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Re: [xmca] a minus times a plus

In answer to Jay's question, how advanced a level is used so widely that it
is efficient to teach it to ALL students in schools?, I have to admit that
the level required is indeed pretty low (note that I focus on work, not on
citizenship). What ALL students need does not go beyond much of basic
arithmetic, including some sense of proportion, percentage, common
fractions, area, data handling (tables, graphs). I agree that few will
need to factor polynomials and solve quadratic equations. One problem of
not maintaining some fluency with these topics is that students forget
about them after grade 6 (as we have seen in the Netherlands).

It gets more interesting if we connect math to IT and finance, and other
areas of work and daily life.

Many employees use software (Excel is widely used) and appreciate some
fluency with Excel spreadsheets and formulas to make their work faster and
more precise (fewer mistakes). Much of the mathematical knowledge needed
at work is mediated by technology such as computer software (see
Techno-mathematical Literacies project, by Hoyles, Noss, Kent, Bakker).

And then finance: when I took a mortgage, I was happy to understand some
of the maths behind it. When I hear people struggling with their debts,
tax, credit cards, mortgages etc, I often think: more insight into the
mathematical structure of loans etc can lead to more sensible action and
more confidence, though I don't know research backing this up.

In my research I have heard many examples of how people benefited from
knowing some maths (related to  work): being able to work more
efficiently, offering a better deal because more precise estimates can be
made, making fewer mistakes because of a sense of numbers, communicating
better to customers, convincing customers of a better deal etc. Much of it
is also confidence - and the attitude not to shy away from numbers, being
able to ask something about numbers etc. Many employers complained their
employees have bad number sense - a very broad notion covering many
different things.

The big question that keeps nagging me is how students can develop this
number sense. Knowing your arithmetic and some math certainly helps, but
then people still need to learn to recontextualise that knowledge and
relate it to new situations. I often envision people living in a space of
reasons, some of which are mathematical and many of which are
non-mathematical. The question is, how do students learn to coordinate
these different reasons in a sensible way? I sometimes think that math
textbooks should have a section on finance, one on using Excel
intelligently etc. rather than structuring the math according to the
disciplinary structure of math.


> I'm pleased there has been uptake from Ivan's and my concerns about
> school mathematics, and science.
> A few miscellaneous responses --
> I don't doubt that there are many, even semi-technical, occupations
> where mathematical procedures are employed, and even where (though I
> suspect much less often) some judgment is required about how to apply
> them (thus requiring at least some theoretical understanding). I will
> be interested to see what Bakker's research shows about the most
> prevalent of these -- how advanced a level is used so widely that it
> is efficient to teach it to ALL students in schools? I really don't
> imagine very many people factor polynomials or solve quadratic
> equations, outside of higher level specializations. But these are
> empirical questions, whereas the content of the curriculum is NOT
> based on empirical findings of this sort, but rather on traditions of
> dubious validity.
> How to teach the mathematics that is widely used is then a separate
> question, and I think there is growing agreement that more realistic
> contexts are better for gaining wider success. There is still the very
> fundamental issue of whether translating such contexts into school
> activities can work well and generally (which I tend to doubt), or
> whether the learning needs to be taken outside of classrooms, or at
> least into mixed settings that combine experience and experience-based
> intuitions from non-school settings with some reflection and analysis
> work in classrooms, etc. Obviously SOME math can be taught
> successfully in classrooms, like some literacy skills, and some
> translation strategies are of value.
> But I would agree that the abstract approach to math and science, and
> the overstuffed topic curricula in these fields, is there more because
> (a) we know how to segment it and test it, and (b) it's a good way to
> keep a lot of people out of universities and professional jobs, while
> seeming to be completely objective and fair about a rigged system.
> Does it benefit the mind in more general ways? I am a splitter ... I
> don't believe much in transfer, generalization, general intelligence,
> multi-purpose higher mental functions, etc. I tend to think that
> almost all practices and procedures are highly context and content -
> specific, with what appears otherwise resulting mostly from certain
> sorts of cultural habitus acquired by individuals. Classes of tasks
> and practices have historically developed with certain kinds of
> similarities within particular cultures. Cultures have preferred
> strategies or ways-of-doing that are implicit, and members who have
> the appropriate trajectories of cultural experiences tend to develop
> dispositions fitted to these.
