Saturday, March 1, 2008
1. Analyse the following, and parse words in italics:
I cannot tell if to depart in silence,
Or bitterly to speak in your reproof,
Best fitteth my degree or your condition.
2. Write the following in prose, and parse the verb awaits:
The boast of heraldry, the pomp of power,
And all that beauty, all that wealth e'er gave,
Awaits alike th' inevitable hour:
The paths of glory lead but to the grave.
3. Give a brief example of a compound and a complex sentence. Give the rule for the use of the subjunctive mood.
4. Define and give etymology of verb, pronoun, conjunction and adverb. Give example of a defective, an auxiliary, an impersonal and a redundant verb. How many kinds of pronouns are there? Give example of each.
5. Prior has the following sentence. State if it be good grammar. If not, why? If it be, parse the word than:
Thou art a girl as much brighter than her,
As he is a poet sublimer than me.
6. Give rule for forming plural of nouns ending in "y," with examples. Give plurals
of staff, radius, miasma, Miss White, rendezvous, talisman, loaf, grief, seraph, Mussulman, forceps, spoonful, who, beef, s, x, 6, and madam. Also singulars of kine, ashes, banditi [sic], swine, animalcula.
7. Compare chief, much, former, far, forth, next, round, up, ill, full.
8. Give the feminines of abbot, earl, duke, lad, marquis, hero, tiger, nephew, testator, bachelor, wizard and ox.
9. Write the past tense and past participle of these verbs.
Lay, Seek, Sit, Get, Dare,
Thrive, Lie, Set, Light, Loose,
Fly, Flee, Chide, Overflow, Catch,
Lose, Swim, Climb, Drink, Slay,
Leap, Quit, Swell, Burst, Eat.
10. Define metonymy, catachresis, and hyperbole; and state difference between a
metaphor and a simile.
11. Punctuate the following lines:
But when I ask the trembling question
Will you be mine my dearest Miss
Then may there be no hesitation
But say distinctly Yes Sir yes
12. Parse the three "thats" in the following sentence:
He that fears that dog thinks that he is mad.
Also parse the word "but" in each of the following:
There was no one but saw him;
We ran, but he stopped;
All fled but Peter;
If you did but know it.
13. Correct the following:
(a) Although I persuaded the old man, he refused to yield, and I expect he divided his estate between his 3 daughters. His example, though he meant well, is calculated to have a bad effect.
(b) As I laid down I seen the smoke raising over the way.
(c) Whom do you say that I am? or who do you take me to be?
(d) John and James were both there, though neither were invited.
(e) As water is froze easier than alcohol, so riches are easier acquired than a good name.
(f) Between you and I, there is some mystery about that fire last night. Did you hear where it was at? I am glad none of my friends were in the house. I should be sorry if either James or William were inculpated in setting it on fire.
(Letter 194, p. d May 23, 1877)
Prepared by Jos: Crosby, Examiner, for June 2/77
1. Give etymology of orthography. What are mutes, labials, and liquids, and why so called?
2. Give meaning of the prefixes, ante, anti, circum, quad, proto, oct, trans, sym, and con.
3. Form derivatives of prefer, begin, stop, run, defy, abridge, tie, and die, with the suffix ing or ed.
4. Write a word containing a diphthong, one containing a digraph, and one containing a trigraph.
5. Define accent, and mark the accent on the words inverse, diverse, adverse, reverse, obverse, calcine, piquant, orthoepy, abdomen, acclimated, area, salutatory, accessary, gondola, illustrate, prolix, portent, inquiry, contemplated, expert, extant.
6. Spell the words (given orally)
(Letter 194, p. d May 23, 1877)
Prepared by Jos: Crosby, Examiner, for May 26/77
Put all your work on the paper and make it explain itself.
1. Define integer, fraction, interest, discount, power, and root.
2. What effect has multiplying both terms of a fraction by the same number, and why; and why in dividing one fraction by another do you invert the divisor and multiply the terms together?
3. If A's age were increased by its 3/7 its 4/5 and 19, the sum would equal 2 1/2 times his age; required his age. [sic]
4. Multiply 718 by .000018 and divide the product by 27 millionths.
5. 32 men agree to construct 28 miles 4 furlongs and 32 rods of road; after completing one-half of it, one-fourth of the number of men left the company, what distance did each man construct before and after one-fourth of the men left.
6. A man drives 97 pegs on a straight line and spaces them 3 ft. 8 in. apart. What is the distance from the first to the last peg, lowest terms?
7. A man receives $65 interest for the use of $600 for 3 years, 7 months and 15 days. What is the rate per cent.?
8. What is due on the following note?
Zanesville, O., Dec. 10, 1871.
One year after date I promise to pay to the order of Richard Roe twelve
hundred dollars, value received.
9. Give the rule for obtaining the difference of time, having the difference of longitude, and vice versa, and give the reasons for the rule.
10. A square lot containing 54,756 square feet is surrounded by a close board fence 12 feet high. What would the boards cost at $13 per thousand?
(Letter 195, p. d May 27, 1877)
Prepared by Jos: Crosby, Examiner, for May 26/77
1. Where does the earth have the greatest diameter?
2. Why do we reckon 180 degrees of longitude and only 90 of latitude?
3. What is meant by the equinoxes?
4. Locate the Crimea, Bombay, Bay of Fundy and the Capital of Mississippi.
5. lnto what three functions is the government of the United States divided?- define each function.
6. Describe the lndus and Niger rivers.
7. Through what waters would a ship pass in going from Duluth to Odessa?
8. Bound France and give five of its chief cities.
9. Name the New England States and locate their capitals.
10. Define equator, zone, latitude and longitude.
11. lnto what bodies of water do the following rivers flow: The Danube, Rhone, Volga, Tiber, Rio Grande, Jordan and Mahoning.
(Letter 195, p. d May 27, 1877)
Sections of a Certification Examination, 1877. Note the civics question ( # 5 ) in a geography examination. The hlahoning River is in northeastern Ohio. Crosby appears to have re-used questions: see "swine" (grammar question #6) and "Lie" (gramnlar question #9) and compare Crosby's comments about the examination given in the summer of 1876 (Letter 133).
Certification in "The Basics" One Hundred Years Ago
John W. Velz
The English Journal, Vol. 66, No. 7. (Oct., 1977), pp. 32-38.
these pages: 36-38
Stable URL: http://links.jstor.org/sici?sici=0013-8274%28197710%2966%3A7%3C32%3ACI%22BOH%3E2.0.CO%3B2-C
Friday, February 29, 2008
Most people can't do mental arithmetic any more. What a pity indeed :-(
Almost everyone relies on calculators these days. So today I'm going to tell you how to do an apparently difficult feat of mental arithmetic. Today you will learn how to take integer Cube Roots in your head! It's really dead easy, believe me!
