Wednesday, June 29, 2011

Don't you get it?

"What's not to like about the flipped classroom?"

I get it. Believe me, I get it. I understand why teachers like me are so enamored with the latest instructional craze. I saw it last night reading the Tweets from an #edchat on flipped classrooms. The attraction for me is that I get to have my cake and eat it too.

The model is based on the idea that teachers can flip the traditional way things are done in the classroom. Whole-class instruction, in the form of lectures or demonstrations or whatever, is now done after class. The application of what was taught, often thought of as homework, is done in class. Technology can support this shift but I am told it is unnecessary.

I like the idea that class time becomes more collaborative in this model. Instead of spending time disseminating information, the teacher can connect with learners one-on-one or in small groups. This offers teachers an opportunity to assess learners' progress in ways that are impossible during whole-class instruction. When learners struggle with a problem, they are not on their own (which is what typically happens when the same problem is assigned as homework). The teacher or their peers are available to support them through their struggles. Essentially, this represents a much more relationship-orientated approach of educating learners.

Outside of class, learners are assigned podcasts, videos, reading, or some other resources that prepares them for the next day's problems the way whole-class instruction used to prepare them for homework. Although technology is not required, it often comes up as a way to replace a lecture. I like the idea that learners might be given some choice as to when, where, and how they watch a lecture/demonstration of the content or skill being covered. I also like the idea that I can create the perfect talk. There will be no more forgetting my place, misspeaking, or being interrupted for a bathroom pass.

This flipped classroom model is a dream come true for teachers like me. It increases the amount of time I can spend working directly with learners without having to give up covering or controlling the content (and doing it perfectly, I might add). But I have spent the last twenty years of my professional (and personal) life working at moving beyond being a teacher like me.

As I see it, the flipped classroom tugs at both my instincts as a learner and my experiences as a student. The learner likes the collaborative approach that views learning from a social constructivist perspective. It fits with what I have learned about natural learning. I embrace this aspect and hope other teachers will as well. It is what the student in me likes - consumption, control, and perfectionism - that causes me concern. My successful experiences in the traditional school setting makes this attractive to me, especially since it "addresses" shortcomings like the ever increasing amount of material to cover and the need to differentiate instruction. But I do not believe that I can have it both ways - satisfying my learner and my student. While others may be able to walk this fine line, I have found that I always end up erring on the side of consumption, control, and perfectionism.

So I get it. I just don't want it - at least not all of it. I will take what works for my learners and leave the rest. And I trust all of you to do the same.

Wednesday, June 22, 2011

What's GRR?

Yesterday afternoon, I engaged in a Twitter discussion about the Gradual Release of Responsibility. Because of the confines of the medium (maximum of 140 characters in its pure form), we quickly condensed the phrase to GRR. This resulted in the following tweet between John and Frank
I am a nosy Tweeter so I butted in and shared this video on GRR by Jeffrey Wilhelm:

My first introduction to this teaching and learning model was at a workshop on content literacy put on by The Learning Network. Margaret Mooney was the keynoter and she described the model using a figure similar to the one shown to the right. I was resistant because the leftmost stages reminded me too much of lecturing - a method I had decided was ineffective when it came to constructing understanding. I was what my wife calls a "constructivist gone wild."

Then I read about GRR in Debbie Miller's book, Reading with Meaning. She writes:
Chances are that if you think back to a time when you learned how to do something new, the gradual release of responsibility model (Pearson and Gallagher 1983) comes into play. Maybe you learned how to snowboard, canoe, play golf, or drive a car. If you watched somebody do it first, practiced under that person's watchful eye, listened to his or her feedback, and then one fine day went off and did it by yourself, adding your own special twist to it in the process, you know what this model is all about. (p. 10)
I was not completely convinced, but I was beginning to see the appeal of this more natural approach to teaching and learning.

Finally, I watched a first-grade teacher use GRR during a series of reading comprehension lessons. She modeled for me how she used formative assessments to evaluate where learners were in the model and how that informed the level of support she needed to offer during instruction. This demonstration helped me to see GRR as a framework that supports teacher decision-making during lessons focusing on processes. It has been an essential framework in my teaching ever since.

