Delta Scape

When were we engaged?

2012-03-02T08:25:00.000-08:00

Teach an engaging lesson and kids are engaged for a day. Teach them how to engage and they are set for a lifetime.

Teaching students how to engage entails: (1) identifying what engagement looks like; (2) setting the conditions under which engagement can occur; and (3) providing opportunities for them to reflect on their engagement. I described the first point here and the second here. In this post, I address the third point.

Confessions of a "bad" math teacher: I often confuse engagement with conformity. In the past, I have given a participation score based on how engaged my students were in doing the assigned work. If they did it all, then they got 100%. If they did half the work, they got 50% of the points. I have come to understand, however, that participation is not necessarily engagement. The Engagement Continuum shown below highlights the problem. While the students may have demonstrated interest in getting all the participation points, there is no evidence that they were truly engaged in the content.

Instead of offering participation points (faux engagement), I now ask students to demonstrate and reflect on their ability to engage. At the end of the semester, they turn in a predetermined number of workshops that provide evidence that they know what it takes to engage themselves. Their engagement score is dependent on their ability to identify their level of engagement and explain how they achieved that level.

Here is an example from this semester. At the end of our first unit, we make anchor charts about what it means to do mathematics. (I have written about this before.) I rarely give them enough time to complete their chart - just enough to have something to share with their peers. This is a draft completed by a group trying to use a blueprint for a school as an analogy for what we have been discussing in the unit.

After the class, the group stayed to complete their anchor chart. They didn't have to do any more work on it, but they wanted to finish it. It was interesting to listen in as they continued to attend to small details and talk to one another about their vision for their teaching.

One of them asked me if he could use this workshop as an engagement exemplar. I told him that any workshop, in class or assigned as homework, was available.

It was clear to me that this group was committed to completing the task, but where it fits on the Engagement Continuum is up to them to demonstrate. Only they know what was going on in their minds as they worked on this project. I am through trying to attach my own judgements regarding engagement to students' efforts. They need to be able to identify when they are engaged and determine how to replicate the conditions that led to their engagement.

I must say that I am interested in what they submit for their engagement exemplars - much more interested than when they simply recorded how many of the workshops they had completed. It is sort of funny; this approach has increased my engagement as well.

Can we just flip the homework?

2012-02-29T09:10:00.002-08:00

The lesson has been taught and there are a few minutes remaining in the class period. Of course, this means that students can get a head start on their homework. I sit in on a lot of lessons in my role as an instructional coach for preservice and inservice teachers and the scene is nearly always the same. As I walk around the class during these final minutes, I typically see the following:

Why do they start at the beginning? Is it because "it is a very good place to start," or is it because that is how they have been trained?

The way most assignments are structured, the items get harder the farther in you get. I understand that this progression can provide support to students as they build on prior success, but I do not think they see it that way. Ask them and I believe you will find that they do it in order because the assignment implicitly suggests that they do it in order. They have been disempowered to make meaningful choices about what items to try.

What if instead we asked the students to look over the items and identify those that they might need peer or teacher support on? These are the items the students could work on during the closing minutes while they have support available. They can finish the easier problems on their own. Dare I say it? We could flip the assignment.

This could also serve as an assessment for the teacher. Knowing which items students considered challenging could offer insight into the effectiveness of the lesson or inform future instruction.

A change like this represents one of the subtle shifts we encourage teachers to make in their practice. We do not always need to make big moves to offer students a chance to make choices and, therefore, take more responsibility for their learning. A slight change can increase the likelihood of student engagement without requiring a lot of extra planning or preparation. This is another example of educational sustainability for both teacher and learner.

How will I stay engaged?

2012-02-24T11:49:00.000-08:00

Based on prior posts, it might seem that I do not want teachers to consider engagement when planning lessons (here). In this post, I intend to describe how teachers and learners share in fostering the conditions that contribute to engagement. This is reflected in Cambourne's Learning Plan shown below.

Again, engagement is at the bottom of this plan because it is a foundational condition to learning. Immersing learners in the discipline is placed on top as a reminder that it influences everything else that is done in the classroom. The remain conditions are split between those that focus on learners' behavior (purple) and those that start being the teachers' responsibility (green). The ultimate goal, however, is that over time the learners will also take on these behaviors as well - setting expectations for themselves, offering one another feedback, and providing their peers with models of success.

Helping learners to take responsibility for their own engagement is also a gradual process that requires teachers helping learners move through awareness, acceptance, and adjustment. This starts with helping learners to be aware of what being engaged looks like (as I described in this post). Having this anchor experience allows us to continually monitor our engagement during learning activities.

I try to design learning plans with Cambourne's Conditions in mind, but I leave learners to evaluate their own engagement throughout the activities. Early in the semester, I provide a table like this:

The learners are asked to reflect on their level of engagement after each phase of the workshop (schema activation, focus, activity, and reflection). At the end of class, they answer three questions that hopefully support their awareness (What did you notice?), acceptance (So what does this mean?), and adjustment (Now what will you do to improve your engagement in the future?).

Their answers reflect the shared responsibility learners and teachers have for fostering engagement in classrooms. The learners accept that they need to be better about things like making connections, asking questions, and monitoring their own engagement. They also describe how it is hard to be engaged if expectations are unclear or if they are not sure how to complete the task. This is on me, as the teacher, and it has reinforced my need to keep in mind my initial responsibilities regarding Cambourne's Learning Plan. Consequently, all involve adjust their behavior in order to improve engagement.

There is one more piece to encouraging learners to take responsibility for their engagement that I have just added this year. I have replaced Participation in my course grading with Engagement. This change will be the focus of the next post. Please stay tuned.

What does engagement look like?

2012-02-18T07:03:00.000-08:00

The first post in this series suggests that teachers must practice caution when designing engaging lessons. Disempowerment and distraction are just two potential problems that may result from teachers trying to own the responsibility of student engagement. Another problem is distorting what engagement looks like. Addressing this last problem is the focus of this post.

I begin nearly all of my college courses with a workshop that asks, "How do we stay engaged?" The activity is based on the rubric shown on the right. My wife developed this rubric with her first-graders to help them to monitor their engagement during independent reading time. The class brainstormed the ideas of what different levels of reading engagement might look like and Kathy wrote them down. If during independent reading time a child is not engaged with their reading, then all Kathy has to do is ask the child where he is on the rubric and what he needs to do to re-engage. In most instances, this works extremely well. I figured that if first-graders could do this, then so could college students.

The phases of the workshop look like this:

Schema Activation Turn and Talk: What engagement does and does not look like.
Focus Setting Expectations

History of Rubrics (providing direction)
Kathy's example

Activity Group Work: Creating Engagement Rubric Rough Draft

Reflection Gallery Walk

Use sticky notes to identify "likes"

Develop a personal engagement rubric

Sometimes these rubrics confuse engagement with compliance. This is related to their distorted thinking about what it means to be engaged. An example that comes up often is texting in class. Student often want to put "texting" in the frowny face category because it means that the person "isn't doing what they should be doing." I understand their rationale and accept that typically it is an example of being unengaged, but I warn them that they need to be careful judging based solely on appearances. What if the person is texting to someone about the awesome activity that we just did? Maybe it fits perfectly with an upcoming project they are doing and the person was so excited they could not wait to share it. I explain that I do this all the time using Twitter at conferences - my tweets become my notes that I am willing to share with anyone who wants them.

The idea that communicating our excitement about something we are learning represents a high level of engagement is reflected in the work of Morgan and Saxton (beginning on page 27). I was first introduced to their Taxonomy of Personal Engagement through Jeff Wilhelm's 2007 book, Engaging Readers & Writers with Inquiry. He modified the Taxonomy in order to develop the Engagement Continuum shown below.

For students who have a hard time separating compliance and engagement, this provides a template for thinking about what pure engagement might look like. I will describe ways I use this continuum to support students taking responsibility for their own engagement in the next post.

But what does engagement look like to us as teachers? Can we distinguish between compliance and engagement? I must admit that it has been a bit of a struggle for me as I try to put theory into practice. My colleague, John Golden, suggested that we (including you, dear reader) might collaborate to design an engagement rubric from the perspective of teachers. Consequently, I put together this google document (here) that we can work on together. I hope that you are engaged enough with this topic that you are willing to share your ideas. Thank you in advance for your participation.

Teach like Adele

2012-02-13T09:33:00.000-08:00

Following a post on how entertaining/engaging lessons are antithetical to sustainable learning with the idea that I want teachers to employ an entertainer's approach to singing may seem inconsistent. I hope you will bear with me, though, because I believe two performances from last night's Grammy Awards actually illustrate the point. As I watched Katy Perry and Adele sing, I was taken by the different approaches taken by each.

Now, I am willing to concede that what I am about to write may be based on generational preferences (I turn 50 this year). While I found Katy Perry's performance entertaining, it distracted from the song. I ended up focusing on surface features - the stagecraft. Adele, on the other hand, seemed focused on the song. There were no distractions. I knew what her performance was about. It felt authentic. And her power and passion affected me.

This is what teaching ought to embrace - focus on the song, not on the performance.

Updated February 18, 2012

I woke up this morning to this piece on NPR's On The Media examining Adele's song Someone Like You. The piece made two things abundantly clear. First, I am not alone in finding Adele's songs moving. Second, I do not know enough about musical performances to make statements like I did above.

It turns out that there is a science to musical performance and Dr. Daniel Levitin is doing the research. He has found that listeners want novelty, just not too much of it. He says, "If the music was completely surprising we would be disorientated. On the other extreme, if the music was completely predictable we'd grow bored with it and it would seem banal. And what the composer has do is find that balance and get it just right - the Goldilocks' Zone." This is accomplished both through writing the music and singing the music. An example of the latter is creating tension by singing slightly out of key but sliding to the correct note (appoggiatura).

Dr. Levitin explains how Adele does all of these things in her Grammy Award winning song. She starts with something simple and familiar. This is followed by several surprises: (1) a tempo change corresponding to the message of the lyrics; (2) an octave change (perhaps representing a change in perspective?); (3) using appoggiatura to suggest instability; and (4) a lower register meant to convey the emotion creeping to the surface. And host, Brooke Gladstone wraps it up saying, "It isn't just about the song. It's also about the performance."

Busted! Like many people who think they can distill teaching into a few "best-practices," I thought I could take two musical performances and identify what works and what doesn't. My efforts failed to take into account all that goes on in the writing of the song and the planning of the performance - even when it seems to be so simple. I hope I learned my lesson, but probably not. I am a sucker for a good teaching simile.

What have you got against engaging lessons?

2012-02-10T12:58:00.000-08:00

I would like you to please take a moment and think about the last time you were engaged in learning. It might have been in a math class or listening to a TED Talk or reading a book. Whatever it was you were doing at the time, take a step back and create a mental picture of what this engagement looked like. We will be revisiting this image later, so please keep it in mind.

Engagement is central to learning. In his research on language acquisition, Dr. Brian Cambourne found it to be one of the conditions necessary for learning to occur. In fact, he saw engagement as the fundamental condition on which all the others are related. That is why it is at the base of his plan for teaching and learning that is shown below.

Teachers recognize the centrality of engagement and often talk about the need for writing engaging lessons. Recently, I have become uncomfortable with this view of engagement. I am afraid that we are confusing engagement with entertainment, and that we are disempowering and distracting students as a result.

