The n-Category Café

In lower-level courses, where the students are still learning formal mathematical style (e.g. meaning of unique, distinct, …), I have found it useful to award 20% of the points for writing style. Then I try to make clear what this means. The two most important words to tell them are precise and concise. This seems to help them break the habit of writing down every single thing that occurs to them in the hope that somewhere in there the grader will find something worthwhile.

I also like questions that ask them to give either an example of X or a proof that no such example exists. I find them much less soul destroying to grade than the standard question that asks them to prove Y. For example, give a complex number that generates a degree 11 Galois extension of Q.

Maybe they’re more fun to grade because you often get unexpected but correct answers, so there are many ways for their mathematical personalities to shine through. Also you can often tell instantaneously if they got it right or not. I even think they are better measures of understanding. But I don’t know how you could make the proof of the infinitude of primes into something of this format, and it’s hard to get more important than that. But with that question, they can only either meet your preconceived ideal or come up short, so you’re almost setting yourself up for disappointment, in my opinion.

At least where I live, one problem is that too many students decide to become teachers not because they enjoy teaching, but because they fail in what used to be called “Diploma” studies here and what is now, following the Bologna process, called the “Master” you all know and love.

That leads to a vicious circle, disastrous for all those involved.

Because, let’s face it: math and physics are dry on first sight. That’s why there’s a national geographic TV channel but not a math TV channel.

The thing about math and physics is that these are esoteric subjects: only those initiated will be grasped by its fascination.

We had a funny story here about a student who went to a professor for advice and told him: “I am trying to become a teacher for physical education [since there are too many applicants, it’s hard to get into this program]. If that fails, I can still become a math teacher.”

Comparing teaching at the university since some time with years of private tutoring for (university) students, my impression is that:

- college/highschool pupils asking for private math tutoring are usually brighter than students in the math lecture.

- school trains students to learn by memory and that only fools try to learn by understanding. They are encouraged to play ‘learning’ .

- declined language skills (e.g. the average vocabulary fall by ca. 30% here the past 2 decades) causes problems in understanding math statements.

- the communication in private tutoring is much better than in the university.

I taught 5 semesters of Elementary Algebra and Intermediate Algebra at Woodbury Universaity, in Burbank, California. My students were mostly majoring in Architecture, Business, Fashion Design, Interior Design, Graphic Arts, or Animation.

Hence I developed a curriculum, approved by the Math Coordinator, which played to the visual strength of the students.

I gave half credit on exam questions under the system: “If you don’t know the equation, draw me a good picture. If you don’t have the equation or a picture, write me a paragraph or so of narrative. I just want to know what is going on in your head, so that I can help you.”

I worked into the course slides of Leonardo da Vinci drawings, dozens of ping pong balls and whiffleballs and party hats cut and pasted to cones of differing angles and with various conic sections, and the breakdown of the Federal Budget as reported and graphed in the New York Times.

Best narrative answer on one of my exam questions:

Q: “How would you use a programmable calculator to solve the following problem…”

A: “I would go to someone who took this course before and say, ‘Dude, if you tell me the answer to Professor Post’s question, I’ll give you this programmable calculator.’”

I gave full credit for that. Obvious Management material.

This summer I am teaching, technically as a Substitute Summer School teacher, at a local High School, to predominantly poor Hispanic and African American students, many of whom have been unable to pass the Califonia High School Exit Exam.

In my interview, I asked the charismatic Principal (who’d been awarded “California Princcipal of the Year” in 2005): “If your car breaks down, do you go to someone who has read some books about cars, or to someone who fixes cars every day? I do Math every day because I love Mathematics.”

I like teaching the best students (my son just earned his double B.S. in Mathematicas and Computer Science, at age eighteen), and the worst students. Anyone can teach the ones in between.

I tell students, and demonstrate it every day: “Every number shows a picture. Every picture tells a story. So every number tells a story.”

Here is why Department Chairs get grumpy:

Student number 1 who has not been advised for Fall semester walks in and does not understand that his advisor is on vacation during the summer term. Student has all but flunked out because during his first year in school he worked 3 jobs. He reports that he will take nothing but PE courses next semester to bring his GPA up. He wants to teach math at the high school level.

I explained to him that by taking English 101, elementary statistics course, and whichever math course he already flunked out of, and by studying, he could bring his GPA up. I guess the idea of studying while in school was too strange.

Student number 2 came in to complain about her instructor and wanted to change sections. Summer school here is well into the 3rd week. I was unwilling to do this for the student.

According to the student, the instructor told students that the material on the first test would not be taken directly from the homework and that there would be problems that they had not seen before. I explained as patiently as I possibly could that this meant that the student should look at the homework problems that HAD BEEN assigned, and then try to work the problems that HAD NOT BEEN assigned. I also explained that the instructor was trying to get students to think on their own, and to learn to synthesize the material.

