Lesson Planning in Tanzania - Poa

Collaborating at the Outpost Lodge

For the last post in this series on what I learned from my Study Abroad experience in Tanzania, I try to combine the themes from the previous three posts - resourcefulness, patience, and acceptance. In order to do this, I want to tell you a story about lesson planning in Africa. Each night, Sunday through Thursday, the teachers gathered to plan for the following day's lesson. The teachers were encouraged to collaborate, and the professors were available for consulting, if needed.

One night, a teacher came to me with a question about a log table.

She was teaching the Tanzanian students how to use the table in an upcoming lesson but she was unfamiliar with how to use this particular version. This made sense, since she had no experience with this type of log table. To be honest it took me a few minutes to understand how the table worked; it has been awhile since I did logs without using a calculator.

It would have been tempting to dismiss the table as ancient history and focus on applying logs in some real-life situation using available technology. I certainly have argued this before - making a point that "there's an app for that." In this situation, however, the teacher accepted that this was not ancient history for her students. They would be expected to know how to use tables like this for the national exam. And since the textbook was the available technology, she said "no thank you" (hapana asante in Swahili) to simply relying on our method of mindlessly plugging numbers into a calculator.

Another thing you should know is that there was only the single textbook for the entire class. Copies were difficult to make, so the teacher had to be resourceful. She took her hamna shida attitude (Swahili for "no problem) and began thinking about how we had come to understand the table. A breakthrough had occurred when we noticed the relationship between the log (2) and log (4). The teacher could write those values on the board and ask the students to consider how the table might be used given what they knew about the laws of logarithms. Then the students could share and critique the different ideas.

MO Snow Plow Convoy

Sure, the teacher could have sped up the lesson by focusing on the process (Skemp's Instrumental Mathematics), but we wanted to take it slow (pole pole). Making time for students to struggle and persevere with a problem is worth it. Recently, I heard someone share the term "snow plow parents" - people who make sure that no obstacles get in the way of their kids. In my opinion, teachers who focus on teaching Instrumental Mathematics are practicing the same principle and do students a disservice.

In this situation, focusing on what Skemp calls Relational Mathematics put logarithms into a context: reading a table. Sure it might be an out-of-date skill for us, but it was real for these students. Also, the students could use this experience of decoding the next time they had to understand something difficult in a mathematics textbook. We hoped that by combining all of these elements the students would experience a cool (Tanzanians might say, "Poa!") way to think about understanding the table and what it means to do mathematics. 

Did I get my money's worth?

The following is not intended to be an endorsement of any product.

UPDATE: 2017 - This year the tumbler cost $42.40 (including tax) and a grande is $2.20 (including tax).


In December, I got a tumbler from Starbucks that allowed me to get free coffee every day this January. The tumbler cost $30 plus tax. My last free grande is shown to the right. I still get a $.10 discount when I use the tumbler but from now on I need to pay the regular $1.95 plus tax for brewed coffee. This got me wondering: did I get my money's worth from the purchase of this tumbler?

Will you (or your students) help me answer this question?

Wait. You need more information - such as? Go ahead and share what you need in the comments.

Does this make sense?

The following guest post comes from my friend and colleague, Robert Talbert. It is thanks to the success of the University of Michigan (MS '92) in the March Madness tournament that I earned this honor. I do not know if Robert is aware that I also attended Northern Michigan University (Teaching Certification '87) and that my grandfather was the registrar there, but these connections make this post all the more special. Thanks to Robert and GO BLUE!

*****

“It makes sense when you do it, but when I try it, I have no idea where to start!” In other words, it never made sense. The sense-making was only an illusion and did not actually happen.

Anyone who has taught mathematics to a real human being for more than 30 minutes has encountered the above statement. I think the sooner we realize that sense-making is not something that you feel but something you do, and the sooner we mathematics instructors shift our focus away from providing feelings (can I say “sensations”?) of sense-making and onto the actual making of sense in the classroom, the better off we will all be.

I’m currently at the annual meeting of the Michigan section of the Mathematical Association of America, where a number of the talks so far have centered on sense-making, none moreso than a plenary talk by Peggy House of Northern Michigan University titled “Reasoning and Sense-Making”. I appreciate Peggy’s perspective on reasoning and sense-making, since she is a veteran classroom educator and math education specialist. Regular readers of my blog over at the Chronicle of Higher Education know that I spend a lot of time thinking about various instances of the flipped classroom, including the adaptation of Eric Mazur’s model of peer instruction to university mathematics. The more I work with these pedagogical models, the more I realize that sense-making is really what they are all about.

A good peer instruction question, for example, is one that is aimed squarely at the biggest misconception on a fundamental concept — which is to say, it unearths the very thing that students most urgently need to make sense of. And then we ask them to make sense of it. And these days, when I discuss the flipped classroom with someone and they ask, “What do you do with the class time once the lectures are moved out?”, I just say: sense-making activities. I want students not only to make sense of the things they are learning but also learn how to make sense of things.

