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.
Tuesday, January 28, 2014
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.
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:
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.
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.
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.
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.
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.
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 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.
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.
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 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?
Sunday, January 19, 2014
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.
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.
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:
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.
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).
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.
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.
Wednesday, January 15, 2014
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.
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.
Tuesday, January 14, 2014
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:
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.
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.
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.
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.
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.
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?
Finally, a bonus problem in case anyone wants to see my problem-solving strategy applied to a second problem.
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.
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.
Friday, January 10, 2014
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.
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.
Tuesday, January 7, 2014
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:
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.
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:
- 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.
- 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.
- 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.
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