Separating Curriculum and Instruction

A friend of mine invited me to a webinar featuring Aleta Margolis, Founder and Executive Director of the Center for Inspired Teaching. Since I had never heard of the Center for Inspired Teaching before, I decided to check out their website. What I found there was very impressive.

The mission of the Center for Inspired Teaching is to build a better school experience for children through transformative teacher training. An inspired teacher teaches students how to think, not just what to think, ensuring that students successfully build intellect, inquiry, imagination, and integrity. Their website also mentions a Wonder-Experiment-Learn cycle, but no details are provided.

This represents the state of the art in student-centered instruction, but they take it one step further. If students learn best when they are active, empowered, and inspired, then teachers learn and teach best when they are active, empowered, and inspired. You'd be surprised at how many proponents of student-centered learning believe that they can impose that model on teachers from the top down.

But the people behind the Center for Inspired Teaching aren't hopeless idealists either. It is clear that they are applying lessons learned from the failed reform efforts of the past few decades. They have set up a demonstration school, which indicates that they know that the state of the art isn't good enough. They offer teacher certification through a 24-month residency program, which indicates that they know that you can't shift core beliefs and practices in a one-day workshop; it takes time for new habits to take hold and for skills to be refined and concepts deepened. At the same time, they are forming district partnerships to keep themselves grounded and to demonstrate real-world results.

However, with all that said, I still don't think that they have a snowball's chance in hell of making a serious impact on teaching and learning. When I talk about a serious impact on teaching and learning, I mean a new normal. When I wake up in twenty years, I want to see that every teacher is an inspired teacher and that the thought of being an uninspired teacher is unfathomable. (It's sad that it isn't already.) But we are still talking about inputs at this point. I also want to wake up and see our lowest-performing students outperforming what we consider high performing today, and for anything less to be unfathomable.

When I started blogging in December about my strategic plan, my goal was to dump a bunch of stuff on paper so that I could start processing it without the overhead of holding everything in my head. One realization that bubbled to the surface was that I was using the development of a sense-making curriculum as my compass. As I iterated and refined my practices, I constantly asked myself: "Is this making more or less sense to my students?" If I had even one student who couldn't make sense of something, then it was time for me to go back to the drawing board. Sense-making is the single metric I use to gauge my progress; it is how I close my loop. It isn't the be-all and end-all, but when you are wandering in the fog and the mountain that is your true goal is out of sight, you need something to orient you.

Once I realized that sense-making was my compass, I also realized I wasn't doing anything innovative on the instructional side. Immersive learning, complex problem solving… everything that I do instructionally is being done by someone else. Everything that I do instructionally went mainstream in this country a hundred years ago with John Dewey and the progressive education movement. What I do and what the Center for Inspired Teaching does may be state of the art, but the state of the art is a century old.

After ruminating on these two realizations for a couple of weeks now, I have now had a third realization: curriculum and instruction work together, but the crappy curriculum we have now is acting as a bottleneck in the system. This means that performance scales with instruction up to a certain point, but once you've hit that bottleneck, improving instruction no longer improves performance because the curriculum is holding you back.

One result of this bottleneck is that proponents of progressive methods and proponents of traditional methods are never be able to convince the other side that their methods are superior. The Center for Inspired Teaching has research that shows that "students taught by teachers participating in Inspired Teaching's professional development show greater overall growth in academic achievement than students taught by non-participating teachers." But if I put teachers through professional development that helped them improve their traditional methods, you'd see a similar bump in academic achievement. Any growth in academic achievement you can achieve through progressive methods can also be achieved through traditional methods, especially when you also get to cherry-pick the metric being used. As much as the Center for Inspired Teaching wants to cite research, their choice of teaching methods is driven by educational philosophy, not data. This is why we've been in a stalemate for the last century.

A second result of this bottleneck is a focus on inputs and not outputs. How does an inspired teacher close the loop? The Center for Inspired Teaching lists five metrics:

• A climate of mutual respect
• Treating the student as expert
• Creating a sense of purpose, persistence, and action
• Instilling joy in learning
• Providing wide-ranging evidence of learning

As an inspired teacher, I should use these five metrics when designing or evaluating my curriculum and instruction. But I would argue that the first four metrics are inputs that the Center for Inspired Teaching believes will lead to learning, not outputs that are the result of learning. These metrics lead teachers to focus on student choice, appealing to student interests, hands-on learning with manipulatives, open-ended play, and other progressive methods. But how do I decide which strategy to use when? Which metric do I optimize for? How can I tell if one strategy is better than another in this context?

I want my students to have a sense of autonomy, to become resilient problem-solvers, and to take joy in learning, but I lean on sense-making as my compass. I believe that if students are able to make sense of things, everything else will follow. Will my compass lead me astray at times? Yes, but that is why I re-calibrate it against the mountain every chance I get. So far, I see the mountain getting closer.

I said earlier that I am not innovating on the instructional side and that the Center for Inspired Teaching and I are both just practicing the state of the art of student-centered instruction. That's not quite true. Because of my compass, I do have an effective way of evaluating whether or not a specific strategy is working within a specific context, so I am continuously refining my judgment and technique. I feel like this has given me a deeper understanding of these instructional methods, especially in terms of how they fit and work together. Because I have removed the curricular bottleneck, I'm able to push performance past anything that anyone with the old curriculum could ever achieve. I have broken the stalemate, but no one knows it yet.

How do I know that the Center for Inspired Teaching isn't doing the same work that I am? I actually started in the same place they are starting. I walked into the classroom with some progressive ideas and a desire to help students build understanding. Math and science were intuitive to me, and I felt that they should be intuitive to everyone. I didn't have any curricular ideas. I came in and iterated. But I quickly hit the curriculum bottleneck and realized that I needed something better. Once you have that realization, you can't un-have it. The Center for Inspired Teaching hasn't had that realization yet because all they talk about is inputs and instruction. They still have time to get it right, but until they do, they are running the same playbook that John Dewey ran… and we can all see how that turned out.