A few years ago, one of our teacher assistants suggested that a rubric describing the roles of teachers and learners in this model might help him in identifying where learners were at in their understanding. Working with my colleague John Golden, this pdf draft was developed. You will see that the role of the teacher moves from being a model, to being a mentor, to monitoring the learner.

There are many other resources available on this model. Fisher and Frey discuss it in Better Learning through Structured Teaching. @mrsebiology recently tweeted these resources (a framework and a matrix).

I hope this helps to answer the question, "What's GRR?" If you're interested in learning more, I would suggest: (1) find an expert to watch; (2) collaborate with the expert in your own classroom; and (3) modify it to make it work for you. At least that is how I learned to apply this model.

Monday, June 20, 2011

Did they guess?

The following is a workshop used in a class for preservice K-8 math teachers. The goal of this activity is to use a simulation model to determine an approximate probability for a compound event. It is based on an example found in The Art and Techniques of Simulation. I updated the activity to reflect my current understandings and the Common Core State Standards (CCSS).

Grade 7 CCSS in Probability and Statistics [7.SP]
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Schema Activation: Pop Quiz
Take the following - record how sure you are of each answer on the righthand side.
Pascal's Triangle Quiz
True or false: The following number sequences can be found in Pascal's Triangle
  1. Triangular Numbers
  2. Square Numbers
  3. Pentagonal Numbers
Focus: Monte Carlo Procedure
from M3RP
Activity: Making a Guess
A class is taking a three-guestion true-false quiz. What are the chances that a person making a random guess on the quiz would get: (1) all of the question correct; (2) two question correct; (3) one question correct; or (4) no questions correct. You should start by making a prediction and then follow the steps outlined in the Focus to conduct 50 trials.

Reflect: Math Congress - Problem Solving
  • What did you understand about the problem?
  • How did you create and carry out your plan?
  • Looking back, where did you learn something new?
  • Looking forward, when could you apply your learning to another situation?

Friday, June 17, 2011

How much ground can I cover?

Last week I shared this visual metaphor for end-of-the-year teaching and asked, "Do you see what I see?" Three commenters (Sandi, Erin, and Manzo) answered the challenge and each presented a plausible vision of how the picture reflects what might be happening in classrooms as the school year ends. It was not my intent to disparage the adult peddling the bike or teachers facing the final days of school. As I said, a snapshot in time requires context in order to develop a complete picture of what is happening - whether it is a bike trip around the lake or a mathematics lesson. Therefore, I want to focus on how the scene reminded me of my own experience as a novice teacher.

My second year of teaching was my worst year in the classroom. The previous year I had been hired just three days before school started, and I felt like I was always trying to catch up. I vowed that I would never be that unprepared again and spent the summer planning my lessons for the coming year. 

I was assigned an eighth-grade general math class and an algebra class. The eighth-grade class was easy since it was the class I taught my first year. The algebra class was a new prep and I spent most of my time planning those lessons. By the time school started, I could have told you exactly what I was teaching on any day during the year. For example, on October 16th I would teach decimal division in the general math class and coin mixture problems in algebra.

School began and I put my plans into action. Pretty soon, it became obvious that the eighth-graders did not appreciate all the work I had put into planning my lessons. This was especially true of the algebra students who were accustomed to being successful in math. No matter - if I was going to cover all the material in the textbook, I needed to stick to my schedule. And so I did.

As I taught the last few lessons of the year, a few things became very clear. First, I had followed my plans and covered all the content presented in the textbook. Second, I had lost nearly all the algebra students around February. Their scores on the cumulative final were abysmal and there were a lot of angry parents. Finally, I understood that it would be a long time until the principal would ask me to teach the algebra class again - even though I had learned my lesson.

In an effort to cover content I had ignored learning. I had ignored assessments. And I had ignored common sense. A truer picture would show the entire class being dragged behind me as I struggled to finish the textbook by the last day.
I am not suggesting that teachers should not make learning plans during the summer. Only that we recognize that they are plans and not scripts. So plan away but remember they will need adjusting based on your learners. And also make time to recreate this summer. A bike ride might be nice.

