Self-critisicm

It is not common for a teacher in this country to be observed by other teachers. It is a shame, as I think, and apparently lot of others as well, that it is a great practice. Here is another excerpt from The Teaching Gap:

As researcher Catherine Lewis found, teacher collaboration can create a profound motivation to improve. A young teacher she interviewed recalled that after watching a lesson by her fellow first-grade teacher, she burst into tears: "I felt so sorry for my own students. I thought their lives would have been so much better if they'd been in the other teacher's class."

Past semester it has become apparent to me that not only majority of people are not critical of themselves, but are not critical of others either. There is a constant fear of hurting other people's feelings. Criticism can be expressed in a manner that is not offensive (ok, maybe I am not the best example, but I am sure it can be done :) ), but rather constructive. Not to be blunt, but I'd rather have one person's feelings hurt, than thousands of children not learning what they are capable of.

In any event, this book is truly amazing, and I recommend it to everybody who aspires to be a teacher (or is one). One of the major points in my view is that majority of U.S. teachers believe that having finished their studies they are ready to teach and the only area in which they might seek improvement is in HOW they teach. But not in WHAT they teach.

Technorati tags: improvement, math, teaching

More on typesetting

I tried to write more about this mimeTeX, but the frustrations with blogger made me erase the whole thing. Turns out that you can use a public server that was generously provided by Mr. Forkosh to generate gifs of LaTeX code. Here is an example


< img src="http://www.forkosh.com/mimetex.cgi?c=\sqrt{a^2+b^2}"
alt="" border=0 align=middle >



Unfortunately, I think one needs very light background, for these to be visible, so I need to make that happen (I'm guessing it'd be enough to make a white box around the gif). Maybe this'll work:




Gotta go walk the dogs. I'll try to make it work later :)

The Teaching Gap

I have often heard the following statements from teachers:

"There is no way my students could handle this."
"This is too hard for my students."
"Good luck teaching that. That's way above their heads."

If I complained that I didn't think that particular topics under discussion were that hard, I would get these answers:

"You don't have experience with (insert your favorite k-12 grade) students."
"You haven't been in our school."

These statements are correct. I do not have yet that much experience with K-12, but I am fairly positive that students respond to unstated expectations very well (and stated ones even better). What I mean by that is: if your students believe that you think they can not learn math or that the math is just too hard for them to grasp, then they do not have much incentive to prove you wrong. After all, you are the expert. However, if you set the bar high for them, their performance and in the process you let them know that YOU think they are able to become good in math, then they will try harder to prove you're right. I do think that the attitude teachers have influences greatly their students. The reason I am writing about this is the book I just started reading: The Teaching Gap by Stigler and Hiebert. I have a couple of quotes that I'd like to share:

One of the most striking impressions when watching the videotapes is that students in teh United States encounter a different kind of mathematics from that encountered by their peers in Germany and Japan. The content appears to be less advanced and is presented in a more piecemeal and prescriptive way.

As it turns out there were NO mathematical proofs in U.S. lessons. In contrast, there were proofs in 53 percent of Japanese lessons and 10 percent of German lessons.

Incidentally, German students did not perform significantly better on the achievement test then American students. The following figures also made quite an impression (I hope they will look decent):

Average percentage of topics in eight-grade mathematics lessons that contained topics that were DEVELOPED or STATED.

Percentage of lessons rated as having low, medium, and high quality of mathematical content (as rated by a team of mathematicians who did not know which lessons came from which countries).

Technorati tags: education, math, content quality

Typing mathematics in blogs

I was talking to couple of friends about blogs in mathematics classrooms and they both asked "How do you typeset math in HTML". I had to say I had no idea. We use LaTeX, but that doesn't translate to HTML, unfortunately. So, instead of asking around, I tried to look it up. The is the first thing I discovered, courtesy of Clark Grubb , and I guess it should be accessible to high school and college students. Scroll down to find LaTeX lover's solution :)

The character entities can all be invoked via their numeric or alphabetic names. For example, both

x &#8712; A and x ∈ A

will display

x ∈ A

A radical can be done. The HTML

&radic;<span style="text-decoration: overline">a + b</span>

will display as

√a + b



Below is a partial list of the character entities specificied in the W3C Character Entity References for HTML 4. It includes most of the characters of interest to the mathematician.

Generic Character Entities

#	alpha	sym	IE6	FF	description
38	amp	&	Y	Y	ampersand
160	nbsp		Y	Y	no-break space = non-breaking space
8194	ensp		N	?	en space
8195	emsp		N	?	em space
8201	thinsp		N	?	thin space
8204	zwnj	‌	?	?	zero width non-joiner
8205	zwj	‍	?	?	zero width joiner
8211	ndash	–	Y	Y	en dash
8212	mdash	—	Y	Y	em dash
8230	hellip	…	Y	Y	horizontal ellipsis = three dot leader

