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Re: [xmca] a minus times a plus

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 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, 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

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?


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.


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
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 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
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
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
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
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,
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 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
believe his method of introducing youngesters to math has some extra
As I understand it, others on xmca are dubious and look to other sources
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
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
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.

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.   :-(


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 this

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


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


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


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



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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