Originally written 4/30/18.

In a recent math class, we were doing some introductory explorations about lines. We did an activity that Fawn Nguyen shared, called Staircase Steepness. It has students rank 6 drawn staircases by steepness, from least to most steep.

At first, students had to rank them without measuring things. They struggled with how to explain how they determined which was more or less steep. We talked about real staircases. We talked about the rise and tread of staircases. We talked about how we could “eyeball” the staircases and see which seemed more steep.

They then had to compare their rankings with peers and make decisions on which were more or less steep. Then, they got to measure. They did all sorts of different measurements. Some measured a diagonal line in different locations. Some measured the “tread” and the “rise” and then did various operations with them.


It was an interesting conversation.  One student, MV, asked if we could just do ramps (which is kind of where a “slope” lesson ends up).

As an exit ticket, I had students share something they learned or something they wondered. Some noted that there was a possibility to use the Pythagorean Theorem if you made a triangle.  A couple of them used protractors to look at angles from the base of the staircases to where the top step began. One asked, “Does it matter if the tread and rise are proportional?” One asked, “Why would we measure just one step at a time?” These are awesome questions that, when explored further, will allow for a deeper understanding of slope and rate of change, and they came about by students looking closely and noticing things they would definitely have overlooked if I had shared the “rise over run” or “change in y over change in x” explanations right away.

Another asked, “Do we use the measurements of the steps or the length and width of the whole stairs?” and “Does slope have anything to do with the activity we just did?” and “If so what are the different parts of slope and can we learn it soon?” (This student has been working ahead with her dad and still has many gaps in her understanding, but she does have a LOT of information that she often tries to share, even if I hold her back a little. I worry that jumping ahead to the procedure and the information she wants to lead us to would force us to miss out on greater conceptual understandings. I appreciate her eagerness to be there, but not everyone is where she is! Yet.) So, YES, the answer is that we ARE doing slope, but I’m not teaching procedures at this time.


The next day, we looked at the Desmos activity, “Which is steepest?” which was the next step and connected to MV’s question about doing ramps. It looks at lines and has students select the steepest, ask for information that would be helpful, then asks them to create a line with a steepness between two other lines’ steepnesses. (At this point, a student said, “If we call deepness ‘depth’, why can’t we call steepness ‘stepth’?” So… now we refer to stepth, which will soon turn to “slope” and then be inexplicably written as ‘m‘.

Students dragged that red dot to try to make a line with a slope that’s between the steepnesses of the green and blue lines.


Here is our class overview:

What we noticed about our attempts:

  • One person went in another direction, which might be a really cool way to think about this, outside the box.
  • There is a darker area, where some people may have had similar ideas.
  • There are some longer lines, which *might* have greater slopes.

We wondered:

  • Is the line that goes downward from left to right negative, while the others are positive?
  • Is there only one correct answer?
  • Does it really matter how long the line is?

AB said that she tried to take the tread and the rise and multiply or divide to find what the rise would be if the tread were 1. So if we had a line with a tread of 2 and a rise of 4, she would divide both by 2, so the tread would be 1 and the rise would be 2. This brought up ideas of proportional reasoning and equivalent fractions.

We took that to the Smartboard and found our rises for the green and blue lines if their treads were both 1. We found that the blue one would be 3 and the green one would be 2. With a little trial and error, students figured that 2 ½ would work, but when they tried to move their red line, it snapped to whole numbers only. With a little practice, we were able to double the tread and move our point up to 5, so we had a 2:5 ratio, which ended up as 1:2 ½.

It was just a very interesting conversation with a lot of exciting ideas to continue to explore. (We never really touched the negative slope line, so that will come up again!)

Moving Forward

This past week, I got some difficult news. A friend, a colleague, the parent of one of my delightful students, has accepted a job that is a wonderful fit for her. This is the perfect opportunity for her and will likely benefit her family in countless ways. In addition, the job will ultimately benefit students across the country, so I know in my heart that this is right. So why is it such difficult news? Because I am selfish. It has been with her that I have been able to teach math the way I’ve had to hide in the past. It has been as though she has given me permission to teach math in a way that makes sense. It is also difficult because she has pushed me, asked me good questions, challenged me to try even more new protocols and routines. I’m not sure who will be that person now.

As I began to teach math many years ago, I was fortunate to be able to work with the Maine Department of Education and Maine Math and Science Alliance on various projects. Each of these helped me to move away from the way math has traditionally been taught. I worked on a statewide project where we prepared and led workshops about making linear equations and expressions visual. I tried to bring some of these techniques into my classroom, and I was pleasantly surprised at how deeply my sixth graders could think about math. But I was in a district where the rationale for the newly purchased textbooks, as told to me, was “there are 162 lessons and 175 days; that’s time for all the lessons with a few days left over for testing.” All of my graduate coursework in mathematics education and special education, as well as the opportunities with the DOE and MMSA and with the experiences I’d created for my students solidified my belief that “a lesson a day” was NOT the best way for students to understand math. I tried to share with colleagues and administrators, inviting them to read Paul Lockhart’s A Mathematician’s Lament, and attempting to engage them in discussions about the research. After five years of hiding and feeling like a bad teacher for not complying and not fully doing what I believed was right, I moved on to become a technology integrator in another district.