Go to this blog post for the technique
Do teachers need to utilize more culturally responsive pedagogy? (Banks et al) Do kids and families of group X need to start acting more like the kids and families from group Y, and like, y'know, get their act together? (Cosby & the staff lounge) Do we put our efforts into wide-scale social transformation, because schools are not powerful enough to overcome such a pervasive inequity? (Rothstein) Do we stop talking about poverty, because it's not about poverty, but about innate factors out of our control? (silly race-based IQ-gap people)
Or do we say, here is School A which has the same demographics as School B, but kids at School A learn and those at School B do not -- just why is that exactly? (Folks who scribble the word educator in front of every published utterance of the phrase "achievement gap.")
That's my thing. Don't engage an endless debate that may or may not get us closer to bringing a better education to more kids.
For those of you who don't read his blog, Teaching in the 408, TMAO teaches in a low-SES, high-percentage-of-English-Language-Learners (ELLs) district in the Silicon Valley.
We must reject the ideology of the "achievement gap" that absolves adults of their responsibility and implies student culpability in continued under-performance. The student achievement gap is merely the effect of a much larger and more debilitating chasm: The Educator Achievement Gap. We must erase the distance between the type of teachers we are, and the type of teachers they need us to be.
The Well-Trained Mind has an excellent review of several math programs, here's a short excerpt:
Which method is better? In my opinion, the one which the student understands most clearly. In both cases, it is possible for the student to learn the mental trick without thoroughly understanding why it works, although the sheer amount of repetition in the Saxon method makes it easier for this lack of understanding to escape detection. But the strongest mathematical training of all would come from a combination of programs – in which the student is taught to do a mathematical process using several different methods and mental procedures.
Currently, Singapore and Math-U-See are “thought-oriented” math programs available to home school parents; Saxon and A Beka are “skill-oriented” programs. A combination of Saxon + Singapore, or Saxon + Math-U-See, or Singapore + A Beka, or A Beka + Math-U-See, may come closest to fulfilling the goals of classical education. Math-U-See + Singapore would also be an excellent combination, as long as you use MUS’s supplementary drill sheets. Treat one program as primary and the other as secondary; when you cover a concept in the primary program, look it up in the secondary program and see whether it is explained and illustrated differently.
I like her idea of using 2 programs. I do that all the time with phonics--I know what is best about each program and pick and choose accordingly, I have dozens of phonics programs from my tutoring. Also, if a student is struggling with a certain area, it's good to have a few books that explain it in slightly different ways. I hadn't thought to do the same with math.
There is also an interesting review at Sonlight, a basic review of their math choices from grade school through calculus, and then, they explain their choice for algebra and geometry--teaching textbooks:
- CD-ROM-based "whiteboard" lectures and step-by-step explanation of how to solve every practice problem taught by a tutor who has been teaching homeschoolers for years...since the days he tutored probability and statistics at Harvard . . . and . . .
- CD-ROM-based "whiteboard" solutions guide that works every step in every homework problem, so you see exactly how to solve each problem . . . and why you want to use the methods the instructor (or, more accurately, personal tutor) uses.
We're currently using Math-U-See with our daughter for Kindergarten math. I personally like Singapore or Saxon better, but this is what is working for her. The good thing about Math-U-See is that they have a DVD with each lesson so that if you get that "blank stare," (yes, it is even possible with kindergarten level math!), you can just plop in the DVD and have an actual math teacher explain it in field tested verbage.
Math-U-See's approach to fractions is also interesting, I'm not sure what I think about it. The rectangles do make more sense to me than the traditional circle/pie method. From their downloads page, in sample lesson pages, click on Epsilon.
For an even more interesting and thought-provoking review, read Wild About Math!'s Calculus in 4th grade? I'm really not sure what I think about that one. Luckily, we're still doing kindergarten math so I have time to figure it all out.
Things are clearly quite different than when I attended Catholic school. For starters, when I was in school, almost every teacher was a nun. You just don't see that anymore.
The principal of the school I just visited seemed pleased that the diocese had recently (in the past couple of years) required the school to amend their curriculum to conform with our state standards. Prior to that, Catholic schools were more concerned with meeting national expectations of learning for all Catholic schools. I don't see this change as a positive, however. Our state standards are woefully low (comparing results of the CMT to the NAEP makes clear just how low they are). Projecting forward and over the long haul, this is really not a good thing for Catholic schools.
The other red flag was the pride in their Lucy Calkins style "writer's workshop" approach to teaching writing. They do teach grammar in every grade K-8 and offer Honors English for students who are capable.
The math, on the other hand, was still much more traditional than anything in our public school district. It's neither Singapore nor Saxon, but it's not Everyday Math or Investigations either. They have honors algebra available in eighth grade. They have a significant percentage of the graduating eighth graders place out of algebra at the high school level. That's definitely something.
The other silver lining: I did observe the order, respect for others, respect for self, discipline and self-assured, smiling faces I remember from my own Catholic school days. I contrast this with the disorder, disrespect, and stress I recently observed at a visit to a magnet school in our county. I believe that children thrive in a structured environment because it feels safe and predictable. I still believe this learning environment/school culture element is an important aspect of most Catholic schools today.
I also think that that this type of school culture is what makes schools like KIPP, Green Dot, Stuyvesant, and others so successful.
Thursday, February 28, 2008
Great, committed teacher/adviser/mentors, high standards, a focused curriculum, a culture of achievement, and plenty of hard work by students well aware that real consequences attach to their performance--what more does a successful school need? Yes, I'm talking about the Knowledge Is Power Program and Amistad, the Academy of the Pacific Rim and Stuyvesant, and others of today's super-schools. But I'm also talking about the Catholic schools of the 1960s and my own time at Phillips Exeter Academy, where by senior year I was awakening at 3 a.m. to study. It paid off, for me and lots of others. (I was able to skip my freshman year at Harvard.) But it was sink or swim--and those who treaded water were sometimes invited not to return for the next semester.
Are the Catholic schools of the 60s definitely gone?
Do we know?
We were in high spirits. The physicist laughed as he told his story. "Everybody had agreed on the proposed plan. The mayor had the support of both the citizens and the city council. Because the volume of traffic downtown and the resultant noise and air pollution had become intolerable, the speed limit was lowered to twenty miles per hour and concrete "speed bumps" were installed to prevent cars from exceeding it.
