kitchen table math, the sequel: Father of Algebra assignment

Thursday, February 16, 2012

Father of Algebra assignment

From an Algebra 2 class.  Thumbs up? Thumbs down? Would you guess this comes from an IB classroom or a Classical School classroom?
Anything like this in college level math courses?


33 comments:

Anonymous said...

This seems like classic IB stuff to me. It's pretty well intentioned. Whether the students are able to constructively pull it off is a different story. I had to do a comparable essay in my BC Calc class years ago. Something between Newton and that other guy. (Leibnitz? Spelling?)
In general, I think students should be doing more writing like this in school--responding to prompts from someone other than an English teacher. I'd like to see what the KTM commenters say.

Michael Weiss said...

I teach a 400-level college math course called "History of Math", and essays in that class are a routine matter. On the other hand I also expect my students to answer questions with the following structure:

Solve the following equation using the method(s) of Diophantus / al-Khwarizmi / Viete.

In my opinion it's impossible to assess the historical significance of the mathematician unless you actually understand his or her mathematical contributions and how they differed from what came before.

Also, the 'father of algebra' topic seems sort of goofy, but maybe that's just a matter of taste, and perhaps it's appropriately goofy for high school students, who are usually better of argumentative or persuasive essays than at purely expository ones.

Jen said...

It was never one of my two kids' IB program, but it does read like one (especially the having to use books or periodicals!)

They have math projects, but they are more like what's described above in that it centers around a problem -- it has to be solved (which makes it sound like a single problem, generally it seems to be a tangled web of several problems that all need to be solved) and then written about.

Bear in mind that while IB programs have guidelines to follow and certain courses to give (especially so in the Diploma Program which is the last two years) -- there is leeway in books, projects, etc. which are assigned.

The teachers are trained within the program so that the lessons they create (yes, the teachers creating lessons, very old school!) meet the standards of IB and that they cover the topics needed for the exams.

Jen said...

http://www.ibo.org/diploma/assessment/exampapers/

There's a link to examples of exams given in the past.

Cassandra Turner said...

Actually, I have one son in the IB diploma program and another at a rigorous charter school. This assignment comes from the Classical, College Prep charter school. It read IB to me, too.

On the one hand, kids don't write enough at the school. On the other, I'm not sure the math head is a good judge of persuasive writing, making it more an exercise in...what?

Jen said...

Maybe the math head always really wanted to be an English teacher?!

I'd have to agree that keeping it much mathier would make it a better math assignment.

Crimson Wife said...

That seems similar to the type of term paper we always had to do once each year in science. I always felt like it was an okay assignment.

Math was always just straight problem sets and exams, however.

Allison said...

I would enjoy the content of this assignment, but still think it's inappropriate for Algebra 2, because it is crowding out other stuff. This is a lousy assignment. Student would be better off learning DeMoivre's theorem or analytic geometry if the teacher has this much time to waste.

Glen said...

I *totally* agree with Allison. Curricular decisions should not be made on the basis of whether something is important or not. You only have time for a tiny fraction of all the things whose importance is > 0.

You maximize the value of the curriculum by filtering, organizing, and sequencing work in such a way that at every stage the thing you are doing is the one thing that is, at that point, the single most valuable piece of work you could do.

Being of some value is irrelevant. It has to be of MORE value than any of the countless other valuable things you could do with that time.

Of course, in the real world, you almost never know for sure which piece of work would be literally the #1 best use of the next hour of class/home work, but you certainly can ask yourself if, on careful reflection, this assignment is more important than *any* specific alternative you can come up with.

If you can think of anything else that would be even more important---something that you COULD do if you chose---then wouldn't you have to ask yourself why you weren't doing that instead?

So, I would wonder whether, 1) this teacher doesn't understand opportunity cost and optimization principles (bad for a math teacher), or 2) hasn't bothered to use those principles to maximize the value of his teaching (bad for any teacher), or 3) honestly can't think of a better use of this significant chunk of math study time than the study of classical history (this is bad for a math teacher in my opinion, though this one IS a matter of opinion about which reasonable people might disagree: variety can improve learning efficiency, breaks improve efficiency, some science is best understood by recapping the discovery sequence, etc. But I think this one was just a sub-optimal use of math time.)

