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.

Right. It wasn’t my objective for the lesson. My objectives were for students to model with mathematics and to use summation notation to describe a given series.

I’m going to answer your question with reference to the unit I just completed, leaving the pyramid question to the side for a moment. I’ve just completed a unit on Rational Functions with my Algebra 2 students. Here is a list of some of the things students learned:

1. how to add and subtract rational expressions
2. how to simplify, multiply, and divide rational expressions
3. how to simplify complex fractions
4. how to solve equations with rational expresssions
5. how to analyze the graph of a (simple) rational function

My lessons were completely devoid of pyramids, pennies, photos, videos, context, pseudo-context, or anything remotely interesting. Just plain old, boring, “nitty gritty algebra stuff.” I am not proud of this, but it’s true. I would be interested to know what a Dan Meyer version of this unit would look like. (Part of the problem here is the limitations of my own knowledge of rational functions. I can do all of the above tasks in a pure math setting, but have little knowledge of why rational functions are useful for solving interesting problems.)

Back to my main point about your lesson on pyramids: for me, the mathematical high point of the task is *developing* the formula for the sum of squares. In theory, all that “hook” you invested — showing the video, getting them to generate questions and guesses, etc — is all working towards getting them invested in doing the deep mathematics. So as I said, I was disappointed that this didn’t receive any attention. Yes, you discussed the key features of sigma notation, but that misses the point I’m driving at. Analogy: you’re teaching a lesson on the area of a triangle. The teacher can stop to make sure the students understand the *features* of the formula — what does b stand for? what does h stand for? can you point these out in the figure? etc — but if the teacher never makes the students *understand* the origins of the formula, then that is a travesty.

Maybe this is a matter of taste. I tend to favor deriving *everything,* but I’ve learned over the years that this is not necessarily best.

Summary of math required of the participants in the video:

1. Lots of modeling stuff. All important. Asking questions. Figuring out what info is needed. Making estimates. etc.

2. Plug n = 40 into formula that came “from the clear blue sky,” provided by the teacher.

So, again, if the purpose of the lesson is to practice math modeling, then great! But if the lesson were given in a context of exploring and learning about sequences and series, then it leaves a bit to be desired in that area.

Looking forward to your response if you care to write one!

After watching the first 20 minutes of that address, I am wondering if the educational “problem” is really a problem at all?

I wonder if the poverty problem is less a problem of school achievement and more a problem of decreased purchasing power of the currency?

I wonder if a call for educational reform is just a means for a group of individuals to maintain their status in the machine?

I wonder if we allowed ourselves to let loose of our need for public control, that we would in turn see greater rewards of self-control?

Lastly, I wonder what would be some of the unseen consequences of actions that never happen?