Wednesday, June 15, 2011

How can we communicate our thinking?

The math teachers I work with often express frustration with their learners' inability to communicate their thinking when it comes to solving problems. If learners are stuck, they often struggle to articulate what they have tried. If the problem has been solved, learners have trouble explaining their efforts. Too often, learners respond to a teacher's question about their thinking with, "I don't know." This does not present teachers with the assessment data necessary to evaluate what learners can do or are trying to do, which makes it difficult to plan what comes next.

I encountered this same problem while working with fifth-graders on fraction computation. They were practiced in giving answers and even showing work but when I asked them to share their thinking they often said, "I don't know." This provided me with an opportunity to try a response suggested by Ellin Oliver Keene in a session at the 2009 MRA Conference: "Pretend that you did know - what would you say?" The fifth-graders found that this framing supported their communication efforts by freeing them to take a risk because they were "pretending."

While this got them talking about their thinking, they still needed more support to organize their efforts. I thought it would help to demonstrate what a reasoning recount might look like. I was introduced to the recount text form through Margaret Mooney's book, Text Forms and Features. Here is the model reasoning recount I wrote based on the prior efforts of the class to think about adding fractions.

Recently, I have been collaborating with Jennifer Brokofsky via Twitter and email about ways to connect reading, math, and writing. The figure below represents our current thinking.  I hope this vignette further demonstrates the link. We recognize that this is work in progress, and your support would be appreciated. Please share your thinking in the comments.

Monday, June 13, 2011

Are they related?

The following mathematics activity was originally written for middle school learners. It was intended to introduce  this  seventh grade, Michigan Grade Level Content Expectation: "Create and interpret scatter plots and find line of best fit; use an estimated line of best fit to answer questions about the data." While lessons on this expectation typically include middle schoolers measuring things like their own arm span and height to see if they are related, my experiences with adolescents suggest that they are often self-conscious about their bodies and uncomfortable with such activities. Instead, I came up with the idea for them to measure dolls and action figures to see if certain relationships fit what we know about the human body. Here is the activity using the Connected Math Project's Launch-Explore-Summarize learning plan format.
Needs: One hour; learners in groups; The Da Vinci Code; The Vitruvian Man handout; something that measures distances; set of dolls or action figures (preferably one for every pair of students); and a method for collecting and graphing data

Launch: The Divine Proportion
Read "The PHI Section" from Dan Brown's The Da Vinci Code. Here is a paragraph to give you a taste:
Despite PHI’s seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fundamental building block in nature. Plants, animals, and even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of PHI to 1. 
Test out the theory on The Vitruvian Man (modified for seventh graders)
Explore Data from Dolls
Assign each expert group to explore a relationship described in "The PHI Section"
  • Total height to belly button height
  • Distance of shoulder-fingertip to elbow-fingertip
  • Distance of hip-floor to knee-floor
Here is the data collected from one set of dolls using Fathom:
Summarize Jigsaw
New groups will be formed that include one person from each expert group. Learners will share what they found during their exploration. Teacher walks around gathering information to summarize the activity with the whole group.
In the Common Core State Standards, this topic is found in the high school statistics section:
Summarize, represent, and interpret data on two categorical and quantitative variables
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Friday, June 10, 2011

Do you see what I see?

We live right next to the bike path that runs around Spring Lake. The other day I witnessed the following scene that gave me a laugh. A parent and child were riding on a tandem bike (that I have since learned is sometimes called a tag-along). The funny part was that the parent was exerting a great deal of effort to keep the bike moving while the child had his head on the handlebars and his feet off the pedals. I did not have my camera handy, so I took a mental picture and later drew the following:

Of course, like any snapshot, it is difficult to tell what happened before that led to the scene or what happened after. But as almost always happens, it got me to thinking about education. I saw the picture as a metaphor for what seems to happen this time of year in a lot of classrooms. Before I go on, I wonder if you see what I see.

Wednesday, June 8, 2011

When does it work?