This commentary comes from one of the U.S. science lessons (US4 Rocks) from the TIMSS videos. It highlights what is happening in a lot of classrooms:

Motivating activities were defined as those activities likely to capture many students' interest and included jokes and humor, games, puzzles, role plays, artistic projects, dramatic events, physical activity, prizes or other rewards, anecdotal stories, or outdoor excursions. Motivating activities accounted for 23% of the U.S. lesson time (Video Report, figure 10.6). They occurred in 63% of the U.S. lessons, which was more frequent than in all the other countries except for Australia (Video Report, figure 10.5).

I have a couple concerns about teachers' efforts to create all of these engaging/motivating lessons.

First, are we creating expectations for students that if a lesson does not engage them it is the teacher's fault? I sit on a lot of airplanes and I am always amazed at how many people are unengaged when the crew goes over the safety information. This is potentially a matter of life and death, and they still have a hard time getting passengers to pay attention. It may be my pessimistic nature or my interest in engagement, but I try to take in everything they say. I worry about my fellow passengers, but not as much as I worry about the students who may not have a teacher capable of designing the engaging lessons the students have come to expect. Cambourne calls students who have lost the ability to take responsibility for their learning "disempowered" and I would agree.

Did this picture draw your interest and distract you from the message?

I am also concerned that students are confusing entertainment with engagement. On one regional flight, I remember the crew ad libbed certain points of the safety talk. "In the event of a water landing the personal flotation device is your's to keep as our gift." I still remember the joke. I do not have a clue where that device was located, however (this was before I began engaging with the talk). In this case, they got my attention but distracted me from the important information. How often are students missing the point of the lesson when we try to gain their interest in extraneous ways?

Consequently, I see the development of engaging lessons as a possible impediment to sustainable learning. Engaging lessons make it more difficult for students to identify what engagement actually looks like. They come to view it as some external factor rather than a personal choice. Why is this a problem? Students who know what engagement looks like can self regulate their own learning; they can find ways to engage when important information is being shared in uninteresting ways. Students who cannot envision themselves engaging in "boring" material have ceded control of their learning to others.

Which brings me back to my original request. Please retrieve your image of engagement from the beginning of this post. Was it really engagement or merely entertainment? And if it was engagement, what did it look like? In the next post, I will share some ways other learners view engagement.

How am I doing? 2012

2012-02-04T12:20:00.000-08:00

It is that time again - annual review. The time of year when I am asked by my employer to demonstrate how I have met the College's expectations regarding teaching, scholarship, and service. This includes providing an up-to-date vita and the past year's faculty activity report. The Department of Mathematics also asks for a self-evaluation that focuses on teaching. As I did last year, I decided that I ought to share this evaluation in order to make my efforts visible to all.

***

Technology, especially as it relates to social media, has been the focus of my teaching journey this past year. Social media provides me with the opportunity to connect with my students and colleagues in ways that were previously unimaginable to me. It has also introduced me to educational technology that has enhanced my instruction. This self-evaluation will highlight my efforts but it also includes hyperlinks if you wish more details.

At the end of last year I joined Twitter. My main reason for joining this micro-blogging site was the success I saw Dr. John Golden have with it during teacher assisting in the fall 2010 semester. It allows me to maintain contact with the teacher assistants by providing a window into their thinking and their classroom experiences. As I explored Twitter, however, I found that it had many more educational benefits.

Twitter is a platform that allows teachers from around the world to connect and share ideas. I have encouraged our students to use these educators as resources and mentors. Several teacher assistants and student teachers have sought advice from and used lessons shared by Twitter teachers. Even I have been exposed to new technologies as a result. In modeling iPad use for the Woodrow Wilson Teaching Fellows, I have incorporated several applications that I learned about on Twitter. You can read how I used the iPad here.

Chats are another aspect of Twitter that I have benefited from and incorporated into my lessons. These chats allow participants to share ideas with one another around a particular topic. I have used them for class discussions on instruction, classroom management, and assessment. The benefits of these Twitter chats is that everyone is given the opportunity to speak and it is possible to archive the chat for assessment or further discussion. An example of one of these chats is here. These class chats have been so successful, I have also started hosting chats in sessions at conferences. Here is an example.

Some students remain hesitant to use Twitter to its fullest as a professional tool. This is a challenge I want to address this year. I do not want to force compliance. I want to model Twitter's benefits. As a part of the student teaching pilot program that I am helping to develop, I intend to include a survey that will ask about Twitter and ways to improve its implementation.

I also began the year by creating a blog. This has been an opportunity to make available to students the resources, examples, and concepts that are addressed in class. The posts have become a kind of anchor chart that students can review long after the lesson. I have also used it to reinforce and extend topics presented in class.

In addition, my blog has supported my scholarly and service activities. The information shared on the blog contributed to my being asked to be a keynote speaker at the Mathematical Council of the Alberta Teachers' Association annual conference. A "transcript" of my talk is provided here. The entries on the Teaching-Learning Cycle have provided a framework for a book that has generated some interest from a publisher. I am considering making the writing of this text a part of a sabbatical proposal. Finally, several teachers have commented on how the lessons I have shared have been of service to them as they teach. Providing an open source resource to inservice and preservice teachers around the world is important to me.

I continue to do teaching, scholarship, and service in the typical ways. My efforts are enumerated on my vita and faculty activity report. While these are important, I am passionate about the opportunities technology has for addressing these areas in the future. I have only just begun. There is a lot left for me to learn.

When do we stop chewing their food?

2012-01-23T07:52:00.000-08:00

The birds have found the feeder. We moved it and a suet holder closer to the house this winter since the trees that used to shelter them were cut down over the summer. I was afraid that their proximity to the house might frighten the birds, but it has not been a problem. In fact, it has made it easier to watch the birds as they feed.

During the spring and summer, I watched as the adult birds fed the babies. The adults would grab food from the feeder and place it in the mouths of the babies waiting on nearby branches. I do not anticipate seeing much of this behavior over the winter. I could be wrong, but I think the birds that visit our feeder and suet cakes have outgrown the need to be beak-fed.

When does this happen in education? In other words: when do we quit feeding learners information and expect them to fend for themselves? This came up this past week as I talked with university colleagues about student evaluations. We have all had comments that our students want more lecture because that is how they "learn" best. I make up that these comments are from students who have come to expect that the teacher's role is to gather and chew up educational information for students to consume. Is that too harsh?

For the sake of completing this post, let us assume that this learned helplessness is indeed the problem. What can we do about it? This is where I try to apply the Teaching-Learning Cycle and the Gradual Release of Responsibility. The Teaching-Learning Cycle provides a framework where I can identify the information and the processes learners need to make it on their own, monitor learners' progress toward these goals, and plan and implement appropriate supports. The Gradual Release of Responsibility represents an instructional approach which helps learners to "fend for themselves" through a series of lessons that begin with demonstrations, move to collaboration, and eventually result in independent practice.

I am fortunate that my colleague, who teaches the prerequisite course for the one I am teaching now, uses these frameworks in his practice. Even after only a few days I have seen a difference in my learners. Not everyone of these learners had John's section, but those who did are able to share with the rest what is expected of them in and out of class. I get the feeling that there will be a lot less gathering and chewing on my part this semester. And for that, I am grateful.

What is your dream?

2012-01-16T12:00:00.000-08:00

Today, Grand Valley held a Silent March to commemorate Martin Luther King, Jr, whose birthdate was January 15, 1929. Along the way, placards lay out a timeline of the civil rights leader's life and some of his quotes. I find Dr. King's messages inspiring, and I believe that they are timeless. So I was grateful when Valerie Strauss shared several excerpts from his speeches and writings related to education in her column, "MLK's prescient thinking on education reform."

I especially connected with his piece for the Morehouse College student newspaper. In this writing, Dr. King discusses the purposes of education. Part way through the piece, he writes:

Education must also train one for quick, resolute and effective thinking. To think incisively and to think for one's self is very difficult. We are prone to let our mental life become invaded by legions of half truths, prejudices, and propaganda. At this point, I often wonder whether or not education is fulfilling its purpose. A great majority of the so-called educated people do not think logically and scientifically. Even the press, the classroom, the platform, and the pulpit in many instances do not give us objective and unbiased truths. To save man from the morass of propaganda, in my opinion, is one of the chief aims of education. Education must enable one to sift and weigh evidence, to discern the true from the false, the real from the unreal, and the facts from the fiction.

The function of education, therefore, is to teach one to think intensively and to think critically. But education which stops with efficiency may prove the greatest menace to society. The most dangerous criminal may be the man gifted with reason, but with no morals.

Reading this, I immediately thought of phronesis - the idea of knowing what's available to do and what's worth doing. Of course, what is worth doing to one person may seem foolhardy to another. And another person might consider the "doing" just plain wrong. In the article, Dr. King writes about former Georgia governor, Eugene Talmadge. Dr. King considers Governor Talmadge intelligent but wonders, "yet he contends that I am an inferior being. Are those the types of men we call educated?"

How can education help us address such unintelligent presepectives? We must provide experiences that challenge unfounded beliefs. We must provide opportunities to reflect on those experiences. And then we must start over again because beliefs can be a difficult thing to change. But change they must, or it will be Dr. King's closing words in the article that will turn out to be most prescient:

If we are not careful, our colleges will produce a group of close-minded, unscientific, illogical propagandists, consumed with immoral acts. Be careful, "brethren!" Be careful, teachers!

Who controls the volume?

2012-01-09T07:49:00.000-08:00

For that special date 11-11-11, NPR's All Songs Considered did a podcast based on the this clip from This is Spinal Tap:

No surprise - my thoughts went to education. I wondered about the ways I try to turn my teaching volume up to eleven. There are certainly times in my classroom where I want to REALLY impress upon my learners the importance of what we are exploring. While this might result in literally turning up the volume, it is more likely to manifest itself in elaborate planning and over-the-top presentation. It might get the learners' attention, and maybe even their interest, but this line of thinking seems too self-centered to be sustainable.

The podcast was really about the songs that listeners want to turn up to eleven, though. This was evident in the following exchange:

Bob Bollen: "I think most people, when they think of 'cranking up', they think of pop music. They think of turning up the radio that they’re listening to when that song comes on."

Jacob Ganz: "Yeah, that’s the instinct. You’re listening to the radio and something comes on and you either reach to change the channel or you reach to crank it."

BB: "It was the original 'like' button. Only there was no feedback. If the station could have gotten that information that you turned the dial up …"

In my way of thinking, the listener represents the learners in my classroom. While I certainly have wished that I could receive immediate feedback on how engaged learners were in the lesson, when it is put this way I am uneasy. It too closely resembles a consumer approach to education - find out what they want and then give them more of it. This seems representative of some of the student-centered approaches that I find unsustainable.

Instead, I would like to have learners be more reflective about what they like or don't like about a lesson. Here is where the analogy breaks down because listeners do not need to connect with every song, but I want learners to be prepared to engage with every lesson. This is a metacognitive act that will allow them to participate more fully in boring lessons that they might otherwise tune out.

I still want to develop lessons that showcase my enthusiasm for the topics being addressed. Enthusiasm can be contagious. As the course goes on, however, I need to be ready to gradually release responsibility to the learners. They must find ways to connect to the learning regardless of their level of interest. After all, learners ultimately control the volume of what they want to hear. I just want to help them to make critical and sustainable choices.

Whose problem is it?

2012-01-03T11:15:00.000-08:00

"Education systems, teachers, school districts all over the world are going crazy about problem-based learning - nothing like a good problem to solve. But they are looking at the wrong bit of it. The thing we're neglecting is to find a generation of problem finders."