During the course of the interview, I tried to help the student learn how to read the book, understand that chapter and section titles contained words that she was expected to understand, and I repeatedly asked her to explain concepts that had been introduced in the book. She did not do well at this task. The student repeatedly thought that the instructor’s job was to teach her — which it is — but she was not taking the personal initiative to learn the material. Apparently, reading the book and working a substantial number of non-assigned problems was thought to be unreasonable.

Since she was concerned about passing the test, I did everything that I could to tell her how to pass the test: Work the assigned problems, work the unassigned problems, read the book, and understand the concepts. I showed her where to find problems in the chapter review, and we went over some of the problems in the most recent section on which she had done homework.
While I was not sympathetic to her complaints about the instructor, I did tell her, as well as I could, what the instructor might be expecting. She thought that she had purchased instruction, and that her teacher was not teaching her. When she received instruction on how to study from me, she did not want to learn what I was teaching.

Her request was to change instructors; I learned after she left my office that the other instructor did not want her in his class. The other instructor had enough boneheads already in his class already.

Student number 2 was majoring in international business. I guess she won’t need much math there, will she?

A plethora of students who major in elementary education can currently get certified to teach with a D in precalculus algebra. “I won’t be teaching little kids THAT.” I keep wondering why not. When *I* was in college, Herstein’s remark in the Preface to Abstract Algebra seemed prescient, “The last few years have seen marked changes in the instruction given in mathematics at American Universities. This change is most notable at the upper undergraduate and beginning graduate levels. Topics that a few years ago were considered proper subject matter for semiadvanced graduate courses in algebra have filtered down to, and are being taught in, the very first course in abstract algebra. Convinced that this filtration will continue and will be intensified in the next few years, I have [written this book].”

Well gee, I don’t know about you all, but I am still waiting for group theory to make it into the high school curriculum. Why can’t it? Not because the students are particularly dumb, but their teachers think that students can’t understand abstraction.

Most of the children I talk to can handle abstraction pretty well. It is the adults who have trouble.

Every third grader that I have encountered can understand modular arithmetic. Most can learn to multiply using differences of squares, and with a little patience and a few floor tiles, they can understand that 1+3+5+…+ (2n-1) = n².

College students after a semester of work still cannot compute the slope of a line, nor complete the square, and administrators are asking why more are not passing.

One local administrator, currently retired from the university and now an elected official on school board wanted me to articulate some set of skills that I thought every college student should have aquired. I thought but did not say, “Adding fractions would be nice.”

There is a culture against mathematical thinking because it is thought to be too difficult. Personally, I always found baseball to be much more difficult — a GOOD hitter gets on base less than 30% of the time. Coaches can yell at the team because it builds character, and it is for the good of the team. A teacher who tells students that they are not working is being condescending. What about the good of society? Bridges that don’t fall down, a careful analysis of planetary temperatures, and the ability to solve some problems?

That felt good!

Speaking as an undergraduate student, I think the most beneficial course structure I have encountered was from Dr. Gierz in both, Math 145B, and in Math 172 this quarter. Although it may seem to some that he is making it a bit easy on us by taking all of the quiz and exam questions directly from the homework, with no alterations to the problems, it was actually very helpful. The reason I say this was beneficial is because he graded primarily on the writing of our proofs. We still had to learn the concepts and theorems, but it was our proof writing ability that he was focused on, and I feel as though I have improved ten-fold in that area.

I’ve just graduated with a BS in Math while tutoring calculus and differential equations students during my final semester, so my experiences with these same issues are very fresh.

I saw instructors making two key mistakes. The first was to assume that assigning homework makes the students get practice solving the problems. The second was in not using a variety of methods to explain what is really going on when teaching crucial concepts and expecting the algebraic proof to also be a sufficient explanation.

–As a tutor, I saw how networked the students are. You can make a simple model in which each student has a probability P of being able to solve each problem on the first attempt. A student with a P close to 1 has little incentive to join a study network, but may choose to join one for social reasons. A study network has little incentive to accept a member with a P close to 0, but may choose to accept such a member for social reasons. What you end up with is a network with a lot of unconnected A and D students who know either all or none of the material, and clusters of B and C students who collectively know the material but do not know it individually.

