Amateur survey of mathematics teachers

There is a lot of talk about what kind of knowledge mathematics teachers need in order to teach well. I am curious as to what practicing teachers think about this. Here are few questions for those of you who may stumble upon this page:

1. Do you think that mathematics courses you took as part of your preparation program (whether it be undergraduate, or certification, or anything else) are relevant to what you do in the classroom? In other words, do you think that what you learned there is directly or indirectly applicable to your profession?
   
   
2. Same questions about methods courses you my have taken.
   
   
3. In light of your experience in the classroom, if you had a say in what should be taught to future teachers as a part of their preparation, what would it be? Or, what do you wish you learned before you started teaching?
   

It is possible that I am asking wrong questions. If you think that is the case, then offer your own questions. And answers :)

Doing the steps and not going anywhere

If I made an actual step every time my teacher said "step" in two classes I observed today I could have had a nice walk.
I don't see them learning, I just see them stepping. Everything is done for them and when they make mistakes and do procedure incorrectly, she says "Oh, you just forgot to ______________". No, they didn't forget. In order to forget, you must have known it at some point.

It seems to be a common practice to have students work on their homework in class. This, I believe, is an American construct. I find it useless, but people feel attached to it (also, which part of the word homework says that the work is supposed to be done in class?). I saw a teacher give kids and upward of 20 minutes to do their homework. About 75% of the students, if they even bother to attempt the homework, are out of their seats and ready to head out the door within 5 minutes. Now, there is all this talk about not having enough time to go through the material that one is required to cover, but there seem to be so much wasted time. Am I observing an extreme case, or is this practice common? Here is a questions for all the teachers out there: How do you feel about homework time, how much do you give and is it really beneficial?

Research vs. Practice

I just spent a few days in to me foreign environment: mathematics teacher educators' conference. Until few months ago I didn't even know that these things existed and now I know that there is a professional development for professional developers!

I don't think I made this clear so far, but I am in between two worlds right now: mathematics and mathematics education. I feel I'm in a no man's land, and there aren't too many of us hanging out there (or we just don't know each other?). Until a year ago I was a semihappy postdoc working on my own little projects, proving my little lemmas, was happy and felt guilty when I taught. Why semihappy? I guess I didn't feel that my contributions to the field or the world at large would be big enough to justify spending my life picking at the mathematics pie and hanging out with people who think that what they do is earth shattering (especially since most of the time I disagreed). Where I did think I could be more useful was with my teaching. I like to think that I am a pretty good teacher, I certainly love doing it, although I may not see myself in an objective light (see the post and post below, and I plan on revisiting this later). In any event, I taught a course for teachers this past semester, and as I was getting ready for it, I started realizing that I don't really know much about teaching, or methods, or pedagogy, or anything really. I decided I ought to learn and started talking to people over in education. It didn't take me too long to realize that they are more similar to mathematicians than either group would like to admit. It seems to be a widespread opinion, especially in education circles, that mathematicians don't care about teaching. That may be, but I would venture to claim that education faculty do not care about teaching any more than mathematicians do. This may need clarification, apart from the one where I say that I haven't actually conducted any research on this and everything I say are observations only and the samples aren't too huge :)

Educators as well as mathematicians largely think that teaching takes the time away from scholarly work. From the professional development programs that I saw, I can not claim that I am convinced that the benefits for teachers are what is on the education faculty's mind, but rather data that they may collect along the way. There also seem to be a big issue of who teaches content courses. Education stance: people who teach it (a.k.a. mathematicians) know the content, but not in the way that is relevant to teachers, so it should be taught by education people. Mathematics stance: the teachers need to know lots more content than they do, and all they get from them (a.k.a. math educators) is this touchy-feely, hold each other hand and reflect nonsense, and consequently they should never be allowed to teach the content courses. Am I exaggerating? Maybe, but only slightly. So what is is missing in this picture? Could it be conversation ? But who is to start it when everybody is busy with their research??? The obvious answer to me is the people who do not want to do research, but want to practice. If they are diplomatic enough (hmm, guess that rules me out) they could bridge the divide between the two worlds. Mathematicians who are also knowledgeable about results of research by mathematics educators (slight misnomer: mathematics education researchers would be more precise) seem to be a natural choice. So, if you are out there, and are reading this, or know somebody who is, let me know.

Too much information?

All thing considered ran an interview with Rafe Esquith, a fifth grade teacher in an LA public school. He was talking about his book Teach like your hair is on fire. The title sounded great (hmm, should it be "don't judge the book by its cover, nor its title"?), so I thought I'd google it. First clicked link was a review from Teacher magazine. I got little taken aback by the second sentence in the article:

“I’m only here,” Esquith announces at the outset, “to share some of the ideas I have found useful.” But most of the things he shares aren’t all that useful and barely qualify as ideas.

The rest follows similar pattern. I have to admit that most of the time I don't doubt NPR much, so this came as a surprise. I decided to look some more. Next was a blog entry , and I list it only because it pointed me to Washington Post's article America's best classroom teacher ?! Its first sentence could not be further from the above quote:

Rafe Esquith is the most interesting and influential classroom teacher in the country, but he is not getting nearly as much glory as he deserves.