The above quote comes early in Ewan McIntosh's talk at TEDxLondon. This really connects with my goal to foster sustainable learning. Here is the entire talk (it is well worth the eight minutes):

I want learners to come up with their own problems - to be able to answer, "Now what?" for themselves. Most times when I try to implement a problem finding curriculum, however, two issues interfere: trust and control. You see, I know what they need to know because I know what I learned and how it has helped me. How can I be sure learners will follow the correct path, find the right problems, if I do not lead them either explicitly or implicitly?

Here is a good example. Over the winter holiday break, I went on a hike through a state managed forest. Along the trail were a variety of signs describing interesting facts about the trees and forest management. The sign below was of particular interest to me.

I thought it had a lot of potential for use in a course on teaching and learning middle school mathematics that I am scheduled to lead this semester. It would provide a great context for the geometry section as I asked my learners to make Biltmore and Merritt Rule Sticks using the information provided. The problem was perfect, but as Ewan points out, it was also mine.

Given my interest in sustainable learning, I would be better off owning the problem myself and using it as a demonstration. It would offer an opportunity for thinking aloud about identifying problems in contexts that interest me - the first step in the gradual release of responsibility. Then, with my support, the learners could begin to find their own problems in whatever math content we must address. By the end of the course, hopefully, the learners could find problems for themselves.

With awareness and effort, I have gotten better at letting my learners lead the way. Every success allows me to trust them a little bit more and give up trying to control the curriculum. Maybe 2012 will be the year I learn to really let go.

What have you been doing?

2011-12-30T17:59:00.000-08:00

To wrap things up for 2011, I thought a recount of the year's posts would be in order. This will be my 90th post on this blog this year, which is pretty amazing considering I thought one a week would be a reasonable goal. But I ended up finding a lot to write about. So much, in fact, that I forgot some of what I wrote. This is the main reason I wanted to write this recount - to refresh my memory.

In January, I introduced my blog and the rationale behind it. It was also my intention that some of the posts might inspire me to write a book on the Teaching-Learning Cycle. So the next few posts began a series on the Cycle - assessment, evaluation, and planning.

The instruction post of the Teaching-Learning Cycle started off February. A self-evaluation that I wrote next was the first unplanned post. This was followed by my yoga story, which I share in nearly every class as a metaphor for effective teaching. Another unplanned post shared my students' views of what they thought it meant to do math. The post on frameworks was the last one of the month and probably my favorite one of the year.

I wrote 12 posts in March. The first was an email I sent to our student teachers about time management. Second, I wrote a commentary to accompany my TEDxGrandValley talk. In the third post, I explain why as a math teacher I attend the Michigan Reading Association Conference. In the next two posts I began a series that looked at testing from the Teaching-Learning Cycle (TLC) perspective (assessment and evaluation). The series was interrupted, however, by a pair of unplanned posts: a letter to the editor in support of teacher unions and a description of how I used a Twitter backchannel in a class. Next was the third TLC/testing post focusing on planning. Two more unplanned "interruptions" addressed a Tweet by Alfie Kohn and a simile survey related to learning math. The conclusion of the TLC/testing series (instruction) was next. The twelfth and final post outlined the workshop I used to have my students reflect on what it means to learn math.

April was my most prolific month with 15 posts. A pair early in the month used Monty Python skits to look at teaching and learning. There were four posts that had been in my files for awhile waiting to be shared with the world: Process Standard math centers; an early elementary math poem; and a pair of posts on Pass the Pigs (the first post was the most popular of the year). There were a couple of posts related to work my students on what it means to teach math. I also introduced why I started the Learning Museum (another blog that need more of my attention). Three posts were related to the NCTM Conference I attended in Indianapolis. A mind map reflecting our student teachers experiences over the semester took up another post. Near the end of the month, I wrote a post that responded to an article suggesting that students learn best from direct instruction. The last post was a copy of a final exam I gave my preservice middle school math teachers.

I kept up the pace in May with 13 posts. The first three were a series based on a student teacher's portfolio project. Next was an impromptu post based on an #anyqs tweet. Another Twitter inspired post considered alternatives to traditional math homework. After attending TEDxGrandRapids, I also wrote a synopsis of the event from an educational slant. I did another three-part series on EdCamp Detroit: introduction, my session, and my reflection. There was another response to the article on direct instruction (which also introduced the clock model for adding and subtracting fractions) and two more posts from my files on grading. And then there was the one where I explained why all the titles of my posts are questions. I'm not sure I'll continue that theme next year.

In June I wrote three more posts associated with the clock model for adding and subtracting fractions. There were also three more posts on activities from my files: data transformations; scatter plots; and Monte Carlo simulations. I wrote a post comparing teaching to training horses, and a pair relating teaching to bike tag-alongs (what I saw and what I thought about it). I ended the month writing about the gradual release of responsibility and my thoughts on the flipped classroom.

I began teaching a graduate class in July and a couple of my posts focused on that experience. The first was based on memories I had as I planned the course. And the second was based on my desire for my grad students to mutiny.
I also wrote about teaching as story telling in honor of the last Harry Potter movie's debut, the importance of trusting oneself in teaching and golf, and a letter to the editor on education reform.

August began with one more post about my grad students - their blog addresses so others could offer feedback. I then wrote about how I use the workshop model. Then I blogged about how a TED Talk encouraged me to take the 30 day challenge. Finally, I talked about why I ask questions in class for which I don't have answers.

September was the start of classes, and I began the month writing about a workshop I use on the first day. Then I wrote about how design thinking might apply to education. A post on teacher pay came next. These were followed by two more math activities from my files: one on story problems and the other on empowering students (this became part of a series called, Now What).

In October I finished the Now What series on empowerment with three more posts. I also shared my session workshop on Metacognitive Memoirs from the MCATA Conference. My big post was a 'transcript' of my keynote at MCATA.

Two unplanned posts bookended November. The first post of the month was on phronesis. The post at the end of the month was about running. In between, I wrote about my session on Twitter at EdCamp GR and what makes math real. I also began a series on using action plans to improve teaching. The first introduced action plans and the second gave an example of one in use.

December included two more activities from the archives: fostering creativity and snowman glyphs. There was another impromptu post based on several Twitter interactions. The third in the action plan series focused on an example related to evaluation. I shared a video reflection from one of my learners. And last, but definitely not least, was my blog recount that you might still be reading.

Well, that's all folks. Thanks to anyone who made it this far. And thanks to everyone who supported me through this first year of blogging. I am excited to see what new learning 2012 has in store.

What did we learn?

2011-12-24T08:33:00.000-08:00

At the end of each semester, I try to make sure that my learners have time to reflect. If their learning is to last, then they must have a chance to consolidate it. This past semester I used three workshops to support their efforts to look back and look forward.

The first workshop asked the learners to focus on the course objectives.

They identified those in which they considered themselves proficient and what contributed to their proficiency. So much occurs during a semester and too often learners do not recognize what was learned. This task represents a metacognitive activity that makes it more likely that the learners will be aware of all they have accomplished. It also provides me with important assessment data that will inform how I teach the course in the future.

In the next workshop, learners write letters to future students as a way to prepare and encourage them to be successful in my course. Here are two from this semester:

I will share the letters with future learners because people often accept advice more easily from their peers than their teachers. This is not the sole reason for the activity, however. It also provides the letter writers with the chance to reflect on their own growth as learners over the semester. I hope that both current and future learners will benefit from this wisdom borne out of experience.

The final workshop asks them to create a 3 minute (or less) TED Talk about what they have learned this semester. This year I found out the power of showing the Student vs Learner clips early on. Many of them commented on how these videos helped to frame the entire semester. I will now try using them the first day in all of my courses.

The presentations are ungraded and usually very informal. Amazingly, this sometimes leads to more preparation on the part of the learners. Regardless of the time they put into them, I find their sharing to be quite profound and at times moving. Here is an example from this semester:

I remember teachers saying to me, "You'll thank me for this later." This typically accompanied some particularly distasteful tasks. I bristle at this notion and hold the belief that I am not in this for people's thanks but because I believe in what I do. Still, I hope I am getting better at accepting people's gratitude as well as their criticisms. After all, both are a part of the learning process.

What's your problem? Part III

2011-12-15T09:48:00.000-08:00

Previously in this series, I shared about action plans (here) and how one teacher used an action plan and observation to improve her use of questions in assessing learners (here). In this post, I provide another example - this time focusing on evaluation. (I have explained before that our framework treats assessment and evaluation as different phases of the Teaching-Learning Cycle.)

The teacher being coached was using entry slips to gather data on her learners but was unsure how to interpret the data. Her question to develop her understanding literally asked, "What do I do now?" It was with this in mind that I entered the class and saw the following on the board in the front of the room.

After about five minutes, the teacher collected the sixth-graders' efforts, looked over them, and, satisfied, moved on with the rest of her lesson.

During a lull in the lesson, I asked if I could look over the slips. With her consent, I began to analyze the assessment data. I organized it using the table shown. As you can see, there were no incorrect answers, however, there was a problem. Approximately 22% of the learners were unable to complete the task in the time provided. Fortunately, the teacher had asked the learners to show their work so that some of their thinking would be made visible to us. This would provide further insight into the problem.

I began to apply the evaluation framework by asking myself, "What can they do? What are they trying to do? What comes next?" Looking over their work, it became clear that they were fluent in comparing fractions using common denominators. Below is an example of how nearly all of the learners went about finding the first answer.

This also seems to suggest where they were approximating. By trying to apply this method (finding a common denominator by multiplying the two denominators) to all the comparison problems, some learners had run out of time. It seemed clear to me that what came next was considering alternative approaches to comparing fractions that might be more efficient. I shared the following list with the teacher:

Using benchmark fractions like one-half and one;
Comparing like numerators;
Finding least common denominators; and
Converting to decimals.

We talked about ways to introduce these strategies and how to make subtle shifts to the worksheets that she was expected to assigned. By asking the learners to look over the worksheet and match each item with a preferred method, they would be engaging in more meaningful work than simply applying a particular approach over-and-over again. As always, I left it open for this shift to be a part of the teacher's next action plan.

How do you celebrate the holidays?

2011-12-10T14:58:00.000-08:00

The following is an activity I included on a final exam I gave to preservice teachers at the end of the fall semester years ago. The activity began the class period before the final with the students filling out this Holiday Survey:

During winter (on a typical day), are we likely to see you wearing a hat?
During winter (on a typical day), are we likely to see you wearing a scarf?
What is your favorite winter activity?

Skating
Downhill Skiing/Snowboarding
Cross-country Skiing
Sledding
None of the above

Which is your favorite holiday television show?

Rudolph the Red-nosed Reindeer
Santa Claus is Coming to Town
The Little Drummer Boy
Frosty the Snowman
None of the above

Do you like winter or not?
Which is your favorite holiday treat?

Sugar Cookies
Candy Canes
Pumpkin Pie
Fruit Cake
None of the above

What is your favorite color for wrapping paper?

Their answers were used to create snowman glyphs based on the key (here) and the template shown below.

Here are some examples of how the students' snowman gyphs came out:

There were typically 24 students in the class and they used the 24 different glyphs (here) to answer this problem-set on the final.

You should use the data found in the glyphs to fill in the contingency table. In class, we explored two methods for determining independence. You should use both to determine if wearing a hat and wearing a scarf are independent events.

You should make a graph to represent your favorite holiday treat. You may use grid paper if you wish.

Disregarding wrapping paper color, how many distinct snowman glyphs are possible based on the key?

And, in case you were wondering, here's my snowman glyph.

Happy Holidays!

How do you foster creativity?

2011-12-06T13:02:00.000-08:00

The following was on a bulletin board in the art class at the middle school where I taught.