> In these terms, experiences with mathematics CAN support developing
> dispositions that make mastering other kinds of abstract reasoning
> practices come more easily. Symmetrically, mastering mathematics is
> easier if you've already had success with other implicitly similar
> kinds of tasks and strategies. Learning abstract decontextualized
> mathematics, however, seems to me one of the hardest ways into such a
> cultural complex of similar practices. And any benefit from working at
> mathematics seems to me to accrue only if (a) the work is enjoyable or
> at least has a supportive relationship to a desired identity, and (b)
> you are successful at it, preferably early on.
> All this applies to conceptual understanding of sciences equally as
> well.
> JAY.
> Jay Lemke
> Professor
> Educational Studies
> University of Michigan
> Ann Arbor, MI 48109
> www.umich.edu/~jaylemke
> On Jul 7, 2009, at 1:59 PM, A.Bakker wrote:
>> Interesting discussion! Let me dwell on two projects in response to
>> Jay and
>> Andy.
>> 1. what kind of math do we need at work?
>> We have analyzed the mathematical knowledge required in 239
>> intermediate-level professions (think of service engineering, florist,
>> baker, low level analyst in science labs, builders, car mechanics,
>> salary
>> administration, secretarial work, hairdresser etc). Some of these do
>> not
>> have to do any calculations at all (butcher in a factory just
>> selecting good
>> and bad parts of meat), but the vast majority of professions face
>> simple
>> arithmetic, geometry (area, volume), data handling and risk, and
>> sometimes
>> formulas. Even at the lowest level of education, lab analysts face
>> some
>> high-level statistics (F-test, t-test, correlation etc) in method
>> validation, precision of instruments etc.
>> Although there is some truth in Andy's comments, this analysis gives
>> a more
>> nuanced image. Moreover, there is more than math at work and in
>> daily life:
>> math required for higher-level education. Vocational students
>> without enough
>> mathematical and scientific baggage have trouble getting through their
>> higher vocational education (nursing, teaching, management etc).
>> however, I
>> should note that our Dutch school system differs drastically from the
>> American one because our vocational education is big (60% of the
>> students)
>> and starts early (pre-vocational education at age 12).
>> 2. basing science units on authentic practices
>> Indeed, many math and science problems at school are not very
>> realistic. It
>> is in fact quite hard to design good ones. Over the past years we
>> have tried
>> to base educational units on authentic practices in which science or
>> math is
>> used (with activity theory in mind as well). We have 'translated'
>> authentic
>> goals to learner goals, adapted ways of working and knowledge
>> required to be
>> manageable to students (grade 10-12). The idea was to use meaningful
>> relationships between goals, tools, knowledge etc in outside-school
>> practice
>> as sources of inspiration for school units. Although we have had some
>> success, there are still many challenges in designing good units -
>> even if
>> we allow the learning goals to be drastically different from the
>> Standards
>> (say: insight into health and nutrition rather than say DNA,
>> evolution, cell
>> biology...).
>> So I agree with Jay that content is a major problem, but even then
>> we have a
>> lot of work to do in terms of designing good alternatives.
>> Chevallard has written interesting papers on didactic transposition,
>> adapting knowledge as used in the 'real world' to school situations.
>> He
>> describes a contradiction that cannot be resolved completely:
>> education
>> promises to prepare kids for their future and for society. At the
>> same time,
>> education cannot really fulfil its promise. What students learn is
>> often
>> something that teachers can easily test. Chevallard argues that the
>> main
>> reason that we still teach math and science is NOT that they are so
>> useful,
>> but they can be rolled out nicely in stages over the school grades
>> and can
>> be tested in objective ways. A lot of things that are very useful to
>> learn
>> do not make it to the curricula, simply because they are so hard to
>> teach
>> and test (medicine, psychology, sociology, social skills etc.)
