this appears to be more or less obvious
to all parties not believing with steve
“abolish pencil and paper” leinwand’s
riotously funny remark that
“adoptions are democratic”.

now: what’s the solution?
more corporate rule evidently.
geez guys. i’m glad dan’s got
a good job and everything but
can we reimagine some of this
kind of discussion with issues
of scale and diversity and whatnot
somehow not-entirely-ignored?

it’s like the whole explain-budget-fiasco-
-by-appealing-to-ward-and-june-fantasies-
-of-family-life thing. oh, look. another
hard-working teacher prepared another
fine lesson and the students learned a lot.
let’s find some way to make all these
prisoners of the EDU-wars *pay* (and
pay and pay) for some sick *parody*
of that lesson by jamming it down the
throat of entire states with governments
so corrupt they make one nostalgic
for richard frigging nixon. everybody
(that counts) will make out like a bandit.

i mean, i’m glad dan’s got a good job and all.
but (philosophically dodgy concept alert)
the “median math teacher” of USA has
seen the job get worse and worse and worse.
and promoting a few talented superstars
into the stratosphere isn’t likely to do
much about it.

good on ya, dan. i expect you know
i mean it. i’ve admired your work
for years. “TED”, though? massively
*part of the problem*. i’d sell out to
the money myself in a heartbeat of course
but haven’t been able to stir up any interest.
in much the same way as i’d work in
a “defense” plant making WMD’s even though
i’ve been anti-every-war since v’nam.

indeed i *did* get a couple of those
“submit your carefully-prepared notes
to our user-defiant humans-are-drones
interface and get a small check while
we ignore your work and print your name”
gigs and would gladly perform this
humiliating work full-time: baby needs
new shoes. but my networking skills
evidently peaked when i was back home
in grad school ignoring “real life” like a man
possessed; ever since i graduated, the
world has offered me less-and-less satisfying
work and less of it pretty doggone steadily.
these guys only wanted me for my (tenuous)
affiliation of course: you need a job
to get a job.

and, please understand. i’m hoping… against
my usual talking-about-politics-is-useless
strategy… here to speak for *many others*:
maybe even a majority of “adjuncts” and probably
a pretty goodly portion of the brave footsoldiers
slogging it out in public schools. *however* many
it is, and however much i hate when people
say “we” (meaning almost anything *but*
“you and i” when politics is the subject):
we math-teachers-caring-about-math-
-but-hating-and-fearing-big-money-power
haven’t done very well since i’ve been on board
(~1985).

so. i know, i know. my whining is very
unpleasant to many readers who are sure
i deserve much worse. so my message is:
you’re gonna hear a whole lot more *of* it
(and not just from me) now that the
EDU wars are in the streets. viva santiago.

The page is already cluttered. I need THIS?

You need to know what questions to ask as hints to students when they stall out for 10 minutes and aren’t getting anywhere. Those who have seen this kind of thing in action before will probably be able to figure something out, and will have questions 6-9 and (more importantly) 11 in their back pocket already. Those who haven’t done this before and didn’t have teacher training getting the concept of Socratic questioning in their heads will probably benefit from a cheat sheet of what they *might* need to ask and when/why.

If it doesn’t make sense to have a second “hint” resource, maybe some of these questions could be collapsed and expanded by clicking on a “Hints” subtitle, so they’re there if kids need them but it’s more clear that they aren’t necessary.

I think the “extra questions” are too helpful and make the task worse. Also, #9 is a “what’s in my head” question that some students will read as “What does the teacher want me to say now?” instead of a real question to be answered. I like the alternate suggestions by Jason and Luke.

I wouldn’t lose the extra questions altogether, since they are useful as guiding questions for teachers to use if students are off task or heading in fruitless directions. Guiding questions would also help teachers with task implementation, since there’s always the worry of “What do I do if it’s not working?” These questions should not be immediately available to students: if this task were for a textbook, those questions would go in the teacher’s edition and not the student book.

I don’t know your grand plan, but if students work this in an online / interactive resource, the resource could provide the guiding questions behind a curtain, maybe a curtain that reveals itself only after 5 or 10 minutes of playing around with the problem (or solving a painful math-related CAPTCHA, heh). Then the guiding questions would only appear to students who need them.