The remedy to this “efficient” homework computation seems to be to give short weekly quizzes in class so that students are forced to grapple with the problems individually within your sight and so they feel immediate pressure to know that week’s material. Students then change focus from telling each other homework solutions to telling each other how to solve each type of problem. The few classes I took with weekly quizzes seemed to produce a few more drop slips, but also a lot more As at the end of the semester. Sure, grading quizzes on the weekend is annoying, but hopefully the pleasure of grading the better midterm and final exams makes up for the misery of grading the quizzes. (there’s also the altruistic payoff of helping students to learn math, if you’re into that)

–I had about a dozen different tutees last semester. A large proportion of them said in our first meeting that they were totally lost in lecture. Their notebooks would typically be full of notes from the first quarter of half of each lecture and then doodles after that point. (a clue that they are visual learners?) Some of them had found lecture so uninformative that their attendance had become sporadic. They had different lecturers but similar complaints about a lecturer who did little but fill the board with symbols without ever interacting with the class. Some students came in with no idea how to even start working on the problems. After I explained that week’s concepts using pictures or English and answered a question or two, all of my students were able to solve the problems related to those concepts. The lecturers thought that their job was to prove theorems to the students, while I thought my job was to explain what the theorems mean and answer their questions.

I know that the Math department here is trying to get professors to accept that students have different learning styles. They are supposed to try to present the concepts to visual learners, experimental learners, auditory learners, et cetera. I don’t think most professors really believe that there exists more than one valid way to learn mathematics: the way that they learned it. I don’t want to sound like I’m blaming the professors, but I also don’t think it’s useful for professors to blame the students or become bitter. I had a very old and grumpy professor in Linear Algebra wait until the meeting after students had filled out the professor evaluation bubble sheets to tell everyone that math education students were ruining his classes and forcing him to dumb them down so much that teaching wasn’t any fun.

Sadly, there seems to be nothing that professors can do directly about the problem of ill-prepared students. Universities, under financial pressure, are increasingly using TAs and lecturers in place of full professors for lower level courses. These replacements may not really understand the theorems themselves! They would then be understandably reluctant to expose themselves to questions. Indeed, the problem has roots deep within the public school system. From what I understand, the teacher’s union has repeatedly rejected calls for differential pay based on area of expertise. This has led to a shortage of teachers with lucrative math and science degrees such that fully 50% of public school teachers of these subjects do not have the relevant degree! Why shouldn’t students of these teachers give their professors headaches when they finally arrive at their junior-level classes with little more than low level symbol manipulation skills and lots of unintegrated bits of knowledge? I have no choice but to agree with my bitter Linear Algebra professor. The students who don’t have a deep understanding of the subject are usually the ones who go into math education. But to give up on these students is to perpetuate the problem and make future professors become even more grumpy! As an ancient Roman poet could have said: who teaches the teachers?

I actually think they get grumpy as a defense tactic. It makes perfect sense and it is a very strong strategic move. Imagine there are two sections in an undergraduate modern linear algebra class: a class that is generally required for math majors but would be considered by boneheads to be on the higher end of difficult undergraduate courses. The two professors teaching the two sections have very different reputations. “You’re taking linear algebra next year? AVOID PROFESSOR XYZ! If you value your gpa sign up for the other section. Professor ABC is so much nicer and more fair of a grader (i.e. he is more likely to give you a decent grade despite how poorly you perform because he is jaded and/or simply doesn’t care as much). Professor XYZ is an embittered and unfair tyrant in the classroom (i.e. he doesnt take bullsh*t from slackers and has a higher standard for learning). Then what happens is that all the slackers sign up for Professor ABC’s class and get C’s. The minority of people who have an actual interest in mathematics will sign up for Professor XYZ’s class along with a few slackers who were too lazy to register for classes in a timely manner, and the overall experience will be more pleasurable for Professor XYZ.

I suspect this is probably not something easily fixable in the blogging software but: is there a reason why the comment box appears to be a fixed size in pixels? It’s a bit annoying because I’ve got a largish screen and use larger than default-size fonts but the comment box resolutely sticks to its size, which means I get about 36 characters on a line in the comment box. This makes seeing larger stretches of the text for editing annoying.

I asked 9th, 10th, 11th, and 12th grade summer school students at a Pasadena high school to write a paragraph on (their choice) “Why I Love Math” or “Why I Hate Math.”

Their responses (names removed for privacy, no spelling or grammar corrections):

=====================

I wouldn’t necessarily say that I hate math, I just dislike it a great deal. Algebra is currently not my best subject and has never been. The reason as to why
I do not like math is because it is confusing and I get headaches from trying to solve all of these equation and word problems. I am very good at basic math, but when it comes to Algebra, I am terrible. If teachers found a way to make math more fun and interesting for their students they would be more interested in learning it. And not all the students
have the same learning style so you would have to put that into mind too. But as of now, it is hard, boring, confusing and at our age we just don’t see the need for more of it in our future. But then again,
that is my personal opinion. I’m sure that there are many people who love math and find it very easy and fun to learn. We’re all different. There will be
things that you like in life and things that you don’t like. There will be things that you find extremely easy and things that you find extremely hard. That is
just the way life is. Think of it as an obstacle that you have to overcome to get where you want to be.

