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. 

Lessons Learned in Tanzania - Pole Pole

Start of our climb up Kilimanjaro

While teaching in Tanzania taught me a direct lesson about being resourceful, there were also some indirect lessons associated with my time in Africa. Probably the most powerful came while climbing Mount Kilimanjaro. We did not climb the entire mountain, only part of the Marangu Route, but this brief hike (ascending more than 2,500 feet over 5 miles and then back down),  made a lasting impression. In particular, the guides' exhortations to "pole pole" (Swahili for go slowly) got me thinking about our constant efforts to improve education.

The guides wanted us to go slowly for a couple of reasons. The first was to ensure that we did actually make it to our destination - the Mandara Hut. The climb is steep and plenty of people do not make it because they expend their energy early on or because of some injury incurred from inattention to the climb.

This reminds me of the point made by Stigler and Hiebert in The Teaching Gap that our efforts to improve education are often too frenetic and unfocused to be sustainable. Instead, we need to consider slow, purposeful (sometimes subtle) shifts that ensure we reach our final goal - student learning.

The other reason to take the climb slowly was so we did not miss anything along the way. Although this portion of the route is mostly forest, without any scenic views, that does not mean there is nothing to see. It is easy to miss some of the flora or fauna if one moves too quickly or without intention.

The same thing happens in education. We do not take time to notice the flowers let alone stop and smell them. I know my own teaching experience is much more positive when I focus on enjoying the journey rather than simply getting to the destination.

So I left Mount Kilimanjaro with a bracelet to commemorate the climb and the lesson.

Hopefully it will remind me to take my teaching practice slowly - to make it sustainable and enjoyable.

Do you have a boring worksheet that you want to make more interesting?

Yesterday, I shared a worksheet I was using in my math course for preservice elementary teachers on Twitter. It got enough of a positive response that I thought I would share it here with a bit more context. Here is the Tweet:

Got a boring worksheet? Make it more interesting by adding 2 sentences. #mathchat pic.twitter.com/UebsbuCjJP
— David Coffey (@delta_dc) April 17, 2014

I made the point that this combined two worksheets. The original worksheet came from fractions4kids.com. This one reminded a lot of the future teachers of worksheets they did in elementary school. Some of the teachers had bad memories about those times. I told them they were not alone. A lot of students become disenchanted with mathematics once they encounter the way we teach fractions - rules without reasons. 

We also talked about how pointless it seemed to do all the problems. How much practice did they really need? How much proof did the teacher need in order to know whether the kids could follow the procedure? I shared (confessions of a bad math teacher) that sometimes I might only assign the odds or evens. Still, the only choice I was offering students was the choice to do it or not do it. And many chose the latter.

Fortunately, I learned from Brian Cambourne the importance of providing learners with choice.

Learners need to make their own decisions about when, how, and what "bits" to learn in any learning task. Learners who lose the ability to make decisions are disempowered. p. 187

This lead me to begin altering my approach to assigning work, which is evident in the second worksheet. I began adding a line or two asking the learners to pick the problems they did or did not want to do and why.

There was nothing special about the first worksheet. It could be on just about any topic. But the extra instructions, the two sentences asking learners to make choices and explain those choice, seemed to make the task much more engaging. And not just for the learners. I found reading their rationale behind their selections much more interesting than simply checking their answers.

Finally, teachers could have fun with the extra instructions. A group of student teachers came up with the idea of asking their high school students, "What items would you assign your best friend? Your worst enemy? Why?" So what questions might you add and why?

Why so tense?

from What I Really Do Meme

It's inevitable. Whenever I work with student teachers, there comes a time when we have to talk about the tension they are experiencing between their ideal classroom and the reality of their placement. The discussion goes something like this...

Think of your vision of teaching (the ideal) and what is actually happing in your classroom (the reality) as two points. Now, stretch a rubber band between these points. Many of you will notice that the ideal and the reality are quite far apart resulting in tension between these two points of view.

The tension often results in one of two outcomes. New teachers quit (the rubber band breaks) because the distance between their ideal and the reality is so far apart. Or a new teacher abandons their ideal and snaps back to the reality (i.e. they teach as they were taught).

In order to deal with this tension, I have found it helpful to expand both my vision of the ideal and my perspective of the reality. This means that I maintain my core teaching philosophy while considering what might be negotiable.  I say to myself, "I would be willing to do x as long as I don't have to give up y." When it comes to the reality, I continue to be honest about my situation but I try to be aware of potential areas I might be able to build on. I remind myself, "If I concentrate only on what I dislike, then that's all I will see." As both expand, I look for areas where they overlap; this is where I find the subtle shifts that allow me to survive the situation while gradually subverting the current system.

Here are some examples of how I and other have attempted to make subtle shifts in our teaching.

So this is part of what I do as a supervisor of secondary math student teachers; I try to help them to be aware of what teaching is and what they want teaching to be. And then I try to support them in improving teaching, which is where the fun begins.

Which way is ... ?

A preservice teacher leading a review for a quiz on rational number computation invited me to watch the lesson and work with her to improve on it. The objectives addressed in the review activity were from the Michigan Grade Level Content Expectations (GLCE):

• N. FL. 07.08 (GLCE): Add, subtract, multiply and divide positive and negative rational numbers fluently 
• B. N. FL. 07.09 (GLCE): Estimate results of computations with rational numbers

But the preservice teacher was also interested in developing conceptual understanding - especially around the idea of how multiplying and dividing by numbers between 0 and 1 impact the result. A good activity from NCTM (pdf) was found and modified in an effort to achieve these goals.

