Teaching & Learning Blog

Learn more about PS1's academic program, philosophy, and curriculum through the lens of Nancy Harding, our Assistant Head for Teaching & Learning. The Teaching & Learning Blog features posts published several times throughout the year. See the latest posts below.

The WHY Behind Multi-Age Groupings

Most of us grew up in age-segregated classes, as did our parents and perhaps, our grandparents. This history makes it easy to assume that such a school structure is both natural and universal. The age-stratified culture in which we educate our children is actually a product of the 20th century.

Early in the history of the US, schools were one-room schoolhouses with age diversity. In the dedicated one-room school building that emerged in the eighteenth century, a full-time teacher would use individual and tutorial methods to instruct a group of 10 to 30 pupils ranging in age from 6 to 14 years.

This one-room classroom practice started to end in 1843 when Horace Mann, the Secretary of the Massachusetts Board of Education, visited Prussia and saw schools in which children were “divided according to ages and attainments." This type of segregation seemed to him an excellent model for preparing a populace for the growing factory economy. By 1852, classrooms in the US were more narrowly segregated by age than ever before. Ability grouping, which is so much a part of how we envision classrooms, gained popularity after about 1920. This further reduced the variety present in classrooms.

In 1963, Goodlad and Anderson looked at the current research in child development and proposed that the rigid age/grade system was not designed to accommodate the realities of child development, including children's abilities to develop skills at different rates and at different levels. The graded system does not take into account differences in children's achievement patterns. Goodlad understood that learning is not linear and children typically progress at different rates in different areas of study and at different times in their development. A traditionally graded school assumes that all children will progress through each area of study at the same pace. In this system, a child has no freedom or flexibility to develop at the pace that is optimal for their needs. 

In a non-graded school like PS1, there is a longitudinal concept of curriculum and planned flexibility in grouping. We describe a student’s trajectory at PS1 as one seven-year experience. The curriculum includes continual and sequential learning, with behavior and content running vertically through the curriculum. Grouping is flexible and changes to meet student needs. Groups are organized around interest groups or work-study groups or achievement, or a combination of the three with some groupings being heterogeneous (mixed levels) in skills and other groups being homogeneous (similar levels) in skill levels. Teachers adjust their lessons to ensure that students grasp concepts, skills, and content through their entire educational journey.

Multi-age groupings are (and always have been) an integral part of the structure of the PS1 learning experience. Just as the research suggests, we see how our multi-age groups enhance learning on a daily basis. Year after year, and now generation after generation, parents come back and tell us that a two-year age range was the most important piece that made their children the well-rounded, well-spoken, confident, comfortable, agile people they have become.


*A note added by Head of School, Joel Pelcyger:

I have two points to share. First, 25 years ago, I attended a national conference and went to a session on the benefits of multi-age grouping. The presenters were in their second year of their program. I asked a question and mentioned that I had nearly 25 years of experience running a school where all the classes were multi-age. The focus of the session then changed - all questions were directed to me and I became the de-facto "presenter," sharing our experience at PS1. Second, another aspect of our successful multi-age approach concerns the overlapping of grade equivalents; viz. our intentional approach to assure that each child is clustered with a different group of children each year and has the opportunity to be both an older and a younger student at various points of their elementary experience.

Source: Author (2002). Title. In Daniel Schugurensky (Ed.), History of Education: Selected Moments of the 20th Century [online]. Available:  http://fcis.oise.utoronto.ca/~daniel_schugurensky/assignment1/  (date accessed).


Mathematics Education at PS1

The mathematics curriculum at PS1 consists of three components that interact to create a comprehensive and motivating approach to learning and understanding mathematics. The graphic above depicts the three components in a triangle with CGI on top, the Bridges math program at one corner and the Common Core Standards for Mathematics at the other corner. Together, the three components provide a conceptual framework that implements a variety of tools for teaching, and clear standards for monitoring students’ progress. In this blog, I will describe each of the components and how they interact.

Cognitive Guided Instruction

CGI is at the top of the triangle since it provides the underlying philosophy of mathematics teaching and learning at PS1. The goals of CGI go beyond teaching students to calculate. CGI creates students who are problem solvers, can communicate mathematically, and have mathematical reasoning ability.Below I explain what each skill means and what it looks like in the classroom.