Making Sense of Solving Word Problems

At around 5th-grade, there is a transition from learning to read to reading to learn. Before the transition, students are taught explicitly how to read. After the transition, students use their reading skills to read and learn new things. Students who don't have the reading skills they need when this transition occurs struggle and often fall behind.

There is a similar transition in math at around 10th-grade. After Algebra II, we start using algebra to model and solve problems not just to learn math, but to learn other things. Unfortunately, only about 10% of us make this transition successfully. I talk to a lot of adults about their math experiences, and many highly intelligent people who got A's and B's in math in school report being absolutely traumatized by word problems in high school. I also talk to a lot of college professors, and many lament that they can't rely on students being able to do math to learn even in technical fields such as pre-med or computer science.

There are two basic approaches for teaching students how to solve word problems using algebra. In the first approach, the student literally memorizes recipes for solving a handful of classic word problem types, such as coin or train problems. (I can feel some of you tensing up just at the mention of train problems.) The hope is that, at some point in the future, students will generalize from those recipes and figure out how to solve word problems on their own.

In the second approach, the student is immersed in a single context and asked to solve a set of problems within that context. Instead of giving the student a set of recipes to use, the student is expected to use his or her deep understanding of the problem context to solve the problems. A premium is placed on using multiple representations, so the student will work with tables, graphs, and equations. Again, the hope is that, at some point in the future, students will generalize what they learned from this problem context and figure out how to solve word problems in other contexts on their own.

I prefer the educational philosophy behind the second approach, but neither approach has been successful and both suffer from the same fatal flaw: students are left to generalize on their own. Starting from a deep understanding of a single problem context is an effective way to help students generalize if the learning experiences and tasks follow a logical progression, but what we always get instead is a shotgun approach where the curriculum designer throws everything against the wall and it is up to the teacher to see what sticks and then build on it. What's worse, generalization is suppose to happen at some future and undetermined date, so no one can assess if the curriculum or instruction are working or not.

I feel like it is the job of the curriculum designer to provide opportunities for sense-making within the curriculum itself, and it is the job of the teacher to monitor the progress of each student's sense-making and adjust the curriculum and instruction as necessary.

When I examined my own strategy for solving word problems, I realized that I use the same basic strategy I use when solving word problems by guess-and-check.

Kirsten bought 36 cans of soda on sale for 20¢ each. She sold some of them to Luigi for 50¢ each, she sold some of them to Melania for 40¢ each, and she drank the rest. Melania bought 8 more cans than Luigi. If she made a profit of $5.90, how much soda did Kirsten drink?

To solve this problem using guess-and-check, I would start by guessing the number of sodas that Luigi or Melania bought.

Let's guess that Luigi bought 5 cans of soda.

If that's the case, then Luigi paid Kirsten $2.50 and Melania bought 13 cans of soda.

If that's the case, then Melania paid Kirsten $5.20 and Kirsten drank 18 cans of soda.

If that's the case, then Luigi and Melania paid Kirsten $7.70.

Hmm… Kirsten made a profit of $5.90, how would I figure out her profit here? Oh! I need to know how much she paid for the soda, which is $7.20. So Kirsten's profit is $0.50.

My guess is incorrect, so I need to guess again.

By starting with a guess, I am able to work forward through the problem. Each time I figure out a value, I scan through the remaining unknowns to see if any of them are easy for me to figure out next. As a curriculum designer, you can't know for sure if this step is intuitive for students or not until you test it out. I have found that every 6th- and 7th-grader that I've worked with has been capable of starting with a guess and then determining if that guess is correct or not. Students have years of experience solving this type of problem (and generally love doing it), so I am leveraging a fairly robust mental model here. However, if I ran into a student who struggled with this step, then I would have to figure out how to start at a more basic level.

I should also point out that I had to do some problem solving in order to figure out how to calculate the profit. To get to the profit, I had to find how much Kirsten paid for the soda first, which is not obvious. The guess-and-check strategy works if the student understands the problem context and is willing to do some simple problem solving. We shouldn't expect students to solve word problems in unfamiliar contexts.

Now that I've done one round of guess-and-check, I have a roadmap I can follow if I want to solve the same problem using algebra. Instead of guessing a random value for the number of sodas Luigi bought, I can guess the correct value and call it x.

Luigi bought x cans of soda.

Luigi paid Kirsten 50x and Melania bought x + 8 cans of soda.

Melania paid Kirsten 40(x + 8) and Kirsten drank 36 − x − (x + 8) cans of soda.

Luigi and Melania paid Kirsten 50x + 40(x + 8).

Kirsten made 50x + 40(x + 8) − 720 in profit.

Since x is the correct guess, then 50x + 40(x + 8) − 720 = 590.

Solving for x, we get x = 11.

So if Luigi bought 11 cans of soda, then Melania bought 19 and Kirsten drank 6.

This is the basic approach that I have used to teach students how to solve word problems using algebra. I have tested this curriculum once in a classroom setting with a heterogenous group of sixth-graders. It was my first time teaching this curriculum, but I was still able to get about 60% of the students to mastery, which means that they could solve any word problem that was limited to linear equations and understand what they were doing. The other 40% of the students could also solve any word problem, but they were less confident in their understanding and more prone to making errors. It was probably a combination of less automaticity with the algebra (more cognitive overhead), the tendency of 6th-graders to make a careless error somewhere in a multistep problem (which can affect confidence and cause second-guessing), and an incomplete understanding of one or more of the subtasks. I would have to refine and re-teach this unit to sort all of that out.

Before I wrap up, I want to point out a few things that trip students up. First, there is the question of which unknown to guess. There are six unknowns in the problem, and I can solve the problem by guessing any one of them. But I know that my life will be much easier if I guess the number of sodas Luigi or Melania bought than if I guessed the number of sodas Kirsten drank or (god forbid) how much Luigi and Melania paid Kirsten. At the beginning, I usually tell the students which unknown to guess, then ask them to choose later once they are comfortable with the guess-and-check process. I tell them that if they guess an unknown and run into complications, then they should just start over again. Over time, this will help them develop an intuitive sense of which unknown to use as their starting point. That's what I'm doing, but I'm doing it so quickly that it seems like I just know which unknown to pick.

