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.
The Long-View Math Block is typically two hours and within that time we engage young mathematicians in two Thought Exercises, a Concept Study (our version of a lesson), and Studio (work time that typically involves partners collaborating at whiteboards, but also sometimes involves individuals working on paper). 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.
Thought Exercises provide serious cognitive challenge for learners while simultaneously facilitating the growth of important learning skills. Thus, by their very design, our six Thought Exercises engage learners in ways that press them to consider understandings that they have already acquired in order to participate in an exchange that will deepen their understandings, as well as enhance their abilities to think critically, clearly communicate their thinking, challenge and/or building on the thinking or ideas of others, and recognize, refine, and/or devise novel, mathematically sound approaches to problems.
As Thought Exercises are about the work that learners must do to solidify their understanding and increase their skills, a teacher’s role, then, is that of a thoughtful mediator of discussions. A teacher provides prompts that are substantial enough in content to elicit robust conversation and asks the questions necessary to help learners when they appear stuck, encouraging learners to conduct a more thorough search of their stock of knowledge or drive deeper critical analysis of the information that is under investigation. Also key in his/her role as a mediator, a teacher remains cognizant of, among many things, the models learners use to help illustrate their thinking, the precision of language in bringing clarity of their thinking for the listeners, the way in which learners challenge and/or offer support of ideas, and the mathematical justification of the ideas presented.
The architecture of a Concept Study is often 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. 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.
In our Math Block we typically move into Studio after a Concept Study. Two or three learners are in a partnership and work together at a whiteboard; transparency is key. Partners explore problems from a set that require a variety of mathematical understandings; this helps to make differentiation possible. With these problem sets, among many things, learners practice the new ideas discussed during a Concept Study in order to begin to make the ideas their own. This is not “worksheet mastery practice” but an opportunity to try out new ideas, rehearse thinking out loud, with support from a partner. This approach is the similitude of a beginner swimmer in a pool or an artist trying out a new technique.
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. Our 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.