>> Arthur Bakker
>>> -----Original Message-----
>>> From: xmca-bounces@weber.ucsd.edu [mailto:xmca-
>>> bounces@weber.ucsd.edu] On
>>> Behalf Of Andy Blunden
>>> Sent: dinsdag 7 juli 2009 13:30
>>> To: eXtended Mind, Culture, Activity
>>> Subject: Re: [xmca] a minus times a plus
>>> Your key claims are beyond challenge Jay; you can get by
>>> perfectly well in all aspects of life without mathematics,
>>> apart from a basic understanding of the notion of quantity
>>> and some elementary arithmetic, except for a very small
>>> group of professions. It has annoyed me, this need to invent
>>> pseudo-problems that seem to demand mathematics, to
>>> "justify" the need to learn maths. It seems to me that it is
>>> requirement to pass maths exams to gain entry to a very wide
>>> range of jobs etc., which is the only real motivation for most.
>>> But can you tell me, is there no evidence that going through
>>> the process of learning maths in some way benefits the mind?
>>> in the same way that (as I understand it) learning to read
>>> and write has a permanent and effect on how people think?
>>> that mathematics is a kind of mental gymnastics.
>>> Andy
>>> Jay Lemke wrote:
>>>> Thank, Ivan, for responding in part to some of my concerns re
>>>> teaching
>>>> math-as-math in schools.
>>>> It's a big, old debate in education whether we should teach ideas,
>>>> concepts, and disciplines as abstract systems, in the hopes they can
>>>> then be used as tools to think with ... or whether that usually
>>>> doesn't
>>>> work, puts kids off from the subject, and it's better to let
>>>> concepts
>>>> appear more naturally in the context of real-world problems, issues,
>>>> activities which are not about math or science, but in which math-
>>>> using
>>>> and science-using activities and practices can play a helpful part.
>>>> The academic, and intellectual answer, as part of a cultural and
>>>> institutional tradition, is that we cheat students out of the
>>>> power of
>>>> math and science if we don't give them the systems of abstract
>>>> concepts,
>>>> and that other approaches tend to degenerate into second-rate
>>>> practicalism that avoids theory, critique, alternatives, creativity,
>>>> etc. My own view, after a long time participating in, observing, and
>>>> trying to analyze the teaching of science, and to a lesser degree
>>>> mathematics, is that the powerful systematic conceptual tools are
>>>> a very
>>>> advanced stage of membership in one or more very specialized
>>>> communities, and are simply not of much use to most people.
>>>> Maybe my view is a bit extreme. But I think it remains true that
>>>> it is
>>>> not just a failure to find the magic method of teaching that is the
>>>> problem with math-as-math and science-as-science in the
>>>> curriculum. It
>>>> is the content itself. Or, really, the lack of content, the lack of
>>>> engagement with real life activities that are meaningful and
>>>> important
>>>> to the students, in the modern math and science curricula. And I
>>>> do not
>>>> see the solution as inventing clever artificial problems and
>>>> topics that
>>>> seem to be relevant to real-life, but which are in fact just
>>>> excuses to
>>>> do more math-as-math and science-as-science.
>>>> A mathematician or a scientist can find, show you, highlight, apply
>>>> their conceptual tools to nearly anything. Some reasonable level of
>>>> abstract awareness of those tools can emerge from encountering, in
>>>> some
>>>> detail and depth, several domains and examples or projects in
>>>> which the
>>>> concepts have been highlighted for their usefulness (and that
>>>> includes
>>>> usefulness for critical thinking, for imagining alternatives --- not
>>>> just for engineering practical constructions or solutions). But this
>>>> comes at the end of a long learning process, and almost as a kind of
>>>> side-effect, and not at the beginning or as the primary purpose or
>>>> goal-of-activity.