Second, some students will want to start adjusting values within a guess instead of starting over with a fresh guess if a guess is wrong. For example, if they guessed that Luigi bought 5 cans of soda and figured out that the guess is wrong because Kirsten's profit is only $0.50, they may try increasing the number of sodas Melania bought without recognizing how that affects the number of sodas Luigi bought and the number of sodas Kirsten drank. They are focusing on one relationship at a time. You need to show them how each variable is linked to every other variable so it isn't possible to isolate a single pair and tweak them. If their guess that Luigi bought 5 cans of soda is wrong, then they need to start fresh with a new guess for Luigi. Seeing how everything is interconnected will help students understand how a system responds to change and why they need algebra to model it.

Once students are organizing their guesses in a table, they can look for patterns and guess more efficiently.

Luigi soda	Luigi paid	Melania soda	Melania paid	Kirsten soda	Kirsten cost	Profit
5	$2.50	13	$5.20	18	$7.20	$0.50
6	$3.00	14	$5.60	16	$7.20	$1.40
7	$3.50	15	$6.00	14	$7.20	$2.30

Here, they can see that each time the number of sodas that Luigi bought goes up by one, Kirsten's profits go up by 90¢, making it possible to determine the correct guess without any more guesswork. This is a nice intermediate step before going straight to guessing x and using algebra.

Third, I like to isolate the subtask of translating computations into algebraic expressions using order of operations. I call it solving word problems without computation.

Kirsten bought 36 cans of soda on sale for 20¢ each. She sold some of them to Luigi for 50¢ each, she sold some of them to Melania for 40¢ each, and she drank the rest. Melania bought 8 more cans than Luigi. If Luigi bought 5 cans of soda, how much profit did Kirsten make?

This problem does not require algebra or guess-and-check to solve. Basically, I am supplying what would have been the guess and removing the check. I ask students to solve the problem without doing any computations, which means writing the answer as a numerical expression.

Luigi soda = 5
Luigi paid = 5 × 50
Melania soda = 5 + 8
Melania paid = (5 + 8) × 40
Kirsten soda = 36 − 5 − (5 + 8) or 36 − (5 + 5 + 8)
Kirsten cost = 36 × 20
Kirsten profit = 5 × 50 + (5 + 8) × 40 − 36 × 20

Besides practicing their metacognition, students are also practicing writing expressions with order of operations and finding values for a series of unknowns.

Wrap Up

So what should you take away from all of this? Learning how to solve word problems using algebra is a serious pain point for both students and teachers. A 90% failure rate is abysmal. It is also a major pain point for anyone who thinks that we don't have enough people entering STEM careers. Most students who lack the skills to transition from learning to do math to doing math to learn opt-out of math completely at this point. There should be some serious urgency around fixing this problem.

Despite all of this, the two approaches we have for teaching students how to solve word problems using algebra are utterly ineffective, and instead of trying something new, we keep trying to refine them. It seems obvious that if we want students to come up with a general problem-solving strategy, then we should figure out the strategy we use and try teaching them that. But I don't see that happening anywhere. Instead, we continue to expect students to generalize and make sense of things on their own outside of the classroom and the curriculum.

By identifying my own problem-solving strategy and trying it with students, I've been able to help over half of my sixth-grade students understand how to solve word problems using algebra. Imagine what those sixth-graders could be doing by the time they are in high school taking Algebra II. I still need to refine the curriculum and my instruction, but that is a simple matter of closely observing students and helping them overcome stumbling blocks to understanding. It is a logical and straightforward process that, again, I don't see anyone else doing. Why is that? And why aren't effective innovations bubbling up to the surface?

Bonus Problem

Finally, a bonus problem in case anyone wants to see my problem-solving strategy applied to a second problem.

Humphrey leaves Chicago at 2:00 pm and drives west at a speed of 55 mph. Iris leaves Chicago at 6:00 pm and drives east at a speed of 65 mph. When Iris finally stops for gas, she and Humphrey are 580 miles apart. When did Iris stop for gas?

Start by guessing a time: Iris stops for gas at 8:00 pm.

If that is the case, then Iris has been driving for 2 hours and Humphrey has been driving for 6 hours.

If that is the case, then Iris has driven 130 miles and Humphrey has driven 330 miles.

Since Iris is driving east from Chicago and Humphrey is driving west from Chicago, then they must be 460 miles apart.

The guess was wrong (460 miles ≠ 580 miles), so guess again.

Using algebra, we can guess correctly on our first try: Iris stops for gas at x pm.

(There are issues that students will have to deal with when using time, but they should be able to handle it if the context is intuitive to them. The traditional approach is to guess hours instead of time, but either will work as long as the students are thinking about what they are doing and not just operating on cruise control.)

Iris has been driving for x − 6 hours and Humphrey has been driving for x − 2 hours.

Iris has driven 65(x − 6) miles and Humphrey has driven 55(x − 2) miles.

They are 65(x − 6) + 55(x − 2) miles apart, so 65(x − 6) + 55(x − 2) = 580.

x = 9

Iris stopped for gas at 9 pm.

Function Junction, Part 3

I stumbled into an amazing opportunity when I started working in Holliston in 2007. The standards movement was in full swing by this point and most districts were purchasing and implementing monolithic math programs from major publishers. Buying a math program from a major publisher gave a district political cover (just like nobody ever got fired for buying IBM, nobody could point a finger at you if you bought the newest and most popular textbook series from Pearson) and it was an expedient way to ensure consistency across classrooms. But before making a purchasing decision, Holliston had decided to conduct a thorough curriculum review of all content areas first, and the math review would not take place until 2010.

When you adopt a monolithic math program from a major publisher, you are basically handing a script to the teachers. The program will come with supplementary materials that you can plug into the program, but none of the core materials are ever designed to be swapped out. The publishers learned to make their programs as idiot-proof as possible after the standard-based math programs of the 1990s were plagued by steep learning curves. But asking teachers to gain a deep understanding of the standards and the curriculum, and then handing them a script, does not make sense. It is also highly disrespectful. To do the work we needed to do, the teachers needed room to experiment and explore, and the district had to be open to the possibility of teachers developing a better curriculum themselves.

