What does a modern mathematics classroom look like?

Hannah Starbuck, Mathematics Teacher, Leysin American School.


The Mathematics department goal this year was communication. This was perfect because it tied into a large part of my vision of a modern classroom. The areas I mainly focused on this year were Problems of the Week, Socratic Seminars, and reading articles.


A Problem of the Week (POW), is a large, fairly open-ended maths problem, with an elegant solution or solutions. These questions encourage multiple pathways of thinking and provide students ample opportunity to look for and identify patterns and trends, a useful habit for mathematicians. I’ve given POWs the last three years I’ve been teaching. Students tend to enjoy them and enjoy collaborating with one another. The biggest pushback I get is my expectation that students produce a formal write-up of the problem.


Students are asked to explain in detail the problem and their process of solving the problem, convincing readers of their methods and solution(s) and reflecting on their learning and the task. The question I get: “Why do we have to write in maths class?”.


From the student’s perspective, the purpose of maths class is to “solve problems and get right answers.” But now I tell my students that we write in maths class because it is good for our communication skills, that mathematical writing is just as important as writing an essay on a novel or a lab report AND that it helps to articulate mathematical thinking that might be difficult to express orally.


Students will sometimes go out of their way to complain about writing, but their reflections and mathematical writing always amaze me. For starters, their reflections are insightful and honest. They provide evidence and defend their claims as to why they deserve a certain mark, or what should be changed about the problem. Their thoughts are generally mature, well thought-out, and genuine. Students go into detail as they describe how hard they worked on this problem, and discuss the different maths concepts they applied. It’s important students embrace the idea of convincing a skeptic. Students are very thoughtful in defending their processes and answers and have detailed work and proof to support their thoughts. Even if the solution isn’t correct, I don’t penalize students for trying to defend their work.


Additionally rewarding for me is to watch students interact with one another as they solve these questions. My role is to provide clarity or define words that might be confusing. The problem solving is left to the students. Naturally they want to work together and bounce ideas off each other. The resulting mathematical discussions are truly rich and exciting to listen to. My role is to be the skeptic, asking a lot of how and why questions, encouraging students to be more convincing, and to make strong arguments.


The resulting discussions are impressive. That is the exciting part of POWs. You have opportunities to listen to students from all over the world discuss and debate maths to solve these problems, as well as watch their confidence grow when they are successful.


The second focus area for me was Socratic Seminars centered on math education. I chose to do this because I had 12th graders who were experienced with debate, had good English levels, and I had small enough classes to make sure everyone’s voice was heard. Some of the ideas we discussed were student visions of a modern maths classroom, the necessity of homework, how students are assessed, theoretical and application maths, and mental maths, and many more. Students were expected to prepare questions and opinions to contribute to the discussion, demonstrate respectful and polite body language during the discussion, and reflect on the seminar after it concluded. The majority of the time, students were prepared and contributed interesting questions and thoughts. Students also demonstrated a genuine interest in what others had to say. They maintained eye contact, sat up straight, and showed appropriate signs when they were ready to contribute to the discussion.


What these seminars taught me is that our students are a lot more able than we sometimes give them credit for. The 12th graders I taught had been in school for 13-14 years; they’ve seen just about everything. “As an IB student, I find homework to be very burdensome”, “I strongly dislike routine”, “It is an effective way for students to make projects based on the topic passed, or to make a quiz right after you finished the topic”. These are direct quotes from my students discussing the seminar themes. Their language is honest and shows they’ve played the game of school for a long time and KNOW how to play the game. I try to incorporate a lot of their ideas. To me, this is an obvious motion to listen to students more than we do. Teachers definitely possess the training AND the experience of being both a student and a teacher, but let’s not forget we also had ideas on how to make school more fun for us. When given the opportunity for students to use their voices, they clearly can make educated and insightful observations.


