Teaching maths is a subject that is often put in the ‘difficult but necessary box’. Over the last fifty years, UK governments have left a trail of initiatives, ranging from “New Maths” to “Maths to 18”, with little or no improvement in maths skills. What if the philosopher Wittgenstein designed a new maths curriculum?

After the publication of the Tractatus, which he felt solved all of the problems in philosophy, Wittgenstein spent six years as a primary school teacher in Lower Austria. Wittgenstein’s style of teaching had an emphasis on precision and clarity; he believed that clear language reflected clear thought. He preferred “hands-on” learning, especially in maths. Wittgenstein treated maths as a language, and he wanted pupils to see how numbers work, not just remember formulas.

Wittgenstein would have used objects, diagrams, and practical examples to make maths ideas tangible. For instance, he’d connect arithmetic to counting apples or drawing lines, which would ensure the children grasped the meaning of number relationships. This would align with how children naturally learn: through seeing and doing before abstract reasoning. They would build strong mental models of quantity and pattern, which would be the foundations for later abstract thought. Today’s teaching sometimes moves too quickly to symbols and abstract problems, especially under exam pressures, which can leave many pupils behind.

Wittgenstein viewed maths as a language, with a set of rules governing meaning and use. Misunderstandings were treated as language mistakes, not failures of intelligence. Pupils would gain clarity in expression and reasoning and learn to think and communicate mathematically. Current teaching often separates literacy and numeracy, even though language shapes how we understand numbers and can reduce maths to rote learning.

For Wittgenstein, teaching maths would have been about cultivating truthfulness, precision, and self-correction - in other words, habits of mind, not just skills. He wanted pupils to care about getting things right because they valued truth, not because of marks or fear of failure. Pupils would develop resilience and intellectual honesty, which are qualities useful beyond mathematics. Compared to the current systems, which often overemphasise grades and targets, Wittgenstein’s curriculum would encourage the moral quality of careful thinking.

Wittgenstein believed that struggle is part of learning. He resisted giving easy answers and expected pupils to work things out themselves. This approach fosters independence and deep satisfaction in discovery. Pupils would then experience the joy of insight and become self-reliant problem solvers. Currently, in classrooms, which are under time pressure, teachers often guide too quickly through difficulty.

If Wittgenstein had designed a curriculum for maths, then it would have generally improved numeracy skills, which are a part of everyday living, from paying bills taking out a mortgage or ordering enough paint for a wall. The curriculum would give students greater confidence in maths so that they could go further in STEM subjects and give greater career opportunities. Finally, it would improve critical thinking skills, which are one of the core skills to actively participate in society. Next time you are attending a parents evening and discussing the progress of your child with maths, why don’t you ask why they are not adopting a curriculum that Wittgenstein would have developed?

The fear of numbers can crop up at some unlikely times. The most recent was over a coffee with a friend, and it came as a surprise.

He ran a successful business which relied on data analysis. He navigated the financial ups and downs of bringing up a family, and made some shrewd house purchases. Numbers were a key tool in his life, and yet he was frightened of them! He is not alone. One in five parents report they are anxious when using numbers.

The ability to handle numbers gets a bad press. Celebrities proudly boast they ‘can’t do maths’, which is a contradiction, because they must understand numbers when it comes to the amount of money they want from their latest appearance in the media or on the socials. Celebrity mathematicians provide an intellectual curiosity about maths, but don’t help with the people trying to understand the latest mortgage deal. All of this gives the impression that the ability to handle numbers is beyond most people’s reach.

I suspect that the fear of numbers is inherent in the education system. In my day we had to stand in front of the class and recite times tables or perform calculations in our head, it was more a test of character than developing numbers skills. The education system has moved on since then but it is estimated that about 30% of young children in the age group 8 - 9 reported feeling anxious when doing maths tests. The government’s Early Years Foundation Stage is an attempt to reduce this anxiety by setting standards for learning about numbers which focuses on learning through play and exploration. For example, counting toys, sorting objects, singing number songs, and using everyday situations such as counting snacks, steps, or friends. Integrating numbers into everyday living and not making them something mysterious is a good step forward. But, for it to be successful, parents must provide support for their children at home. The challenge is to get those parents who are struggling with maths, to come to terms with their lack of numeracy skills in a positive way so that they are motivated to seek help. They need to move from “being bad at maths” to building their confidence.