Students who are mathematical problem solvers question, find, investigate and explore solutions to problems. They demonstrate tenacity and stick with a problem to find a solution. They understand that there may be different ways to arrive at an answer and apply math successfully to everyday situations. In the classroom, students will use different strategies to solve problems and be encouraged to teach each other their strategies to expand their problem-solving repertoire. It is essential for the teacher to “hear” what students are thinking.

Students who communicate mathematically go beyond simply answering a question. They use mathematical language, and numbers, charts or symbols to explain things and to explain their reasoning for solving a particular problem in a certain way. CGI requires that students employ careful listening to understand others' ways of thinking and reasoning. In the classroom, students model the actions or relations described in the problem. +, =, >, <, etc. so that symbols become actual words and pictures with meaning for the students. 

The student with mathematical reasoning ability is applying logical thinking, seeing similarities and differences in objects or problems, making choices based on those differences and thinking about relationships among things. This occurs when students apply what they are learning in math to their work in The Studio (for example) and visa-versa. Students who are learning measurement in the classroom use this knowledge to estimate and plan what they will need to build a birdhouse.  The “measure twice, cut once” adage has meaning to them.

Bridges Mathematical Program

This year, PS1 implements the Bridges Mathematics Program. This program is a philosophical fit with both PS1 and CGI. A task force of teachers worked with the entire faculty over a two-year period to look at math programs that fit the philosophy and structure of our school. In total, approximately eight different programs were examined. We looked for a program that promotes learning as a collaborative and social endeavor, that supports problem solving, skill-building and provides excellent professional development support for teachers. Bridges does all of this. The Bridges philosophical underpinning is that mathematical learning is a process of constructing meaning to make sense of concepts and requires perseverance and willingness to encounter new problem-solving challenges. This is an excellent match to PS1.  Students learn to apply problem-solving strategies new and different ways and in different contexts. This is an essential preparation for higher mathematical thinking.

From the Bridges website:

“Bridges focuses on developing deep understanding of math concepts, proficiency with key skills, and the ability to solve new and complex problems. Learning activities tap into the intelligence and strengths all students have by presenting mathematically powerful material alive with language, pictures, and movement. Students in a Bridges classroom talk about math, describe observations, explain methods, and ask questions. They are encouraged to find multiple ways to solve problems and show different ways of thinking. This is a vital way to help students build more flexible and efficient ways to solve increasingly complex problems. Hands-on activities engage them in exploring, developing, testing, discussing, and applying mathematical concepts.”

In August, all of the teachers attended a two-day (16 hour) professional development training at PS1 to learn about how to implement the Bridges program. Bridges will continue to provide support to teachers live and online throughout the year.

While Bridges is the tool that we have adopted school-wide, we always supplement our curriculum to meet the diverse needs of our students. Olders teachers have been using the Eureka program as a supplementary math tool. Eureka has a very strong 6-12 curriculum. Olders teachers will continue this practice.

Common Core Standards for Mathematics (CCSM)

The Common Core Standards for Mathematics were developed in partnership with the National Council for Teaching Mathematics. They are detailed and developmentally appropriate. CCSM provide a clear road-map for teachers to monitor student progress. The Bridges tool is aligned with the CCSM. Additionally, the CCSM have a foundation of eight practices that pervade learning mathematics at all grade levels. As you will see from this list, these practices complement and support CGI. The Mathematical Practices are:

  • Make sense of problems and persevere in solving them. 
  • Reason abstractly and quantitatively. 
  • Construct viable arguments and critique the reasoning of others. 
  • Model with mathematics. 
  • Attend to precision. 
  • Look for and make use of structure. 
  • Look for and express regularity in repeated reasoning. 

These three programmatic elements of CGI, Bridges and CCSM create a robust mathematical curriculum for students at PS1. 