I have been sharing my experience teaching a group of fifth-graders how to problem-solve around fraction computation and using it as an opportunity to demonstrate the Teaching-Learning Cycle in action. Previously, I wrote about how I planned for a problem-solving lesson and then described my instruction during the following lesson. In this post, I want to discuss how I used assessment and evaluation to monitor the learners' progress and inform future planning and instruction.

We were using the clock model as a context for adding fractions and I wanted to gather data about whether or not the learners could determine when this model was an effective approach. I used an existing set of textbook items and asked the fifth-graders to: "Look at the expressions shown below - circle the ones that you think you could use the clock model to solve and place an 'X' through those you could not."
from Scott Foresman – Addison Wesley Math [5th Grade]
Once the kids had completed this task, I asked them to solve one of the problems they had circled. As they worked, I gathered data on whether or not they were able to determine when the clock model could work.

In analyzing my observations, I noticed what the fifth-graders could do and what they were trying to do. First, they all recognized that fractions involving ninths and sevenths were poor candidates for the clock model. Those who chose to solve #2, #9, and #10 were also fluent in applying prior experiences with the time context. Some learners thought eighths could work (circling #4) and others struggled to see that fifths could work ('X'ing out #3, #5, and #8). These last two areas of approximation gave me some ideas about what to focus on next.

The last assessment I gave was intended to gather data about how the fifth-graders might apply the idea of context to a problem that could not be easily solved using the clock model. As a ticket out the door, I asked, "Now what could you do to solve a problem you put an 'X' through?" Based on my evaluation of these assessments, I was prepared to plan for future lessons.

What would you do next?

Monday, June 6, 2011

How might this change things?

Here is another statistics workshop used in a class for preservice K-8 math teachers. The goals of this workshop are: (1) develop familiarity with the lists on the TI-84; (2) practice applying Polya's problem-solving phases; and (3) develop conceptual understanding of the measures of center (mean, median, and mode) and variability (mean absolute deviation). It has been updated to reflect my current understandings about teaching and learning and the Common Core State Standards (CCSS).

Grade 7 CCSS in Statistics and Probability [7.SP]
Draw informal comparative inferences about two populations
(3) Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities...
Standards of Mathematical Practice
(1) Make sense of problems and persevere in solving them.

Schema Activation: Teacher Recount
"Last class, we were introduced to the Penny for Peace contest going on between the middle school math classes at a local school. We used this data to develop a procedure for finding the mean absolute deviation (MAD). This involved: (1) finding the mean of the data; (2) finding the distance (absolute value) between the mean and each data point; and (3) averaging these distances to determine the MAD."

Focus: Problem Solving
"Today, we are going to use Polya's problem-solving phases to explore, 'What happens if...' Recall that our modified phases are: Understand the problem; create a plan; carry out your plan; and look back and look forward. Also, remember that the phases are not intended to be linear. Effective problem-solvers jump between the different phases as they work toward a solution. Please be sure to record all of your work and your thinking so we have an artifact that we can refer to during our reflection. Are there any questions?"

Activity: Transforming Data
Understanding the problem

It turns out that the principal, Ms. Sanchez, decided to add $10 to all the class totals.  How do you think this will affect the three central tendencies (mode, median, and mean) and the mean absolute deviation?  Start by making a prediction.


Create a plan
With your group, work on a plan to use your calculator's list feature to explore this problem.

Carry out your plan
Keep a record of your efforts and your thinking.

Looking back & looking forward
  • What happened to the central tendencies and the MAD when you transformed the data?
  • How did you go about obtaining your results?
  • Why do you think this happened?
  • What do you think would happen if the original amounts were doubled?

Reflection: Problem Solver's Chair
The teacher selects a learner to recount his/her group's problem-solving exhibition. This learner can be a volunteer, selected at random, or chosen based on something interesting observed by the teacher during the activity portion of the workshop. After the recount, other learners offer observations or ideas based on their own experiences

Friday, June 3, 2011

What did you do over summer vacation?

I thought about education, of course.