Move down or sideways (never up) through the maze from Start to Finish. You may not retrace any steps. Begin with 10 and as you move along a segment do the indicated computation. Record your steps on the scorecard. Your goal is to find the path that results in the largest (or smallest) value when you reach the Finish.

After the lesson, we discussed ways of using the activity more effectively. The first idea was to model what a path looks like. Because this kind of activity was new to the students, it took them a while to understand what was expected of them. For example:

What if we just followed along the left-most edge?

1. 10 x 0.9 = 9; 
2. 9 x 1.75 = 15.75
3. 15.75 + 5 = 20.75

Next, without doing any computation, we would ask the students to predict the path that would result in the greatest result or the least result. Making predictions is a great way to develop interest in a task. Student would share their predicted path and the rationale for their choice. This would provide some insight into the students' number sense related to multiplication and division of rational numbers. 

Then the students would estimate the results of several paths. This would allow them to check out their predictions and refine our list of which paths might represent the largest (or smallest) value. Also, this would address objective B. N. FL. 07.09 from above.

Finally, the students would be asked to compute the path they believed would result in the largest (or smallest) value [N. FL. 07.08]. Calculators are not allowed in this classroom but we decided that we might allow students to use calculators on up to half of the calculations. That way they would be exposed to the idea of using calculators strategically instead of with an "all-or-nothing" mindset.

As an extension, we might ask the students to find the easiest or hardest path to follow without using a calculator and why. We thought this would offer the students an opportunity to be metacognitive. It would also provide us with information on areas where students could improve on their fluency.

The preservice teacher was able to apply some of these subtle shifts to her later class with success. She writes:

...they did much better!  They were excited to do something "more fun than boring problems."  I was really happy with the responses I got...

What are your thoughts? How would you improve on this activity? Why?

What if we gave them the answers?

My experiment of teaching a course where preservice and inservice teachers share two hours of class time has been going well. (I introduced the concept here.) In fact, one of the preservice teachers said recently, "I wish all of my education classes had classroom teachers in them." I believe the following example explains why he feels that way.

That same preservice teacher was part of a small group (along with an inservice middle school math teacher and a community college instructor) who were analyzing middle school students' work on an algebra assessment. They were talking about how difficult it is to get students to share their thinking especially once they assume they have arrived at an answer. I concurred and explained that this was one of the reasons I focused on using metacognitive memoirs, saying, "I know the answer but I don't know what you're thinking." This gave the inservice middle school teacher an idea.

He wondered what would happen on the next test if he gave the answers and asked the students to focus on their thinking. A few days later, I (along with the preservice teacher and the community college instructor) received the following email:

Hi, I gave a test yesterday in my 8th grade math class and I gave them all of the correct answers at the beginning of the test to see if it would improve the work that they showed and how well they explained their thinking.  They were shocked, but they actually caught onto the idea quickly, I didn't even have to tell them why I was giving them the answers, they came up with it themselves.  While the test responses weren't perfect, students did a MUCH better job sharing their thinking than they ever have before.  I am excited about how this turned out and I anticipate doing this more often in the future.

I asked the teacher if he would mind me sharing this experience and the test on my blog and he agreed. Not only that - he also provided how he implemented this new approach, a sample of students' work, and students feedback.

After handing out the test, the teacher began:

• Teacher: "Listen closely. This is a test. You know the rules as far as talking, etc."
• Teacher begins reading off answers.
• Students are following directions, no questioning until after the page flip.
• Student 1: “Why are you telling us all the answers?”
• Student 2: “I like this!”
• Student 3: “Don’t stop him.”
• Teacher keeps reading answers. There is no contesting of getting the answer and the kids keep filling in right answers for remainder of test.
• Student 4: “I don’t understand this...”
• Student 5: “Why did you just give us the answers?”
• Student 6: “Do we have to explain what we did for the answers you just gave us?”
• Teacher: “You’re not going to get any credit for having the right answers. You’re only going to get credit if you can explain how you get the right answers. So all of you are starting right now with all the answers and a 0%.”
• Student 4: “I like the other way better.”
• Teacher: “Let me just say one more time...You all have the right answers, so the explanations are where you can earn the points. With that in mind, go ahead.”

Here is what the test looked like after the teacher had read the answers.

And here are some examples of what students wrote:

After the test, the teacher asked for students' feedback on this approach to assessment. These represent some of their responses:

• "My head hurts because I actually had to think."
• "I realize now that I've never done a very good job explaining my answers."
• “This was like an English test!”
• “It took forever...like, I know what I want to say but I can’t explain it.”
• “Didn’t like volume of writing and repetition.” (Felt like there was too much writing and they were answering the same questions over and over.)
• “Didn’t see the point of giving out the answers because you have to do all that thinking to get the answer anyways.”
• “Liked it. I always spend time figuring out the problem so I don’t explain. This helped cut out the calculation step.”
• “Didn’t like because I don’t like explaining myself.”
• “Would have prefered to find answers instead of trying to explain because sometimes I can just get it (in my head).”
• “Liked having answers, otherwise I spend a lot of time trying to get the answer. This way I know the answer is right.”

I hope that I was able to adequately articulate this approach to assessing students' mathematical thinking. If you have questions or ideas, please leave them in the comments and I'll be sure to pass them along. We have 8 more weeks together in this course. I'm looking forward to whatever else they come up with in that time.