Anonymous said...

This teacher may be using this assignment to comply with a directive to include "writing across the curriculum". This kind of assignment is more work for a math teacher. It would be easier to assign more traditionally mathematical materials. And since we are not as experienced evaluating such work, I can tell you from personal experience that grading them is a real challenge. I'd rather grade 10 proofs than one essay. But I do agree that if this were my assignment, I would include a "mathy" component -- "Provide a specific example of one of their mathematical developments and show how it is used" -- or something like that.

Cassandra Turner said...

Allison said: ...it is crowding out other stuff. This is a lousy assignment. Student would be better off learning DeMoivre's theorem or analytic geometry if the teacher has this much time to waste.

This was all done at home.
In addition to nightly homework.
It was assigned to my foreign exchange student from Spain.

My Pre-Calc son hasn't had to do one of these since his report on Pythagoras in Geometry - which involved proving the Pythagorean Theorem.

Rebecka Peterson said...

This is fascinating. I teach College Algebra, and I don't think I would ever have the guts to assign something like this (nor do I think it's a great use of time). This is more like something I would expect to see in a College Math History class (taken by math majors), but, I also agree that it would be much more advantageous if there were more math involved.

On the other hand, I'm all for incorporating more history into math classes. If you have a story to go with the discovery, it seems easier to remember the discovery. For example, thinking about schoolboy Gauss adding up the first 100 natural numbers makes me...well, sick...but also helps me remember the formula for an arithmetic series.

Also...if the idea is to engage students, I'm not sure this assignment quite does the trick. It looks like it was created just to torture. Just sayin.

Allison said...

A math term paper over and above the normal curriculum? No wonder homework has a bad name.

who thinks these things up? What math teacher is remotely qualified to grade such a paper?

SteveH said...

This is clearly a "writing across the curriculum" assignment and oh so politically correct.

If my son had this in Algebra II, you can be sure that the school would hear about it. Are they doing math in English?

Barry Garelick said...

Not sure which is worse; this assignment or the bromide that algebra is all about patterns.

Anonymous said...

This seems like overkill. Sure, it's good to use history of math as a hook from time to time, but as a required assignment it does not make a lot of sense. And the "many moving parts" aspect of it would be developmentally hard for many teens.

Anonymous said...

But if it is an assignment that was in response to the "writing across the curriculum" requirement, can we please cut the teacher some slack? A directive is a directive -- you have to at least look like you are following it. Besides, is it really so terrible to suggest to students that the ability to write clear, correct prose is a skill that is important -- even outside of English class? Even in a STEM career, writing ability is an asset. One assignment like this in a full year of math class does not merit the outrage. And done well, it might lead to some interesting discoveries and increased enthusiasm for math. I wouldn't make a staple of it, and I would have tweaked it a little, but really no big deal and worth a shot...

Glen said...

is it really so terrible to suggest to students that the ability to write clear, correct prose is a skill that is important -- even outside of English class?

No, it's not terrible, and it wouldn't be terrible to have the math teacher take them out to run laps to suggest that exercise is important, even outside of PE class. There are countless activities that might be of some value.

As I said, being potentially of some value is irrelevant if optimizing your results is your goal. If your goal is to optimize educational achievement, then the only justification for a task is that, after careful analysis, it appears to be the MOST valuable thing you could do at that point. Reasonable people can disagree about what their analysis concludes is most valuable, but if it's not even your goal to optimize educational achievement, that merits outrage.

it might lead to some interesting discoveries and increased enthusiasm for math. ... [This assignment is] really no big deal and worth a shot

A research paper including multiple drafts plus, if your high school math grade matters to you, a poster, extending over nearly two months, a task that might even turn out to have some sort of benefit (relative to nothing, not relative to better tasks that this prevented you from doing) is "no big deal" and "worth a shot"?

Consider this mindset. Suppose I approached a venture capital firm here in Silicon Valley and made that pitch. "If you invest in my project, there might turn out to be some financial benefit for you but, regardless, I consider this much of your money to be really no big deal, so it's worth a shot."