Help is at hand. The UK government has a Multiply programme which offers free or fully funded numeracy courses for adults who do not already hold a qualifications in maths. The charity, National Numeracy, whose ambassador is the financial celebrity Martin Lewis, offers resources and support for adults. Free online tools like the National Numeracy Challenge lets adults test and improve their skills anonymously and at their own pace. This builds confidence before joining classes or talking to others. Many local colleges and adult learning centres offer free or subsidised maths/numeracy courses for adults.

Numbers are part of the language of everyday living. They let us measure, compare, and understand the world. With improved numeracy people will gain freedom, confidence, and control. Next time somebody mentions they are frightened by numbers why don’t you buy them a coffee, and encourage them to contact an organisations that could help.

There is a danger when writing a blog that the words can turn into clichés, overused and worn out. Looking at how other authors use words, such as poets, can add energy to a tired looking post.

Poets take the words that we use every day, and when successful, turn them into memorable poetry. They are economical with words, and rely on the careful use of metaphors to express the emotion they are trying to capture. For example, Seamus Heaney uses metaphors to explore the creative process in writing poetry. In The Forge, the blacksmith’s work in the forge becomes a metaphor for poetic creation. Just as the smith shapes metal in heat and darkness, the poet hammers out words into form. The “anvil’s short-pitched ring” stands for the sound and rhythm of poetry. In “Personal Helicon” wells and echoes are metaphors for poetic inspiration. Peering into the depths of wells is like plumbing the depths of language and memory to find poetic voice.

A metaphor is a word or a phrase used to describe something as if it were something else. There are many techniques that can be used to help in developing them. For example, define what you want to describe, list words from other areas, and then look at the shared features. For example, the impact of AI on society could be compared to a river, then a metaphor would be “AI is a river quietly changing the shape of the banks.” Another approach is to take unrelated items and see what connections can be made. Trying to connect democracy to a flock of birds would produce “Democracy works better when its voices flock together, not scatter”. Even Wittgenstein can help. He believed the meaning of a word comes from its use. To form a metaphor in his style you need to show a scene. Instead of saying people are easily influenced, it can be written as “Crowds lean like sunflowers towards the brightest noise”. A quick search on the web will provide many more techniques.

Imagine using metaphors in your next post to explore changes in technology, politics, climate change or anything else that you want to write about. For example, a post written about the influence of AI on politics. Picking up on the river metaphor: “AI behaves like a fast, silent river that divides as it flows.
 As people step into it, the current gently directs them left or right in a way that they hardly notice. Soon, each side sees only those drifting the same way.
 By the time they reach a delta, it feels obvious that everyone is with them.
 Politics becomes a landscape eroded by invisible currents.” Is this metaphor more lively than the many sleep inducing posts in the socials at the moment?

Learning from poets and how they use words and phrases, particularly in metaphors, to express emotions can be adapted to freshen up opinions and arguments written in a post. Maybe one day a post will be written that will be memorable and have an impact on how we live.

When I came across The Barista Paradox, I started to imagine a queue of puzzled people standing outside my favourite café. Would it’s door open or not?

The Barista Paradox is as follows: in a small town there is one barista, where the rule is that they make coffee for those people, and only those people, who don’t make it for themselves. But, who makes coffee for the barista? There are two situations:

1. If she makes it for herself, then because she only makes it for those that do not make it for themselves, she can not make herself a coffee.

2. If she does not make it herself, then because she makes it for all those that don’t make their own coffee, she makes herself a coffee.

Therefore, the barista only makes herself a coffee, if and only if she does not make herself a coffee. A paradox! Most philosophers would say the answer is the barista  doesn’t exist. No door opening soon!

The paradox has a sprinkling of the terms ‘all’, ‘only’ and “if and only if” which signals it’s a problem in logic rather than a real life problem. They force the description into saying that all coffee drinkers who do not make their own coffee can only get it from the barista, and nobody else. Not realistic, if the café has a future.

The barista paradox shows what happens when buying coffee is forced into a logical problem. Logic has many applications; electronics, computer science and mathematics. Wittgenstein would show the words from logic have lost their meaning, when used in buying coffee, and we have been sucked into a philosophical vortex. Rather than spending time puzzling about the paradox, we should see it as nonsense, and have another cup.

