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Re: [xmca] a minus times a plus

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 Lemke
Educational Studies
University of Michigan
Ann Arbor, MI 48109

On Jul 7, 2009, at 1:59 PM, A.Bakker wrote:

Interesting discussion! Let me dwell on two projects in response to Jay and

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

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.


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

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 Lemke
Educational Studies
University of Michigan
Ann Arbor, MI 48109

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,
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

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
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
real big struggle to get to the top. And we're talking Mt. Washington,
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
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
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
is what is at issue.

ZPDs are value agnostic.  Mike and his team at LCHC are currently
to create ZPDs that can instill basic arithmetic in kids whose daily
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,
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
aid in its acquisition.  So, might not Jay's concern (if I read him
that mathematics (and the whole lot of techno-science) becomes
surreptitiously reified in our frustrated attempts to teach it be
from a different direction?

Jumping-jacks anyone?


On Sat, Jun 27, 2009 at 11:00 PM, Andy Blunden <ablunden@mira.net>

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
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

I presume the hesitancy about speaking up is probably the cause of
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,
it impossible for the father to get Marissa talk about it to him, or
Marissa in games of 20 Questions or something to lead her to the joys
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
help if I was there, but I'm 1000 km away.


Mike Cole wrote:

SO glad you are interested in this, Jay.
I have just made contact with Karen Fuson who has, lucky for us,
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
and god knows, of practrical importance to lots of kids.


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
think we want to teach seems to depend on deeper (e.g. 4000-year
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
(like 15 minus 8) seems to have gone so wrong that it's hard to know
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
it's time to seriously re-think the whole of the what, why, and how
education. Math is a nice place to focus because at least some of it
universally agreed to be useful by almost everyone, because
mathematicians and most people, including teachers and mathematics
educators, seem to hold radically different views about what the
and because success in teaching it, measured in almost any way, is
near the bottom of the heap.

Yes, we can find somewhat better ways to teach the same stuff, but
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
still get a minus.


Jay Lemke
Educational Studies
University of Michigan
Ann Arbor, MI 48109

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
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

Could we consider four years, just for whole numbers? Davydov
starts with
Algebra as the gateway arithmetic. Jean Schmittau, Peter Moxhay and
believe his method of introducing youngesters to math has some extra
As I understand it, others on xmca are dubious and look to other
difficulty. Karen Fuson, in her article on "developing mathematical
ins whole number operations" focuses on introducing number
through very simple, familiar, imaginable,
events where exchange is involved.

Its odd to me experiencing the cycle of time, the "coming back to
beginning and recognizing it
for the first time" that is happening for me right now with
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
15 and know enough to treat the
second zero as a 9. But the child, even understanding that the task
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
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.

On Sat, Jun 6, 2009 at 3:11 AM, Ng Foo Keong <lefouque@gmail.com>

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.   :-(


2009/6/4 Mike Cole <lchcmike@gmail.com>:

I am currently reading article by Fuson suggestion by Anna Sfard on

number operations. I also need to study Anna's paper with exactly

example in it. Not sure what moment of despair at deeper


me. Now that I am done teaching and have a whole day to communicate


are looking up!! Apologies for doubting I could have deep


why minus x minus = plus and minus x plus = minus. At present my

understanding remains somewhat bifurcated. The former is negation of

negation as david kel long ago suggested, linking his suggestion to



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.


And member book links are coming in. Nice.


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