Except Michelle, I'm not willing to doom all children with unmotivated or unskilled parents. Or all children of parents who struggle with English or with reading or who need to work 4 minimum wage jobs between them to survive.

If it is truly "all about parenting" then we need not have schools at all, and we would simply accept inherited position as the path to success.

I don't mean to suggest that parents should not do all they can, but this is the US where parents get no support from their society - no guarantees of vacation time, limited work hours, paid parenting time, or even health care - the things which allow families to properly function in social democracies. So we have to decide if we believe - in the UK phrase - that "Every Child Matters" - and if so, are we going to act on that through schools or through a dramatically improved vision of society?

I'd like to point out a distinction. "Racist" is about hate. It involves purposeful discrimination against a particular group. "Bias" is a very different thing; it involves a weighting of values that may be largely unconscious. Bias can often be corrected simply by identifying it and pointing it out. But very few racists say "OMG, you're right! I am a racist... Well HOW did THAT happen to me? We'll have to fix that."

You seem to be purposely confusing the two (presenting them as somehow synonomous) as a rhetorical ploy...

Unfortunately, the GT identification/service route has gone so neglected for so long (how many GT certified / GT specialist teachers are in any given building?) that it's easy to confuse what often passes for GT education and what GT education really needs to be.

It sounds like you are describing great opportunities that all kids deserve. But there's nothing necessarilly 'GT' about those opportunities. They're actually just the opportunities for learning that we SHOULD be offering on a regular basis to all our students.

Where schools often miss the boat with regards to gifted ed is that they treat GT as if it's just a bunch of smart kids who need enrichment activities.

That's not the case.

GT kids of every race, class, and background are regularly unidentified because folks are looking for numbers and grades when the research suggests that (especially by high school) many true GT kids have 'dropped out' (in the old sense of dropping out of the local society mentality).

GT kids don't necessarily get good grades. GT kids are often problem-behavior kids. They are often especially prone to perfectionism, obsessive behavior, self-abuse, isolation, etc.

Many GT kids have severe LDs, as well. Asperger's is common among GT circles. As are all sorts of social and physio-social problems likely related to hypersensitivity.

Forget the Stanford Tests, try visual batteries. The results some high-end GT kids attain there are nothing less than breath-taking. And it's not because they do so 'well'; rather it's because in reading the results of the tests, you can see that their minds are working in ways that are so far removed from what we consider 'ordinary'.

And that can be both a blessing and a curse.

A good GT program would understand this and treat its children on an individualized basis not unlike the best of any Special Ed program.

Shelley,

I'm with you all the way on this. Like I said though, it is not what I see in the programs across Michigan.

But this just adds to my desire for IEPs for every student. We miss so much because we are not looking, because information doesn't carry from teacher to teacher, grade to grade, school to school, because we rarely think that students can be both talented have special needs.

As long as we stick to age-based grades, grade-level expectations, and the industrial model of schooling, any child not absolutely average will get less than he or she needs, and both edges of the spectrum will flee school - quite justifiably.

Hmmm... I think you're confusing gifted and advance placement. Lot's of perfectly normal students take advance placement classes. From an administrative point of view, the idea that we might have to start the year prepared to teach trig to 55 kids (two large classes) who WANT the challenge of trig, and that 18 of them may need to be moved by October to a business math class (still leaving more than one full class of students in trig) has some personnel and planning implications.

I'd be interested to know how you'd work model this at the elementary level, where gifted students are often pulled and transported off campus for part or all of a school day.

No, I'm thinking of challenging classes in general. It seems to me that the logistics need not be as prohibitive as some schoole make them.  Why not simply allow each student to work through a math curriculum at his/her own rate?  As a student, I attended a classes like this (in regular public schools, with large class sizes) and they were highly successful classes, precisely because they engaged each student at his or her level.

The idea of providing a curriculum that challenges and/or meets the needs of all students sounds to me like differentiated instruction (DI), which has become a major focus of professional development and curriculum design in the last few years. Carol Tomlinson is one of the more popular authors on the subject. I've been sent to three or four workshops on it in the past three or four years - and the general ed teachers I work with have been required to attend with me.