Luckily, the curriculum review schedule gave us the window of opportunity we needed. The middle school math department was using an old textbook series from Scott Foresman, but no one really liked it, everyone knew it was on the way out, and most teachers were already heavily supplementing it with other materials. And a replacement textbook series was still several years away. We had time to experiment with new units and instructional strategies, and then inject this new learning into the curriculum review process.

In 2006, the year before I came onboard, there had been a major, district-wide push to identify power standards. This work was designed to pave the way for the curriculum reviews to follow. It did not go well. Power standards are suppose to point you to a guaranteed, viable curriculum. The ones we identified pointed in random and contradictory directions; instead of bringing clarity, they brought confusion. The district had hired high-priced consultants to lead the work, but now those consultants were gone. Jessica, the principal of the middle school, hired me to get things back on track.

Throwing away a year's worth of work that the teachers had done and rebooting the entire power standards process in the middle school was hard, but that is what we did. It was hard, but it was also necessary if the curriculum reviews that we were starting were going to be guided by a deep understanding of curriculum and not deeply-held beliefs based on nothing more than superstition and old wives' tales. The second set of power standards we identified in 2007 at the middle school were much better. Teachers could see how the power standards helped them understand the curriculum and they appreciated why we had asked them to do the work over again.

Imagine that a software company decides to implement a code review system to make sure that the code it produces is easy to scale and maintain. What would happen if the system didn't work? Would they just move on or try again until it did work? Schools routinely implement systems that don't work. The power standards identified by schools don't work 99% of the time. Professional learning communities implemented in schools don't work 99% of the time. The reason this happens is because the administrators in those schools don't expect those systems to make a difference. A system is implemented to implement a system, not to achieve a specific outcome.

At the middle school, we worked on our power standards until our teachers gained a much deeper understanding of the standards and the curriculum. In the end, our teachers found the power standards useful and wanted to continue building on them. The principals, curriculum specialists, and department heads at the high school and elementary schools did not expect anything from their power standard work, so they took one pass at it and were ready to move on. Their power standards ended up in a binder on a shelf in someone's office, never to be looked at again.

Instead of confronting the other schools and asking them to redo their power standards, perhaps with guidance from the middle school, the superintendent and assistant superintendent for curriculum both decided to take what they could learn from the power standard experience and apply it to a second initiative. Since power standards hadn't work, the district would now use Understanding by Design and Atlas Rubicon (a collaborative curriculum mapping tool) to drive the curriculum review process. We were still encouraged to use power standards at the middle school, but we had to integrate them into these other systems and we would have to work on power standards on our own time.

To do this work, you need someone who can provide coaching to teachers on a daily basis and can read situations on the ground well enough to tack when necessary. But you also need someone who can champion the work at an administrative level and work with fellow administrators when necessary. Everything that I was doing with teachers also needed to happen with the administrators, but I wasn't in a position to do that. For most of my time in Holliston, I had carte blanche in my role as curriculum specialist. My bosses trusted my judgment and trusted me to do my work well. They supported the work that I was doing. Unfortunately, when you have carte blanche and no one is looking over your shoulder, your bosses don't always know what you are doing. And because no one really knew what I was doing, no one could help me when I needed to interact with the high school or the special education department. For the work to move forward, more and more people needed to come onboard. I could do that within my departments in the middle school, but not beyond them.

Function Junction, Part 2

When I started as curriculum specialist at Robert Adams Middle School in Holliston, my goal was to develop and deliver a curriculum that made sense to students. The raison d'être of math and science is to help us make sense of the world around us, so it is pretty crazy that most students experience math and science as subjects in school that don't make sense and need to be memorized. The students that do well in math and science are the ones that take what they learn in the classroom, and make sense of it on their own. I strongly believe that the curriculum should be designed to encourage and enable all students to do that.

If something makes sense to you, you feel confident about it. You feel like you are standing on solid ground and that you are capable. That is an awesome feeling, and when you experience it, you want to experience it again and again. So, instead of memorizing things in isolation and relying on procedures to solve problems, you start integrating things into conceptual frameworks and reasoning through problems on your own. This changes the entire dynamic in the classroom and how you experience learning in school. Instead of being taught, now you are constructing your own understanding and driven by a sense of autonomy.

This is what most teachers want. We don't want to stand up and lecture, and spoon feed little isolated facts to students all day. Doing that day after day sucks the life right out of you. We teach that way because we don't know what else to do and it seems like the only way to get students to learn anything. Over time, the walls go up and we convince ourselves that this is the only way to teach, that students need it. This, of course, warps how we see and think about students. It also warps how students see and think about themselves.

Developing and implementing a curriculum that makes sense to students may seem like an easy and straightforward enterprise, but it isn't. First, there isn't a curriculum that you can just buy off the shelf; we would have to develop something from scratch. Second, there isn't a model that you can point to and follow for creating a sense-making curriculum; no school that I'm aware of has developed one. Third, no one at the school (except me) believed that a sense-making curriculum was even possible because we had all developed the core belief that some students simply aren't capable of making sense of some math and science concepts. We didn't want to believe that, but our experiences had led us to that conclusion.

This means that, as we set off on this journey to a sense-making curriculum, we were blazing a new trail through unexplored territory without a mountain or a compass to guide us. Using a sense-making curriculum as our goal wouldn't help; it would be like setting out for Xanadu without having any idea where Xanadu is, what it looks like, or even if it exists. A mountain needs to be something that you can see on the horizon, and a compass needs to be something that you can and will rely on when you are lost and can't make out the forest for the trees.

In Holliston, I was hired to help the staff develop a math and science curriculum based on Power Standards. This was tremendously helpful because it set the expectation that we were going to be writing our own curriculum, and I felt that Power Standards could serve as an effective compass. However, to be an effective compass, everyone would have to trust in the compass. There would be times when we'd be convinced that the compass was guiding us in the wrong direction and want to turn back, but we'd have to trust in the compass enough to keep going. Developing that level of trust was going to take time and evidence.

