Three sisters sat in a cluster in my prealgebra class. I was puzzled. Two of them clearly did not need to be there. They casually aced all the quizzes and exams, looking bored as they did so. The third sister was doing fine, but she was not gliding above the fray. She had to get dirty occasionally, struggling with the concepts and the calculations. I expected her to get a good or excellent grade in the long run, but it was clear that she was properly placed in the class that she needed.
As for her sisters, they didn't seem to match the two classic cases of under-assigned students. Most students who are in classes below their preparation level are either timid, afraid to take the class they should be taking, or opportunists, deliberately aiming low so that they can pad their GPAs with easy A's. It turned out that the girls were traveling as a cohort to ensure the success of their youngest sibling. The two older girls had no need of prealgebra (in fact, they were simultaneously enrolled in algebra!), but they were watchdogs shepherding their younger sister through the travails of math.
Well, that's family unity, I guess. I've seen other cases where siblings take charge (or try to take charge) of a student's education. If the sisters were willing to go slumming in prealgebra for the sake of their little sister, who was I to protest? (Even if they were taking up spaces that could have gone to students who actually needed the course, of course.)
There was another little problem. The middle sister loved to participate in class. Oh, my, did she love to participate! I could not get her to shut up.
“Okay, class. Let's start the unit on percentages with a simple example. Suppose we take a look at seventy-five percent.” I wrote it on the board. “What if we wanted to express that as a frac—”
“Three-fourths!”
“Uh, thank you, Linda, but perhaps you could wait till I actually ask the question. Okay?”
I was going to put 75% on the board, tell them (or elicit from them) that “percent” means “per 100,” rewrite the percentage as 75/100, and reduce it to 3/4. And then I could mention the noncoincidence that seventy-five cents is equal to three quarters of a dollar, as they well know, so we're tapping into knowledge they already possess in part.
Sure grateful to have been spared all the bother.
It also happened when we were solving a simple equation and we were about to divide both sides by 2 (to eliminate the coefficient of 2x). Linda yelled out, “Why don't you just multiply both sides of the equation by the reciprocal of the coefficient?”
And why don't you just shut up? (No, I didn't say that. I just thought it very loudly.)
This went on all semester. I tried everything.
“Please let others have a chance, Linda.”
“I distinctly said that people should raise their hands if they have the answer, Linda. You pay such close attention that you should have heard that.”
“How about letting someone else have a chance, Linda?”
“Linda, you know that I know that you already know this. Try giving it a rest.”
“Linda, I've looked into the matter, and it appears that modern-day teachers are discouraged from stashing their students bound and gagged in the back of the room. I'll keep checking to see if there's a loophole.”
Yeah, I really said that. It actually kept Linda subdued for the remainder of the period. And a little more restrained in subsequent class sessions, but by then the semester was all but over.
We were probably both glad that it was over. Linda had no impulse control and I was rapidly losing mine!
And now for a bit of entirely pertinent entertainment from Stephen Fry and Hugh Laurie:
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20 comments:
Re: Laurie and Fry. There are times when I regret that they grew up and went on to bigger and better things.
I keep being astounded by your posts on the subject matter of your courses. This is 4TH GRADE math!! I think my 10 year old cousin has passed these subjects already. I don't think anything anywhere near even the dumbest colleges I know are at this level, regardless of subject. Basically, I can't imagine this kind of education being given to anyone over 15 in Israel, unless they are foreign workers or for some reason had no education (very old)...
Is this common in USA?
On another note, in University, in calculus and other difficult courses, I always find the process of questions asked by the prof a bit bizarre. The result is the same nearly 100% of the time - half the people are still writing frantically what he wrote on the board and aren't listening at all, a whole bunch of people are just sitting there with a dumb look on their face, having completely lost track an hour ago, and a small few know the answer as completely obvious, and don't bother replying. (I'm almost always in the latter group, though on a few occasions, in the 2nd group)
Basically, most of the question asking is just idle time, giving those who write on their notebooks a little buffer time to catch up. Until the teacher just ignores the silence and moves on.
(Sometimes, if I'm feeling particularly pricky, I'll answer the questions as quickly as possible, just to keep the lesson going, which totally upsets all the people writing :)
I do enjoy the rare, really difficult question, that does come up, and I get right, and everyone turns out to look at me, bewildered where the hell I got that from :) "So, whats the total amount of combinations of bla bla bla?" "2^(5^2), about 32 million!" (I though of the question and the answer well ahead of when he asked)
Oded, open-admission institutions like my community college in California routinely offer remediation classes that start at A-B-C and 1-2-3. In my case, as a math prof, that means arithmetic for people who don't even know the multiplication table. It's difficult to believe that there are adults who can't figure out 4 times 6 without a calculator, but I've seen it often enough to make it less amazing. These are people who are older and forgot things or -- in most cases -- never learned it in the first place. It can be a depressing class to teach, although one learns to take comfort in the gratitude of the innumerate who learn their lessons and gain a modicum of math knowledge.