DI comes in flavors. There's DI intended to cater to different learning styles to ensure that kinesthetic learners don't have to rely on auditory or visual processes alone. There's DI intended to address the level at which students get challenged. The math lesson may be on probability; the curriculum we use will provide the teacher with a variety of tasks (of varying difficulty) that can be used with their students.

Both the math and the reading curriculum my district uses in the elementary grades build DI into each lesson.

Rate is a different issue. The adoption of a spiral concept in curriculum design makes *rate* seem like a problematic concept. Instruction is cyclical: we may spend a couple of weeks working on learning and using central tendencies in statistics with the fourth or fifth graders, then the curriculum moves to addition of improper fractions for a week, then it spends some time reinforcing student knowledge of geometric shapes and their properties, and a few weeks later it cycles back around to central tendencies.

Chances are good that the second and third graders are working on exactly the same concepts (though with simpler problems) if their teachers are on track with the pacing guides.

If a student came to me at any grade level (I've taught K-12) and said that they understood chapter 11 and had finished the work in it, and they asked me if they could go on to chapter 12 WITHOUT the rest of the class, I'd say no. The reasons are simple. I need that student to participate in cooperative learning and group activities for the sake of the other students (whom they can help in ways that I can't) and for their own sake (because being a peer tutor has been show to produce a more secure set of skills in a student).

It's not that there is material to cover and the student could finish early, it's that there are standards (content standards) to meet. I'm happy if the kid can excede the standards for central tendencies; I'm not happy for the kid to move on to geometry while the rest of the class is still in statistics. In addition to math skills, I have to think about the student's social skills - their ability to work with with others.  

DI can and should lead to engaging each student at their level. But it doesn't (and shouldn't) lead to anyone finishing the year's math curriculum sometime in March...

I know a lot of talented math buffs (the subjets of my forthcoming book) who are extremely frustrated by the practices you describe (which are unique to the U.S., and certain schools in Canada, Britain and Australia).  They are severely underchallenged, especially with today's Reform Math (where the actual math is much, much easier than it used to be), hate working in groups, and resent being asked to teach math to other students.  We are at risk of marginalizing (and under-preparing with respect to students from other countries) the next generation of potential mathematicians.

There are no randomized studies showing that social skills improve when students are forced to work in groups; if there were, then presumably my generation (which didn't do much work group at all), and students in non-Anglophone countries, have weaker social skills than younger Americans do.

Social skills (I'll have to search the literature on that sometime) are a smaller part of the picture than academic skills. I think there's considerable support for the idea that participating in cooperative learning activities results in a stronger set of academic skills for the students. I'll quote Johnson, Johnson, and Stanne (2000):

Cooperative learning has been around a long time (Johnson, 1970; Johnson & Johnson, 1989, 1999). It will probably never go away due to its rich history of theory, research, and actual use in the classroom. Markedly different theoretical perspectives (social interdependence, cognitive-developmental, and behavioral learning) provide a clear rationale as to why cooperative efforts are essential for maximizing learning and ensuring healthy cognitive and social development as well as many other important instructional outcomes. Hundreds of research studies demonstrate that cooperative efforts result in higher individual achievement than do competitive or individualistic efforts. Educators use cooperative learning throughout North America, Europe, and many other parts of the world. This combination of theory, research, and practice makes cooperative learning one of the most distinguished of all instructional practices.

http://www.co-operation.org/pages/cl-methods.html

Its not, but you insisted that there's nothing "cultural" about math, I was trying to show you that you were wrong.

All learning, all testing,is culturally constructed. To pretend otherwise is absurd.

*facepalm*

Education is obviously a cultural construct. It differs from culture to culture, and we talk constantly about changing the culture of education in America - from the level of individual schools all the way to the national level.

Is math, as an academic field, a cultural construct? I'm certified in math, but I wouldn't call myself a mathematician by any stretch. That said, all academic fields are culturally constructed. What makes one person a biologist and another a chemist is culturally defined - and it changes from place to place, university to university, country to country. Math is no different. We may all agree that irrational numbers can't be expressed as fractions or that "pi" is a useful concept in geometry. But culture impacts what gets studied and how it gets thought about. The same cultural characteristics that helped the Chinese excel at arithmetic and develop the abacus impeded their development of algebra.

