All sorts of numbers are used to describe us. The following numbers (with a few alterations to protect the innocent) can be used to describe me: 70, 50, 44, 145, 18, 2013, 176, 12438756, 51008420, 3, 0.25, 2, 26, 6439801802, 36942, 120000, 7,12. They include:

- number of children born to me,
- bank number,
- tax number,
- age,
- street number,
- postcode,
- clothing size,
- phone-number,
- number of years married,
- height,
- weight,
- employee number,
- level of education,
- IQ,
- number of weeks pay in my redundancy package,
- amount owing on my mortgage.

Some of the numbers uniquely describe me, such as the IRD number, or employee number. Some of them are arbitrary and serve only as an identifier; others are meaningful and describe some aspect of my life. Some are exact and others are approximations. Some are permanent and some are temporary. Some have meaning in my life, and affect the way I think about myself, whereas others are value-neutral. In my previous post I talk about models, and how the disciplines of statistics and operations research rely on the concept of a mathematical model. For some people, the application of numbers or a diagrammatic model to solving a problem is obvious. It was to me. I have always had a visual representation of the year in my head. As a child, the six week holiday over summer was represented in my mind as the base of a trapezium, with the three school terms as the other three shorter sides. The lengths represented their importance to me, rather than their actual length. One year I was teaching Critical Path, a technique I find appealing and enjoyable as it uses diagrams and numbers to bring order to a messy problem. One student commented that she had never thought of representing a real-life problem in that way. That was an outrageous thought, though I hid my astonishment. Some people do not think in terms of mathematical models! Go figure!

# Lost in Transnumeration?

Maxine Pfannkuch and Chris Wild coined a useful term, transnumeration, meaning, in their terms, “changing representations to engender understanding”. Transnumeration occurs when a physical or abstract idea is expressed in terms of a number, or vice versa. There are several instances where transnumeration takes place during a statistical exploration. To begin with measures of the characteristics of interest are taken, either with physical measurement, or categorisation or an ordinal scale approximating opinion or similar. Each of the numbers listed above constitutes an enumeration of some aspect of my life. The raw data is then expressed in graphical form, or undergoes some numerical transformation to produce, for example, a confidence interval or prediction. This is done in order to obtain meaning from the data, and fits with the formal definition of transnumeration. Each datum has little meaning individually, but the emergent property of the collected data provides insight and information. However, in order for the meaning to be communicated, there must be a further transnumeration, expressing the bare outcomes in terms of the original problem situation. This occurs in the writing of a statistical report.

Transnumeration is essential to operations research. Most OR solution processes involve converting a real world problem into a mathematical model. Aspects of the situation may be transnumerated several times to enable insights, and at times, optimisation. Data include objective measures and subjective values such as weights of importance for aspects of a problem, probabilities of outcomes, and priorities for objectives. I suspect that some people struggle with the idea of transnumeration, and it may be helpful to teach this explicitly. Commonly the people who teach statistics and operations research are natural transnumerators. (Another new word). We naturally move between numbers and words. Or maybe not! Could it be that truly great mathematicians also have trouble transnumerating, preferring to stay with the numbers? However it is, we should be aware that students may not be like us, and may struggle to see the links between the numbers and the entities, or any reason for it!

# Teaching suggestion

I don’t know about you, transnumerating readers, but I find this whole topic intriguing. After first thinking about it, at several times I would think of different numbers that can be used to describe me. I suggest that this would make for an interesting lesson in a mathematics class at the beginning of a statistics unit of work. Students are asked to think up what numbers can be used to describe them. They could have a competition to see who can think up the most, or the most that no one else thought of. There could be a class target – can they come up with fifty such numbers. Can we tell who people are from their numbers? They can go home and ask about numbers their parents use. (With a proviso that NO ONE tells PIN numbers!). You might even like to look at numerology. Then when you have your collection of “number-labels” you can then discuss their properties, which leads naturally to explanation of levels of measurement – nominal, ordinal and interval/ratio, or quantitative and qualitative. (depending on the level of the class). From here, graphical representation is an interesting step. This would include collection of data from the class, and creating different types of graphs, and each time getting the individuals to see where they fit in the graph. I wish I were teaching a class at an appropriate level to use this, as it sounds fun and useful. I’d love to hear from anyone who does this – or who may have already thought of it and do it already. I hope my favourite maths teacher blogger, Fawn Nguyen will do it, if she hasn’t already.

# I am not a number; I am a free man!

People can have quite an aversion to being labelled as a number. We like to think of ourselves as individuals and unique. As a closing thought, here is a tribute piece to Patrick McGoohan and the 1960s UK television series, Prisoner, which coined the phrase, “I am not a number; I am a free man.

Someone (I forget who) characterized mathematics as the discovery of patterns. Those of us in OR, stats or analytics need to quantify those patterns with numbers, but in other areas of mathematics I suspect that “transnumeration” plays a much lesser role. I seem to recall topologists (or algebraic topologists) drawing planar diagrams such as rectangles, sticking arrow heads on edges to indicate orientation, and visualizing them as torus, Mobius strip, Klein bottle etc. No numbers required.

Thanks for the clip of the late Patrick McGoohan. As a youth, I watched at least three TV series (or, in one case, miniseries) in which he starred.

My husband recently watched all of “The Prisoner” on DVD. I found it really odd and disturbingly Kafka-esque.

It seems real mathematicians abandon numbers altogether except as subscripts. Further evidence that stats and OR are not just “subfields” of mathematics as some believe.

I too thought “The Prisoner” was a bit odd. McGoohan as “The Scarecrow of Romney Marsh” (https://www.youtube.com/watch?v=JDYId2Ab1o8) might have been a bit more disturbing, especially as I was a pre-teen when it was on. I understand the books are actually scarier than the Disney version (not shocking).

As to mathematicians and numbers, the number theorists of course still pay attention to them (mainly integers, I suppose), and the numerical analysts still rely on them. The rest of us live and die by the Greek (and occasionally Old German) alphabet.

I’ve occasionally wondered what the Greeks use.

Hi Dr Nic. I had to read the section “Lost in Transnumeration?” twice as I was lost at first with this new term. But its meaning is not new at all, rather it seems natural to take raw data and produce their graphical equivalences.

You’re right in both these sentences: “We naturally move between numbers and words. Or maybe not!” When older kids have trouble placing fractions and decimals on a number line, we must find a better way to help them grasp and make sense of it for themselves. Any visual model of a number is powerful — and we can start with having kids picture triangular, square, and cube numbers.

I love your whole idea of having kids use numbers to describe themselves. We can learn about sets and subsets too. It would be fun to guess which set of numbers belongs to whom. One version could be that the guesser may ask for any 3 numbers about the mystery person — like the person’s number of brothers, last digit of cell phone number, and height in centimeters. I’m already thinking of lots of ideas with this!

You are so kind to comment about me in this post. I’m flattered beyond words. Thank you for sharing your wonderful idea with us!

That’s great. I look forward to hearing how it goes. I love hearing about what you do in your classes – it makes me wish my children could have had you for a teacher.

I think I’ve mentioned Rogo before, but you might like to take a look at that as another possible enrichment activity. It is based on the travelling salesman problem and kids really like it. You can see it at http://www.rogopuzzle.com.

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