Because I wanted my math class to be more like that art class - learners exploring the boundaries of their knowledge and skills - I posted a copy in my classroom.

Twenty years later, a copy is now thumbtacked to my office door. It reminds me that fostering creative problem solvers is one of my primary goals as a teacher of teachers. Some of the ideas may not be reasonable, but creativity sometimes requires a certain suspension of rationality.

Why do you like Twitter?

2011-12-02T13:02:00.000-08:00

People not on Twitter ask me this a lot - usually as they back away slowly. Last night's Twitter stream provided two answers. But I'm still not sure they would get it.

First, Erin Ochoa tweeted that she had posted her reflection on EdCamp Edmonton. We did a joint session together on using Twitter that day, although we were nearly 1,500 miles apart. (I wrote my reflection on that session at EdCamp Grand Rapids here.) It was good to be reminded of how Twitter supports professional collaboration.

Second, John Spencer started a Twitter meme with #pencilchat. He explained how it started here. Seventeen hours later and it is still going strong. Whatever the reason behind #pencilchat, it provided a lot of teachers with a laugh. And I for one could use the comedic relief.

PS The link in the tweet above pointed out another positive aspect of Twitter - connections. You can find out why here. I don't want to ruin the "rest of the story" surprise.

What do you think about while running?

2011-11-29T09:07:00.000-08:00

My colleague, Robert Talbert, wrote a post today connecting teaching and running. The analogies he points out are effective and I have little to add of any substance from this perspective. His post did rekindle my interest in sharing what I think about during a run. And it should come as no surprise that I am usually thinking about teaching.

During the first mile of today's run, I thought about the following idea that I wanted to tweet:

This is related to a concern I have about attendance issues in my Introduction to Learning and Assessment class. I am trying something new this semester by not using in-class participation as part of their grade. In fact, I showed this video the first day of the semester and encouraged the preservice teachers to be learners instead of students. Perhaps I ought not be surprised when they take advantage of their new found freedom. The tweet was intended to remind me that learning has no bounds and that I need to trust a process that values intrinsic rather than extrinsic motivation. I tweeted it because I know I might be wrong and hoped that my Professional Learning Network would challenge me to refine my vision.

Halfway through my second mile, the podcast I had on got my attention. I like listening to All Songs Consider while I run. The combination of new music and analysis supports my pace and offers plenty of distractions as I consider, "How might this apply to education?" Today, I was listening to a question and answer session with Wilco about their new album. About 14 minutes and 45 seconds in, a listener asked, "How do you know when a song is finished?"

How would students respond to this question as it relates to their work? I would guess the answer would typically be, "When the teacher says so?" whether that means the teacher deciding when the work is good enough or when they call "times up." But Jeff Tweedy said that during the making of the album, it seemed to be when all the band members were satisfied with their personal contribution. This idea of learner autonomy is exactly what I wanted to get across in my #TeachingReminder tweet.

It also reflected the focus of last night's outside observation with four of our student teachers, which is what I thought about during the third mile. We do outside observations to remind our teachers in training that instruction is only one part of the Teaching-Learning Cycle. Last night, the question the student teachers shared on their action plan asked, "How can we make assessment and evaluation less stressful for students?" Based on their experiences over the semester, they were concerned with the unhealthy relationships many of their students seemed to have with tests and grades.

I know that some people have suggested that doing away with tests and grades would solve the problem. In fact, we just read an article by Alfie Kohn on this very topic in Introduction to Learning and Assessment. While this might be a solution, it is one that is curently beyond the student teachers' control. Therefore, we tried to focus on what we could do to help students have a healthier relationship with those grades.

When I finished my run, I saw this common thread in my thinking: supporting learners in developing autonomy. Now, how do I go about doing that? Maybe that will come to me during another run (or in the comments).

What's your problem? Part II

2011-11-22T08:13:00.000-08:00

In this series of posts, I want to share an approach we use with student teachers to support their development as reflective practitioners. The first post introduced the idea of using an action plan as a way for teachers to identify an area of challenge and seek out support. In subsequent posts, I plan to share examples of this approach in action.

After reading a teacher's action plan, I assemble any resources that might come in handy and then head to the observation. During the lesson, I keep notes on what is going on and ideas, questions, and concerns related to the challenge identified by the teacher. In the example given below, the teacher asked me to focus on formative assessment.

More specifically, she asked, "What types of formative assessments would be beneficial to student learning and how can I use these assessments?"

The notes are a combination artifact of what I saw and stream-of-consciousness of what I thought. I try to keep everything related to the focus provided by the teacher. (Unsolicited advice is an insult.) All of this is shared with the teacher during the debriefing after the observation but I try to attend to a few important points - highlighted in pink.

When I sat down with the student teacher after this particular observation, she started by saying that she felt there was very little formative assessment in her lesson. She was frustrated that because there was so much material to cover there was little opportunity to check for understanding. This seems to be a common challenge this semester.

I responded by pointing out a particular question she asked early in the lesson. The class was discussing issues with story problems on the homework. The teacher asked, "Is your problem the set up or the solving?" There was a resounding "set up" from the students. So the teacher focused on setting up several problems, leaving the students to do the solving on their own. This was a significant moment where the teacher used formative assessment to make an effective instructional decision but she had not recognized it as such.

Part of my responsibility as a coach is to help teachers to recognize things that they do intuitively and make them more intentional. We spent the rest of the debriefing time identifying places during the lesson where using formative assessment could focus instruction. This would provide more time for the formative assessment.

Finally, the teacher is asked to reflect on the experience. Here is part of what this teacher wrote after the observation and debriefing:

For the in-class observation, I was most concerned about assessing my students informally during class. After the debriefing, I feel I was able to recognize moments in my lesson that could have been altered to include more assessment. I feel that I have learned to consider the one or two important question within the lesson. If I recognize the few points that I really want my students to focus on, I can be sure to form my lesson around those specific ideas. Plus, if they are learning new material that progresses from previously learned mathematics (as it almost always does), I can 'skip' the information that they may already be very comfortable with to focus on the more difficult material. ... I feel this observation was very helpful in giving me ideas to use within my classroom to assess my students, as well as learning to focus more on the 'big picture', rather than material they may already be comfortable with.

These reflections often support teachers in developing their next action plan.

What's your problem? Part I

2011-11-17T11:23:00.001-08:00

My problem is that I tend to teach as I was taught. I know that research shows that I am not alone in this, but I thought I had gotten over this hurdle. Since 1990, I have been teaching math differently - and I have the student comments and parent phone calls to prove it. The changes I made as a math teacher were one of the reasons I became interested in mathematics education. Unfortunately, these changes did not transfer to all aspects of my teaching.

Early in my career as a math educator, I began doing observations of novice teachers in their first practicum experience. I remember going into classrooms and watching lessons that failed to meet the principles of good mathematics teaching suggested by the NCTM. After an observation, I would sit down with the novice teacher and play "fix the lesson." I would share with the novices everything that was wrong with their teaching and what they could do to improve it. I left feeling as though I was making a difference in math education, much as my university supervisors must have felt after filling me with their ideas.

Then, one day I took a deep breath. I had just watched an awful lesson where the teacher read the overhead to her students, who were sitting in rows, and then had them work independently on 30 problems from the textbook. I was getting ready to share my fixes when the novice teacher spoke up.

"That didn't go the way I wanted it to go," she said. "If it were my class, we wouldn't be in rows but in groups so that students could learn from one another. And I wouldn't assign all those problems. I would ask the students to pick out the ones they think they needed practice on. But, you know, I am a guest in this classroom and I need to follow the cooperating teacher's plan. Also, I normally don't read the overhead slides but I saw that Jamal didn't have his glasses and I wanted to be sure that he could participate."

I do not remember how I responded but there was the sense that my points of judgment were being ticked off one-by-one - check, check, and check. My problem had reared its ugly head once again but this time in terms of teaching teachers. I was doing what had done to me. It was time for another change.

Fortunately, I was introduced to a literacy coach from The Learning Network at about the same time. When I shared my problem with her, she responded with two pieces of information. The first was how her motto, "Unsolicited advice is an insult," influenced her practice. The second was the book, Literacy Coaching: Developing Effective Teachers through Instructional Dialogue by Marilyn Duncan. These led to the thing that most affected my teaching of teachers - action plans.

In chapter two of her book, Marilyn Duncan describes action plans as follows:

The action plan is also a tool to focus the support provided by the coach. It allows the coach to see where the teacher needs feedback. It provides the coach with a window into what the teacher already knows and has tried. It becomes a planning tool for their job-embedded work. (p. 20)

Here is her example:

With colleagues from the GVSU Mathematics department, I adjusted the action plan to meet the needs of our novice mathematics teachers. Our form asked:

What is my current challenge in teaching for mathematical literacy? Four areas were suggested (assessment, evaluation, planning, or instruction), based on our work with the Teaching-Learning Cycle.
What do I already know about this?
What questions do I have?
Which one of these questions do I need to focus on to develop my understandings?
How will I develop my understandings?
What support do I need to enact my action plan?
How will I monitor my progress?

This framework provided a way for novice teachers to ask for help, which meant that our advice would nurture their developing practice rather than insult it.

My experience in student teaching 'taught' me that observations were intended to be dog-and-pony shows where I was expected to impress the observers. Consequently, I was actually concealing my flaws from the person best situated to help me address them. I cringe when I think about all the teachers I passed that same lesson on to early in my career. Action plans have been my amends and I have been amazed by the results.

This action plan from one of our student teachers demonstrates the power of the approach. The plan helped him to self-identify his "problem" and articulate where he wants to be. It provided me with something to focus on. Without this focus, it is easy for me to fall back into my old pattern of judging lessons based on what works for me. It is interesting that by asking teachers to identify their challenges, I have been addressing my own.

In future posts (here and here), I plan to share other examples of how these action plans have aided our efforts to support the development of effective mathematics teaching.

Is this real?

2011-11-11T07:49:00.000-08:00

Today, part 2 of Harry Potter and the Deathly Hallows is available in stores. When the movie opened in theaters this past July, I used it as an opportunity to write about teaching as storytelling. In this post, I want to take a look at a particular exchange between two main characters and consider what it might mean for teaching and learning mathematics. Warning - spoilers ahead.

The following happens near the end of the story. Harry has sacrificed himself to save his friends and finds himself in an ethereal version of King's Cross Station talking with his deceased mentor, Albus Dumbledore. The conversation is nearly over when this brief exchange occurs.

"Tell me one last thing," said Harry. "Is this real? Or has this been happening inside my head?"

Dumbledore beamed at him, and his voice sounded loud and strong in Harry's ears even though the bright mist was descending again, obscuring his figure.

"Of course it is happening inside your head, Harry, but why on earth should that mean that it is not real?"

There seems to be a strong push in mathematics education to attach some real-world significance to nearly every topic in the curriculum. I appreciate the effort and agree that content a learner can connect to is more likely to be engaging. For me, however, the search for context to wrap around content can sometimes be distracting and inauthentic.

How can real-world examples be inauthentic? Let me provide a personal example. As a middle school mathematics teacher, I often extolled the practicality of learning to add and subtract fractions with unlike denominators. I would say, "You'll need this skill in cooking and building." Now maybe someone else can pull this off in an authentic way, but I cannot remember ever adding or subtracting fractions outside of an educational setting. Granted, I do not cook or build anything from scratch so I may be missing something. This only reinforces that this purpose was inauthentic to me and a distraction to my learners.

