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. 

How would you describe someone successful at math?

I try to start each semester with some activity that brings to light my preservice teachers' beliefs about mathematics. This typically entails using a modified version of a Simile Survey used in my doctoral work, but last August I attended a session at the Michigan Council of Teachers of Mathematics Conference that introduced me to some new tools. This is one of the tasks that Shannon Sweeny shared with us during the session:

Shannon gave me permission to use this instrument in my teacher preparation  courses, and I combined it with an article from Educational Leadership to create a productive workshop that got at beliefs about success in mathematics.

The teachers picked the five words their "elementary-self" would have chosen to describe someone who was successful at math. Then I asked that they write out the experiences they had during elementary math lessons that might have contributed to their choice of words. Next, I had them read the first part of the article, Afraid of Looking Dumb, that includes the following exchange between the author and a second grade student, Brenda.

Brenda: I feel like I'm not good in math, and I get scared. I feel like I'm the dumbest in the class in math.  

Me: In just math or other things, too? 

Brenda: Other things, too. 

Me: Why do you feel that way? 

Brenda: I can't do as much as other people. 

Me: When you start math, what do you tell yourself? 

Brenda: I can't do it. I'm scared. 

Me: What if you couldn't do it? 

Brenda: I wouldn't be smart in that subject. I should be able to do it. I'm scared that people will tease me. 

Me: What would it look like if you were smart? 

Brenda: I'd be a fast thinker, quick learner. The first time I try things, I'd get it right. 

Me: What do you tell yourself when I tell you that you're better than you think in math? 

Brenda: I don't believe you. 

Me: Do you want to change? 

Brenda: Yes, but how? How are we going to change our thinking? 

I asked the teachers to: (1) predict the five words Brenda might have selected regarding success; (2) identify the evidence in the dialogue that supports their choices; and (3) infer what experiences Brenda might have had that fostered her beliefs. As an example, I said, "I think Brenda might pick 'arrogant' because of the line, 'I'm scared that people will tease me.' This might be the result of having experiences where kids or teachers teased her or her peers when they made math mistakes." A fair number of the teachers remarked that they thought Brenda would have picked a lot of the same words as their "elementary-self." Consequently, in predicting the experiences that might have contributed to Brenda's view, many shared their own experiences in elementary math classes as possible reasons for Brenda's perspective.

This awareness that not much has changed in the teaching math since they were elementary students is an important step if we are going to begin doing things differently. I told them that I do not want to be having this same conversation with their students when they get to my class in 20 years. Don't get me wrong, it's an interesting discussion, but the semester would get off to a better start if everyone entered class believing they could be successful in math.

What's wrong?

• Make sense of problems and persevere in solving them
• Construct viable arguments and critique the reasoning of others
• Attend to precision

These are three of the eight Practice Standards associated with the process of doing math that will now be assessed in states that have adopted the CCSS. Some teachers I have talked to are concerned about how they will incorporate the Practices in their math classes. I decided to share an activity that addresses these three Standards and can be adapted to nearly any content - a Crit Session.

I first heard of Crit Sessions while reading Jonah Lehrer's Imagine. Despite the controversy surrounding this book, I found his description of the Crit Sessions held at Pixar to be particularly interesting and began to consider how to apply this idea in my classes. So I developed the following workshop.

+++++

Crit Session for Presentations of Mathematical Thinking

This can be used for any content where students, individually or in groups, are presenting mathematics. The work does not need to be completely thought out. In fact, it helps if the work is "in progress" as the feedback provided by peers can help the students to move forward.

Schema Activation [5 minutes]: Review the information you want to share.

Focus [5 minutes]: What is a Crit Session?

Read the following edited extract from Imagine:

Concentrate on the second paragraph:

• Learning from other people's mistakes
• Distributed responsibility for everyone's success

Activity [40 minutes - assuming 4 group presentations]: Crit Sessions

• 5 minute presentation
• 5 minutes for feedback
• Critical groups (audience) take a minute to organize critiques
• Members of the presentation group splits up to meet with critical groups - this makes the critiques more manageable and seem less harsh than in the whole class setting

Reflection [10 minutes]: What; So What; and Now What?

Back in presentation groups, after everyone has presented, the members share:

• What feedback did we hear?
• So what made the feedback important?
• Now what should we work on?

The information does not necessarily need to focus on their own presentation since we sometime learn from other people's mistakes.