Fast forward several years and a few jobs later when I began my current job at Maranacook Community Middle School. Having worked in a different school, where the math coach encouraged teachers to prepare a packet for every unit for every student that they could work through independently (in what they called “customized learning” and was only customized in pace, as long as you were at “teacher pace or better”), I was nervous about Sarah, this new district’s math coach. I soon realized that my response that “I’m all set; I’ve taught math before” to her offer of support did not offer her much hope. Generally, people with experience teaching math seem to be very traditional in doing so.

I was excited to teach math again, and I was looking forward to doing it in a more meaningful way than what had previously been expected. As the year progressed, and we had grade-level math meetings where we would plan together, co-teach a lesson, and then debrief, I learned and grew and knew that I was in the right place. Sarah and I shared the belief in allowing students to construct meaning, to notice and wonder, and to build conceptual understandings. Every time I’ve worked with her, in any capacity, I’ve been energized to push myself and my students a little bit further.

In addition to collaborating and supporting me and my peers at the middle school, Sarah has worked with my daughter’s class. Audrey would come home excited to share the tasks they had worked on in class, and she shared not only her thinking, but that of others’ in the classroom. Her enthusiasm for math and the various ways one could think about patterns and numbers has been infectious. Audrey adores Sarah, and she begs me to invite her to come do math with us. So this fall, when Audrey came home reporting that she’d been doing timed tests (“called ‘drills,’ Mom!”), I contacted Sarah for support. I do not want Audrey to start to dislike math or feel as though she needs to do it one “right” way or quickly. After assuring me that she and the math interventionist in Audrey’s school will work with the new teacher, she broke the news that she was leaving. Being the optimist, I asked her, “How will you do your new job and this one, too?” But I knew the answer.

I am happy for Sarah. She will continue to spread this love of math to many others. She will question and encourage and notice and wonder. She will tweet and blog and find math in the real world. She will continue to speak up for those who struggle and demand that the same be asked of them as those who excel. She will express her enthusiasm for unusual strategies and student work.

And I will do these, too. I will just miss her challenging, probing questions, her encouragement to dig a little deeper, her forcing me to wrestle along with her about difficult issues… her belief that I can do more.



In the Spotlight

On April 4, 2018, Maranacook Community Middle School was recognized with the renewal of a Spotlight award from the New England League of Middle Schools. This award recognizes MCMS’s record of effective teaching and learning for young adolescents and the consistent implementation of middle level best practices. In addition, Spotlight Schools are “recognized for developing strong effective programs that reflect concepts contained in Turning Points 2000, and This We Believe, and current middle level state recommendations” (

Of the thirty-one NELMS Spotlight Schools, only one is in Maine. Maranacook Community Middle School is described as “a dynamic learning environment that recognizes and values student voice in decisions regarding their own learning. It is a school where learning and leading are evident in all that [they] do” ( MCMS was recognized as having an exemplary advisor/advisee program, an exemplary alternative education program, and for the exemplary practice of the James Beane Process for developing curriculum units.

Maranacook Community Middle School is located on Maranacook Lake in Readfield, Maine. Students come from Readfield, Manchester, Wayne, Fayette, and Mount Vernon elementary schools. The school has five multi-age (6th, 7th, and 8th grade) teams, the majority of which use an integrated curriculum for their core classes. These teams have graadde-level math classes and multi-age reader’s workshops, as well.


Teams at MCMS use the James Beane Process to ensure that student voice and choice are at the center of our curriculum. Teams determine each trimester’s theme by gathering together and generating questions about themselves, their community, and the world. They then share their questions in small groups to find common threads. From there, they begin to look for topics that can fit together, and a list of possible themes is generated. Students then vote on a theme and create a list of more specific questions related to that theme. From there, team teachers get a planning day to sort them into three core classes, incorporating learning standards from science, social studies, writing, reading, math, and other content areas that support student questions.


Our advisee program is the hallmark of the Maranacook Community Schools. The spring before students begin middle school, they are matched with an advisee group and an advisor who will remain together for the three years there. Sixth grade advisees begin school a day before their seventh- and eighth-grade teammates to learn more about the school and to participate in teambuilding activities. It is important to MCMS that every student has at least one adult advocate, and that they meet at the beginning and end of each day. Advisee groups are organized by grade level, and there are different expectations for each. Sixth grade focuses on the transition to middle school and teambuilding. Seventh grade focuses on community service. Eighth grade focuses on healthy decisions and preparations for high school. During the year, there are several “Advisee Days Out,” when advisee groups participate in different activities, based on their grade level goals. At the very end of the year, advisee groups plan a day-long excursion to a destination of the group’s choice. Groups have gone to Monkey C, Monkey Do; escape rooms; overnight camping trips; and more.


The alternative education program at Maranacook Community Middle School has evolved over the years. At one time, it was where kids who had behavior problems went. Over the last few years, however, the criteria for who is placed on the team has changed. Students who are unable to behave appropriately are no longer eligible for the team; instead, students who learn best in an alternative setting–small group, hands-on learning can request or be referred to join the team. There is a cap of 20 students, and the RTI and alternative education teams determine who is a good fit. Four adults work with this group of students to keep numbers low and to offer opportunities to meet a wide range of needs.

Maranacook Community Middle School was built on a strong commitment to what students need. There are many opportunities for students to have voice and choice and control over their days and their years. There is a strong support system from advisors, teachers, the wellness staff, and the community. As times change and mandates and expectations force staff to evaluate programs and philosophy, there is uncertainty. As the school and staff continue to adapt, there are bumps in the road, but by putting students at the center of decision-making, we can continue to be a strong school, recognized as a leader in the field.