The third item I focused on was having students read various articles on global maths education. This seemed appropriate because my students were from all around the world and could provide insight and perspective on each topic we read about. Some of the topics we read about were: counting on fingers, math anxiety, mathematical geniuses, and teaching math as a language. Similar to Socratic Seminars, I asked students to write their initial opinions based on the headline of the article and give an explanation on why they felt this way. Then they read the article and shared their opinions, touching on whether their previous ideas had changed, why or why not, and tried to connect the article back to their home country. Again, their responses are incredible to read. Students provide so much detail and perspective, and what’s even more interesting is that a lot of them had never thought about some of the topics we read about.


For example, we discussed the benefits of counting on one’s fingers and encouraging teachers to allow it. “Many people, especially children, are using the fingers for counting. The brain structure is working by counting fingers, it is such an instinct. I think that using fingers before school is fine, but step by step you need to get rid of it. I think that taking notes is more efficient”, “Most people, myself included, are visual and tactile learners to some extent. So using our fingers to count quickly is a good way to do mental calculations”. These quotes from my students confirm many things for me. The primary one is each student comes from a unique background and maths has its own identity and culture in their homeland.


Maths takes many shapes and forms depending on location and culture. And once again, students have shown me their strong abilities to read, interpret, and form a thoughtful and relevant opinion on a concept that might be unfamiliar. This is the power of giving students the opportunity to use their voices: they provide insight, perspective, and ideas to those around them, including their teachers.


Where do I go from here? There are a few areas I’d like to experiment with. The first is using a portfolio as a way to assess students. Ideally the portfolio would contain projects, reflections, assignments, and other pieces of “evidence” that students would use to show their development throughout the year. The second is to buy a class set of books and incorporate literature into maths class. I have a few titles I’d be interested in using. The next step is to discuss best practices with an English Literature teacher. I’ll also change some of the topics for the Socratic Seminars to be more mathematical. Zero is a popular choice, prime numbers, calculus, etc, all have potential for an interesting discussion. And finally, I’d like to incorporate more projects into maths class. Maths is not a one size fits all subject, it has so much room for creativity, problem-solving, and growth. Projects give students more choice and authenticity in their learning and what students create is truly interesting! The best thing we can do for our students is to empower them by giving them opportunities to speak, write, listen, read, and create pieces of work that demonstrate learning and also have meaning and personality.



Fendel, D., Resek, D., Aler, L., Fraser, W. (1997). Integrated Mathematics Program (IMP) Curriculum. Integrated Mathematics Program (IMP) Curriculum, Key Curriculum Press: Emeryville, CA.



Hannah Starbuck is a maths teacher, volleyball coach and resident scholar at Leysin American School. She is originally from Colorado and is finishing her sixth year of teaching.


Paul Magnuson, Director of Educational Research, Leysin American School.




As I understand it, kaizen is Japanese for improvement, but the use of the word in agile contexts has come to mean the type of gradual improvement that comes about when everyone keeps their eyes open for each little opportunity to make things just a little bit better. Crowdsourced improvement, more or less.


A few years ago we converted a run of the mill classroom located in the center of campus into a flexible meeting space for our research center. Café size tables, comfortable chairs, rugs, plants, a great view of the Alps. There is no projector, and originally, no whiteboard (though we later decided the utility of a whiteboard outweighed our goal to keep the room looking unschoolish).


We named the room Kaizen to emphasize that anybody can use the space and that any small improvement that comes out of that space helps the overall quality of the school.


A few months after naming the room, I got into the elevator directly across the hall and was greeted with a pleasant surprise. A professional looking set of stickers had been affixed next to the buttons marking each floor of the building. Kaizen Research Lounge jumped out at me immediately. Floor 1. Someone had improved the name of our room. A perfect example of the spirit of the room, baked right into its name. The Kaizen Research Lounge.


We probably all have our own examples of kaizen, if you stop to think about it a bit. I’ve seen our Head of School stoop to pick up a candy wrapper or piece of paper as he walks between buildings. I’m sure others have learned from him to do the same. The combined effort of many faculty ensures a clean campus.