So, until the Power Standards compass came online, I'd have to function as the staff's mountain and compass. One of the first practices that we implemented at the middle school were weekly grade level department meetings. Once a week, all of the seventh-grade math teachers would sit down during a 45-minute common planning period to discuss curriculum, share practices, and write, score, and analyze common assessments. Our teams actually had 90 minutes of common planning time a day, but by contract, the administration could not mandate what the teachers did in that time. It was rare for teachers to meet to discuss curriculum, but through the Power Standard work we were doing, we were able to convince them to meet voluntarily.

I would drop by some of these meetings and listen in on the discussions. I would listen for any issues that the teachers were having (pain points), and occasionally offer suggestions. Some of these suggestions were based on curriculum units that I had developed in the past. Initially, about a third of the teachers were willing to try some new curriculum, and about a third of them were willing to let me come in and teach a model lesson or coach them. Few school districts employ coaches, and most coaches can only work with teachers that volunteer. Imagine being a member of a professional sports team and telling management that you know what you are doing so you'll be skipping team practices and just showing up for the game. That is the norm in schools. Coaches are told that they need to build trust and relationships with teachers before working with them.

While we were working on developing Power Standards as our internal compass, I wanted teachers experimenting with some new units in order to establish a mountain. My hope is that teachers would try a new unit and then be surprised by how well students performed in it. This would begin shifting some of their core beliefs about what students can and cannot do, and give them a goal to work toward. Unlike the goal of creating a sense-making curriculum, which is too abstract and distant for them to see, this goal would sit on the horizon and lead us to the more distant goal. This work was fairly successful because, by my third year in Holliston, about two thirds of the teachers were volunteering to use my curriculum and about half of them were open to coaching. I had established a goal that most teachers were willing to step outside of their comfort zones in order to achieve.

Selecting the appropriate curriculum for teachers to try in that first year was critically important. The curriculum had to be short and it had to be relatively aligned to their current beliefs and practices. In other words, it couldn't be too radical. But it also had to be effective; they had to be surprised by the outcomes they achieved with it. Administrators often talk about going for low-hanging fruit in order to develop trust and confidence with the staff. You need short-term wins to increase buy-in. But I believe that those short-term wins need to be eye-popping, and the results need to exceed what the staff feels is possible with orthodox methods. If you accomplish something that the staff knows they could have done just by rolling up their sleeves and working together, then you aren't really providing any cognitive dissonance. You build some trust and confidence in the leadership team, but you aren't convincing them to wander into unknown territory with you.

I'm going to throw some numbers at you. These numbers are just a way for me to establish some relative scale, so take them with a huge grain of salt. I have curricula that I have used and refined over the years to the point where my student outcomes look like this: 0% don't get it | 10% get it, but only enough to pass the test | 90% really get it and hold onto it. The normal breakdown is 33% | 33% | 33%. If teachers take my curriculum and use it, 1% will achieve a breakdown of 15% | 10% | 75%, 9% will achieve 25% | 25% | 50%, 40% will achieve 25% | 35% | 40%, and half will achieve 33% | 33% | 33%. This means that half the teachers won't see any gains from using my curriculum and 40% will see some gains, but it will be within the margin of error so treated as noise. 10% of the teachers will see student outcomes that rise above the noise, but only 10% of them will be inspired enough to come bug me for more curriculum. If that is your first tack into the wind, you're going to find yourself sitting in a dead zone.

If you are going to build momentum and make progress, then teachers really need to see better results from these initial experiments. The key is coaching. You need to provide it and they need to accept it. I had two teachers who were open to coaching in my first year. I was also able to teach some model lessons in front of the entire department at mandated professional days and monthly curriculum meetings. Teachers feel more comfortable observing model lessons because they don't feel as though they are being critiqued and evaluated, and they can pick up some valuable instructional strategies just by watching you. It is harder on the coach or the curriculum specialist because you are essentially putting yourself on stage and saying watch while I show you how it is done. The teachers will nitpick your performance, but suck it up. Through word-of-mouth and a new hire, I was able to establish or maintain coaching relationships with four teachers in my second year.

If your coaching and curriculum are effective enough, then you will start to win over teachers and move them to higher levels of buy-in. But you won't move the teachers who start out as resistors and saboteurs and don't experience the results of your coaching and curriculum firsthand. And surrounding them with peers who do want to work with you won't help. When you work with individual teachers, they will start to take ownership in this work if the results are there, but they will see it as a way for them to become better teachers and for them to provide better learning experiences for their students within their classrooms. You may want them to recruit other teachers or to speak positively about your shared work in public forums, but they don't have any incentive to do that, and doing that puts them way out on a limb they don't want to be on. Instead, they will work with you enthusiastically in private, but do nothing in public. It will drive you crazy.

(I actually had a great relationship with the entire science department after my first year. We did some great curriculum work together, including a seventh-grade chemistry unit, and they really saw the potential of Power Standards. The conversations that we had around curriculum at our curriculum meetings were at an incredibly high level. But if there was one teacher present who wasn't in the science department, they would clam up. The one exception was a science meeting where we were meeting with science teachers from the high school and elementary schools. The Power Standards work at those schools were not going well and those science teachers wanted the district to change course. At that point, the middle school science teachers stood up for Power Standards. I was blown away by how articulate they were and how deeply they understood the work we were doing. Unfortunately, we had some key retirements and lost momentum the next year.)