+++++

The students I have tried these Crit Sessions with have been positive about the approach. They appreciate the permission to be critical (critique the reasoning of others). Most times they feel as though they have to be nice, non-critical, in order to have a safe classroom environment. Yet, no one felt attacked or belittled and the classroom community seemed even stronger after the workshop. After all, they were working toward a common goal - success.

I am also impressed with how these Crit Sessions have supported a growth mindset in regards to doing math. The students seem more attuned to the need to put effort into working toward precision in their work and the work of their peers. Also, there is a built in expectation that they will need to make improvements (persevere) to their work. No one expects to get it entirely right the first time.

If you try this approach in your class, please share your efforts and any adaptions you needed to make in the comments. Thanks!

What does it mean to do mathematics? IIc

Previously, we were introduced to the Doing Math Anchor Chart task (here). Then I shared a recent exemplar that used the metaphor of riding a bike to communicate the preservice teacher's vision of what it means to do mathematics. In this post, I will offer a teacher's more traditional concept map representation.

Chartists Statement

My Anchor Chart is in the form of a cycle or a process because I see the act of doing math as a cycle with the central goal of deepening our understanding.  When we do math we begin with a problem that we want to solve.  Using prior knowledge and problem solving skills that we have developed (which could include using representations) we work on the problem to get a response.  We then must evaluate our response and evaluate the process we took to get the response.  If our method isn’t working or we don’t feel our response is correct then we go back using this knowledge we gained to implement a different strategy or to see if it’s possible we responded to a different question.  If we don’t need to go back to rethink about our question or our problem solving method, we make conjectures or generalizations about the responses we obtain.  Many times we have questions about the conjectures about whether or not they work for all cases.  This gives us a new question that we may want to explore.  Regardless of exploring new conjectures we somehow share our thinking and responses with other people to gain their insight.  Because we share our responses, conjectures, and thought processes, we allow other people access to these things so they can ask questions and do math themselves.  In addition, the conjectures we develop may allow us to make connections to problems in other contexts or we can use the problem solving skills we developed in other aspects of life, so the cycle of doing math is not closed, and because we go back and retry different problems and form new questions based on the work we are doing the cycle does not go one way. 

Key

I choose to do develop my chart without specifically writing down the Process Standards in any area because in my cycle of doing math, different aspects of each standard are included in different steps of this cycle.  Here I will explain how the different standards fit into the chart and where. (Note: the numbers correspond to the labeled boxes):

1.     Communication:  the problem we have to solve maybe to analyze and evaluated someone else’s work.

Content: The problem we are working on is based on the content we want to learn, for this unit that that would be Algebra.

2.     Problem Solving: The problem that we have to solve requires that we implement a variety of strategies that are appropriate to the context of the problem.

Reasoning and Proof:  If our problem is to prove something then the strategies we implement will be the different methods of proofs and determining which proof method is most appropriate for the problem presented.

3.     Problem Solving: Implement a variety or strategies determining which one will work best as we work towards a response.

Connections: Recognize similarities between the problem that’s presented and previous problems we have solved.

Representations: Use representations to help you think about a problem and translate between the representations to help solve problems.

4.     Problem Solving: Reflect on the strategy we used.  Did the process we used allow us to effectively find a response?

Connections: Think about how the problem we are solving connects to and builds on other mathematical ideas.

Representations: Use representations as a way to organize our thinking.  Look back that the ones we used when solving problems and think about how they helped us.

Metacognition: interpreting our response and strategies require that we reflect on our process and how we thought about the problem. 

5.     Reasoning and Proof:  Make conjectures about the response we obtained, this may occur though connecting our work with previous work we have done.

6.     Problem Solving: It’s possible that the knowledge we gained and would like to share is the skills we used to solve the problem.

Reasoning and Proof: We can share a conjecture that we have developed or we can share our thinking in the form of a proof.

Communication: Clearly expressing the response and the process to others either formally through writing, possibly a formal proof (Reasoning and Proof), or through discussion with others. 

Connections: It is possible that we use examples and make connections to other problems to help our audience understand what we are communicating.

Representations: Maybe we choose to organize our thinking into some form of representation as way to communicate our thinking with others.

Metacognition: We may decide to share our thinking process; inorder to do this we must think about our own thinking. 

7.     Reasoning and Proof: Throughout the entire process of doing math, we are building an argument.

Connections: We build on the knowledge we gained in order to solve problems in the future.

