.

Other teachers in my primary schooling demonstrated a poor grasp of not just numeracy, but also of reasoning (the next step up in the field of mathematics). It was this that caused me to rant.


I have heard two main reasons for why people find maths less than appealing. The first is that “it is all too complicated”. This may be due to the intellectual bias of the learner, but most of the time it seems to come down to the teacher lacking either or both of two key skills: a thorough grasp of the material being taught, and an ability to put it across in such a way that the student can learn. The other reason is that students don’t engage with the subject: they do not see it as being relevant to their lives. Again, this comes down to the teacher: the relevance of numbers in our lives is not communicated.


I cannot remember a time when I was unable to do simple multiplications in my head, so working out the total cost of three bananas at $2.00 each is trivially easy for me. At the other extreme, I have seen people being totally confused when I have tendered $12.30 for an item costing $7.30 . Having said that, I can share an experience where the real-life relevance of what I was learning was not to become apparent for 20 years.

There is something called “constrained optimisation”, where the basic idea is to make as much money as possible from any given situation, usually some sort of manufacturing. The favourite scenario used by maths teachers seems to be breweries. I can also remember this being used in my own schooling, but I soon forgot about it in the flood of mathematical techniques that I was learning. In terms using of those techniques in the abstract setting of a high school maths class, I was entirely comfortable

It was not until 20 years later that the importance of those mathematical techniques in the human context was to become apparent to me. I was told by my employer that the results of my mathematical labours were saving them around $10 million each year. Had it not been for my own fascination with patterns, I would not have remembered what I had learned 20 years previously.


The human cost of innumeracy should not be underestimated. I see it every time I care to look (which is not very often, as I find it distressing). I see innumeracy as being as disabling as illiteracy. If this post is about anything, it is a plea to decision makers to drag the teaching of mathematics up to the level that applies to literacy.

An Essay on Technical Writing


I have often wondered what technical writing was all about, and my curiosity finally reached the level where I decided to find out.


Technical writing is about the transfer of knowledge and understanding from one person to another.


The foregoing definition hides a multitude of details. This post outlines my own understanding of technical writing.


My first move was to see how other people were describing technical writing. This included a Wikipedia article at (accessed 10 March 2013), as well as course summaries from four training organisations. While the levels of detail differed in every case, they all had a number of common themes.

Drawing on my own experiences in both education and information technology, I was able to recast those themes in my own terms.


A useful starting point is to identify who is involved in the technical writing process. There are three main groups:

  1. The subject matter expert,
  2. The technical writer,
  3. The people who need to acquire the subject matter expert’s knowledge and understanding (the target audience).

When viewed like this, the technical writer is acting as a conduit between the subject matter expert and the people with the need to know.

This has a number of implications for the technical writer:

  • A good interview technique
  • The ability to record the subject matter expert’s knowledge and understanding accurately
  • A good understanding of aspects of the target audience:
    • existing knowledge and understanding,
    • vocabulary,
    • culture,
    • expectations.
  • A thorough grasp of language
  • A facility to present the knowledge and understanding using words that are readily understood by the target audience (“plain English” if you like)
  • An ability to organise all the material in a logical and coherent fashion
  • The ability to use authoring tools efficiently and effectively, including word processors, web page builders, and image editors
  • A good understanding of the strengths and weaknesses of different communications technologies, and how this impacts on the target audience’s learning experience
  • An ability to work well within the client’s documentation styles and conventions
  • A strong desire to produce a first-class document, this requiring persistence and patience.

With the exception of the first two items, educators who prepare their own delivery materials can be expected to have all of the above. Even so, the ability to read and fully understand a curriculum document is similar to the first item, and consulting colleagues on any of points of concern bears a resemblance to the second item.


I must leave it to others to decide for themselves how well this post meets the requirements listed above. If you are interested in also evaluating my work with images, you might want to critically review this article on managing images.

Zombie code: the ghost that bites you.


My thanks goes to Julian Sims (web site, twitter @juliansims) for prompting me to write this post.


Julian and I had a long and wide-ranging discussion on Twitter about some of the causes of the recent disasters experienced by banks when it came to delivering services to their customers. Examples include the Lloyds Banking Group and Bank of America. This was underpinned by Leo Kelion’s article entitled “Why banks are likely to face more software glitches in 2013” which examined the causes of such events. One of the key phrases in that piece is “dead code was brought back to life”. This is often known as “zombie code”, hence the title of this post. The problem with such zombie code is that it was created for another age – it is much the same as asking a stagecoach driver from 1750 to drive a modern day Ferrari: the result is guaranteed to be a disaster.


Julian and I were were tweeting each other about possible causes of zombie code, and one of his tweets “@philhart in the UK Aspergers is labelled ‘learning difficulty’ rather than ‘learning specialty’ – could that be part of the problem?” jolted my thinking out of its usual ruts. My short answer to that question is “No”, but that answer needs justification.

Before I go too far, I should mention that Asperger syndrome covers a range of symptoms, one of which includes “Intense preoccupation with a narrow subject” (Wikipedia, accessed 3 February 2013). It is entirely conceivable that this would manifest itself in the way such an individual would write software. I have seen examples where software writing was done in such a way that it was obvious that the software writer had an intense preoccupation with over-applying a particular design style. It is not obvious to me how the software writer’s style would have been influenced by his educators’ use of the labels “learning difficulty” or “learning specialty”.

Once a piece of software has been written and properly tested, it will continue to do the same thing forever, regardless of whether or not its author had Asperger syndrome. We now need to look at the business context in which this happens. Business rules change over time, and financial products come and go. In particular, a new financial product with a strong resemblance to a previous financial product can arise. The key point here is “a strong resemblance”, meaning that it is not identical. In particular, there is something different, and the piece of software that worked correctly for the old financial product is not guaranteed to work with the new product: it has been resurrected, and has turned into a zombie.

By way of a thought experiment, imagine that “cash account” has been changed into a “ready cash account”, and with a change to at least one of its rules, such as what happens when an attempt it made to go overdrawn. It is a new product, but the decision maker, usually somebody high up the organisational tree, is unaware of its impact on the piece of software that is about to become a zombie. The individual responsible makes the assumption that the old software will work correctly with the new financial product. The behaviour of the piece of software then becomes undefined. At the lowest level, cash could simply disappear – this would be picked up by an internal audit and rectified, and the general public would be none the wiser. At the other extreme, it can cause the whole system to crash, and everybody knows about it.

To the Future

Re-casting Leo’s words, I think it is something that we may need to learn to live with. :(