My strategy for overcoming this was to work with an entire grade level instead of individual teachers. To do this, you need to be lucky enough to have an entire grade level of teachers working with you as individuals. I was lucky enough to have this working relationship with all three sixth-grade math teachers, so we implemented a sixth-grade functions unit. If you can pull this off, you accomplish three things:

1. Teachers are able to experience what it is like to work with students who walk through the door with a solid conceptual understanding of the content. Helping them arrive at that conceptual understanding is rewarding, but it is a little hard to know how much they truly understand at the end of the unit. Building on conceptual understanding that is already there takes things to the next level and confirms that what the teachers in the previous grade level thought they saw and assessed was really there.
2. The evidence becomes stronger if the gains generated through a sixth-grade functions unit are then amplified through a seventh-grade functions unit. The signal becomes so strong that it is hard to ignore even if you don't experience it firsthand. I had strong relationships with most of the seventh-grade math teachers at this point, but I elected to start with the sixth-grade teachers because I knew the seventh-grade teachers would be able to amplify the signal. My goal was to win over a few eighth-grade hold outs with this new evidence.
3. Having a consistent curriculum demonstrates the kinds of outcomes you can achieve when all students are developing a solid conceptual understanding. If the students in one sixth-grade math classroom are building understanding, but the students in another sixth-grade math classroom are not, then the seventh-grade math teachers cannot build on anything. Once you see what is possible, you are going to want the teachers at the grade level below you to build understanding so that you can build on top of it, and you are going to want the teachers at the grade level above you to build on what you built so that it isn't wasted. Instead of taking ownership of the curriculum because it helps you be the best teacher that you can be, you start taking ownership of the curriculum because of what it can do for students. And this forces you to publicly advocate for this shared work.

This is as far as I got in Holliston. I left because another opportunity came up and I didn't think that my tacking was going to continue building momentum. As the work progresses, the level of trust and risk-taking required also increases, so you need to be able back things up with increasingly compelling results. To get those results, you need more people working together, and I wasn't getting the traction I needed at the administrative level.

I also feel like I made at least one critical mistake in the tack I took. The new units and instructional practices we were implementing were designed to establish a mountain, a goal for us to take ownership in and shoot for. That mountain was the idea that curriculum and instruction could be built for understanding, and that if we could get all students to develop a solid conceptual understanding, then we could do amazing things together as a school community. I was guiding the way through the first leg of the journey until everyone could see the mountain and we could find a compass, but there was still a really long way to go, and the compass is how we would get there. Remember, no one has walked this trail before, including me.

Jessica was the principal of the middle school when I arrived and she left after that first year. She was the driving force behind the Power Standards work. In the second year, I started having trouble figuring out how to make Power Standards into an effective compass, and when I couldn't figure it out on my own, I pretty much abandoned them. Without an effective compass coming online in the near future, the middle school math and science teachers were relying on me to guide them. That worked initially, but it began to feel that I had conned them into surrendering some of their autonomy and that I had no intention of giving it back. I had taken some expedient steps and steered us into a corner I wasn't sure I could steer us out of. Part of what I've been doing these past two years is figuring out the compass that I use to design curriculum so that I can hand it to someone else. I won't make that mistake a second time.

The Five Levels of Buy-In

I hate the term buy-in, but decided to use it for this post because it is a term most people know. While googling to see if I should use a hyphen in buy-in or not, I happened to stumble upon this rant contrasting buy-in with ownership. It's worth a read and I think it is spot on. (I especially enjoy how he also tears apart the idea of best practices.)

When you do use the term buy-in, you are essentially talking about convincing people to adopt your idea as if it were their own. That should raise red flags. It is disingenuous and more than a little patronizing. But, by definition, a new normal isn't something that people are going to arrive at on their own or through any kind of incremental analysis. You have to start by convincing them to try a new practice that they don't believe in. As you are tacking up wind, you need to start with buy-in, but you eventually want to transition to ownership. If teachers don't take ownership of these new practices, then you aren't shifting core beliefs and those new practices are going to disappear once you stop pushing them. That's not a new normal.

The level of buy-in is one of the compasses I rely on to do this work. If buy-in is increasing, then I am making progress. If buy-in is decreasing or stagnant, then I'm in trouble. Here are the five levels of buy-in (in reverse order) that I look for:

Resistance

This level is fairly self-explanatory. I am flat out refusing to do what you ask.

Sabotage

On the surface, I am doing what you ask. In fact, on the surface, I am cheerleading for the change. But behind the scenes, I am undermining everything you are doing. I have two goals here. One is to co-opt the change process so that I can mitigate it. The second is to insulate myself from any blame when the change process fails. This way, I can say, "I did everything that you asked me to do and it didn't work. Now what?" This makes you look bad and makes it less likely that you'll try to change something else. Failing to detect sabotage and call it out also makes you look weak and desperate for any kind of win.

Compliance

I'm not trying to make the change fail, but I don't actively support it either. At this point, I'm just doing the minimum I can to keep you off my back. If you accept compliance, then it indicates that you are also going through the motions and don't really believe the change will have a lasting or significant impact either.

Professionalism

I have a lot of personal integrity and want to do the best job that I can. This means that I will do whatever you ask me to do. It also means that I will continue doing anything on my own that I think a professional should do. However, I don't necessarily believe that the change is going to have a lasting or significant impact.

It is hard to be critical of this level of buy-in, but it is insufficient when trying to change core beliefs and establish a new normal. Lots of teachers feel an affinity for the Understanding by Design process. As a professional educator, I should be designing backwards from enduring understandings. But when the process does not yield better outcomes, they don't question it. They don't say, "This should have worked, but it didn't. What did I do wrong? I need to dig in and figure this out, and keep at it until it does work." Instead, they feel like they did their jobs and did it well, but things just didn't pan out.

A teacher that goes beyond professionalism and expects a change to yield better outcomes will engage in inquiry and take risks, similar to what I did when my gratitude journal wasn't quite working.

Ownership

This change is one that I want, I have some say in what this change ultimately looks like, and I am part of the group trying to bring it about. I tend to be driven by some goal rather than the change itself. I am trying to implement a change to bring about some outcome. And I am committed to making this happen, so I am prepared to make sacrifices and put myself at risk.

This is the level you need to see if you are going to bring others onboard. If it is just you driving things, then it feels like you are doing this for yourself and not for the organization or some greater good. You can't reach a tipping point if others don't take ownership.

