Every time Benezet comes up, or Davydov, I think we should also mention Caleb Gattegno, founder of the equivalent of the NCTM in the UK, and an exponent of using Cuisenaire Rods to teach the four operations in algebraic expressions, including fractional notation.
His reliance on color coding was extreme and would put color-blind students at a disadvantage. Studies were done, videos made, and to this day his memes propagate and re-inspire, much as Benezet's do -- but lets be clear they're not advocating the same thing.
[ I think Benezet's + Haim's solution would be a fun experiment: no math up to 6th grade and then no math higher than 6th grade level i.e. get all the arithmetic you need in one year and you're off the hook -- opt in if you want more but it's your choice, not Nanny State's. ]
I like how they do it in PER (physics education research) a little better: they give a pre-test and a post-test, with the experimental and conventional programs in between, both designed to impart X (e.g. Newtonian mechanics say). Who scores higher after six months or a year or even just a week of learning regimen Y?
Six year longitudinal studies are great, but those who don't believe in your approach will always explain away the data.
In Benezet's case, we say "they learned at home" -- a hole they can't plug. Significance drains away. So was it worth six years to (a) preach to the choir and (b) recruit zero defectors from the opposition?
A better approach is to have experiments that are easier to implement (six days vs. six years). Give me six weeks with 8th graders to both teach basic Python and apply that to rational numbers and operations with rational numbers (Q), versus another teacher with six weeks and only scientific calculators for technology. Give a pre and post assessment tests.
In my class, we "construct" (note "constructionism" -- a spin-off of "constructivism") a rational number as a Python type:
class Q: * initialize / reduce to lowest terms * add * multiply * add inverse (subtract) * multiply by inverse (divide) * return reciprocal * return negative * exponentiate
Each of the above starred items corresponds to a Q class method, with "reduce to lowest terms" relying on Euclid's Method which must be clearly explained, preferably by means of cartoons involving architecture / construction (bricklayers need lengths to come out even i.e. with no remainder). [math-teach 1]
We're clearly setting up for abstract algebra latter because even when the type changes from Q (rational number) to P (permutation) we'll still have:
class P: * initialize * multiply * multiply by inverse (divide) * return multiplicative inverse * exponentiate
A permutation would be like a bijective function of A to itself, where A is the set of alphanumeric characters, or just the 26 English lowercase if applicable / relevant to our students' other work. A Python dictionary serves beautifully to hold the permutation data while the surrounding type apparatus allows them to be multiplied and inverted.[recap on edu-sig 2]
We may also look at class M, M for Modulo. These would be integers that do everything modulo N.
After Python, we probably transition them to Mathematica and reinforce a lot of the same concepts using more traditional notation and piling on the nuances. In 10th grade we do a lot more with spatial geometry with calculus as neither the apex nor principal be-all-end-all goal of a strong STEM education at this level. Analog math is still important, but not the only kid on the block. You still should know it, its notation. Get to it through Big-O notation when expression algorithms' powers.[Knuth 4]
When you get to the college level, that's when to decide how much more calculus to take on. If you're going into a discrete math or statistical subject, or one that needs trig, you may go with a different sequence than the physics major does (to the extent we still call them "physics majors" in this future). In high school, we have a stronger STEM footprint than ever, but not so weighted towards calculus. Those that do calculus are probably also using Mathematica.
Some of our students dive into web development right out of high school and enjoy the rest of their educations as paid professional development time. They might take more economics and get more math through that window (except in this future we've replaced "economics" with "general systems theory" which is more robust and and pays more attention to the biosphere aka "motherboard Earth"). Use of simulations is key (ala SimCity / SimEarth), as well as programming simulations. Cellular automata were a core topic in high school, making use of those new programming skills. The PSU Systems Science degree program is pioneering this curriculum today, with schools like LEP allowing students to access college content from within a high school matrix.