"But the results were hardly what the planners anticipated. The lower speeds forced cars to travel in second rather than third gear, so they were noisier and produced more exhaust. Shopping trips that used to take only twenty minutes now took thirty, so the number of cars in the downtown area at any given time increased markedly. A disaster? No--shopping downtown became so nerve-racking that fewer and fewer people went there. So the desired result was achieved after all? Not really, for even though the volume of traffic gradually went back to its original level, the noise and air pollution remained significant. To make matters worse, during the period of increased traffic, word had gotten around that once-a-week shopping expeditions to a nearby mall on the outskirts of a neighboring town were practical and saved time. More and more people started shopping that way. To the distress of the mayor, downtown businesses that had been flourishing now teetered on the verge of bankruptcy. Tax revenues sank drastically. The master plan turned out to be a major blunder, the consequences of which will burden this community for a long time to come."
The fate of this environment-conscious town demonstrates how human planning and decision-making processes can go awry if we do not pay enough attention to possible side effects and long-term repercussions, if we apply corrective measures too aggressively or too timidly, or if we ignore premises we should have considered.
The Logic of Failure: Recognizing and Avoiding Error in Complex Situations
by Dietrich Dorner
For some odd reason I'm tickled by the idea that it is apparently not possible to sidestep the conundrum of unintended consequences by simply doing nothing, or by doing just a tiny bit of something.
Too much of nothing is just as bad as too much of something.
There's no sure thing.
image from reflective design
"The Education industry is burgeoning," says Deven Klein, vice president of Kumon franchising. "Tutoring is a widespread option for an increasing number of families in the United States."
Gee, lucky for us that we have this option.
Over the past five years, the average Kumon Center enrollment in the United States increased by 55 percent, and overall, the company has 80,000 more U.S. students than in 2002.
Kumon opened 84 new US centers last year, and expects to open 120 in 2008. Business is booming.
I am paying a bit more than $1.20/minute for one of my graduate school classes. How do I know this? I divided the tuition cost for the class by the number of minutes the class is nominally in session.To date, on the $/minute metric, I've spent $864. Personal valuation of information received, to date? $125, max.
Well, I've made some friends and/or future colleagues. That's worth something.
On the other hand, it has really sharpened how I think about the airtime I take up in class. Before I open my mouth (which usually lasts at least a minute), I think, "Is this worth $1.20 of my classmates' money?"
On the gripping hand, there's the Ditz Boredom Index. My urge to talk is directly related to how dissatisfied I am with the rate of [information/data] transfer. I realize I am hogging the airtime....
Grumpity grump. But you know, in every training program there are the slog-through bits.
Hmmn, maybe I should write the syllabus I wish I the class had offered, including a more meaningful reading list.
Wednesday, February 27, 2008
works if life is about submitting
to silly rules handed down from on high.
& g-d knows: that's what school is *for*.
but i want no part of it myself;
do what thou wilt shall be
the whole of the law.
if they give you ruled paper,
write the other way. etc.
yours in the struggle.
Tuesday, February 26, 2008
1) I teach middle school, and it's just scary how much of success at this level is linked to organizational abilities and work habits, not intelligence. I've got crazy-smart kids who do terribly because they do the homework and can't find it, or (more likely) never did the homework because they don't have some consistent way of writing it down, or couldn't do it correctly because they didn't have the notes, or, or, or...All organization. I don't believe in notebook grades, but that does put me in a bit of a bind...
2) Joan Sedita's book on study skills is the bible. I don't remember if it touches directly on organization, but it does a great job identifying and breaking down study skills things that students may be having trouble with, and its general approach is illuminating.
3) As a basic template for organizing notebooks, you can seldom go wrong with this:
Get a three-ring binder with as much looseleaf paper as will fit. (Notebooks where you tear out paper are the devil; paper, once torn out, cannot be put back in any helpful way.) If at all possible, get one which doesn't have folders in the front or back, because then all the paper will end up *in* the folders, getting dogeared and disorganized.
Make four sections: notes; homework; tests and quizzes; handouts. Or maybe five: blank paper.
Everything in those sections is to be chronological. It doesn't matter if the most recent is first or last, as long as it's consistent. This gives you an easy organizing principle: whatever you are working on should be placed directly after the last page with writing on it (or as the very first page of the section, depending on whether you prefer chronological or reverse). Actually, having all the blank paper segregated in a blank paper section will probably benefit the very disorganized (who tend to have random quantities of blank paper interrupting stuff).
HAVE A HOLE PUNCH which lives in the binder. (You can get little thin three-ring punches which have holes so they can live in three-ring binders.) If the teacher gives you anything unpunched (which I think is unacceptable but no one made me god), hole-punch it immediately so you can put it in the appropriate section.
The virtue of the notes/homework/tests and quizzes/handout layout is that pretty much every paper you will ever get in class goes in one of those, and it's generally obvious which. There are a couple of odd cases like syllabi, but they are rare.
As long as we're on the topic of notebooks, rereading your notes within a day of taking them is also a great habit to be in -- it's spooky how much that aids retention (and especially the retention gained/time spent ratio).
THANK YOU, Andromeda!!!!!!!
(And tell us more!!)
At least, I think I saw something about an identified need on a slide in the Middle School
Then today my sister explained their high school's block scheduling.
So naturally I've been on a Google mission from God looking for the good stuff.
Didn't take long:
Effective Instructional Strategies for Block SchedulingIn 2000 Jenny Burrell, Stephanie McManus, and I identified and reviewed several instructional strategies that are suitable for blocked classes.48 However, we admit that, just as a 90-minute lecture is inappropriate, a 90-minute discussion session is probably too long too. We found that teachers should change activities every 10 or 15 minutes. blah blah blah
Cooperative learning. blah blah blah
Case method. blah blah blah
Socratic seminar. blah blah blah
Synectics. In the early 1960s, William Gordon developed an approach that, through the use of analogy, enabled students to associate a new topic with prior experience.50 The teacher asks students to describe the similarities between a given topic (the concept) and some unrelated item (the analogue). For example, a biology teacher might ask her students to describe the similarities between the parts of an animal cell and the parts of a city. After reviewing these similarities, students are asked to "become" the concepts and analogues by using first-person statements of feeling. The teacher may elicit such statements as "I feel strong when my cell membrane keeps out impurities." If obvious differences exist between the topic and the comparative element, the teacher can address these differences while being careful not to destroy the links previously made. Finally, students create their own new analogies to enable them to better retain the original concepts. This method serves well as a review activity and can be a valuable tool in assisting students to retain facts and concepts.
Concept attainment. blah blah blah
Inquiry method. blah blah blah
Simulations. blah blah blah
Block Scheduling Revisited
That's a new one.
Here’s how he ran his class.
Every week his students have 3 chances to take a test on that week’s material. The test is short, perhaps 5 key problems. If they ace it the first time, they’re done. If they don’t ace it, they take it again; if they don’t ace it the second time, they take it one last time. The final grade stands.
The cool thing: the tests are cumulative.
