Where have I been?

I haven't posted on this blog for nearly two years and nine months. It's not that I lost confidence in the power of blogging as a way to share and reflect on my thinking. I just got busy.

First, I took a position as director of the Design Thinking Academy [DTA] at Grand Valley State University. The goal of the academy is to support the use of design thinking methods and mindsets across the campus community. My work entailed visiting classes, scheduling pop-up courses, facilitating semester-long design challenges, managing creativity kiosks, and organizing the GVSU Design Thinking Speaker Series.

Ela Ben-Ur was our first speaker. She held a series of workshops where she introduced participants to a design thinking tool she created - the Innovators' Compass.

Ela Ben-Ur: Design Thinking and You (GVSU DT Speaker Series)

I learned about Ela's work on this episode of the Design Thinking 101 podcast.

This introduction to the Compass lead to the second project that has been occupying my time - writing a book. Kathy and I have been wanting to write about the Teaching & Learning Cycle for some time, but we always felt like something was lacking. When we learned about the Innovators' Compass and how it can help people to get unstuck or explore uncharted territory, we thought it would be a good resource for teachers engaging in the Cycle.

Innovators' Compass - design thinking cycle (hexagons) - Teaching & Learning Cycle (purple)

Now that a draft of the book is done (#DesigningMathAdventures) and sent to some publishers, I'm not as busy. Because I had set aside an hour each morning to engage in creative writing and didn't want to lose that momentum, I decided to fire up the old blog. I figured it would not only give me something to do while we wait to hear from the publishers but also allow me to share some parts from our writing that didn't make the latest cut.

Thanks for indulging me in this exercise of creativity. As always, if you have any questions, please post them below or reach out on Twitter (@delta_dc). The comments are open.

How can we assess 3.MD.B.3?

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. [3.MD.B.3]

Teaching-Learning Cycle

Teachers use #MTBoS as a way to find interesting and effective math lessons. Recently, some of us have noticed that assessments are often lacking as this community shares in the work of teaching. So I am proposing the #MTBoSAP (Math Teacher Blog-o-Sphere Assessment Project) as a way to pass along our wisdom and experience assessing students in our mathematics classes. I figured I could start by sharing some work I am doing in one of my courses for preservice elementary teachers.

The teachers are currently researching assessment items related to the Measurement & Data Domain (focusing on Data) from the Common Core State Standards (CCSS) in Mathematics. One of the third-grade standards in that domain is written at the beginning of this post. Because we are partnering with a school district that uses EngageNY, we looked for assessment items that already existed within that curriculum. This is an exit ticket that we found in the Grade 3, Module 6, Lesson 4. 

We thought that this item did a fair job of assessing the last part of 3.MB.B.3 but completely missed the first part, "Draw a scaled picture graph and a scaled bar graph..." So we looked through the rest of the lesson and thought this item from the problem set looked promising.

This item has students "draw a scaled bar graph" (still no scaled picture graph) and asks that they solve "two-step 'how many more' and 'how many less' problems." As we thought about it further, however, we were concerned that it was not clear whether students would use the chart or the graph to answer the questions.

Therefore, I decided to try to modify the original exit ticket in order to assess more of the standard. I got rid of two of the bars and wrote questions intended to have them "draw (a part of) a scaled bar graph" and answer multi-step questions that require some "information presented in scaled bar graphs."

Please complete the bar graph using the following information. 

• The number of books checked out on Thursday was 10 more than the number of books checked out on Tuesday. Draw the Thursday bar.
• 1,480 books were checked out Monday through Friday. Draw the Friday bar.

What are your thoughts about using this modified assessment item to gather data on students' mathematical understanding related to 3.MD.B.3? Do you have a good item for this standard that you'd be willing to share? If so, please share the item or a link to the item in the comments. Also, If you have any other effective CCSS assessment items, please share them on Twitter using #MTBoSAP. Thank you in advance for all you do to advance the profession.

Not so easy, is it?

Deborah Ball, Dean of the University of Michigan's School of Education, chairs the Michigan Council for Educator Effectiveness (MCEE) which was tasked to make recommendations to the Legislature about teacher evaluation. As part of her testimony before a joint meeting of the House and Senate Education Committees, she took some time to make the point that the work of teaching is often misunderstood and harder than most non-educators imagine.

The example she gives comes from fourth grade mathematics - multiplying two two-digit numbers (4.NBT.B.5). First, the teacher must be competent in the content. For example, what is the product of 49 and 25?

From Dr. Ball:

Obviously one wouldn't want anyone teaching third or fourth grade who couldn't do that. But, in fact, what skilled teaching involves is responding when students don't understand the material...

Knowing the mathematics is not enough. Teaching requires more than simply marking students' answers right or wrong. To be effective, teachers must be able to diagnose students' misconceptions in order to provide the support necessary for students to develop mathematically. Dr. Ball provides three examples of possible errors teachers might encounter from students multiplying 49 and 25:

When she asks the legislators how the students might have arrived at these answers, ... watch for yourself (it starts at about 2:30).

