Excellent summary of what your lesson, thank you.

I think your textbook version of the problem is missing something like “the formula for finding the sum of a sequence of square numbers is S = n(n+1)(2n+1)/6.” For whatever reason, the formula for finding the sum of a sequence of square numbers is typically not done, at least in any of the units on sequences I’ve been told to do (I always include some discussion of it anyway), and so a textbook author might be compelled to include it, forgetting all about the Internet… and/or spreadsheets.

“How is it more boring?”

As soon as I know I have all the data, the exploring side of my brain just checks out. I go straight to my brains list of formulas and start looking for ones that will fit together to solve the problem.

When I don’t have the numbers yet, I can almost feel synapses firing all over my brain. I’m looking at the pyramid and actually seeing it. My pattern recognition parts of my brain are drawn to the splotches of newer and older coins. I question briefly whether the sides are truly equal in length, and size things up for a moment for deciding whether I buy that or not. I look at whether the edges and corners are crisp or sloppy. I’m reminded of the feel of sorting cold coins, and the smell (taste?) of copper. I think of Scrooge McDuck and Egypt and pranks and the Statue of Liberty and cash registers and the glass coin jar in my parents’ bedroom and my own attempts to count coins.

All those things happen in just a few moments, and all those connections bring me deeper into the problem. Now I really want to study this pyramid of pennies.

@Dan: “What’s Missing: Formalize the math. Because I’m working with adults, I gave the math a brief treatment here…” That was my only complaint about this video (as I mentioned in my comment under Act 2.) I gather that your purpose was to show how you roll re: 3-act stuff. With regards to that, I’d say: mission accomplished. If someday you would be inclined to show how you *teach math* — i.e. the boring, nitty-gritty stuff — I would be very interested to watch, as I’m sure many others would be as well.

There’s a moment at the end where one of the attendees praises you for using their questions.

You kind of playfully dismiss this saying that you could have predicted the top four questions and you go on to make the point that students hate when teachers try to get them to guess the questions we want asked.

I think, though, that you too quickly dismiss one of your strongest assets. You captured the students questions in their own words before making any adjustments. That was huge. You also kept a question that would not have been on my top four and wasn’t in the top four that you mentioned: “Why would anyone do this”. The answer to this widely held question was very interesting and it rewarded students who thought they were getting a jab in.

You really did value the questions you got and you really did try to answer them. No book or software can do this.

I enjoyed your lecture a lot.

1. You did some nice, quick work on notation. “Would anyone know what it looked like if we were multiplying” was a great question. Also liked “This tells me what I want, this tells me how to get it.” Wonder if you initially thought that there was more background knowledge on that formula. Great adjustment, if so.

2. “But don’t do nothing.” Curious: how would/did this play out in one of your old classrooms? Say a student finished at the 15:00 mark, but 80% needed until the 33:00 mark. I agree that “the billion pennies” prompt is so compelling and is probably better than the alternative (“do eight more textbook problems”).

But I had a tendency to conflate “does work quickly and correctly” with “is interested in more cognitively demanding work.”

3. “Say more about that, please.” This is a highly underrated teacher move, no? You got people talking with this prompt.

4. Love everyone’s comments (both in the video and on this thread) comparing this problem to the textbook version.

My ah-ha moment: when there’s a discussion on whether the next stack has 38 or 39 and you show them that footage of half a penny covered up. That would never, under any circumstances, be discussed or talked about in the context of that textbook problem. It would be an unchecked assumption.

The textbook problem fails as a modeling task. It’s not all bad though and may be useful for other purposes. It’s quicker. It has “high literacy demand” which, in the age of close reading, might be beneficial. Kids would have to attend to precision in the reading of words like “stack” and “layer.” There’s potential to reverse processes: A teacher could easily slide a part “b” in there that stated “What would this look like with a billion pennies?”

@James:
I don’t mean to speak for @Dan, but I don’t think the penny modeling problem was intended to assist in deriving the formula for finding the sum of squares.
However, I too would love to hear or see what a @Dan version of the algebra 2 unit described above would look like.
Finally, I agree with you that it is important that we turn at least some of our focus to deriving the formulas we use. I feel the Common Core calls for us to do so. Your example of the area of a triangle is perfect. I just completed a unit in my geometry class. I had questions on my assessments that asked students to explain how some of the area formulas were derived, almost like informal proofs – triangles, parallelograms, trapezoids, etc. I’ve never asked students to explain where these formulas come from before. I was very happy with the results.

Hi @James:

I don’t see any part of learning something “new” in math, including the 5 items on rational functions, as boring and nitty gritty at all. What is boring is having to repeatedly do procedural stuff. There is necessary procedural practice, then there is overkill procedural practice. I think kids would WANT to learn all that rational expression/equation stuff if there was a HOOK for them to learn it. “Work” problems might provide the hook to learn how to solve rational equations, although I don’t teach it using equations, I do it visually first (http://fawnnguyen.com/2012/12/11/20121211.aspx)

We were talking about irrational numbers recently in my algebra 1 class, and I started by asking them about the diagonal of a 1×1 square. The fact that its diagonal is sqr(2) is mind blowing — this shook the followers of Pythagoras, they HID this fact because they were scared of it (just as the Greeks banned the idea of zero)! Kids love to hear stories. I’d done a lesson on constructing irrational numbers on the real number line (without telling them how), but they remembered this and it made all the subsequent “boring” stuff not so boring because there was a reason for learning it.

We just did Penny Pyramid and the heart of our lesson was actually in trying to find the equation: http://fawnnguyen.com/2013/05/18/20130514.aspx

But I have 8th graders, we ventured in that direction and we learned what we learned.

First of all a HUGE thank you to Dan for his session at the NRich/PRIMAS day in Cambridge in March. It was hugely inspirational & I finally “had a go” myself today using the Yellow Starbursts resources.

I have a tricky Year 9 (13-14 yrs) class and was rather apprehensive as they don’t work well with open ended tasks where the structure is not immediately apparent. They were great! The maths talk & debate that came out of the lesson was brilliant. They lost their way when it came to the actual calculations but nevertheless their involvement in trying to solve the problem was fantastic. I’m looking forward to the next time I use this type of lesson.