Having lived on four continents and in 14 time zones, the most obvious interface between culture and math is in concepts of time. And I can't help be think that THAT impacts discussions, for instance, in calculus in the more rarified academic atmosphere of math theory...

The work of pure mathematicians, by which I mean the ones you find in math departments, abstracts away from particular units (hours; meters; horsepower, etc) and systems of measurement (base 10, base 8, etc).  While all academic fields are cultural constructs in the ways you cite (definitions; career choices; pedagories), I can't imagine a less culturally-charged field than pure mathematics, given its level of abstraction.

Do you have a reference for what you say about  Chinese cultural characteristics and the abacus impeding algebra?  It sounds interesting, and I'd like to read more.

Hi Katherine, I've thought about that statement and done a little Googling (on a dial-up modem - staying with relatives at the moment). I don't have a cite.

The statement I made was from a workshop a few years ago, and after considering it for a while I think that what was actually said was that China's preoccupation with arithmetic (not the abacus) impeded the development of Algebra.

Zhu Shijie's 1303 book (Jade Mirror of the Four Unknowns)  was the peak of algebra's development in China. Egypt, Greece, Persia, and India developed algrebra well before the Chinese. But then, Europe was till behind the curve...

If prime numbers are socially constructed, so is the speed of light.  Your argument applies equally to this concept.  What makes it "the speed of light"?  Yes, you've got the answer, but that requires you to believe that say, the speed of 299,792,458 is somehow fundamentally different from, say, 200,000,000 meters per second, or 400,000,000 meters per second.  Which is not a "fact" but a cultural choice.  (For a much more sophisticated argument on the social constructedness of the speed of light and other physical concepts, see Sokol's "Transgressing the Boundaries").

Cheers!

There are 10 types of people in the world: those who know binary and those who don't....

You're saying, Ira, that prime numbers are prime partly because we've decided to think in base 10? And if we didn't think in base 10, we'd have different prime numbers? After all, how many prime numbers are there in binary? Two?

I tell my kids that NUMBERS are values that have a place, an address on the number line. On the other hand NUMERALS are the symbols (or "letters") we use to write numbers. The numeral is a disposable, cultural defined concept. How different would our math theories be if we thought in a system with 16 numerals (like with HTML color codes - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F; where A is greater than 9 and F is greater than A) or if we still used Roman numerals? I'm not sure...

I won't pretend to have any serious grasp of special relativity. But does the math in question all still work if we measure the speed of light in feet and inches? (That's a sincere question.) I grew up thinking light traveled about 186,000 MILES per second. I'd never conceived of it in meters.

I recently saw a CBS news piece on the question of whether the speed of light was slowing down. It was dated back it 2002. And I understand the idea has religious implications for some reason. Talk about cultural baggage in math - or physics, whichever we're in (I suppose that's a cultural question). Where in this does math become theology or philosophy? (That's rhetorical.)

Greg,

I appreciate all of your questions - I guess I'm saying that we understand prime numbers because we give special status to points on the number line which correspond to numerals. And I say that as someone who, as a kid, was confused by this. "Of course I can divide 13 by something other than 1: 10x1.3=13 Of course, I was classed as "retarded." As Silvia suggests, our simplistic K-12 view of math punishes creative math thinkers (as we do in every subject) or those who bring any different viewpoint to the topic.

It goes much further. Why is the Fibonacci sequence so essential in the UK and Ireland and barely taught in the US? Does understanding the golden mean" have an absolutely different value depending on hemisphere (as Pullen might suggest - based on his comments here), or is that choice cultural? Isn't the 1.06 multiplication table the most important to memorize in Michigan (sales tax rate)? or the 1.0975 table most important in New York City? And, as you say, considering the speed of light - is there not indeed a conceptual change when you switch between feet, meters, and nautical miles?

All of this creates confusion for those "not from "normed" backgrounds." I wrote a blog post a year ago about the confusion created simply because the US writes sports scores "backwards" from the rest of the world, or writes the dates reversed from the rest of the world. Strange little cultural decisions which make life harder for certain groups. Our world is full of them. Our curricula are full of them. Our tests are full of them.