=====================

Math is my least favorite subject. I really do not like math one bit. Although I think math is unique because it is everywhere you look and is the same in
every country, I can not seem to understand it. All my life I have not been able to understand the math. I honestly don’t think I should have passed
Pre-Algerbra but, I am very grateful that I did.

I have never been good at math… why? It could be because I’m just useless when it comes to math, or I just don’t try hard enough.

=====================

I didn’t always hate math, in a matter of fact I used to like it a lot. It wasn’t till I started 8th grade that I began to lose my love for the subject. Up till
now, math has become very troublesome now that I’m having such a hard time with it almost all the time. What I think I need to do is to slowly go thought all of it, to improve my overall understanding

=====================

Well I don’t hate math I just don’t understand. How to do it Sometimes but sometimes I do hate math maybe because of the teacher, Classmates, or because I
didn’t understand it and the teacher call on me then I don’t answer question quickly but yes I don’t hate math I just don’t understand it but I’m willing to learn How.

=====================

The reason why I hate math because it really doesn’t interest me anymore. Math used to be my favorite subject when I was younger but not it’s whatever to me. Math can be very easy or it can be very hard to
you. But personally math isn’t my subject anymore. The reason I hate math, because it’s no spark to it. I just hate math period.

=====================

Why I hate math its because all the fractions and the ather things and all the Decimals and metric system, percent thats why Hate Math.

==========================

Mathematics is the most challenging subject for me. I don’t like math because I get lost so quickly. When I don’t understand it after the teacher just explains it I get frustrated. When I ask for the teacher to repeat it all the kids get upset that makes me feel embarrased. I sometime like math when I understand
it and get good grades. But when it comes to fractions or memorizing formulas I completely hate math.

==========================

I hate math because there is to much thinking and that makes my head hurt. It also gets me tired because it is boring. I never like math because when I was a
little kid I wouldn’t like to work with numbers. I also hated the way they teached it. Math to me is very boring.

==========================

The reason I don’t like math is because you have to memorise alot of things. You have to memorize alot of formulas and alot of ways of doing the things. I don’t like math also because some problems are very
hard to understand, or to do Math just makes you think and work to much.

==========================

Everyone has a Subject that they dislike. I hate math Just because it’s a group of numbers put into one problem. And Just the chances at getting a probelm Correct is low chances. And it affect people
emotionally they start thinking and well it can not be over with.

==========================

I hate math because math is really hard to do. Math has so many rules to follow to be able to solve a problem and if u don’t follow them the problem will be completely wrong. The problems are very complicated
and sometimes can be confusing. I don’t think that I would need to use this kind of math besides just the basics in our lives. All these fractions and decimals
may be common but math is my worst subject.

============================

I hate math because it confuses me. Similar to girls, it confuses me more when I try to understand it (them). I have never been at ease with the subject. I feel intimidated by it, and as a result, I tend to second guess myself when solving problems.

============================

I hate math because it make’s you think too much and it hurts my brain and I don’t like thatabt.

============================

I don’t like math because it is boring I really dont like math It makes my head hurt and math doesn’t really make me think. Math makes me sleepy. I use to
like math when I was in 5th grade but when I hit middle school I started hating math I don’t seem to get math know to me math is hard.

============================

Hi my name is ******* ********. and I really don’t care for math. I have a hard time understanding certain parts of the math. When I get stuck on a problem I lose focus and then give up. But I would
like to get help so that being able to understand math will make it more fun and easy.

============================

First. I don’t like math because never passt this class and I don’t understen Nothing I want passt math in summer school because I don’t want repid more math. Second. The math little bet is my faborite class but want understen math this class is Hart. Thirth. matematic I don’t like but make my major idea for understen and passt the math class.

=================================

Math is the worst subject Invented. Its so hard and intricate. URCTH. It involves way too much thinking and brainwork. I rarely do the work w/out a calculator. And I cheat a lot every chance I get now
don’t get me wrong. I do good at everything (SUBJECT) else I do. So maybe I could learn to discipline to myself to focus and excel at math, cuz I’ve BEEN
failing since about 2nd grade. So do what you need.

=================================

I honestly dislike math because of of one good reason, and that is all because I have taken Alegbra 1 for about 3 years now. It is the same stuff over and over
again. Having to go through the same course so much has gotten on my nerves. I used to like math, until my 8th grade teacher did not want to transfer me to a
higher level of math because I had gotten a C average. That upset me in such an enormous way because I PASSED!

=================================

Math is like whatever to me I like it when I get what their teaching me. And when I don’t get it, I hate math. Overall math is OK, I don’t necessarily hate it
or like it.