The first week students take a test on the the first 5 problems; the second week students take a test on those five problems along with the next five problems; the third week they take a test on the first week’s 5 problems, the second week’s 5 problems, and the 3rd week’s 5 problems, and so on.
By the end of the semester they’ve worked up to a final exam covering everything they’ve learned in the course, which they remember because they’ve been re-tested on it every week of the semester.
The teacher also sends each parent a weekly email laying out in detail his child’s progress in the course. Parents can see whether or not their child mastered that week’s material, and they can see when he mastered it: 1st test, 2nd test, or 3rd. Parents can also see any set of 3 tests where his child did not master the material even with three tries. My sister says you could glance at the email and see exactly which material your kid barely squeaked by on.
The teacher explained his system on Back to School night and told parents not to panic when they got the initial emails because early on in the year the point total could suddenly drop 50 points when a student blew the first or second test. He told parents not to panic & not to yell at their kids because they’d have two more chances.
The emails don’t function as a veiled request for parents to kick in with reteaching and tutoring. They are information. Parents know what their kids are doing in the course. I assume that his emails do function as an invitation for parents to kick in with oversight and homework monitoring. Which is fine by me. Parents of students in his class know exactly what they need to know to manage the situation at home.
This may be especially important in my sister’s school because some kids intentionally blow off the class due to a complicated CA system whereby they get credit for “Math 1” even if they flunk Algebra 1. My sister and I agree that the problem these kids pose is school level, not teacher level. In my view (not necessarily my sister’s) the school needs to kick in with supervised homework sessions and the like. (See: LaSalle High School.) Working in a system that rewards kids for flunking algebra, this teacher deals with it by making sure parents know their kids have decided to flunk algebra, providing them with a weekly update on just how much algebra their kids have flunked to date.
The teacher is available every lunch hour and frequently after school for Extra Help. And: Extra Help actually helps. My niece went in twice when she wasn’t getting something. The reason she knew she wasn’t getting it was that she had barely squeaked by on the first two tests and still came up with a low score on the 3rd test.
She went for Extra Help after the 2nd test. My sister says the 3-test format taps into the Magic Number 3 that is embedded in the hearts and minds of children everywhere, as in: “I’m going to count to 3 and when I get to 3 you better be factoring trinomials or else.”
Clearly, the three tests serve as formative assessment. The teacher knows, the student know, and the parent knows whether the kid has or has not mastered the material covered in the course to date. That doesn’t happen in a normal math class. In a normal math class, as my sister points out, “Since no one grades homework, you don’t find out if they know anything until they flunk the test.”
This math class is far from normal because, as it turns out—and this came as a surprise—this teacher also grades homework. The way my sister and my niece found that out was that one day my niece blew off her homework: she just wrote down whatever came to mind and turned it in.
The homework came back with an “F” on top. The teacher had read her homework, corrected her homework, and graded her homework.
She went to see him and apologized. It had been years since a teacher had so much as looked at her homework and she’d assumed he wasn’t going to look at it, either. She asked if she could do it over again & the teacher said yes.
It will probably come as no surprise to learn that the homework sets weren’t burdensome. Perhaps because this teacher read and graded all the homework, or perhaps because he knew exactly how much homework the kids needed in order to master the concepts, he gave small problem sets. My other niece, whose teachers never so much as glanced at anything the kids did outside class, would be assigned dozens of problems every night; she’d sit and slave over her math and no one at the school would give it a second thought. As a result, the stuff they turned in was “the crappiest sh** you’ve ever seen.”
The kids in this teacher’s class, because their teacher was a collecter and correcter, learned to produce neat, readable solution set with the answers circled.
I would put money on it the kids in this man’s class have some of the highest math achievement coming out of a public school Algebra 1 course in the country.
They better have, since it'll be 9 long months before any of them looks at the inside of a math book again.
the Gambill method
I'm sure it's not true. My own belief now is that TA quality absolutely, positively, can matter in college courses. This is true in math courses, computer science courses, physics courses, etc. I'm sure it's true in the humanities as well.
Now, for a while as a TA I thought the profs were right, that it didn't matter, because I didn't see any TAs whose students significantly outperformed others. But I went on to lecture that very same intro sequence course, and had 6 of my own TAs. My results looked very different: one TA's students were better than the rest, and two were markedly worse. Those TAs were outliers--and those outliers had effects.
After that, I began to believe again, and I began to improve my own TAing ability in other courses, until it became clear I could, in fact, remediate students, and if I could, then so could others. In one course, I was able to pull up grades of students who were in danger of flunking through my hard work and theirs on at least half a dozen occasions. This wasn't about motivation: it was about reteaching the material differently, and starting with tiny, baby, solvable problems until class level problems were clear. I was no extraordinarily gifted teacher. I was simply a student who had almost flunked the same course as an undergrad, and I understood what was wrong with the presentation after I had remediated myself in it years later.
So why did the profs think that TAing didn't matter?
TAs were basically given no guidance. What guidance they had was at the discretion of the prof, and most profs felt that since TAing didn't really matter, then there was no point in structuring or teaching the TAs. So they were just told what the weekly material was, and expected to fill an hour of section. They weren't even expected to attend the course. So, the TAs weren't taught how to teach the material at all. Self fulfilling prophecy. So "how well or how poorly you teach" was really a misnomer in the first place: most didn't teach well or poorly; probably, they didn't actually teach at all.
Next, neither the students nor the TAs had any idea what they did not know. Of course the students don't know what they don't know, so they don't know they need remediation. But neither do the TAs: they did well enough to get into grad school, or get a good grade in the course, and that's that when it comes to examining their knowledge base. Students in section seldom speak up to stump a TA; there's nothing good that comes of that, so TAs aren't challenged on the material. Even if they know the material, that doesn't correspond to having the skills to remediate the students. At best, they attempt minorly to remediate those who say "I'm deficient HERE". How many students can do that?
But the outcome, that another generation of soon-to-be-profs believes that teaching doesn't matter, is huge.
From my perspective, this attitude in college courses--not just in ed school, but in college at all, is another big source of this hard-to-break-myth that teaching doesn't work, can't remediate, can't "get out what God didn't put in", so to speak.
It means even college educated parents don't believe teaching matters--because they didn't see teaching matter for them. It means the talented members of their fields don't think remediation works, so they don't bother to do it. And it means that even before teachers enter ed school, they've already been primed by a system that told them teaching didn't matter.
University attitudes trickle down, and not just through ed school.
Should I go ahead and install the wavy-letter-thingie that prevents spam?
I hate to do it because of the FWOT aspect, but I think we're at that point.
Speaking of the wavy-letter-thingie, Jeff Hawkins' book explains why it works.