>>>> There is math and science in jumping jacks and football, in mountain
>>>> climbing, in raising a pet or growing some food, in figuring the
>>>> cost of
>>>> better garbage collection in the neighborhood, in organizing a block
>>>> party, in understanding when to go to the hospital or what counts as
>>>> evidence in a court case. It might be better to say that there are
>>>> issues of quantity and degree, of probability and risk, of
>>>> nutrition and
>>>> cause and effect in all these domains and phenonena, and that the
>>>> workarounds and tricks and mnemonics and practical methods
>>>> accumulated
>>>> across them all tend to implicate some more general strategies ---
>>>> which
>>>> we could just tell you, but then the odds are you wouldn't
>>>> understand or
>>>> remember or know how to use them for yourself.
>>>> I am not talking here about advanced levels of education, but about
>>>> introductory ones ... up to about the age of 15 or 17, or up to the
>>>> point at which interest and possibility tend to focus students
>>>> toward
>>>> some preferred specialization. Then the balance shifts, again not
>>>> all
>>>> the way toward abstract disciplines (as, for example, medical
>>>> education
>>>> has struggled to sort out for a long time now), but a bit more
>>>> toward
>>>> the justification of more emphasis on theoretical learning, as
>>>> part of
>>>> membership in a specialist community of knowers/doers.
>>>> What are the practical situations in which you need to multiply a
>>>> minus
>>>> times a plus? not textbook imaginaries, but for real? If you had
>>>> some
>>>> broad and in-depth knowledge about such a situation, would it then
>>>> be so
>>>> hard to make sense of how signed numbers multiply there? And how
>>>> far a
>>>> step is it, and how necessary a step for all to take, from an
>>>> induction
>>>> from several such well-understood situations to the pure
>>>> mathematicians'
>>>> abstract arguments about how signed numbers multiply everywhere, or
>>>> really, nowhere??
>>>> JAY.
>>>> Jay Lemke
>>>> Professor
>>>> Educational Studies
>>>> University of Michigan
>>>> Ann Arbor, MI 48109
>>>> www.umich.edu/~jaylemke
>>>> On Jun 30, 2009, at 6:50 PM, Ivan Rosero wrote:
>>>>> Here's a familiar exhortation:
>>>>> "We need as many engineers as possible.  As there is a lack of
>>>>> them,
>>>>> invite
>>>>> to this study, persons of about 18 years, who have already
>>>>> studied the
>>>>> necessary sciences. Relieve the parents of taxes and grant the
>>>>> scholars
>>>>> sufficient means."
>>>>> According to my brief cyber-sphere search, these are the words of
>>> Emperor
>>>>> Constantine.
>>>>> So, anyway, we all know what road that empire took.  I doubt it was
>>>>> lack of
>>>>> engineers though :)  So, given the very similar verbiage spilling
>>>>> out
>>>>> of NSF
>>>>> these days, I agree with Jay, perhaps slowing down and taking a
>>>>> minute
>>> or
>>>>> two to rethink this wouldn't be bad at all.
>>>>> If I read you correctly Jay, one big worry you have is that we
>>>>> don't
>>>>> end up
>>>>> reifying mathematics (in the sense Constantine seems to be doing
>>>>> with
>>>>> engineering) in the frustration we experience with our almost
>>>>> complete
>>>>> failure in teaching it.
>>>>> It reminds me of mountain-climbing.  For me at least, this is one
>>>>> hell
>>>>> of a
>>>>> difficult sport, and the few times I've ever participated, it has
>>>>> been
>>> a
>>>>> real big struggle to get to the top.  And we're talking Mt.
>>>>> Washington,
>>> a
>>>>> measly ~6000 ft peek.  Anyway, I struggle, sweat, almost pass-
>>>>> out, and
>>>>> finally I'm there.  It is AWESOME, the joy is overwhelming.  20
>>>>> minutes
>>>>> later, as my muscles cool down and my adrenaline levels-off, I
>>>>> stare
>>> down
>>>>> the thing and feel a creeping dread, even if the way down is many
>>>>> times
>>>>> easier than the way up.
>>>>> This story can go in many directions from here, as many as there
>>>>> are
>>>>> people
>>>>> who have made it (oh, God, this is cheesy) mountain-top.  They
>>>>> are not
>>>>> universally happy stories however.