How would that cavalier attitude toward someone else's resources go over? Are the hours of my children's educations a less precious resource?

Anonymous said...

I think you are really overstating the case..."extending over two months" does not mean that this it two months of work. "Multiple drafts" means "help them to not leave this until the last minute".

But to your bigger question: yes, every lesson, every activity, every homework is a decision to do THIS and not THAT. And there is no way to know which path offers the most "benefit". That goes for absolutely EVERY curriculum decision you make. And these decisions are being made continuously. If there is another way to decide other than intuition and best guess, I haven't seen it. But I will defend this particular one a little further (but not much more than that).

As a math and science teacher, one thing I have to accept is that I am not going to teach everything in one year. So part of doing my job includes presenting the subject in away that creates further interest so that my students will want to continue learning more in future classes and on their own. A project like this may inspire some students in a different way than some of the other things we do in and out of class. And yes, there is a trade-off. I'll have to spend say 1.5 class periods on this. So I won't get to teach Descartes' rule of signs. Or Heron's formula. Or some other topic. Something will have to go. (Notice I picked two topics that are usually presented WITHOUT PROOF...not sure what the long term benefit of that investment is.) But this is not venture capitalism. There is no balance sheet to look out to find out if you made the right choice.

I do agree that we should keep an eye out to avoid ridiculous homework loads. It would be great if an assignment like this could count for two classes, with an English teacher grading the writing and a math teacher grading the math content.

And, one more time...if it's a directive, then the teacher has to make it work somehow.

Glen said...

This project hangs over students for two months, a full 20% of the school year. Knowing how limited a student's math time budget was, the teacher knew that all the logistics of going to the library to hunt for books, going to the store for poster board, reading sources, gathering notes, documentation, drafting arguments, writing and rewriting, coloring, cutting, pasting...., all of that could not come out of a mere couple of days, or even a couple of weeks, of math time. There wouldn't be any time for any actual math at all.

So what did he choose to do? Turn it into a much smaller---say one-night---interesting challenge for variety? Turn it into a real math project?

No, he (perhaps forced by EdD's above) chose a two-month, non-math project.

How many two-month math projects do you imagine there were that year? If it's "no big deal" and "worth a shot," why not have lots of them? Because there's not enough time, which implies that the one or two you can afford had better be the best you can come up with, after a lot of careful thought.

I need to say that your point about the decision being driven by directives from above is both very correct and very important. My comments about "the teacher" aren't about teachers per se, but about curricular decisions. By "teacher," I really do mean the system that produces the decision. I'm arguing against a system, not blaming teachers, who would do things very differently if the system were different.

Regarding your "no way to know which is best other than best guess" statement, systems that obsess over optimizing their results, obsess over measuring their results. VC's never know what's going to happen either, so they are constantly looking for better ways to do "due diligence." Their intuition is informed by as much data as they can get their hands on.

Japanese factories tweak, measure, tweak, measure. If their goal is the unmeasurable, say the beauty of a painted surface, they FIND a way to measure, so they can optimize, and you see the results. They take the same approach in their schools.

So, how well does this obsession with more measurement and optimization match American educators' mindset? I think we all know, and this math-writing nonsense assignment is a typical product of the intuition informed by a "nobody can know, but no big deal" mindset.

SteveH said...

"...can we please cut the teacher some slack? A directive is a directive .."

Is somebody blaming the specific teacher? But then you go on to defend the assignment.

So now, the argument is not about this specific assignment, but the general idea of "writing across the curriculum". Why don't they have math or PE across the curriculum? Is a math teacher capable of giving proper feedback on the writing part of the assignment? Is it just about learning math history? If so, then it's one stab in the middle of history and its goal is to express one's opinion, not necessarily learn much about math history. Neither task is done well.

There is nothing stopping the English class from using their time and effort allotment to get kids to write a paper on science or math. That might be good, but then I would disagree with the opinion portion of the assignment. The math history learning part is minimal. You would be better off separating the two tasks; writing persuasive essays and learning about math history.