Today's Reform Math punishes the most creative of all mathematical thinkers:  our budding mathematicians.  It repeatedly requires them to explain, in words, their solutions to calculations that they solve, nonverbally, in their heads, withholding full credit for unexplained answers. And it systematically deprives them of problems that actually challenge them (by making them work at the same rate as, and in groups with, the other students in their class). 

I had a long conversation with my significant other about mathematics in education.  As a computer science graduate I can tell you with certainty that at an upper-level math elective, you are taught right away that the foundation of grade school math is based on assumptions.  I remember the first time we did a proof dealing with polar math that proved everything we learned about geometry in grade school is simply based in baloney.

The basis of all geometry is the rule that for any two points, you can create two parallel lines.  When you get to a high-level mathematics curriculum, You are challenged to prove this and it is simply not possible to prove.  It is just an agreed upon assumption so that we can teach geometry.

I may not necessarily see it as a cultural difference, but mathematics is taught in a context that is vastly different dependent upon who teaches it or where you are in the world.  Values hold constant, but numbers do not.

What you're calling "assumptions" mathematicians would call "axioms".  Euclidean geometry uses one set of axioms; Lovbachevskian geometry (where there are no parallel lines) uses another. Mathematicians use both systems; there's no contradiction or baloney. I learned both systems in 9th grade geometry, and was intrigued by their conceptual coexistence.  It's actually rather beautiful.  Ask a mathematician!

Ladson-Billings is not a cognitive scientist; I strongly suspect that cognitive scientists involved in empirical research on math acquisition, e.g., Stanislas Dehaene, would disagree with her general thesis about the "profound impact" of culture on cognition.

Did you read the study that was above quote? The quote was from a paper by Dr. Steven Guberman - University of Colorado at Boulder. Other research includes Supportive Environments for Cognitive Development. - The above research "Cultural Aspects of Young Children's Mathematics Knowledge" was used by NCTM.....all of this doesn't matter huh?

As I read it, the above paper mostly discusses cultural variations in pre-K mathematics exposure.  These don't constitute a "profound impact" of culture on cognition, but rather the not-so-surprising fact that pre-K math preparation affects k+ math preparedness. 

Are you familiar with Dehaene's research?

I am not familiar with Dehaene's research..which article do you recommend?

Are you familiar with LiPing Ma and Deborah Ball?

I'm familiar with LiPing Ma; she's highly critical of current math practices. 

I haven't read Deborah Ball's work, but have just looked at her CV.  She's neither a cognitive scientist nor a mathematician (she publishes almost exclusively in education journals) and therefore I'd be dubious of what she says about math and cognition.   Her ed-school and NCTM affiliations suggest that she's part of the current power structure that supports today's unfortunate Reform Math programs (about which LiPing Ma is rather critical).

Oh, a good place to start with De Haene is his recent book, "The Number Sense"

Hi Becky, As a member of the Council for Exceptional Children and the ACLU let me make a couple of points. First, being gifted is not a disability. Second, identifying kids who have real disabilities and taking steps under the Individuals with Disabilities Education Act is a civil rights issue. It is tempting to take offense at yourimplication, that we don't give "every child the best we have" simply because we've recognized the fact they have autism, or some behavioral disorder, or a learning disability, or fetal alcohol syndrome and we've taken steps to adapted their curriculum and their educational environment to meet their needs.

Let me rephrase your question. You seem to be asking, "What if we repealed the Individuals with Disabilities Education Act and just left it to schools to *do their best* with kids with disabilities?" Answer: Kids with disabilities would face discrimination and their right to be educated with their peers in the least restrictive environment possible would probably deteriorate quickly.

In any event, what your suggesting WOULD require repealing the IDEA...

Just an interesting aside...

'Disability' is a relative (and not very useful term) especially with regards to numerous LDs; unfortunately, that's the language the folks writing the initial legislation decided to go with.

GT itself, at the time of the creation of IDEA, was only not included under its umbrella for a mixture of political reasons combined with the inability to come to a definition suitable for psych diagnosis.

The problem is the variability of presentations and manifestations of giftedness -- from radically high functions in specific areas of MI to intense hypersensitivity and debilitating perfectionism. In other words, in GT you don't have a continuum along which to diagnose severity, but rather a host of semi-related outliers which must be considered individually.

The mistake many schools and school districts get into is trying to use a 'standard' -- which usually amounts to testing and academic performance -- to identify GT students.

