> > On Sep 2, 2013, at 2:25 AM, Richard Strausz > <Richard.Strausz@farmington.k12.mi.us> wrote: > > > I was agreeing with your first paragraph. I agree > that I have to go to the board and teach the content, > proving that the Pythagorean theorem is correct and > that the law of sines is correct. You may be > surprised to learn that just because I teach it some > of the students don't learn it... :-) > > True, I don't expect all students to get it. I am not > sure how that leads to doing activities that forgo > the math altogether. What do the students that get it > do during this time? Keep their mouths shut? But I am > still mystified that in all the years of posting here > you have never posted about the "content", yet you > state "Yeah, sure, I teach all the content."
I have never found it necessary to discuss 'usual' content and pedagogy. What I find interesting and worthy of discussion are different approaches. I presume that the readers of this forum are more interested in those.
> What I see in these activities is evading "teaching the > content". In an actual class, whether it be algebra > or geometry after algebra, the "popcorn picker" > problem would be nothing more than a word problem, > maybe accompanied by a figure, and solvable by a > student, using algebra, in a few minutes.
I see 3 types of students in class. First there are those who didn't 'get' the content I taught. Second, there are those who can do the math but don't 'really' think that what the old guy teaches connects with the actual world outside of the classroom (more about them in a moment). Finally there are the 'Bob Hansen' students who get the math and of course see the real-world connections. Doing 'popcorn picker' - which takes 5 to 10 minutes maximum out of a class - has most of its value for type 1 and type 2 students.
I don't think you can appreciate the existence of type 2 students because you were a strong student. These are students who know the Pythagorean theorem and its proof and have done in-class and homework problems on them, BUT if they see a piece of graph paper with 3 dots marked at (0,0), (10,0) and (10,4) and they are asked the length of the hypotenuse they are surprised that it isn't exactly 10.
> > But that is not how it is presented by Dan because > Dan clearly is not presenting this activity in the > context of algebra. > > Let's recall the "acts"... > > http://threeacts.mrmeyer.com/popcornpicker/ > > 1. Which container will hold more popcorn? > > If this was an actual algebra class, this would have > been the end of it.
My students have all had algebra and have seen and used the volume and surface area formulas in my geometry classroom. The type 3 students make that connection immediately. I would estimate that less than 20% - even in honors geometry - make the connection without any prompts from me. That's one reason why I think the activity is worth some of my valuable time.
> > 2. Write down a guess. > > Guess? Guessing isn't algebra. Guessing isn't even > mathematics. > > 3. What information would be useful to know here? > > This sounds very much like asking the students to > guess again. Maybe if question 2 wasn't there, this > might be a sincere question. > > 4. Can a rectangular piece of paper give you the same > amount of popcorn no matter which way you make the > cylinder? Prove your answer. > > After the student answers question 1 (algebraically) > this question is moot. > > 5. How many different ways could you design a new > cylinder to double you popcorn. Which would require > the least paper? > > This is guessing again. > > 6. Is there a way to get more popcorn using the exact > same amount of paper? How can you get the most > popcorn using the same amount of paper? > > More guessing. > > > Not only does this activity not include a lick of > "teaching" algebra, it evades algebra, because had > the properly prepared student established the > algebraic relationship between the dimensions of the > paper and the volume of the cylinder, all of these > questions, except the first one, are moot. > > Richard, why must "regular" students suffer this > fate? > > Bob Hansen