Similarly, several years ago a few of us began helping the dining services staff clear the tables after school-wide banquets, instead of leaving directly after the closing words. Every banquet since there have been a few more of us. No one was ever told they must stay to help, but the combined small efforts of a few have created a new culture. I can guarantee that after the next banquet a large number of faculty will help clear tables. The work will be quick, distributed, satisfying, and in collegial collaboration with the non-academic staff working the dining hall. Kaizen.


You can see how this same principle applies to one’s personal life. Don’t fret about having to make major changes. Just make one small change, and one small change again after that. It adds up. You can also see how this principle encourages an increasingly professional teaching corps. If there is something to try that might be just a bit better, try it. If there are enough teachers making small improvements to their teaching, the overall teaching quality rises, and with it student learning. If teachers start to talk to each other about small improvements, sharing their successes, the effect is accelerated.


Kaizen creates a culture of improvement. One starts with oneself and it rubs off on others. People start noticing and join in, in small ways. Still others experience the positive modeling and they, too, shift their actions and thinking just a little bit. Change comes about, and continues coming about, because the culture is one of continuous improvement.


Students may notice, but even if they don’t, they are likely to benefit. It may be worthwhile, however, to make them aware of kaizen, to let them know that improvement is a distributed process, created through distributed leadership to which everyone has access and for which everyone is responsible.


Kaizen Research Lounge. Can you believe it? What a great name.



Paul Magnuson is the director of Educational Research at Leysin American School and adjunct faculty for the International Education Program of Endicott College. His interests include student agency and self-regulated learning for students and teachers.

What if?

Paul Magnuson, Director of Educational Research, Leysin American School.


A student wrote to me yesterday with a request that has really got me thinking.


For some context: in the final weeks of our sudden shift to school online, I convened a group of a dozen eleventh grade students, twice, with the goal of talking through their remote learning experience and what we can learn from it moving forward.


The students reflected on their ability to self-manage and self-motivate, their established study practices, and how they might like to reorganize school in the future, based on our collective redo of how school works.


We called the two sessions blue sky thinking. As we wound up the second session, a group consensus emerged that the students would like to have more blue sky time next academic year – time to come together and talk creatively about academics, alternatives to how we do things now, and how they might learn best.


I suspect any educator’s heart should beat a little faster when students express interest in rethinking pedagogy. But perhaps in reality that faster heart rate is the realization that talking what ifs is the equivalent of a crack in the dam or a loose thread in the sweater. Something could burst or unravel pretty quickly.


So a student wrote to me. One of the blue sky thinkers. A smart, reflective, serious student. She asked me this:


I wanted to know … if I could have the option to not go to class and instead go to a supervised study place … where I could work alone like we did during the online schooling period … Maybe I would go to class at least 2 times a week … I would also agree to turn in everything on time … I think this would help me keep the efficiency that I had when I was forced into learning by myself. Do you think the school will let me do this? 


Blue sky, definitely, and not unreasonable. But the questions this request raises! Say the student did only go to class twice a week (that’s half the scheduled classes in this case) but turned in all her work on time and did well in class – and perhaps in less time. Is that an acceptable outcome for the school? It seems to be acceptable – preferable even – for the individual student.


I can hear the collective, hypothetical voices of my colleagues and I as we discuss the request: You can’t just skip class! (Knee jerk reaction). Who would supervise? (Practical constraints.) Wouldn’t this open the doors to other students asking for the same thing? (Are we talking about building a whole new program here?) What about students who wanted to do this but weren’t able to handle it? (Further administrative complications.) If it’s only necessary to attend half our classes to succeed, what will parents think? (Public relations.) 


And so on. 


I’ve heard colleagues talk about school reform in terms of aspirational goals, which feels a bit like code language for blue sky thinking that we know will never happen. I’ve also heard plenty of pundits say education will never be the same, post-Covid. But as I open the mail to respond to this student, I can’t think of any response that doesn’t cloud that blue sky. She is onto something, but I do not know how to support her thinking best.