Not only is this a disservice to the actual kids who are gifted and regularly go unidentified for academic and behavior reasons, but it creates a false sense of understanding across the school community. It suggests that 'gifted' kids get their work done, do well in school, and behave in society.

The reality couldn't be further from that 'truth'.

Greg, you say that to follow Becky's thinking, "Kids with disabilities would face discrimination and their right to be educated with their peers in the least restrictive environment possible would probably deteriorate quickly." And I agree. This really isn't a matter of "giving each child the best we have to offer"; there has to be science and law backing it up. But, by-and-large, with regard to GT education, we already see the situation you are fretful of on a daily basis in innumerable schools: there is GT dicrimination and those kids do suffer the inability of teachers and admins lacking GT training to provide an adequate learning situation.

How do we go about changing that?

Hi Shelly,

The disabilities law, IDEA, covers a number of categories of disabilities. I have to disagree with you and say that for the most part it is both useful and concrete. If a child is autistic, has a psychosis, is legally blind, has some inconvenient medical condition (for example, epilepsy), has Down Syndrome, or has an IQ of 65 and difficulty coping with their environment - in all of those cases the concept of disability seems both useful and appropriate.

You are correct regarding learning disabilities. IDEA's 2003 reauthorization left the concept of "learning disability" in a state of foggy flux. Without going through the issues in any detail, the old discrepancy model based on comparing IQ and achievement has been put out of its considerable misery and replaced with, well, nothing in particular. We're encouraged (but not required) to use a model called RtI (Response to Intervention) and the education community is feeling its way through just how that should work at the moment. IDEA 2003 basically leaves individual states to define "specific learning disability" as they see fit - and in some cases the states have passed that on to individual school districts. The result is the possibility of literally hundreds slightly different definitions of learning disability - and a situation where a child has a disability in New York (and in some parts of New Jersey) but not in Pennsylvania.

To be fair, there is more smoke (and heat) than light surrounding the scientific research on learning disabilities. Dyslexia, for example, is an ill-defined concept that I don't think is even mentioned in the DSM-IV. In the last few years there've been controversial assertions that some of the genetics underlying dyslexia has been identified. One group promised that in less than a year we'd have a cheek swab that could be used to test newborns for dyslexia; they made that promise in early 2006 and their year is over, but I haven't heard of the swab yet.

The growing field of cognitive neuropsychology is shedding a little light on reading disorders by using MRI results. And it's hard to deny that there's some real neurological disorder behind many reading problems. At the moment, that knowledge doesn't help a school psychologist on the ground here in my part of Central Appalachia to sort out which kids qualify as having that disability and which kids don't.

Do gifted kids face discrimination? I've said already on this page that giftedness is not a disability. The idea of discrimination implies a status, a legal standing that I have no qualms about extending to blind children, or to kids with autism, or to students with measurable and significant cognitive deficits, but that I'm not at all sure should be extended to students who qualify as gifted.

We've used the word "best" a lot. Let's just give kids "the best we have." That's a noble sentiment. And when I enter a classroom my goal is to give the children in it the "best" I have that day - in that room, and in me. That's different than giving a particular child the best possible education. In a world where we all share limited resources and where education is funded by society in general, the courts have repeatedly ruled that no one is entitled to the best possible education - just to an appropriate one, at public expense.

As educators struggle to bring pedagogy and educational environments into the 21st century, students don't have to be gifted in order to be bored. The changes that are needed for gifted students are needed for MOST students. Central themes in that include: technology integration, more rigorous content standards, increased professional development for teachers, and a focus on higher level thinking skills in the curriculum. It's a problem that's faced by most kids, not just gifted ones. And (to answer your closing question) we need to restructure education in general, not just gifted education.

http://educationalissues.suite101.com/article.cfm/digital_natives_and_digital_immigrants

Re 2+2, please provide the names of the mathematicians who would not simply answer "4".  I know many mathematicians, and I'm curious whether the ones you say would give a "range of answers, questions, and demands for clarification" are people that I know.  I can absolutely guarantee that MANY of the mathematicians I know would simply answer 4; but I have not surveyed all of them.

