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[Xmca-l] Re: off list on mathematics and the social
Peg and Andy
Let me say that I rather like how Vygotsky thought about development (whatever the discipline) and I like doing some as Davydov recommended, but curricula-wise I am always a bit skeptical. US kids, and I would disagree as to all, do have this problem, but I am not, at all, sure it is because of the mathematical focus on number. I can easily imagine the following conversation
Children start with counting numbers
Alyosha’s number is greater than Borja’s. Anyone can
perceive the difference. The mathematical recording of that is A>B.
In the very same situation, one is less than the other,
**It is important here to make sure the notion of fair is ties to equal
It isn’t nice or fair that one child’s object should be
greater than and another less than, mathematically recorded as A ≠B and as
B≠A. How to get to B=A and A=B?
How to have a nice, fair situation? The teacher and
children work it about and discover the important operations that
mathematics has for working on >, <, ≠ and = and the mathematically
recordings with + and -.
**The key notion here is counting on
The whole situation of transformations takes this
nice set mathematically recorded as:
Then of course there’s more fun when Katya’s in on it and
transitivity pops in so that even without direct perceptual comparisons
mathematics comes to the rescue so you can figure out stuff you wouldn’t
know otherwise (do I smell motivation here?):
And they work out proudly that you keep the ? (don’t know)
answer in the following situation
** I suspect if asked why they (re Davydov) might say something like "Well, A is bigger than B and A is bigger than K., but that doesn't mean B is bigger than K. It doesn't say about B and K." I suspect that US children might say the same.
There is, in my opinion, a very strong element of teaching here that makes lessons like this happen. Mathematics teaching has always been much weaker in the US than elsewhere. However, that doesn't mean there aren't substantial exceptions even with a counting number curriculum.
There is another issue and that is why I mentioned Benezet. Skipping formal operations with number until later grades seems to have benefits as to later learning and, of course, as you mentioned informal operations with number are happening in the background. I wonder if the informality around number might contribute to a more relaxed view of mathematics (this has been argued to a degree elsewhere).
Unfortunately, it is somewhat likely any of this will catch on any time soon because of the social forces in motion, but that doesn't mean one can't agitate (smile). I choose to do so on the side of teaching rather than curriculum.
On Nov 3, 2014, at 8:06 AM, Peg Griffin wrote:
> Hi, Ed,
> I barely touched on the example-- sorry! It comes up when Katya gets into
> the situation.
> Essentially, given the following situation, most US fourth graders do quite
> well with just a little introduction:
> Response A>K
> BUT, most US fourth graders do not do well with the following situation:
> US kids answer with some symbol or other or say they haven't learned it yet
> or they can't do math or they have to go to the bathroom; the Davidov even
> younger kids say "don't know, no one could know."
> As far as I can tell, the US kids are doing the first one by relying on
> knowing the > < among numbers under ten and transfer for each little
> porblem; they don't seem to have a firm grasp of situations with greater
> than, less than, and transitivity.
> I hope you and Andy get back to the issues involving what's social with
> I'm glad folks are dealing seriously with mathematics on the list and that
> the readings have been posted but I still hope you will get back to that
> when you have time.
> -----Original Message-----
> From: firstname.lastname@example.org
> [mailto:email@example.com] On Behalf Of Ed Wall
> Sent: Sunday, November 02, 2014 3:03 PM
> To: eXtended Mind, Culture, Activity
> Subject: [Xmca-l] Re: units of mathematics education
> By ''formal arithmetic' I mean the usual US curriculum to which you
> refer to below; I wasn't talking about 'formal mathematics' when I mentioned
> Benezet. The point Devlin makes (and I'm not sure I entirely agree) is that
> the Davydov curriculum is about real number versus counting number. While
> Devlin and I both have problems with the usual US curriculum it is not
> entirely evident mathematically why one approach (counting number versus
> real number) is better than the other.
> I am confused by the statement below concerning an example you gave
> 'earlier about US fourth graders.' The only example I remember was the one
> using the Davydov approach with participants Alyosha and Borja.
> I would appreciate it if you would say a bit more about why "I don't
> know" is a 'mathematically' correct and 'impersonal' answer in some 'little
> systems.' I would tend to think otherwise about "We can't know.' in some
> little (and some large) systems; however, I may misunderstand.
