Drawing Area

On Monday, Drawing Area won the WGBH Interactive Math Challenge and will eventually become part of the Math at the Core: Middle School collection at PBS LearningMedia.

Derek, a co-organizer of an instructional design meetup I attend, asked me if I'd be willing to share my thinking behind Drawing Area with his instructional design students. I replied by joking that there wasn't much thinking behind it. But, of course, that isn't the case.

In Drawing Area, students find the area of a polygon by drawing the polygon on a grid with rectangles and right triangles. This happens in a browser window. The idea for Drawing Area came from a series of lessons I developed when I was a classroom teacher. For those lessons, students drew their polygons on sheets of graph paper.

To find the area of a polygon, it helps if you can decompose the polygon into rectangles and triangles. However, I found some of my students doing things like this:

This student has decomposed the polygon into three triangles, but the area of the middle triangle isn't going to be easy to find. This student either hasn't understood why he or she is decomposing polygons into rectangles and triangles, or is unable tell at a glance when a rectangle's or triangle's area is going to be easy to find.

You could ask the student to find the base and height for each triangle. And then, when the student can't find the base and height of the middle triangle, ask the student to try again by decomposing the polygon another way. However, that is going to get old pretty quickly.

You could give the student a rule to follow: only make horizontal or vertical slices starting from a vertex. Over time, the student may make sense of the rule and internalize it.

I elected to take a different approach. I decided to draw the polygons on a grid and ask the students to find the number of squares the polygon covered.

When you are counting squares, it is natural to divide a polygon along grid lines. You aren't decomposing the polygon into rectangles and triangles so that you can apply the area formulas for rectangles and triangles; it is simply easier and more efficient to count squares when they are grouped into rectangles (an array of squares) and right triangles (half of a rectangle). This instantly makes sense to students and is something that they come up with on their own.

Once a polygon has been decomposed into rectangles and triangles, some students have a hard time finding dimensions that are not given. This is not an issue when a polygon is drawn on a grid (the student can simply count the number of squares along an edge), but it is an issue when a grid is not provided. What is the length of the base of the triangle?

It is not uncommon for students to pick two numbers in the drawing and add or subtract them. They may see the 12 and the 7 and think: 12 − 7 = 5. It doesn't matter that this makes no sense whatsoever. They know they are suppose to either add or subtract two numbers to find the unknown length.

You could guide the students to focus on the relevant information and eliminate distractors. Since the student is trying to find the length of a horizontal line, have the student highlight all horizontal lines and eliminate the dimensions from all vertical lines:

Is there a way to use the information provided to figure out the length of the middle red line? Some students will make sense of and internalize this over time, but you are still subtly reinforcing the notion that you should be looking for things to add and subtract.

Since the students in my classroom are already comfortable finding the area of polygons drawn on a grid, I decided to leverage that. Can you take this polygon and draw it on a grid? Most students think that's easy. And once the polygon is drawn on a grid, they can find the area.

A student may start by drawing the rectangle to the left, since those dimensions are given.

The student knows that the height of the second rectangle is 6 cm, but the length of the base and its exact position relative to the first rectangle are unknown.

But once a rectangle with a height of 6 cm has been drawn, it can be moved so that its base is aligned with the base of the first rectangle.

Then it can be resized so that the width of the entire polygon is 15 cm.

The right triangle fits right in between the two rectangles, and now any dimensions that the student needs can be determined from the drawing.

After a while, I would encourage the student to predict the dimensions of the individual rectangles and right triangles before drawing them, and then confirm those predictions with the drawing. At no point is the student following rules or doing something that doesn't make sense. The emphasis is on figuring out the number of squares a polygon covers. Using this process, all students can reason through increasingly complex problems:

Where to Next?

My career has been a twenty-year journey toward a new normal. On the first leg of my journey, I developed a sense-making curriculum. On the second, I formed a learning community in the classroom. On the third, I discovered what students could do if they were immersed in a sense-making curriculum and a learning community for three years. On the fourth, I coached teachers to use a sense-making curriculum and begin forming learning communities in their own classrooms. On the fifth, I tried forming learning communities among teachers.

I haven't quite finished that fifth leg yet. One thing that I've learned during my journey is that you need to prepare for a leg long before you get to it, and I wasn't prepared for the fifth leg. First, I couldn't articulate how to design a sense-making curriculum, so the teachers couldn't really share in ownership of the work. That is a killer. Second, a learning community among teachers needs a supportive environment. Ideally, it would be part of a larger learning organization, with learning communities at all levels. I never figured out how to engage administrators in the work that teachers and students were doing, so there was always a lack of buy-in at the leadership level.

I fully intend to tackle that fifth leg again. I can't complete my journey without it. But before I do, I want to survey what lies beyond it, so that I can start preparing for the sixth and seventh legs at the same time. This is why strategic planning is so important.

It seems clear to me that, at some point, I need to create a school to demonstrate that a new normal is possible. This school would be a learning organization (adaptive, focused on continuous improvement, learning communities at all levels) and have a sense-making curriculum. By enabling students to make sense of the curriculum, students will climb faster and farther, take ownership of their own learning, and develop the habits of mind and skills they need to continuously test and revise their mental models. These students wouldn't just go on to study at the four-year colleges of their choice, they would go on to become the best thinkers of their generation.

Performance talks, and the performance of this school and its graduates would be off the charts. This would be an Olympic stage: a signal so unmistakably strong that it would cut through the noise. I firmly believe that I can create this school and achieve that level of performance. And I don't say that lightly. My entire career has been a series of tests to prove or disprove this hypothesis.

But I can't create a school by myself. I need a team. Fortunately, I don't need an Olympic stage to form a team; I can go out and enlist people one by one. The hard part is figuring out who the right people are and what evidence will convince them to take me seriously. The right people are few and far between, so I've been following a two-part strategy. While I'm looking for them, I'm providing ways for them to find me.

Vertical Learning Labs is producing curriculum in order to generate income so that I can sustain my work. But these curriculum products are also a trail of bread crumbs leading back to me. If you can look at Drawing Area, Chocolate Chip Cookie Factory: Place Value, Chemistry from the Ground Up, or Teaching a Robot How to Dance without being immediately blown away by what these pieces of curriculum can do for students, then you probably won't fit on my team. These efforts are crude, but I'm looking for the people who can see the genius in them and can see what they can be with refinement. If those people are out there, they should be as hungry to meet me as I am to meet them. That's my working theory for now, anyways. I sure hope I hear more than crickets out there.  :)

You Are Here

So, where am I now? That is an excellent question.

