Robert Hansen (RH) posted Sep 3, 2013 5:32 AM (http://mathforum.org/kb/message.jspa?messageID=9252270): > > On Sep 2, 2013, at 2:25 AM, Richard Strausz <snip> > > 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." 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. > > 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. > Rarely - if EVER - do we reach "the end" of anything. Life's like that. Algebra is like that. There are ALWAYS further questions to explore. Each question explored enhances the knowledge gained. After 'basic' algebra, there is 'Linear Algebra'. And then there is 'Homological Algebra', 'Galois Theory', a whole variety of algebras, such as von Neumann Algebra, and so on and so forth. (But perhaps that is a later stage: the first step is to get the student to understand the power of 'abstract, symbolic reasoning' - and that is just what Dan Meyer has done in his 'Popcorn Picker' activity). > > 2. Write down a guess. > > Guess? Guessing isn't algebra. Guessing isn't even > mathematics. > The 'guess' isn't (by itself) algebra, I agree: but it is a useful tool nonetheless in real life - AND in mathematics (including algebra). It can teach us plenty.
The way Dan Meyer presented his 'Popcorn Picker' activity, the guess is very useful indeed - except for those who're stuck in 'traditional, by-rote teaching'. > > 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. > Well, perhaps this might have been question No. 0 (before question No.1). It IS a 'sincere question', with or without Question 2 (IMHO). What, in your opinion, is an 'insincere question'? Was this one such (because of question 2), in your expert opinion?
By the way, I seem to recall that George Polya strongly recommends something along these lines to help the mind organize needed and available information. > > 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. > No. See above. > > 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. > No. It IS a very useful activity of the mind. (For anyone who is aware abut how the human mind actually works. Perhaps it is not terribly useful for 'traditional, by-rote teachers'). > > 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. > Nope. It's an activity called 'exercising the mind' on an issue. This is a useful activity for anyone who is aware that he/she does not know everything. > > 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. > It involves and includes plenty of algebra: if you do not see the algebra in the 'Popcorn Picker', well I guess you don't. See above (and also below). > > Richard, why must "regular" students suffer this > fate? > > Bob Hansen > The 'fate' is called 'thinking about a specific problem set'.
When presented effectively, the 'Popcorn Picker' would be very useful indeed, IMHO.
I have no reason to believe that the person who thought it up, Dan Meyer, would not present it effectively. If GSC were to set it for a school class, there is every reason to doubt that it would be effectively presented - but GSC is not a teacher. When Dan Meyer or Richard Strausz (RS) presents it, there is every reason to believe it WOULD be presented effectively, so that the student's learning capabilities are really engaged - and that the student may understand a bit about the potential power of 'abstract, symbolic reasoning'.
To my mind, the underlying attitude is to try to interest the student in 'abstract, symbolic reasoning' - relate such reasoning to the concrete: the 'Popcorn Picker' certainly does that. The idea is NOT to teach the student by way of commands like:
"DO THIS! DO THAT! THIS IS THE RIGHT WAY!"
Effective teachers should always be aware that 'teaching' does not exist by itself - it is part of the 'learning+teaching' dyad - and that is what Dan Meyer has recognized and has successfully accomplished in the 'Popcorn Picker' [PP], IMHO. He is actually working on the mental model:
"To create a concrete example" MAY CONTRIBUTE "to indicate the power of abstract, symbolic reasoning" (which in turn MAY CONTRIBUTE "to interest the student in abstract, symbolic reasoning").
(This model may mean something to anyone who is capable of going beyond PERT Charts).
The Popcorn Picker is, IMHO, an excellent example of just how to bring home the awareness of the utility (and perhaps the power) of abstract reasoning to students, particularly those students who may not have 'got' the idea of algebra.
The PP certainly IS a VERY useful way to engage the student in the rather abstract issue of algebra, symbolic reasoning and the like. Even if just 5% of the students 'taught' come to understand the power of 'symbolic reasoning' thereby, Dan Meyer has done something very useful indeed; I would guess that a lot more than 5% of the students exposed to the PP would at least start 'getting' it.
I shall be recommending the 'Popcorn Picker' to any teachers of algebra that I may encounter. (I shall, of course, be informing them that RH does not share my positive opinion about the PP).
I note that RS has adequately responded, with the explanation that he is NOT trying to engage with the Robert Hansen type of student (who 'gets' everything instantaneously), but rather with the type of student who may not so readily understand the power of abstract, symbolic reasoning, who may need algebra to be related to something very concrete, such as the PP.