Philosophers spend most of their careers thinking about our lives. Questions about what we know and the world that we live in, including how we should live with people. What they find is wrapped up in books that most of us don’t open. However, they can influence decision makers who are always grasping for a banner to wave as a gathering point, such as monetarism, classical liberalism, securonomics etc. Maybe we should have a closer look at what they are waving, just in case it is riddled with paradoxes which lead us nowhere.

Next time you are in your favourite café and the barista is crafting a delicious coffee, why don’t you buy them a coffee to make sure that they don’t fall into the The Barista’s Paradox.

Increasing swings in extreme weather are happening globally: it is clear that climate change is underway. If I was a gambling person, what odds would be favourable enough to place a bet that we would reach the global target of net-zero greenhouse gases by the end of the century?

Taking up a pen and paper I started carry out a back-of-the-envelope calculation to estimate the odds of reaching net-zero greenhouse gases by the end of the century. As Sir David Attenborough constantly reminds us, greenhouse gases are driving climate change which is a product of human activity. To get a handle on calculating our impact, I broke it into the following factors: population growth, technology development, climate policy, energy transition, and geopolitical stability.

The current global population is around 8 billion people. The best estimates give a spread of between 7 billion and 10 billion by the end the century. Therefore, I would give it a probability of about 50%, that the population will stabilise at around the current size and therefore its contribution to green house gases stabilise. Next is technology development which covers AI, biotechnology, green technology, and probably technology that we haven’t dreamed of yet. However, based on the speed of adopting new technology, which is always slower than we think because of high initial costs and the lack of skills to implement it, then it could take tens of years to stabile greenhouse gases, therefore I put it at a probability of 30%. Now for the one that is in the headlines most climate policy. The Paris Agreement set the global climate goals: limiting global warming to well below 2°C above industrial levels, preferably limiting the increase to 1.5°C and achieve net-zero global greenhouse gases in the second half of the 21st century. As of 2024, 140 countries have set net zero targets including: EU, UK, USA, Japan, and South Korea by 2050, China by 2060 and India by 2070. The current status is that the global greenhouse gases are still not declining fast enough to stay below 1.5°C, and it is currently estimated at 1.2°C, therefore it would be fair to conclude that climate policies are not working effectively, and therefore the probability of success is around 30%. Next is the energy transitions from oil based to clean sources such as electric sources of energy. Currently, only ~15% of global primary energy, that is, total energy used across all sectors, including heat and transport, comes from clean sources. Most applications such as transport, heating, and industry still rely heavily on fossil fuels. Although many businesses are giving out optimistic messages about the transition, I am not as hopeful that it will be fast enough to stabilise greenhouse gases, so I put the probability at 50%. Finally, geopolitical stability which is difficult to assess. If people are busy fighting for their survival then greenhouse gases and climate change isn’t a priority. There is more conflict in the world than reported in the media. For example, in 2023 about 14% of the world’s population lived within 5 km of a violent conflict. Therefore, I gave the probability of 30% that geopolitics will distract from work on reducing greenhouse gases and therefore it will not stabilise by the end of the century. Pulling all of the factors together to calculate the overall probability of reaching net-zero by the end of the century gives: \(0.5 \times 0.3 \times 0.3 \times 0.5 \times 0.3 = 0.00675\) or approximately 1 in 150 ^*chance that we will reach net zero by the end of the century - not good odds!.

Of course the above are all guesstimates supported by a lot of hand waving. But the key point is that there are many complex factors involved in reducing the greenhouse gases to net-zero. For example even if we were very optimistic about the success of each factor and recalculated e.g. \(0.9 \times 0.9 \times 0.9 \times 0.9 \times 0.9 = 0.59\) or which is only an evens change of reaching net zero. Global leadership needs to be made more accountable at reaching net-zero, honest about the success of reducing greenhouse gases to net zero, and put more effort into mitigating the effects of green house gases that are already built in to climate change. What bookmaker would give us favourable odds for achieving that?

^* To convert from a probability to odds, divide the probability by one minus that probability. So if the probability is 0.00675, then the odds are 0.00675/(1-0.00675) or approximately ‘1 to 150’