=================================

What I think about Math. There are some things I like about math and Somethings I don’t like. I like learning new math consepts. I also like the feeling
of getting my answers right. I do not like taking math test. I do not like the tests because of the memorizations.
=================================

Right now I hate math because it’s complicated for me, back then it wasn’t because it was all basic for me and easy, but not when I entered Pre-Algebra that’s
when I said to myself “damn this isn’t basic’s any more”. After that it just started to get complicated in Algebra or I just started to forget the equations or lesson the teacher told me. Another reason I hate math is because I always get low grades on tests no matter what I just get a D- or F. Another reason I hate math especially when it is mathematical word
problems, man they are so confusing, I don’t even read the problem or equation.

==============================

Although I’m in summer school to take math again, I like math. I like doing math sometimes. Sometimes I don’t like it. I used to be good at math, really good. Then I just froze, and since then I haven’t really been good at it any more.

==============================

I don’t hate math or like math I just find it unfavorable.

For me it is always been harder to understand than most subjects.

There is just something about numbers that is confusing especially fractions.

I know that I need Algebra 1 i just wish it was easier to understand.

==============================

The things I hate about math is this doing homework. I mean whats the point of doing it at home. I mean where not in the classroom. Another thing I hate doing is working out long problems, I really don’t
feel like doing it I mean who does.

Now lets talk about things I like about math is the learning experience. I mean even though I’m very smart, and intelligent individual. And when I feel
like working I and really focus.

So that’s the things I like and hate about math.

==============================

I love math because when you get job you have to no math to count the money you get. Another reason why I love math is because it is more easy then something
else and you will need it if you want a good paying job.

==============================

I love math because without math I can’t count my money.

And also the more math you learn the more money you make. You use numbers everyday, without knowing math, Algebra, how are you gonna know if your getting the right change back from the money you spend. So that’s why I take math.

But I still don’t like to sit in a math class.

==============================

Copyright (c) 2007 by Computer Futures, Inc.

A study about the be published in Science, and my subjective reaction.

Contact: Steve Bradt
617.496.8070

College Science Success Linked To Math And Same-Subject Preparation

Cambridge, Mass. - July 26, 2007 - Researchers at Harvard University and the University of Virginia have found that high school coursework in one of the sciences generally does not predict better college performance in other scientific disciplines. But there’s one notable exception: Students with the most rigorous high school preparation in mathematics perform significantly better in college courses in biology, chemistry, and physics.

The work will be published this week in the journal Science.

Authors Philip M. Sadler of Harvard and Robert H. Tai of Virginia say the findings run counter to the claims of an educational movement called “Physics First,” which argues that physics underlies biology and chemistry, and therefore the traditional order of high school science education – biology, chemistry, physics – should be reversed.

“Our findings knock out one of the primary claims of ‘Physics First’ advocates,” says Sadler, F.W. Wright Senior Lecturer in Astronomy in Harvard’s Faculty of Arts and Sciences and director of the Science Education Department at the Harvard-Smithsonian Center for Astrophysics. “Taking more physics does not appear to improve students’ subsequent performance in either chemistry or biology courses.”

“Many arguments have been made for chemistry and physics preparation to benefit the learning of biology,” says Tai, an assistant professor in Virginia’s Curry School of Education. “On the scale of single cells, many processes are physical, such as neurons ‘firing’ electrically. Also, the complex molecules at the root of life obey chemical laws that are manifested in macroscopic processes. Yet our analysis provides no support for the argument that physics and chemistry principles are inherently beneficial to the study of biology at the introductory level.”

Sadler and Tai surveyed 8,474 students enrolled in introductory science courses at 63 randomly selected four-year colleges and universities across the U.S. The students reported on their high school coursework (0, 1, or 2 years) in biology, chemistry, physics, and mathematics; this data was then correlated with their ultimate performance in their introductory college science courses. Sadler and Tai subjected this raw data to robust modeling to correct for socioeconomic factors that may advantage some students, including race, parental education level, and mean educational level of students’ home communities, as defined by ZIP code.

Not surprisingly, the controlled data indicated that high school preparation in any of the scientific disciplines – biology, chemistry, or physics – boosted college performance in the same subject. Also, students with the most coursework in high school mathematics performed strikingly better in their introductory biology and chemistry courses in college; introductory college-level physics performance also benefited. Conversely, little correlation was seen between the amount of high school coursework in biology, chemistry, or physics and college performance in any of the other disciplines in this trio.

“The link between math and biology is not exactly an intuitive one, but biology has become an increasingly quantitative discipline,” Sadler says. “Many high school students are now performing statistical analysis of genetic outcomes in addition to dissecting frogs and studying cells under a microscope.”

