Before I begin, I want to thank David Martin, Rod Lowry, and the rest of the MCATA conference committee for inviting me to share with you tales from my educational journey. I also want to acknowledge their hard work pulling this conference together. I have been on several conference steering committees, co-chairing a few of them, and I can tell you that managing a conference can be stressful. So let us to take a moment to recognize their efforts.
I stand before you today experiencing a combination of excitement and nerves. I am excited to be here for your fiftieth MCATA conference. Congratulations on this milestone. I am nervous because it is has been a long time since I have talked at anyone for 60 minutes straight. My wife, Kathy, assures me that’s not true. She says she has seen me go on for hours about educational issues. I do not question her reality but I see this differently.
Still, I believe that I have come up with an interesting way to share with you my ideas about transforming mathematics education. However, I know that 60 minutes is a long time and it will be easy to disengage along the way. You do not need my permission but you have it to take notes, discuss ideas with those near you, or tweet your thoughts to the world – whatever it takes to stay engaged.
I think it was Jeff Wilhelm, an education researcher from Boise State, who wrote that Vygotsky saw analogy as one of those supports that helps learners to successfully explore the edges of their understanding – their zone of proximal development. This is affirming to me because analogies are a mainstay in my teaching practice as I urge learners to move from the known to the new. Consequently, I apologize in advance if I take too far the analogy between educational reform and roads. My intent is that the analogy will provide us with a common context that can help you connect with my efforts to improve my practice – that the roads I have traveled and the methods I have used to ensure that I am headed in the right direction will be more recognizable to you.
Before we explore those roads less traveled, let us consider what traditional math teaching looks like. I would like you to please take a moment to visualize the roads traveled in most math classrooms. Now, would you please turn and talk, sharing your vision of the traditional math class with someone sitting near you.
This is my vision. It represents an efficient model for covering content someone else has deemed important. A group of students follow behind a trained adult to some destination. The adult is more tour guide than bus driver, however, pointing out the highlights and shortcuts as they zoom along. I imagine Chevy Chase up front saying, “Look kids, invert and multiply” This is essentially what I see when I visit math classrooms to do student teacher observations.
Recently, a student teacher complained to me that he was unable to do any meaningful formative assessment because of the amount of material he had to cover. I argued that there are ways to fit formative assessment into any lesson. He responded that it would not matter since the lessons were already set for the marking period regardless of whether or not the students were learning.
When teachers are expected to introduce a new objective every couple of days, it leaves lessons feeling like expressways where it’s unsafe to let learners take the wheel. But there are casualties, nonetheless. I want to tell you the story of one of those casualties.
This is my step-son, Andrew. When he was in elementary school he would ask Kathy to read him stories at bedtime. He would ask me to give him algebra problems. We would do problems until he got stuck on one and then he would go to bed to sleep on it. Flash forward to middle school and a conference with his seventh grade math teacher. Right off the bat she told us she was concerned about his ability to do math. “Really?” I said, “How so?” (I don’t think she realized that I was a mathematics educator.)
She gave the example shown here. “We are working on raising a negative number to an even power and I shared this trick with the class. If you draw a line between the base and the exponent without hitting a parentheses then the answer will be negative.” Here, she held here forearm horizontal to reinforce the relationship between the line and the negative sign. She continued, “If the line between the base and exponent hits a parentheses, then the answer is positive.” Now, she added her other forearm to represent the vertical parentheses and make the sign associated with positive numbers.
She was clearly pleased with this trick and I understood why. This little shortcut would be easy to apply on a test as long as the test-taker remembered all the conditions associated with it. Quick and easy and onto the next stop.
Andrew had responded by asking what if he drew the blue line. The teacher was not amused and expressed her concern that Andrew was not taking his learning seriously. I bit my tongue because Andrew had asked us to not make a scene, but I often wonder what would have happened if I had challenged this teacher to teach mathematics not gimmicks. Andrew ended up failing the section. Worse, yet, he was turned off by mathematics for the rest of middle school and high school. All was not lost, however.