This is a real bugaboo in textbooks: if there’s any help you think students may want (important things like “Remember: the origin is the point (0,0)”), you either decide to provide it to all students by putting it in the book, or not provide it to anyone. The only in-between is leaving a note for teachers, then crossing your fingers that it gets read and used appropriately. This stinks, and nothing can be done about this. It easily leads to texts being too helpful in bad ways. But an interactive resource could make the right amount of support appear, and only when necessary.

Lastly, I felt you should put question #2 (“Guess as close as you can”) after #3 and #4 (“too high” / “too low”), so I’m curious if you could say something about the decision to use the ordering you did. Thanks for posting it, and good luck with Project X!

Are you trying not to be “too helpful” by giving the students three rates to think about and letting the students decide which rates to use. I feel like asking “what is the rate in pages per second” might be “too hurtful” for some student as it involves a rate of less than 1 (0.3 pages per second).

I don’t like the question about which rate they think is more useful. The rate I used was the one I thought was most useful — duh. Or, why are you asking me about rates, I am using the caveman method: 10 seconds for 3 pages, 20 seconds for 6 pages, etc. until I have about how many pages I think are in the stack.

I think the pages per minute question could be a good hint in this case because the timer ran for a convenient amount of time (10 seconds). A student might get a feel for the problem, and start to sense how the proportion or rate works, as they ratio up by a factor of six.

Maybe these would go under the “too helpful” category, but I might ask a student that is stuck things like: What would happen if the video clip ran twice as long? Three times as long? How much time would it take to print 12 pages? How much time would it take to print 61 pages?

Sequel: This style of open ended questions, presumably designed to elicit creative responses, usually flops for me — maybe that says more about me than the question. I suspect many students would just give 3 more or less arbitrary numbers of pages with no justification (none was requested), which technically is as good a way to answer as any since the detail of the copy will affect the speed, and we don’t know the speed of the other three printers anyway. I would rather turn it into the house painting problem — a second printer is twice as fast or 2/3 as fast or whatever. Working together how long will it take to make 1000 copies.

6. What is the rate of the printer in seconds per page?
7. What is the rate of the printer in pages per second?
8. What is the rate of the printer in pages per minute?
9. Which of those measurements of rate is more useful to you?

Which I’m not wild about (especially #9) because it gives the notion there is One Right Way of doing the problem. I guess most students would reach for 7 or 8; I’d use 6 because I mentally have the units cancelling (88 pages * X seconds / 1 page causes the unit of pages to cancel) Usually with this sort of problem I have some students use seconds and some students use minutes and everything goes fine.

I would either rephrase #9 to make it clear this is a preference of the student, or rewrite that portion entirely to be a more general hint (ex: “Hint: Relate the number of pages to the time by a ratio.” — how this would read exactly depends on the grade level you’re aiming at here.)

A more open ended hint would also allow more strategies. There’s no shame in the students starting with a raw table before jumping to rates; they could solve the problem without thinking of rates at all except implicitly (something the teacher could then leverage; that’s the sort of thing you’d want in the teacher guide). Also from my experience of students watching videos with timers there’s a bit of disagreement as to when the cues happen. A table would make clearer this is an inverse problem with no absolute “right” answer given the information the students have.

(Some problems are inverse problems through-and-through — is something mystical going on in the Bermuda Triangle? — and some are problems you can make inverse problems by withholding information — is Mark Twain is the author of this text? The printer problem is the latter.)

Jason: What hints would be most helpful? What understandings do they need? What generalized techniques apply here (so they can recognize they are using a technique that will occur elsewhere)? Does everything need to be done in-class or is there a portion that would work as homework? Can it be used to learn a particular mathematical technique for the first time or does it need to be implemented after they are already experienced with whatever math is needed?

Yeah, really good questions. I think Mr. Mathalicious has also thumped me a few times for being too unhelpful. My sense of math curriculum right now is that it’s overly helpful, too mealy, and too structured. I know I can throw in the towel at any point and develop worksheets and teacher guides and add in lots of supporting questions. Some of those are probably even advisable. I’m just being cautious, I guess. In my pursuit of the least powerful framework for this kind of curriculum, I’m adding power very, very slowly.

If you get a sec, I’d be interested in your sense of these two lessons. One of them is for Still Undisclosed Project X and the other is stripped down to the chassis. Are the additions too helpful? Not helpful enough?

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