The current order of high school science education was established in the 1890s, in an attempt to standardize what was then a system of wildly disparate science education in high schools across the U.S. Biology was given primacy in that ordering in part because the late 19th century experienced a flowering of interest in the natural world, and also because it was perceived to be less daunting intellectually than either chemistry or physics.

Sadler and Tai’s work was funded by the National Science Foundation’s Interagency Educational Research Initiative.

===============

This is good news indeed.

What my wife and I found, teaching Science courses in colleges and universities, was the problem of students being under-prepared for Science, particularly because of weak Math.

The Harvard University/ University of Virginia study is, to me, another justification for my nearly completed time in the trenches of an inner city high school, teaching Algebra. A few of my students have a chance, now, if they go on to college.

In early September, I’ll start teaching Chemistry, Earth Sciences, and Biotechnology at a school-witin-a-school (Health Careers Academy) at that high school. This will allow me to asses, for this school, the level of hands-on lab technique, and qualitative abstract reasoning, with the dreadful level of math that I observed.

Social Promotion seems to be one culprit. Some of my summerschool students explicitly, in writing, wondred why they had been allowed to pass “pre-algebra.” As a result, I spent more time reviewing “pre-algebra” – i.e. how to add, subtract, multiply, and divide fractions and decimals, than I did hammering home what equations were, how to manipulate them, and how they related to the physical and social world.

Yesterday being the anniversary of Syncom 2 (26 July 1963), I took my class back to the Age of the Beatles, told them stories about how Arthur C. Clarke lost a billion dollars in his spare time, and guided the kids through finding diameter, and circumference, of cricular orbits of various radii, and the speed a stallite needed to go to be synchronous, and how fast the Earth was moving around the sun.
In the process, I had to confiscate a skateboard and 3 iPods being used in violation of posted classroom rules. had I moved faster, I’d have had 2 skateboards in use (inside!) and 6 iPods. These get locked in the Principal’s office safe, and need a parent to retrieve.

Those students who cooperate get to (quietly) play Chess and cards during class.

Carrot and stick.

But Math seems to be at the root, beyond behavioral problems, for these poor kids. Some live in Group Homes. Some are technically homeless. Many do not see a father.

If I cannot save my students, with all my passion and expertise, I don’t know who can. If I can make a difference, then I can have hope for Western Civilization.

Public education is badly dysfunctional in America. Better to light a candle than curse the darkness.

===============

– – – – –
definition → theorem → proof
the words “obvious” and “trivial”
general statements
jargon
facing the board

++++++
pictures
examples
curiosity
questions
handouts
personification
applications
answering the question “Who cares?”

Students don’t need someone to copy a proof from a book to the blackboard in front of them, or to be told that an eigenvector is “defined as a vector that contains all the eigenvalues of a linear transformation”.

They need someone to get them excited about why eigenvectors are important, what you do with them, what they’re LIKE, and then tell them what they are.

Start with questions, or examples. Then build to the general.

The conventional style of writing proofs up on the board is a waste of students’ time. Every theorem taught in undergrad has been looked over by 10^n mathematicians who are more likely to see mistakes than someone just hearing it.

Students need explanations, not proofs. Pictures, stories, characters.

Chemistry teachers talk about bonds and chemicals having personalities. In physics you might say a ball “wants” to roll into a potential well. Science teachers also lie to their students. Economists say there is “an” interest rate, and chemists say orbitals are circular. Electricity “flows” and, you know what, it’s all a metaphor anyhow. No one needs to waste their attention

Math teachers need to perform more, lie more, draw more, and face the board less.

Always start with either a motivating question, a curious puzzle, or an applied example, and move to general proofs later.

And, ALWAYS give students print-outs rather than making them copy down funny symbols.

Lastly, the following question should never be dismissed:

“So? Who cares? Why should I spend my time on this?”

I have just graduated with a BA in mathematics, with a straight 4.0 over 190 credit hours and a math honors thesis (in Knot Theory, working on a follow up paper). I am also a retired software developer, returned to college, which was never possible before I retired.

The reason for stating the above is that I an a older adult and am not a student complaining about instructors because I fail to understand the material or get a poor grade. I will also say that every instructor that I have had has been a very nice and helpful person. I have enjoyed virtually every class. So, here are some of the problems that I see and have had with education in general and math education in particular …

1. First, if I have to take notes, I can either take notes or listen to the instructor. I cannot do both. Once I start writing I don’t hear anything. Perhaps that degree of focus is unusual, but nobody is truly capable of multitasking. If there is not an associated textbook covering the same material, then the instructor needs to provide handouts. I try to use a phone to take photo notes, but is not ideal and very, very slow to transcribe. Not to mention that my writing is bad enough that you could probably take something I wrote to the pharmacy to get drugs.

