Atul Rana - Online Tutor

New blog post! by Complete Mathematics ticked all the boxes.

"And then I go to another one and come away thinking about helping struggling pupils grab their first mark, triangle centric worldviews, probability as measurement, using mathematical surprise to provoke curiosity, and the thousands of tiny decisions that make up good tutoring."

MathsConf41 Liverpool - With a Little Help From My Maths Friends My MathsConf41 Liverpool review: five workshops on GCSE maths, connected revision, probability, mathematical surprises, tutoring and AI.

"Society rightly celebrates mathematicians, scientists, engineers and people capable of operating at extraordinary levels of mathematical sophistication. But there is another kind of mathematical expertise that receives far less attention: the ability to take mathematics that has become incomprehensible to another human being and make it intelligible again."

Dyscalculia Tutoring: The Hidden Intellectual Work Behind Great Maths Teaching Why does dyscalculia tutoring need real expertise? See how expert maths teaching diagnoses gaps, rebuilds foundations and develops confidence.

Totally agree.

Love this question from Corbettmaths Emerald 5-a-day. It gets surprisingly close to the heart of what measurement actually means.

At first glance, this looks like a very basic perimeter question. But underneath it, there is a remarkable amount of mathematical thinking going on.

The pens are functioning as a non-standard unit of length. The child has to understand that measuring something means iterating an equal-sized unit along a length. Two pen-lengths make the long side; one pen-length makes the short side. The important thing is not really the pens themselves, but the repeated length that each pen represents.

This creates a subtle distinction between counting and measuring. There are only three actual pens visible in the diagram, yet the answer is six. Simply counting the objects in front of you will not work. The child has to mentally extend the unit to parts of the shape where no physical pens have been drawn. In other words, they are not counting pens; they are reasoning in pen-lengths.

There is also conservation of length. A pen placed vertically represents exactly the same amount of length as one placed horizontally. Rotating or moving the unit does not change its magnitude. This sounds obvious to an adult, but it is part of the conceptual architecture that allows measurement to work at all.

The properties of the rectangle matter too. The child has to recognise that the opposite sides are equal. If the top side is two pens long, the bottom side must also be two pens long. If one vertical side is one pen long, the other vertical side must also be one pen long. The diagram therefore asks the learner to combine geometric structure with measurement rather than treating them as separate topics.

And then we arrive at perimeter.

The perimeter is not simply a rule that says “add all the sides”. More fundamentally, it is the total length of the boundary. The child is composing separate lengths into one complete length:

2 + 1 + 2 + 1 = 6 pen-lengths.

That additive composition of lengths is the underlying mathematical idea from which the perimeter calculation emerges.

There is even the beginning of multiplicative thinking hiding inside the problem. Instead of seeing four unrelated additions, a child may eventually see two long sides and two short sides: two lots of 2 and two lots of 1. Much later, that same structure becomes the familiar formula 2(l+w). But the formula makes far more sense if the underlying structure has already been understood.

Perhaps the deepest measurement idea is that the number itself is not the quantity. The rectangle has a fixed perimeter, but “6” only makes sense relative to the chosen unit. If the pens were half as long, perhaps twelve would fit around it. If the measuring unit were twice as long, perhaps only three would. The physical length has remained unchanged; the numerical measure has changed because the unit has changed.

So hidden inside this tiny primary-school question are ideas about unit iteration, equal units, conservation of length, orientation, the distinction between counting and measuring, properties of rectangles, spatial structuring, additive composition and perimeter.

https://youtube.com/shorts/LM1sJxdWWiY?is=aneDukotDXNxO7tB

Donald Duck Explains the Secret Math Behind Music! The film demonstrates how Pythagoras linked math to music through s...

https://www.atulranatutors.co.uk/mathematics-deeply-understood/mathematics-music-art-patterns/

"We need patterns. They help us make sense of the world. They create continuity. They allow us to learn. They give us something to return to. But we also need variation. Without variation, a pattern becomes a prison. Without pattern, variation becomes noise. Somewhere between the two lies expression."

Mathematics, Music, Art, Poetry, Dance — It’s All Patterns Mathematics, music, art, poetry and dance all share a hidden language: patterns, repetition, variation and surprise.

Why do teachers keep telling students to “show your working” in maths?

It is not just about getting marks for the method. Writing things down reduces the strain on working memory, makes mistakes easier to spot, and helps students think more clearly.

In this article, I explain why the page is not just a place to record mathematics. It is part of the thinking process itself.

Why Writing Things Down Makes You Better at Mathematics (And Why Teachers Insist on Showing Your Working) Why should students show their working in maths? Writing reduces mental overload, reveals mistakes and helps learners think more clearly.