>>>>> I DO think it is useful to know some mathematics and have a host of
>>>>> scientific concepts to think with and through at our disposal.
>>>>> None
>>>>> of this
>>>>> is Bad (or Good for that matter) in and of itself.  The Purpose, of
>>>>> course,
>>>>> is what is at issue.
>>>>> ZPDs are value agnostic.  Mike and his team at LCHC are currently
>>>>> attempting
>>>>> to create ZPDs that can instill basic arithmetic in kids whose
>>>>> daily
>>> (and
>>>>> arguably far stronger) ZPDs pull them in many other (sometimes
>>>>> directly
>>>>> opposite) directions.  Some of those ZPDs, however, are not in
>>>>> direct
>>>>> conflict with math.  That is my hunch, or assumption.  The task,
>>>>> then,
>>> is
>>>>> perhaps a bit simpler than creating new ones.
>>>>> Is it simpler to find and then piggy-back on, ZPDs that contain
>>>>> kernels of
>>>>> arithmetic in them?  Susan Goldin-Meadow has pretty convincing
>>>>> evidence that
>>>>> specific motor activity can not only presage basic arithmetic,
>>>>> but can
>>>>> even
>>>>> aid in its acquisition.  So, might not Jay's concern (if I read him
>>>>> right)
>>>>> that mathematics (and the whole lot of techno-science) becomes
>>>>> surreptitiously reified in our frustrated attempts to teach it be
>>>>> addressed
>>>>> from a different direction?
>>>>> Jumping-jacks anyone?
>>>>> Ivan
>>>>> On Sat, Jun 27, 2009 at 11:00 PM, Andy Blunden <ablunden@mira.net>
>>> wrote:
>>>>>> I hope people won't mind if I continue to pick the brains of this
>>>>>> list on
>>>>>> the problem of my niece's progress in maths, or lack of it.
>>>>>> It seems that the suggestion last time - that Marissa may have
>>>>>> missed
>>>>>> important lessons while on holiday - may explain her poor
>>>>>> performance
>>>>>> last
>>>>>> year in maths, even though maths has always been her weak subject.
>>>>>> She has
>>>>>> caught up a bit but she is still badly behind.
>>>>>> It seems that the issue Mike has raised also applies: she is
>>>>>> getting
>>>>>> homework that seem to presume she know things that in fact she
>>>>>> doesn't. The
>>>>>> only other negative in her school reports is that she doesn't
>>>>>> participate in
>>>>>> class discussion or ask questions when she doesn't understand
>>> something.
>>>>>> I presume the hesitancy about speaking up is probably the cause of
>>>>>> failure
>>>>>> to correct her maths problems and the teachers giving her
>>>>>> homework she
>>>>>> doesn't understand.
>>>>>> She is now 15 and her maths homework is also beyond her father! :)
>>>>>> and the
>>>>>> crisis of the transition from childhood to adulthood around this
>>>>>> age,
>>>>>> makes
>>>>>> it impossible for the father to get Marissa talk about it to
>>>>>> him, or
>>>>>> engage
>>>>>> Marissa in games of 20 Questions or something to lead her to the
>>>>>> joys
>>> of
>>>>>> asking others. Discussion over the dinner table is apparently also
>>>>>> unconducive to her participation.
>>>>>> Does anyone have any ideas? I've run out of suggestions. I could
>>>>>> probably
>>>>>> help if I was there, but I'm 1000 km away.
>>>>>> Andy
>>>>>> Mike Cole wrote:
>>>>>>> SO glad you are interested in this, Jay.
>>>>>>> I have just made contact with Karen Fuson who has, lucky for us,
>>>>>>> "retired"
>>>>>>> from Northwestern and moved nearby. She is away for a week or so
>>>>>>> but then we are getting together. This is a problem that just
>>>>>>> may be
>>>>>>> tractable, theoretically interesting for sure, attractive of
>>> experience
>>>>>>> collaborators,
>>>>>>> and god knows, of practrical importance to lots of kids.
