Early in the year, Mr. Mann brought a challenging concept to Math Block in Sage and Crimson Bands. It wasn’t exclusively a math concept, but an invitation to think differently about risk-taking. What does it mean to be intellectually fearless?

The phrase "intellectual fearlessness” comes from the book Thinking, Fast and Slow, by psychologist and economist Daniel Kahneman. When Mr. Mann introduced the concept, learners recognized fearlessness as a sort of bravery that, to the observer, presents as a lack of fear. They observed that while a person might have the experience of feeling afraid but taking action anyway, that same person might seem “fearless” to others.

Sage and Crimson learners also grappled with the term “intellectual,” which they defined as “of or relating to the mind.” They connected the attribute with learning and conjectured that “the growth of the mind is learning.” One learner posited that intellectual fearlessness was simply “the willingness to learn.”

Ideas kept building as another young learner suggested that being intellectually fearless would include being vulnerable….

Guest Post: Astrid’s Essay About Leadership in "Big + Little Math"

Every fall our 8th graders begin their journey of looking at high schools. We hate to think of these amazing kids leaving us, but they are ready for the next step and always go off to rigorous high schools across Austin. Recently one of our 8th graders, Astrid, wrote a high school admissions essay that highlighted her experience as a math mentor in our program called “Big + Little Math.” Astrid gave us permission to share this essay with you once it had been turned in as part of her application process for a particular high school — Enjoy!

I never thought I would get to be a math teacher in seventh grade, but in late fall of 2019, I suddenly found myself standing at a whiteboard teaching a third grade student how to simplify fractions. This unexpected opportunity not only taught me more about myself as a leader and as a friend, but also ended up impacting the entire school….

math, generalLisa ZapalacFebruary 6, 2021Long-View Micro SchoolGuest Post

The Long-View Math Block

Our Math Block is more of an experience than a class. It is rich and multi-dimensional. There are multiple goals at any one time, and the content isn’t explored in a strict linear trajectory. To us, the traditional math class pattern seems to be more about filling time and covering content than providing an experience that will transform thinking. In contrast, we think of the development of mathematics knowledge as multiple threads that must be braided together over time to create strong and lasting understanding. We pull on threads related to multiple concepts and ways of thinking, then bind them together. We aim for deep conceptual understanding, bearing in mind that the way learners understand an idea can have major implications for how, or whether, they understand other ideas. Thus, we are very purposeful in the way we build concepts, and we also work to emphasize the conceptual continuities among different number forms…

mathLisa ZapalacOctober 21, 2020Long-View SchoolMath Block, Thought Exercises, Studio, concept study

Field Study Days: Visiting Educators in Long-View Math Classrooms

Long-View supports the learning of both our amazing student population, as well as the learning of schools from across the world. Through the number lab, our joint venture that is focused on teacher education, Long-View Micro School students and faculty share in the mission of helping other teachers meet the challenge of educating children at high levels in mathematics….

mathLisa ZapalacFebruary 26, 2019Long-View Micro Schoolmathematics, field study days, the number lab

The Culture of a Long-View Math Classroom

At Long-View, we spend a significant amount of time investing in the learning culture of our school, with particular attention paid to how this translates within our mathematics classrooms. As Harvard educator Dr. Richard Elmore has so often made clear, the “default culture of American instruction” contains “certain robust patterns of instructional practice that are unique to the US and that are highly destructive to higher level student learning.” From our standpoint, these highly destructive instructional patterns are easily observed within American math classrooms and at Long-View we seek to disrupt these and nurture…

mathLisa ZapalacOctober 24, 2018Long-View Micro Schoolculture, math, studio time, integers, subtraction, navy band

Metacognitive Coaching in Math Classrooms

Having been fortunate to observe a variety of math classrooms, especially within the independent school sector, we’ve come to learn a number of things, which we will write about across several blog posts. Today we are focused on the lack of metacognitive instruction and coaching within many math classrooms.

Metacognition research has assisted educators in changing the way we view learning. We know high achieving students usually apply metacognitive processes in their learning and problem solving. And students who apply metacognitive processes tend to be higher achievers. Recent studies have shown that even young children can apply metacognitive processes when the tasks fit their interests and capabilities. And any age-related development may likely be due to lack of appropriate exposure in school....

mathLisa ZapalacJuly 30, 2017Long-View Micro Schoolmathematics, metacognition, growth mindset

Watch Your (Mathematics) Language!

If you step into a Long-View mathematics class, you’re apt to hear students reading mathematics expressions using language that seems a little different.

In reading an addition expression such as 41 + 17, for example, you’d hear a Long-View student say “forty-one and seventeen” rather than “forty-one plus seventeen.” While this may seem like a subtle substitution, there is a great deal of deliberate thought underneath this use of language that supports our young mathematicians as they develop strong conceptual understandings that will transfer across all of arithmetic to Algebra. Language is actually one of the most under-utilized models in school mathematics. Not at Long-View....

mathLisa ZapalacJanuary 25, 2017Long-View Micro Schoollanguage, math, models, manipulatives, addition, integers

The Japanese Math Problem Experiment

This past week Long-View mathematicians were presented the following problem:

9 – 3 ÷1/3 + 1

Do you know the correct answer?

This expression went viral in Japan a few years ago and only 60% of Japanese 20-year-olds could solve it correctly.

So why is it that most of the Long-View students solved this problem correctly (during the 6th week of school, no less)? Long-View kids know to look at a complex expression and....

How is it an 8-year-old can do that?

Visitors in our Long-View math classes often inquire about the ages of the children, because it is surprising to see such young children engaged in high-level content. How is it that an 8-year-old can do that?

The answer begins with letting go of developmental constraints previously imposed on young learners. Because our classrooms are multi-age and our instructional planning does not start from an assumption that children can only do as much as a generic textbook says their grade level can accomplish, we move much further and much deeper, traversing some very interesting mathematical terrain…

mathLisa ZapalacSeptember 26, 2016Long-View Micro Schoolculture, curriculum, high-expectations

Math: Here's How We Do It

The first step in our math program is to build what we call "Number Reasoning." In order to do this, we will go back through major concepts and rebuild them so that the kids have a much deeper understanding that will promote stronger reasoning and "number sense." Additionally, the deeper conceptual understanding will build a connection between arithmetic and algebra, providing a strong foundation for algebraic thinking…

mathLisa ZapalacSeptember 11, 2015Long-View Micro Schoolnumber reasoning, arithmetic, algebra, deductive lesson