Teaching Number Sense

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KingdomBuilder
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Re: Teaching Number Sense

Post by KingdomBuilder »

ohio jones wrote:... as evidenced by Arabic numerals, al-Jabr, and the fourth book of the Torah, for example.
The ole abacus.
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temporal1
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Re: Teaching Number Sense

Post by temporal1 »

hoping Neto, Adam, george will see this request.
they may have had first-person experiences with different remote cultures.

Peregrino, if they don't see this thread, i suggest you try PM or MN email.
i hope you can find some help.

i'm glad the children are coming. :D

i'm trying to imagine not thinking of numbers .. it's not easy!
we do begin teaching numbers very early! i hadn't thought of another way.
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Neto
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Re: Teaching Number Sense

Post by Neto »

As some others have commented, numbers may not be a cultural value, unneeded, in fact. Not all cultures have number systems, because not all cultures need them. Where we were, they have a word for 'one' (two words for that, in fact) and 'two', then it's 'many'. They can count using 'one' & 'two', and 'hand' up to 10 w/o any difficulty, and up to 20 if necessary. Beyond that they use notched sticks to mark off days. As each day passes, you cut off the little 'mountain' for that day. Longer periods of time are counted in moons, and they know a great deal more about exactly what phase the moon is in at any time than Westerners who have to have a calendar. I do not mean to imply that these children are in any way incapable of learning the numbers, but if there is no cultural value ascribed to it, then the efforts will be unsuccessful, or will be no more than a curiosity. Classroom study & learning may or not be a cultural value, and if possible, it may be better to drop the school environment and look for ways to incorporate this new knowledge into the culture, so that it will be sustainable & on-going, a new cultural value.

But if you are to teach the numbers, my question is whether you are using objects to teach the concept of counting, or if you are teaching the symbols used to denote each number. The latter is much more difficult. Think for a moment about the letters 'b', 'd', 'p', & 'q' in the English alphabet. They are all the same, just turned differently. In an oral society the concepts of the meaning of shapes is more concrete. A knife is a knife regardless of which way it is pointing, or how it is turned. (Fortunately for us, our language only had need of 'b' & 'd', but when copying a word off of the black board, some of the people would lay the letters down, or turn them any which way, because shape has meaning, but not position, or orientation.)
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PeterG
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Re: Teaching Number Sense

Post by PeterG »

Josh wrote:My question continues to be why it is essential to indoctrine another culture with our western concepts.
Peregrino can clarify, of course, but I suspect that these folks are about to confront Western civilization one way or another whether they like it or not, and they may as well be prepared for it.
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temporal1
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Re: Teaching Number Sense

Post by temporal1 »

Neto wrote:As some others have commented, numbers may not be a cultural value, unneeded, in fact. Not all cultures have number systems, because not all cultures need them.

Where we were, they have a word for 'one' (two words for that, in fact) and 'two', then it's 'many'.
They can count using 'one' & 'two', and 'hand' up to 10 w/o any difficulty, and up to 20 if necessary.

Beyond that they use notched sticks to mark off days. As each day passes, you cut off the little 'mountain' for that day. Longer periods of time are counted in moons, and they know a great deal more about exactly what phase the moon is in at any time than Westerners who have to have a calendar.

I do not mean to imply that these children are in any way incapable of learning the numbers,
but if there is no cultural value ascribed to it, then the efforts will be unsuccessful, or will be no more than a curiosity.

Classroom study & learning may or not be a cultural value, and if possible, it may be better to drop the school environment and look for ways to incorporate this new knowledge into the culture, so that it will be sustainable & on-going, a new cultural value.

But if you are to teach the numbers, my question is whether you are using objects to teach the concept of counting, or if you are teaching the symbols used to denote each number.
The latter is much more difficult.

Think for a moment about the letters 'b', 'd', 'p', & 'q' in the English alphabet.
They are all the same, just turned differently.
In an oral society the concepts of the meaning of shapes is more concrete. A knife is a knife regardless of which way it is pointing, or how it is turned.

