How does it fit?

At the beginning of the semester, I wrote a post about using the history of the LEGO company as a cautionary tale for innovation in education reform. The idea came from a Diane Rhem interview with the author of Brick by Brick. But there was another idea presented in book that also caught my attention - clutch power.

When a child snaps two bricks together, they stick with a satisfying click. And they stay stuck until the child uncouples them with a gratifying tug. And therein lies the LEGO brick's magic. Because bricks resists coming apart, kids could build from the bottom up, making their creations as simple or complicated as they wanted. … it is clutch power that makes LEGO such an endlessly expandable toy, one that lets kids build whatever they imagine. (page 20)

It was the last sentence that got me thinking. How might I design learning experiences that use clutch power - allowing learners to "build whatever they imagine?" In particular, I had one of our math courses for preservice teachers in mind. It explores a variety of mathematical topics that students often see as disconnected: algebra, geometry, measurement, and data.  Was there a way to treat concepts in that course as building blocks that learners could use across the domains to build new understandings instead of simply following the instructions I provide?

I am still working on that question, but I did start experimenting with simple problems that allow learners to add their own ideas while demonstrating their content competency. The following is one problem (modified) from the final I just gave. I would be interested in your feedback.

Plot the points (2, 2) and (2, 4) on the coordinate plane. Pick two other points that could combine with these first two to be used as the vertices of a trapezoid. Use this context to develop items that would align with the following targets (be sure to justify your alignment using the indicators):

• Graph points on the coordinate plane to solve real-world and mathematical problems (5GA1, 5GA2)
• Classify two-dimensional figures into categories based on their properties (5GB3, 5GB4)
• Solve real-world and mathematical problems involving area, surface area, and volume (6GA1-A4)
• Draw, construct, and describe geometrical figures and describe the relationships between them (7GA1, 7GA2, 7GA3)
• Solve real-life and mathematical problems involving angle measure, area, surface area, & volume (7GB4, 7GB5, 7GB6)

Because this was a new assessment for most of my students, this time around I offered some example items to pick from:

1. Define the term "trapezoid" and then use that definition to identify what other names apply to your shape.
2. Find the area and perimeter of your shape.
3. Extend each side (creating 16 angles) and find all the angle measures.
4. Write your own question for this context.

I still expected them to align the item with the targets, justify their alignment, and come up with a correct response.

What's your plan to improve?

One of my favorite activities to do with teachers, both preservice and inservice is the Marshmallow Challenge. I view it as an excellent metaphor for planning and improving lessons. If you are unfamiliar with this activity, then you might want to watch this TED Talk before going any further.

Often, teachers fall into the same trap as others described in the talk. The teachers develop elaborate plans, create tall spaghetti structures, and wait until the last second to put the marshmallow on top (use this bomb timer to add suspense). Usually, the structure cannot support the added weight and it falls or breaks. 

So what does this have to do with lesson planning? I see the same thing happening in developing and implementing lessons - especially among new teachers. There is a great deal of planning beforehand in order to make things perfect. They work until the last minute, trying to get everything ready, but when the lesson finally encounters the learners it falls apart. Novice teachers are not the only ones to experience this phenomenon, however. Personally, I have had my fair share of lessons crash and burn despite massive preparation. In fact, I have wondered if the preparation was sometimes the problem.

Near the end of the talk, Tom asks, "What is your marshmallow?" For me it is a focus on student learning goals. With that in mind, and an awareness of how building on and improving prototypes increases the likelihood of success during the challenge, I have decided to try to plan differently. I want to start by designing a lesson with the least amount of support necessary to achieve the learning goals. Then I will try it out, gather data on whether or not the structure supports my goal, and make the necessary adjustments. If things go according to plan, I can create the next lesson/scaffolding using the same approach - start simple and add on as necessary.

This past week, it was interesting to watch two groups that contained members who had preformed the Marshmallow Challenge in other settings. Each followed the kindergarteners' approach of making successive prototypes. Where they differed was in their acceptance of their circumstances. 

One group kept adding on until the very last minute - continually tinkering with the prototypes. Unfortunately, they ran out of materials and the final structure was unsteady and fell over as the timer went off. A few minutes earlier it was standing tall:

The other group "finished" with about four minutes left. Given more time and resources, they might have been able to build something taller, but under the circumstances they were satisfied with their effort. They ended up besting the only other standing tower by about 6 cm.

It is only a metaphor, but I think the Marshmallow Challenge has a lot for us to think about as we consider improving education. Michigan is in the midst of considering plans for overhauling school funding (and therefore schools) but these plans have the feel of untried spaghetti towers not tested prototypes. As Stigler and Hiebert point out: 

Traditionally, Americans haven been more willing to accept dramatic failures than to applaud or even appreciate, small successes. (page 139 of The Teaching Gap)

If we are to succeed at this challenge of education reform, then we ought to heed the lessons of the Marshmallow Challenge.

How much ground can I cover?

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


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


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


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


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


In an effort to cover content I had ignored learning. I had ignored assessments. And I had ignored common sense. A truer picture would show the entire class being dragged behind me as I struggled to finish the textbook by the last day.

I am not suggesting that teachers should not make learning plans during the summer. Only that we recognize that they are plans and not scripts. So plan away but remember they will need adjusting based on your learners. And also make time to recreate this summer. A bike ride might be nice.

Why all the questions?

“The important thing is not to stop questioning.”

Albert Einstein

By now you may have noticed that each Delta Scape blog post title is in the form of a question. Why is that? I am glad you asked. Like most of what I do as an educator, it is intentional and built around ideas gleaned from multiple sources.

In the reading comprehension literature, asking questions is identified as one of the core strategies effective readers use. The questions readers ask serve two purposes: (1) Questions help readers to monitor if what they are reading makes sense; and (2) Questions propel readers deeper into the text. A person who reads a passage and asks, “What just happened?” or “What happens next?” is likely to be more highly engaged than someone who is just reading the words.

Questions are also an essential part of the Understanding by Design approach to unit planning. These essential questions take three forms: (1) big-idea questions; (2) key-content questions; and (3) making-sense questions. The goal is to design the unit by determining the essential questions that will frame an authentic and engaging learning experience.

In How to Solve It, Polya uses questions as a means to support problem solving in mathematics. Examples of questions a mathematician might consider during each phase of the problem solving process are provided in the book’s introduction. If you get a chance to watch Polya’s video, Let Us Teach Guessing, you will see him modeling the use of questions to work through a problem and make sense of it.

Asking Better Questions by Morgan and Saxton is another good resource. On page 27, they write: “Learning springs from curiosity, from the need to know.” The questions learners ask contribute to this need and their level of engagement. In order to support teachers and learners in increased involvement in any learning experience, the authors introduce the Taxonomy of Personal Engagement. I use a version of this taxonomy in the courses that I teach as a means of supporting my learners in monitoring their engagement and considering questions they could ask that would improve their involvement in the task at hand.

I hope this answers the question why all my posts have a question in the title. If not, feel free to ask your questions in the comments. Sorry, I couldn’t resist.

How do I plan for problem solving?

Almost a month ago, I wrote a post "Is direct instruction a better approach to teaching math?" that got a lot of attention (relatively speaking). My post was in response to an article which used one poorly constructed (my opinion) study to suggest that problem-solving or inquiry-based lessons were less effective than lecture-style instruction when it comes to standardized-test results. What seemed to get the most attention/ire was a comment by the article's author, Paul E. Peterson. 

"I, too, like those problem-solving classes. They require less preparation and are easier to teach."

This might be true for a tenured university professor who: (1) enjoys academic freedom; (2) has no "accountability" to a standardized, national test; and (3) does not believe in the problem-solving lesson as an instructional approach. But for the rest of us, a problem-solving lesson requires a great deal of effort. I want to share my process of preparation for such a lesson in this post.

Because I am also a university professor who enjoys those first two perks, I want to focus on a fraction lesson that I planned and taught a few years ago at a local elementary school. When the fifth grade teacher contacted me for help, she was very specific about the content that I needed to addressed in the unit. In Michigan, the driving force for most K-8 teachers is the Grade Level Content Expectations (GLCE) and for my series of lessons I needed to get at this standard: 

N.FL.05.14: Add and subtract fraction with unlike denominators of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 100, using the common denominator that is the product of the denominators of the 2 fractions.

My first step in planning was to gather data about my learners. Through a series of emails, I found out that this school grouped fifth graders by ability for math and that this teacher taught the lowest group. This included several learners with special needs supported by a special education teacher. While I am not a fan of ability grouping, this was not my fight. I was just grateful for the information.

Mathematics understanding is about experience not ability. It was up to me to plan a problem-solving lesson that offered learners an experience that would support their development of a relational understanding of fraction computation. Fortunately, I was familiar with an excellent resource that provides just such an experience. Planning a problem-solving lesson is not about developing activities from scratch (but if you have time and training, then this can work). Still, the planning does require effort in identifying appropriate resources and structuring them in such a way that they support learning.

Cathy Campbell wrote about this excellent resource on her blog. In particular, she discusses the clock model used for adding and subtracting fractions, which can be found in Minilesson for Operations with Fractions, Decimals, and Percents. Cathy does an great job describing this resource, so there is no reason for me to say much more except that I find its use of context and connections to prior successes very supportive for learners.

I am also fortunate to have the professional development packet that goes along with the series. This packet includes videos of teachers modeling some of the lessons. Before planning my lesson, I watched Joel teach the clock model, and it gave me some ideas of how to organize the lesson. In particular, it showed that he introduced the model in a whole-group setting.

Finally, I was ready to write out the plan. I decided to use a slight modification of a lesson planning framework Debbie Miller shared at the Michigan Reading Association Conference in 2008 and described in her excellent book, Teaching with Intention. You can view my plan here. As you can see, it is quite detailed, yet I do not consider it a script. I am a firm believer in Jon Stewart's approach to planning, "Creativity comes from limits not freedom ... When you have a structure, then you can improvise off of it..." (I still wish I had remembered to share that quote during my TEDx Talk). This detailed plan allowed me to make necessary adjustments as I taught the lesson, but that is for another time.

I hope this makes the point that planning for problem-solving is not easy. "Where's the problem-solving?" you ask. Let's compare the plan with the National Council of Teachers of Mathematics Process Standard for Problem Solving:

• Build new mathematical knowledge through problem solving;
• Solve problems that arise in mathematics and in other contexts;
• Apply and adapt a variety of appropriate strategies to solve a problem; and
• Monitor and reflect on the process of mathematical problem solving.

Please let me know in the comments if any of these are unclear in my planning.

When will they be ready to go on?

from Teaching-Learning Cycle

Assessments play a key role in the Understanding by Design approach to unit planning. They represent the way that I plan to monitor learners’ progress toward a goal. Therefore, I need to think carefully about what I am going to assess (objectives) and how I am going to assess (approaches) before the unit even begins. In other words, effective assessments require planning.

My first attempt at planning with the end in mind was a middle school math project I called the “Dream House.” I was also interested in using alternative forms of assessment and this project seemed a natural place to start. The summative assessment was a class presentation where each middle schooler shared a scale model of his or her dream house along with specific details like perimeter, area, and cost. This was a 9-week project that covered many of the Michigan state standards in geometry, measurement, and number & operation. And the grade on the presentation was their grade for the marking period. I made this video to model what the final product and presentation might look like:

In order to increase the likelihood of learner success, I set up assessments (I called them benchmarks) along the way. Again, I worked backwards in selecting and designing these assessments. I just kept taking a mental step back and asking myself, “What should they be able to demonstrate in order to do the next task?” and “How will I assess/gather data on their level of understanding?”

As you can see in the example to the right, my plan included some traditional forms of assessments (highlighted in yellow). Learners needed to demonstrate proficiency on the skills assessed by these quizzes before they were ready to move on to the next benchmark. Our school determined that 80% or better was required, but you will remember this did not figure into their final grade; it was merely to monitor progress and ensure success.

This video represents a sample of the eighth graders' efforts on the Name Plaque benchmark:

The planning and preparing for this unit took a great deal of work. Fortunately it was the first marking period in the fall, and I had all summer to work on it. Still, many of the quizzes I used were not developed from scratch but modified from our school’s math text. There was no reason to reinvent the wheel and slight modifications offered the data I needed to analyze understanding and make instructional decisions. After all, these were formative assessments.

I point out my use of textbook quizzes because I had a teacher assistant ask me if using existing curricular resources is “cheating.” Another education class had suggested that using such resources is shameful because teachers should and can create better materials for their learners. Maybe with more time and experience we could make better tests and quizzes from scratch. But why would we if what’s available requires minor, meaningful modifications in order to accomplish our goal? It just depends whether or not it fits within our plan.

How do I plan for success?

Before I was a teacher, I was a computer programmer. It wasn’t a long career. An unsatisfying summer internship saw to that. But to this day, I am amazed at how much my programming experience impacts my teaching – especially when it comes to planning units.

My computer classes at Pepperdine University taught me to write programs using an approach called Top-Down Design. I start by developing a clear picture of what the program is meant to accomplish. Next, I put into place an output protocol that allows me to ensure that the program is working correctly. This is quality assurance. With this end in mind, I begin the process of writing code that meets the desired outcome.

Depending on the size of the program, this can be an arduous chore. Therefore, the Top-Down Design approach suggests thinking of the program as a series of smaller tasks. I simply write “black box” procedures or functions to accomplish these tasks that I fill in later. Each of these “black boxes” includes some output protocol that allows me to monitor the progress of my program and make adjustments as needed.

Now I am ready to write the code for the “black boxes.” As I make my way through writing the program, I execute test runs during which I check the outputs I put in to monitor the program’s progress. That way, when problems arise I can address them immediately. If I wait until the end to see if the program works it is often too late or too unmanageable to make necessary corrections. (This reminds me of Tom Wujec's TED Talk.)

Once the program is complete it is time to put it to the test with real, messy data. Chances are there will be some bugs that I didn’t anticipate. Fortunately, if I wrote the monitoring outputs correctly, then I have feedback regarding where the problem is and what I need to do about it. More often than not, this results in a successful program that accomplishes its goals.

What does this have to do with unit planning? Those of you familiar with Understanding by Design probably see the connection between the Top-Down Design approach to programming and the essential elements of the unit planning design developed by Wiggins and McTighe. Anyone struggling with the connection between programming and unit planning might replace “program” with “unit,” “output” with “assessment,” “black box” with “lesson,” and “real, messy data” with “learners.”

If not, maybe this diagram might help. The summative assessment is the goal of the program. Formative assessments represent the output protocols that I will use to monitor learners' progress toward the goal. And the experiences are the lessons that I will develop that supports this progress.

I learned to write unit plans after I learned to write programs, but I initially did not see the connection. I planned my units like I was planning a parade - one lesson after another until it was time for a test. And then the test only checked to see if the learners had paid attention to the parade. Planning with the end in mind from the start makes so much more sense and results in a much more coherent curriculum.

I hope you plan for and find success.