## Friday, September 9, 2011

### Geometric Sequences Reductions

Here is an activity I created to introduce geometric sequences.

1) Watch the movie below:

2) Determine the dimensions of the last image if the original image was 210 × 297 mm.

**I had a copy of the original and final image and had students measure the original on their own, and then check their answer by measuring the final image.**

3) Extenstions: If we combined all the images, determine the total area.

If I extended this sequence to infinite, what would be the limit of the sum of the sequence?

1. I saw a recent post on Dy/Dan's site where he timed how fast a printer printed. An introduction to rates I suppose. I then observed days of replies with praise or nitpicking over details of his presentation. Finally, they got around to actually going past the brief movie into actually teaching the mathematics and mathematical analysis that is the expected outcome of all this. The thread died.

Another blogger I know of captured this phenomena rather well. He pointed out that these "activities", even according to Dy/Dan, represent only 25% of the teaching effort and it's the other 75%, the math itself, that is the stumbling block. Yet Dy/Dan (and others like yourself) seem to be entirely focused on the 25%. What lies past these movies Dave?

Bob Hansen

2. Bob what I am suggesting is that activities, such as the one above, spark the interest of students and allow for them to see the relevance of the other 75%.

Using your numbers, which I may not agree with, we need to make sure that the 25% motivate the students (intrinsically) to discover the other 75%. I would like ask you how do you introduce geometric sequences?
In the past, I have used worksheets or numbers without context and I have found my students truly don’t understand the WHY we are learning this concept. I would hope you agree that the WHY part behind mathematics is much stronger than the “JUST DO” way.

3. To me, the big questions about the "other 75%" are 1) how do you learn it, and 2) what happens when you don't know/forget/encounter something newer/harder.

Math stuff is made up of concepts/understanding, methods/strategies, and procedures/algorithms.

If we think of the "other 75%" as just the procedures (and maybe algorithms) then the videos feel like window dressing.

If we think of the "other 75%" as the concepts, methods, and procedures and the understanding story that connects all three, I think the videos are powerful tools.

What does exponential decay mean. It has to do with getting smaller and smaller, but not in a linear way? What makes it unlinear? Why is it more like reducing by a percent on a copy machine, than, say, repeated cropping? I think having a story, something tangible, helps the concept be well understood and articulated.

The methods for solving problems come from a mixture of using concepts to bootstrap new methods (I know how to reduce to 64% once, what can I figure out using tables, graphs, and patterns about reducing by 64% an arbitrary number of times?) and being taught new tools (how do I represent repeated multiplication efficiently? How do I represent sequences so everyone knows what I mean?)

Procedures come from wanting to get fast and efficient at methods, and may require teaching, practice, etc. But they should come as part of a story that starts with, "remember the copy machine? What was going on there? How did we deal with the copy machine? What's a similar story/problem? How could we use the copy machine work as a tool for that problem? What tools do we want to take away?"

So... the 25% shown here is a fundamental part of making sense of and doing the other 75%, not a gimmicky way to introduce it, at least in my book.

That said, I have a burning question! How can teachers share and support each other with the "other 75%" How can we help each other think about the methods our students might come up with, how to facilitate and teach problem-solving strategies and tools, and how to help students transition from concepts to methods to procedures.

Dan Meyer has really helped us form teacher communities to get feedback about our launches. Can the same be done for the student exploration and summarizing stages of problem solving?

4. Robert says:

"I then observed days of replies with praise or nitpicking over details of his presentation. Finally, they got around to actually going past the brief movie into actually teaching the mathematics and mathematical analysis that is the expected outcome of all this. The thread died."

@Robert, if we're talking about the same post, you seem to have basically made all that up.

@Dave, fun application. I've had the same one on my to-do list for awhile and now I feel totally scooped. Unsolicited advice? Get the camera on a tripod, use some editing software, and pose the problem in under a minute. You've already got all my other favorite ingredients.

5. Dave wrote...

"Bob what I am suggesting is that activities, such as the one above, spark the interest of students and allow for them to see the relevance of the other 75%."

No argument there. But the key interest we are after is an interest in the mathematics itself and its intrinsic truth and beauty and that is not accounted for in these videos. I don’t have an issue with the video as an act opener but what is sorely needed is to get the students to think mathematically. This is what we mean by the other 75%. This video may spark interest but is not a proxy for that and I want to see how you transition from an act opener into the act itself. That is my only beef. I see these activities around on the internet and they always end before the math starts. Like watching a bunch of trailers. And naturally, I am asking myself “What’s really going on here?”.

Dan wrote...

"@Robert, if we're talking about the same post, you seem to have basically made all that up."

Dan, which part did I make up? What I said seems to be happening all over again…

"Unsolicited advice? Get the camera on a tripod, use some editing software, and pose the problem in under a minute. "

Nitpicking (but relevant to the 25%).

I still await the other 75%.:)

Bob Hansen

6. Robert:

"Finally, they got around to actually going past the brief movie into actually teaching the mathematics and mathematical analysis that is the expected outcome of all this. The thread died."

Me:

"If we're talking about the same post, you seem to have basically made all that up."

Robert:

"Dan, which part did I make up?

In which comment of that post, exactly, does someone kill the thread by introducing mathematical analysis?

The issue Bob raises is an important one, but it's hard for me to get excited about engaging it with someone who flagrantly makes stuff up.

7. I agree with Robert Hansen in that I would like to see more discussion on how the lesson proceeds, what the student work would look like, and what the next couple of lessons would look like. There doesn't seem to be a lot of discussion about the details involved in these lessons.

Dan seems super defensive on this. I assumed Robert meant that the thread just fizzled out with no posters wanting to discuss nuts & bolts details of this or subsequent lessons - which seemed true to me - not that someone "killed it" with a post. I think Robert's whole point is that this type of discussion just never seems to get started.