(Fortunately for us, our language only had need of 'b' & 'd', but when copying a word off of the black board, some of the people would lay the letters down, or turn them any which way, because shape has meaning, but not position, or orientation.)
Neto, it seems, teachers will be most effective if/when they (first) learn the culture, as possible, then they can find ways to introduce new ideas that will relate and have value? .. this correlates with what many teachers have long described, that they learn more from their students than the other way around (?) .. i find this true as a parent, also.
PeterG wrote:
Josh wrote:My question continues to be why it is essential to indoctrine another culture with our western concepts.
Peregrino can clarify, of course, but I suspect that these folks are about to confront Western civilization one way or another whether they like it or not, and they may as well be prepared for it.
i imagine, through both government and cartels, these folks do cross paths with Western civilization, likely not to the positive. "numbers" are of primary importance to both groups; both groups may understand very well how Peregrino's folks think+live, thus, how they may be exploited.

i have read a little about how Brazil tries to afford legal protections to remote tribes. i've never read of anything similar in Mexico (?)
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Josh
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Re: Teaching Number Sense

Post by Josh »

Numbers are complicated. I was having to deal with answering the question today of "how can I figure out if something is a number?" Is ١٠٤ a number? Is "\N{MATHEMATICAL SANS-SERIF DIGIT ZERO}"? The answer might surprise you.
How to detect a valid integer literal

There are hundreds of questions on StackOverflow that all ask variations of the same thing. Paraphrasing:

lst is a list of strings and numbers. I want to convert the numbers to int but leave the strings alone. How do I do that?

This immediately gets a half-dozen answers that all do some equivalent of:

lst = [int(x) if x.isdigit() else x for x in lst]

This has a number of problems, but they all come down to the same two:

"Numbers" is vague. You can assume it means only integers based on "I want to convert the numbers to int", but does it mean Python integer literals, things that can be converted with the int function with no base, or things that can be converted with the int function with base=0, or something different entirely, like JSON numbers or Excel numbers or the kinds of input you expect your 3rd-grade class to enter?
Whichever meaning you actually wanted, isdigit() does not test for that.

The right answer depends on what "numbers" actually means.

If it means "things that can be converted with the int function with no base", the right answer—as usual in Python—is to just try to convert with the int function:

def tryint(x):
try:
return int(x)
except ValueError:
return x
lst = [tryint(x) for x in lst]

Of course if you mean something different, that's not the right answer. Even "valid integer literals in Python source" isn't the same rule. (For example, 099 is an invalid literal in both 2.x and 3.x, and 012 is valid in 2.x but probably not what you wanted, but int('099') and int('0123') gives 99 and 123.) That's why you have to actually decide on a rule that you want to apply; otherwise, you're just assuming that all reasonable rules are the same, which is a patently false assumption. If your rule isn't actually "things that can be converted with the int function with no base, then the isdigit check is wrong, and the int(x) conversion is also wrong.
What specifically is wrong with isdigit?
I'm going to assume that you already thought through what you meant by "number", and the decision was "things that can be converted to int with the int function with no base", and you're just looking for how to LBYL that so you don't have to use a try.
Negative numbers
Obviously, -234 is an integer, but just as obviously, "-234".isdigit() is clearly going to be false, because - is not a digit.

Sometimes people try to solve this by writing all(c.isdigit() or c == '-' for c in x). But, besides being a whole lot slower and more complicated, that's even more wrong. It means that 123-456 now looks like an integer, so you're going to pass it to int without a try, and you're going to get a ValueError from your comprehension.

Of course you can solve that problem with (x[0].isdigit() or x[0] == '-') and x[1:].isdigit(), and now maybe every test you've thought of passes. But it will give you "1" instead of converting that to an integer, and it will raise an IndexError for an empty string.

One of these might be correct for handling negative integer numerals:

x.isdigit() or x.startswith('-') and x[1:].isdigit()
re.match(r'-?\d+', x)?

