Edit: @SolveMyMaths has written an excellent blog on this

Edit: a talk by Conrad Wolfram – thanks to RF for sharing

Edit: Vsauce’s compelling argument that exploring Maths is a function of our curiosity – the thing that sets us apart from Neandertals. See video, 17.45 in, or start earlier to learn about the Ross-Littlewood Paradox.

**Noble Thoughts**

In a maths meeting this week we spoke about why people should learn maths, and what key things we should ensure are covered in the lessons. Things that came up were:

- To teach core wellbeing principles, such as how to fail and how to work with others (present throughout all schools)
- An appreciation of truth and beauty (maybe in sixth form)
- Learning to problem-solve generally, for use in 21st century jobs (maybe in the middle school)
- Understanding links between maths and other subjects, between maths and things in the real world (maybe in secondary school)

Different teachers placed emphasis on these differently, but broadly we agree that these four are important. We think noble thoughts, but we do do noble deeds?

**Noble Deeds?**

In a typical lesson, students will be given carefully and excellently planned tasks, controlled by the teacher, to progress through a topic (*To clarify, I wish I could plan lessons as deeply thought of as this one, here is one by me that I am less pleased with*). The focus will be on doing a rule, and possibly applying it to worded problems. Assessments to check understanding are skills-based, with minimal problem-solving (at least in my lessons, despite my best efforts). The majority of projects link maths to art, sometimes tenuously (for example I am now running a project where the students are using their geometry to design a gate).

An example. This week in Yr7 we have been adding fractions (very thoughtful planning by Alberto). I have not had to add fractions since I was at school myself. If I needed to, I would probably use a calculator anyway. Why do we teach adding fractions? (*Edit: there might be better examples of topics that are hard to justify. For example, polynomial equations, volume of cone, negative indices…)*

- Because it is on the exam.
**We are meant to be re-inventing education, this is simply not good enough.** - Because it might be useful to some students in their future careers.
**So is gardening, or coding, but we don’t make everyone learn those.** - Because it is a building block of more difficult maths.
**It might be a building block to more difficult, but even more irrelevant maths (trigonometry, probability of picking two red balls from a bag…).** - Because it offers a glimpse into the structure of maths.
**Not the way we teach it (here is a method, now practice it).**

I currently cannot give a satisfactory answer for why students should be able to add fractions (please tell me I am wrong!). Differentiating between real and fake news, understanding how an infographic might be persuading you to think a certain thing – these are far more obviously useful.

Learning how to stay healthy (by 2030 a **third** of adults in the UK will be obese) is surely more important than finding the length of the hypotenuse. Currently we cannot teach children either.

**Conclusion**

Our teaching practice is at a mismatch to our vision. Either we change our vision, or we change our practice. If we are to change our practice, we should do it one step at a time, making each other accountable for small achievable differences. (This is not yet a satisfactory conclusion).

**Postscript**

By the way, there are plenty of other models out there for what every lesson should include. We should walk before we can run (by implementing our vision), but for interest they are:

Don Steward says every lesson should include a glimpse of infinity – a chance to generalise beyond specific examples.

Mathwithbaddrawings says every lesson should involve the following process:

Rosie and I sketched another process:

- Do the rules
- Apply the rules
- Understand the rules
- Break the rules

Dan Meyer says that the headache (a difficult and novel problem) should be presented first, before giving the aspirin (a new mathematical technique).

I’m still thinking this out, so bear with the rambling comments.

I think fractions are incredibly useful, first of all. Once you understand them, they are way easier to deal with than decimal notation (of course, this is not helpful when trying to convince your students to use them). But my point is that in the real world, we consider ratios, comparisons, percentages, etc all the time, and these are (in my opinion) best understood through the lens of fractions. Being able to add them together is a way to drill understanding of fractions. Can I think of a real world example where I have sat and calculated the sum of two fractions (well yes, because I do math, but other than in my work….)? No. But I can definitely think of times when I have used fractions as a tool of approximation, and needed to be able to add them together – cooking is the most obvious example, and this ties in nicely with your comments about obesity. Similarly, reading nutritional data requires a good understanding of percentages (fractions!).

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Agreed. I chose fractions hastily, but my point applies more in general. Pythagoras? Enlarging a shape about a point? Equation of a straight line? Volume of a cone? Negative indices? Really?

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Pythagoras is an example of the beauty of math, or it should be. Why is there a relationship between side lengths of the triangle???? This is a rigidity theorem, and should actually be surprising. Lengths and shapes are connected???

I haven’t got a clue what enlarging a shape about a point even is, can’t comment.

Volumes go to estimation again.

Equations of lines is an interesting one. So geometry should simply be beautiful – it’s one area that can be taught without much (any) need for calculations. It is theoretic, yet entirely approachable. And things suddenly become calculations when we apply the Cartesian plane. So for people who will be using graphs, the equations are important. An argument could probably be made that people should know how to read graphs to be informed citizens. But yes, the equations separate us from the idea. Why do two lines with the same slope never intersect? Because there is no solution to that set of equations, but also because they are parallel.

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