Katharine, you keep asking. Each time you've asked I've not just reached for my bookshelf but put "culture and mathematics" into Google Scholar, pulling up tens of thousands of articles on this issue - most by, yes mathematicians.

It is nothing new. I have a friend who wrote his 1962 Masters Thesis in Math on this subject, and I know it is spoken of constantly in our math education program (which is tied to a 'fairly reputable' mathematics department).

I'd again encourage you to read Morris Kline's work - going back to the early 50s - or to look at the "NYU School" exploring the "fictions of mathematics." I think you will enjoy the conversations.

Ah, but have you searched "culture and mathematics" & "2+2=4"?  That's specifically what I was asking about...

Also, there may be a bit of a selection bias in the theses of the articles that turn up under your search...  I know many mathematicians (algebraists, analysts, number theorists), but I'm guessing that not a single one who would say that mathematics is, in any way that is deeply significant to math itself, a cultural construct.

Katharine,

PJ Davis in Mathematics (1988) provides a primer for you http://www.people.ex.ac.uk/PErnest/pome22/Davis%20%20Applied%20Mathematics%20as%20Social%20.doc

"Take any statement of mathematics such as ‘two plus two equals four', or any more advanced statement. The common view is that such a statement is perfect in its precision and in its truth, is absolute in its objectivity, is universally interpretable, is eternally valid and expresses something that must be true in this world and in all possible worlds. What is mathematical is certain. This view, as it relates, for example, to the history of art and the utilization of mathematical perspective has been expressed by Sir Kenneth Clark ("Landscape into Art"): "The Florentines demanded more than an empirical or intuitive rendering of space. They demanded that art should be concerned with certezza, not with opinioni. Certezza can be established by mathematics".

"The view that mathematics represents a timeless ideal of absolute truth and objectivity and is even of nearly divine origin is often called Platonist. It conflicts with the obvious fact that we humans have invented or discovered mathematics, that we have installed mathematics in a variety of places both in the arrangements of our daily lives and in our attempts to understand the physical world. In most cases, we can point to the individuals who did the inventing or made the discovery or the installation, citing names and dates. Platonism conflicts with the fact that mathematical applications are often conventional in the sense that mathematizations other than the ones installed are quite feasible (e.g., the decimal system). The applications are of ten gratuitous, in the sense that humans can and have lived out their lives without them (e.g., insurance or gambling schemes). They are provisional in the sense that alternative schemes are often installed which are claimed to do a better job. (Examples range all the way from tax legislation to Newtonian mechanics.) Opposed to the Platonic view is the view that a mathematical experience combines the external world with our interpretation of it, via the particular structure of our brains and senses, and through our interaction with one another as communicating, reasoning beings organized into social groups.

"The perception of mathematics as quasi-divine prevents us from seeing that we are surrounded by mathematics because we have extracted it out of unintellectualized space, quantity, pattern, arrangement, sequential order, change, and that as a consequence, mathematics has become a major modality by which we express our ideas about these matters. The conflicting views, as to whether mathematics exists independently of humans or whether it is a human phenomenon, and the emphasis that tradition has placed on the former view, leads us to shy away from studying the processes of mathematization, to shy away from asking embarrassing questions about this process: how do we install the mathematizations, why do we install them, what are they doing for us or to us, do we need them, do we want them, on what basis do we justify them. But the discussion of such questions is becoming increasingly important as the mathematical vision transforms our world, often in unforeseen ways, as it both sustains and binds us in its steady and unconscious operation. Mathematics creates a reality that characterize our age."

Other "open" readings on the same issues

http://www.economics.pomona.edu/widner/courses/econ58/ps/whatmath.pdf

http://markelikalderon.com/wp-content/uploads/2006/12/EpistemicRelativism.pdf

http://www.members.tripod.com/~jan_dejnozka/peano_russell_quine_number.pdf

And I'd urge you to go the the library, and if you are resisting Kline, read Reuben Hersh's "What is Mathematics, Really?" (1997)

But he doesn't actually deny that 2+2=4.  Find me a math professor (a mathemtician, not a philosopher!) who says, specifically, that 2+2 doesn't equal 4 in all possible universes, and I'll eat my hat.

I'm a big fan of foundational problems (computation theory; model theory; incompleteness the unprovability of certain mathematical statements--have you taken courses in any of these?) And I'm all for openness, as long as its meaningful!