But is it obvious that either one is correct? The whole reason you wanted to use isdigit is to have something simple, obviously right, and fast, and you already no longer have that. And we're not even nearly done yet.
Positive numbers
+234 is an integer too. And int will treat it as one. But the code above won't. So now, whatever you did for -, you have to do the same thing for +. WHich is pretty ugly if you're using the non-regex solution:

lst = [int(x) if x.isdigit() or x.startswith(('-', '+')) and x[1:].isdigit() else x
for x in lst]

Whitespace
The int function allows the numeral to be surrounded by whitespace. But isdigit does not. So, now you have to add .strip() before the isdigit() call. Except we don't just have one isdigit call; to fix the other problems we've had two go with two isdigit calls and a startswith, and surely you don't want to call strip three times. Or we've switched to a regex. Either way, now we've got:

lst = [int(x) if x.isdigit() or x.startswith(('-', '+')) and x[1:].isdigit() else x
for x in (x.strip() for x in lst)]
lst = [int(x) if re.match('\s*[+-]?\d+\s*', x) else x for x in lst]

What's a digit?
The isdigit function tests for characters that are in the Number, Decimal Digit category. In Python 3.x, that's the same rule the int function uses.

But 2.x doesn't use the same rule. If you're using a unicode, it's not entirely clear what int accepts, but it's not all Unicode digits, at least not in all Python 2.x implementations and versions; if you're using a str encoded in your default encoding, int still accepts the same set of digits, but isdigit only checks ASCII digits.

Plus, if you're using either 2.x or 3.0-3.2, and you've got a "narrow" Python build (like the default builds for Windows from python.org), isdigit is actually checking each UTF-16 code point, not each character, so for "\N{MATHEMATICAL SANS-SERIF DIGIT ZERO}", isdigit will return False, but int should accept it.

So, if your user types in an Arabic number like ١٠٤, the isdigit check may mean you end up with "١٠٤", or it may mean you end up with the int 104, or it may be one on some platforms and the other on other platforms.

I can't even think of any way to LBYL around this problem except to just say that your code requires 3.3+.
Have I thought of everything?
I don't know. Do you know? If you don't how are you going to write code that handles the things we haven't thought of.

Other rules might be even more complicated than the int with no base rule. For different use cases, users might reasonably expect 0x1234 or 1e10 or 1.0 or 1+0j or who knows what else to count as integers. The way to test for whatever it is you want to test for is still simple: write a conversion function for that, and see if it fails. Trying to LBYL it means that you have to write most of the same logic twice. Or, if you're relying on int or literal_eval or whatever to provide some or all of that logic, you have to duplicate its logic.
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Re: Teaching Number Sense

Post by lesterb »

trying to teach the concept of 'zero' may be the most difficult...
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temporal1
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Re: Teaching Number Sense

Post by temporal1 »

toddlers can begin to grasp the idea of 0 when they are refusing to eat something their parents believe they should.

food is a way to think about numbers .. "1 more bite?" "3 more green beans?" .. etc.
in a family, children learn about fractions .. "1/2 stick of gum" .. "the bigger half" .. etc.

Peregrino,
i wonder if your teacher might begin approach numbers in context of snacks/lunch?
food is an important motivator.
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Re: Teaching Number Sense

Post by Adam »

temporal1 wrote:hoping Neto, Adam, george will see this request.
they may have had first-person experiences with different remote cultures.
Minority cultures in the world are changing rapidly. 40-50 years ago it was still possible to find remote minority cultures that were completely monolingual and who basically had no contact with the outside world. I don't think there are many, if any, cultures like that now. Even the most remote cultures have contact with the outside world, have a number of speakers who know a trade language (in addition to their vernacular language), and deal with things like money, which requires knowledge of more westernized ways of dealing with numbers. Papua New Guinea is one of the most remote places in the world, and I know people who live in some of the most remote places in Papua New Guinea, and they are trilingual and are familiar with and use more westernized number systems.