The thematic idea of combining subjects seems like it should be more engaging and meaningful, but I don't see that. I see this assignment as being very contrived. I see students learning little about writing a persuasive essay and little about learning the history of math for the amount of time spent.

This strikes at one of the fundamental themes at KTM. It's the direct attack on learning. Don't circle around wasting time hoping that students will be engaged or motivated while they are learning naturally. Many students can't stand this. Just dive right in and get to work. Don't waste students' time. Students know this assignment is a lot of work for little gain.

If you want to learn about writing persuasive essays, follow the examples in the SAT and start doing a lot of them. Don't burden the process with a lot of research in the middle of a history that has a lot of depth that can't possibly be understood properly. If you want to learn about math history, tackle that directly. Offer the class as a one-semester elective course.

This assignment clearly shows the inefficiency of thematic or top-down approaches. They think this assignment is a good way for students to learn basic skills of writing persuasive essays and math history. It does neither well. Students are not stupid. They can see that they really need to first learn more skills about writing and they need to have some framework of the history of math.

Some might see this assignment as engaging and unifying, but I see it as frustrating kids who don't have enough basic skills or knowledge in either area.

If a math teacher is required to give a writing assignment, then the opinion part could be dropped and the students could be expected to write an historical paper where they learned so much more. I assume that the teacher wasn't given the assignment. It's bad pedagogy and it's a bad assignment.

Anonymous said...

You may be surprised that I agree with you here about much of this. As a teacher and as a parent, I do not like to see time wasted in class or out of it.

Part of what you are up against not just the administration but the entire education commumnity who believes that including the opinion component elevates the assignment to the highest level of Bloom's taxonomy (where the lowest level is of course, mere knowledge). That structure is presented in teacher-school as received wisdom. And then you are supposed to write lesson plans, tests, curricula -- all keyed to this structure. And you are not supposed to question its validity. It's always seemed to me that to make it through these classes, you have to turn off the questioning, analytic part of your brain.

But one more question (though I think I can guess the answer): does the outrage you feel about this assignment permeate your feelings about just about every class lesson and every assignment that your children are subjected to in every subject? Because they are all based on intuition and best guess.

As for long term assignments (and I've watched my gifted but not artistic son spend all too much time with posterboard and hot glue), it would be good if it was at least conceivable that some part of the assignment could plant the seed for further interest and learning. At least this one has a chance. Before (or while) you condemn it, look at the other long-term homework in all of the other classes -- do they meet that standard?

But Ok, now I'll let it go...

Anonymous said...

"As for long term assignments (and I've watched my gifted but not artistic son spend all too much time with posterboard and hot glue), it would be good if it was at least conceivable that some part of the assignment could plant the seed for further interest and learning. At least this one has a chance. Before (or while) you condemn it, look at the other long-term homework in all of the other classes -- do they meet that standard? "

I home school my child, so your question only sorta applies to me ... but I'll take a crack at answering it anyway.

I can actually see myself assigning something like this to my child as part of a high school level curriculum. Find out who did what when and who influenced other people. And define the terms a bit more tightly. And then write about this persuasively.

But ... and this is a fairly large but ... I would score this as "history", not as "math." I would *not* dial down the equations/functions/proofs/etc part of my curriculum to teach this. It would come out of the what happened when, and why and how and how did this influence *other* events part of the curriculum.

-Mark Roulo

Cassandra Turner said...

Ok, for those of you following this, here's a link to the research outline & bibliography turned in by my student on Feb. 17th.

Keep in mind, the student is an exchange student from Spain, so for that reason, (as he always says) his grammar may seem unusual.

The teacher said the above linked paper had the most info of any turned in.

Glen said...

No, Anonymous, I'm not fooled by the "they are all based on intuition and best guess" argument, implying that one guess is as good as another.

In industries where "results matter" is more than a platitude, they know the difference between best guesses based on careful due diligence and best guesses based on careless "no big deal" negligence.

I'm not just mindlessly outraged at "every class lesson and every assignment in every subject." Like many teachers and many people here, I have the ability to make reasonably informed judgments of how much a task is likely to increase my child's expertise vs. how much time it will use up.