My problem was that I was confusing context with purpose. I now explain to learners that the rationale behind our lessons on fractions are related to the NCTM Process Standards and not some potential future use. I do not know if my learners will ever need to add or subtract fractions while cooking. I am sure that their future success will depend on their ability to problem solve, reason, communicate their thinking, use representations, and make connections. Yes, even connections to the real-world.

Let me be clear that I am not suggesting context is not important. In order to learn something new, we need to connect to something known. Looking back, I am not sure how many 8th-graders have experience with cooking and building. They are familiar with time, however, which is why I like using the clock model. They also have worked with whole numbers, and I often try to connect to these computational experiences. Sometimes the mathematics itself is the context.

Essentially, it is about finding what is real for my learners. Certainly, there will be times when their reality is at odds with that of the mathematical community - that is when the fun begins. I can only hope that my learners, like Harry, will begin by asking, "Is this real?"

Are we the first?

2011-11-09T10:57:00.000-08:00

I admit it. I am a bit on the competitive side. But the question I tweeted to Dan Callahan about hosting a Twitter chat between two edcamps was also about gaining perspective. If it had been tried before, then I wanted to be able to build on previous successes and avoid prior pitfalls. Dan's response meant that we were on our own (and that we were first, but that's not what's really important here.)

The idea was the result of a Twitter conversation I had several months ago with Erin Ochoa. We were chatting about assessment and wishing that we could talk face-to-face about it at Edcamp Edmonton. It was then that I realized that Edcamp GR was being held the same day. Initially, we thought about a Skype session. I had seen it done at Edcamp Detroit and thought it went well (I wrote about it here) but wondered if we could get things started the same way Erin and I had begun our conversation - via Twitter. Therefore, we organized an "edchat" between the two edcamps around the following questions:

We worked out a schedule that took into account the two-hour difference between time zones. The first 10 minutes would be spent getting participants acquainted with Twitter, the next 30 minutes engaging in the Twitter chat, and the last 20 minutes Skyping, which would allow us to extend the conversation beyond 140 characters. So on November 5th, John Golden and I signed up to lead the following session:

It worked out well. I lead the conversation and highlighted some of the nuances of Twitter chats while John tweeted links supporting our discussions. You can see a transcript of the chat here. At about 12:40 pm EST, we Skyped between the sessions. Fortunately, Janet Bell, in Edmonton, took a picture and tweeted it:

Interestingly enough, the Skype conversation we engaged in was mostly about Twitter, not assessment. Participants wanted to talk about how we use it inside and outside the classroom. It was not what I expected (real learning rarely goes as planned) but I would say the first ever edcamp edchat was a success. But being first is not what is really important here. This is what is important:

Phronesis - What is that?

2011-11-01T12:03:00.000-07:00

This was the question guest host, Susan Page, asked author, Eric Greitens, on an episode of The Diane Rehm Show. Greitens responded:

Phronesis is so interesting because it's a concept that we really don't have a word for in English. You could say that it's something like common sense, but it is really more than that. What phronesis means is practical wisdom. And phronesis is the ability to both figure out what to do in any given moment while also knowing what is worth doing. So the idea is that it's a practical wisdom - that you are wise about your intentions, wise about your ends, and at the same time you have a very clear understanding of the means that you need to actually get there.

Upon hearing this, I immediately said, "That is what I want teachers to have - phronesis!"

Teacher effectiveness is an important topic to me and I am not alone. There are centers devoted to teacher quality (TQ Center) and programs popping up dedicated to improving how teachers learn how to teach (Teaching Works). Distilling teaching to a few measurable points is beyond my current level of comprehension, however. Which is why I wanted to write about it - to try and, as Cambourne suggests, "talk" my way to understanding.

As I read the description above, it suggests four things for teachers to consider in supporting the development of their educational phronesis.

Can I identify my learning goals (ends)?
Can I provide purpose for my instructional choices (intentions)?
Am I aware of the resources available to support teaching and learning (means)?
Am I able to make decisions about what is "worth doing" in my educational setting?

It is tempting to affix to this list the things I stress in teacher preparation and professional development. In fact, I have written and deleted numerous sentences that use these four questions to support my use of particular teaching frameworks or instructional approaches. (I think Grace would say I was trying to make sense instead of developing understanding.) But are these things truly necessary for all teachers or are they just the resources available to me that I have deemed worth doing?

I'm done talking for now. This has certainly helped me develop a better understanding of phronesis but I am left with many questions. Is it enough for teachers to be working to answer the bulleted questions above or does effective teaching require more? And how do we design teacher preparation to support the development of phronesis? I look forward to your ideas.

Metacognitive Memoirs - what are they?

2011-10-26T11:55:00.000-07:00

At the recent Mathematical Council of the Alberta Teachers' Association [MCATA] Conference, I facilitated the following workshop: Making Mathematical Thinking Visible – Metacognitive Memoirs. Here is the workshop's description:

Metacognition is the awareness of one’s thinking. Memoir is a genre usually referring to a piece of autobiographical writing focusing on some problematic event. Together they represent a powerful tool for helping learners experience what it means to do mathematics by thinking about and communicating their efforts to others. In this session we explore how creative writing supports creative thinking in mathematics – certainly a road less traveled.

This was a breakout session based on a portion of my earlier keynote.

Schema Activation: Cambourne's Conditions of Learning [10 minutes]

In an effort to connect to their prior learning, participants looked over Figure 1 from Cambourne's article, Toward an educationally relevant theory of literacy learning, and identified what was important, what they could connect to, and what questions they had. Cambourne's Conditions were one of the themes of the keynote and I had promised that we would consider them in more detail during this workshop.

Focus: Metacognitive Memoirs [10 minutes]
Participants read the following information which allowed them to concentrate on the important aspects of this style of writing in mathematics.

Objective(s): The learner will develop the ability to monitor their thinking as they engage in mathematical problem solving. [According to the book, How Students Learn: Mathematics in the Classroom (National Research Council 2005), this is an essential principle to learning.]

Time: The first demonstration of the approach usually takes a class period. Shared practice, between the students and teacher, may require another couple of class periods. The goal is that after this initial period of scaffolding learners can use the metacognitive mathematical memoirs on a regular basis throughout the remainder of the school year. During the scaffolding, the mathematical content is still being addressed as it is the topic of the memoirs.

Activity: Have the learners look over a group of problems and identify one that represents a ‘just right’ problem – not too hard and not too soft; this might be a problem set they have worked on previously or it might be the first time they have encountered the problem. The idea behind selecting a ‘just right’ problem is that the learners want to be able to tell an interesting story of successfully solving a problem including struggles they encounter along the way.

The learners are asked to describe how they went about solving the problem and what they were thinking as working on it; this represents the metacognitive aspect of the activity. Because memoirs can include embellishments to make the story more interesting, the learners are encouraged to be creative (yet realistic) in describing the “challenges” they encountered along the way. In this way, they are anticipating problems associated with the content.

Pros: (1) I have found these memoirs support learners’ in both a better understanding of the mathematical content and a clearer picture of the process of problem solving; (2) The learners’ writing also provides insight into their thinking – something I often found lacking in their previous assignments, even when they were required to show their work; and (3) The memoirs have been much more interesting for me to read than a solution set – and I only read one or two instead of an entire set.

Cons: (1) Like any new approach, learners often balk at having to write in math class; (2) Also, if learners don’t have experience with metacognition, it can be initially difficult for them to write about their thinking beyond, “I just know it.” It will take time and modeling by the teacher for the learners to become more comfortable showing their thinking as well as their work; and (3) It takes longer to grade a memoir than correct a traditional assignment.

Assessment: For simplicity sake, I often use these four Cs in evaluating learners’ efforts. Learners must have met the expectations at one level to move onto the next. There are opportunities for revision in order to support success.

D	C	B	A
The work is Clear – you have communicated your work with clarity.	The work is Correct – your efforts demonstrate an understanding of mathematical content and/or processes.	The work is Complete – all required aspects of the project have been addressed. There are no gaps in your thinking.	The work is Creative – you present a unique perspective and show you can extend your thinking.

Lessons Learned: Do not try this alone – ask for support from our writing colleagues.

Activity: Choice [30 minutes]
Each workshop participant made a decision regarding how to proceed in order to further develop his or her understanding of Metacognitive Memoirs. Some people chose to look at existing models: High School Student (This is the result of a web search from many years ago. I would appreciate any help identifying the source so I can give credit where credit is due.), Preservice Elementary Teacher, and Preservice Secondary Teacher. Others worked on solving and writing their own Metacognitive Memoir using one of these problems: Which is 19/24? or Sowing Seeds.

Reflection: Author's Chair [10 minutes]
I asked one of the participants who had been working on writing his own Metacognitive Memoir to share with the group using an approach that I have adapted from literacy instruction.
What did he do?
He selected to work on "Which is 19/24?" because it related to struggles he was aware of that some of his current students were experiencing. He explained how he used what he knew about common denominators to narrow down the computations he would ultimately have to do. As it turned out, this approach arrived at a solution without having to do any fraction addition or subtraction.
So what did he learn?
While he wrote in great bit detail how he went about solving the problem using the common denominators as a sorting mechanism, he had not completely shared his thinking. Upon reflection, he recognized that he had thought about and dismissed using other approaches (compute all the items or use a benchmark of 1 to sort the items) because they would not be as efficient. In order to make his thinking visible, he realized that he needed to include this information in his writing.
Now what will he do with his learning?
Besides being more aware of his thinking (metacognitive) and sharing it with his learners, he also discussed how he might modify the problem so that his learners would have to do some computation. He acknowledged that he had done some deep, conceptual mathematics, but he wanted the problem to address some content skills as well. The other nice thing about using Metacognitive Memoirs is that you can tailor them to the needs of your learners.

Now what? Part IV

2011-10-19T06:52:00.000-07:00

So far in this series I have discussed the need to empower learners by getting them to ask and explore their own "Now what?" questions (here), considered possible answers to a messy learner-generated word problem (here), and identified implicit conditions associated with the different answers (here). In this final post of the series, I share my preservice teachers' efforts to extend our understanding of one of the possible answers to this word problem:

In the Community, you get two pets. The Elders pick the pets for each family. There were six choices of pets to have: dog, cat, fish, snake, bird, and hamster. What was the probability of getting a dog and a cat?

Three of the possible responses represent combinatoric approaches that the preservice teachers are already familiar with: combinations, permutations, and the multiplication principle. The 1/21 answer (duplicates are allowed but order does not matter), however, represents a new approach.

Many of the learners decide to explore, "How could I generalize this result?" They quickly realize that the 21 comes from adding the combination, 6 choose 2, with the 6 pairs. Consequently, they hypothesize that the general case would be:

n choose 2 + n

In order to check this rule, they try out some simpler problems. 1 pet results in 1 possible pair. 2 pets result in 3 possible pairs. 3 pets result in 6 possible pairs. 4 pets result in 10 possible pairs. 5 pets result in 15 possible pairs.

While this satisfies many of them, a few embrace the idea of extending the problem and notice that the sequence 1, 3, 6, 10, 15, 21, ... looks familiar. In fact, it can be thought of as:

(n + 1) choose 2

This fascinates them and me. I knew this result going into the lesson but I resisted the urge to explore it further before the lesson. "Why does this work?" was a question I did not have an answer to ahead of time. I wanted to work with them instead of guiding them to the answer. (This is an important instructional approach that I wrote about here.)

We are able to show that the two approaches are equivalent fairly easily.

But this still does not explain why (n + 1) choose 2 works when selecting 2 pets from n animals allowing for duplicates.

Finally we begin listing possibilities which leads to us designing the following table. The first entry would be a pair of dogs.