Changing Beliefs and Practices

My goal is to establish some new normals for teaching and learning. This means adopting some new practices. But the old practices that I am hoping to replace are rooted in core beliefs. So, in order to get people to adopt new practices, I also need to get them to adopt new core beliefs. Which do I work on first?

I have had this debate many times with my friend and former colleague, Ginny. I think that we both agree that you need to change practices before core beliefs. To change a core belief, you need firsthand evidence, and you will only get that firsthand evidence if you try something different and get unexpected results.

One of the milestones in Algebra I is learning how to find the equation of a line given two points. The summative assessment for the seventh-grade functions unit we developed in Holliston included this problem:

Find the slope, y-intercept, and equation for the line that passes through (-9, 2) and (6, -4).

Start by finding the slope:

Then substitute the slope and one of the points into the slope-interecept form of the equation for a line to find the y-intercept:

Finally, use the slope and y-intercept to write an equation for the line in slope-intercept form:

Most teachers firmly believe that finding the equation of a line given two points is hard. They regularly see high-performing eighth-grade students struggling with it. Learning it requires a student to memorize a number of steps, and many students are either unwilling or unable to do that.

I, on the other hand, believe that this process is fairly intuitive and accessible to most sixth-graders. You may have noticed that the summative assessment for the sixth-grade functions unit we developed in Holliston contained this problem:

Find the rule to predict the area of the colony on Day x.

Day	Area (cm2)
0
5	32
9	52
13	72
17	92

Okay, we are giving the students four points on the line here, but that doesn't change the nature of the problem.

Start by finding the rate of growth. The bacteria colony is growing at a rate of 20 square centimeters every four days, which is 5 square centimeters per day.

Then find the area of the colony on Day 0. To do that, we work backwards. From Day 0 to Day 5, the colony grew for five days. At 5 square centimeters a day, it grew 25 square centimeters. So the area of the colony on Day 0 must have been 7 square centimeters. (A student also could have counted backwards by 5, finding the area of the colony on Days 4, 3, 2, 1, and then 0.)

Finally, write a rule to predict the area of the colony on Day x.

If the colony starts with an area of 7 square centimeters and grows at a rate of 5 square centimeters per day, then on Day x, the colony will have an area of 5x + 7.

Within a week, we had close to 100% of all sixth-graders doing this type of problem, including special education students in our substantially-separate math class. More importantly, they viewed this problem as trivial and did not feel as though they needed to memorize anything to figure out how to solve it.

By convincing the sixth-grade math teachers to try a new curriculum with new instructional practices, we were able to generate firsthand evidence that caused some of them to reconsider their core beliefs about the difficulty of tasks and what students can and cannot do. It was a small shift, but it was enough for me to build on and introduce even more new practices. Presenting this evidence at the professional day in January to the rest of the department caused a stir, but it did not really move any of the seventh- or eighth-grade math teachers. Because feedback is so noisy in educational systems, it is very easy to dismiss secondhand evidence.

My overall strategy in Holliston was to build working relationships with the math teachers and to encourage a few individual teachers to try some new practices. Because teachers would be experimenting with new practices without a change in core beliefs, none of those new practices could be that radical. However, the new practices had to be powerful enough to generate short-term evidence that would register above the noise. If the firsthand evidence was too powerful to ignore, then a small shift in core beliefs would occur and I could then suggest a few slightly more advanced new practices. This would lead to another small shift in core beliefs and more advanced new practices. I would essentially be tacking into the wind. As evidence mounted and core beliefs shifted, more teachers would take ownership in the work and, together, we'd be able to convince more teachers to take that small first step.

Function Junction, Part 1

I was talking to my coach Sarah this week when I realized that I haven't been doing a very good job of telling my backstory. There are things that I've seen, done, and learned in life that no one, not even my closest friends, really know about. These are things that make me who I am and fuel my vision.

I was the math and science curriculum specialist at the Robert Adams Middle School in Holliston, Massachusetts from 2007 to 2010. In the fall of 2008, I worked with the 6th-grade math teachers to develop and implement a new functions unit. In most middle school math programs, students learn a little about slope in pre-algebra. But their first real introduction to functions is in Algebra I, where they are expected to master a number of discrete skills, including:

• Using the slope and y-intercept to graph a line
• Writing the equation for a line in slope-intercept form
• Finding the slope and y-intercept of a line given two points
• Working with linear equations in point-slope form
• Working with linear equations in standard form

All of these skills should sit on top of a single conceptual framework, but because very few students enter Algebra I knowing anything about slope, there really isn't time to develop a conceptual framework and then get students to mastery in the time allotted. Instead, we are forced to ask students to learn each skill in isolation.

In Holliston, about 60% of our students take Algebra I in eighth-grade, and only half of those students do well on the functions chapter test. I would describe functions as a significant pain point for the eighth-grade math teachers; they really wanted their students to do better on this key concept, but they didn't know how to make that happen.

I decided to target the functions pain point in 2008 for five reasons:

• I didn't have a very good working relationship with the eighth-grade math teachers and alleviating this eighth-grade math pain point would be a big step toward establishing one.
• Functions is something that we could do completely in-house in the middle school. We could introduce functions in sixth-grade without relying on any help from the fifth-grade math teachers based in the elementary school, and we could have everything wrapped up and tied in a bow by the end of the eighth-grade, so we wouldn't need the high school math teachers to do anything different to take advantage of what we had done.
• The math teachers all recognized that procedural approaches weren't working and wanted students to have a conceptual understanding of functions.
• If we could help students develop conceptual understanding and double the number of students reaching mastery, then people would notice and applaud the work we were doing. It would also go a long way toward changing some of the core beliefs held by the middle school math teachers themselves.
• The functions pain point isn't something that any one teacher can successfully tackle on his or her own. This would reinforce the fact that I wasn't pointing my finger and saying that individual teachers weren't doing their jobs. It would also highlight the good things that could happen if we set aside our egos and worked together as a department.

I spent about a month working with the sixth-grade math teachers to design a four-week functions unit for sixth-grade. We did this work after school and during their prep periods. As curriculum specialist, I had no say over how they spent their prep periods, so this was a purely volunteer effort. When we finally implemented the unit in November, I spent some time modeling lessons and coaching teachers in their classrooms.