The more likely an assignment is to maximize learning per unit time, the more I like it. The shorter it is, the less I care. I'm annoyed in proportion to the degree to which a project is obviously a sub-optimal use of time times the amount of time it uses.

Put aside the "everything is a best guess" smokescreen. The bigger the assignment, the more justification there had better be for it. If there is not enough evidence that it is an outstanding use of time, change to something that is or shrink it down to the much smaller scale appropriate for an experiment.

"it would be good if it was at least conceivable that some part of the assignment could plant the seed for further interest and learning. At least this one has a chance."

"At least this one has a chance"? How low are the standards in education that "it's not as bad as it could be" constitutes a justification for a large use of resources?

Imagine taking a large project idea to a CEO and promoting it with the claim that some vague benefit was "at least conceivable".

(cont'd)

Glen said...

(cont'd)

As for possibly motivating interest, what if you took just ONE DAY, and introduced them to the German Tank Problem. Have them be spies and let them risk capture as they try to collect randomly selected tank serial numbers. Should they stop at what they have or take even more risk to get more samples? Have them compete to see who comes up with the closest estimate of the actual strength of the Nazi blitz capability (but if you're captured, you lose). SHOW them that a person with the math skills to come up with that formula was as dangerous to the real Nazis as a weapon. You could even have the class work as a team. If their collective effort produces a good enough estimate, the Nazis lose. Otherwise, they win.

Do something similar with a different math tool on "Mad math skillz Monday" each month. Make careful note of which ones elicit followup math questions, which ones fall flat, which appeal to artists, which to mathy types, which to math haters. Replace the boring ones and evolve a better set each year.

Or, yeah, since nothing is certain, so one guess is as good as another, there's no good reason to expect that TEN DAYS of Mad Math Mondays each school year would be a more efficient way to inspire math interest than a two month writing chore, right? If I can't be absolutely certain, and I can't, then there can be no valid basis for condemning this as a poor choice.

If I knew nothing about uncertain reasoning, and had never worked in an industry that did, I might even be fooled by that argument.

Anonymous said...

The exchange student's submission is a great example of how some assignments sound more intimidating than they actually are. Also, given that this student's work was the most complete in the class, of the reality of pushback from students. A huge feature of HS curriculum that can only be mitigated slowly, and at the margins. If all the students are aware that none of the rest is going to try to deliver what the assignment allegedly requires, they are all free to deliver something less. And they do. In a well-functioning HS, this happens, but not too often. In a less well-functioning one it happens all the time, and is the reason why some teachers give up on assigning homework at all.

Anonymous said...

"The exchange student's submission is a great example of how some assignments sound more intimidating than they actually are. Also, given that this student's work was the most complete in the class, of the reality of pushback from students."

Yeah :-)

I just read Cassandra's link and then went to Wikipedia.

I *think* I could build an outline (and argument) from the Wikipedia article alone and then use the sources in the Wikipedia notes section to meet the "works cited" requirement.

My question is now: Could/would I have done this in high school? I'm pretty sure that I could/would have ... wasn't starting from the Encylopedia pretty much standard practice back then?

So ... now I'm thinking that this is a "5 - 8 hours" assignment spread out over two months. Basically:
    (1) Read the "History" subhead of the Wikipedia article on Algebra.
    (2) Read the Wiki articles on Diophantus, al-Khwarismi, and Viete.
    (3) Pick a position (at random, if you like).
    (4) Write four pages defending your (possibly arbitrary) choice.
    (5) Find a few of the books/whatever referenced by Wiki. Or use the library card catalog to find books on the three guys (and maybe a history of math book).

Skip the poster unless you are doing poorly in the mathy part of your Algebra II class.

Have I missed something obvious?

-Mark Roulo

Anonymous said...

"So ... now I'm thinking that this is a "5 - 8 hours" assignment spread out over two months."

I agree. So it would NOT be interchangeable with a series of one-class actvities that added up to 10 class periods (and I am not convinced that the tank-id activity could reach meaningful conclusions in one period).