And, at the other end, we offer classes in calculus, differential equations, and linear algebra that transfer to any college or university in North America (at much lower cost than four-year institutions can manage).
That's incredible.. How do these people even use money? You can't really survive in society today without basic arithmetic.
What's usually the reason they never had an education? Too poor, and they had to drop out?
In Israel 12 years of education is mandatory, and I figure there's at least something similar in US too
Enjoyed both the story and the video link. Math is not my thing. What I have learned mostly was after school when I started self educating in history, science and other subjects that caught my attention. Math came up when I started playing with basic programing and aspects of science in general.
That said, you tell a great story whether about education or family, and I enjoy your writing.
Thanks.
FWIW, my FIL taught community college in Ontario, and has stories about students who couldn't do things like take 10% without their calculator....
Love everything you write, Prof Z., but it's the weird student stories that I really look forward to.
A treat to get two in a row this week!
While I sympathize with Oded's incredulity, I'm also the father of a Student Who Struggles With Math.
My 15-year-old has a learning disability that interferes with abstract thinking, to the extent that he finds it difficult to round numbers to the nearest arbitrary power of 10. The difficulty persists despite hours of explanations, re-explanations, exercises and drills. It just doesn't stick.
He is currently enrolled in algebra. When we complained that the energies spent browbeating him about the Distributive Law might be better spent on sums, products and word problems we were told, "But Algebra is the lowest level math we offer."
So thank you: to your institution for offering basic mathematics, and to you for teaching it patiently. My envy is not small. As for the girls who sit through the class in support of their sibling: goodness bless them, and somebody really should tell them about auditing courses.
Billy C, I sympathize with your situation...
My point was that I was surprised that this level is taught at college. I of course express much gratitude for Zeno's and other such professor's efforts, and I think it is wonderful that they offer this education with patience for those who need it.
But I am still extremely surprised that this is needed at all. If your point is that such schools and courses are made for people with learning disabilities, then I am less surprised, it does make a lot of sense to have such courses. I somewhat would not expect them at a college, but that's a minor point.
But I still think these courses are not intended for such people? According to Zeno, most of his students simply have never even been given this education, or forgot it. In that case, I am extremely surprised! How did they get to that situation? Why have they never learned it before? How do they go about their daily lives without knowing these things? (How can you even understand an advertisement, "sale! 25% off!")
(How can you even understand an advertisement, "sale! 25% off!")
You can't, of course. And they don't, which is why some of them grit their teeth and sign up for arithmetic or prealgebra at a community college.
How did they get this way? I detect two cohorts in my "developmental" classes. ("Developmental" is the currently popular euphemism for "remedial.") The older cohort consists of people who have bogus high school diplomas from the era when "social promotion" kept moving them through the grades K-12 despite flunking everything. (No one would be held back at a grade level more than once.) Nowadays the state of California mandates an exit exam that you must pass before being given a high school diploma, but that's been in force only a few years. Back in the seventies, my cousin got a high school diploma despite being unable to read. It was that much of a farce.
The younger cohort consists of those who flunked the exit test, didn't get a diploma, and are trying to reboot their educations. A few are high school dropouts who outgrew the secondary school system and, as adults, have to resort to the college system.
In my opinion, a big problem with the American system of education is our willingness to give teaching credentials to future elementary school teachers who know very little math (or actively hate it). Our credential candidates are the most innumerate group in our college population. This exacerbates and prolongs the problem. Of course, if we clamped down with a stronger math requirement, we'd have even greater shortages of credentialed teachers than we do now.
Sigh.
Our credential candidates are the most innumerate group in our college population.
And why is this? Because except for the rare duck who really does love teaching in public schools, being a public schoolteacher is a terrible job. The pay is lousy compared to other things like engineering people with strong math skills can do, for one.
But, worse, at least half the job is classroom management and has nothing to do with teaching. There are skills to getting kids to do what you want them to do, and nobody seems to teach them. As a result: if you don't already feel comfortable in front of a class of teenagers, and you can get paid far more to program a computer, why would you ever want to take a job teaching in a public school?
Yes, unapologetic, you're right, as least where K-8 is concerned. High school teaching doesn't look like a whole lot of fun either. I'm glad I teach at the collegiate level of public education. I feel both lucky and appreciated (most of the time).