Dave & Kathy at Mary's Bistro
Mackinac Island is a summer vacation destination located between Michigan's Upper and Lower Peninsulas. The island is home to numerous resorts, a state park, and various historic landmarks. You might recognize its Grand Hotel as the backdrop for the 1980s movie, Somewhere in Time. A unique feature of the island is that motorized vehicles are prohibited. It was in this setting that Kathy and I spent a beautiful day two summers ago taking in the sights as we rode our bikes around the island.

Photo courtesy Todd Van Hoosear
Because no cars or trucks are allowed on the island, horse-drawn vehicles are found everywhere. There are horse taxis, delivery carts, and even street cleaners. During our biking, we happened upon a garbage "truck" being pulled by a team of horses. As we passed, we noticed that the driver was fast asleep. It didn't matter. The horses knew where to go - probably after years of following the same path. I was impressed.

I have been thinking about that experience recently (and not just because summer is once again upon us). Current education reform efforts have me wondering if what policy makers have in mind is a system that trains kids to be like those horses we encountered on Mackinac Island. Seth Godin might call it training cogs. Cogs that can follow a predetermined path while those in charge feel safe enough to take a nap. Probably the most frustrating part is that this is not real reform, just reorganizing a system that already rewards conformity and penalizes creativity. In fact, much like transportation on Mackinac Island, educational reformers seem satisfied using "vehicles" (high-stakes tests, teacher-proof curricula, back-to-basics resources ...) from the past. (Read Linda Darling-Hammond's commencement address at Teachers College for more examples)

In researching this post, I found the following. 
Where Mackinac Island Horses Spend the Winter - Pickford, Michigan
Photo courtesy Kate Ter Harr
This photograph represents some of the ideas I have for education reform: (1) freedom to explore; (2) opportunity to collaborate; and (3) open space that allows for choice. The metaphor is not exact but it doesn't have to be. It's got me thinking. And I cannot imagine a better way to spend my "summer vacation."

Thursday, June 2, 2011

What support do learners need?

Last week I shared a lesson plan used to introduce the clock model for adding and subtracting fractions to a class of fifth-graders. This post focuses on the follow-up lesson, which concentrated on developing an anchor chart that learners could lean on as they solved progressively more difficult fraction computation problems. In particular, I will discuss how the instruction attempted to offer support so new learning could occur.

Learning is about moving from the known to the new. Therefore, we began by activating our schema regarding how we had used the clock model to add fractions in the previous lesson. Then we considered other fractions that could be represented using time as a context and began building an anchor chart based on our experiences with clocks in and out of the classroom.

An anchor chart supports learning by creating a record that learners can refer to as cognitive demand increases. In Reading with Meaning, Debbie Miller writes, "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)." This anchor chart offered the fifth-graders support both as a representation for thinking as they computed the fractions and a representation of thinking that they could point to as they communicated their thinking to their peers.

We worked through the number string together, with me starting the computation and learners offering advice as we went along. A number string is a series of progressively more difficult problems that build on the success of prior solutions. Being sure to use "I language," I started each problem by saying, "This reminds me of ..." As we went on, I asked more and more often, "What should I do next?"

From the perspective of the Gradual Release of Responsibility, my approach for this lesson would be considered Shared Practice [WITH]. (In the previous lesson, I had relied on Demonstration [TO] in order to support the introduction of the clock context - something new. Based on learners' efforts in that lesson, I was confident that they were ready for more responsibility.) With Shared Practice, the teacher supports learners by reinforcing how problem solvers get started, but there is still room for exploration and approximation as learners offer their suggestions for what comes next. This came in the form of the fifth-graders telling me what to do to complete the fraction problems. Not all of their suggestions worked, but we thought through them together and used our prior knowledge and anchor chart to get back on track. Any "mistakes" were used as an opportunity to foster a learning community that could support each other through difficult problems.

At the close of this lesson, I offered a final support - time to reflect. Without an opportunity to consolidate their experiences from the lesson, it is very likely that the learning would not last. I asked the fifth-graders to write in their journals a recount of the day using the What, So What, and Now What framework. Their responses would serve as a formative assessment used to inform future lessons. But that is for another post.