I got a B.S. in chemical engineering at Carnegie Mellon in 1991, and went to graduate school at the University of California at Berkeley for a year before dropping out to pursue my passion in education. After getting a M.A.T. in math education from Boston University, I started teaching middle school math and science on a two-person team in Attleboro in 1995.

Attleboro was an incredibly creative time for me. The superintendent had banned math textbooks, so I was making up my own curriculum on the fly. I remember designing a unit on fluid pressure and the particle theory of matter. In this unit, students learned by conducting a series of hands-on investigations. One of the initial investigations involved using a digital camera to record video of collisions. By studying the video frame-by-frame (we were able to capture grainy video at 320 x 240 and 5 fps), students applied vector concepts to discover the conservation of momentum. I was appointed to the District Math Curriculum committee and hired to run a physics workshop for teachers over the summer.

In 1998, I started teaching 7th-grade math and science in Brookline. Technically, Brookline was using MathScape for its middle school math program, but I didn't particularly care for it. I managed to convince another new teacher, Charles, to join me in creating our own math curriculum. The parents in Brookline were very demanding, but they were pleased with the results, so we were supported by the principal. Charles and I also shared a classroom, so we got used to observing and critiquing each other's lessons. We refined the curriculum over two years.

While we were getting good results, we were also doing our work in isolation. Imagine that this is a graph of student performance over time. Our students would make higher than normal performance gains using our curriculum, but those performance gains would largely be erased once they were back to using a normal curriculum.

This was still better than a typical intervention. In a typical intervention, there is an initial bump in performance that quickly levels out, and then disappears once the intervention is over. At least with our curriculum, students would make performance gains the whole time they were using it.

On the science side, I was learning to take my curriculum vertical. The math curriculum that Charles and I developed still introduced concepts in isolation; in science, the concepts built on top of each other. I had done this in Attleboro with the fluid pressure unit, but that was a two-month long series of investigations. In Brookline, I was connecting all of the earth and life science concepts in the 7th-grade curriculum in a yearlong series of investigation, using the Big Bang and chemistry as the starting point.

Brookline was partnering with the Virtual High School at the time, and my principal asked me to design an online science course for them. The science course I designed was based around the challenge of programming a spaceship to navigate through a two-dimensional course (up-down, forward-back). The spaceship was essentially a tank of high-pressure gas. You piloted the spaceship by opening and closing nozzles. When a nozzle was open, gas would flow through the nozzle, providing thrust. Over time, the pressure in the tank would drop and the mass of the spaceship would decrease. Students would use experiments to learn the science concepts, and they would use spreadsheets to model the spaceship. All of their calculations would be based on discrete time steps, so calculus was not required and all relationships would be linear. (Students would be able to adjust the size of the time steps, so they would be working with limits.)

The other significant development that occurred while I was at Brookline was the formation of my first learning community. In my first year at Brookline, the students in my homeroom also had me for math, science, and study hall. I didn't know it at the time, but these students had two teachers walk out on them in 4th- and 5th-grade. So, they tested me. They tested me to see if there was anything that they could do to push me away. Somehow I hung in there, and managed to break through with them in December.

For me, a learning community is a community whose primary focus is learning. That means that learning is more important than worrying about making mistakes or looking bad. It means taking ownership of your own learning, approaching new experiences with a sense of inquiry, and pulling together to help other members in the community to learn. These students regularly entered the flow state and tested their own understanding. They wanted to make sense of what they were learning and believed that they could.

In 2001, Charles told me that his wife's school, the Jewish Community Day School (JCDS) was looking for a middle school math teacher. The middle school had less than forty students in it at the time, so I would be the only math teacher in the middle school. Charles didn't think I'd be interested, but I jumped at the chance to see what our math curriculum could do when scaled across three grade levels.

JCDS was very progressive and student-centered, so they were using Investigations in the elementary school and Connected Mathematics (CMP) in the middle school. But they were also based in Newton so expectations were very high. The expectation was that all students would take Algebra I in 8th-grade and some students would take Algebra I in 7th-grade, but students had much less time for math because they had to take so many additional subjects, such as Hebrew and religious studies. This meant that using CMP in the middle school was going to be unworkable. Most public schools have a hard time getting through the CMP curriculum even with extra instructional time and without having to complete Algebra I by 8th-grade.

I took the math curriculum that Charles and I had developed in Brookline and adapted it for JCDS, using the vertical learning principles I had applied to the science curriculum in Brookline. Being able to build concepts across three grade levels enabled me to design my first sense-making math curriculum. By going vertical and investing in student understanding early on, I helped students climb faster and farther in the long run.

After working with the same group of students for three years, I saw impressive performance gains for the entire three years. When those students left JCDS for high school, their rate of performance growth returned to normal levels, but they didn't lose the gains they had made at JCDS. This indicates that these students were able to build on the understanding they had developed at JCDS, even in environments that didn't natively support sense-making.

After three years of making sense of things in math, these students expected to make sense of things in all subjects, but they were still dependent to some degree on their teachers helping them. Ideally, students should be able to develop the sense-making skills they need to make sense of things completely on their own.

By this point, I had concluded that a sense-making curriculum works. The next step was to see if I could bring it to scale. I left JCDS in 2004 to become middle school math curriculum leader for the Groton-Dunstable Regional School District (GDRSD). The administration at GDRSD wanted the middle school math teachers to adopt a standards-based math curriculum and I wanted to see if I could lead the change process.

The math department was bitterly divided and none of the previous curriculum leaders had been able to achieve any form of consensus. My job was to guide a committee of teachers in selecting a new math program for the middle school by the end of the year. I started capacity-building right away. I wanted everyone to have a deeper understanding of how students learn and how standards-based math programs work (the math wars had kicked up a lot of FUD). We established the criteria we would use to evaluate math programs and started collecting data. As part of the process, I asked teachers to try some different approaches. One of the 7th-grade math teachers who was firmly in the "traditional" camp ended up raving about a functions lesson I had designed for him. He observed students actively taking ownership and reasoning through problems instead of passively waiting for him to tell them what to do. I also secured a commitment from the administration that the committee would select the new math program and not simply make a recommendation to the administration.

In the spring, two clear favorites had bubbled to the surface: CMP 2 and Math Thematics. Both programs were standards-based math programs, and I thought both were good choices. The traditionalists preferred Math Thematics because the program was structured more like a traditional textbook. The progressives preferred CMP 2 because it focused more on open-ended problem solving. The administration wanted CMP 2, but the committee ended up deadlocking 4-4. With both sides digging in, the administration felt justified in breaking the tie and choosing CMP 2 itself.

I begged the administration to give the committee one last chance; I felt that we were on the verge of consensus. So, we gathered for one last time at the end of June and I ran through the possible outcomes. We were about to make a decision that the teachers were going to have to live with for the next ten years. We could make the comfortable choice and hope that Math Thematics would encourage students to become independent problem solvers, or we could go all-in and take a risk on CMP 2. The traditionalists had already acknowledged that the primary benefit of Math Thematics is that it made the transition to a standards-based math program smoother for teachers (which is important). The 7th-grade teacher who had tested the functions lesson stood up and decided to change his vote. His 7th-grade colleague stood up and agreed. If CMP 2 gave them a slightly better chance of encouraging students to become independent problem solvers, than it was worth the risk and greater discomfort for teachers. The committee voted 7-0-1 for CMP 2.

We were doing a staggered roll out and building in lots of professional development, so there was still lots of work to be done. I was drawing up budgets and three-year implementation plans, presenting in front of the school committee, working with vendors and consultants, and coordinating everything. It was fun and exciting, but the most exciting thing for me was the potential of creating a learning community among the teachers. The breakthrough we had in June felt similar to the breakthrough I had had with my students in Brookline.

Unfortunately, the administration didn't share my enthusiasm and optimism. My boss, the Director of Curriculum, hadn't experienced the same things that the committee had experienced because she hadn't been there day-to-day, so she assumed that the teachers' buy-in was temporary. She didn't realize that their buy-in was actually at the ownership level; she probably couldn't even conceive of that as a possibility. So, instead of building on their commitment and establishing a learning community where we would all work together toward a common goal, she seized on their commitment to leverage her own agenda.

In my mind, we were removing one script (the traditional textbook) and exchanging it for a slightly better script (CMP 2). But how can you believe that teaching from a script is good for teachers when solving problems from a script is bad for students? Many people are able to live with that level of cognitive dissonance, but I can't. I felt like I had betrayed the teachers by unwittingly participating in a bait-and-switch on them.

In 2007, I became middle school math and science curriculum specialist in Holliston. I won't go into a ton of detail since I wrote about my experiences in Holliston here, here, and here. My goal was to see if I could implement a sense-making curriculum designed by the teachers themselves and to create the learning community I failed to create in GDRSD.

In my first year in Holliston, I became fast friends with Jessica, the principal of the middle school. We worked closely together and she valued my insights. It was a highly collaborative relationship driven by inquiry. When Jessica became assistant superintendent for the Freetown-Lakeville Regional School District (Freelake), she immediately hired me as her K-8 Math Program Coordinator. I was highly impressed by the team that John, the superintendent, was putting together and I relished the opportunity to work with Jessica again, but this time, at a district level.

Unfortunately, Jessica had other plans. She admitted to me that she only listened to me in Holliston because she was about to flame out there and needed me to bail her out with the staff. Now that she was at Freelake and had carte blanche from the superintendent, she wanted to do things her way. While I can respect that on some level, she should have told me that before luring me away from Holliston. I would have never have gone to work with her under those conditions.

The whole year in Freelake was a complete disaster. Jessica, the superintendent, and the consultant they brought in from UMass Lowell spoke passionately about forming a learning community. I think that they, to this day, genuinely believe that they are true believers of learning communities. But the administrative team at Freelake was no learning community. John and Jessica had their agenda and they used the pretense of a learning community to ram it down our throats. Their goal was to win and then leverage commitments from us and then the teachers. When I objected, the learning community ran roughshod over me. I left when John and Jessica used me to get a group of young teachers to go out on a limb, and then undercut the initiative behind their backs. It was GDRSD all over again, and all though I was wiser and more experienced, I couldn't do anything to stop it. That's when I decided to abandon schools for a while and start up Vertical Learning Labs.

Wrap Up

So, where am I now? That is the question that opened this post and it is where I'm going to end. I haven't reached a new normal, yet. I don't even think it is visible from my current vantage point. But I've been to the foothills of the new normal and I'm as confident as ever that it's there. And if it isn't there, I already know that the foothills I'm on are much taller than the tiny cluster of hills that everyone else is on. So this is a good place to be.

I have developed a sense-making curriculum for middle school math and components of a sense-making curriculum for science. The sense-making curriculum for math works for kids. It's been tested and refined. It's also been designed to fit within the constraints of the current system: 50 minute classes/5 days a week, Common Core standards, high school math courses, etc. Even when I designed a sense-making curriculum for JCDS, my priority was preparing students for a seamless transition to the elite high schools in the area. As we move together toward a new normal, those constraints will fall away and I'm excited to see what people can create from scratch.

Other teachers can and will use the curriculum that I've written, and get similar results. A small number of teachers will fly with it right away. The majority will need coaching from someone like me.

I'm working toward articulating a process that other teachers can use to create their own sense-making curriculum. That is a work in progress and I've got a long way to go.

I can consistently create learning communities. I stumbled onto my first one in Brookline, but I've been consciously creating them ever since. It's fairly easy to do in the classroom when I have students in front of me 50 minutes a day/5 days a week. It is harder to do with teachers, whom I may have in front of me only 50 minutes a day/one day every two weeks (10% of the contact time I have with students). I haven't actually managed it with a group of teachers yet, but I have gotten close in GDRSD and Holliston. Doing it with administrators is much tougher because I get much less contact time with them, and when we do have contact, it is usually around their agenda instead of mine.

I'm not yet at a point where I've started to articulate how I create learning communities. In the classroom, I think that a sense-making curriculum is a great start, but other components are definitely necessary.

Finally, I know that I can't convince anyone through words. I've always suspected that, but banging your head against that particular wall for over 15 years brings another level of clarity. The flip side to that is that anyone who has worked closely with me for any period of time walks away with a much deeper understanding and appreciation of what I do. It is rare for me to not win someone over when they experience firsthand what I can do and what I'm about. The question, as always, is: How do I bring that to scale?

Toward a Compelling Argument

In Pulling It All Together, I laid out my vision for teaching and learning, and what I believe it takes to get there. The first step is creating enough cognitive dissonance to convince a few hardy souls to join me on a journey. I don't think that words are going to be enough; I think that you need to experience something profound before your perspective widens enough to see the mountains I see. I have a couple of things in the pipeline that might accomplish that. But words help. They especially help after someone has had that profound experience and is eager to learn more.

Getting someone to hear, never mind accept, all of the arguments in Pulling It All Together is unlikely. I want to start by making one compelling argument. I'm confident that, if I can make the case for one argument, then the rest of the arguments will follow like dominoes. But which one? That's where I need your help. Which argument was the most compelling or created the most cognitive dissonance for you? Which argument resonated the most with you and made you want to learn more? That's the argument I'd like to refine.



Argument #1: We need a new normal. We need a shift in core beliefs and some of the things that we accept today need to become unthinkable tomorrow. Creating oases of inspired teaching is not good enough. We need wide-scale change that touches everybody and everything.


Argument #2: We need a sense-making curriculum. It is completely unacceptable to force students to endure a curriculum that makes no sense to them. Students cannot take ownership of their own learning, engage with the content, or problem solve if they can't make sense of the curriculum. All students can make sense of the curriculum and relying on instruction alone is not good enough.


Argument #3: Incremental progress is not good enough. We are trying to roll massive boulders up small hills without recognizing that the peaks of those hills are not nearly high enough to get us where we want to go. We need to find another set of hills that are taller. None of the current reform efforts are working.


Argument #4: You need cognitive dissonance to get someone to stop what they are doing and widen their perspective. If they don't experience something that makes them consider that the impossible might just be possible, then you aren't going to get them to set off in a new direction. They might follow you to be compliant or professional, but they won't take ownership of the journey.


Argument #5: We need a compass. We need a compass that will give us a clear heading when we are lost in the daily grind and don't have time to look up and gain perspective. The compass should be a good proxy for all of the metrics we care about. It should have a strong and compelling signal. Everyone in the party should have a compass so that we can work independently while all moving in the same direction. We need to internalize the compass over time.


Argument #6: We need to follow a line of mountains. Each mountain enables us to set a heading and recalibrate our compasses. The line of mountains enables us to move from one peak to a taller peak. Each time we scale a taller mountain, our confidence and capabilities grow and more people join our party.


Argument #7: We need hands-on and shared leadership. Each leg of the journey prepares us for the next leg. Each leg is arduous and requires strong leadership and good decision-making along the way. School leaders need to be on the ground the entire time or else they won't be prepared themselves. You can't just check in periodically. You can't turn things over to consultants. We have to be in it together.

Pulling It All Together

Over the past two months, I've attempted to lay out my thoughts on teaching and learning. Today, I am going to try to pull it all together.

The Need for a New Normal

I believe that we need a new normal. A new normal is reached when things that are accepted today becomes unthinkable tomorrow and when things that are unthinkable today become normal and expected tomorrow. New normals are not reached through incremental progress. They are disruptive and occur suddenly and without warning.

Here are some things that I'd like to see in a new normal:

• All students making sense of the curriculum.
• All students taking ownership of their own learning.
• All students reasoning through complex and meaningful problems.
• All students and teachers engaging in learning communities.
• All teachers seeking out, receiving, and applying constructive feedback from peers, supervisors, and coaches.
• All teachers engaging in inquiry, collaborating, and sharing practices.
• All school leaders acting as instructional leaders.
• All schools focusing on continuous improvement.
• Everyone engaging in an open and honest dialogue on teaching and learning.

This list is far from comprehensive. It is impossible to know what a new normal is going to be like until you get there. But it does remind me of how far we have to go. Today, most principals are prevented from acting as instructional leaders because that isn't how their jobs are structured; most students are prevented from making sense of the curriculum and solving meaningful, complex problems because that isn't how the curriculum is written; and most teachers toil in isolation because we believe good teachers are born and we only need them to teach from a script. If your child does get an inspired teacher or a piece of curriculum that makes sense, you count yourself lucky. This is all backwards. We won't get better until we expect better.

Making Sense of the Curriculum

I believe that, in any new normal, all students must be able to make sense of the curriculum. Imagine that you are a third-grader sitting in math class, and nothing makes sense to you. You study as hard as you can and your parents get you a tutor, but it all seems random to you. Meanwhile, some of your classmates get everything right away. Now imagine that you experience this 50 minutes a day/180 days a year for the next ten years. How corrosive is that?

Some schools attempt to insulate you from the corrosiveness. Not making sense of things is easier to cope with when you are surrounded by adults who know and care about you and when you are striving toward a personal goal, such as college. But you can't take ownership of your learning and learn independently if the stuff you are learning doesn't make sense. You are still relying on an authority to tell you what to know instead of knowing for yourself, even if that authority is a journal article and not a classroom teacher. Without an expert mental model, you won't be able to integrate what you know and solve complex problems.

Schools should be in the business of helping students make sense of stuff so that those students can make sense of more stuff on their own. Protecting students from the corrosive effect of a nonsense-making curriculum isn't enough.

The Need for a Sense-Making Curriculum

I believe that, to help students make sense of the curriculum, we need a sense-making curriculum. Learning is a two-step process: you experience something (external) and then you try to make sense of it (internal). If you can't make sense of something using your existing mental models (cognitive dissonance), then you revise your mental models and learning occurs. Most curricula focus on the learning experience, leaving students to make sense of the experience on their own outside of the classroom.

It is possible to make sense of a curriculum that doesn't make much sense. I did it. I hated not understanding the stuff I was learning in school, so I kept turning things over in my mind until they did make sense. Making sense of stuff takes time and effort, but it also requires certain skills. Those skills can be learned.

Right now, most educators are focused on amping up student engagement in order to get students to make sense of the curriculum. The theory is that if students are more engaged, they will put more time and effort into making sense of learning experiences. However, this theory presupposes that all students already have the required sense-making skills, which, based on my observations, is not the case.

In my opinion, we need to increase student engagement and lower the barriers to sense-making that are built into the existing curriculum at the same time. Relying on instruction alone is like trying to do a job with one arm tied behind your back. The curriculum itself should also support sense-making and encourage students to develop those skills. Student engagement is strongest when the curriculum makes sense and students feel capable and powerful.

Current Reform Efforts Are Not Working

Most current reform efforts rely on making incremental improvements. This is reasonable since most progress is incremental. Incremental progress is like climbing a hill. To make incremental progress, all you need to do is study the terrain in front of you and keep going up.

I believe that current reform efforts aren't working because we need a new normal and incremental progress won't take us to a new normal. To reach a new normal, we need to widen our perspective. The terrain tells us to move in one direction, but we need to ignore it and head for a point on the horizon instead. This means going downhill at times and trusting in our heading if we go downhill and lose sight of the horizon.

I find it interesting that it is now 2014 and no one is talking about how 2014 is when all public school students in the U.S. are suppose to be proficient or higher on state tests. On the 2013 MCAS (Massachusetts' state test), 55% of our eighth-graders were proficient or higher in math. I don't think that we are going to make it. (In tenth-grade, 80% of our students were proficient or higher in math, but standards were lowered on the tenth-grade tests when they became graduation requirements; it would have been political suicide to deny half of our high school students diplomas.)

The No Child Left Behind Act (NCLB) imposed severe sanctions on schools that were not making adequate progress toward the 2014 goal. If a school did not make adequate progress five years in a row, then the state could take them over and fire the entire staff and administration. These sanctions were designed to motivate school districts to take drastic action. In the early days of the Act, Massachusetts did take over a number of failing schools, but the state could not turn them around even with massive infusions of resources. State officials now know that they can't turn failing schools around, yet they are still expecting school districts to do it themselves. Unfortunately, since almost all of the middle schools in the state are now failing, no one is that worried about being sanctioned and the stigma from being a failing school is gone.

Is anyone doing any soul searching over this massive failure? One high profile proponent of NCLB, Diane Ravitch, has publicly admitted that accountability and competition from charter schools aren't working and aren't going to work, but everyone else seems to be doubling down. Instead of relying on each state to design their own curriculum standards and state tests, this will now happen on a national level with the Common Core and PARCC. But wait a second, which state is one of the primary models for this national effort? Oh, that's right, Massachusetts. D'oh!

One other thing that the federal government is doing is tying teacher evaluations to state test results, which means that teachers will get fired if their students perform poorly on the PARCC. This is the graduation requirement thing all over again. Policymakers are trying to raise the stakes for accountability hoping that schools will respond. They are doing this even though they have no idea how to do what they are asking the schools to accomplish. They are basically playing a giant game of chicken with the schools for the third time, without realizing that the car the schools are in does not have a steering wheel in it. No one genuinely believes we are going to get anywhere close to 100% of our students to proficient or higher on state tests, but we keep our heads down and keep trudging up the slightest incline instead of recognizing what we don't know and looking around for a new direction.

I wish I could say the progressive movement was in better shape, but their incremental approach isn't working either. The state of the art in progressive education hasn't advanced much, if at all, since the early 1900s, and the progressive movement has never been able to expand its base by writing educational philosophy, publishing research, and opening demonstration schools. Progressive educators will flock to a progressive school, creating a temporary beacon of student-centered instruction, but it is a zero-sum game. The progressive movement had more momentum, influence, and flagship schools in the mid-1900s than it does today, but that still wasn't enough to convince traditional educators to cross battle lines. Because of community backlash during the math wars, I would say that there are fewer self-identified constructivist math teachers today than when I first started teaching in 1995. If your goal is to create a few oases in the desert, that's fine, but it isn't going to help you reach a new normal.

The Need for Cognitive Dissonance

To reach a new normal, I believe that we need to create cognitive dissonance first. Cognitive dissonance is what causes us to re-consider and then revise our mental models. Without cognitive dissonance, we tend to repeat existing patterns without reflecting on or evaluating them.

But cognitive dissonance isn't easy to create. When we experience something that doesn't fit our mental models, our first reaction is to make it fit, even if we have to distort and ignore evidence to do it. The threshold to create cognitive dissonance varies, but it tends to be very high when you get anywhere close to the instructional core. I usually start with the impossible: getting a group of students to learn something that no one thought they could learn. That is the threshold I need to cross in order to get a staff to even consider making a change and moving in a new direction. If you are asking someone to take on the impossible, it helps if you can show them that it might just be possible.

When states first started taking over schools that weren't making adequate progress, they had an opportunity to do the impossible. All they had to do was demonstrate that they could take a failing school and get 100% of the students to proficient or higher. They couldn't do it, which only confirmed everyone's belief that no one could do it.

The result is a climate where all schools buy the same curriculum, hire the same consultants, and launch the same initiatives. This herd mentality is the opposite of what NCLB was intended to accomplish. Accountability was suppose to generate urgency, which was suppose to generate innovation. But who is going to take a risk when there is no upside because there is no chance of success? When the goal is an impossible one, the best thing to do is to not stick your neck out. And without variation and experimentation, there's no chance for a new normal.

The Need for a Single Metric for Performance

Okay, the staff has agreed to set off in a new direction in the hopes of reaching a new normal, but how do we get there? No one has been to or even seen this new normal before, and the path we'll be blazing descends into a fog-enshrouded valley and will include numerous switchbacks and detours to get through the rugged terrain. How do we orient ourselves and keep everyone together?

To reach a new normal, I believe that you need a compass. The compass is a single metric of performance that functions as a proxy for all of the metrics you care about. We want all students to make sense of the curriculum. We want all students to take ownership of their own learning. We want all students to reason through complex and meaningful problems. We want all students to engage in learning communities. But if we try to pursue all of those goals at the same time, the party will split as people head off in different directions. Someone will shout, "The path is this way! This increases ownership." And someone else will shout, "No, the path is this way! These problems are more meaningful." People will get lost and end up walking in circles as they switch between different metrics.

Choosing a single metric gives you a consistent heading, but which one? The one I've settled on for the moment is sense-making. I feel that sense-making is a good proxy because it is so tightly coupled with the other metrics, either as an input, output, or both.

• I believe that students will take ownership of their own learning once they have made sense of the curriculum.
• I believe that students will have to be able to reason through complex and meaningful problems in order to make sense of the curriculum, and that students will be able to reason through complex and meaningful problems once they have made sense of the curriculum.
• I believe that students and teachers will have to engage in learning communities in order for all students to make sense of the curriculum, and that students and teachers will choose to engage in learning communities once students are making sense of the curriculum.
• I believe that teachers will have to receive and apply constructive feedback in order to help all students make sense of the curriculum, and that teachers will seek out constructive feedback once they see that some, but not all, students are making sense of the curriculum.

I could go on, but I think you get the point. In my opinion, it would be impossible to reach the goal of all students making sense of the curriculum and not reach all of the other goals at the same time.

I've also settled on sense-making as my compass because it has a strong and compelling signal. Feedback in educational systems tends to be extremely noisy, so you want a metric where you can see clear and significant improvements. In my experience, small increases in sense-making yield large performance gains on student outcomes. Another factor is the level of confidence a staff has in the compass. If you try to use the development of collaboration skills as your compass, some teachers won't get on board with that because they don't think that collaboration skills are important enough to risk such a long and perilous journey through the unknown. Remember, there is going to be some point where it is dark and pouring rain, you can't see more than two feet in front of you, everyone is convinced the party is lost, everyone is hurt and close to collapse, and people are crying to turn around to go home. That is when everyone is going to have to suck it up and trust in the compass.

Finally, a compass enables everyone to take shared ownership in the journey. Instead of relying on a single navigator, everyone will have a compass on them. Small scouting parties can go out to explore potential routes and forage for food without getting lost or getting separated from the main party. To reach a new normal, you need shared ownership and you need everyone working in the same direction. Innovation always occurs from the bottom up, never from the top down.

Following a Line of Mountains

Relying on a single metric for performance has some obvious risks. If it is a poor proxy for your other metrics, it can easily lead you astray. You don't really want to cut the entire arts program just to squeeze out a few more points on the math MCAS, do you?

The compass helps you orient yourself when you are in the middle of your daily grind, but you still need to make time to reflect and think about where you are and where you are going. Whether it happens once a month or once a year, the staff should come together periodically to review the entire journey and make course adjustments if necessary. This is when you can, and should, recalibrate your compass. Using a proxy is a bit like using magnetic north to find the north pole. The closer you get to your destination, the bigger the adjustment you'll have to make. However, it's really handy when you are first starting out.

Cognitive dissonance is used to get people to widen their perspective. Unfortunately, a new normal is too far away to be seen by even the most eagle-eyed observer from the current normal. So how do we pick a heading for our compass and recalibrate the compass along the way? The key is finding a line of mountains that will guide you in the direction of the new normal. The first mountain is close enough to be seen from the current normal. The second mountain, which is a little taller, can be seen from the first mountain.

For the first mountain, you may try to get a few more students to understand a concept. When you reach that mountain, you may try to get enough students to mastery so that next year's teacher doesn't have to do as much re-teaching (the second mountain). As you were scaling that second mountain, you may have noticed that students got into the flow. Can we get them solving more problems independently (the third mountain)? Now that they are so good at problem solving, can we teach the entire curriculum through problem solving (the fourth mountain)? Eventually, the new normal looms into view.

While the compass gives you a clear heading, it doesn't give you a sense of accomplishment as you follow that heading. Scaling a series of higher and higher mountains does. You'll also find that, each time you scale a new and higher mountain, your party gets a little larger as stragglers and colleagues who didn't want to leave the base camp rush to catch up.

However, once you've scaled that first mountain, some teachers will want to rush straight toward the second mountain, forgetting about the compass. That is something to guard against. Someone will say, "Hey, we accomplished this impossible task of getting more students to understand this concept, let's build on that and try to reduce how much re-teaching we are doing." And then someone else will say, "Okay. How about everyone focuses on drilling students until they memorize their math facts. That would be a huge help." You need to remind them that sense-making is the compass that got us to the first mountain and that sense-making will be the compass that gets us to the second mountain. Over time, teachers will internalize the use of the compass for themselves once they see how effective it is.

The Need for Hands-On and Shared Leadership

I don't believe that you can, or should, identify a line of mountains to follow in advance. Those goals need to be established collaboratively based on the situation on the ground. This illustrates the need for hands-on and shared leadership.

To reach a new normal, school leaders need to take an active role in the journey. When the party gathers around a map hastily drawn in the dirt, trying to figure out which route to take crossing a river, school leaders need to be there providing leadership and guidance. It takes time for teachers to internalize the compass and take ownership of the journey. It takes time for a group to learn how to create spaces so that all voices can be heard and to make decisions collaboratively. School leaders also need to internalize the compass themselves and learn how to share leadership, otherwise they will end up making decisions that sidetrack the journey and damage the very culture they are trying to nurture.

Finally, the journey takes time. You won't get there in a year. You won't get there using consultants that drop in for professional days. You won't train "trainers" at a four-day institute over the summer. This journey means rolling up your sleeves and sleeping in tents with the rest of us. Are you ready?

The Sense-Making Curriculum

Here is one way to visualize the traditional math curriculum:

The learner climbs a series of ladders. The lower ladders are fairly easy to get onto, but there are gaps that the learner has to traverse to reach the higher ladders. The ladders are grouped in isolated sets. To climb a new set of ladders, the learner has to drop back down to the ground and start climbing from the bottom again. The ladders don't go very high. Some learners can't figure out how to cross the gaps and get stuck. Some learners get frustrated at getting stuck all the time and quit climbing all together. Other learners get sick of climbing short distances over and over again, and also quit. It takes years before the learner climbs and reaches someplace interesting.

Here is a standards-based math curriculum viewed through the same lens:

The lowest ladders are a bit higher off the ground. It is easier to jump from ladder to ladder and the learner has more freedom to explore. The ladders also reach a bit higher. But there are still large gaps so few learners are able to reach the higher ladders. Some learners aren't able to pull themselves up onto the lowest ladders. Some learners get frustrated and give up when they hit a dead end, especially if they are used to a traditional math curriculum where the path from ladder to ladder is clearly marked. After clambering on a cluster of ladders for awhile, the learner still has to drop back down to the ground to climb a second cluster.

Just to clarify, a standards-based math curriculum is a curriculum based on the Curriculum and Evaluation Standards for School Mathematics published by the National Council of Teachers of Mathematics (NCTM) in 1989. Examples include Investigations in Numbers, Data, and Space (TERC), Connected Mathematics (CMP), the Interactive Mathematics Program (IMP), and Everyday Math. The "math wars" is a debate over traditional and standards-based math philosophies.

Here is a math curriculum that uses vertical learning:

Like the traditional math curriculum, the lowest ladders are easy to climb onto. Like the standards-based math curriculum, the ladders are linked together and interesting to explore. However, there are no difficult gaps to cross, there are solid platforms to stand on (making navigation clearer and easier), and the ladders go much higher. More importantly, once the learner is on the ladders, there is no need to drop all the way back down to the ground to reach other ladders. The learner is free to climb onward and upward.

When most people see a math curriculum that uses vertical learning for the first time, they love it because it enables learners to climb much higher. But they assume that the result will look like this:

But having taught math with vertical learning before, I know that the result looks like this:

If you give learners an accessible ladder that goes someplace interesting, they will climb it. And they will become stronger, faster, and more confident climbers in the process.

Mental Models

The standards-based math curricula of the 1990s grew out of a desire to make learning in math classrooms more constructivist. According to constructivist learning theory, we build our own understanding by developing theories (mental models or schema) of how the world around us works. Some people associate constructivism with discovery-based learning, but I am still building my own understanding whether I am reading a book or playing with blocks. First, I experience something. Then, I try to interpret it (make sense of it) using my existing mental models. If I have to modify an existing mental model in order to make sense of the experience, then learning has occurred.

In a constructivist classroom, the role of the teacher is to probe the mental model of the learner. When you ask the learner to read a book or play with blocks, you can't know what meaning the learner will make from it. You need to probe their mental model to see how they see things. If the learner's mental model is faulty, the teacher then needs to create learning experiences that generate cognitive dissonance, causing the learner to recognize the fault in his or her own mental model and fix it.

The mental models that we want students to construct for themselves should be functionally similar to the mental models that experts have.

An Expert Mental Model of Order of Operations

In this expression, there are two operations (one addition and one multiplication):

If I evaluate the addition first, I will get one value for the expression:

But if I evaluate the multiplication first, I will get a different value for the expression:

To make sure that an expression only evaluates to a single value, mathematicians created rules for deciding which operation should be evaluated first. In sixth-grade, these rules are:

• Parentheses
• Exponents
• Multiplication/Division
• Addition/Subtraction

If there is a tie, then you go from left to right.

These rules are somewhat arbitrary, but everyone has agreed to follow them. Two common mnemonics for remembering the rules are PEMDAS and "Please excuse my dear Aunt Sally."

In this expression, there are five operations:

We do multiplication/division before addition/subtraction, so we are going to start with either the division operation or one of the multiplications. To decide which of those to do first, we go from left to right and start with the division operation.

Now there are only four operations left. Again, we do multiplication/division before addition/subtraction, so we are going to do one of the multiplication operations. Going from left to right, we do the first multiplication operation.

Using the rules for order of operations, we evaluate each operation in the expression until we arrive at a single value.

For most sixth-graders, order of operations is used to decide which operation to evaluate next when evaluating an expression. But that is not how math experts understand and experience order of operations. For a math expert, order of operations is used to group parts of an expression.

So, when I see the expression:

I know that I can multiply 5 times 4 first. How do I know that? I know that because I know that when I do follow the rules for order of operations and get to that multiplication operation, I will be multiplying 5 times 4, so multiplying them out of order doesn't change anything.

Likewise, I know that I can't add 2 + 10 because before I ever get to that addition operation, I've done:

So I am really adding 18 ÷ 3・2 plus 10, not 2 plus 10.

This is also how I know that I can add 2x + 7x first in this expression:

Technically, I am performing operations out of order, but I know that I can get away with it because, if I did each operation in order, I would be adding 2x + 7x at some point anyways.

Building an Expert Mental Model

A sixth-grader doesn't need to have the same mental model for order of operations that I have. But a seventh-grader does need to be able to chunk expressions in order to add like terms, apply the distributive property, and identify the bases of exponents. Without that foundation, a seventh-grader would have to memorize separate rules for each of those skills (jumping all the way down to the ground just to climb a new ladder).

By tenth-grade, anyone who is fluent in algebra has figured all of this out for themselves. I would say that there is an extremely high correlation between students who can use order of operations to chunk an expression and students who are successful at algebra. Yet, we don't invest any effort in helping students go from point A (a sixth-grade understanding of order of operations) to point B (an expert understanding of order of operations). We don't even bother assessing whether a student is building the appropriate mental model or not, which is the primary role of a teacher in a constructivist classroom.

A curriculum that uses vertical learning is explicitly designed to help students build expert mental models. In order to access this curriculum, students need to be able to pull themselves up onto the first rung of the lowest ladder. From there, they are able to climb anywhere. In the case of order of operations, that first rung is applying the rules for order of operations to identify the next operation to evaluate in an expression.

When I was teaching sixth-grade, I liked to start the year off with order of operations because it let students know that they weren't in Kansas anymore. You may have gotten C's and D's in math all through elementary school, but once you had one foot on the ladder, it was a quick and easy climb to evaluating expressions that look like this:

I have worked with hundreds of students on order of operations over the years, and I have yet to find one that wasn't able to climb to this point in two or three days. To evaluate this expression, you don't need to be able to chunk it. But it does help if you can isolate or ignore parts of the expression. For example, students quickly realize that when they are evaluating an operation inside of a set of parentheses, they can finish evaluating all of the operations inside of the parentheses without having to go back out and look over the entire expression.

Chunking the expression also makes it easier to copy the expression down to the next line. (Think about copying a long sentence that is gibberish versus copying a sentence made up of simple phrases.) As long as students want to evaluate these complex expressions (which they do — it gives them an immense sense of accomplishment), then they will start chunking.

Another place in the curriculum where students can apply order of operations is when simplifying expressions with exponents. Before they know or are fluent with the laws of exponents, my students expand these expressions out with multiplication:

In order to expand an exponent using multiplication, you have to be able to identify the base. While that might seem trivial, it really isn't. Some students never understand how to identify the base and guess each time.

If a student isn't sure what the base of an exponent is, he or she can figure it out by applying the rules for order of operations.

The first operation you would do is the exponent in the inner parentheses. This means that the base of that exponent is just the a.

You would then complete the operations inside of the inner parentheses (two multiplications) before doing the exponent on the inner parentheses. This means that the base of that exponent is the contents of the inner parentheses.

After completing the operations inside of the outer parentheses (two more multiplications), then you would do the exponent on the b. This means that the base of that exponent is just the b.

Over time, students begin to notice patterns and soon they are chunking the expression. It helps if you can recognize that the base of the first exponent is just the b without having to even think about the contents of the parentheses. Then you can tell at a glance that there are six b's as factors in this expression:

And thirteen a's as factors:

In a curriculum that uses vertical learning, students don't develop fluency and automaticity by doing repetitive practice, they develop fluency by applying skills and concepts in a wide-range of contexts and they develop automaticity in order to reduce cognitive load (the more they can do on autopilot, the more they can focus on problem solving).

Wrap Up

The structure of an expert mental model is similar to the structure of a curriculum using vertical learning. This means that students learning math using a traditional or standards-based curriculum must do a significant amount of re-mapping in order to build the understanding they need to be successful. But, regardless of the type of curriculum you are using, you can't just assume that a student will absorb a curriculum as it was designed and presented; everyone builds their own understanding as they go. Therefore, it is the responsibility of the teacher to assess the robustness of each student's mental models and to provide learning experiences that will generate cognitive dissonance if a student's mental model is faulty.

In a traditional curriculum, the emphasis is on procedural learning. Students are expected to develop conceptual understanding later. Conceptual understanding is important because that is how a math expert understands and experiences those procedures, not in isolation but as part of a larger conceptual framework. Unfortunately, in a traditional curriculum, students are generally left alone to develop a conceptual understanding outside of the curriculum and without any help from the teacher. Evidence suggests that this doesn't work for a significant number of students.

In a standards-based curriculum, the emphasis is on conceptual understanding. Critics complain that these curricula do not enable students to achieve automaticity because there is not enough of an emphasis on procedures. I disagree. I believe that the lack of automaticity arises because these curricula are generally unfocused. They provide a wide-range of learning experiences in the hopes that every student will get what they need, but students and teachers are given little guidance in terms of what they should be getting out of them. Teachers are not assessing a student's mental model against any kind of standard. Again, evidence suggests that this doesn't work for a significant number of students.

In a curriculum that uses vertical learning, the emphasis is also on conceptual understanding, but the curriculum is designed to help students develop expert mental models. Just because there is a specific target in mind does not mean that students are taught procedures. Students still learn by doing and problem solving, it simply means that the teacher is assessing those mental models and providing targeted learning experiences. This leads to a curriculum with ladders that are easy to climb onto, but still reach great heights and can be explored freely. In fact, the ladders go higher and provide more room to explore because the teacher is guiding the students to develop mental models that we know lead to a greater understanding of math.

In turn, ladders that are easy to climb onto but go high and reach interesting places quickly are incredibly motivating and empowering for students. Once something makes sense, you want more things to make sense and you never look back.

Scripts and Actors

Yesterday, I wrote a post on Separating Curriculum and Instruction. I meant to include a wrap up at the end of the post, but I forgot. Then, I thought about updating the post today to include a wrap up, but now I think that the wrap up works better as its own separate post. Phew! So, here we are.

Think of learning as a play where the curriculum is the script and the instruction are the actors. I am arguing that, in general, we have better actors than scripts. Sure, there are still good actors and bad actors, but the average actor is better than the best script, and the best scripts are pretty terrible.

If the best scripts are pretty terrible, then the best plays are also going to be pretty terrible, and hiring better actors isn't going to help. In fact, there is little incentive to go out and hire the best actors since a slightly above average actor will be almost as good in the role.

Now, a better actor is going to get more out of a bad script than a bad actor will. Even if the scriptwriter didn't do it, the better actor will infuse the character with nuance, motivation, and maybe some kind of backstory. But imagine that the script is so bad that the character is constantly saying and doing things that the audience finds ridiculous and out-of-character. There is only so much that the actor can do without re-writing the script from scratch. In the same way, the best teachers will massage the curriculum, but there is only so much they can do without re-writing the curriculum from scratch.

Once all plays are terrible and hiring better actors doesn't help, the craft of acting starts to devolve. Why push the state of the art if mediocre is good enough to land any role and the audience can't easily distinguish between mediocre and best? What's worse, how do you push the state of the art? Imagine that there are two schools of acting. Given a terrible script, the actors who subscribe to school X give performances just as good as the actors who subscribe to school Y. Which school is better? You can't tell. School Y may be better, but you'll never know it until the actors from school Y have a chance to strut their stuff with better scripts.

I had lunch with my coach Sarah today, and we talked a little bit about the progressive education movement. So many people are rushing to place their ideas and methods under the progressive education umbrella that the term has lost some of its rigor. This is natural, but at some point you need to prune things back. Unfortunately, that is hard to do when you can't measure instructional methods or acting techniques against performance because the curriculum or the script is holding back that performance. When that happens, everything becomes personal preference and nothing gets resolved. This is why educational researchers are constantly rebadging old ideas with new names in order to escape all of the baggage.

I'm going to make one last point about curriculum, instruction, and new normals. It doesn't really belong in this post, but I'm going to make it anyways. In Separating Curriculum and Instruction, I pointed out that we accept uninspired teaching as normal, and that I wanted to wake up in a future where uninspired teaching is unfathomable. I'm guessing that most people would nod when reading that. However, I then pointed out that we accept a low standard of performance as normal, and that I wanted to wake up in a future where our lowest-performing students outperform what we consider high performing today. I'm guessing that most people would feel kind of awkward reading that because they don't truly believe it. This highlights our beliefs about instruction (embodied in teachers) and curriculum (embodied in standards). We need a Fosbury Flop on the curriculum side because no one believes it is possible. It's not a communication issue. I have to prove it in a way that the truth is undeniable. The instruction to support the curriculum will then follow, and a lot of the instructional crap we have now will get pruned away.