Apropos of this discussion, a recent Atlanta Fed Bank study found that people with deficient math skills were much more likely to have their mortgages go into delinquency and foreclosure. See here (pdf). The questions they used to assess numerical/financial literacy are on pp. 11-12.
The "numerical abilities" they refer to are as simple as figuring out how much something that lists for $300 costs if the item is on sale for half price!
Wow. That is just very sad, and very bad. And I thought the education system in Israel sucked. At least dropouts only become dropouts at 11th grade, and know all the basic reading and math. By law, you can't drop out before 11th grade.
Didn't I hear that US spends more than any country per child in education? Where is all that money going if teaching is still such a poorly paid profession? Or is this a California problem, where I've heard there's a specific problem of the state being broke?
Teaching is very poorly paid in Israel as well. And I can assure that dealing with students here is no fun as well. But the teachers, usually, still know their stuff... I remember especially the math teachers were the best. (At the other end, the computer science teachers were the worst. Either slackers who just played video games instead of teaching, or clueless idiots who never learned anything beyond the material for the exams)
Apologies for the nagging, I am fascinated by this... I had no idea such a situation existed in US education system.
Oded Shimon, I first became aware of the stupidity of grade school teachers many years ago in 4th grade when the teacher assigned a homework problem from the textbook and then canceled it the next day. She never tried to explain the problem (not a hard one) and there was no possible explanation for her actions other than that she didn't know how to do the problem herself. My 5th grade teacher told us that the Earth went around the Sun, and then ruined her pretense of minimal intelligence by asking me a question about it and then saying I was wrong (I wasn't). And of course all the grade school teachers skip most of the "word problems" in the math book since they don't know how to do them. It's amazing that anyone survives the intellectual abuse of K-8 "education" with a functioning brain. High school is only a bit better.
I am sending that to the entire university of morris, zeno...
shall I send it to yours as well?
you missed the point of life...
Coming on to this conversation late, but what the hell....
I also have tremendous difficulty with math and I struggled through the math requirement for my degree. When math is in numbers, I fail amazingly, but when it is in words, I excel at it. I am now a database administrator, believe it or not, and I succeed at it because while SQL follows a mathematical pattern it is words, not numbers.
I was fortunate to have professors like Zeno in remedial math classes who were willing to help me learn to convert math to words so I could pass the regular college level math classes to get my degree.
I work closely with programmers who don't know the difference between there/their/they're but they can do the most complicated equations in their heads.
I'm another person who'd probably be considered "innumerate" by a lot of people's standards. Dyscalculia sucks, but on the other hand, it gives me some interesting insights on, say, history -- it's really easy to understand the thought processes of an innumerate or illiterate culture when you've got a similar deficit.
The only branch of mathematics that makes any sense to me at all is statistics, and that's basically because it's observable and works in predictable ways; I find much of math to be totally counterintuitive. It doesn't seem to operate according to any system that's congruent with anything in the world; it's just kind of this Platonic thing that functions in its own self-contained space. I have trouble even conceptualising numbers larger than about 20, which more or less makes me the equivalent of an early Bronze Age person. :)
I don't even know how it's possible to "know the times table" in the sense that you mention it. I don't actually really get the concept of a "times table" in the first place, even though I have one printed on my desk. It's just an arbitrary arrangement of information to me. I just sort of have a few key ones memorised and then I derive the rest by counting upward or downward.
I can handle money, but I have to think of money in terms of "units of labour," because otherwise, pft. Abstract stuff is abstract.
I suspect that if I were ever a student of yours, you probably wouldn't even understand a lot of my questions; it happens a lot with me and math...
Late to the party, only because I just found you (via a round-a-bout way from Phayngula).
Mr. Shimon is probably confused in that in the USA all children are required to get an education. This is not true for many countries (like India). From the CIA Worldbook the literacy rate in Israel is 97% with school life expectancy of 15 years (6.9% GDP on education expenditure), in the USA it is 99% with a school life expectancy of 16 years (with 5.3% GDP of education expenditure).
I suspect the innumerate ones in Israel get to leave school earlier, with no chance to catch up in community college. A place where my very learning disabled son is getting a chance.
Community college is also the place I went to return to the working world. Now I am a graduate non-matriculated student who pulled high "B" in advanced engineering mathematics.
My daughter will be taking a class at community college next year while still in high school.
I just discovered a TV sitcom that is actually enjoyable. It is called "Community", has Chevy Chase and Joel McHale, and takes place in a community college. Perhaps I enjoy it because I and my kids experience have some of the same experiences.
Where else can you take a computer science class with classmates who are between the age of 16 and 60?
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