Also, claiing that current assignments are based on guess and intution is not the same as claiming that one guess is as good as annother. Certainly, part of what makes one teacher better than another is their awareness of (even obsession with) effective use of time. You could call the intution and guesses of the teachers you agree with "informed judgement" but I don't know of any colleagues in any department who are measuring the long-term impacts of their long-term assignments. So maybe you should be enraged more frequently.

I also think the earlier post about push-back is a great insight as to what often happens with
these assignments.

Anonymous said...

"...I don't know of any colleagues in any department who are measuring the long-term impacts of their long-term assignments. So maybe you should be enraged more frequently."

Um ... I home school my child.

If I don't like the mix of assignments and the use of time I can do a lot better than just being enraged :-)

-Mark Roulo

Anonymous said...

I feel like my wife and I have always home-schooled...afternoons and weekends :)

Glen said...

I don't know of any colleagues in any department who are measuring the long-term impacts of their long-term assignments. So maybe you should be enraged more frequently.

I should say a couple of things about this. One is that "enraged" is the wrong word. I'm not enraged. You were right when you called it "outraged" before. The difference is that, as I said, I don't blame teachers for this. They adapt to the attitudes and incentives of the environment they work in, as we all do. Enraged, to me, is when I see villains and want to choke them. Outraged is when the people are fine (I love my kids' teachers, my mom was a teacher, three of my grandparents were teachers, you seem like a good guy/gal, I'd rather socialize with most teachers than with people from a lot of other industries...), but the system itself seems just hopeless.

This, "I don't know of anybody measuring long-term impacts" stuff is what I'm talking about. They don't (and, again, I mean the system in which individual teachers operate), because they don't have to.

I was in public schools in Shanghai recently, observing classes, talking to teachers, principals, students, parents. I've tutored math in Korea, and spent a lot of time talking with teachers, students, and parents when I lived in Japan and Singapore.

Educators in these places obsessively tweak, measure, track, and optimize. They don't talk like you and your colleagues. They talk like factory managers in Japan, like logistics officers at Walmart or the US Navy, like Olympic trainers: "find another 5% improvement somehow and, when you have, then find another 5%...."

Take a look at the first part of this video from today's WSJournal. This is what I see with my own eyes in Asia. (The second part of the video, the iffy "but we don't have to worry, because they're not really creative" part, is a conversation for another time. I'm talking about measuring and optimizing for proficiency the way organizations do when results really do matter, not the interesting question of, proficiency at what?)

Tiger teachers: Asia's schools outperform US.

The Shanghainese didn't produce the world's best math students, getting more proficiency per unit time each year, nor did Steve Jobs produce the world's best electronics products, with a "relax, it's all just best guesses, nobody really measures this stuff, no big deal" approach.

And, Anonymous, you don't have to agree with me for me to be interested in what you think. I hope you'll comment in future threads.

Anonymous said...

"And, Anonymous, you don't have to agree with me for me to be interested in what you think. I hope you'll comment in future threads."

Thank you. Not necessary to have said, but still welcome. Though people who don't enjoy a good argument should probably stay off the boards...Also, please don't worry that an attiude of "hey, whatever, man" is part of my own teaching approach!

I just think that there are so many decisions to be made when teaching, more than can be measured. That's why we do need teachers with really good judgment. Because even without long-term measurement, there are certainly some assignments that make me scratch my head with wonder -- "What chain of thought led you to believe that this is worth doing?" It's just that this particular assignment did not set off that response (for me).

Also, I wish that there were a structure in place that could give students some options without introducing extra credit (which I oppose as a form of currency devaluation -- a separate discussion). I am imagining something where the subject teachers agree that 3rd marking period is project period. Each teacher maps out a project that will count as one major test that quarter. Students would then be required to pick ONE subject area and do that area's project. There could be some common elements required in terms of outlining, sources, etc.. but that way, students would be more likely to explore something that they are interested in. This might reduce some of that push-back.

There are probably 18 reasons this can't ever happen. But I have experienced times when SOME students have embraced an assignment and run with it while many others slogged through. I wish that they could find a project that they could embrace. (And if none of their subjects are interesting them in this way, well that's kind of depressing.)