Mathematics at MTS

The Process is as Important as the Solution

Raising Mathematicians

The philosophy behind the way we teach math at Mount Tamalpais School is exemplary of one of our community values, “Celebrate the Journey.” As Nick Wilsey, MTS Head of Middle School and math teacher, describes it, how our students derive their answers is as important as the solution. Our focus on math at MTS centers on raising students as mathematicians, which expands beyond how math programs were designed in the past. The way that we support students to identify and develop as mathematicians begins with a focus on number sense.

Not Your Parents' Math Class

In traditional programs, what mattered most was getting the right answer. Students were taught to memorize algorithms, repeat, and reproduce. Today in “new” math, students are taught different strategies to solve a problem and to think about which solution would work best in a given situation. It is not about memorizing algorithms. It is about understanding how and why algorithms work. What do numbers mean? “We want our students to struggle with real problems because that’s how you learn,” says Nick. “If you don’t fail at it, you’ll never learn it.” 

According to Cathy Fosnot, a leading voice in mathematics education, in developing numeracy, “students learn that mathematicians look at the numbers first and then choose the most efficient and elegant strategy that works best for the problem.” At MTS, students are taught different ways to break down numbers – that numbers can be composed and decomposed in a multitude of ways. For example, the basic facts, the ones we all memorized growing up, are simply a starting point. Say, 8 + 2 = 10, but also 4.5 + 5.5 = 10 or x + y = 10. Or, in another example, 13 x 7 is also 10 x 7 combined with 3 x 7. Our students learn that there are multiple ways to approach and solve problems. Flexibility in thinking, creativity, debate, and risk-taking are all valued and considered inherent in new math. 

Teaching Kids How to Think, Not What to Think

Many parents have asked me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics. – Jo Boaler, Stanford Mathematician

Why use strategies and models instead of algorithms? “When you’re working from this philosophy and approach to math, it takes the ceiling off what students can learn and be taught in a single year,” says Meagan Andrews, MTS Head of Lower School, and previously a math instructor. In other words, so-called “low floor/high ceiling” math problems allow students of different abilities to engage in the same problem at their own level. When students are focused on the process and not on getting the right answer, there is also rich learning in making mistakes. Students then have the opportunity to learn why and where things went wrong. And in gaining the ability for problem analysis - not just in math, but in all problems - students develop perseverance and develop a growth mindset.  

Project Highlight: Inquiry-Based Project Highlight: First Grade Adventure to Antarctica

In first grade, students embark on a virtual trip to Antarctica as research scientists to study penguins. This thematic unit is centered around measurement: finding the height and weight of different types of penguins, feeling how cold various weather temperatures feel, recording daily air temperatures in Antarctica, and comparing it to other places around the world. Throughout the unit, students reference the number line and gain an understanding of the scale of numbers from -30 (Antarctica temperatures!) to 100. 

In a typical first grade math standard, students work with numbers through 200. However, because of a connection to the real world, such as in the Antarctica Adventure unit, students are exposed to negative numbers. The students have a purpose for utilizing those numbers with a concrete example of the daily air temperatures in Antarctica. They need to know geography - where penguins live and why - and this analysis and depth of thinking applies to the real world. Students are engaged and excited when their curiosity is piqued, and the lessons can be designed around the questions they have; this is when powerful learning happens. 

Inquiry-Based Project Highlight: Third Grade “Muffles’ Truffles”

Each year our third grade class works on a unit titled, “Muffles' Truffles.” This unit focuses on developing the open array as a model for multiplication and division through investigations based on the context of the Muffles' Truffles Shop. The question posed in the first investigation allows students to explore place value - the multiplicative structure of our base-ten system and quotative division. In the second and third investigations, students build two-dimensional blueprints of one-layer boxes and use these arrays to explore some of the big ideas in multiplication (the distributive, associative, and commutative properties.) In the fourth and final investigation, students work with open arrays in the context of labeling and pricing wrapped boxes of truffles. To figure out the dimensions of the wrapped boxes (open arrays) and the cost, students need to apply several big ideas previously developed in this unit. The unit culminates with making and decorating truffles to take home. 

Curiosity and Joy in Learning

My students transform mathematical thinking into daily discovery. This culture of curiosity and exploration provides lifelong tools and a confidence students take beyond their time at MTS. - DJ Thistle, Math Teacher

MTS teachers hear and see the joy of rigorous thinking when MTS students, in finding a solution that works and gaining an ability to describe that thinking because of a deep understanding of the strategies available, are engaged and excited. They see the purpose and the way that it connects to their lives. Math is astrology, math is dance, math is the play structure and the beauty of nature. “When you see kids start viewing their world through a lens of math through the way math is being taught today with opportunities to be curious, it’s motivating,” says Meagan Andrews.  