2. The above is made worse when so many of the instructors have very, very heavy accents. US students generally are not exposed to many accents (and I, in particular, grew up in a very homogeneous area), and so our ears are not trained to understand accents. Just the difference between a Texas and Jersey accent can be problematical. Usually, those same people have writing as bad as mine. So it is important for the handouts to not be handwritten. LaTeX is wonderful there. Even Word Equations works ok. For the first couple of years I did all of my homework in Word Equations and switched to LaTeX after that. (I am also the author of the logix package on CTAN.)

3. Fully working examples on the board is very important. For example, in one class we were given principles. The instructor was a very nice person – but new to the teaching game. We were given problems to work, and he worked directly with anyone who had questions. I did not have questions because I made an unwarranted assumption and thought that I understood the problems. A single worked example would have nixed that problem in the bud. This was not math class, but in motors and controls (Electronics Engineering). Regardless of subject, working examples helps the students to understand how things fit together. Quite possibly having individual students do the problems on the board in class (with help from both the instructor and class) might help even more.

4. In a different Engineering class, I was astounded to discover that the instructor pretty much hated mathematics – even as a person who worked in that field before teaching. Even basic arithmetic. Forget Algebra or Trigonometry. While this was community college, Algebra was a requirement for the AAS degree and to enter the college students had to pass basic tests in mathematics (so that, if necessary, students could be assigned to remedial math classes).

5. Even when not taking the course load that I did, to graduate with 120 credit hours in 8 semesters requires 15 credit hours a semester. That is typically five courses. Unless those are fluff courses, many instructors do not seem to understand that theirs is not the only course being taken, and time flexibility is important. I had one semester where all of my seated classes were on Monday (a twelve hour day, before commuting) – and it seemed like every instructor always thought that their assignments were “easy” and could be done by Wednesday – or even Tuesday. I found that time flexibility was one of the most important features that helped. When I transitioned to the university, I then had to commute and had other overhead. I could not predict in advance the amount of work for each class so time constraints prevented me from doing as good a job as I would have liked. Easy for the instructor is not always the same thing for the students. Even good students.

6. In computer science classes I was introduced to the horror of “flipped” classes. This is where you do a week’s of programming during class, and study outside of class. Working on anything during class is simply a really, really bad idea. Because it is the worst possible working environment. Very noisy. Uncomfortable seating. Unnecessary time pressure. Working with play computers (laptops). Working during the day when I am night person. And no programming assignment can be done in an hour. Usually a full day is required, sometimes two.

7. Almost as bad are working in “groups”. I encountered that nonsense in all kinds of classes. If you are a good student, that just slows you down and either you do all of the work anyway (my usual solution) or your grade is held hostage by those who either don’t understand or don’t want to do the work. And if you are a bad student, you get a grade you don’t deserve. My grade is my grade. Period. Almost everyone in every group. And group assignments are far, far from working in “teams” in industry. They just don’t work like that.

8. As a mathematics major, I was good in every single class – usually the top student. However, I doubt that I could solve any complex Calculus or Differential Equation problem at all. Why? Simply because I haven’t used either of those in over two years. As one class follows another, there is no chance to retain and practice the previous material. And not just those topics. Every topic. Sure, I can pick them back up quicker – and, in fact, I will need to do that to take the GRE to apply to graduate school. I have a couple of years waiting for my wife and daughter to finish their degrees, but ideally, that should not be necessary.

9. Many instructors really love their subjects and want to teach as much as possible. I have had quite a few instructors that say “time permitting, we will cover xyz”. That never happens. But, they try to cram in more that is useful. Often the last several weeks worth of material are not covered in the final, because the instructor rushed that part too much. I actually had one instructor who had homework due before class even started. But, to really learn the material, you do need to work extra problems as others have stated. This is great when you can take one class per semester. Then you can really do it right. But, alas, that is not the case. Students will be very, very lucky and skilled just to complete the assigned problems. It always takes students much, much longer to do a new class of problems than the instructor thinks. Not only is each class of problems new, they really don’t have a firm grasp of the previous set of problems, or the set before that, etc.

10. Many classes really require prerequisites that are not stated. In my university almost every class had to start with basic logic and set theory – but there was no actually class for either. There was a class in deductive logic taught by the philosophy department, but the successor class for predicate logic was never offered while I was there.

11. That was another problem. Classes in the catalog never being offered. My university was very intensely focused on applied topics. So critical foundation classes were omitted in more than one department. I know that is not universally the case, but there are many similar universities. Because that is where the money is. But, those classes are necessary, and those interested in the actual science or in pure mathematics and logic have a tough road to carve out a meaningful degree. Part of that is not offering classes that only have one student, or two or three. At least offer them online.