Mathematical Character Entities

#	alpha	sym	IE6	FF	description
60	lt	<	Y	Y	less-than sign
62	gt	>	Y	Y	greater-than sign
176	deg	°	Y	Y	degree sign
177	plusmn	±	Y	Y	plus-minus sign = plus-or-minus sign
215	times	×	Y	Y	multiplication sign
216	Oslash	Ø	Y	Y	latin capital letter O with stroke = latin capital letter O slash
247	divide	÷	Y	Y	division sign
8226	bull	•	Y	Y	bullet = black small circle
8465	image	ℑ	N	Y	blackletter capital I = imaginary part
8472	weierp	℘	N	Y	script capital P = power set = Weierstrass p
8476	real	ℜ	N	Y	blackletter capital R = real part symbol
8501	alefsym	ℵ	N	Y	alef symbol = first transfinite cardinal
8592	larr	←	Y	Y	leftwards arrow
8593	uarr	↑	Y	Y	upwards arrow
8594	rarr	→	Y	Y	rightwards arrow
8595	darr	↓	Y	Y	downwards arrow
8596	harr	↔	Y	Y	left right arrow
8709	empty	∅	N	Y	empty set = null set = diameter
8711	nabla	∇	Y	Y	nabla = backward difference
8712	isin	∈	Y	Y	element of
8713	notin	∉	N	Y	not an element of
8715	ni	∋	Y	Y	contains as member
8719	prod	∏	Y	Y	n-ary product = product sign
8721	sum	∑	Y	Y	n-ary sumation
8722	minus	−	Y	Y	minus sign
8730	radic	√	Y	Y	square root = radical sign
8734	infin	∞	Y	Y	infinity
8736	ang	∠	Y	Y	angle
8746	cup	∪	Y	Y	union = cup
8747	int	∫	Y	Y	integral
8764	sim	∼	Y	Y	tilde operator = varies with = similar to
8773	cong	≅	N	Y	approximately equal to
8776	asymp	≈	Y	Y	almost equal to = asymptotic to
8800	ne	≠	Y	Y	not equal to
8801	equiv	≡	Y	Y	identical to
8804	le	≤	Y	Y	less-than or equal to
8805	ge	≥	Y	Y	greater-than or equal to
8834	sub	⊂	Y	Y	subset of
8835	sup	⊃	Y	Y	superset of
8836	nsub	⊄	N	Y	not a subset of
8838	sube	⊆	Y	Y	subset of or equal to
8839	supe	⊇	Y	Y	superset of or equal to
8853	oplus	⊕	Y	Y	circled plus = direct sum
8855	otimes	⊗	N	Y	circled times = vector product
8869	perp	⊥	Y	Y	up tack = orthogonal to = perpendicular
8901	sdot	⋅	N	Y	dot operator
8968	lceil	⌈	N	Y	left ceiling = apl upstile
8969	rceil	⌉	N	Y	right ceiling
8970	lfloor	⌊	N	Y	left floor = apl downstile
8971	rfloor	⌋	N	Y	right floor
9001	lang	〈	N	Y	left-pointing angle bracket = bra
9002	rang	〉	N	Y	right-pointing angle bracket = ket

Greek Letter Character Entities

Both IE6 and Firefox implement all of these characters.

#	alpha	sym	description
913	Alpha	Α	greek capital letter alpha
914	Beta	Β	greek capital letter beta
915	Gamma	Γ	greek capital letter gamma
916	Delta	Δ	greek capital letter delta
917	Epsilon	Ε	greek capital letter epsilon
918	Zeta	Ζ	greek capital letter zeta
919	Eta	Η	greek capital letter eta
920	Theta	Θ	greek capital letter theta
921	Iota	Ι	greek capital letter iota
922	Kappa	Κ	greek capital letter kappa
923	Lambda	Λ	greek capital letter lambda
924	Mu	Μ	greek capital letter mu
925	Nu	Ν	greek capital letter nu
926	Xi	Ξ	greek capital letter xi
927	Omicron	Ο	greek capital letter omicron
928	Pi	Π	greek capital letter pi
929	Rho	Ρ	greek capital letter rho there is no Sigmaf, and no U+03A2 character either
931	Sigma	Σ	greek capital letter sigma
932	Tau	Τ	greek capital letter tau
933	Upsilon	Υ	greek capital letter upsilon
934	Phi	Φ	greek capital letter phi
935	Chi	Χ	greek capital letter chi
936	Psi	Ψ	greek capital letter psi
937	Omega	Ω	greek capital letter omega
945	alpha	α	greek small letter alpha
946	beta	β	greek small letter beta
947	gamma	γ	greek small letter gamma
948	delta	δ	greek small letter delta
949	epsilon	ε	greek small letter epsilon
950	zeta	ζ	greek small letter zeta
951	eta	η	greek small letter eta
952	theta	θ	greek small letter theta
953	iota	ι	greek small letter iota
954	kappa	κ	greek small letter kappa
955	lambda	λ	greek small letter lambda
956	mu	μ	greek small letter mu
957	nu	ν	greek small letter nu
958	xi	ξ	greek small letter xi
959	omicron	ο	greek small letter omicron
960	pi	π	greek small letter pi
961	rho	ρ	greek small letter rho
962	sigmaf	ς	greek small letter final sigma
963	sigma	σ	greek small letter sigma
964	tau	τ	greek small letter tau
965	upsilon	υ	greek small letter upsilon
966	phi	φ	greek small letter phi
967	chi	χ	greek small letter chi
968	psi	ψ	greek small letter psi
969	omega	ω	greek small letter omega

Immediately after stealing all this code from Clark (thanks a bunch), I ran into Flip Tomato's Blog. This was an absolute gem, as it contained the following link: mimeTeX. Score! I better get this running on my computers. If anybody has better solutions, let me know.

Technorati tags: tex, math, type math