Kline and Hersh both say this. Read, it'll give you something to chew on.

and

"Believe it or not, sometimes 2 + 2 does not equal 4. It depends on what type of measurement scale you are using. There are four types of measurement scales - nominal, ordinal, interval, and ratio. Only in the last two categories does 2 + 2 = 4."

Thus, even on number lines, 2+2=4 is only sometimes true. You need a formal set of, yes, culturally applied, rules to make 2+2=4.

I understand the desire to have something absolute and "always true" in the world. And I am sure it is stunningly frustrating for a pure rationalist to argue with a post-modernist like me, but I challenge you to dig deeper into this.

As Bain's book demonstrates, the shattering of the knowledge system created by K-12 math and science teachers is the first task of professors at top universities.

apples and oranges!  "nominal scales" use a different definition of number from that used in the statement "2 + 2 = 4"

From the paragraph that follows the one you cite from:

"Each number merely represents a category or individual. For example, numbers on baseball or football uniforms are only nominal. Having the number "1" on your uniform does not necessarily mean you are "numero uno" (the best) in your sport. Social security numbers are also nominal. All they do is name or classify the individual."

In making statements like 2+2=4, people are not referring to numbers on uniforms.  If they were, then the statement would be no more meaningful than "too plus too equals for"!

Speaking of postmodermism and math, have you read Sokol's "Fasionable Nonsense"?

Wait! A student would have to know your definition of "number" in order to answer that question? I thought this was "absolute" and "culture free"?

But no, I don't read intentionally fraudulent academic writing. I have other things to do with my time.

I've taught math for a number of years, to a number of different students from different cultural backgrounsd, and have *never* encountered a student who needed to know my definition of number (i.e., that I wasn't referring to numbers on uniforms) in order to answer 2 + 2 = 4.

Have you?

Ira says that 2+2 doesn't always have to equal four: it's a qualified truth.

Katherine wants Ira to find a math professor who will stipulate to the idea that 2+2 does not equal 4 (which doesn't strike me as quite what Ira said). And Katherine says that if Ira finds such a math professor, it proves only that the math professor is a closet philosopher. If I were Ira, I wouldn't spend much time looking... 

Without investing the time to read Ira's citations (I have a life), I thought he was pretty convincing. And I thought Katherine was circular: all mathematicians agree with her; if they don't, they're not pure mathematicians.

I think we've all agreed that education (its value in a society, its pedagogy, etc.) is culturally defined. The math discussion has been entertaining; I just can't decide if it has a point in the context of Jon's article.

I will say this: the ability of pure mathematicians to articulate great truths abstractly (which I take to mean hypothetically, in the absence of any real context) is something that I see as a cultural exercise in itself. Most great truths can be articulated concretely or abstractly. You can talk about materialism and the nature of reality or you can talk about Plato's Cave. You can talk about Grace or about the Prodigal Son. Some cultures prefer the concrete presentation. Most Western European cultures prefer abstraction. And that preference is in itself cultural. Even for mathematicians, I think...

To clarify, what Ira said was:

"If you asked mathematicians what 2+2 is, you would get a range of answers, questions, and demands for more clarification. It's hardly cut and dry. I can absolutely guarantee that NO mathematician would answer "4" without qualifying the answer with additional information."

I then said:

"Find me a math professor (a mathemtician, not a philosopher!) who says, specifically, that 2+2 doesn't equal 4 in all possible universes, and I'll eat my hat."

Note that Ira is making a claim about what "NO mathematician would do;" I'm asking him to find ANY mathematician who believes that 2+2 doesn't equal 4 in ANY possible universe. 

You're also reading too much into what I said about philosopohers.  There are some really good mathematician/philosophers out there.

Fair enough.

So you'd take an answer from a mathametician-philosopher?

Yes.  The question, again, is whether the statement "2+2=4", when it is a mathematically well-defined rather than a mathematically nonsensical statement (which was implicit in my initial question but which I now feel the need to state explicitely!), is true in all possible universes.

I thought the question had something to do with culture and education...

I've said that this discussion was entertaining. I don't really have a problem with the idea that 2+2=4 (or that if I give Sue 2 counting bear and I give Bob 2 counting bears and I ask them to tell me in their own little kindergarten words how many counting bears they have together, they should say "four"). I'm not sure how that impacts my understanding of Jon's article.