Dr. Ball does an effective job of pointing out that teaching is more than knowing and sharing content. However, this example focuses almost entirely on the "upper-half" of the Teaching-Learning Cycle: Assessment and Evaluation. Developing a plan for addressing the student-misconceptions and implementing that plan with a class of fourth graders raises the level of difficulty even higher. I hope these lawmakers now understand that there is more to teaching than what they experienced from the student-side of the desk.

Did you watch TEACH?

Friday night I watched the documentary TEACH with two other educators. Dad taught high school math and science for over three decades. My wife was an elementary teacher for 15 years. I taught middle school math and computers for seven-and-a-half years, making me the least experienced of the bunch, but we could all empathize with the stories being told.

And that seemed to be the point - to elicit emotions from the viewers. From the film's website:

Davis Guggenheim's award-winning documentary reveals the human side of the story: showing what it takes to survive the first year teaching in America's toughest schools. (emphasis mine)

Based on many of the Tweets posted to #TEACH, the movie accomplished its goal and touched many of those who watched.

It is important to recognize this as Mr. Guggenheim's vision for the film and understand that it must have influenced how he edited his footage. Just as a director who is adapting a book for a movie must consider what to keep and what to cut, Mr. Guggenheim could not hope to communicate the complexities of teaching in 35 minutes. Consequently, he presents a picture of teaching that diminishes the importance of certain aspects of the practice in order to focus on the teachers' stories.

I started this blog as a way to chronicle my teaching practice. In particular, I wanted to focus on the framework known as the Teaching-Learning Cycle. To Mr. Guggenheim's credit, his documentary acknowledges each of the Cycle's phases. However, it gives too much attention to assessment. Here are some examples:

• Several scenes where teachers and administrators wait by Scantron machines for testing results;
• A teacher expressing her concern about whether the test will "validate" her efforts; and
• Another teacher sharing her frustration that students were bubbling in the wrong answers when they should have known better.

Given the current national obsession with test scores, it is understandable why assessment became the focus of the film. However, it is also important to see how the assessments added tension to the storytelling. We want to find out, "Will the students pass the big test?"

While this plot device was effective in driving the stories in TEACH (see also Stand and Deliver and Lean on Me), it is unnecessary. As Kathy said, then Tweeted, at the end of the film:

As I recall, these themes have been successful in many other stories.

Part of the problem is that the assessment storyline is also taking over the current education reform narrative. The intent again seems to be to create unnecessary tension to manipulate people's emotions. My concern is that we are being swept up in that emotion, thereby, attending solely to teaching and testing while diminishing what's really important - learning.

What do teachers do?

The above Tweet was my take on a comment made at the Michigan Council of Teachers of Mathematics (MCTM) Conference. I agree with the President's point. An unintended consequence of compulsory education in the United States is that everybody thinks they know what it takes to teach but these people have only experienced education from the student-side of the desk. It is important to remember that those interested in becoming a teacher also fall into this category. Teacher preparation needs to make explicit what it means to be a teacher.

I am in the process of redesigning a probability and statistics course for preservice elementary teachers and I want to be sure that the activities in the course reflect the actual work of teachers - especially those aspects that go on behind the scenes. In order to set the intention that they will be doing the work of teaching, I am going to call my students teachers and group them according to grade levels and schools. Furthermore, the course will concentrate on the three areas of the Teaching-Learning Cycle that are often invisible to the casual educational observer: Assessment, Evaluation, and Planning.

Because I just finished reading Hattie's (2011) Visible Learning for Teachers, I want the overall theme of the course to be "Teaching is Learning." Too often, people interested in teaching think it is about telling or controlling or managing or ... because that is what they saw. And while teaching may involve these behaviors, if the teacher is not learning about the content and the learners along the way to inform instructional choices, then the teacher's actions will be haphazard and likely ineffective.

The course is separated into three projects. Each project is worth 30 points toward the teacher's final grade. Engagement Exemplars make up the last 10 points.

In the first project, schools will create 6-8 curricula using the Standards for Mathematical Practices (SMP) and the content standards for Probability and Statistics from the Common Core State Standards (CCSS). This is intended to make it clear that the CCSS do not represent a curriculum but require teachers to create units appropriate for their students. The 6-8 curricula will build on a K-5 curriculum using the SMP and Data and Measurement Standards that we will create together beforehand for practice.

The second project will require each teacher to demonstrate competency in the content associated with the standards in the curriculum. It is important that teachers are fluent with the mathematics they are teaching in order to assess understanding and select appropriate learning trajectories. Sometimes teachers encounter new content during their planning that they have to learn for themselves first. I remember having to teach myself about box-and-whisker plots when the topic showed up on the Eighth-grade Michigan Curriculum Framework. In order to demonstrate competency in the content, my teachers can choose between creating a problem portfolio or taking traditional exams.

For the final project, teaching pairs will conduct formative assessments on middle school students. They will gather data related to the students' understanding in Probability and Statistics and ability in the Standards for Mathematical Practice. In order to determine students' fluency, the teaching pairs will use the evaluation framework from the Teaching-Learning Cycle: What can they do; What are they trying to do; and What comes next?