Here are some of the problems we used in the summative assessment we gave at the end of the unit:

3) At the start of an experiment, a bacteria colony has an area of 22 cm2. It grows at a constant rate of 3 cm2 per day. Fill in the table and come up with a rule for predicting the area of this bacteria colony on Day x.

Day	Area (cm2)
0
1
2
3
4
28
55

4) Bob uses the following rule to predict the area of a bacteria colony on Day x:
area = 21 + 4x. How fast is the bacteria colony growing? What was its area at the start of the experiment? Explain how you can tell by looking at the rule.

5) Find the rule to predict the area of the colony on Day x.

Day	Area (cm2)
0
5	32
9	52
13	72
17	92

Use the graph to answer problems 7-10 and the bonus:

7) Which bacteria colony is growing at a constant rate? How can you tell just by looking at the graph?

8) When is Bacteria Colony B growing faster than Bacteria Colony A? How can you tell just by looking at the graph?

9) What was the average growth rate per day for Bacteria Colony B between Days 4 and 7?

10) Describe the growth of Bacteria Colony B over time. Explain when it is growing faster or slower, when it is shrinking, and when it is staying the same size.

11) Bonus: Write a rule for the growth of Bacteria A over time.

We spent a professional day in January analyzing the results of the summative assessment as a department. 85% of the students scored a B or higher. The seventh- and eighth-grade math teachers were floored by both how well the students did and the quality of their written responses. We estimated that 75% of the students had a solid conceptual understanding of linear functions and could fluently translate among tables, equations, graphs, and written descriptions as long as functions were presented in a real-world context. 55% of the students completed the bonus correctly even though we had never asked them to write an equation from a graph with a fractional rate of change before, and another 20% understood what they needed to do, but just made a procedural error somewhere along the way.

One of the sixth-grade special education teachers worked closely with us as we developed the sixth-grade functions unit. She used a pared down version of the unit with the students in her substantially-separate math class. (A substantially-separate math class is for students with math learning disabilities so severe that it has been determined that their needs cannot be met in a general math class.) Her students loved the unit and got 75% of the pattern, relationship, and algebra questions on the state test correct when they only got 45% of the questions in the other four strands correct. The state test was administered in May, about five months after she taught the functions unit.

In 2010, I developed the seventh-grade functions unit with the seventh-grade math teachers. By this time, the first group of sixth-graders that had taken our sixth-grade functions unit were now seventh-graders. The seventh-grade math teachers had been excited by how well these students had done on the sixth-grade summative assessment, but they were anxious about how much those students would actually retain. We decided to start the unit with a warm up activity designed to re-activate what the students had learned sixteen months ago.

Although the students said that they had forgotten everything about functions from sixth-grade, the warm up activity demonstrated that the conceptual framework for functions they had developed was still there, and everything came flooding back to them.

Here are some of the problems we used in the summative assessment we gave at the end of the end of the seventh-grade unit:

A plant is growing at a constant rate. Use the table to answer problems 1-5.

Day	Plant Height (cm)
0
20	17
28	23
36	29
44	36

1) What is the rate of growth of the plant? Explain how you found it.

2) What is the starting height on Day 0 of the plant?

3) Write a rule for the height of the plant (y) after x days.

4) Use the rule to find the day when the height of the plant will be 80 cm. Write the rule and show all steps.

5) Create a Plant Height vs. Time graph.

Use the following rule to answer problems 6-8: w = 50m/3 + 100 where w = gallons of water in a tank and m = minutes.

6) Use the rule to fill in the rest of the table. Show all steps.

Minute	Water in Tank (gal)
0
10
15
	620
	800

11) Graph: y = 3x/4 − 2

13) Write an equation for the given graph.

In problems 14 and 15, graph the two given points, then find the slope, y-intercept, and equation for the line that passes through those two points.

14) (-9, 2) and (6, -4)

15) (-8, 2) and (-5, 10)

16) Explain how you found the slope for the graph in problem 14.

17) Explain how you found the y-intercept for the graph in problem 15.

18) Challenge: Find the point (x, y) where the two lines in problems 10 and 11 would cross if they were on the same graph.

We found that the 75% of the students who had developed a solid conceptual understanding of functions in sixth-grade were able to pick up immediately where they had left off and build on what they had learned. In fact, the seventh-grade math teachers found that they had to do very little teaching in order to get students to generalize from real-world contexts to abstract x's and y's, or to introduce fractional rates of change. However, the 25% of the students who were shaky after sixth-grade got left in the dust.

I left Holliston after the 2009-10 school year, but the plan was to develop a new functions unit in eighth-grade. The 75% of the students with a solid conceptual understanding should have easily been able to extend what they knew to linear equations in standard and point-slope forms. This means that all of the eighth-grade Algebra I students (instead of less than half) should have mastered linear functions before moving on to Algebra II in ninth-grade, saving the high school math teachers 3-4 weeks of re-teaching time. And the students that didn't take Algebra I should have been much better prepared for Algebra I in ninth-grade. All of this should have then opened the door to a district-wide conversation about collaboration and curriculum and instruction.

Within the middle school math department, I was hoping that the success of this conceptual approach to functions would lead to a willingness to work together and to try new instructional strategies. Up until this point, getting teachers to try some new curriculum hadn't been too difficult, but getting them to accept coaching was like pulling teeth. And new curriculum was substantially less effective without the necessary coaching and shifts in practices. My next goal would have been to analyze what we were doing and then to try to increase the percentage of students developing solid conceptual understanding. If you've been reading my blog, then you know that the first step is to design a curriculum that breaks down tasks to intuitive subtasks. I think we did that in the sixth- and seventh-grade functions units. The next step is figuring out how to help students learn how to perform those tasks. We got to 75%, which is excellent for our first try, but now we needed to push that up to 95%+. Doing that would mean challenging even deeper core beliefs and forcing more changes in how we operated as a middle school and district.