Maths from a Picture

The inspiration for this post comes from Brian Marks (Twitter: @Yummymath, web site with this tweet:
It seemed to me fairly easy to use that picture as a basis for writing a number of maths related questions, so I produced this document in Google drive. It has 13 questions related directly to the picture, and another 23 questions which use that picture as a starting point. The questions are in no particular order, and cover a broad range of topics and levels. While that document is a response to the question posed, I would hesitate to give it to any but the most competent and confident student of maths at year 11 or 12 level. It also goes well outside the area of pure mathematics, and provides some support in the area of physics. Having said that, it may provide an idea or two to anybody who cares to read it.

I would be very interested to hear your comments.

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.

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.

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.