Monday, February 25, 2008
The fundamental problem with learning mathematics is that while the number sense may be genetic, exact calculation requires cultural tools—symbols and algorithms—that have been around for only a few thousand years and must therefore be absorbed by areas of the brain that evolved for other purposes. The process is made easier when what we are learning harmonizes with built-in circuitry. If we can’t change the architecture of our brains, we can at least adapt our teaching methods to the constraints it imposes.
For nearly two decades, American educators have pushed “reform math,” in which children are encouraged to explore their own ways of solving problems. Before reform math, there was the “new math,” now widely thought to have been an educational disaster. (In France, it was called les maths modernes, and is similarly despised.) The new math was grounded in the theories of the influential Swiss psychologist Jean Piaget, who believed that children are born without any sense of number and only gradually build up the concept in a series of developmental stages.
Piaget’s view had become standard by the nineteen-fifties, but psychologists have since come to believe that he underrated the arithmetic competence of small children. Six-month-old babies, exposed simultaneously to images of common objects and sequences of drumbeats, consistently gaze longer at the collection of objects that matches the number of drumbeats. By now, it is generally agreed that infants come equipped with a rudimentary ability to perceive and represent number. (The same appears to be true for many kinds of animals, including salamanders, pigeons, raccoons, dolphins, parrots, and monkeys.) And if evolution has equipped us with one way of representing number, embodied in the primitive number sense, culture furnishes two more: numerals and number words. These three modes of thinking about number, Dehaene believes, correspond to distinct areas of the brain. The number sense is lodged in the parietal lobe, the part of the brain that relates to space and location; numerals are dealt with by the visual areas; and number words are processed by the language areas.
And if evolution has equipped us with one way of representing number, embodied in the primitive number sense, culture furnishes two more: numerals and number words. These three modes of thinking about number, Dehaene believes, correspond to distinct areas of the brain. The number sense is lodged in the parietal lobe, the part of the brain that relates to space and location; numerals are dealt with by the visual areas; and number words are processed by the language areas.
Nowhere in all this elaborate brain circuitry, alas, is there the equivalent of the chip found in a five-dollar calculator. This deficiency can make learning that terrible quartet—“Ambition, Distraction, Uglification, and Derision,” as Lewis Carroll burlesqued them—a chore. It’s not so bad at first. Our number sense endows us with crude feel for addition, so that, even before schooling, children can find simple recipes or adding numbers. If asked to compute 2 + 4, for example, a child might start with he first number and then count upward by the second number: “two, three is one, our is two, five is three, six is four, six.” But multiplication is another matter. It is an “unnatural practice,” Dehaene is fond of saying, and the reason is that our brains are wired the wrong way. Neither intuition nor counting is of much use, and multiplication facts must be stored in the brain verbally, as strings of words. The list of arithmetical facts to be memorized may be short, but it is fiendishly tricky: the same numbers occur over and over, in different orders, with partial overlaps and irrelevant rhymes. (Bilinguals, it has been found, revert to the language they used in school when doing multiplication.)
The human memory, unlike that of a computer, has evolved to be associative, which makes it ill-suited to arithmetic, where bits of knowledge must be kept from interfering with one another: if you’re trying to retrieve the result of multiplying 7 X 6, the reflex activation of 7 + 6 and 7 X 5 can be disastrous. So multiplication is a double terror: not only is it remote from our intuitive sense of number; it has to be internalized in a form that clashes with the evolved organization of our memory. The result is that when adults multiply single-digit numbers they make mistakes ten to fifteen per cent of the time. For the hardest problems, like 7 X 8, the error rate can exceed twenty-five per cent.
Our inbuilt ineptness when it comes to more complex mathematical processes has led Dehaene to question why we insist on drilling procedures like long division into our children at all. There is, after all, an alternative: the electronic calculator. “Give a calculator to a five-year-old, and you will teach him how to make friends with numbers instead of despising them,” he has written. By removing the need to spend hundreds of hours memorizing boring procedures, he says, calculators can free children to concentrate on the meaning of these procedures, which is neglected under the educational status quo.
This attitude might make Dehaene sound like a natural ally of educators who advocate reform math, and a natural foe of parents who want their children’s math teachers to go “back to basics.” But when I asked him about reform math he wasn’t especially sympathetic. “The idea that all children are different, and that they need to discover things their own way—I don’t buy it at all,” he said. “I believe there is one brain organization. We see it in babies, we see it in adults. Basically, with a few variations, we’re all travelling on the same road.” He admires the mathematics curricula of Asian countries like China and Japan, which provide children with a highly structured experience, anticipating the kind of responses they make at each stage and presenting them with challenges designed to minimize the number of errors. “That’s what we’re trying to get back to in France,” he said.
It makes for fascinating reading. I still don't agree with the give a calculator to a five-year-old stuff, though. Nope. I just don't buy that.
Are Our Brains Wired for Math?
The New Yorker
March 3, 2008
Harvard Study on teacher effects
A brilliant piece of research and analysis appeared in the 1978 Harvard Educational Review. Authored by McGill University sociologist and administrator, Prof. Eigil Pedersen and colleagues, "A new perspective on the effects of first-grade teachers on children's subsequent adult status" is masterfully written and caps more than a decade of extremely sophisticated data collection and analysis. It attracted little mainstream media attention at the time.
Pedersen and his team were sociologists, looking for factors that made children from impoverished backgrounds successful in later life. Following in the wake of Coleman and Jencks, who both detailed the powerfully negative predictive value of low SES on academic achievement, Pedersen's insights and meticulous research were probably not given the attention they deserved, and their most powerful implications were missed.
Pedersen selected for his study a low-income, chronically low-performing school in an urban industrial area situated in the article "in a large city in the Northeast." It was, in fact, Royal Arthur School in the former Protestant School Board of Greater Montreal -- a mammoth brick edifice with windows barred like a jail, situated across the street from a brothel, and sandwiched between toxic industrial factories.
This school had by far the lowest levels of academic achievement of the 80 elementary schools in the school district at the time, and a high rate of discipline problems. Fewer than ten percent of pupils went on to graduate from high school; more than half dropped out before completing Grade 10.
Pedersen had access to pupil records covering a 25-year period, from about 1935 to 1960. Along with detailed achievement, attendance, and discipline records, the school also had scores of standardized IQ measures on every pupil, given in Grades 3 and 6. While tabulating other data (from which he gathered many useful and interesting conclusions, adverted to in the article), he noted a strange, and unexpected, phenomenon. There were a number of cases in which individual students' IQ scores changed dramatically, either up or down, in some cases by more than 2 standard deviations -- the difference between being "low average" and "gifted." This is a huge differential, and something that is not supposed to occur except under extraordinary circumstances. Pedersen's curiosity was tweaked. In addition to analyzing his other data about student performance, he decided to take a large set of pupils in 3 categories: IQ's that had risen substantially, IQ's that fell substantially, and IQ's that remained more or less constant, and examine these three cohorts in more detail.
