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 within Long-View math classrooms instructional practices that support high cognitive demand and high levels of learning among all students. This starts with deliberate work cultivating an environment that attends in an intentional manner to attitudes, beliefs, customs, and ways of interacting that resist the “default culture of schooling.”
When visitors walk into a Long-View math classroom, it is evident immediately that something is different. It is clear that argument is encouraged, sound reasoning and mathematical justification is lifted up, and the kids seek deep understanding. Whiteboards are covered with mathematics, the excitement is palpable, questions hang in the air, and feedback is constant.
Yesterday, as a means of an example, a visitor joined Navy Band and watched as these young elementary students discussed ways they could tackle simplifying 36 - 71. With a growing understanding of integers and the conceptualization of the definition of subtraction as “adding debt,” the band discussed how their knowledge of range could be employed as a means of simplifying 36 + -71. After, they leveraged their understanding of like terms to simplify it another way. Incidentally, in the granular instruction that is typical of US textbooks, a child would likely not be asked to wrestle with this problem until around 6th grade. This problem, for example, hits a 6th grade Texas Essential Knowledge and Skills standard. At Long-View we don’t shy away from discussing numbers below zero and then suddenly, at 6th grade, reveal to kids that some “mysterious” numbers exist below the number line. How confusing to know about the existence of negative numbers from experience with thermometers but to be told they are so complex that one can’t grapple with them until middle school?? How very destructive to the goal of cultivating high levels of learning!
Our visitor to Navy Band continued watching as the group collectively discussed 36 + -71 as equivalent to [36 + (-36 + -35)] (this understanding was related to their work determining the range of two numbers on a number line) and as equivalent to [(30 + -70) + (6 + -1)] (this work was built on their understanding of like terms). After the discussion, these six through nine-year-olds moved to rolling whiteboards spread across the room to work with partners on a set of “Studio Time” problems that contained problems that looked similar to those those that anchored the group discussion, as well as problems that were novel and required what we call a “cognitive leap.”
The Long-View math experience starts with our educators cultivating a culture that stands on different fundamental assumptions than those that undergird the traditional math classroom. These assumptions are then nurtured to foster very different norms of participation and values within the student community.
Two key assumptions at Long-View are:
Children have significant capacity to understand many of the complexities that are part of the world in which they live.
Thus, as our math teachers planned the content for the lesson as described for Navy Band, they worked to shape the broadest and deepest understanding of the concept of subtraction as possible, by connecting it to the concept of addition and by asking the kids to grapple with a range of number types (the set of integers on this day, but the set of rational numbers on another day).
Deep understanding derives from being a producer – making, doing, and creating – rather than simply being a consumer of knowledge.
Thus, the teacher in the above episode did not teach through the method of “watch me and then do exactly what I do,” but rather strategically asked questions, required clear reasoning and justification, and scaffolded the kids toward a deeper understanding of subtraction and integers. His goal was not teaching the kids how to solve today’s problem, but teaching them how to solve all future problems by leveraging deep conceptual understanding. Studio Time then followed the group discussion, and the children worked at whiteboards to create their own algorithms simplifying the set of Studio Time expressions.
These assumptions and others are then nurtured to foster very different norms of participation within the student community. Norms, or ways participants act and behave within a culture, can radically affect learning outcomes.
Some of the norms of participation that researchers and visitors notice while observing in Long-View classrooms are:
We listen critically
We take charge of our own learning by asking questions
We know that there are many learning opportunities and that collaborating with peers or following a peers’ reasoning is one of those crucial opportunities for learning
We take intellectual risks, actively and thoughtfully contributing reasonable ideas
We seek and accept challenges, persevering and understanding that challenge prompts new learning
We are open to feedback as a crucial part of the learning process, accepting it readily and giving it often
We bring intellectual curiosity and seriousness to our community
We discuss, debate, and converse to grow both individual and collective understanding
When visitors walk into a Long-View math block, it is clear many things are different. Math is not a “class” but an experience, and we typically work for two hours at a time so that we have time to move into deep work. Lessons are focused on concepts and not procedures, and productive discourse is the key pedagogical tool that brings individuals and the collective group towards new learning. Kids work in Studio Time at rolling whiteboards and experience several Thought Exercises during each block. Feedback is constant. But below the surface, the culture and norms support these productive ways of working and learning that “resist the default culture” of schooling and instead produce a high level learning environment.