The "+ 1" is the repeat column in the table. This satisfies the "Why?" question but leads us to consider what would happen if 3 pets were selected from 6 animals - allowing for duplicates. Time is up, however, meaning this will be something we can think about on the drive home. Enjoy!

Now what? Part III

2011-10-12T07:10:00.000-07:00

Thus far we have considered ways middle school learners can extend their learning by generating their own problems based on young adult literature (here) and how preservice teachers can extend their understanding by considering alternative solutions (here). Given the four different answers they usually come up with (1/15, 1/30, 1/21, and 1/36), the preservice teachers attempt to revise the original problem to match each answer.

For 1/15, the clearer question might be:

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family without any duplication (e.g. no cat-cat pairs). What is the probability of getting a dog and a cat if the order doesn't matter (i.e. cat-dog is the same as dog-cat)?

Typically, this is how the preservice teachers read the original problem even though the original lacks many of the specifics. They see it as a combination problem and so add the necessary conditions in their head. It helps these future teachers to be aware of the implicit conditions hiding in many problems.

The next answer that my learners usually address is 1/30. The clarified question might read:

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family without any duplication (e.g. no cat-cat pairs). What is the probability of getting a dog first and then a cat?

Because the preservice teachers are familiar with permutations this revision is fairly simple for them.

1/36 is usually the third answer they choose work to find the question for:

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family. Duplications, cat-cat pairs, are possible. What is the probability of getting a dog first and then a cat?

This, too, comes quickly since they are comfortable with the multiplication principle.

Finally they address 1/21:

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family. Duplication, cat-cat pairs, are possible. What is the probability of getting a dog and a cat if the order doesn't matter (i.e. cat-dog is the same as dog-cat)?

While they come to this version of the problem easily based on the prior revisions, this is a new context for them and they are eager to explore it further.

Before we look at the math, I engage the preservice teachers in a discussion of the pedagogical worth of having "messy" problems with many possible interpretations. They are inclined to want to clean up problems before sharing them with students but they recognize this is often based on their own experiences in math class. Fortunately, there are usually some voices that identify how considering different points of view made the problem much richer. This ability to identify underlying conditions and considering the alternative problems supports the "Now what?" stance I hope to foster.

We explore the "Now what?" question regarding how to generalize the "1/21 case" in the next post.

Now what? Part II

2011-10-05T09:10:00.000-07:00

In the prior post, I introduced a problem written by a seventh-grader as both an example of what middle school students could do when deciding what comes next and an opportunity for preservice teachers to develop and explore their own "Now what?" questions. This was the student-generated problem:

In the Community, you get two pets. The Elders pick the pets for each family. There were six choices of pets to have: dog, cat, fish, snake, bird, and hamster. What was the probability of getting a dog and a cat?

Typically, the preservice teachers came up with an answer fairly quickly. After all, since it comes after a unit on combinations the solution method seems obvious. Still, I ask them to explore the problem further by using one of the extension questions we collect over the course of the semester.

It is sometimes difficult for the preservice teachers to consider alternative answers, however, because of their own experiences with math problems having a single correct answer and the fact they think this problem is so cut-and-dry. Fortunately, I have examples of alternatives to their expected answer of 1/15 that were identified in previous classes. If no one comes up with these alternative answers in the current class, I offer them as other possibilities we ought to consider. I say, "A group came up with an answer of 1/30. Another was pretty sure that it was 1/21, although they also considered 1/36 after they hear the 1/30 rationale."

The preservice teachers' initial reaction is, "Those answers are wrong." I remind them that as educators we must consider that learners are not wrong but they may have answered a question different than what we expected. (I wrote about this here.) Therefore, the natural "Now what?" question that a teacher can consider is, "What question does this answer?"

To be continued...

Now what? Part I

2011-09-28T08:06:00.000-07:00

Learners who lose the ability to make decisions are disempowered.

Brain Cambourne

This quote comes from Cambourne's theory of learning as it relates to responsibility. I see this "disempowerment" in the secondary math classes that I observe and the college courses that I teach. Students are constantly waiting for someone else, usually the teacher, to tell them what to do. The time when this is most evident is when a student finishes an assigned task and sits back waiting for the teacher to answer the question, "Now what?"

Disempowerment has tremendous consequences. It removes from the student the responsibility to be a self-directed learner. Once a task is complete, they fail to consider what might come next which results in a loss of cognitive momentum - disengagement. Their need to be directed by others to explore beyond the assigned task is unsustainable. What happens when a teacher is not available to tell them what comes next? They sit and wait.

This is why I spend a great deal of time in my classes, especially those populated with teachers-in-training, encouraging them to ask the "Now what?" question to themselves and not wait for me to tell them what comes next. Granted, this is not easy at first since it runs counter to years of training, but with time I have found that I can gradually release the responsibility of identifying extensions to my learners. It usually starts with me explicitly identifying the disempowerment issue and modeling what self-directed learners might do with extra time on their hands. After modeling this behavior multiple times, I share with my learners the responsibility of coming up with ways to maintain our momentum. My goal is that eventually learners will develop their own approach to extending their learning and thus empowering themselves.

An example of sharing this responsibility comes from a probability and statistics course I have taught for preservice K-8 math teachers. The activity is based on an article from Ann Lawrence, "From The Giver to Twenty-One Balloons: Explorations with Probability." I like this article because it models several possible extensions students might consider to further their learning. For example, after working on several probability problems related to The Giver, the middle school students in the article are asked to write their own problems given the story's context. Here are the results of this extension:

Novice teachers are sometimes hesitant to turn over this responsibility of generating problems to students because they feel unprepared to deal with what might be messy stories. Therefore, I ask my preservice teachers to look at Mark's story and consider possible solution methods and then what comes next. With practice, these future teachers can become more adept at dealing with unpredictable situations. Also, I explicitly state that their "Now what?" question can focus on either pedagogy (responding to Mark) or content (exploring the math) - it's up to them. Empowerment!

Before I share the typical results, I want to provide you the opportunity to explore Mark's problem. A chance to ask and answer your own "Now what?" question. I hope you will share your learning in the comments.

What would Charlie Brown do?

2011-09-21T06:24:00.000-07:00

Josh earned $1,962 over the summer. Hilary earned $129 more than Josh during the summer. How much money did Hilary earn over the summer?

When I taught middle school math in the 80s and 90s, this was the type of word "problem" that filled the curriculum my district used. Furthermore, the "problem" was typically at the end of a worksheet entitled Adding with Two Addends that was filled with 12 other addition exercises. Even back then I knew that these were not true problems for my 8th-graders, and that if I gave them a reading assessment they would have no idea who Josh and Hilary were and what they did over the summer. They had simply added the two numbers. And who could blame them?

It was during this time that I began to explore the idea of subtle shifts. At that point, I did not have the time nor the expertise to develop a whole new curriculum. Instead, I decided to rewrite these exercises in order to make them problematic for my learners. How? I received my inspiration from the Charlie Brown cartoons.

I replaced every syllable in the story with "wu" - as if the teacher in the cartoon were reading it aloud. It removed the real-world context but forced the 8th-graders to rely on their number sense. They had to decide which operation (addition, subtraction, multiplication, or division) made the most sense and why. Here are a few for you to try:

Wawa wa four wa wawawa. Wa wawawa wa wa wa 150 wa. Wa wa wa wa Wawa wa?

Wawa wawa $519 wa wa wawawa, Wawa wa $267 wa. Wa wa wa wawa wa wawa wa wa wa wawawa?

Wa Wa Wawa, wa wa 936 wa wa Wawa Wawa wa 1,443 wa wa Wawawa. Wa wa wawa wa wa wa Wawawa?

Wa wawa wa wa wa eight- wawawa wawawawa wa wa wawa 232 wawawa?

Unlike Peppermint Patty in the clip, my learners were much more engaged in these stories. Maybe it was the novelty of them. I suspect, however, that it had to do with the fact that they were more challenging, and therefore more interesting, than the originals. And the 8th-graders were usually fairly successful predicting the operation required in the original story.

Nearly 30 years later, I still use these Charlie Brown Word Problems with the preservice and inservice teachers that I work with. They help to represent the difference between exercises and problems. And they demonstrate how subtle shifts can enhance learners' mathematical experiences.

Is it really about the money?

2011-09-14T11:32:00.000-07:00

US Secretary of Education, Arne Duncan, was a part of a panel discussion held at the University of Michigan School of Education recently (watch it here). Part way through, the discussion turned to teacher pay and Secretary Duncan again called for a substantial increase in starting teachers' salary. (To be fair, the issue was not raised by the Secretary - see around 46 minutes.) For some reason, the idea that paying teachers more is a solution to our educational issues rubs me the wrong way.

Don't misunderstand, I would accept a pay increase no questions asked. It would help me to get some things done around the house and might allow me to pay someone else to do my chores so I can focus on teaching issues - maybe. Here's the thing, though. I didn't get into education for the money. In fact, I got into education in spite of the money.

My first real job was as a computer programmer. It was only a paid internship but it was enough for me to know that it was not the kind of work I wanted to do for the rest of my career regardless of the money. Contrast that with my internship as a teacher (what we call student teaching). Not only did I not get paid for that work but I had to pay to do it.

Still, I loved it! I got up early, stayed late, and volunteered for lunch duty. And I know I am not alone. Nowadays, I work with teachers doing various internship experiences and I see all the extra responsibility that they willingly take on at their placements. Many times they do this along with some other job that pays them so they can turn around and pay to do what they love - teach.

Because of this first teaching experience, I struggle attaching my paycheck to my teaching. Each time I get paid I am grateful for the support I receive. However, getting paid more or less would not change the way I go about my professional efforts. Most of the teachers I work with seem to have the same mentality.

The idea that more pay would get current teachers to work harder to improve their teaching or get better people into teaching is insulting. From my perspective, many of the best and brightest are already in education or training to be a teacher. Increasing teacher pay simply to attract someone attracted to more money does not seem wise.

Should teachers be better compensated for their commitment to our most precious national resource? Absolutely! Does our educational system need to be improved? Constantly! Let's pay teachers more and let's improve education. But these are two separate issues that seem to be getting confused.

What's TED?

2011-09-08T07:02:00.000-07:00

I'm asked me this question a lot recently. Now I know that TED is an acronym for Technology, Entertainment, and Design, but as I have written before I tend to relate everything to my practice. Therefore, I often think of TED as being Teaching, Education, and ... well, Design.

How does design relate to education? I was first introduced to the connection when Mickey McManus from MAYA (Most Advanced, Yet Acceptable) talked at TEDxGrandRapids about Innovating Education. In particular, he described how MAYA developed the LUMA (Looking, Understanding, Making, & Advancing) Institute as a sort of bootcamp for teaching Human-Centered Design. Based on LUMA's successes, they began considering whether design thinking is a basic literacy for everyone and how they might bring their curriculum to kids. This seemed to nibble around the edges of educational reform, however, as it treated design as an extracurricular activity and not a part of school.

Then at the College of Education start-up meeting a colleague asked me if I had ever heard of REDlab at Stanford University. I said that I hadn't and asked what it was. She said RED stood for Research in Education & Design - there's that word again and that connection. With the beginning of the semester at hand, I did not have the opportunity to find out much about REDlab but according to the website:

REDlab was founded in 2009 to study the impact of design thinking in education. It grew from the Taking Design Thinking to Schools Project, which was funded by Stanford’s K-12 Initiative. It sparked a partnership between the Stanford School of Education and the Hasso Plattner Institute of Design (d.school). The goal was to partner with local schools to explore the feasibility of design thinking as a new way to teach and learn.

All of this intrigued me, so when I had an opportunity to sit down for breakfast with an actual designer I grilled him for information. Unfortunately, Rob was unfamiliar with either Mickey McManus or the REDlab. Instead, he shared his own educational journey on his way to becoming a designer.

Rob went to a traditional university before transferring to a design school. As far as he was concerned, these two institutions offered him vastly different educational experiences. The university provided a structure while the design school offered guidance. I quickly interjected that this was understandable because the university would need to have a proven/vetted curriculum and a clear way to assign grades. He could see that and added that some design schools had done away with grades and focused on competencies instead. In fact, design-based businesses seemed less interested in grades than in authentic examples of the designers work.

I asked how this work was shared and Rob said through a portfolio. But the portfolio was not just samples of final products; it contained artifacts of the design process from start to finish. When I asked if this meant it had examples of where mistakes had been made, he hesitated. After some thought, he assured me that someone looking over the process would certainly see where his thinking had changed, but he did not seem to consider these mistakes - just part of the normal creative process.

Breakfast was served and I had taken enough of Rob's time, so we joined the rest of the group in their discussions. I keep thinking about our conversation, though. How can design thinking apply to education?

What do you want to know?

2011-09-02T08:09:00.000-07:00

Grand Valley started this week. And, as usual, I began my Introduction to Learning and Assessment course with a workshop called "A Piece of Me" (handout). This involves an activity that was introduced to me over 14 years ago during an Integrated Thematic Instruction class. I wish I could remember the presenters so I could give them credit but that information has long since vanished from my memory and my files. If anyone knows the origin of this activity, please let me know so that I can give credit where it is due. Anyway, here is how I implement the activity.

***
Goal: The learner will ask questions in order to get to know the instructor/course.

Schema Activation: Whole Class Discussion
When you meet someone, What do you want to know about him or her?

Focus: Expectations - Small Group

Develop one personal and one professional question to ask the instructor.
During the activity, keep a record of each question that is asked.

Activity: Asking and Recording Questions

I begin this portion by saying something like, "I reserve the right to decline answering any question. I have been doing this for many years, in K12 and college classrooms, and I have never had to refuse answering a question. Now this is not a challenge for you to be the first group to come up with the unanswerable question. It is intended to demonstrate that this can be used at any level - even with middle schoolers who are known for their inappropriate questions."

We then move on to the groups asking their questions. I respond providing answers and other related information that I think they'd like or need to know. After all the groups have asked their questions, if time permits, I ask if there's anything else anyone would like to know about me or the course.

Reflection: Looking Back - Small Group Analysis

From your list of questions, pick one that you think was an effective question. What made it effective?
If you finish early, pick another effective question and describe what the two question have in common or consider an ineffective question and rewrite it or ... (What comes next for you?).
How engaged were you during this activity?

I end the workshop by getting some feedback from the participants. The consensus seems to be that this activity is much more engaging than what they imagine it would be like for me to talk at them for forty minutes about what I think they need to know. They appreciate that they control the activity with their questions. They acknowledge that they are much more interested in my points because I am responding to their questions.

Because they are teachers in training, I let them in on a few secrets behind the activity: 1) I am still in control of what I share and usually get to say what I think is important regardless of what they ask; 2) I use the questions they ask as a formative assessment that provides data on what they find interesting; and 3) The activity is much more engaging for me as well because I do not know what I will be asked. (One time it was what my favorite sandwich was.)

I follow this workshop up with a home workshop where they apply the same strategy of asking questions to reading the course syllabus (handout). This is another activity that I have decided I won't waste time doing in class. They post their questions on our course Blackboard site and I address them directly on the discussion board or during subsequent class meetings.

***

Now it is your turn. Any questions?

Have you ever ... ?

2011-08-27T19:30:00.000-07:00

The #mathchat topic for August 18th was "How do I get students to take the first steps in problem solving?" During the course of the chat, I asked the following:

Here are some of replies:

With the exception of @kassiaowedekind, very few details were provided. This is understandable given the limitations of Twitter, but I thought it might be worthwhile to continue the conversation using a venue that allows for more explanation. I will get us started with a couple of ways I have approached this question and then others can add their ideas in the comments.

The first approach I use is trying to avoid asking questions to which I already know the answer. I learned to do this while working with the first-graders in my wife's classroom. When I asked, "What is four plus three?" the first-graders typically responded with the expected answer and then waited for me to validate it. This was unrewarding to everyone involved. They were becoming reliant on me as the authority, and I really wasn't getting any insight into their thinking. Maybe they just knew the fact. Or they could have counted all, counted on, used doubles plus or minus one, or some other strategy. Or maybe they answered seven to every addition fact.

Instead, I decided to ask the first-graders, "How would you figure out four plus three?" Their answers to this were much more interesting to me. Although I had some ideas of how they might respond, I could not be sure and there were times I was surprised. Also, the first-graders always enjoyed sharing their thinking with me - much more so than just giving an answer.

The second approach is to share a problem I have not solved. I have some ideas of how to solve it but I have resisted the temptation to actually work on it. That way, when my learners ask if their solution is correct, I can honestly say, "I don't know." Then, they have to share their thinking, not just their answer, in order to convince me.

One of my favorite problems to ask the preservice teachers in my Teaching and Learning Middle Grades Mathematics course is, "When does 3/19 repeat?" We have tried graphing calculators and spreadsheet to no avail. As a result, we end up solving simpler problems looking for patterns and making conjectures. (I am told Wolfram Alpha can compute it but I refuse to spoil the fun - for now.)

Now it is your turn. Have you ever given a problem you didn't know the answer to? And if so, what was it and how did it go?

What are you waiting for?

2011-08-18T11:21:00.000-07:00

I was catching up on watching some TED Talks this past week and found this one by Matt Cutts to be particularly appropriate for the start of a new school year.

The idea to try something new for 30 days is appealing to me as a teacher. It offers the opportunity to inventory my current practice and consider things I would like to add or subtract. I like the idea that the 30 day challenge represents an activity that encourages me to focus on the present moment. It reflects the ideas associated with the growth mindset and building trust in ourselves. Finally, the idea that the changes ought to represent small changes in order to be sustainable (what I call subtle shifts) really resonates with me.

Today, a group of middle school teachers came up with a list of things they want to work on this year to set the stage for continual improvement at their schools. This was the result of our discussion of chapter 8 from The Teaching Gap. Here is their list:

Volunteering more to be part of district teams developing exemplary lessons and assessments;
Engaging in more peer coaching;
Sticking to practices that are effective based on formative assessments even when standardized tests do not show immediate improvements; and
Spending more time reflecting on practice using a structured approach.

Given Matt's TED Talk, I suggested they consider focusing on these practices for 30 days in order to develop these as habits.

And what are my 30-day challenges?

Professional:

Add: making ShowMe think-alouds
Subtract: grading any work

Personal:

Add: practicing my flute
Subtract: eating potatoes

I will get back to you in 30 days and let you know how I did.

How do you use the workshop model?

2011-08-10T09:03:00.000-07:00

My vision of teaching and learning is summed up in my six-word teaching philosophy: engagement that fosters capacity and agency. This is based on my current understandings of the research of Vygotsky, Cambourne, and Johnston. It is also reflected in the National Resource Council's book How Students Learn. Putting structure to this vision has resulted in my implementation of the workshop model in the courses that I teach.

Planning and Instruction Portion of Teaching-Learning Cycle

I first read about this model for planning and instruction in Cambourne's book, The Whole Story. He saw it as a framework for providing the Conditions of Learning that he had identified in his research. Next, I became aware of a slightly different version of the workshop model used by various teachers from the Public Education and Business Coalition. Authors like Keene, Miller, and Conrad explained how this structure supports learners in developing the comprehension strategies they can use to make sense of the world. Nowadays, Lucy Calkins is a big name in Readers' and Writers' Workshop.

With my colleagues, Esther Billings and John Golden, we have modified these models to represent our theoretical framework of how to support learning. The specific structure is a combination of four key components that I have come to label connection, concentration, construction, and consolidation. We have also used schema activation, focus, activity, and reflection, but I like the alliteration.

Connection: The teachers or learners make connections between previous experiences and the present workshop. This provides a cognitive foundation on which to build new ideas. Schema activation was our original label because we liked the comprehension aspect - moving from the known to the new.

Concentration: The teacher sets the expectations for the workshop during this phase. By letting the learners know exactly what the focus is for the lesson, they are more likely to be engaged in the workshop because they know its purpose. Often, this looks like a mini-lesson based on the gradual release of responsibility that highlights the work the learners will be doing.

Construction: Learners are provided an opportunity to employ the idea presented during the concentration. This is the bulk of the workshop. An attempt is made to immerse the learners in authentic tasks so that they are learning in context and see the complexity surrounding the idea they are concentrating on. This usually includes some compulsory work, and perhaps a limited number of predetermined choice activities. While the learners work, the teacher is free to observe their progress, confer with individual learners, or provide a small group lesson.

Consolidation: This is the phase that attempts to ensure that learning lasts. It is imperative to provide learners with time to reflect on their experiences otherwise the response to "What did you learn today?" is a predictable "Nothing." We ask learners to look back at what they did and what it meant to them and to look forward to how their experience might transfer to other situations. This might entail writing (exit ticket), small-group sharing (round table), or whole class presentation (mathematician's chair). Typically, we use the formative assessment data collected during this phase to inform planning and build connections to future lessons.

In order to ensure that all phases are attended to, I often use an online stopwatch to manage our classroom time. This sometimes means interrupting learners during the construction phase before they are done, but usually they have engaged in enough of the activity to be able to reflect on it. When someone complains that they haven't finished, I respond, "Feel free to work more on it when you get some free time." Some take me up on this and some don't - it's their choice. I like to think that this builds what Ellin Keene calls, "learning lust."

I use the workshop format for in-class activities and for assignments. It even provides the framework for some of my assessments. Ultimately, the learners come to a point where they want less structure and more freedom. Then I turn the responsibility of designing workshops over to them. I provide the objective and the resources and they plan their time - keeping in mind that learning is supported by connections, concentration, construction, and consolidation.

The workshop approach may not be for everyone, but it works for me.

Can you help a Fellow out?

2011-08-05T12:47:00.000-07:00

The first cohort of W. K. Kellogg Foundation's Woodrow Wilson Michigan Teaching Fellow program descended on Grand Valley this summer. I have been teaching one of the many courses they have been taking in preparation for a year-long student teaching residency program beginning in the fall. My course, as the name states, is about facilitating learning environments. While I can offer them theories and my experiences teaching a middle school math class nearly two decades ago, I have relied on current K-12 teachers in my Professional Learning Network to fill in the gaps.

Already, the support has been amazing. Some of that support is chronicled here. Also, a few bloggers have posted their ideas about classroom management (here and here) and included the course hashtag, #FLE11, in their tweets so that the Fellows would have access to their experience and wisdom.

You all have given so much already but I have another request. As a part of a home workshop, Fellows were asked to come up with a list of questions they had about facilitating learning environments. They have posted these on their blogs. I would appreciate it if you would take some time to read these posts and provide answers to their questions when possible.

These are the Fellows who have completed the workshop.

I will add more links when they become available. Thank you in advance for your time supporting these teachers in training.

"What if we mutiny?"

2011-07-30T10:45:00.001-07:00

That was the question a graduate fellow asked me when a workshop we were doing in class took an unexpected turn. The fellows had been working in small groups to identify important information read in one of the course textbooks. Each group was assigned a different chapter, and they were preparing to share with their peers what they had gotten from their reading. When preparation time was up, however, I changed things up. Instead of having them share with the whole group, I asked that they stay in their small groups and develop a rubric for evaluating collegial sharing.

My request disrupted the anticipated flow of events - do assigned reading, plan in expert groups, and share with whole group. As I have said before, effective teaching is a lot like storytelling. And I wanted to shake up this "story" (lesson) in an effort to make it more memorable. I also knew the larger plot of this particular "story" and that it involved several opportunities to share these chapters, while incorporating more readings in future lessons.

It was not a surprise to me when the fellow asked, "What if we mutiny?" This is a group that is highly motivated and wants to know as much as they can about facilitating learning environments. The fellow's peers had important information that was expected to be shared with the group and that ought to happen now in the typical chain of classroom events.

When the fellow asked, "What if we mutiny?" I responded with something like, "I would love that!" But there was no mutiny and I went ahead with my own storyline anyway. The following day, I thought more about the exchange and decided that a mutiny fit well with the next lessons' theme - responsibility. So, I replaced a learning workshop dedicated to reading groups with time for a mutiny (what Daniel Pink has called "20% time").

The day arrived and I explained that for the next workshop how they spent the time was up to them. I reminded them of the goal of this particular journey (collaborating in development of effective learning environments) and then said, "I'll be in the brig." For the next forty minutes, the fellows took responsibility for their learning. They organized their time. They monitored their progress. And they negotiated their needs. As they worked, I took notes on their collegial efforts that I could use to evaluate their progress on this project.

At the end of the time, I broke free of the brig just before the fellows began a second round of sharing. I affirmed their efforts and said that I was pretty sure there would be another chance to mutiny during the next class. A fellow asked, "What if we mutiny now?" I smiled.

Will this "reform" really improve education?

2011-07-20T12:43:00.000-07:00

This is a letter to the editor that I am considering sending to papers in Michigan. Your feedback would be appreciated.

***

Dear Editor:

I am tired of reading how the new tenure legislation recently signed by Michigan's Governor Rick Snyder "makes it easier to fire bad teachers." A more accurate description would be that the law potentially makes it easier to fire ANY teacher since what is meant by "bad teaching" is not well-defined. Consequently, the protection of effective, experienced teachers has been severely compromised by this new law, and it will have a detrimental effect on education in Michigan.

The argument is that the old tenure law protected bad teachers. Here we have the false assumption that led to the flawed logic behind this supposed "reform" effort. Originally, tenure protected teachers, all teachers, from being removed without justification. This did not prevent bad teachers from being removed; it just meant administrators had to be diligent in documenting abuse.

Were there bad teachers who gamed the system and made it difficult for administrators to remove them? Certainly, but that is the nature of a system meant to protect teachers and predicated on their presumed innocent. Closing loopholes that bad teachers used to game the system would be a reasonable response. What the new law seems to do instead is react in a way that goes beyond the problem of bad teachers.

The new tenure law certainly makes it easier for administrators to get rid of bad teachers. It also makes it easier to get rid of the experienced teacher who is high on the pay scale. It also makes it easier to get rid of the teacher committed to using promising practices that an administrator does not agree with. It also makes it easier to get rid of the teacher who holds kids accountable for their actions even though it ruffles the feathers of the kids' parents.

If you trust administrators, then there is nothing to worry about. I talk to a lot of teachers, however, who are concerned. They say they will hesitate to volunteer to teach in a high-needs classroom because their evaluation may depend on these students' test score. They describe how they will be more reluctant to address bad behavior because student and parent evaluation may play a role in determining their effectiveness. They explain that they will no longer take risks on new techniques and technologies because there seems to be little room for error. This does not sound like educational reform to me.

Whether or not these issues come to pass (the evaluation piece is still up in the air), the reality is that teachers feel attacked by such overreaching measures as the new tenure law. The message appears to be that teachers now have to take the education of their students seriously - as if they didn't before. Under such circumstances, the likelihood of educational innovation seems small.

What's the story?

2011-07-15T08:42:00.000-07:00

Hogwart's Express?

I am a huge fan of J. K. Rowling's work. She is a fantastic writer who has introduced a new generation to the magic of reading through her Harry Potter series. My guess is that she was also an excellent teacher. While I have never seen her teach in a classroom, I believe effective teachers are essentially storytellers.

First, there is Ms. Rowling's phenomenal ability at writing backstory. This is the part of the narrative that provides background supporting a better understanding of story. She flawlessly weaves this information throughout her writing. It is akin to what effective teachers do when they help learners to activate schema during a lesson - providing a firm foundation on which to construct new ideas. The ability to make connections is fundamental to teaching and learning.

Could this hat do more than sort?

Foreshadowing is the literary device used by authors to provide readers a subtle peek at future plot points. Again, here Ms. Rowling is a master. Once I was aware that her books included clues about the upcoming story, I began reading (and rereading) the text even more deeply. Imagine teaching a lesson that has learners engaged with every moment because it has purpose - they don't want to miss something important that they can use in later learning. I see this as being related to Cambourne's ideas of engagement and immersion.

Another thing I like about the Harry Potter series is that there doesn't seem to be any wasted effort. The story always moved forward. And while there were times that we were introduced to characters, settings, or events that seemed superfluous, they almost always ended up playing an important part in the resolution of the story. This reminds me of what I have read so far in Mike Schmoker's book, Focus. In order to be effective, teaching needs to maintain momentum toward a clear set of manageable goals.

This car plays a part in several books

Certainly, storytelling is not the only simile that can apply to effective teaching. It just seems fitting to explore it the day the second part of Harry Potter and the Deathly Hallows is released in US theaters. Maybe I'll write my "teaching is like kayaking" post after my next trip to the UP. It is the rationale supporting the simile that is most important - that is where we find the characteristics of effective teaching.

(If you want more evidence of Ms. Rowling as an effective teacher, then consider her 2008 graduation speech at Harvard and her characterization of effective teachers in her writing.)

Do you trust yourself?

2011-07-06T12:02:00.000-07:00

I went golfing today with a friend, and as you can see by the scorecard (generated by Golfshot GPS), we played horribly. Now 45, or bogey golf, is not a bad score for me but we played a scramble format which means this score was a result of our best shots throughout the round. Let's just say we have played better.

What does this have to do with the Virtual Conference's Prompt? I know that part of my problem today was that I was not trusting myself as a golfer. On almost every shot, I hit harder or softer than I knew I needed to because I was over thinking based on the previous shot. This is a poor way to play golf. And after reading Golf and the Spirit, I know that these lessons can translate to other aspects of my life, like teaching. So my answer to, "What is at the center of my classroom?" is two-fold: reflection and trust.

In Golf and the Spirit, M. Scott Peck writes:

Forgive yourself each and every bad shot - as long as you have learned what it had to teach you - and then get on with it, free and unencumbered. (p. 266)

As a golfer and a teacher, I spend a great deal of time analyzing my practice, but I struggle with the letting go of mistakes part. This past year I created the following qualitative graph to remind me of my issue.

Personal reflection certainly improves my professional growth but there is a point where it seems like more reflection actually causes me to go backwards professionally. The extra reflection sucks my time and my confidence. I was convinced that I needed to find that reflective sweet spot and then stop.

Fortunately, I shared this visual metaphor with one of my teachers in training, who asked, "Does that mean I get to a point where I never get better?" To which she quickly responded, "I don't think that's true." (Have I mentioned how great our teaching candidates are at GVSU?) Instead, we thought about how adjusting the graph from a quadratic to a cubic might make better sense.

The idea is that when I find myself questioning my ability as a teacher, I need to be aware of this, accept it as part of my nature, and then adjust by letting go, moving on, and trusting myself (the Yoga Story strikes again). Essentially, I want to identify what I am doing instinctually well and make it intentional.

I have been thinking about this a lot this week because I am putting the finishing touches on a new course that begins next Tuesday. While I worked hard with my colleagues to design this course (see the Understanding by Design plan here), I am now second guessing myself and thinking about ways to do things differently. I almost skipped golf this morning to put even more work into the course. Lucky for me I trusted my instincts instead.

If reflection and trust are at the center of my classroom, then it must begin with me (Jason was right!). My learners will look to me as a model whether we are talking about doing mathematics or teaching. Therefore, it is important to demonstrate a reflective practice that is built on trust - trust of my learners and trust of myself.

Who's doing the work?

2011-07-01T09:04:00.000-07:00

I am in the process of planning a course for GVSU's experimental program involving W. K. Kellogg Foundation - Woodrow Wilson Michigan Teaching Fellows. The course is called Facilitating Learning Environments. One of the assigned texts is Harry Wong's The First Days of School.

I received a signed copy of this book from my mother early in my teaching career and read it over a summer. I still remember the impact the following passage had on my career:

But have you ever noticed what happens at 3 o'clock when the students leave? "Yea, yea, yea!" Why are they so full of energy? Because they have been sitting in school all day doing nothing while the teacher is doing all the work.

The person who does the work is the only one doing any learning! (page 205)

One way I related to this passage was my habit of taking students' pencils when they encountered a problem. There, I admitted it. I was a serial pencil grabber. A student would raise his or her hand and say, "I don't get it." Like a trained dog, I would grab the pencil, do the work, and ask if they got it now. After the obligatory nod by the student, I would walk away feeling like I was the best teacher in the world. Actually, I was psuedoteaching but I was unaware of it until I began to reflect on who was doing the work.

I vowed that I would never grab another pencil. When circling the classroom, I would have my hands in my pockets or clasped behind my back in order to resist the urge to do the students' work for them. I got better at guiding them through problems using questions.

Last year, I shared this story with a pair of teacher assistants after observing them grab students' pencils. They made a good point that these students were completely lost and required a demonstration not merely guidance. (The gradual release of responsibility strikes again.) We discussed how this might look.

The demonstration would start with the teacher saying, "Watch what I do, listen to what I say, and keep a record of it so we can talk about it later." Next, the teacher would model solving the problem, using his or her own pencil, being sure to think aloud while working through the process. Once the problem was solved, the teacher would ask the learner for a recount of the process. This could serve as an assessment - did the learner attend to the important details of the process? If the learner was ready, the teacher would ask the learner to apply the process to another problem as the teacher now served as a guide. If all went well, the teacher would move on taking the work done during the demonstration. After all, it was the teacher's not the learner's work.

I appreciated that these teacher assistants were able to question my "wisdom" (and authority) regarding this issue. It reminds me that I have a lot to learn from my learners - as long as I am willing to let them do the work.

Don't you get it?

2011-06-29T09:04:00.000-07:00

"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.

What's GRR?

2011-06-22T12:32:00.000-07:00

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.

Did they guess?

2011-06-20T11:40:00.000-07:00

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

Triangular Numbers
Square Numbers
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?

How much ground can I cover?

2011-06-17T12:32:00.000-07:00

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.

How can we communicate our thinking?

2011-06-15T07:56:00.000-07:00

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.

Are they related?

2011-06-13T09:58:00.000-07:00

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.

Do you see what I see?

2011-06-10T07:30:00.000-07:00

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.

When does it work?

2011-06-08T10:37:00.000-07:00

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 [5^th 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?

How might this change things?

2011-06-06T11:12:00.000-07:00

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.

Classes	6.1	6.2	6.3	7.1	7.2	7.3	8.1	8.2	8.3
PfP	$76	$76	$76	$74	$73	$71	$69	$68	$65
PfP+10	$86	$86	$86	$84	$83	$81	$79	$78	$75

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

What did you do over summer vacation?

2011-06-03T14:00:00.000-07:00

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."

What support do learners need?

2011-06-02T08:17:00.000-07:00

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.