I've reread Sylvia's comment at the top of this subthread from August 2. I take her to be saying that the emphasis we place on memorizing math facts (2+2=4, 6x5=30, etc.) is cultural. And that it is problematic because simple memorization, which for so long has been the measure of academic success, is no longer a sufficient skill in isolation. The value we place on it has changed.

Students need to be required to manipulate those math facts within some context. If I've been in my classroom four years and every year they bring a new computer, someone might expect me to have four computers now; but they have to understand the idea that in the third year they began taking the oldest one with them when they left, so that no matter how many years in a row they add a computer, I'll never have more than three. That's context. Sylvia's example is perfect. The track may be a mile around, but walking around it twice and then walking around it two more times just gets you back to where you parked; it doesn't leave you four miles away (even if an abstract 2 plus an abstract 2 is four is "all possible universes").

And Sylvia is absolutely right about testing compliance. If a kid puts five on the test and has a brilliant explanation (one that shows that she can manipulate the numbers just fine) about how she was imagining cans of juice and her friend always comes over unexpectedly and she has to get an exit glass and divide the four cans of juice up five ways - well, the State education people couldn't give a rip and the answer is still wrong and I have to explain to the kid that she should have put "four" because that's what the people in Charleston expect her to put. Tests have their limitations. We only give the tests for cultural reasons - to measure something we value.

Sylvia is guilty of hyperbole. But memorization of isolated abstract facts is being downgraded in state content standards all over America. Not many people care about other possible universes. And none of my students at the elementary school where I teach at the moment are mathematicians. I don't even have one whose parents are mathematicians. But a couple of my school's kids have been identified as "gifted."

As for the argument between Katherine and Ira regarding the predictability of the behavior of mathematicians (how they'd answer a particular question) - I suspect the answer you'd get would depend on how the question was phrased. And the line of argument seems to have become an exercise in refining the question. And now, in order to be sure of my own answer I have to go study other universes. But since I'm not a mathematician, I answer probably doesn't matter...

It's hard to keep track of these moving goal posts! But, for an excellent conceptual math curriculum, consider Singapore Math.

I spent two years in a flat on Balestier Rd. Singapore's education could be a model in almost any area...

Whatever you're suggesting now, your original suggestion was that we stop "sorting, selecting, and labeling kids." That would require a change in law, and that idea has significant implications for how federal resources are distributed to fund meeting those specific needs you're taling about.

That said, I whole-heartedly agree that teachers need to know the individual student and address the individual needs of their students (whether they have a disability or not). For children with disabilities, the process of identifying their disability and acknowledging its existence SHOULD facilitate this process, and only becomes a hindance when staff at a school don't *get* that...

Abstraction (in particular, from measurement systems, everyday use of language, everyday use of math...) does not imply "objectivity", as you put it, but it does imply abstraction from everyday culture. 

I've always thought that the notion that "Westerners" are more into abstraction than non-Westerners is a rather Western-centric idea, particularly since most of the people who push it seem to be Westerners.  Do you know any ethnographies conducted by non-Westerners that reach such conclusions?

I'm aware of my own ethnocentricity. That said, I was speaking mostly from personal experience, having lived in the Caroline Islands, The Mariana Islands, and a sprinkling of SE Asian's locations.

I don't think abstract thinking and concrete thinking are by any means mutually exclusive. But I do think some cultures value abstration more than others. You seem to have just agreed with me on that by suggesting that the idea was Western-centric. But then, I read too much into your words...

I don't really feel the need to produce an cluster of ethnographies to sustain the idea that some cultures value abstraction more than others. But I'll have to keep my eyes open now for support for that idea.

It's specifically the(potentially patronizing) Western/nonWestern dichotomy I'm wondering about; not the much more likely possibility that some cultures are more into abstraction than others.  (For example, it seems likely that to me that the subculture of U.S. academics is more into abstraction than other U.S. subcultures).

Even if we have lived in other cultures, it's hard to know how abstract the thinking is unless we've mastered the native language.

Without having surveyed much of the literature, my [ersonal impression is that it's more of a rural-urban thing these days.