Main Subjects of Exploration
While I like to teach around topics and activities of interest to the student - whether that’s arts and crafts or astronomy and physisc - my teaching and exploration includes three inseparable areas of focus:
Numbers, Measurements and Calculations
English, Languages and Linguistic Processing
Reasoning, the Learning Process and Dialogue
Below I’ve written a few words on each to give a sense of some of the areas that I’ve found to carry great significance, and how they can be explored.
Numeracy, Measurement and Maths
If you were born a million years ago, without words and numbers, how would you tell me that there were 11 sabretooth tigers behind me?
I like questions like this as they are fun doorways to making sense of the number system and symbols that form the basis of what most of the world thinks of as maths today. You can see how the Roman’s started to use the letter ‘X’ to represent 10, and the letter ‘I’ to represent 1 for example - a bit like a flash of a pair of hands with 10 fingers followed by a single finger - before the adoption of a system being used in India and Arabia to develop symbols to represent how many ‘X’’s and how many ‘I’’s according to their position was used.
The number systems and the techniques most commonly used these days aren’t the only ones - in Japanese education, calculations are more pictorial; in Alaska, the counting system common amongst Inuit’s groups things in 20 rather than in 10; while in many Indigenous cultures, numbers and arithmetic are often secondary to that understanding of the relationships between moving parts. Afterall, sundial and compass become less important if you can read the skies.
Nevertheless, you’re probably not reading this from the starlit plains, but rather from a place where discomfort with the basics of modern day maths may close some doors; and where a little familiarity could be useful; and where some curiosity and enjoyment might open new doorways. Understanding the language of maths and how it has evolved gives a good grounding to build on.
Why is a meter the length it is?
Surely an important part of understanding maths is understanding of the things being counted or measured. Think of France in 1789, and you probably wouldn’t be thinking of the group that set off to measure the distance from Paris to Barcelona armed with telescopes and measurement sticks in the midst of the French Revolution. But this measurement became the basis for the Meter - a million meters (10,000km) their estimation of the distance from the North Pole to the Equator - and the metric system, which related length to weight and volume using water: one cubic meter of water becoming the measure of a tonne (1000kgs), and 1000 litres.
And so the world moved - somewhat begrudgingly in part - from counting in feet, steps, fields and whatever else, towards a universal measurement system that the World could relate to.
Where will the next leaf grow?
Understanding the system that underpins the mathematics being used not only gives sense to the techniques taught in school, but opens up freedom to use alternative techniques, tricks and shortcuts, especially ones that make maths useful on the go, away from an exam room.
So bringing their design and evolution to life is a good starting point, and the same can be said for algebra and more advanced mathematics. I’m confident that I can bring mathematics into pretty much any topic of interest to the student, and at the very least, I’d like to share a level of comfort, ability and practical understanding with fundamental mathematics.
English, Languages and Linguistic Processing
While maths is brilliant for counting things and working with quantities, language has the difficult role of describing the things to be counted and their qualities.
Consider the expression ‘I’m not good at maths’. If I said I wasn’t good at speaking Japanese, that would be true, but it might not carry the implication that I couldn’t do it. Mathematical ability seems to be considered differently, like it’s an ability that is predetermined. Shakespeare’s Julius Ceasar said ‘I am constant as the northern star’ - only it turns out that when Julius Cearsar was alive, the northern star would have been different to the northern star when Shakespeare was around (given the Earth’s spinning like a top, gently wobbling around as it rapidly spins).
I can’t be certain, but I have seen that for people who say they’re not good at maths, this idea - often linked with ideas of intelligence - is often the most limiting factor, that once challenged can open new possibilities. The association with intelligence is an interesting one also, as the original meaning of the word - around reading (legere) between the lines (inter) - perhaps wouldn’t have be so closely associated with something as technical as maths in years gone by. Maths these days seems to be increasingly about following clear lines, or connecting them, not necessarily reading between them.
So my renewed interest in languages comes largely from an increased awareness of the relationship between language and culture. This has led to building relationships with learning centres dedicated to the deep explorations of the likes of David Bohm at the Pari Centre, and Jiddu Krishnamurthi at the Krishnamurthi Centre. And yet, it has been the communal explorations with students, including those who have struggled with verbal communication, that have revealed the most about the significance of languages and the meanings and understandings they carry.
While some of these subjects may appear more adult in nature, I’ve found topics like how the meanings of words have changed over time, and how meanings vary across cultures, to be a starting point that sparks curiosity and offers a deeper understanding of language as a tool. It invites exploration into the nature of words, their etymology and meaning, grammatical frameworks, and the cultural implications of languages, and can be helpful in learning English according to the UK curriculum; English as a second language; or other languages (I also teach French).
Reasoning, the Learning Process and Dialogue
Earlier on this page, I asked the question ‘Why is one meter the length that it is?’. The answer is based on the attempt to have a universal measurement system, based on a universal thing… like the size of the world. The length of one meter is based upon the decision to describe the distance from pole to equator as 10,000km); and from this we also have notions of weight and volume: 1cm cube of water weighs 1 gram, and is 1ml.
These endeavours were breakthroughs in measurement, and contributed to technological progress. Thing is, as knowledge develops, it seems we discover as much about new stuff as we do about things we missed or got wrong. For example, as nice an idea as it may be to have the earth as a universal measurement, it’s actually quite hard to measure, not least because it’s not perfectly round, but flattened at the poles. So the meter is now based upon a distance travelled by light in a vacuum - a constant speed, apparently. Mind you, the meaning of the word “atom” is indivisible, and then they split the atom…
So how about this one. Let’s say the rope is tied around the equator, which is been nicely smoothed out for us, such that the rope is 40,000km long. If you were to cut the rope, and add an extra meter in length, and then stretch it out again all the way around, so that any slack is evenly distributed into a big circle… how far off the ground would this 40,000.001km new rope be when pulled out tightly?
Are we talking nanometers, micrometers, millimeters, centimeters, meters? Would you see a visible gap between the Earth and this new rope, 40,000,001m long?
It turns out, that single extra meter would create a well-placed tripwire all the way around the Earth...
…about 16cm off the ground. When I first heard this, I didn’t believe it. But it’s hard to argue with the fact that if you increase the circumference of a circle by 1 meter, the radius extends by 1 meter / 2π. And that’s why I like it - because it’s a great example of the case for flexibility in judgement.
When I was 7 or so, my claim to fame was that I drew a chess match with a Grandmaster. Unfortunately, that doesn’t tell the full story - he was playing 7 other people at the time; I was clinging on; and my Dad snuck in to point out a stalemate.
And to borrow a line from The Strokes, life ain’t chess, and many aspects of life don’t seem appropriate to consider as individual pieces on a board, planning several moves ahead. So while I teach some chess, and enjoy a game and the soft skills that can come with it - like patience, perseverance, learning from mistakes and sportsmanship - really I’m more interested in the times when the learning process is seen to have limitations, and whether these limitations can be better revealed and understood.
I’ve worked with many children and adults with SEN, including diagnoses of dyslexia, dyscalculia, ADHD and autism. Personally, I am a little wary to use labels that are so open to interpretation, so while diagnoses may provide some context, I like to come in with as clean a slate as possible, and the possibility that behaviour may not be as fixed as the language used around it.
And, a hobby of mine if you like, is participating in what those of us involved call Dialogues - group inquiries where the limitations of language, perspective and learning processes are taken into account, to offer potential for exploration further into the unknown.