But What Use Does Maths Have Anyway?


The inspiration for this post comes from Lee Finkelstein (Twitter: @leefink) with a tweet that resulted in the following exchange:
and it raises the question of why mathematics is taught in schools. The rest of this post shares my ideas on this question.

Assumptions Revisited

It seems to me that one of the purposes of schooling is to equip students with the knowledge, skills and attitudes to function effectively as adults once they leave school. It also seems to me that different students have different aptitudes and interests, and this has impact both on their performance at school and their learning. There is also the observation students will be “going into the world” where there will be jobs that do not exist at the time of their schooling.

Using Lee’s cri de coeur about why his 11th grade daughter needed to memorize things about the unit circle as a starting point, it is worthwhile commenting that every subject at school has relationships with other subjects, even though they tend to be taught in isolation from each other at high school. By way of example, mathematics is an essential component of the high school subjects chemistry, physics and engineering drawing. Mathematics is also an enabling tool for vast numbers of jobs in the workplace. Similar comments apply to the study of language its relationship with the workplace.

Speaking from personal experience, there were subjects that “bored me rigid” at school. It was not until well into my own adult life that I developed an interest in some of those subjects.


The above raises the thorny and perennial question of what, and perhaps to a lesser extent when and how, educators (and by this I include non-teachers as well) should decide what should be learned. I have no answers to this question. All I can do in this context is to mention ideas and raise issues for other people to ponder and perhaps come up with their own answers.

It now seems appropriate to offer some personal experiences. When it comes to mathematics, its relevance to me was that I found it engaging for its own sake. There were three areas in particular (quadratic equations, linear algebra and statistics if you must know) that proved to be invaluable 20 years later, and saved my then employer in the region of US $10M/year. I found history and geography to be as dry as dust, and it was not until I explored the countries of western Europe as an adult that what little I had learned as a child served as an invaluable basis for learning about, and more to the point understanding, what it was that I was looking at.


I must now leave it for you, dear reader, to respond with your comments below. I look forward to hearing from you.

Maths Wars


The inspiration for this comes from Lina Zampichelli (Twitter: @micky_lina) with her post in the FS Teach Facebook group. (It is a closed group, so no link, sorry.) It links to an article in Canada’s National Post headlined “Math wars: Rote memorization plays crucial role in teaching students how to solve complex calculations, study says“.

I have been conscious of the recent shift towards “exploration learning” of mathematics, and I have felt uneasy about it, which unease I put down to a sense of nostalgia. Reading that article has made me re-appraise my unease, and the time has come for me to put my own viewpoint about how mathematics might be effectively taught so that it can be effectively learned.

I expect that I might upset a few of today’s younger teachers.

My Use of Mathematics as an Adult

Before describing my own experiences of learning mathematics, it might be useful for me state how I use mathematics in daily life. I do so as a matter of routine, and it is for on a par with reading, writing, speaking and listening: I could not function effectively without my skills in mathematics.

I use my estimation skills when shopping. I do so to check the reasonableness of the total payable. People “behind the counter” sometimes make mistakes, and I will discuss this when I think that I am being overcharged or undercharged. The estimation skill also applies when planning a major expense: what are approximate balances on the credit cards and bank account, what roughly is our monthly income, and what is the cost of the proposed expense. I can then make a yes/no/defer decision.

Basic counting along with addition and subtraction skills can be a time-saver, particularly in shops. At shops where I am known, rather than having to scan six or more items with the same bar code, or wait for the shop assistant to count them, I simply tell them the number, this saving time for both parties. When it comes to presenting cash, I often find that it is quicker and simpler to present money with the cents and low dollars value to the amount where I can receive a single bank note in return. Presenting $20.70 for a total of $10.70 means that I have less metal weighing down my wallet, and the giving of change is also quicker. (It also often brings a smile to the cashier, some of whom seem chronically short of change.)

When it comes to photography, the numbers associated with shutter speed, aperture and focal length always come into consideration when I compose a shot. Here we have gone away from simple counting, and moved into the areas of multiplication, division, and geometry.

Stepping up another gear, a grasp of statistics means that I am able to move from “average care” given by my local doctor, to “better care” with my doctor’s support. Whenever I consult him, I turn up with a printout of all the relevant data presented in statistical form, which allows him to make much better informed decisions. It also allows me to adjust my own medication very rapidly to respond to changes in my own body. A bout of influenza had a very strong impact on some of my vital signs, and ceasing one of my medications for the duration made my life a lot more survivable.

Raising the ante yet again, I use something called “vector matrix algebra” to solve problems with real-world impact. There is a bank of mains power outlets on the island bench in our kitchen. That bank looks a bit like a toblerone chocolate. The quickest and most effective way to find out which angles I needed to use to cut the pieces of wood was by using vectors. Anybody who writes software to display three-dimensional objects effectively on a computer screen needs a thorough understanding of vectors and matrices.