Noting that the first measure of IQ was taken early in Grade 3, he wondered if students' achievement in Grade 1 was a predictor of their Grade 3 IQ. So that led him to examine and tabulate student data by Grade 1 class. There were three Grade 1 teachers who were on staff throughout the period when the pupil records were sampled. Pedersen called them Miss A, Miss B, and Miss C; others who came and went after varying short terms, he presented in aggregate form as "Others." Studying his matrix of data, Pedersen was astonished to observe consistent and striking differences between the three long-term teachers, compared to random variation apparent in the aggregate "Others."
Miss A had taught most of the students who showed a large IQ increase between Grades 3 and 6, and it didn't matter whether they were male or female. Miss C taught most of the students who showed a large decrease in IQ, and again, it didn't matter whether they were boys or girls. Miss B taught many female students who showed an increase in IQ, and males who showed a decrease. When they investigated further, the researchers found that Miss B had always evidenced a preference for girls, seating them in the front rows, holding them to a higher standard, and engaging them more frequently, while boys were banished to the back of the room.
This was all interesting, but not what Pedersen's study was about. He and his fellow-researchers went on to a more ambitious project, locating a large number of adults in their late 30's who were graduates of the school, and interviewing them and getting data about their current lives. He developed a complex dependent variable he entitled "adult status" which involved level of education, income, type and cost of home, employment status, etc. He got independent analysts to develop a classification system from 1 to 9 to represent "adult status." An example of 1 would be "on welfare, has never worked" and 9 would be "professional" (dentist, university professor, etc.). The bottom 3 were grouped as "low status," the middle 3 as "medium" and the top 3 as "high status." This team collaboratively assigned each participant an adult status ranking from 1 to 9.
So, Pedersen et al were looking at a number of variables, and studying data from a number of sources. However, when they had tabulated their findings, one things stood out so stunningly they had to stop what they were doing and change direction entirely. It was what they saw when they looked at "adult status" vs. "first grade teacher."
The mean adult status for the aggregate "Others" was 4.6 -- smack in the middle between 1 and 9, and exactly what , with random variation (and a large sample) would be predicted.For Miss C, it was 4.3. For Miss B., 4.8. All within the range expected. For Miss A it was a stunning 7.0 ! Even more remarkable was the percentage of pupils in the various groups. The other teachers averaged 29% "high status" adults and 40% "low status." For Miss A, an amazing 64% were in the "high status" category......and the percent in the "low status" category was --- ZERO.
The researchers went back to their data collection to rule out alternate hypotheses. They found that classes were not stacked in Miss A's favour, there were no significant differences between her assigned pupils and other class groups in terms of parents' occupations, families on welfare, behavior problems or other identifiable issues. All classes were heterogeneous. Miss A. had retired by the time Pedersen had reached this point in his study but he was able to interview former colleagues and pupils to identify factors in her success. She loved her work, she expected every student to learn and succeed, she gave extra help to kids who needed it, she was a low-key but firm disciplinarian, she showed affection to the children and had an amazing memory for their names and biographies -- remembering them in department-store chance encounters 25 years later.
But a major reason for her results is probably summed up by a former colleague who went on to be an administrator with the PSBGM: "It did not matter what background or abilities the beginning pupil had; there was no way that the pupil was not going to read by the end of Grade One."
SHE TOOK RESPONSIBILITY FOR STUDENT LEARNING.
If all, or even just 99.9% of, students learned to read in their year with Miss A, they were a long step ahead along the road to success.....and their IQ increase between Grs. 3 and 6 is likely due to the increased language aptitude engendered by their reading skill (kids who can't read consistently show an IQ decrease over the elementary years).
In her 34 years at this one high-poverty school, "Miss A" taught about 1200 students. While the researchers were unable to follow up all of these students, the cohorts they studied showed such consistent patterns that we can be sure she had an immeasurable effect on countless lives which could easily have gone nowhere, due to such unfavourable beginnings. The difference between the achievements of her students and the "average" of the other teachers' students is HUGE.
When I read and studied all of this, all I could think was, WOW.
Of course, it is an inspiring story. But also a cautionary, and disturbing, one. Why?
Because it was only because of Pedersen's work that anybody NOTICED what an extraordinarily effective teacher this person was, and how her effectiveness changed lives for decades to come (there's more about that in the article). In the "real world" of urban school systems, effectiveness is not only not recognized, it is rarely even noticed. But for Professor Pedersen's serendipitous research and meticulous data analysis, Miss A's. astounding effect on generations of low-income students would have gone unremarked, except by those students and their families. The "Miss As" among us today -- and they are out there, perhaps not in every school, but certainly not isolated phenomena -- fly similarly under the radar screen while initiatives to "improve schools" come and go with predictable regularity and minimal overall effect.
Not all education professionals would like Miss A. Not all share the view that it is the teacher's responsibility to TEACH every student, never mind what the student brings to the table. Successful learning does not depend upon intrinsic factors in the students (motivation, self-esteem, nourishing home environment, et al) OR in the teacher. It can be MADE TO HAPPEN. Michael Pressley's work on motivational behaviors of effective primary teachers clearly demonstrate this.
Miss A proved it, and her successors among us today continue to do likewise. Alas, I doubt the message her accomplishment sends -- that teachers can be far more effective than anyone dreamed -- will be taken to heart any more now than it was 28 years ago. If we collectively had a commitment to ALL students learning, what we do in schools would look a great deal different than it does.
A belated thank you, Miss A -- you put the lie to the persistently proffered argument that students' background, SES, IQ or previous experience is an insurmountable barrier to success in school and adult life. Perhaps some day we adults will be "developmentally ready" to learn from you. I hope so.
For the whole article, check interlibrary loan and look for Pedersen, E., Faucher, T. A., & Eaton, W. W. (1978). A new perspective on the effects of first-grade teachers on children's subsequent adult status. Harvard Educational Review, 48(1), 1-31.
* "Miss A" was in fact named Iole Appugliese (YO-lay AP-poo-lee-YAY-zee), and she was born in Montreal to Greek immigrant parents. Her pupils, who could not pronounce her name, called her "Miss Apple Daisy.") She retired only when she discovered she had terminal cancer. An interesting finding of Pedersen's was that more ex-pupils remembered being in her class than actually were. While many could not remember who their other early grade teachers were, every one remembered "Miss Apple Daisy" 's name and had memories and stories to share. Judith Harris hypothesized that part of the reason for her consistent student results was due to her creating a community culture of learning and achievement that carried her pupils through elementary school. By Grade 7, the largest predictor of student achievement (outweighing IQ, SES, family status, etc.) was which first grade teacher a pupil had had. Amazing stuff.
