The Long-View Math Block
ARTICLE
INTRODUCTION
The Long-View 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. 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.
THOUGHT EXERCISES
Thought Exercises are multi-concept challenges designed to provoke thought and discussion and to expand and deepen understanding. Each component is multi-dimensional, not simply covering a singular goal, objective, or standard.
Thought Exercises are routine in structure, but not routine in content. Because a Thought Exercise is delivered in the same way every time, learners know just what to expect and how to interact with the Thought Exercise. This predictability supports learners in focusing on the mathematics content and their thinking. The six Thought Exercises that cycle across the year enhance learners’ abilities to think critically; communicate this thinking clearly; challenge and/or build on the thinking of others; and recognize, refine, and/or devise novel, mathematically sound approaches to problems.
The six Thought Exercises are:
Thought Exercises, as the name implies, are meant to develop the thinking that is crucial to the full development of mathematicians. To truly access the conceptual terrain of Long-View’s mathematics, teachers need to work to develop the thinking skills of their learners—both generally and with specific regard to mathematics. Thus, these are exercises in thought and require the teacher to deliver them in ways that prompt learners to grow more sophisticated ways of thinking.
During Thought Exercises, the teacher’s role is to mediate discussions. The teacher provides prompts that are substantial enough in content to elicit robust conversation, asks the questions necessary to help learners when they appear stuck, and encourages learners to problem-solve through a more thorough search of their knowledge or deeper critical analysis of the information that is under investigation.
The teacher must also remain cognizant of the models learners use to help illustrate their thinking, the precision of the language they use to explain their thinking to others, the way in which they challenge and/or offer support for their ideas, and the mathematical justification of the ideas presented.
And not unlike Thought Exercises, the expressions and questions presented in Concept Studies facilitate discourse intended to create new knowledge, understanding, and experience — building upon the prior knowledge, understanding, and experience of those in the conversation.
When our learners are engaged in discourse, they form, communicate, and listen to arguments. We invite, encourage, and coach learners to communicate and grapple with conflicting claims and counterclaims. The teacher may ask the band whether they agree with the argument. If anyone disagrees, the teacher or a member of the band will ask for evidence or reasoning that supports the counterclaim. In this way, we normalize the messiness of learning. We recognize that struggle is indeed intrinsic to learning; thus, we embrace struggle.
The Long-View Math Block is typically two hours and within that time we engage young mathematicians in two Thought Exercises, a Concept Study, and Studio. There’s also always a series of brief “brain breaks,” where kids go outside for fresh air, eat a snack, and just generally prepare themselves to re-engage in academic study.
CONCEPT STUDY
The Concept Study is the “lesson” segment of Long-View Math. It is devoted to constructing or deepening understanding of a mathematical concept. Concept Studies are thoughtfully prepared, focused community discussions.
The architecture of a Concept Study is typically a series of expressions combined with very purposeful questions. Learners are led from some mathematical understanding with which they are quite confident to new mathematical terrain. A new idea is constructed, and learners make cognitive leaps when taken through a purposefully designed series of prompts that helps them deduce this new idea. Questions provide scaffolding; the teacher does not “show” or “tell” learners what they must do.
STUDIO
Studio typically follows Concept Study and allows learners to immerse themselves in working as mathematicians. This happens at whiteboards where partners collaborate and present their work to their peers for inspection and discussion. Partners explore problem sets that require a variety of mathematical understandings, which supports differentiation. Through these problem sets, learners practice the new ideas discussed during a Concept Study and begin encoding these ideas into their stock of knowledge. This is not “worksheet mastery practice” but rather an opportunity to try out new ideas and rehearse thinking aloud with support from a partner.
During Studio, a learner's ability to collaborate is likely more evident than in any other element of the Math Block. As partners work through a problem set, one partner is the designated scribe, yet both partners must reach consensus on the approach to a particular problem as well as the way in which the thinking is illustrated. The norm for this collaboration during Studio is captured in a common refrain: “One Marker, Two Minds.” Thus, good collaboration, which we essentially define as creating together, is constantly modeled, coached, and commended. Learners are encouraged to habituate listening critically to one another so that learners are aware that learning comes from many sources, especially their peers. Additionally, learners are supported as they attempt to use strategic questioning to scaffold for their partners. All of this coaching that learners receive is lean: teachers drop in and listen to the exchange between partners and give minimal feedback, being careful not to take away the learning opportunity and to provide just enough feedback so that the learners can incorporate it into their practice. Lastly, learners work with their partners over the course of several weeks in order for them to have ample time to develop productive partnerships.
As the Math Block allows learners to engage in the work of deeply understanding mathematical ideas, learners have the opportunity to interact with each other in many of the ways that professional mathematicians do. Learners methodically analyze situations, question, seek connections, formulate conjectures, debate, test, iterate their claims, etc. As a result of this very deliberate, active engagement, learners enhance their abilities to think critically and clearly convey their thinking. They also become better able to collaborate with their peers in order to devise mathematically sound approaches and solutions to problems.