Effective Communication of Ideas

What is really exciting about how math is taught today is to walk into a kindergarten or third grade classroom and hear the students describing their thinking with accuracy and depth well beyond getting the right answer. Students create learning videos, which is a great way for teachers to hear about how they’re solving a problem. Through these videos, teachers can gain an understanding of where students are in their math reasoning development and the efficiency of their strategies. 

Watch this video of 3rd grader Violet explaining using the strategy called "constant difference." 

Individualized Learning & Technology

Students are able to find personal success since they decide which solution works for them based on the strategies they’ve learned. In middle school, MTS has three math teachers for each grade, allowing for maximum mathematical thinking and confidence building. The goal is for students to be skilled at math, not to be the fastest. Students are regularly asked 1) what do you notice in this problem, and 2) what do you need to solve this/how does it work? Students may come up with five different ways to solve a math problem, but ultimately, they can decide which solution is going to work best in a given situation. 

Our use of technology, especially in lower school, allows for further individualized instruction. Resources like Dreambox develop student skills and fluency by working with different math models and strategies that complement the work happening in the classroom. Dreambox grows in complexity as the students grow in their abilities. It also gives teachers instant feedback on when a student may have gotten tripped up, and it has assignability features, allowing teachers to target extra practice in certain areas on an individual level. In addition, Dreambox and other math apps can give visual representation to concepts students are learning.

Prepared for the Future

The MTS professional community works hard to ensure our students are ready for the world in high school and beyond. Key components of the MTS math program include effective communication of ideas, embracing struggle and mistakes as opportunities for growth, and practicing collaboration. Kindergarten through 5th grade focuses on numeration, operations, computation; data and chance; geometry and measurement; patterns, functions, and algebra. Students in middle school build on these foundational concepts and concentrate on application, problem-solving, data analysis, geometry, and a more in-depth study of patterns, functions, and algebra.

In the school’s Strategic Plan, one of our key Habits of Excellence that we commit to is “Critical Reasoning – valuing and seeking multiple inputs and considering all facets of a topic to understand, analyze, and effect change.” So much of what we do revolves around critical thinking. “An MTS student can question and defend answers. We want them to leave MTS knowing how to collaborate, take risks, and leave school more curious,” says Nick Wilsey. “A resilient MTS student is one who knows how to struggle through setbacks in math. Ultimately, we want our students to think of themselves as mathematicians and to leave MTS with confidence.”

Growth Mindset in the Professional Community

Our MTS teachers are role models for our students by teaching a growth mindset and practicing it themselves. In the last year, lower school and middle school math teachers have engaged in rich, creative professional development. Last spring, Dave Baker, Rob Potter, and Jennifer Adams attended an all-day workshop at Stanford University that was led by Dr. Jo Boaler. The workshop focused on teaching mathematics through big ideas which engage students and increase achievement. At the workshop, MTS teachers explored how to engage all students through open, in-depth, and connected content. 

In the last few years, lower school math teachers have participated in numerous Pam Harris, Math is Figure-out-able workshops and regularly listen to Pam’s weekly podcast by the same name. Pam Harris is changing the way our math teachers view and teach mathematics. She is a mother, a former high school math teacher, a university lecturer, an author, and a mathematics teacher-educator. She believes that Real Math is thinking mathematically, not just mimicking what a teacher does. Pam helps leaders and teachers to make the shift that supports students to learn Real Math.

Last summer Robin Davis, Kevin Markovich, and Jennifer Adams attended several of Cathy Fosnot’s workshops. Cathy Fosnot is a leading voice in mathematics education. She is Professor Emerita of Education at the City College of New York and the founder of Mathematics in the City, a national center for professional development located at the college. Cathy devotes her time to speaking around the world, fostering school change through professional learning and support, and producing new resources and materials. MTS math teachers felt lucky to be at Cathy’s Context for Learning: A Deeper Dive workshops in San Francisco, where they focused on how to engage students in examining structure and regularity, how to support generalization and the development of proof, and how to use tools for formative assessment that truly provide insights for teaching. Jennifer Adams also participated in a Learn-to-Lead workshop with Cathy at her home in Connecticut. The Learn-to-Lead Seminar is designed for coaches, math curriculum leaders, and administrators. The topics of study included how to assess in powerful ways during conferrals when examining students’ work samples, how to record critical formative data, and how to use developmental rubrics to pick up data that will genuinely inform teaching.