Bases.

[catfish_man@mastodon.social]

@futurebird yeah, tricky. “View this abstraction you’re comfortable with as an entirely different category of abstraction” is always a tall order.

It reminds me a little of the confusion I’ve seen in beginners about the distinction between variables themselves and the values of variables.

[mvsamuel@types.pl]

@futurebird maybe https://en.wikipedia.org/wiki/Numerals_in_Unicode

[quasit@kolektiva.social]

@futurebird
Have you ever read "How To Count On Your Fingers" by Frederick Pohl? I taught myself to count in binary on my fingers one night when I was driving on a dark highway after reading that! It's an article, not a short story.

[rk@mastodon.well.com]

@futurebird @dancingtreefrog

In all seriousness maybe different written random symbols like Zener cards?

Be like “oh look, add one to star and you get wavy lines but oh shit you add one to wavy lines and you gotta bring another card in” and later “but what if we wrote the star as (dramatic pause) 1”

(This may be the card sorting exercise you referred to earlier, in that case ignore me.)

But the distinction between sign-and-signified is the single biggest aha moment you can get in CS, IMHO.

[fcbsd@hachyderm.io]

@futurebird while "symbols" might be confusing initially, it seems like the most straight forward, especially as you develop examples, eg you get to hexadecimal 1a = 26 in decimal and your symbols include numbers and letters even though you are still talking about numbers...so symbols would be my approach.

[futurebird@sauropods.win]

@merms @graydon

They don't have place value. And they use... subtraction. But our students are very familiar with roman numerals for some reason (I think the PE staff uses them a lot?)

I want to bring them out when it can be more obvious how strange they are.

Change the symbols and I don't even know if you could do a sorting problem with them.

[catfish_man@mastodon.social]

@futurebird ok EXTREMELY off the wall suggestion: warm them up to thinking like this by having them read a bit of Through The Looking Glass

[willie1foot@theblower.au]

@futurebird Digits?

[e@az.social]

@futurebird
uhm...as a thought experiment, and based on the principle that a rose by any other name would smell as sweet, I wonder how it would go to pick three random actual symbols, for instance ^,&,* and count with them, then repeat with three characters like F,K,T, in hope that they could eventually relax with the specific visual representations of the counting operations

[bumblefish@mastodon.scot]

@futurebird @dancingtreefrog Maybe use 'numbers' until you hit the letters and then see if they can solve the problem of what to call them. Explain that you also had trouble about what to call them, and that their ideas might help next years' class, and hey presto you're also teaching empathy! Win-win!

[fcbsd@hachyderm.io]

@ColinTheMathmo @futurebird glyphs are fancy symbols, so that would be a more advanced approach

[graydon@canada.masto.host]

@futurebird Roman numerals are like that because it's a finger counting system being written down, essentially a record of hand positions. (So four is traditionally IIII, not the compact for monuments IV.)

If I understand the discussion at all, which I might well not, the history-of-writing folks think positional notation and zero are effectively the same concept; it's difficult to do the one without the other. And, oddly, given the Babylonians did have it, Classical Antiquity didn't.

@merms

[meltedcheese@c.im]

@futurebird Symbol. It means “stands for”. The trouble comes because are reusing 1 and 0 together to mean something different. Somehow you need to convey that the symbol “represents” a concept, it is itself not the concept. The symbol alone carries no meaning except that which we agree to assign it.

[futurebird@sauropods.win]

@meltedcheese
I'm hoping this helps:

https://sauropods.win/@futurebird/115896355329222284

[simon@procrastodon.net]

@futurebird Maybe not the answer you're looking for, but this got me thinking about why exactly these words are confusing, and it occrs to me that people are trying to un-learn the decimal system every time they use numbers to express things in a different base. I wonder if it would help to just use actual, unique symbols. If you give people 0, 1 and 2 and tell them to count, their brains are tripping over themselves because they have to fight the natural assumption that "10" is "the number of fingers I have" or "the number that comes after nine". Take away that association until people understand the concept of counting using fewer or more symbols (which look nothing like the numbers they're using in their head), and maybe it will make more sense.
It would be like learning a different language where every word already means something in English.

[robotistry@mstdn.ca]

@futurebird I'd use emoji to introduce the concept. For extra fun, have each emoji relate to the number.

Example: I have a new way of counting. It uses 🍩, 👁️, and 🧦.
(doughnut/circle/zero, eye/i/one, pair/two)

[gooser3000@mastodon.social]

@futurebird there are only 10 numer keys on the keyboard... Imagine of there were more or fewer. Also consider a clock where we count by 12s and 60s and use : to separate.

[timtfj@mastodon.social]

@kw217 @futurebird This does at least make the idea more concrete and intuitive rather than introducing special language. I think people should ideally learn the terminology for a concept *after* grasping the concept itself. Then the fancy word ("symbol", "digit" etc.) is just a label for the thing they're already familiar with, and they can learn one thing at once.

[captmorgan@freeradical.zone]

@futurebird Mathematical

A number in base n is a polynomial

xn^0 + yn^1 + zn^2 + …

where x,y,z, … are whole numbers < n

[futurebird@sauropods.win]

@CaptMorgan

This is great for students who have strong algebra. But that idea of using increasing powers? It's really not obvious that's what's going on when you first start.

But could one do something with... physical cubes and literal flat squares? That's Cuisenaire rods for decimal.

Are there... base 16 Cuisenaire rods? Why not? hmm....