Beauty and Big Ideas: How We Teach Math

“We are all mathematics learners, and we can all develop active, inquiring relationships with mathematics,” writes Stanford math education researcher Jo Boaler in the research paper “Prove It to Me!” “When we do, and mathematics becomes a creative, open space of inquiry, mathematics learners will find that they can do anything, and their mathematical ideas and thinking can extend to the sky—and beyond!”

Boaler and her colleagues are on the forefront of math education research, which shows that with good teachers and self-confidence, all children are capable of achieving in math. According to her research, building a strong foundation in number sense—a feel for numbers and the ability to use them creatively—is key to becoming a successful math thinker, as is an ability to grasp big ideas and make connections between them. 

At Marin Waldorf School, our approach to math is designed to do just that: Through a multidisciplinary, multilayered approach to math, starting at the earliest ages, students learn to see the joy and beauty in numbers, approach math work from many perspectives, and eventually build up to the conceptual ideas that fuel advanced-level math in middle school. Here’s how we do it.

Early Childhood: Encouraged to Explore

“In our preschool and kindergarten classrooms, math education is intentional, but not as directly articulated as you’d find in a traditional classroom,” explains Daniella Baker, MS, early childhood director at Marin Waldorf School. “Math is taught through songs with numbers, counting, jump rope, marching, and other activities.” 

“The approach is layered,” Ms. Daniella continues. “Activities in kindergarten lay the groundwork for math instruction in grades 1, 2, and 3, when students practice multiplication tables by skip counting on jump ropes or in songs, and begin to master math facts.”

They are also given ample time to play, the best way for young children to explore the world and their ability to discover and make connections on their own. In an interview with Ed Source, Gennie Gorback, president of the California Kindergarten Association, explains, “Children learn high-level, intangible concepts such as the laws of gravity, conservation of liquids/mass, mathematical concepts such as more vs. less, all through hands-on, interactive play.”

Elementary School: Building Skills and Enthusiasm

“My task as a second grade teacher is to ignite the fire of enthusiasm for learning. So right now my students are at play in the field of numbers, opening their eyes to the artistic beauty and wonder in math,” says Roland Baril, second grade class teacher, who uses a blend of artistic exercises and math drills to build a foundation for his students. 

At the beginning of the school year, Mr. Baril guided second graders in drawing geometric forms that are created by placing 10 dots equidistant around a circle, numbering them 1-10, then connecting them with straight lines via counting by the various times tables. Building on their first diagram of a decagon, children drew decagrams, pentagons, and pentagrams, finding number patterns in each. 

Later in the semester, students studied the patterns created by magic squares and worked in pairs to create times table charts to hang up at home, a project that requires students to accurately mark along the length of a ruler, connect the dots with a yardstick without slipping, and work together to fill in the chart. “We also do lots of drills around the four processes—addition, subtraction, multiplication, and division—so the children are very strong in calculations,” Mr. Baril.

“Good mathematics teachers typically use visuals, manipulative and motion to enhance students’ understanding of mathematical concepts, and the US national organizations for mathematics, such as the National Council for the Teaching of Mathematics (NCTM) and the Mathematical Association of America (MAA) have long advocated for the use of multiple representations in students’ learning of mathematics,” write the researchers in the Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning.

Mr. Baril’s visual exercises, alongside mental math games, skip counting, and the use of wood manipulatives, help build an awareness of numbers and an enthusiasm for math. With this sense of wonder and play imbued in their math work, second graders are naturally drawn to the more complex problems they encounter in second grade, like regrouping (carrying and borrowing), word problems, and multi-digit operations. In fact, ask our second graders and many will say math is their favorite subject! 

Middle School: Growth Mindset and Big Ideas

By 6th grade, students are ready for more complex and abstract mathematical thinking, and the curriculum meets their growing abilities with big ideas in math. Middle school math teacher Julia McIlroy aims to show students how mathematics is integral to all parts of life by looking at big ideas, patterns, and relationships between mathematical ideas and by grounding all subjects in experiential learning before moving to abstract principles. 

In 7th grade, for example, students measure large circles on campus to discover the relationship between the diameter and circumference (which we know as pi), before looking at it through the lenses of geometry, ratios, and algebra. Group work is also a key component to Ms. McIlroy’s teaching, requiring students to work together to solve unfamiliar problems in both concrete and abstract situations—to find patterns, make conjectures, and test those conjectures, and to understand that mathematical structures are useful as representations of phenomena in the physical world.

“What I’m trying to do is bring more complex tasks, where there are no immediate right or wrong answers. This prepares them for higher education and the workplace and keeps everyone engaged,” explains Ms. McIlroy. “If you have rich tasks, the students who may be behind academically can do something, and the students who are really high-achieving can take it to the highest levels.”

“My number one goal is to keep students interested and engaged in math so they will continue on with it in college and beyond,” Ms. McIlroy says. “I want to have as many of our students go into STEM fields as we possibly can. Our students have a complex and ethical view of the world and their contributions to science and technology would be profound.

Igniting Curiosity in Middle School Science

A spirit of inquiry, experiential learning, and an integration of academic disciplines are cornerstones of our school’s approach to education.

In our middle school classrooms, where 6th, 7th, and 8th graders explore complex topics in chemistry, physics, physiology, biology, and geology, science isn’t presented in concepts and lectures. It begins with observation. Through observation, our students learn to think like scientists, asking questions and posing theories before being given all the answers.

Even art, which is integrated throughout our curriculum, is used as a tool for understanding STEM topics. For example, in their studies of combustion as part of the 7th grade chemistry block, students were asked to observe and then draw a bonfire and a candle. You can see a few examples of our students’ work below.

As a capstone to the study of combustion, teacher Ms. Terziev performed a demonstration for the class, asking them to watch silently and then offer their theories on what they’d observed.

After watching the experiment, students brought their questions and theories to the group, inspiring both curiosity and critical thinking in the classroom. This approach helps students build a meaningful understanding of complex scientific concepts. In Waldorf education, this is called a phenomenological approach to science. In this way, 7th graders at Marin Waldorf School connect and internalize complex topics in chemistry, like combustion, as well as crystallization, acids/bases, and the lime cycle.

The study of science at Marin Waldorf School begins in early childhood, with the simple observation of the seasons and the natural world, and through nature stories. In elementary school, students learn to ask questions and to learn through doing, laying the groundwork for more complex critical thinking that they will need to tackle their studies of chemistry, biology, physiology, geology, and physics in middle school.