8.     Connections:  We can apply the knowledge we built through this process to contexts outside of math. 

What does it mean to do mathematics? IIb

Previously, we were introduced to the Doing Math Anchor Chart task (here). In this post, I share a recent exemplar that used the metaphor of riding a bike to communicate the preservice teacher's vision of what it means to do mathematics. The next post will offer a more traditional concept map representation.

Artist's Statement

Doing Mathematics is like riding a bicycle. It requires all pieces to work together to move forward. When riding a bike one must stay balanced. This is the same for mathematics. We must employ all the processes to complete a problem and understand if fully.

Pieces

Algebra = Bike Rider

As a student, doing mathematics means doing algebra in some cases. The student must employ all of the tools (the bicycle) in order to perform - just as the rider must pedal as the wheels move and control using the handlebars to ride the bike.

Pedal 1 = Reasoning and Proof

When doing mathematics, reasoning and proof is central to the process. When riding a bicycle, moving the pedal is essential to moving the bike along. When we are reasoning we are investigating a problem and developing arguments. We can also select how we want to reason, like we can change the pace at which we are pedaling.

Pedal 2 = Problem Solving

Problem Solving and Reasoning and Proof go hand-in-hand - just as the two pedals work together to move the bike forward. When problem solving you are building on mathematical knowledge and working to use appropriate strategies (like the appropriate pace of pedaling).

Bell = Communication

The bell on the bicycle is used to communicate with others around. in mathematics we use communication to talk with others in a clear fashion about our work. There are precise signals one can use to tell others you are oncoming when using the bell just as mathematicians must use precise language. Also, the rider must evaluate when the best times are to use the bell and evaluate if others will run into them before using the bell - like we evaluate others' thinking in mathematics.

Gears = Connections

The gears of the bicycle work together with the pedals and the wheels to move the bike forward; they are also the pieces that keep the whole process of riding a bike continuous. The fluidity of a bike is similar to the fluidity of mathematics in which we can find connections and then apply them to doing mathematics.

Wheels = Representations

We use representations in mathematics to communicate or record our ideas. Essentially representations are what help us solve problems through their application. Without the wheels on the bike we would go nowhere, thus we need representations to model mathematics like a bike needs wheels to move.

Handle Bars = Metacognition

When riding a bike we balance on the handle bars. While you can take a hand off now and then, we find we are most balanced with both hands resting on the handle bars. In mathematics, we use metacognition to think about and communicate our thinking. It controls the steps we take when working as we analyze what we have done or what we need to do. The handle bars control in which direction we go. The brakes are also located on the handle bars. At times we may get stuck; this is when we stop our work and think about our thinking once again.

What does it mean to do mathematics? IIa

One of this blog's most popular posts describes how a group of preservice teachers envisioned doing mathematics. They combined elements of concept maps with metaphor to create an anchor chart that expressed their views. This activity is typically untaken at the end of the first unit in Teaching and Learning Middle Grades Mathematics - a unit that focuses in on the NCTM's Process Standards. It seemed like a good time to share some more recent exemplars. 

First, here's the workshop:

Schema Activation:  What will it look like?

• Look back over your work from previous Teaching Math Workshops as you determined what was important in the NCTM’s Process Standards. Note any patterns you see in your journal.
• “Students entering a classroom that visually represents the mathematics being studied are more likely to share in that enthusiasm and be willing to create and share their work (Ennis and Witeck, 2008).” So what will your math classroom look like in order to show what it means to do mathematics and encourage learners to do the same?

Focus: Anchor Charts

The following description comes from Debbie Miller’s (2002) Reading with Meaning:

…I do create “anchor charts” after lessons from which I want children to remember a specific strategy or concept. I write a note of explanation at the top of the chart and note snippets of conversation, individual comments, and statements that reflect our work together.

Anchor charts make our thinking permanent and visible, and so allow us to make connections from one strategy to another, clarify a point, build on earlier learning, and simply remember a specific lesson. (p. 57)

An anchor chart is one way to communicate to learners your expectations regarding what doing math will look like in your classroom.

Activity: Create an Anchor Chart for Doing Mathematics

1. Reflect on the Anchor Chart Rough Drafts completed in class.
2. Develop an anchor chart called "Doing Mathematics."

Reflection:  What’s important

Review your “Doing Mathematics” anchor chart. Write an "artist's statement" that highlights what is important in your chart.

