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ProfSmudge

@profsmudge.bsky.social
425 followers 206 following 1.2K posts

School maths should be more than tables and algorithms. I try to write materials that show that. Dietmar Küchemann

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ProfSmudge @profsmudge.bsky.social · 15m
You could try something like 4^10 which they might be able to scrutinise after writing it as 4x4x4x4x4x4x4x4x4x4. I would have thought some nice conjectures might arise (without the debilitating multi-choice), which could then be scrutinsed further.
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ProfSmudge @profsmudge.bsky.social · 22m
A nice task but challenging. And you'd think they would have included 2^2022.
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ProfSmudge @profsmudge.bsky.social · 04/10/2026
Is it the geometry or the algebra that tells us that (a + b)/2 = a + (b – a)/2 ? [Or is this the wrong question?!]
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ProfSmudge @profsmudge.bsky.social · 03/10/2026
@sainsburys.bsky.social What is going on here?
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ProfSmudge @profsmudge.bsky.social · 02/10/2026
Think of all the apples he could have grown
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ProfSmudge @profsmudge.bsky.social · 02/10/2026
Small: 80 tea bags Large: ?
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ProfSmudge @profsmudge.bsky.social · 02/10/2026
A 'baseline' test that a charity here uses for its Year 5 tutees starts with 21 abstract arithmetic items, followed right at the end by just 5 'real life' items (mostly involving money). I fear this might reflect what the children get at school.
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ProfSmudge @profsmudge.bsky.social · 02/10/2026
I think it is very hard to resist the idea that what we need to focus on is procedures, exacerbated by the need to pass high stakes tests. It's a small step from there to thinking that maths is procedures. [And, it's very easy to believe that if you've explained something, they've learnt something!]
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ProfSmudge @profsmudge.bsky.social · 02/10/2026
In England, Cockcroft was all for informal methods, so was the National Numeracy Strategy. But then along came Gove, Gibb and Cummings, who were all for whatever dire experiences they'd had when they were in their little prep schools.
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ProfSmudge @profsmudge.bsky.social · 01/10/2026
Thank you, but I think we lift students out of their world and into an abstract and formal realm very quickly which then only makes sense to some. I fear our radar might not be very good at registering students' thinking.
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ProfSmudge @profsmudge.bsky.social · 01/10/2026
In #MathsToday I was working with a young adult with limited schooling. He struggled with mental arithmetic (he got 27 for 18+10) and explained that he would use a written method instead. This seems extreme, but with our emphasis on formal procedures, is it all that different for some UK pupils?
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ProfSmudge @profsmudge.bsky.social · 01/10/2026
Oaksymoron?
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ProfSmudge @profsmudge.bsky.social · 01/10/2026
- a similar idea:
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ProfSmudge @profsmudge.bsky.social · 30/09/2026
Maths IS a language?? Or Maths employs 'languages', some uniquely mathematical?
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ProfSmudge @profsmudge.bsky.social · 28/09/2026
When people say 'I'm not good at maths', I think they often mean 'I couldn't make sense of those arcane procedures that we were asked to practise all the time'. They are not really talking of maths.
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ProfSmudge @profsmudge.bsky.social · 28/09/2026
That's nicely grounded. I use similar triangles, which gives a paired down version. The areas of the three similar triangles are proportional to 17², d² and 7².
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ProfSmudge @profsmudge.bsky.social · 27/09/2026
Ah, so that's why they chose 7 and 17 in the halve a trapezium task!
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ProfSmudge @profsmudge.bsky.social · 27/09/2026
A fain point - when the parallel edges are roads, rather than one being the river!
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ProfSmudge @profsmudge.bsky.social · 27/09/2026
Or halve it like this:
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ProfSmudge @profsmudge.bsky.social · 25/09/2026
Four eyes... [why are modern glasses so b-b big?]
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ProfSmudge @profsmudge.bsky.social · 25/09/2026
It arrived, not by 3:15 but at 8pm, simply left on the doorstep.
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ProfSmudge @profsmudge.bsky.social · 24/09/2026
@royal-mail.bsky.social What is the point of this from Royal Mail if the parcel doesn't actually arrive?
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ProfSmudge @profsmudge.bsky.social · 24/09/2026
Image of a Grammar School classroom, courtesy of the New Statesman. I wouldn't want one of those things to land on my head....
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ProfSmudge @profsmudge.bsky.social · 21/09/2026