> On Nov 2, 2014, at 9:42 AM, Peg Griffin wrote:
>> Thanks for this and the Hawaii information, Ed. I had looked into the
>> Hawaii work before but I know nothing at all of Benezet, I'm afraid.
>> I'm not sure what you (or Benezet) mean by "formal arithmetic," so I
>> don't know what to make of the implication that the early Davidov
>> mathematics educators were "something like" an approach that lacked it.
>> In my understanding, the Davidov mathematics is essentially all about
>> formal mathematics --symbols and systems of symbols are developed with
>> the children for relations (=≠ ><) and operations (+ =). Ignoring
>> numbers until later allows teachers to avoid an epigenetic byway we
>> often see in US elementary schools where counting relations among
>> number symbols overshadow other aspects of mathematics. The example I
>> gave earlier is about the fourth graders in US schools who seem to
>> understand > and < than relations in a little system of three
>> mathematical statements but they do not understand that "don't know"
>> is a mathematically correct answer in some of the little systems --
>> for them don't know is essentially a personal thing not a mathematics
>> -----Original Message-----
>> From: firstname.lastname@example.org
>> [mailto:email@example.com] On Behalf Of Ed Wall
>> Sent: Saturday, November 01, 2014 10:45 PM
>> To: eXtended Mind, Culture, Activity
>> Subject: [Xmca-l] Re: units of mathematics education
>> Something like this - i.e. lack of formal arithmetic until 7th -
>> (although the details are a little unclear) was done in the US in the
>> 1920s by a Louis Benezet. My impression is that he was building on ideas
> of Dewey.
>> On Nov 1, 2014, at 8:48 PM, Peg Griffin wrote:
>>> No move from numbers to x. No numbers to begin with in mathematics
>>> education. Kids count in everyday life but no numbers in the
>>> beginning mathematics classes. It really is strings! Not even
>>> rulers or tape measures of strings.
>>> -----Original Message-----
>>> From: firstname.lastname@example.org
>>> [mailto:email@example.com] On Behalf Of Andy Blunden
>>> Sent: Saturday, November 01, 2014 7:12 PM
>>> To: 'eXtended Mind, Culture, Activity'
>>> Subject: [Xmca-l] Re: units of mathematics education
>>> Phew! So I was not the only one mystified by that expression.
>>> However, wouldn't the kids have been confused by it as well? Or would
>>> they react by
>>> saying: "Hey, Teacher! That's stupid!"?
>>> But certainly making the move to using letters only when the children
>>> are reaching out for some more convenient symbol seems the right way
>>> to go. I used to teach the first lesson in algebra by playing "Think
>>> of a number, double it, ..., what's the number he first thought of?"
>>> with a classroom of kids and then introducing x for the number you
>>> first thought of. Vygotsky tells us to provide the symbol as a means
>> solving an existing problem.
>>> How did Davydov make the move from numbers to x?
>>> *Andy Blunden*
>>> Peg Griffin wrote:
>>>> The * was an intrusion! The expression is just paradoxical. There
>>>> cannot be a concrete world such that "Alyosha's string is greater
>>>> than Boya's string equals Alyosha's string is less that Borya's string."
>>>> (By the way, in case you want a smile on this November day, my
>>>> favorite paradox is the pragmatic one: " Inform all the troops that
>>>> communication has broken down." Can't remember who is the
>>>> originator of it, though!)
>>>> -----Original Message-----
>>>> From: firstname.lastname@example.org
>>>> [mailto:email@example.com] On
>>>> Behalf Of Andy Blunden
>>>> Sent: Friday, October 31, 2014 7:58 PM
>>>> To: eXtended Mind, Culture, Activity
>>>> Subject: [Xmca-l] Re: units of mathematics education
>>>> Could you elaborate on what is meant by this passage, Peg? I am not
>>>> familiar with this use of * in mathematics, and I am not sure how
>>>>> and < relations are being evaluated here. Andy
>>>> *Andy Blunden*
>>>> Peg Griffin wrote:
>>>>> ... That mathematical model (*A>B=A<B) DOES NOT have a concrete
>>>>> world to rise to! Instead, the children see/feel/perceive the
>>>>> strings and symbols having a relation among relations: A>B = B<A.