I read Washington Post regularly. Along with NPR and BBC, it's one of my favorite news sources. I expect more from them than just stating that somebody is the most interesting and influential classroom teacher in the country. What is the criterion for "interesting teacher"? Or "influential"? Have they visited every single classroom in the country? How many people have participated in determining that he should get the title?

I suppose after seeing such opposite opinions there is no other option but to read the book myself, and make my own judgment. I was going to say that I'd prefer my news to be less grandiose and more specific, but after paying little more attention to where this article appeared, I realize that it was written in a weekly education column Class struggle, so I guess there is no reason to complain.

Observations cont'd

I visited a new classroom and a new teacher. Although there are great similarities between the two classroom, there are some major differences as well. One of the things that I liked immensely in the new classroom was teacher's constant inquiry and/or revelation of alternate methods for solving problems. The beginning of the class was spent by going through some problems from the homework. 4-5 students were on the board at the same time giving their solutions to the problems that other had asked to see. Once they were done writing, each student had to explain their solution to the whole class. Whether the solution was correct or needed fixing, the question that would be asked was whether somebody had a different solution. Sometimes there was, and they would go to the board and write it out. If nobody offered anything, the teacher gave her solution. I am pretty sure that on all but one (sample of 10) they have shown more than one strategy.

One of the things that bothers me and seem to be common across the board: each class was interrupted by something/somebody from the outside anywhere between one time and three times during the class. I am not talking here about student's random talking, but PA/phone/students walking in or out/student council announcements/teacher aids.... This is something I do not remember ever being done when I went to high school. I don't even know if we had AP!

Observations

There are two things that I'd like to mention today. I finally finished The Teaching Gap, and I find it full of very good observations and recommendations. It is sad that it doesn't seem to have been noticed sufficiently. I would like to give one more quote that is on a similar in spirit to one below that compares teaching to acting.

Doctors don't try to figure out a new technique or procedure for every patient who comes into their office; they begin by using the standard techniques and procedures based on the experience of many doctors over the years. Nobody considers this a way of doctor-proofing medicine, although they do have a name for the failure to use standard practices -- it's malpractice. The standard practices that doctors use contain the wisdom of the profession.

Last few days I have been observing a class in a near by high school. In this country teachers are not often observed, and from what I can tell even when they are, they do not seem to consider it an opportunity to improve their teaching. I asked the teacher in question today whether she would like or mind me making a couple of comments about what I've seen. Her immediate reaction was less than welcoming and she told me that she gets little defensive about it. On one hand, I suppose there is really no reason for me to think that what I have to say would improve student learning or her teaching. On the other hand, if I were her, I would at least want to know, so that I can judge myself whether the comments were warranted or not.

Another thing that book mentions, and that I am becoming more and more aware of is the following. A lot of teachers can explain, in theory, how they should be teaching. They also may think that they are doing what they know they should be doing. However, their practice is far from their words. I guess what I am trying to say is that how can one expect teachers to apply their theoretical knowledge if they don't know what the application should look like.

Technorati tags: teaching, math, observation

Over-achievers

We listen about under achieving students, poor quality of education, inadequate preparation and low test scores so much, that it was quite refreshing to read this New York Times article. However, (I think I may have already mentioned this somewhere below) moderation isn't bad at all. A high school kid with a working day 6 am to 1 am is little scary. When do they get to relax and have fun?

Technorati tags: education, students, content quality

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

Michigan Radio and probability

I walk my dogs twice a day. The dogs are not known to be the best conversational partners, and they usually forgive lack of intellectual stimulation from me, so I almost always listen to Michigan radio (local NPR station) as we walk. Half way through our walk there was a weather blurb that, among other things, said: "And there is 70% chance of rain this evening in southeast Michigan including Ann Arbor and Detroit". I look around me and think: It is 5:20 pm, which during this season I believe qualifies as evening since it is pretty dark out here. I am in Ann Arbor. True, it's not pouring rain, but something is falling, and trust me it's not cats and dogs. Last time I checked they called it rain. Now, what needs to happen before they say that the chances of rain are 100%? Or even better, why can't they say "It's raining, and it doesn't look like it's going to quit any time soon"?

Oh, and one more thing: Ann Arbor and Detroit are in southeast Michigan. No need to include them just for the weather report.

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 :)

Gifted students

It appears that nothing in this country can just be average, which makes me wonder where this phrase "average Joe" came from. About the gifted, from Washington Post. It is good to hear about the gifted every once in a while :)


Technorati tags: elementary education, math, gifted

Frustrations with blogger

I've been trying forever to write a post, but this stupid thing keeps popping all sorts of junk when I try to type. I go between "Compose" screen, "Edit HTML" and "Preview" and I haven't been paying attention to patterns, but every once in a while it totally junks my html code with all sorts of additions of stupid symbols and it's a pain to recover it (and every once in a while it also delets parts of what I wrote). Incredibly annoying!!! If you know what I'm doing wrong, let me know.

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