One alumnus Pedersen interviewed remarked that he had a five-year-old about to enter first grade. Sadly he sighed, "I only wish there was a Miss Apple Daisy for him."
I mention Moneyball because of a rondelay amongst Kevin Carey, Matthew Tabor, and Leo Casey over the question of whether teachers can or cannot be an undervalued "commodity" in the sense that hitters with high on-base percentage were before Bill James and the Oakland As demonstrated their true worth.
What value-added research is showing us – and no one seems to have absorbed this yet – is that a good teacher is vastly more valuable than anyone recognizes:
"The biggest factor affecting student achievement is teacher effectiveness," said Sanders, who emphasizes that class size effects and differences in ethnicity, family income, and urban-suburban location fade into insignificance when compared to teacher effects.
Excuses, Expectations, and Learning Gaps
All of us have known or heard of teachers like Miss A: teachers who change young people's lives.
But has any of us suspected that the Miss A's of this world were changing IQs? That the Miss A's in our schools were altering forever the future education, income, and life prospects of the children they taught?
Or that they were doing so for hundreds upon hundreds of children year in and year out?
The answer is no. None of us imagined or knew.
I thought this could be fun. Catherine asked about the infamous "Folding House Poem Project" in another thread. I'll post as many artsy projects as I can remember that I've encountered over the years teaching middle and high school which had little to no redeeming value for the lesson at hand. You see how many you've seen. Add more if you think of them. I'll post more as they come to me.
1. The Folding House Poem Project: A favorite for 11th grade on-level English, until I became team leader and flat out refused to do it, to consider it, to support it, or to give any of the new teachers the resources for it. (I hear there was some black-market trading under the radar, but for the most part the teachers seemed relieved.) Students make about 5 folds in a rectangular piece of paper until it vaguely resembles a house on the outside. They decorate the outside either like their house, or their dream home. When the front flaps are opened, you get the "interior" which is decorated like their room. Open it up all the way to reveal a poem about their house, their room, their life, or some vague, abstract pseudo-poem or loose verse (no format, of course) about anything that might relate to their ideas of what "home" means.
What could be taught: how to write descriptive passages using directions (prepositions), how to follow a poetic format, how to mimic a poetic format based on a poem read in class.
What is not taught: see "What could be taught."
Purpose: killing time in the first 3 weeks of school when class size and enrollment fluctuate due to schedule changes, simply because no one wants to put together a horizontally aligned, level-wide lesson plan that could accommodate students changing classes.
2. The Mandala Project: 10th grade. Students write and discuss opposites, a la yin and yang. Compose copious journals on their personal opposites, animals, colors, objects which represent them and why, etc. They then are shown mandalas, and given a paper plate. They trim the scalloped edge from the plate, and create a "mandala" of their own, which they color in and draw or paste pictures to, to represent all the aspects of their personality. The removed scalloped edge is either thrown away, or used as additional decoration in creative ways.
What could be taught: I'm still flummoxed by this one in an English class, except for personal journaling. Anyone?
Purpose: killing time again. Those first 3 weeks are killers.
3. The 1-Pager: Directions may vary. Students are required to complete a letter-sized page "poster." Used generally as a vocabulary reinforcer. Each student picks a vocab word from the reading. (Just one per student, mind you.) Students must put the vocab word on the page, include a definition, include a quote from the story where the word is used, and then various pictures "attractively arranged" to a) represent the word, b) represent the word's meaning, c) represent the student's interpretation of the word, d) represent what the student thinks about the word, e) represent how the word is used in the story, f) represent the word's importance in the story, g) represent the character related to the word...ad nausea um. (My point being that these projects are usually terribly vague and abstract and difficult to grade EXCEPT on their artistic merit: 1-2 pics +content = C, 3-5 pics +content=B, and etc.)
Purpose: What? You can't tell that's a good 1-2 weeks of
every-other-day classes for prep and production time?
4. The Tri-fold: as seen in Catherine's KTM post about The Project Method (http://kitchentablemath.blogspot.com/2008/02/project-method.html).
Students are required to buy an enormous cardboard "tri-fold" and fill it with stuff. Our AP students know this as the "Author Project." To be fair, they also have to research the author, read two of his/her works, and write some analysis. But then they have to create this tri-fold monster, the best of which go on display in the library, and include 10 or more "artifacts" representing the author's life, works, etc. Every single year someone staples a decapitated Barbie doll head to one of those suckers. Every. Single. Year.
Purpose: myriad - can be adapted for any purpose/genre/subject.
5. The Alphabet Book: Students must create an alphabet book in which they use one word per letter to describe anything and everything in a book they've read/subject being studied. 1 page per word, 1 definition, 1-3 pics, lengthy (and sometimes groping) explanation of how their chosen word relates to the book (especially up in the XYZ category). I have seen some of these which looked like they were worth a small fortune in scrapbooking supplies. Very pretty. Very time-consuming. More points for better artwork presentation, leaving cash-strapped kids out in the cold.
Purpose: Darwinian annihilation of students who cannot buy out the Hobby Lobby's selection of stickers, ribbon, scrapbook paper, and small, adhesive medallions.
Warning: also can be adapted for any subject.
6. The Cell Project: Students must create a 3-d representation of a cell (or something from the periodic table) using "whatever." I don't know much about this one, honestly. It appears to happen around 9th grade. During this time, small Styrofoam balls litter the hallways, being kicked, thrown, punched, hackey-sacked, tossed and speared by cafeteria sporks, bounced off pretty girls' heads in passing (it is 9th grade, after all), rolled for the tripping factor (10 points for teachers who stumble), and all around abused. Generally, they have little to recommend them except some markered cross-section which once had little labels on toothpicks stuck in them. The toothpicks are long gone, no doubt used for nefarious purposes.
Purpose: to annoy, I think.
7. Salt Maps: do I need to explain this one?
So, what have you got?
Just noticed redkudu has added a paragraph on technology!
8. “Interfacing” with technology: any project which has, at its core, the intent to teach students how to interface with technology on a superficially cosmetic level, such as replying to a classroom blog, etc. Has it escaped anyone that teenagers are very, very good at interfacing with technology? I have a colleague with a “Smart” classroom (wired to the hilt, that is), who is having her students make slide shows of books they’re reading. She is struggling to learn the technology as her students zoom past her - thus, it isn’t the students who are learning anything, it’s the teacher. She also has a classroom blog, but when questioned admitted she doesn’t read any blogs, nor know of any to recommend her students read. So what is the point then? What students need to be learning is how to manipulate technology, how to use its myriad programs and capabilities to arrange, display, and explain information. Some good examples of this are dy/dan and his Feltron project, or some of the ideas to be found at 21st Century Collaboration. I’m not against technology in the classroom, just don’t make a project of it if it isn’t teaching students anything about using it.
worst project ever
worst project ever, part 2
the project method
"The Project Method: Child-Centeredness in Progressive Education"
We're all trying to find the right angle or leverage to create the most change. Perhaps [Prof. Wu] can change the thinking at the NCTM level, I don't know, but I don't think he will have much effect at the local level.