After high school, Andrew attended a traditional building school where he learned, among other things, how to design and build post-and-beam structures. He came home excited to tell us about what he had learned and asked me about trigonometry. It turns out he needed to know trig in order to design roofs. I pointed him to some resources and he taught himself the trig he needed to be successful.
Obviously, this is a single story, and one very personal to me. I can easily be accused of being biased. However, given that you are at a conference with the theme of looking for roads less traveled, I would imagine that you are also uncomfortable with aspects of the educational expressway we find ourselves on. So the question is, where and when do we exit?We gave the Emerson quote to Andrew in elementary school. It accompanied a picture of a sled dog team making its way through the wilderness. Andrew loved sled dogs and that was the main reason we gave him the picture, but we agreed with the sentiment as well. We wanted all of our kids to be explorers. Given the message, we were no doubt complicit in his rebellious nature in seventh-grade math class.
My goal for you is the same: to be explorers. So this is my exit. You will need to find your own path to improving your teaching practice. There is no single answer to effective mathematics teaching. It depends on what you teach, how you teach, and who you teach. But if I’m not going to tell you how to teach, then what is this talk about? It is my hope that you will concentrate on the signposts I have used to monitor my progress and maybe find them useful in guiding your own efforts.
Change is a necessary but not sufficient condition for educational reform. We must consider changes with a goal in mind and benchmarks in place to monitor out progress. Otherwise we are just flailing about – one of the biggest problem the US has when it comes educational reform (as described by Stigler and Hiebert in their book, The Teaching Gap.)
In a recent TED Talk, Tim Harford describes how complex problems are often solved using a trial and error process. We try something. We gather data about its effectiveness. We make an adjustment based on some benchmark. We repeat. We make progress. I think we can agree that teaching is a complex system and so we need to be willing to use a trial and error approach in order to make progress. But this means being able to monitor our progress toward our goal. We need signposts along the way.
What do these signposts look like along the roads less traveled in mathematics education? For me, there are four things I use to monitor the effectiveness of my efforts to make changes in my educational practice. Is the effort sustainable? Is it intentional? Is it gradual? Is it natural? If you aren’t too weary of this analogy, I hope you will join me as I describe each of these signposts in greater depth.
Sustainability has nearly always been a concern to me as it relates to education. (Not in the sense of a paperless classroom, although I do want to move in that direction.) I want to know whether or not the teaching and learning associated with my practice is sustainable. Will the learning last? Will I last as a teacher?
As a preservice teacher in the 1980s, I was fortunate to have been part of a teacher preparation program at Northern Michigan University that was already suggesting that we needed to change roads. One of my assignments was to come up with 50 math activities that I could use in my classroom to replace a traditional lecture. I completed the assignment and then used many of the activities during the first years of my teaching. But these tasks represented dead end streets.
The content was disconnected and therefore the learning did not last. Educational research shows that topics learned in isolation are more likely to be forgotten than those built on prior knowledge. Consequently, high school teachers were constantly asking me if their students had gone over fraction computation. I assured them that the students had learned it and passed the test, but I could not explain why their learning had not lasted.
Years later, a math teacher I was working with put it into perspective. “If the students didn’t learn it, then did I really teach it? And if they don’t retain it, did they really learn it?” It is a scary thought. Clearly, teaching disconnected content was an unsustainable approach for me and so I tried a new road.
This picture is the Seney Stretch – a 25 mile straightaway in Michigan’s upper peninsula. It represents the next phase of my journey. I took the dead end activities and tried to put them together into some cohesive whole. These connected ideas formed a straightforward road my students could follow. I have since heard this referred to as funneling or pseudo-teaching. Frank Noschese has done some nice excellent writing about this on his blog.
Richard Skemp might classify this as instrumental understanding. A person knows one way to solve a problem but struggles if new constraints are added. While the content was remembered, they struggled to apply the content to new situations. I can now put words to the concern that I felt, “If they cannot transfer it, did they really learn it?” This approach was an improvement but was also unsustainable.
I also want to point out that I was two lanes away from reconstructing the expressway I had eschewed. Fortunately, fate intervened in two ways. First, I was accepted to a mathematics education masters program at the University of Michigan where I worked with Art Coxford and Joe Payne. They challenged me to think deeper about what it means to do mathematics with understanding. I was also tapped by my district to be trained in outcome-based education – a combination of problem-based learning and standards-based grading. As a result, I began developing units that would ask my students to explore and apply mathematics.
I finally began to allow my learners to engage in authentic experiences. The first unit, a nine-week project, asked students to design and build a scale model of their Dream House. It incorporated content like measurement, decimal computation, and ratios along with the National Council of Teachers of Mathematics Process Standards. In retrospect I wonder about the sustainability of this approach as well since the learners remained reliant on me to set the path. What would happen when I wasn’t around?
Gary Stager loves to say, “Less us, more them.” This was certainly true for my new units when it came to in-class instruction but I still controlled the assessment, the evaluation, and the planning. I was watching a recent keynote where Gary also reminded me that learning is a verb not a noun. Looking back, I realize I was still focused on teaching a product to my students.
Recall that another element to educational sustainability is the question, “Will I last as a teacher?” An often repeated statistic states that roughly 50% of new teachers quit within the first five years. I lasted longer than five years but only two doing these types of intensive project before I moved to teaching at the university level. Because I was single and living in rural Michigan, I could afford to put a lot of time into the design, implementation, and grading associated with these units. I do not think I could have lasted much longer, though.
From both perspectives of sustainability, I was doing too much. Whatever I do, it needs to be sustainable for me and for my learners. And that means I need to focus my attention on what is important.
This means that I must ask myself, “Does my path represent some intentional purpose or is it simply the latest fad?” You see, intention can be a powerful predictor of success - perhaps more powerful than any single educational approach. In The Dreamkeepers Gloria Ladson-Billings describes two teachers with vastly different teaching methods but identical intentions. One is fairly tradition. The other follows a more reformed approach. Both are successful because they are purposeful in their teaching and make building relationships a priority. Because they had a purpose for doing what they were doing, could articulate it, and had ownership in it, the intention informed their practice and led to positive results.
I have also learned the importance of sharing my intent with my learners. A few years ago I was teaching a seminar for student teachers which included classroom observations. I remembered my own experience with these observation as trying to impress my university supervisor. As a result, I learned to hide my flaws rather than reach out for help. I do not want to encourage this behavior in the student teachers I work with. Therefore, I have made the decision to use a coaching model during their observations.
In order to model what coaching looks like, I asked a literacy coach to observe one of my seminars and engage in a debriefing session afterwards. I taught the lesson to my student teachers with Bekki, the coach, watching, and then we debriefed with the student teachers watching. Bekki asked me how it went and I responded, enthusiastically, “Great! I really think I stayed focused on the objective.”
She asked, “And what was that objective?” Again, a bit too emphatically I pointed to where the objective was at the top of my lesson plan. This was going well.
But then she asked, “Do you think they knew this was the intent of the lesson?” Rather weakly, I mumbled that I could not be sure. So she asked the class to raise their hand if they knew what I intended for them to get out of the lesson. Thankfully they were honest and not one hand went up. I knew what I wanted to accomplish but I had not included the learners in on this secret. Somehow, I thought that if I was good enough this would be obvious to them.
Now I make it very clear during the focus of my math workshops what I want my learners to concentrate on. For example, when the task involves exploring conjectures related to whether 3/19 terminates or repeats and when, I make it clear that I do not expect them to complete the task in the time allotted. This is because I want them to attend to the process of making conjectures not the product of an answer. My intent is clear to me and to them. I want them to wander but that does not mean that they are lost. They have a purpose.
Such intense focus means that sometimes we will be facing significant climbs.
This means that I must ask myself, “Am I offering support and guidance that will lead not only to learners being successful but also being self-sufficient?
The picture here is of Walter’s Wiggles at Zion National Park in Utah. These switchbacks were built to make it possible to ascend what would otherwise be a steep, nearly impossible climb. Novices facing daunting mathematical content can become lost and frustrated when not provided the appropriate support.
I must admit that there was a time when I treated student frustration with a task as a badge of honor. After all, isn’t it through struggle that we learn? Yes, but struggle is different than frustration. When learners experience frustration they tend to retreat into what Seth Godin calls the lizard brain – the brain stem. What I realize now is the need to find ways to support learners in ways that make them more and more comfortable exploring the edges of their understanding.
The gradual release of responsibility is an instructional model based on the research of Pearson & Gallagher (1983) and the approach I find most useful in supporting learners to become self-sufficient. It begins with modeling (I do as you watch), followed by mentoring (we do together), and ultimately leading to monitoring (you do as I watch). Most math classes I observe operate at the extremes – lecture (a form of modeling) and independent practice or homework (which would be associated with monitoring) – while ignoring the mentoring.
In our analogy, modeling might be associated with a knowledgeable trail leader thinking aloud as she treks through the wilderness. How is this different than the tour guide on the bus? First of all, it is in the intention. This leader wants to prepare those following her to takeover the responsibility for the journey. The tour guide may have that same hope but rarely turns over the wheel. Second, there’s the delivery. This leader uses “I language” rather than telling others what to do. It is a powerful switch. When coaching teachers who want to do less lecturing, I often suggest this subtle shift. "Instead of telling students what to do I share how I would do it. Afterward, I ask them to recount what they saw me do and say. This becomes the assessment that informs whether or not I can move on."
In the mentor phase, the leader and the learners work together to find the path. Sometimes the leader starts off and turns over the leadership to the learners along the way. When the leader thinks they are ready, she might let the learners start out but she is always on the alert to offer guidance. This is akin to Schoenfeld’s research (PDF) on improving the problem solving abilities of college students. Without support, novice problem solvers often get stuck in trying to solve the problem using a single approach. Schoenfeld found that he could improve problem solvers’ metacognition through coaching during what was essentially the mentoring phase.
Once the learners have demonstrated that they can be successful on their own, the teacher provides opportunities for independent practice. This is more than simply applying what they have learned. The expectation is that they will synthesize the content and the processes in order to forge new paths. And in the long run, the goal is that they will be able to be self-sufficient in any learning environment.
I have begun to use the following rule of thumb to monitor my progress in this area: each lesson ought to be preparing learners for the eventuality of me becoming obsolete in the classroom. In other words, if my classroom practice looks the same in December as it did in September, then I’m doing something wrong. I do the heavy lifting, the intense preparation at the beginning of the school year when I have the most energy, and then later I can concentrate on assessing learners progress as they take the lead.
Still, there is a great deal of work to do, even in this model, and to be sustainable I need to leverage ways that people learn.
The act of learning is a natural activity. I must constantly consider to what extent is my teaching is supporting or inhibiting learning.
When Kathy and I walked the 70 plus miles along the Great Glen Way, a canal that splits Scotland, we noticed how much the builders used the existing rivers and lakes to make the construction more manageable. Built centuries ago, the Scots took advantage of the natural contours of the land.
In my practice, I need to leverage the brain’s natural ability to acquire and use language. In his TED Talk, How language transformed technology, Mark Pagel’s highlights the various ways our brain uses language to learn. He suggests that it is what distinguishes us from other animals. Why not take advantage of this ability?