The goal is that the teachers will leave this course with a better understanding of what teachers do and a set of tools that will empower them to teach effectively regardless of the circumstances they find themselves in during their career. Again, it comes down to phronesis.

So what do you think? Besides drinking large amounts of coffee, have I forgot anything that teachers do?

Why did you do it? Part II

In a prior post, I explained why John Golden and I used satire to critique a Khan Academy video for MTT2K. As Audrey Watters points out in the Tweet on the right and a post at Hacker Education, the approach has worked and started a serious conversation on Khan Academy's role in education reform. (You can add an article on Huffington Post Education and blog posts from Rhett Allain at Wired, Robert Talbert at The Chronicle, and Keith Devlin to the ongoing discussion.) But starting a conversation was only part of what I hoped to accomplish with the MTT2K parody. My primary reason for this project is revealed here.

A few people have attributed MTT2K to my not being a fan of Mr. Khan. This is only partially true. I am a big fan of his idea of moving K-12 education forward into the 21st Century. Anyone who wants to help teachers and students improve learning is someone I would consider an ally. Mr. Khan is not alone in this endeavor to provide online lectures, nor is he the first, but he has an incredible story and over 3,000 videos across several different disciplines.

While I love the idea of Khan Academy, I am not a fan of its implementation. Mr. Khan's videos are reminiscent of the lectures many of us experienced in math class, but they are lacking in some essential elements. As is the case with many people in the U.S. who think they understand education and what is wrong with it, Mr. Khan has a one-sided view of teaching. That side is from behind a student's desk, and because most teachers do not make visible all the work that goes on behind the scene before, during, and after the lesson this picture is incomplete.

As teacher educators, we share the Teaching-Learning Cycle with our preservice teachers in an effort to introduce them to the entire picture of what it takes to be a teacher. The work of assessment, evaluation, and planning often go on unnoticed to the casual observer, so people can be excused for not being aware of them. But in order to foster learning, a teacher needs to not only be aware of these parts of the Cycle, they need to know how to implement them effectively.

Much like the preservice teachers just entering our program, Mr. Khan seems unaware of the importance of each aspect of the Teaching-Learning Cycle. For example, in this Wired piece from last year, he seems to suggest that before his dashboard system, teachers usually "fly blind." This demonstrates a lack of awareness of the multitude of formative assessments that teachers use for evaluation which go far beyond counting video views and the number of right answers.

Furthermore, many of the math videos I have watched from Khan Academy suggest a lack of planning on the part of Mr. Khan. This is confirmed in a recent Time article:

He doesn't use a script. In fact, he admits, "I don't know what I'm going to say half the time."

There is something to be said for a teacher exposing students to an authentic learning experience and expert teachers can make this look easy. But it is not easy. As Mr. Khan showed in the video we critiqued and novice teachers find out on a regular basis, lack of planning can have disastrous results. Teachers who do not know the vocabulary used in the lesson or think the numbers used in the examples can come out of thin air risk fostering misconceptions rather than learning. Novice teachers who make and catch these kinds of mistakes have opportunities to quickly set things back on the correct path. It is unclear how Khan Academy handles this as the video we critiqued was more than a couple years old.

So this is what I hoped people would get out of the video. Yes, Mr. Khan has a good idea - let's improve education by allowing teachers to work more directly with students. He is not a world-class teacher, however. In fact, his videos demonstrate that he suffers from many of the same mistakes that preservice teachers make because they do not understand the complexities of the Teaching-Learning Cycle.

Given this line from the Time article, my greatest concern is that Mr. Khan is satisfied with his current understanding of teaching which perpetuates the idea that teaching is easy:

I think there is an advantage to being an outsider - I'm not colored by the dogma of the Establishment.

Which brings me back to the power of satire. The Time article was done prior to MTT2K hitting the scene. Let's see if it does anything to push Khan Academy to  improve its implementation. There are expert teachers willing to help. All Mr. Khan has to do is ask.

When do we stop chewing their food?

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's your problem? Part III

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.

What do you think about while running?

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 I

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 been 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.

How do you use the workshop model?

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.

How can we communicate our thinking?

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


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


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

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

When does it work?

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


We were using the clock model as a context for adding fractions and I wanted to gather data about whether or not the learners could determine when this model was an effective approach. I used an existing set of textbook items and asked the fifth-graders to: "Look at the expressions shown below - circle the ones that you think you could use the clock model to solve and place an 'X' through those you could not."

from Scott Foresman – Addison Wesley Math [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?

What support do learners need?

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

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

An anchor chart supports learning by creating a record that learners can refer to as cognitive demand increases. In Reading with Meaning, Debbie Miller writes, "Anchor charts make our thinking permanent and visible, and so allow us to make connections from one strategy to another, clarify a point, build on earlier learning, and simply remember a specific lesson (p. 57)." This anchor chart offered the fifth-graders support both as a representation for thinking as they computed the fractions and a representation of thinking that they could point to as they communicated their thinking to their peers.

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


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

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