>>>>>>> mike
>>>>>>> On Sun, Jun 7, 2009 at 3:27 PM, Jay Lemke <jaylemke@umich.edu>
>>>>>>> wrote:
>>>>>>> Yes, Mike and F.K., these are very disturbing issues. Both that
>>>>>>> what
>>> we
>>>>>>>> think we want to teach seems to depend on deeper (e.g. 4000-year
>>> deep)
>>>>>>>> knowledge than it's realistic to expect most people to learn (or
>>>>>>>> want to
>>>>>>>> learn), and that how we teach even the most practical bits of
>>>>>>>> mathematics
>>>>>>>> (like 15 minus 8) seems to have gone so wrong that it's hard
>>>>>>>> to know
>>>>>>>> where
>>>>>>>> to start, especially for those we have most systematically
>>>>>>>> failed.
>>>>>>>> We do indeed need to not give up. But we also need, I think, to
>>> admit
>>>>>>>> that
>>>>>>>> it's time to seriously re-think the whole of the what, why,
>>>>>>>> and how
>>> of
>>>>>>>> education. Math is a nice place to focus because at least some
>>>>>>>> of it
>>>>>>>> seems
>>>>>>>> universally agreed to be useful by almost everyone, because
>>>>>>>> professional
>>>>>>>> mathematicians and most people, including teachers and
>>>>>>>> mathematics
>>>>>>>> educators, seem to hold radically different views about what the
>>>>>>>> subject
>>>>>>>> is,
>>>>>>>> and because success in teaching it, measured in almost any
>>>>>>>> way, is
>>>>>>>> pretty
>>>>>>>> near the bottom of the heap.
>>>>>>>> Yes, we can find somewhat better ways to teach the same stuff,
>>>>>>>> but
>>>>>>>> maybe
>>>>>>>> it's the stuff itself (the content of the curriculum, viewed not
>>>>>>>> just as
>>>>>>>> information, but as activity) that needs to be rethought? along
>>>>>>>> with the
>>>>>>>> ethics and efficacy of who decides.
>>>>>>>> No matter how many times you multiply a minus by any number of
>>> pluses,
>>>>>>>> you
>>>>>>>> still get a minus.
>>>>>>>> JAY.
>>>>>>>> Jay Lemke
>>>>>>>> Professor
>>>>>>>> Educational Studies
>>>>>>>> University of Michigan
>>>>>>>> Ann Arbor, MI 48109
>>>>>>>> www.umich.edu/~jaylemke
>>>>>>>> On Jun 6, 2009, at 6:12 PM, Mike Cole wrote:
>>>>>>>> Hi Foo Keong-- It is so generous of you to even try to
>>>>>>>> explain! And
>>>>>>>> your
>>>>>>>> question re math seems to me
>>>>>>>> relevant to other areas of knowledge as well when you ask,
>>>>>>>> "Can we
>>>>>>>> condensefour thousand years of
>>>>>>>> human development into an easily digestible four minutes for
>>>>>>>> learners."
>>>>>>>> Could we consider four years, just for whole numbers? Davydov
>>>>>>>> starts with
>>>>>>>> Algebra as the gateway arithmetic. Jean Schmittau, Peter
>>>>>>>> Moxhay and
>>>>>>>> others
>>>>>>>> believe his method of introducing youngesters to math has some
>>>>>>>> extra
>>>>>>>> power.
>>>>>>>> As I understand it, others on xmca are dubious and look to other
>>>>>>>> sources
>>>>>>>> of
>>>>>>>> difficulty. Karen Fuson, in her article on "developing
>>>>>>>> mathematical
>>>>>>>> power
>>>>>>>> ins whole number operations" focuses on introducing number
>>> operations
>>>>>>>> through very simple, familiar, imaginable,
>>>>>>>> events where exchange is involved.