The language group I work with is very strong in using their vernacular language, but most people also know at least some Tok Pisin (the most widely used trade language). Also people have largely abandoned their traditional ways of counting and have switched to borrowing Tok Pisin terms for numbers. This is particularly true with numbers past 5 or so. It is still common, however, to use vernacular terms for one, two, three, and four. Although I did have an encounter with a boy about 8 years old who wanted to teach me the vernacular language. He quizzed me on how to say things like pig and house. Then he asked me if I know how to say 'three'. So I replied 'tepoma', which is the vernacular word for 'three'. He corrected me though and said 'sitii', which is the vernacular pronunciation of the Tok Pisin 'tri', which itself is borrowed from the English word 'three'. It was really funny to hear him say that, and I told him that 'tepoma' was the actual vernacular word. (His mother is from a different language group, so he does not have a good knowledge of the vernacular.)

But to say something like 99 in the vernacular, one would have to say, 'on top of nine tens also nine', and that is much longer than just saying 99. I doubt numbers like 99 were used very much in former times because there would rarely be a need to count so high, and so when it became necessary to use larger numbers through contact with the outside world, people became used to hearing such larger numbers in Tok Pisin or English, and so they just borrowed those terms instead. At least that is my analysis of what probably took place.

One interesting fact is that the term 'a couple' can not only be used for count nouns like 'people' or 'sweet potatoes' but also for noncount nouns like 'water'. If someone says, "Give me a couple water", it means "Give me a little bit of water".
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Re: Teaching Number Sense

Post by Neto »

After reading Adam's response (above), I thought of some other uses of 'numbers' in the tribal group where we lived. I put 'numbers' in quotes because what you know of as a number may not be a real number in the speaker's thinking. Money. Brazilian currency during our early years in the village was around 1,000 to one American dollar. (Over the years we were in Brazil, they knocked 3 zeros off of the currency three times. They later went without any currency for a while, then introduced the Real at one to one. It has been the most stable currency since our first arrival in Brazil in 1985.) Anyway, so the native people used words like 'ten thousand', '50 thousand', and '100 thousand' fairly commonly, but the meaning was that particular value in currency, not a real number.
The native method of counting goes like this: ONE {one, or 'only one'} TWO {dual}, Three {dual and another one}, FOUR {dual dual}, FIVE {only one hand}, SIX {only one hand & one more}, SEVEN {only one hand, dual}, EIGHT {only one hand, dual, only one more}, NINE {two hands, lacking only one}, TEN {2 hands}. They can go on up to twenty this way, but the Portuguese borrowed words were commonly used for these, even when we first arrived. I suspect that the vernacular numerals are used much less now that they were even when we moved back to the States (almost 14 years ago). When asked recently by another linguist about the number system of this language, I initially replied that it was confined to 'one', 'dual', and 'many', because the rest are not distinctive single words, they are constructions based on those basic concepts, along with 'hand'. In respect to these basic terms, it should also be reported that their verb phrase system also follows this basic model. That is, in addition to singular & plural forms as we have in English, they also have distinctive forms for 'dual', and this is marked not only in the pronoun phrase, but in the verb as well.

Regarding isolated societies with no current contact with any outsiders - we do know of one such group in the Amazon. They almost certainly had contact with the outside back in the rubber boom years, but none at all for the last several decades or more.

But getting back to the original question, I don't recall that we taught numbers in a classroom situation. Because of their isolation, and the fact that outside goods that did find their way into the are were at an extremely high price, we brought in various goods to trade with them. One item that was of particular value to them was nylon cord, and we sold it in the commonly used measurement in the region, by 'arm spans'. So they may have increased their ability to count through this practical use in every day life. I do think that for counting to become integrated into the culture, there must be a ready practical use apparent to them. And this will probably be more difficult to identify in relating to children than with adults. Some type of game is my best suggestion.
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