Bases.
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@graydon @futurebird Numeral. Yes. As in Roman numerals.
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.
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@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.
@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
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Bases. (decimal, binary etc) are best explained through examples.
"You can only write three symbols in base 3. These are: 0, 1, 2"
So you count: 0, 1, 2, 10, 11, 12 ...
I think the word "symbols" is confusing, but so is "characters"? Students don't think of numbers or letters as "symbols" or characters. The card sorting puzzle helps with this. But I'm always refining the language:
How would you put this as plainly as possible?
@futurebird Digits?
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Bases. (decimal, binary etc) are best explained through examples.
"You can only write three symbols in base 3. These are: 0, 1, 2"
So you count: 0, 1, 2, 10, 11, 12 ...
I think the word "symbols" is confusing, but so is "characters"? Students don't think of numbers or letters as "symbols" or characters. The card sorting puzzle helps with this. But I'm always refining the language:
How would you put this as plainly as possible?
@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 -
Thing is then when we get to hex they are upset that A and F are not "digits" but ... maybe.
@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!
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@futurebird Glyphs ... Then use things other than digits.
@ColinTheMathmo @futurebird glyphs are fancy symbols, so that would be a more advanced approach
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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.
@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.
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Bases. (decimal, binary etc) are best explained through examples.
"You can only write three symbols in base 3. These are: 0, 1, 2"
So you count: 0, 1, 2, 10, 11, 12 ...
I think the word "symbols" is confusing, but so is "characters"? Students don't think of numbers or letters as "symbols" or characters. The card sorting puzzle helps with this. But I'm always refining the language:
How would you put this as plainly as possible?
@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.
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@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.
@meltedcheese
I'm hoping this helps:myrmepropagandist (@futurebird@sauropods.win)
I don't know how well this puzzle will translate to a toot. Imagine each line is on a card: □□▷ □□□ ■■ ■ □■ □▷▣ ▣ □▣ ■▷ ■▣ ▣▷ □□■ □▷ ▣▣ □□ □ ▣□ □□▣ ■□ ▷ □▷□ □▷■ ▣■ □▷▷ Put them in order. (The 5th graders could do it, but they did have a helpful example first... There may be more than one solution, but I think there is ONE really good order. Can you find it?) (I should also mention that every adult I've shown this to gives up. But I only showed it to two rather grouchy teachers.)
Sauropods.win (sauropods.win)
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Bases. (decimal, binary etc) are best explained through examples.
"You can only write three symbols in base 3. These are: 0, 1, 2"
So you count: 0, 1, 2, 10, 11, 12 ...
I think the word "symbols" is confusing, but so is "characters"? Students don't think of numbers or letters as "symbols" or characters. The card sorting puzzle helps with this. But I'm always refining the language:
How would you put this as plainly as possible?
@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. -
Bases. (decimal, binary etc) are best explained through examples.
"You can only write three symbols in base 3. These are: 0, 1, 2"
So you count: 0, 1, 2, 10, 11, 12 ...
I think the word "symbols" is confusing, but so is "characters"? Students don't think of numbers or letters as "symbols" or characters. The card sorting puzzle helps with this. But I'm always refining the language:
How would you put this as plainly as possible?
@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) -
Bases. (decimal, binary etc) are best explained through examples.
"You can only write three symbols in base 3. These are: 0, 1, 2"
So you count: 0, 1, 2, 10, 11, 12 ...
I think the word "symbols" is confusing, but so is "characters"? Students don't think of numbers or letters as "symbols" or characters. The card sorting puzzle helps with this. But I'm always refining the language:
How would you put this as plainly as possible?
@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.
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@futurebird "shapes"? "squiggles"?
@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.
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Bases. (decimal, binary etc) are best explained through examples.
"You can only write three symbols in base 3. These are: 0, 1, 2"
So you count: 0, 1, 2, 10, 11, 12 ...
I think the word "symbols" is confusing, but so is "characters"? Students don't think of numbers or letters as "symbols" or characters. The card sorting puzzle helps with this. But I'm always refining the language:
How would you put this as plainly as possible?
@futurebird Mathematical
A number in base n is a polynomial
xn^0 + yn^1 + zn^2 + …
where x,y,z, … are whole numbers < n
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@futurebird Mathematical
A number in base n is a polynomial
xn^0 + yn^1 + zn^2 + …
where x,y,z, … are whole numbers < n
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....