Paul Magnuson is the director of Educational Research at Leysin American School and adjunct faculty for the International Education Program of Endicott College. His interests include student agency and self-regulated learning for students and teachers.

The Ascending Cognition Principle

Kevin Jennings, M. Ed., World History Teacher, Leysin American School.

Designing Units Systematically and Gradually Through Mastering Skills.

In my 10 years in education I’ve had the good fortune of teaching on three different continents, with colleagues from all over the world, and students from over 30 countries. In that time, I have found a few challenges to be both universal and recurring.  

These problems include: 

  1. I) How do we create a classroom environment that promotes higher-level thinking without overwhelming students?
  2. II) How do we make sure one class period, one week of classes, a 5-week unit aligns meaningfully with the next?

III) How do we create a classroom environment that promotes student leadership and independence?

  1. IV) How do we make sure we practice the skills and develop the understandings that are necessary for success?


Through trial and error, and through conversations  with colleagues, I created a standardized unit outline that helps to address the problems mentioned above. I call it The Ascending Cognition Principle. 


 An Educational Principle 

Think of skill development in the classroom in terms of the proverbs “you must learn to crawl before you can walk” and “you don’t need shoes to run, but it helps.” While developing units, It’s important to target specific skills necessary for student development. The first few units may require students to focus on the mastery of lower level, less cognitively demanding skills that are necessary for success, e.g. Cornell Notes, extracting information from a text, comparing and contrasting resources, etc. These foundational skills are our proverbial “shoes”, preparing students for their next challenge.  


In the beginning of the school year, a class may be asked to practice skills such as Cornell notes, summarize, and compare and contrast, in order to set a foundation of skills to build off in the lessons and units to come. As a student, a group, or a class is able to master these skills, they will then “graduate” to more cognitively demanding skills such as classify, predict, interpret, solve, analyze, cause and effect. Of course these skills are made easier by the use and implementation of earlier, foundational skills. 


As the year progresses, lower-level skills and activities will take less time to complete, allowing more time for higher-level, more cognitively demanding skills to be targeted. For example, while we may take 2 weeks to master Cornell notes in the beginning of the year, writing these notes will (hopefully) become second nature into the latter half of the school year, allowing more time for new exciting and challenging skills. In our case, the next two levels of thinking skills are Application and Analysis.  These two levels allow for students to solve problems, building prototypes, investigate, defend reasoning, develop a thesis with conclusion, etc.. The mixture of interesting and challenging material with constant practice and feedback will allow even these difficult skills to become natural as the school year progresses.  


As mentioned in the initial problems, the ultimate goal of my class is for students to be able to handle higher-level thinking tasks and skills, while becoming more independent and self-reliant. The ascent to the highest level thinking skills, Synthesize and Evaluate, encourage such behavior as they are meant to help students develop something new, or justify a position. Some students or classes may be ready to ascend quicker than their peers. In which case, it would be beneficial for educators to allow these students to embrace more cognitively demanding tasks and activities. Students, who are ready, will benefit from more challenging material as well as more opportunities for creative expression e.g.developing a song, creating a historical fiction, creating a political cartoon or meme, solving problems, etc. 


A unit lasts as long as time allows, or as you (or your school) see fit. The following unit will then build off the foundation of skills you have just developed in order to embrace more challenging material. The ascending complexity of tasks and activities over the course of the school year is represented by the gradual change from red to purple in the figure below.  


Preparing for a Successful Unit  

In order to figure out which activities are less or more cognitively demanding, you may want to use Bloom’s Taxonomy Teacher Planning Kit (Google it) as a guide, then use and/or add what makes sense for your classroom.[Text Wrapping Break] 

As you move from left to right on the Bloom’s scale, from “Knowledge” to “Evaluate” the keywords, actions, and skills will become ever more complex and cognitively demanding, as seen on the figure above. 