>>>>>>>> Its odd to me experiencing the cycle of time, the "coming back
>>>>>>>> to
>>> the
>>>>>>>> beginning and recognizing it
>>>>>>>> for the first time" that is happening for me right now with
>>> arithmetic
>>>>>>>> and
>>>>>>>> early algebra. The source
>>>>>>>> is quite practical with social significance: the unbridgable
>>>>>>>> gap the
>>>>>>>> children I work with face between
>>>>>>>> what their teachers are teaching about (say) subtraction
>>>>>>>> (2005-118
>>>>>>>> is my
>>>>>>>> current keystone example)
>>>>>>>> trying to get their kids to learn that the first step is to
>>>>>>>> subtract
>>> 8
>>>>>>>> from
>>>>>>>> 15 and know enough to treat the
>>>>>>>> second zero as a 9. But the child, even understanding that the
>>>>>>>> task
>>>>>>>> the
>>>>>>>> teacher is focused on is
>>>>>>>> disabled because when asked 15-8 the answer =3 and only
>>>>>>>> painstaking
>>>>>>>> attention to the problem set up with fingers and subtracting
>>>>>>>> one by
>>>>>>>> one,
>>>>>>>> with full compliance and even eagerness by the child, brings
>>>>>>>> her to 7.
>>>>>>>> Now suppose this phenomenon is ubiquitous, affects 100's of
>>>>>>>> thousands of
>>>>>>>> children, and is heavily correlated with social class.
>>>>>>>> Then ....  ??? ....
>>>>>>>> I think my frustration is probably equivalent to yourse in
>>>>>>>> intensity, but
>>>>>>>> the quality is of a somewhat different nature.
>>>>>>>> mike
>>>>>>>> On Sat, Jun 6, 2009 at 3:11 AM, Ng Foo Keong
>>>>>>>> <lefouque@gmail.com>
>>>>>>>> wrote:
>>>>>>>> I was trained in mathematics at the University of Cambridge (UK)
>>>>>>>> for my undergraduate studies, concentrating more on pure
>>>>>>>> mathematics (including algebra).  I am able to roll out a
>>>>>>>> rigorous abstract proof of why "minus times minus" is a "plus",
>>>>>>>> using only the basic axioms of real numbers (actually you only
>>>>>>>> need a few of those axioms).
>>>>>>>> However, abstract proofs aren't likely to be useful for non-math
>>>>>>>> specialists and struggling neophyte learners of algebra.  in
>>>>>>>> order to pull off such a proof, or even just to understand just
>>>>>>>> the few lines of proof, you almost need to be a mental
>>>>>>>> masochist.
>>>>>>>> Who likes to go through mental torture?
>>>>>>>> Can we condense four thousand years of human development of
>>>>>>>> mathematical understanding into an easily digestible four
>>>>>>>> minutes
>>>>>>>> for learners?
>>>>>>>> thus the huge gulf of understanding still persists.  that's why
>>>>>>>> as an educator, i feel so useless being unable to help other
>>>>>>>> people.   :-(
>>>>>>>> F.K.
>>>>>>>> 2009/6/4 Mike Cole <lchcmike@gmail.com>:
>>>>>>>> I am currently reading article by Fuson suggestion by Anna
>>>>>>>> Sfard on
>>>>>>>> whole
>>>>>>>> number operations. I also need to study Anna's paper with
>>>>>>>> exactly
>>> this
>>>>>>>> example in it. Not sure what moment of despair at deeper
>>> understanding
>>>>>>>> hit
>>>>>>>> me. Now that I am done teaching and have a whole day to
>>>>>>>> communicate
>>>>>>>> things
>>>>>>>> are looking up!! Apologies for doubting I could have deep
>>>>>>>> understanding
>>>>>>>> of
>>>>>>>> why minus x minus = plus and minus x plus = minus. At present my
>>>>>>>> understanding remains somewhat bifurcated. The former is
>>>>>>>> negation of
>>> a
>>>>>>>> negation as david kel long ago suggested, linking his
>>>>>>>> suggestion to
>>>>>>>> Anna's
>>>>>>>> comognition
>>>>>>>> approach. The second I think more of in terms of number line and
>>>>>>>> multiplication as repeated addition.
>>>>>>>> Perhaps the two will coalesce under your combined tutelage.
>>>>>>>> mike
>>>>>>>> And member book links are coming in. Nice.
>>>>>>>> mike
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