12. Another issue is the transitions of “prerequisites” from “suggestions” to “implacable mandates”. That takes control of the student’s education out of their hands. Every student is different, has learned different things can has different capacities. Some need to stick strictly to the approved chain of classes. Others could take every class in parallel without problems. Especially for non-traditional students such as myself that have learned a good deal since leaving high school.

13. A class with 300 students is essentially pointless. Only those in the first row or two learn and participate. Even just 30 students is problematic.

14. There are too many required fluff courses. At my university, out of 120 credit hours for a BA in mathematics, only 34 credit hours were required to actually be mathematics (I took a lot more, every available class). Make liberal arts its own major and minor and make both optional. Nobody can learn everything in their field. Not even my auto mechanic. And it is far worse in mathematics. I understand that Gauss was one of the last mathematicians able to do that. I would suggest at least 100 hours of mathematics – perhaps more slowly paced so it is assimilated better, including a broader base of topics, with a few necessary courses in English, writing and computing. Every mathematician should be able to write at least simple computer programs. Writing software was essential for my Knot Theory thesis. More and more, computers will be an essential tool for mathematics, and not just for writing papers in LaTeX.

15. With respect to learning styles, every student is different. I, personally, cannot visualize. That is about 5% of the population. Further, I am dyslexic “light”. Meaning that reading is not an issue. Spelling sometimes, especially if I am tired. But, symmetry is the key. Be cautious about only using a single learning style. For me, not being able to visualize was useful in higher math courses, because I never visualized my way to a solution. Abstract thinking was natural to me. Also, there is a threshold in intelligence where a person can learn on their own. At least 95% of students are below that, and have to have their hand held, step by step. Those above that threshold learn 10 times faster than those below. However, most mathematicians are above that threshold, and so may have difficulty understanding students below the threshold. It really makes a big difference. I can see it in my own family. It is almost impossible to teach those below the threshold when you are above it. Yet, it is a snap to teach those above it. I have tried until I am blue in the face trying to get a trivially simple concept across to one person – and for another, just a quick comment suffices. I don’t know the answer to this, but it may be why, even for some relatively low level math courses, there is a 50% or higher failure rate.

16. Students are never taught how to learn. At the community college, there was an introductory class that supposedly included that. But, it was only covered a tiny bit in the first couple of days. All the rest was devoted to “do not cheat!” in a myriad of different forms. Largely a waste. And to some degree, even wrong. I will assert that you cannot plagiarize your own work. In fact, it is expected in industry. It is called the DRY principle (don’t repeat yourself). If a student has completed a program or essay or whatever, and it fits the bill for another assignment. They are done. However, students are not taught things like smelling Rosemary oil improves long-term memory by 40%. Nor are they taught, that the brain only holds so much new information at a time, until it is assimilated during sleep. Read a new chapter and then take a nap. Taking walks is very beneficial as well. Sometimes, they do teach to “chunk”, but there are many, many different study techniques that can help. There are programs that you can use to create virtual flash cards, where the frequency that the program will show the card to you is based both on your performance and on how long since the last time. This can be very, very effective because of the way that memory works.

17. In the past, every student learned to memorize large amounts of material. There was less material, no computers, etc. However, today, students are not taught to memorize, and generally don’t have that skill. If there are 100 definitions, axioms and theorems that may be needed on a test, there isn’t really a hope in hell that they will remember all of them. Not even the good students that understand the material. You only remember what you use (which is why I am rapidly forgetting Calculus). The more often you use material, the better you will remember it. But, the pace being set to cover a massive amount of material precludes the time needed to memorize large amounts of material. For that type of class, a test should include the necessary formulas and other materials. The student needs to understand how to apply and manipulate that material. They don’t need to memorize. If they use it, they will remember in time. But, otherwise, it is pointless.

18. If I were an instructor, I would not accept hand written work for grading. Grading is bad enough (I have not taught or been a TA, but I was a grader for one class). Word is available on both Windows and Macs. There is equivalent for Linux (which I don’t use for that type of thing, but there are a few students). Today, every student pretty much has to have at least a laptop. Take advantage of that and make it a class requirement. They can use any software they like, but require them to turn in .pdf files. Both Word and LaTeX can do that (actually, I always use LauLaTex, because I insist on Unicode). There is no reason for grading to be harder than necessary. Any student that gets to college has to be computer literate. Or get that way fast. We are not using quill pens any longer.

Some of the problems that I mention can be addressed by individual instructors, others require changes at the departmental level or even at the accreditation level. And, of course, not everything I have mentioned is an universal problem. Different instructors will have varying options on these notes. This is just my experience and viewpoint from the other side – “hot off the press”, so to speak. My experiences are fresh in my mind at this point.

Sorry, I did not expect this post to be this long. I’m afraid that I “ran off at the keyboard”!

I hope that this helps prevent some instructors from being “grumpy”!