Finally, parents often ask what they can do to encourage their child in math. If we are not advocating that you practice drills and flashcards, what else can you do?   Encourage your child to be a good problem solver by including them in routine activities that involve math—for example, measuring, weighing, figuring costs and comparing prices of things they want to buy. You can help your child learn to communicate mathematically by asking them to explain what they must do to solve a math problem or how they arrived at their answer. For example, ask your child to draw a picture or diagram to show how they arrived at the answer. You can encourage your child's mathematical reasoning ability by talking frequently with her about these thought processes.

We look forward to continuing to update you about our math curriculum through Class News throughout the school year.

The “Summer Slide”

You may have heard the term, “Summer Slide.” It refers to the perceived loss of learning that children may experience over the summer months. The assumption is that the only learning that matters occurs in school, and without that daily reinforcement children will be at a deficit in September. This is a very narrow definition of learning, and yet it often goes unchallenged. Grey, 2017, suggests that the most important lessons of life are learned in more unstructured settings; giving us opportunities to make our own decisions, create our own activities, and figure out how to be with ourselves and each other.

Posted Jul 22, 2017, Peter Gray Ph.D. Psychology Today

As we head off into summer break, I reflect back on my own experience as a parent.

In the summer of 2001 my son was five years old and I started to read him the first Harry Potter book. We sat on the couch and read a lot that summer. Sure he went to day camp and swimming lessons and participated in other summer activities, but what I remember most fondly is being curled up together and reading to him for hours on end. Happily, there were seven other books and we got to do this for seven summers. Sometimes as he got older, we would share the reading, but mostly he preferred just listening. And yes, as he got older, the space between us on the couch grew… though he stayed close.

These are deep and lovely memories of being with my boy (who is now grown). Why am I telling you this story? I think that as parents, we want to give our children EVERYTHING. We want to be THE BEST parents and make sure that our children don’t MISS OUT. But the most important stuff happens in the times between all the planning and structure.

Be quiet with your children this summer, lay around, relax, draw, read, listen to music, play! They will be out of the house in what will feel like a minute. The summer slides to think about are the slip n slide or the slides in the playground or the slides at the water park!

​Have Fun. Enjoy your precious children.​


Problem-solving as adults relates to the imaginings of our childhood. Imagination is not typically highlighted as part of the executive function skills, yet we know that the ability to examine various outcomes and possibilities is essential to creating the lives we want as adults. Make-believe or imaginative play requires a considerable amount of intellectual flexibility in the child, and flexibility is a key ingredient in both planning and creative problem-solving.

Theoretically, one term for this is decentration. Decentration explains the link between fantasy play and divergent thinking (Rubin, Fein, & Vandenberg, 1983). Decentration involves the ability to attend simultaneously to many features of one's environment, to transform objects and situations while at the same time understanding their original identities and states, to imagine at the same time things as they are and also as they were. For example, the child engaged in make-believe knows that the object he is sitting in is a cardboard box, but he pretends it is a car; in a sense, it is both a box and a car at once, and perhaps it was a submarine ten minutes earlier. Make-believe play, therefore, provides evidence of a considerable amount of intellectual flexibility in the child, and flexibility is a key ingredient in the creative process.

At PS1, imagination is recognized as an important aspect of children’s development as a way for them to express themselves and follow their unique interests. Classrooms and indoor/outdoor play spaces are intentionally organized with innovative materials to support joy in discovery and engage each learner. From the ground up, PS1 encourages this exploration and imagination.



Learning Expeditions

Field trips at PS1 are an integral part of the learning experience. Our teachers use the rich opportunities afforded us by being an urban school, as well as our access to ocean and mountains and the diverse ecology that exists so close at hand.

Our field trips are actually Learning Expeditions.

The purpose of these trips is to provide students with real-world connections to the curriculum content. From La Plaza de Cultural y Artes to the California Oil Museum to the all-school camping trip, our students are out in the world acquiring and processing information. New research on the power of learning expeditions indicates that enrichment field trips increase critical thinking, motivation to learn, and address multiple learning modes (Neville, 2011). Teachers and students return from these trips rich with anecdotes about connections that the students have made and the thinking that was generated and teachers build on these insights back in the classroom. PS1 Learning Expeditions enrich the curriculum and student learning.


Additional Posts:

PS1 and Thematic Curriculum:


Structure in a Progressive Classroom: