-by Krista Taylor
“Verify that P = (1,-1) is the same distance from A = (5,1) as it is from B = (-1,3). Find three more points that are equidistant from A and B. Can points equidistant from A and B be found in every quadrant?”
I’m sorry, what?
It was the first day of math training, and this was the initial task. I had signed up for this professional development opportunity because Jack said it would be good, and because I have spent a significant amount of time over the past several years trying to improve my math instruction. Because, you see, I was “bad at math,” and that is not something that I ever want to pass on to my students. As a result, I have worked hard at becoming a better math teacher.
Math has never come as easily to me as other subject areas. In the 7th grade, I was “honors-tracked” in all subjects. I only stayed in honors through the end of the 8th grade, but by then the damage was done because my course of study in math had already been accelerated. I skated through Algebra II in the 9th grade, and very nearly failed Geometry in the 10th. At that point, I stopped taking math altogether. My advisor told me that I shouldn’t even consider taking Chemistry “because of all that math,” so science went as well. While that opened a lot of time in my schedule for terrific courses like Art History, and the History of the Theater, ultimately quitting math early did me no favors.
What irony then that teaching math has been a part of my job description throughout my career.
It wasn’t until I spent three years co-teaching math at Gamble with Katie Doyle that I began to feel a sense of competency. This was reinforced by the occasional proclamations of my own children when, upon returning home from work in the evening, I would hear, “I’m glad you’re finally home; I’ve been waiting for you to help me with my math homework!” Every time, I was tempted to turn around and look for whoever most certainly was coming in the door behind me. “I’m sorry, you’re waiting for who to help you with your math homework?”
Which brings me back to the math training. I want to be a better math teacher. I want teaching math to feel as fluid for me as teaching English does. I want to be certain that I am serving my students in the best way possible. To that end, I know I need to keep working at math. So, I committed to four summer days of math PD.
Which brings us back to, “Verify that P = (1,-1) is the same distance from A = (5,1) as it is from B = (-1,3). Find three more points that are equidistant from A and B. Can points equidistant from A and B be found in every quadrant?”
I wanted to raise my hand and say, “Excuse me, but you see, I think I’m in the wrong training. I want to get better at teaching math, not doing math!”
I wasn’t nearly that brave, so instead I did this. (Okay, with the group I was working with, and assistance from the teacher, I did this. It’s still pretty cool.)
I’ll admit it, as an adult learner, the math was interesting. We were working on problems from the 9th and 10th grade math curriculum at Phillips Exeter Academy in New Hampshire.
Yes, THE Phillips Exeter Academy. Arguably the most prestigious 9th-12th grade prep school in the country. Tuition for boarding students at Exeter is $47,000 annually; for day students, it is a mere $36,500, and there are rigorous admission criteria. The average class size is twelve; the student-teacher ratio is five to one. THE Phillips Exeter Academy. Full stop.
- There is no textbook, only a binder full of problems.
- There is no direct instruction.
- There is no modeling.
- There are no examples.
- Nearly every problem is a multi-step, word problem.
The structure of the class is such that when students enter the classroom, they immediately begin putting answers on the board from the homework the night before. Then the group discusses each problem, assessing accuracy, determining alternate methods, revising the work, questioning the results. The teacher’s role is to provide a few prompts and probing questions to deepen the understanding of the group, and to correct inaccuracies. Once all of the problems have been explored, new homework is assigned to be analyzed the next day in class. That’s it. That’s the entirety of the instructional process each day.
But, at the end of Day 1, I found myself feeling aggravated. I wanted to get better at teaching math to MY students. My 7th and 8th grade students at Gamble Montessori High School in the Cincinnati Public School System. How in the world was spending 8 hours a day for 4 days completing Exeter math problems going to help me to do that?!
Exeter students are not my students.
Exeter students pay tens of thousands of dollars in tuition each year. Seventy percent of my students are eligible for the Federal Free Lunch Program. Exeter students must demonstrate academic excellence in order to be admitted to the program. Thirty percent of my students are identified as having a disability; many more struggle with significant skill gaps. Exeter students either live at school full-time or remain on campus until 8:00pm each evening. My students go home at the end of a 7-hour school day, and some of them experience significant stress in those home environments. Exeter students are instructed in class sizes of 12. My students are in classes with 25-35 of their peers.
Exeter students are not my students.
The second afternoon, during a break, I had a casual off-hand conversation with Sami Atif, one of our instructors who is a math teacher at Exeter. We were discussing the make-up of the student body at Exeter.
He said, “It’s a cultural thing.”
Taken aback, I defensively asked, “What do you mean, exactly?”
His response surprised me. “It’s about culture. These kids are empowered. They don’t hesitate to question a teacher or a problem. I don’t think I ever questioned a teacher when I was growing up. I wouldn’t have dared. These kids don’t have that issue. It’s about power. They believe they have the right to question and to speak up.”
The conversation lagged, the break ended, and we went on with class. But his words hung with me. I revisited them on the drive home, and at some point before I went to bed that evening, it hit me like a kick in the gut.
“These kids are empowered. They don’t hesitate to question a teacher or a problem . . . It’s about power. They believe they have the right to question and to speak up.”
I want that! That empowerment? That questioning? I want that for my students. I don’t care how I get them there. If this math strategy is what will give them that, then I want that for them, and I will do whatever it takes to get it for them.
Days three and four of the training were different for me. I was all in. Not just for me, for my students, too. It helped, of course, that I was witnessing my own math development in action. I was thrilled on Wednesday evening to discover that not only was I able to approach nearly every problem assigned for homework, I was able to get to an answer that I was near certain was correct.
Here is an example: “Let A = (-2,4) and B = (7,6). Find the point P on the line y=2 that makes the total distance AQ+BQ as small as possible.”
That was the change in me after three days of practice.
And I started to observe the instructors. This was far more than a curriculum; it was a methodology. The first thing I realized was that they never (never!) provided or confirmed an answer. This prompted more than one person to question whether the instructors actually even knew the correct answers! Instead they met questions with questions and provided guidance in the form of suggestions or references back to previously constructed knowledge.
By this point, I was writing down everything they said because I know from past experience that when looking to make a shift in practice, sometimes you have to “fake it ‘til you make it.” I was seeking a script, so I allowed the instructors to unknowingly provide it for me themselves. Here are some of their prompts:
“It looks like maybe you were thinking . . . “
“Are there any other ways to get there?”
“That’s a step I want to process more.”
“That’s really interesting.”
“Are there any other ways to look at this?”
“Are you convinced that you’ve found the correct answers?”
This strategy is known as “Harkness teaching” as it was first conceived of by Edward Harkness, an oil magnate who gave a significant donation to Exeter Academy for implementation of a teaching style that he described thusly:
“What I have in mind is a classroom where students could sit around a table with a teacher who would talk with them and instruct them by a sort of tutorial or conference method, where each student would feel encouraged to speak up. This would be a real revolution in methods.”
Oh, yes, Mr. Harkness, I, too, see your vision as revolutionary.
And yet I remain haunted by the question of “how.” How can I possibly implement this in my classroom – keeping in mind that many of my 7th and 8th grade students arrive with math skills that are expected from a 4th or 5th grader. What can I do to help them to reach this level of math confidence and comprehension?
What I didn’t realize initially was that this work had already been begun by Savannah Rabal, a junior high math teacher at our sister school, Clark Montessori. Savannah was out of town for the first two days of the training, but when she arrived on the third day, I began picking her brain for how she had done it. Her wise words, “Trust the Process,” provided me with hope in my ability to implement something similar in my own classroom.
Here are some of the expectations that she and her class developed for working with this type of instruction.
So perhaps it is possible to do something like this after all – to provide my students with the opportunity to work collaboratively with their peers solving high-level math problems through exploration, discussion, discovery, and critical thinking. I do not know yet exactly what implementation of this methodology will look like in my classroom, but here are my thoughts so far:
- Begin with just 1 day a week
- Provide direct instruction in expectations for the process; allow for student input and suggestions as we identify strategies for working together
- Establish small groups that would work together consistently
- Groupings could be heterogeneous, allowing stronger students to support those who are struggling
- Groupings could be homogenous with differentiated questions, allowing strong students to work together toward acceleration, while struggling learners would be obligated to take risks and make attempts to approach the task without the support of their typically-relied upon peers.
- Groupings could be a flexible combination of both homogenous and heterogenous groupings, allowing for the benefits of both options
- Develop scaffolded supports to support student exploration and learning
- Teacher prompts
- Written structures such as guiding prompts and organizational supports
- Pre-select appropriately leveled questions or design our own
- Begin the process by working the problems together in class rather than expecting students to tackle them independently as homework in the initial roll-out phase
I’ll be honest. I am nervous as all get-out to even attempt beginning this process. There seems to be so many hurdles in the way. The challenges my students face with math content is just the beginning.
How on earth will I get my colleagues on board? They will not have the benefit of a four-day experiential training to elicit their buy-in; they will only have me (and Rosalyn and Erin, who also attended the workshop) waxing prophetic and showing them the materials that at first glance seem utterly ridiculous.
Even if I only implement this approach one day a week, it will throw us further off the curriculum content pacing that the district expects. How can I demonstrate that this is beneficial enough to make it allowable?
What will the parents think? Savannah already had this experience when a parent contacted her saying, “So, I hear you don’t teach math anymore.” Many parents are already wary of Common Core math, and already feel beyond their ability to assist with junior high-level math. What will they think when we throw this at them?
And what will happen when it doesn’t go as I have planned? In fact, the only thing I am certain of is that it won’t go exactly as I have planned. What then? Will I have the courage to stick with it? Will my students? Will my fellow teachers? Will my administrators?
It helped to discover this excerpt by Elisabeth Ramsey in the Exeter “Introductory Math Guide – Written For Students By Students.” It feels a bit as if she was writing directly to me regarding my apprehension about implementation, “I learned one of the more important lessons about math at Exeter; it doesn’t matter if you are right or wrong. Your classmates will be supportive of you, and tolerant of your questions. Chances are, if you had trouble with a problem, someone else in the class did too. Another thing to keep in mind is that the teacher expects nothing more than that you try to do a problem to the best of your ability. If you explain a problem that turns out to be incorrect, the teacher will not judge you harshly. They understand that no one is always correct, and they will not be angry or upset with you.”
And I continue to hear Savannah’s words echoing in my head, “Trust the Process.”
And Sami’s comment, after I acknowledged him for the powerful impact his words had on me, “Yeah, this is social justice work.”
So, remembering the feeling of: “I want that! That empowerment? That questioning? I want that for my students. I don’t care how I get them there. If this math strategy is what will give them that, then I want that for them, and I will do whatever it takes to get it for them,” I am ready to take the plunge. I’ll let you know how it goes.