Regarding Benezet, I think there's an argument for not starting on Python until 6th grade. They need some typing skills first, which most get from communicating on the Internet (with keyboards at home, not just thumbs on a smartphone). They will learn with more mental drills and abacus before that, with such New Math topics as union and intersection but previewing SQL.
Instead of dreary hands-on work with databases (dreary because typing is still a chore), they'll watch exciting documentaries in which real people both do and explain their jobs.
School is really a time to day-dream about the personality you'll be in your next adult incarnation (even if you're already an adult).
"Career orientation" should not just be left to once- a-year assemblies with booths in the gym. You need those serious documentaries right during school hours, when you're on the meter, earning credits (exchangeable for real goods those credits though not "same as cash" given how their not easily transferrable).
If you have this hole in your own knowledge, consider yourself shot through the head by your own alma mater and seek medical / remedial attention immediately. Not knowing Euclid's Algorithm means you should think twice about voting or registering an opinion on any matter. Don't get up in the morning. Get out your smartphone and learn it instead.
That's what the Math Wars are all about. Those without any knowledge of Euclid's Method by the end of high school are the losers, and that goes for any professors in the same boat. Lets help these poor vanquished back on their feet again shall we? I urge empathy towards the world's least privileged. Lets start a new charity if need be.
On Wed, Sep 25, 2013 at 6:57 PM, GS Chandy <email@example.com> wrote:
> Jonathan Crabtree posted Sep 25, 2013 8:06 AM ( > http://mathforum.org/kb/message.jspa?messageID=9282611): > > > > a) They are lazy. > > b) It's their job to teach existing courses, not > > experiment with new courses. > > C) They don't know about Benezet > > d) They don't care about Benezet. > > e) No retraining has been mandated. > > f) Parents would complain if their child was not > > taught math at the same age they were. > > g) Parents would arrange tuition and undermine the > > principles of Benezet. > > h) The Russian (Davydov) method is better. > > > > Repeat question with Davydov instead of Benezet. > > > > Repeat responses a) to g). > > > > Strike out a) as teachers are usually overworked and > > underpaid. > > > I don't know enough about either Benezet's work or Davydov's to comment > about their breakthroughs. In India, reason 'a' is not on - as you've > noted, teachers are usually overworked and underpaid. > > However, the 'general culture' IS in fact to be intellectually lazy (AND > to punish intellectual 'exploraton'). > > Witness for instance (as instances of intellectual laziness): > > - -- i) Professor Wayne Bishop's panacea to the ills of the educational > system ("BLOW UP THE SCHOOLS OF EDUCATION!"); or > > - -- ii) Haim's ("PUT THE EDUCATION MAFIA IN JAIL!") or > > - -- iii) Robert Hansen's ("Children must be PUSHED to learn math!"). > > On the other hand, I suspect that this intellectual laziness IS in fact > pretty widespread. As Mr Crabtree has indicated, at his reason 'b': > > - -- "b) It's their (teacher's) job to teach existing courses, not > experiment with new courses." > > - -- "C) They don't know about Benezet". > > Many individual teachers do indeed try to think beyond this business of > "teaching existing courses" and do try (and sometimes succeed) to make > their teaching interesting to students and useful, they do their own > research and find out about say, Benezet, Davydof (and, of course, > Montessori to begin with). But none of that can compensate for 'system > inadequacies' [which enable the sloganeering noted at i), ii, and iii)to > prosper]. > > The 'system itself' of course does not effectively allow for 'retraining' > as Crabtree has observed at: > > - -- "e) No retraining has been mandated." > > Often the system would actually punish teachers who try to go beyond the > 'permitted' means and methods. > > Of course, all of the above are various aspects of serious 'system issues'. > > By and large, the teachers often do not themselves have a 'culture of > learning'. Observe various other valid reasons listed by Crabtree: > > - -- "f) Parents would complain if their child was not taught math at the > same age they were." > > - -- "g) Parents would arrange tuition and undermine the principles of > Benezet" > > - -- "h) The Russian (Davydov) method is better". > > All of the above indicate the need for widespread 'system' reform. > > GSC > >