I find this an interesting point to elaborate on. I understand Steve's point, but I don't know if I agree: Steve, what do you think created the perspective at the local level?
Personally, from my outsider perspective, it looks to me like all of the local changes in our school board to disasters like Everyday Math or TERC came from the top level. They came from national boards. They came from national foundations, big things like Gates, etc. And overwhelmingly, the ideas came to them from ed schools' ideas having infiltrated the very top of the food chain, and that food chain is now pushing down down down to the local level.
I cannot IMAGINE a situation where a local public school board change is made without the national ethos changing first. That's why it takes privates or charters or any other structure to just get decent textbooks.
But maybe I'm wrong. What do you all think? I'm not a teacher in the schools. For those who are, where did these changes come from? Top down? Has ANY change come bottom up? What can an individual teacher or even principal do? What would it take to make a change at the local level if philosophies on the national level aren't changed?
I heard it again from another lecturer as well, and this lecturer was famous as the best teacher in the department (he was only a lecturer, not a researcher.) He was beloved for teaching this course, and he too said it didn't matter. Actually he even said "I've straight scaled the class because it doesn't matter. The grades will fit a bell curve. They always do, have for the last 20 years."
Bracket for the moment whether these teachers have given up and are jaded, or have hit a wall, or simply have found some universal truth.
Assume for now that this attitude is the predominant one among faculty in college, at least among those who teach several hunded person courses. Assume it's the correct attitude once you average over all of the various motivations for students in such a class: this is a subject they love, this is required for the major but NOT IN their major, this fulfills a breadth requirement, they have to pass it to graduate, they excel at school, they are taking this because of parental expectations, they have no background, they've seen the material before, they go to college to carouse and do drugs, that they are full time students, they go to school on the side of full time work, etc. etc. etc.
But now do the backwards regression:
At what point is it no longer true? At what point DID TEACHING MATTER? If the college profs say it doesn't matter by the time the 17-22 year olds enter, do they think it matters for high school? Has this idea trickled down to middle school? Grammar school?
Is the in vogue idea of "student maturity" just another way of saying that middle school teachers think EXACTLY the same way that college profs of adults think about their law-of-large-numbers-averages across hundreds of students?
Is this taught in ed school? Is this such a prevalent idea in college that ed school can't undo it anyway? Because if you believe this, then WHAT IS THE POINT of going into teaching?
Sunday, February 24, 2008
Wu gave the plenary talk at the National Council of Teachers of Mathematics (NTCM) Annual Meeting in 2007. The comments here are taken from the slides of that talk (available here) and from recent papers he has written on the same subject, that of the relationship between mathematicians and math educators (see here). The errors or confusions should be considered due to me, rather than Wu. Consider the comments in italics to be straight from his talk or papers, and the rest to be from me.
Wu uses this definition of engineering: engineering is the customization of abstract scientific principles to satisfy human needs.
Mechanical engineering, then is about turning the laws of classical physics into pulleys, cranes, refrigerators, etc. It's how you build your cel phone or IPod so you can drop it without it breaking. Likewise, chemical engineerings is about turning chemistry into the plexiglas for your aquarium, the non toxic antibacterial cleanser you use on kid's toys, the gas you put in your car.
Mathematical engineering, then, is about customizing mathematics for use by students and teachers in K-12. That is the human need of mathematical engineering. Without the customization of math to students and teachers in K-12, math cannot be used by adults. It is as if you were unable to make any use of chemistry or physics. Math education, and its pedagogy, matters deeply. You cannot simply throw adult math at school age children.
Engineering gets better over time, with refinements, but the underlying principles don't change. Engineering must find its way between two principles: the inviolable scientific principle, and user friendliness of the end product.
Likewise, mathematical engineering must improve with better pedagogy, better adaptation to the students, etc. but it can not change violate the principles of mathematics: 1) precision, 2) definition, 3) coherence, and 4) reasoning, 5) purposefulness.
Precision means making clear, unambiguous mathematical statements. There is no unspoken context in math: you are expected to make clear what is known/given, and what is unknown.
Definition: concepts in math have specific definitions. A specific concept as a specific definition--one, and all others can be shown equivalent. But the structure of math demands a definition for all concepts.
Coherence: math builds on prior knowledge. Nothing comes out of the blue, but it unfolds from what is known already.
Reasoning: mathematics cannot proceed without reasoning. Reasoning must be illuminated.
Purposefulness: math is goal oriented. It solves specific problems.
Mathematical engineering, that is, math education, to date as lost these principles. While there is some pedagogy available for teachers, over and over again we see that the teaching of basic ideas (geometry, fractions, etc.) of math violate most or all of the above 5 principles. This is unworkable engineering: it is the equivalent of chemical engineering labs that don't know enough chemistry to create stable chemicals.
In other papers as well as this talk, Professor Wu elaborates on the failures of current pedagogy to address those 5 principles. He does this by showing the failure in the presentation of fractions, geometry, and in what he calls the Fundamental Assumption of School Mathematics. I will address the fractions presentation in another post.
For geometry, he states that the mathematics of Euclid and Hilbert goes from axioms to theorems to proofs in an un-user friendly way. However, the major school presentations either teach it this way, without addressing students' learning capacity, or they present it without definitions, theorems, and proofs, as if it is experimental geometry, and so it lacks precision, coherence, reasoning, etc.
For the Fundamental Assumption of School Math: mathematically, it is true that all arithmetic operations on fractions (i.e. rational numbers) can be extrapolated to work on the reals. Why this is true is nontrivial mathematics. School math is only about rationals, and then presentation of irrationals is done without any such claim or explanation that it can be done. The assumption is left unstated: a vioation of the precision, definition, reasoning, etc.
So mathematical engineering requires both mathematicians and math educators, just as chemical engineering requires engineers and chemists. The math educators are needed because they know the students, and what is needed by the students at their various ages. Math educators know what the school math curriculum is for a given level, if not how to present it. They, too, are familiar with what the maturity level of their students is. Mathematicians are needed because correct mathematics must be taught at each of these levels, even though what is known at each level is different. That means a variety of correct explanations must be made available. That requires deep subject knowledge--deep enough to understand what's true and correct from a variety of view points. And in truth, these issues come together as we attempt to find the best approach for each student: you need both pedagogy and mathematics in order to reach students and still teach them math that adhered to the above 5 principles.