Pulling Out the Bones

In the examples above, as well as in other scenarios, I use the following skills as easily and as unconsciously as I read and write:

  • Counting
  • Addition and subtraction
  • Fractions in all their various forms: a/b, percentage, 12.3456
  • Multiplication, division
  • Geometry, both Euclidean and spherical
  • Trigonometry
  • Time, whether measured in femtoseconds or billions of years
  • Logarithms and exponentiation
  • Algebra and power series
  • Equations
  • Imaginary numbers
  • Base 2 (binary) and base 16 (hexadecimal) numerals
  • Statistical presentation and testing for statistical significance
  • Vectors matrix algebra
  • Calculus

At the bottom of all this lies my ability to count (1, 2, 3, …) which I was taught at a very young age. It was at about this time that I also noticed that not everything could be measured in whole inches, which lay the basis for my later grasp of fractions. I have a very clear memory of learning how to add double and triple digit numbers. Later came the rote learning of times tables, and I am very grateful for having been taught them. Procedures for long multiplication and long division soon followed, whence the concept of a remainder built on my observation of fractions of an inch. I was introduced to simple geometry at about the same time, this built on the concepts of space and measurement. At high school, these ideas were generalised into algebra, which itself served as a basis for learning calculus and the real-world applications thereof. Algebra was also used as the basis for learning vector matrix algebra, as was statistics.

What we have here is a structure of learning that was built on a foundation of a few simple concepts (counting, space, time, and observation) that my teachers helped me to assemble.

Putting this into a historical perspective, I have been the beneficiary of mathematicians whose efforts go back to the ancient Egyptians. The Arabs are credited with the idea of zero as a digit, but this is predated by its appearance in India in about the fifth century CE. Isaac Newton and Gottfried Leibniz are jointly credited with inventing calculus. More recently, vectors came about as the result of about a dozen people over 200 years. Let me put it another way:

If it had not been for the efforts of such people, I would still be counting by using pebbles on the floor of a cave.

Back to the Battlefront

The idea of “discovery-based learning” in mathematics strikes me as being fundamentally flawed. In my view, what is needed is a thorough discussion on what mathematical skills today’s children might want to use in their own adulthoods, and how best to equip them with those skills. Given the developmental benefits described in the article cited at the top of this post, I think it would be madness not to use rote learning to teach times tables.



This post owes its existence to a friend of a friend who asked about how the image below was created. This post is my answer to his question.

Of Cellular Automata in General

A picture is worth a thousand words, so I’ll start with an animated picture from a famous cellular automaton:
(Image courtesy of Nathaniel Johnston who has placed it in the public domain.) This animation is a “period 3 oscillator”. The black cells are “alive”, and the white cells are “dead”. At each stage, the cells that are alive in the next generation are governed by the cells that are alive in the present generation. This example comes from John Horton Conway’s Game of Life. The rules that govern which cells become alive and which cells die can be found here.

This is an example of a “square” cellular automaton, and is one of the three regular tilings of the plane, the other two being triangular and hexagonal:

triangular hexagonal

Cellular automata are not limited to tilings of the plane, nor are they limited to being organised in any particular or regular pattern. Moreover, there is no requirement for the “life” of any cell to be governed by immediately adjacent cells, nor indeed for the state of any particular cell to be binary. However, for the purposes of study it is convenient to limit oneself to regular tilings of the plane, the cell itself, and immediately adjacent cells, all cells being binary in value.

The Game of Life defines a function from one time step to the next, and has three possible values for every cell in the automaton:

  1. Birth
  2. Persistence
  3. Death

This can be expressed in more mathematical terms as:

  1. f(x) = 1
  2. f(x) = x (i.e. the identity function)
  3. f(x) = 0

The Game of Life uses the values of nine cells (eight adjacent plus the cell itself) when calculating f. Generalising this to the other two regular tilings yields:

  Triangular Square Hexagonal
Count 13 9 7

There is also appeal in generalising f as follows:

  1. f(x) = 1
  2. f(x) = x (i.e. the identity function)
  3. f(x) = 1-x (i.e. the inversion, or binary “not”, function)
  4. f(x) = 0

For the sake of both clarity and brevity in the rest of this post, it is convenient to use the word “machine”. The machine for the Game of Life looks like this:
In this scenario, the Game of Life is just one of 49 different machines, and can be notated as “000001I00”. The image at the top of this post is based on 47 different machines, the majority of which turn out to be quite boring. The following comments apply to all machines:

Notation Observations
000 … 000 All cells are “dead” after the first generation
III … III Nothing changes
VVV … VVV Every second generation is the same – the whole automaton merely inverts at every generation
111 … 111 All cells are “alive” after the first generation

Some cellular automata result in an ever growing number of “live” cells. The example below is from the Game of Life. It was made by Kieff, and is reproduced under a CC BY-SA 3.0 licence:

Of Some Particular Hexagonal Cellular Automata

The machines used in the creation of the image at the top of this post started with the following question: “If I start with a single live cell, what happens to it under the complete range of machines?”. It turns that some of them grow forever. The first 17 generations of machine “0000I10” look like this:
The natural extension of this idea is “What do successive (and not ‘boring’) machines look like at the same generation?”, to which this is a typical answer at generation 13:
and you can see machine 0000I10 at generation 13 (filename 0000130.bmp) in both images.