*****

Two exemplars from Fall 2012 are found in the next two posts. (It was turning into a really long post.) The first leans heavily on metaphor as she associates doing math with riding a bike. And the second uses a more traditional concept map to communicate her vision.

Y U no like math?

Last semester, I was fortunate to teach a class that brought together preservice and inservice math teachers in a combined undergraduate/graduate course. The graduate students mentored the undergrads in the areas of assessment, evaluation, and planning. In return, I gave the inservice teachers the freedom to use design thinking to explore any issue they were experiencing in their teaching. This post presents a portion of one of their projects.

Julia Moore is an adjunct at Grand Rapids Community College, and she gave me permission to share her work with you. One of the things she found in the research she read was that students' self-perception related to mathematics impacted their engagement in the subject. She wondered if she might influence these individual perceptions by addressing the intellectual struggle and emotional  feelings typically associated with doing mathematics. In order to tackle these perceptions, she asked students to use Internet memes to describe their relationship to math. They shared the memes in class: 

This is my favorite:

According to Julia, the most valuable part of this exercise was the discussion it started. She writes: 

The conversation had its foundation in humor, so the classroom environment was relaxed and free-flowing. ... They shared stories of:

• Struggles with mathematics or previous teachers
• Successful experiences with mathematics or previous teachers
• Frustration at learning something they may NEVER use again
• Being able to excel in their other classes, but haven't been able to transfer those skills into mathematical thinking

I wish I had TAPED IT!!!!

The discussion presented Julia with the opportunity to explain that everyone struggles in math; it's one's willingness to engage in the struggle that increases one's likelihood of success. She also highlighted where people encounter math in the world. Because of the safe space the original activity created, the students' were more open to this discussion and more engaged in later mathematical discussions.

She plans to add a second piece to this activity in the future. Students will create a new meme at the end of the semester that indicates some mathematical understanding they learned during the course. This is the example Julia shared:

When will it end?

[In the United States] Teachers act as if student interest will be generated only by diversions outside of mathematics. (The Teaching Gap, p. 89)

Schema Activation: Journal Jot

Describe the process of converting three-nineteenths to a decimal

Focus: Changing the Script

We have read in The Teaching Gap how teaching mathematics in the United States typically focuses on preparing students to perform prepackaged procedures. What if we tried to change the focus from mathematical procedures to mathematical practices?

In this workshop, try to refocus your efforts away from simply following a mindless process and explore what new knowledge is waiting to be learned. As you work on the following problem, please keep track of how you are (or aren't) engaging in these practices.

Activity: Find the exact decimal representation for three-nineteenths

[The follow represents how learners have engaged with this activity.]

This seems like a fairly straightforward exercise. Why not just plug the numbers into a calculator? The converting-a-fraction-to-a-decimal procedure requires us to divide the numerator by the denominator until the decimal either repeats or terminates. Three divided by nineteen - easy. Except, the quotient displayed on the calculator screen does not provide enough information since it does not show enough places.

Maybe using a different calculation tool would help. What about an Excel spreadsheet?

This seems to suggest that three-nineteenths terminates. And it matches the calculator's answer. A problem solver would not simply accept these and be done - would she?

What other strategies could apply?

• Work the division out by hand
• See if another related fraction might shed some light on the decimal
• 1/19 since 1/19 multiplied by 3 is 3/19
• 16/19 since this would be the complement of 3/19
• Find x such that 19x = 100 and then calculate 3x
• Multiply 0.157894736842105 by 19 to see if the product is 3

[Learners often explore these approaches with various levels of success. Having monitored their own progress, the learners often turn to other strategies or decide to seek a solution to other related problems. They do not want to be stuck trying the same approach over and over again.

WolframAlpha has added a wrinkle to this activity as it provides an exact decimal representation for three-nineteenths. Originally, we thought this would be problematic as it gives a clear answer to the original question. It is interesting, however, that focusing on a problem solving approach generates similar alternative/extension problems whether or not the answer has been found.]

New problems:

• Which fractions repeat and which terminate? Why?
• Can we predict the period of the decimal representation of a given fraction?

Partial table created by learners in order to look for patterns

[Answers to these alternative/extension question rarely are answered in the time available to the workshop. Learners are encouraged to continue exploring the problems if they are interested. Even though they have not come to any firm conclusions, they usually have some interesting answers to the reflection questions.]

Reflection: Mathematician's Chair

• What did you do as it relates to the mathematical practices?
• So what was important about this work?
• Now what might you do the next time you encounter a problem?