Download Bloom’s scale


Targeting Specific Skills 

As we prepare for a successful unit, one thing we should be cognizant of is the skills that you would like your students to practice and eventually master. If you are using The Bloom’s Taxonomy Planning Kit, these skills and command terms are located in the “keywords”, “actions” and “outcomes” section, and if they are not there, add your desired skills to the appropriate section of the unit. 


The location of the skills along the Bloom’s scale (e.g. under “Knowledge” or “Comprehension” etc.) determines when in the unit the skill will be practiced. For example, any desired skill that appears under “Knowledge” or “Comprehension”, which are the first two levels of Bloom’s, will likely be targeted earlier in the unit, while any skills that fall under “Synthesis” or “Evaluation”, the highest levels of the Bloom’s scale, will only be practiced once that individual, group, or class, is ready for the challenge, which may not be until the latter months into the school year. 


Once we know which skills we would like our students to master, and we know where in the unit the skill will fall, it’s now time to create activities allow students to master these skills. For example, if I want my students to learn how to take Cornell Notes, it’s up to the teacher to come up with an effective way for students to learn this process. This large task will be infinitely easier if the teacher has the ability to collaborate with other teachers of similar grades or subjects. 


Cycling Through the Units 

While the goal of a school year may be for students to master all of the skills necessary for their next year, teachers will also benefit from having a classroom which is (among other things) predictable, flows seamlessly from one lesson to the next, has a clear purpose, offers creative outlets for students, and guides students to higher-level thinking activities. The Ascending Cognition Principle was developed to allow teachers to produce units that do just that.  


Kevin Jennings is a social studies teacher and resident scholar at Leysin American School in Switzerland. He is a native of the Washington DC area and is currently in his 10th year of teaching.

Learning Ecosystems

Sandy Mackenzie, Director, Copenhagen International School.

The concept of a learning ecosystem has never been so relevant as it is today in March 2020. In countries across our inter-connected world, the delicate nature of a healthy ecosystem has been brought into stark focus through the spread of a global pandemic. International schools are resilient beings that have withstood many tests of disease, natural disaster and man-made catastrophes. Never before has such a single, tiny entity had such far-reaching implications – as well as creating a threat to health worldwide, making us question our modus operandi in all aspects of society. 


In our globally mobile profession, we all have friends and former colleagues working in China. Therefore, we had all heard about corona virus, that schools had to close their doors and provide remote and online learning in February. Somehow, in Europe we became apocryphal King Canutes and believed that it would not happen here and so were in varying states of readiness when it was time to write a risk assessment matrix and a remote learning plan. 


Across Europe, in late February and early March strategic thinkers and planners made calm arrangements for procedures in the event of a case of the virus entering our communities. The speed of the spread of this threat quickly overtook the pace of the careful, thoughtful leaders. Countries swiftly brought in restrictions and closed schools to halt the impact, and brought new terms to our lexicon such as “flattening the curve” and “social distancing”. Social media became the super-spreader of information and mis-information, from which Donne’s “no man is an island” is even more relevant than it was 400 years ago. 


International schools often describe themselves as a bubble within an environment, floating in the ecosystem they inhabit. COVID-19 pierced that bubble and illuminated the symbiotic relationship between a school and its surroundings and its neighbours. Moreover, the success of the school to respond to the challenge of remote learning and campus closure is largely dependant on four main factors: 


  • Human adaptability and preparedness for change 
  • Consistency of availability of tools required for remote learning 
  • Clarity of expectation in the local and national environment
  • Leadership and communication 


Presence of all four conditions is required for a sustainable, successful remote learning solution during this uncertain time. Without able, caring, dedicated, versatile teachers any effort to alter the nature of learning so radically virtually overnight would fail. Software solutions and technological tools are fantastic aids to distance learning; for them to be employed equitably and usefully, all students need access to them. We know that stressed and anxious people rarely make good students – those governing the local community and the nation need to provide clear guidelines for operating within restrictions. Otherwise, toilet paper runs out and sane, upstanding members of society become headless chickens caught between two stools!  