(As an aside, the folder in my computer that contains the generation 13 images has a total of 2,282 files and is 13.3 GB in size.)

Monochrome is all well and good, but adding colour adds visual interest. The easiest way to do this is to take successive triples of images, assign each member of a triple to red, green and blue channels, and combine the result:
The image immediately above is composed as follows:

Channel Machine
red 0000IV0 (0000120)
green 0000I10 (0000130)
blue 0000VV0 (0000220)

The final step was to arrange the composite hexagons into the image at the top of this post.

And by way of a bonus image, here is machine 000I11I at generation 50 coloured using a completely different algorithm:

A Mathematical “Deception”


People often use visual cues to decide whether or not to buy something. How big are those bananas, so how many should I buy of them? The same question gets asked when buying chocolate for Christmas. I was tidying up after last year’s festivities, and I noticed something about one of the empty boxes. Now seems like a good time to explore it.

Judging By Its Size

This is a picture the Christmas box, with an ordinary bar of chocolate put on top of it:Top Viewand here is the same thing seem from the edge:Side ViewFrom a visual perspective, you might expect the box to contain more, possibly a lot more, than is in the simple wrapper. You would be mistaken. The simple wrapper contains 350g. The box contained only 320g. I leave it you estimate the ratios of width, height and depth from these pictures.

A look at the internal carrier was also revealing:CarrierThe individual chocolates are held in the widened “bases” as they appear in this picture. Those things that look like upside-down plastic beakers raise the individual chocolates from the bottom of the box. (The carrier is shown upside-down in this picture.) A comparison of the simply wrapped chocolate with all that empty volume is revealing.

Next Time You Go Shopping …

Next time I go shopping for Christmas, I will be taking a very hard look at the weights printed on the boxes and wrappers!

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.

Maths Teaching – a Rant

The Teaching of Mathematics – a Rant


My Experience of Maths as a Child

When I was a child at school, I learned my maths from teachers with a wide range of understanding of the subject. I was fortunate in that one of my teachers in kindergarten taught it extremely effectively.

As my schooling went on, my understanding of maths continued to develop. I still use this as a basis for much of my own work today.

There was a theme that ran through all of my teachers’ delivery: “All the answers in mathematics are known.”, and as a student I was expected to learn them.

Other Sciences in the News

Ever since I can remember, there have been reports in the popular media about progress in the sciences, but that reports of progress in mathematics were distinctly lacking. Recent examples in the sciences include developments in astronomy and information technology. As an impressionable child, the message that I learned was that while the physical sciences was an area of active research (in all its different fields), nothing was happening in mathematics. This, of course, was a completely false impression.

My Own Experiences as a Teacher of Maths

It was my pleasure to work with a youth-at-risk student who had both the desire and capacity to learn mathematics. This was up to Year 10 level. At the end of the course I asked him if he thought that all the answers to all the problems in mathematics were solved, and he answered with an emphatic yes. When I told him that there were myriad as yet unsolved problems in mathematics, he was incredulous.

At the same time I also tutored a year 9 student with her mathematics. From the nature of the feedback from her Mathematics teacher, and also looking retrospectively at my own feedback to her, I expect that she also would come to the notion that all the problems in mathematics were solved. She was not a strong enough student to cope with the alternative notion, so I forebore to mention it.

And So to University

Well, I went to university to read mathematics, and the result was a disaster. Okay, I may not have been bright enough to make the grade, but I am convinced that the culture shock from “all the answers are known” to “here we address unsolved problems” did not help, and I strongly suspect that I am not alone.

A More General Malaise

Mathematics is an enabling tool with which to do science. No year goes by but I hear of university science lecturers bemoaning the lack of mathematical understanding in the year’s undergraduate intake. By way of example, young people go into environmental science courses without appreciating that they need a very good grasp of statistics, and then find themselves in a remedial maths course just to catch up.

Just to make matters worse, the Training Packages used in Australia no longer identify mathematics as an explicit employability skill. The message is clear: numeracy is not seen as being important. I am no longer surprised by shop assisants who are totally thrown when I hand over $20.30 for an item costing $15.30: I am now just saddened. Mind you, this dropping of mathematics from the list of employability skills comes as a result of research done in Australia into the needs of Australia’s employers. I wonder how employers will cope with an even less numerate workforce in the medium-term future.

What Needs to be Done?

The message needs to be put out to the whole of the Australian primary and secondary educational establishment that mathematics is important. We are handicapping ourselves economically as a nation by remaining as innumerate as we are. We are also failing to properly prepare those who go onto university degree courses in whatever science subject: this does nothing for the future development of the country.

And on a personal note, I think that it is high time that Year 12 teachers in mathematics should better prepare those students who a going on to study mathematics at university: there are lots of challenges for the students to take on!