Most anyone who has learned a second language later in life will tell you that it is a difficult endeavor. Yet, we did it effortlessly as children. Why is that? Some of it is the stage of our development, but Brian Cambourne suggests that it also has to do with the conditions associated with language acquisition. In his book, The Whole Story, (and this article) he shares his research on how children learn to communicate and how we might apply this to literacy instruction.
What does it mean for us as math teachers? Well, these same conditions can support our efforts to make learning more natural. These conditions are immersion, engagement, responsibility, use, approximation, demonstration, feedback, and expectations. Let us briefly look at each one.
Immersion: Learners need to encounter a variety of mathematical experiences. They need to see examples of mathematics across the arc of history, how it applies to everyday situation, and even ways their peers use mathematics to solve problems. This needs to be an overarching theme in the mathematics classroom.
Engagement: In Cambourne’s learning plan, this is the foundation of all learning. Learners need to see purpose for what they are learning. They need to believe that they have the potential to be successful. And they need to feel protected from harsh consequences for being wrong.
Responsibility: Cambourne wrote, “Learners who lose the ability to make decisions are disempowered.” This is what I see in most of the secondary classrooms that I visit – a group of students waiting for someone else (the teacher) to tell them what to do. We need to find ways to offer learners a chance to make choices when learning.
Use: Learners need to have opportunities to practice what they are learning in authentic ways. They need to explore new mathematical ideas in order to develop control over their use and construct understanding. This is not mindless practice. It is thoughtful practice.
Approximation: Learning is messy and we need to accept that as a natural part of the process. We see this as children try to master language. A child reaches for a bottle and says, “Da baba!” Most parents accept this approximation as a natural part of learning language. They do not slap the child’s hand telling him it's wrong and to do it again. Pink Floyd had it right, this kind of behavior builds walls.
Demonstration, Feedback, and Expectations: It is much more likely that the parents will respond with excitement to "Da baba." After all, the child has just proven that she is the smartest, most precious child ever. They say, “That’s right, honey. That’s a bottle.” This kind of response both supports the approximation while setting the expectation and demonstrating proper language use. We have also seen the importance of sharing expectation at the intention signpost and the power of demonstrations under the gradual signpost.
Cambourne saw those conditions to the left (in purple) as being primarily related to learners' actions and those to the right (in green) initially being the teacher’s focus. As time goes on, the hope is that learners take on each of the conditions as they take control of their own learning.
Take in Kathy’s session or mine for concrete examples of how we apply these conditions in our classrooms.
Sustainable, Intentional, Gradual, and Natural – those are the signposts I currently use to check whether or not I am making progress as I explore the mathematics education landscape. I know they might not match where you are at. By the end of this conference, I might hear or see something that causes me to reevaluate them myself. A framework ought to be used only as long as it is providing support for growth. Once it becomes restrictive, it ought to be adjusted or abandoned.
For those concerned about what effect leaving the educational superhighway will have on your learners, let me try to reassure you with one last analogy. I learned to drive on roads like this one. My driver’s education teacher thought it was important to have practice on this kind of terrain. It was slow going but I learned a lot about being aware of my surroundings and having a feel for the road. When I went off to college in Los Angeles, the roads were much different than this. Yet I was able to apply what I had learned on dirt roads in the UP to navigate the expressways of LA.
In 1987 Myles Horton and Paulo Freire decided to “talk a book.” The result was We Make the Road by Walking, based on a phrase borrowed from Spanish poet Antonio Machado. “Searcher, there is no road. We make the road by walking.” This quote describes the process they took in writing the book. It is my hope that my sharing will be the start of a long conversation that begins here at MCATA.
So how do you teach math and how do you know if your teaching is successful? It’s your turn to share. And not just those of you who are presenters. If you don’t already, consider writing a blog or sharing your ideas through some other social media like Twitter. Maybe you could participate in the upcoming edcamp Edmonton on November 5th. Whatever you do, start blazing a trial. We all have a responsibility to contribute to constructing these new educational roads. Thank you in advance for your participation.