Norms/Expectation Artifacts
Below are two examples of artifacts from my classroom (Fig. 1 and Fig. 2) which detail my efforts to communicate to students what the subject of math is about and what mathematicians do. Both are examples of how I help students make sense of and understand math problems.
1. Student-Centered Routines to Promote Learning and Sense-Making in Students
Art. 1a and 1b are a graphic organizers that I created to help my students make sense of word problems without taking away the cognitive load by giving them a prescribed procedure to follow.
What stands out is that the Art 1a and 1b attempt to help solve word problems by having students ask themselves questions about the particular problem. What some might find surprising is that there is no list of key words with their associated mathematical operation to help students "decode" a problem. These decisions were intentional to allow the students to continue to do the majority of the thinking because "if the teacher provides too much guidance to students during the lesson, students may be left to simply carry out a procedure, with limited thinking required" (Smith and Stein, 27). This artifact is getting students used to the expectation that effective mathematicians need to make sense of problems by consciously thinking about what is happening in a given situation. This graphic organizer shows that strategies effective problem solvers use may include "describing what is going on in your own words", "asking yourself what is known in the given situation", and checking if a solution is "reasonable". Prior to sharing this organizer with my students, I noticed that a common strategy among my students for solving word problems was to scan for numbers and "key" words (altogether, difference, etc.), then apply the mathematical operation of the key word or the procedure that we recently learned in class to the numbers. Considering what some students prior experiences with math might have been, it makes sense why students might do this. In his article, Kristopher Childs describes a common way teachers teach word problems within a GRR model that can be summarized through the following steps: students (1) search the problem for key words and numbers, then (2) apply the 'appropriate' math operation. Frameworks such as this don't hold students accountable for making sense of the mathematics. This supports what I noticed in the earlier stages of my student-teaching experience. My students would often give answers to word problems as simply numbers without context. Then when I would ask students what their answer represented, they would either say they didn't know or re-scan the word problem. Strategies like this are ineffective, because "research tells us that students learn when they are encouraged to become the authors of their own ideas and when they are held accountable for reasoning about and understanding key ideas" (Stein and Smith, 2) |
Art. 2 is work from a student during a Notice and Wonder (N&W) routine regarding an activity on the dilations. During N&W routines students take time, prior to doing any computations, to write down things that they see (notice) in the problem and things that they have questions about (wonder). What stands out to me in this artifact are the guiding questions at the top. The questions lead students towards actions of mathematicians (which I describe here) such as finding patterns ("What do I notice about shape, size, relationships, or patterns?" and "I wonder if this pattern will continue?") or considering multiple strategies/ways of thinking ("Is there another way to think about this?"). Art. 2 shows evidence of this student making connections and tying in ideas from earlier lessons. For instance, the student connects the idea of congruence to this activity when they write "When you move k to 180, the triangles are congruent", and the student is displaying signs of curiosity when they write "What value of k makes the triangle get bigger or smaller?"
This was not the first time we had done a N&W in class, so I was not surprised to see the high number of noticings and wonderings this student provided. During the first few times I enacted this routine with the class, students were reluctant to write more than a one or two total noticings and wonderings. In the early stages, students often would ask me "How do I know if I'm writing the right thing?" This type of question reflects the notion reinforced to students through the GRR model that math is about getting correct answers. With the Equity Literacy framework in mind, this N&W routine is equitable because it provides all students access into the mathematics. I repeat to my students "You cannot be wrong about what you see." |
When considering these artifacts together, as opposed to separate, common patterns appear and connections to the Norms/Expectations component of my vision are more evident.
The main connection both of these artifacts have to the Norms/Expectations component is that they engage students in work that establishes mathematics as a learning subject, not a performance subject. Computations are not the focus of either artifact. These artifacts give students experience developing what I called "habits of good mathematicians" on the previous page. For example, both the graphic organizers and the N&W have students asking questions and making sense of problems. These artifacts are also connected to Freirean Pedagogy because they position students at the center of the lesson by allowing them to form their own thoughts and ideas about the problem through asking themselves questions like what information is given, what information is missing, etc. Using this information students are able to individually make sense of the task and proceed in solving it in a way that makes sense to them mathematically. The You-do We-do I-do model that I propose has students work on tasks without being shown a demonstration by the teacher first, so being able to make sense of the task is a crucial step. This contrasts with the GRR model where the teacher would tell students the important information to notice as they demonstrate how to do a similar task.
Through my exploration of having students think deeply about problems and make sense of them, I have come to realize a potential downside is the amount of time this process can take. Having students take problems to great depths to build strong connections can take a significant amount of time. Expecting students to get through more than 2-3 problems with this protocol in a given class period using the You-do We-do I-do framework can pose a challenge.
The main connection both of these artifacts have to the Norms/Expectations component is that they engage students in work that establishes mathematics as a learning subject, not a performance subject. Computations are not the focus of either artifact. These artifacts give students experience developing what I called "habits of good mathematicians" on the previous page. For example, both the graphic organizers and the N&W have students asking questions and making sense of problems. These artifacts are also connected to Freirean Pedagogy because they position students at the center of the lesson by allowing them to form their own thoughts and ideas about the problem through asking themselves questions like what information is given, what information is missing, etc. Using this information students are able to individually make sense of the task and proceed in solving it in a way that makes sense to them mathematically. The You-do We-do I-do model that I propose has students work on tasks without being shown a demonstration by the teacher first, so being able to make sense of the task is a crucial step. This contrasts with the GRR model where the teacher would tell students the important information to notice as they demonstrate how to do a similar task.
Through my exploration of having students think deeply about problems and make sense of them, I have come to realize a potential downside is the amount of time this process can take. Having students take problems to great depths to build strong connections can take a significant amount of time. Expecting students to get through more than 2-3 problems with this protocol in a given class period using the You-do We-do I-do framework can pose a challenge.