As well as clear societal guidelines, a calm and reasonable set of expectations for all community members in the face of a dynamic, shifting environment is needed. School leaders set the tone for the response to this novel situation – optimism and confidence are key. In our school, Arthur Ashe’s famous words “Start where you are, use what you have, do what you can” provided a guide star for approaching the challenge of shifting learning to a blend of online and offline activity delivered through sychronous and asychronous methods. While we may not have had a glossy (digital) brochure describing our remote learning provision with virtual bells and whistles attached, we were fortunate to have the four bullet points above. 


The subtitle to the ECIS Leadership Conference due to take place in Madrid 2020 was “leading school communities that thrive”. In this new normal of remote learning, how do school leaders ensure that their community continues to not only survive but to also thrive? At these moments of uncertainty, leaders display empathy and provide reassurance; they communicate thoughtfully and appreciatively. Moreover, they look for opportunities for new learning, not merely a pale imitation of on-campus learning. It is quickly apparent that teaching volleyball or developing skills in using 3D printers and laser cutters are not possible in a home learning environment. Well supported, adaptable teachers make proverbial lemonade from those lemons – PE teachers creating podcasts for a modern sex education programme, Design students taking photos and measuring the height of their tower made of household packaging that needs to support a carrot on top, video challenges that involve the entire family and encourage social interaction, home cooking and human connection. 


Colleagues from China with experience of many weeks of remote learning tell us that students, and their teachers, focus less on the content of learning as time passes; instead they crave the social interaction, collaboration and human connection that school provides in their daily lives. That is evident in week three of our enforced remote learning experiment. Reflecting on the different things we can do, on the things we can do differently and the aspects that we can live without is making educators question what is important. How many conferences have you attended where the keynote speaker expounds a powerful message that it is about time we changed school education, that in the 21st century our content-based, teacher as fount of all knowledge paradigm needs a significant shift? In the age of the answer being immediately available on a screen, is it not time that we asked different questions? Many of us leave those conferences with great ideas in our mental briefcase, brimming with confidence and good intentions to bring in a new initiative only to find that days later, we are back in a familiar groove. 


The retrospective inertia that exists in all schools (also known as the “this is how we have always done it” syndrome) can slow or stifle change. Could it be that the necessary catalyst for disrupting the status quo is this global pandemic of COVID-19 and the international response to lock down countries, restricting movement and enforcing working and learning from home?  


Across the world, in every type of international school, educators are asking what are the important skills for students to gain. Teachers are utilising creativity and employing ingenuity to design experiences that engage students, both in real time and in asynchronous, offline tasks. They are engaging in a different manner with their environment; they are adapting the ecosystem to ensure that communities remain strong. As they do so, even larger questions come into focus that may require a re-examination of the axioms of school education. If the International Baccalaureate and other bodies can cancel all exams, and universities are able to make good decisions about admission, in 2020, are written, timed examinations still fit for purpose? If we truly value collaboration, research skills and project-based learning, do we need to redefine the concept of academic honesty? What is the true purpose of the teacher, and what skills and attributes are necessary to be an inspirational educator in the 21st century? 


As we hear about some of the horrific immediate effects of COVID-19 ravaging countries, our thoughts are with families and communities losing loved ones. The next phase of concern will be the resultant economic changes for organisations and societies. For many of us, the medium term impact may be deep with educators examining the core of how we define school as part of a global, interconnected learning ecoystem. 



Sandy Mackenzie is Director of Copenhagen International School and has over 20 years of experience supporting the education of young people in many parts of the world, including China, Denmark, Scotland, and the United States. Sandy has taught Mathematics, co-authored a textbook and held senior leadership positions in four schools. Empowering and supporting teams to provide an outstanding education to young people that  positively contributes to their academic, personal, social-emotional, and intercultural well-being, learning and growth is his true passion.