In Peggy’s talk, she posed the following problem: Find the area of the octagon in the picture below, which is formed by the intersection of the various segments connecting vertices and midpoints of a square:

One of the first points Peggy made in her talk was that sense-making is in the eye of the beholder. When instructors approach mathematics as pure procedure and then focus on showing students how to perform a procedure, this not only is not sense-making, the thing we are talking about will only make sense to the students insofar as their ways of seeing the problem coincide with our ways of seeing it. Which is to say, most of the time it won’t make sense at all, even though after a clear lecture it will seem to make sense. This is a dangerous position for students. It would be better to have concepts not make sense and for students to know that they don’t make sense, than it would be for the concepts not to make sense although students think that they do.

In the octagon area problem, for instance, I could certainly lead students through a clear discussion of a solution through direct instruction. This would not be all bad. There is some value to seeing an expert learner (that’s me) explicitly model his or her decision-making and thought processes as he or she works through a problem. But the benefit ends there, because this problem is not about “the right answer”. Very few nontrivial mathematics questions are! Peggy went on to give examples of some of the ways her geometry students solved this problem:

Some of them were wildly creative:

And some of them were creative in all the wrong ways — for example, one group got an answer by assuming that the octagon is regular, which as it turns out, it isn’t. And that’s where reasoning comes in to sense-making. When a student is solving a problem, it’s not enough to be creative. There also has to be good reasoning involved as well. If one’s reasoning behind a solution or a conjecture is flawed, then the concept being thought about has not fully made sense yet.

One of the great lessons learned from this talk, something I knew already but can always use encouragement on, is that my job as a mathematics teacher is not to download by brain into the students’ brains, or even to wire their neural pathways to look like my neural pathways. Indeed this is not really even possible, because — thank goodness! — every student is a unique human being, with a unique backstory and ways of looking at the world. Sense-making is in the eye of the beholder. My job instead is to provide them with an environment that is rich in sense-making activities and in which students are held to high standards in the ways in which they argue for their solutions.

Someone in the audience asked, not cynically, how instructors in K–12 schools can be expected to teach in this sort of way while still preparing kids for the content on standardized tests. Peggy’s answer was, you just have to do it, and perhaps there are ways to design classes where you can teach content in this way and not lose much. I have nothing to add to that answer, except that I hope that people everywhere begin to realize that it doesn’t do much good for students (K–12 or university level) to know a lot of content that makes no sense to them.

Epilogue: I’m writing this guest post because I lost a bet that I never made. My wife’s alma mater Indiana University was making a good run of it in the NCAA mens’ basketball tournament back in March. Dave, John Golden, and I thought it would be fun to see whose Big Ten alma mater (or proxy thereof; I went to Vanderbilt, which posted a relatively miserable 16–17 record and didn’t even make the NIT) went the farthest. Needless to say the Hoosiers didn’t get it done. But at least some good can come out it.

Do you really understand?

A version of this post was originally written for my Learning Museum blog. As I continue to consolidate my blogs, the timing seemed right to move this post here. You see, it's ArtPrize time again in Grand Rapids. Also, two of my classes are currently focusing on the differences between making sense of something and truly understanding it.

=====

Grace Chen wrote a very thought provoking post called, Making Sense of Understanding. I was reminded of her points as I attended an ArtPrize 2011 Sneak Peak at the University of Michigan's School of Art & Design and SiTE:LAB. As I reflected on what I made up as I watched the the movement of the piece in the movie below, it became clearer the distinction between making sense and understanding.

When I first saw it moving, I thought that it was a mobile and that the wave motion was the result of the wind. That didn't make sense, however, as we were indoors and there wasn't that much air movement. My wife, Kathy, thought that it might be moving to the music. But when the music stopped, it kept undulating. Then I looked up and saw that levers were moving up and down which resulted in the wave-like motion. Mystery solved - the piece moved in some preprogramed way. This "made sense" to me and so I moved on to look at more of the art.

Before I got to far, Kathy exclaimed, "Cool!" I turned back to see her reading a description of the piece near by. "It's the motion of the ocean," she continued.

Huh? That didn't make sense. But then I read the description too:

Because it might be difficult to read, here is what it says:

The installation draws information from the intensity and movement of the water in a remote location. Wave data is being collected and updated from National Oceanic and Atmospheric Administration data buoy station 51003. This station was originally moored 206 nautical miles Southwest of Honolulu in the Pacific Ocean. It went adrift and the last report from its moored position was around 04/25/2011. It is still transmitting valid observation data but its exact location is unknown. The wave intensity and frequency collected from the buoy is scaled and transferred to the mechanical grid structure resulting in a simulation of the physical effects caused by the movement of water from this distant unknown location.

Now I understand. And understanding did make the piece even cooler.