You will obviously want to read this, on insect flight, by my father and godfather...
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ProfSmudge @profsmudge.bsky.social · 18/09/2026
sorry - 'less direct' !
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ProfSmudge @profsmudge.bsky.social · 18/09/2026
But the rectangles method is less indirect, and I haven't (yet?) developed a feel for it, whereas the two triangles are there in front of me, and I can see they are similar.
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ProfSmudge @profsmudge.bsky.social · 18/09/2026
They're lovely - especially the last one (the triangle)
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ProfSmudge @profsmudge.bsky.social · 17/09/2026
The author (Frank Swetz) claims the ancient Chinese would have used the fact that the areas of R1 and R2 are equal.
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ProfSmudge @profsmudge.bsky.social · 17/09/2026
The article contains this, based on a 2000 year old Chinese problem. We would solve it using similar triangles.
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ProfSmudge @profsmudge.bsky.social · 17/09/2026
In the light of your comment I found this paper. Do you know it?
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ProfSmudge @profsmudge.bsky.social · 17/09/2026
That's very helpful, many thanks
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ProfSmudge @profsmudge.bsky.social · 17/09/2026
What is it and what is its value?
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ProfSmudge @profsmudge.bsky.social · 16/09/2026
I quite like using the number line, as it nicely models both measurement (quotition) and sharing (partition).
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ProfSmudge @profsmudge.bsky.social · 16/09/2026
One doesn't add context, one eventually takes it away. Maths grows out of our actions in the world, we then gradually make these actions more abstract.
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ProfSmudge @profsmudge.bsky.social · 15/09/2026
Wow, thanks. I'm long familiar with the diagram and why this produces 'equal' rectangles, but didn't know of the related Principle.
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ProfSmudge @profsmudge.bsky.social · 15/09/2026
As here:
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ProfSmudge @profsmudge.bsky.social · 15/09/2026
It's a nice (mental?) exercise to draw a (any) rectangle, to represent the bottom-left rectangle, say, and then to draw the others scale, starting with either the top-left or bottom-right rectangle (and based on their areas). The result stays to scale if one stretches it vertically or horizontally.
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ProfSmudge @profsmudge.bsky.social · 14/09/2026
The response to 2/3 is intriguing! It looks as though they perhaps think fractions of the form a/(a+1) are equivalent to 2/3, and have then come up with a post hoc explanation (one more than half) that doesn't really fit. A next step could be to ask for some more examples equivalent to 2/3.
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ProfSmudge @profsmudge.bsky.social · 12/09/2026
I found the diagonal notation quite disorientating at first. Would it help to put the number at the midpoint of the diagonal (and hence middle of its rectangle)?
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ProfSmudge @profsmudge.bsky.social · 11/09/2026
Ah, that's excellent. I'd enjoy it at that speed!
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ProfSmudge @profsmudge.bsky.social · 10/09/2026
It is neat. I enjoyed it. Bu t I don't want to be marked down for taking a breath. I'm please to have (eventually) determined the factors of 135 (my nemesis). I don't want to be classed as mathematically inferior to those who got there quicker.
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ProfSmudge @profsmudge.bsky.social · 10/09/2026
Quite possible (though I don't remember)!
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ProfSmudge @profsmudge.bsky.social · 10/09/2026
Interesting how for ..... – 2(x – 2) the minus signs somehow attach themselves to the numbers which are then called 'negative 2' and 'negative 2 times negative 2' and then morph back into operators, as in 'minus 2x add 4'. Needless sleight of hand, I feel.
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ProfSmudge @profsmudge.bsky.social · 08/09/2026
I think the portions are in proportion....
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ProfSmudge @profsmudge.bsky.social · 08/09/2026
Fraction as measurement.... [Is it 32ths or 32nds?]
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ProfSmudge @profsmudge.bsky.social · 07/09/2026
I don't understand the price reduction. Mensa is clearly not for me.
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ProfSmudge @profsmudge.bsky.social · 06/09/2026
Aargh, I remember tying to teach from one of those books. So formal, so deadly.
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ProfSmudge @profsmudge.bsky.social · 06/09/2026
I'd love to know whether this item was ever pre-tested. From the 2nd NAEP maths assessment (1977-78) and probably the most quoted and worst fraction item in the world.
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ProfSmudge @profsmudge.bsky.social · 06/09/2026
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ProfSmudge @profsmudge.bsky.social · 06/09/2026
What were these strange arcs at 6:15 this morning?
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