I just attended an AMTE session on the Common Core State Standards and math teacher education. The panelists and audience listed many immediate needs as we plan how to implement the standards; the one that caught my ear was the need to align the Mathematical Practice Standards to the work that’s already been done, like the NCTM practices and the “Adding it Up” strands.
I figured that I could at least align the work I’ve already been doing about developing activity structures for teaching problem-solving and problem-solving strategies to the process standards. Here’s my first attempt at that.
The Practice Standards section describes certain behaviors that students who understand the concept in question might engage in:
consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut
The authors hypothesize that without understanding the concept, students will revert to procedures and not do any of above behaviors. I wonder if students who have mastered the process standards might not be able to apply those behaviors to new concepts, and seek to understand and learn the concept through good problem-solving behaviors. To me that suggests that the process standards can become an explicit strand of the curriculum, related to a learning to learn strand. The process standards don’t necessarily need to be applied after learning, as part of application and extension. Instead, mathematical practice standards can be taught with and prior to content standards, through explicit instruction in problem-solving.
So, how do the practice standards align with problem-solving strategy instruction? I’ll try to go practice by practice and think about strategy connections.
1. Make sense of problems and persevere in solving them.
The obvious connection here is the strategy we call Understand the Problem. Included in Understand the Problem is the core activity I Notice/I Wonder, which gives an entry point into the problem to students of all levels, but which can be perfected and improved even by expert problem-solvers (see, for example, the world’s hardest easy Geometry problem). I Notice/I Wonder begins to orient students towards recognizing givens and constraints, identifying mathematical quantities or objects in the problem, and describing relationships among them.
Other Understand the Problem activities help students to sketch or act out the problem, a “lite” version of the Change the Representation strategy that helps students further understand the problem scenario and bring out hidden relationships. Paraphrasing the problem is another key activity here. We also engage students in thinking as much as they can about what the answer can and can’t be. Activities support students to describe as much as they can about the final answer (units, estimates, ranges, etc.) and to work through the problem with an answer they know or suspect is wrong, just to get a further sense of the relationships in the problem.
The Practice Standards also include Change the Representation and Solve a Simpler Problem as strategies that students might employ as they make sense of and persevere in solving problems. Our Change the Representation activities support students to identify “what this problem is about” and use their understanding of mathematical ideas in the problem to brainstorm or select from a provided list of alternative representations. They work in groups to compare representations and try to connect their representations to the constraints and relationships they noticed as they first explored the problem. The Solve a Simpler Problem activities support students to articulate “what makes this problem hard.” We’ve found that students often identify the very things mathematicians notice, but then struggle to understand how they can then simplify the problem or even that they are allowed to simplify the problem. We offer students (and their teacher facilitators) several concrete suggestions for how problems can be simplified or analogous problems can be posed.
In terms of persevering to solve problems, we also offer activities to help students learn questions and techniques to Plan and Reflect and Get Unstuck. To help students plan we offer tips on moving from what you noticed and wondered to possible strategies to try, as well as a structure with group roles and tasks to help students think metacognitively as they select and carry out a plan. For getting unstuck we offer a self-diagnosis based on the guess and check strategy (could I even think of a quantity to guess for? What would be a good guess? Do I know any calculations I could do if I had a guess? How would I know if my guess were correct?) and a series of questions students can ask themselves, a partner, or be asked by a facilitator to help them “unstick” themselves.
Finally, since students who understand problems check their work, we offer the activity “Testing 1, 2, 3, 4″ in the part 2 of the Guess and Check activity sequence.
2. Reason abstractly and quantitatively.
The Problem-Solving and Communication Activity Series takes students through a sequence of activities specifically designed to help students “make sense of quantities and their relationships in problem situations” and introduces the skills of “creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.”
Beginning with the noticing and wondering stages, we introduce a “level 2″ of noticing and wondering that helps students focus on quantities and their values, and relationships or calculations they could do. We stress naming quantities and providing units over identifying values (the numbers in the problem).
Students then use those quantities and relationships to get good at guess and check. Being able to identify the appropriate quantity to guess for, set up appropriate calculations, check results against appropriate constraints, and repeat the process in an organized way are vital to being able to set up a mathematical model of a situation. If the student can’t use guess and check successfully, it’s unlikely that they could set up and solve the problem algebraically.
We then encourage students to employ the Make a Table strategy. Partly this allows them to “take advantage of regularity in repeated reasoning” and notice the patterns in their calculations that could be represented in a mathematical model. It also focuses them again on the naming and organization of quantities. Answering the focal question, “what needs to be organized” further develops the habits of mind in this practice.
Lastly, we move students to the Make a Mathematical Model strategy. It again begins by listing quantities and relationships. Then students are supported to “build up” or “break down” the relationships to develop a complete model for the problem. An alternative strategy is to begin the problem by guessing and checking a few times, organizing work into a table, and then rewriting the calculations with a variable for the unknown quantity in the problem.
3. Construct viable arguments and critique the reasoning of others.
First let’s take a look at the “viable arguments” part. The Look at Cases and Use Logical Reasoning strategies both help students to develop their skills at constructing logical, complete arguments. Look at Cases provides students many examples of the kinds of cases they can explore and break situations into, as case-based reasoning is new to many novice problem solvers. Use Logical Reasoning focuses on the questions, “What must be true? What can’t be true? What might be true?” and then supports students to use if-then reasoning to move items from the “might” column to the “must” or “can’t” column.
Broadening the focus to communication overall, throughout the series we introduce different communication formats and structures that get students writing and talking to learn, to get help, to share, and to present their ideas formally. The activities run the gamut from brainstorming to organizing thoughts to formally presenting. In addition, several activities include structures that have students comparing their work to the work of others and using suggested questioning techniques to help them learn how to give effective feedback.
4. Model with mathematics.
Mathematical modeling is a practice I’d love to do more with. The Make a Mathematical Model strategy has a stronger focus on actually building up the equation, table, flowchart, etc. that students will use to abstract and represent the relationships. We still need some activities that support problem-posing and problem-defining. One way we’re moving in that direction is the posing of mathematical scenarios, without a specific question, about which we ask students to notice and wonder. Every PoW written in the last couple years includes a “scenario only” version, and we give a way a free scenario a week here: http://www.mathforum.org/pow/scenarios/.
5. Use appropriate tools strategically.
We haven’t gone as far as we could to teach students how to select and use appropriate mathematical tools in the activity series, because we didn’t want the reality of technology-poor classrooms to prevent students from using the activities. But in the Activity Series Examples documents, available for all 2009-2010 PoWs, we do model uses of tools like manipulatives, dynamic geometry software, spreadsheets, and graphing utilities. In addition, we’ve done some work to align the activity series with the capabilities of the TI-Nspire here.
6. Attend to precision.
The activity “Testing 1, 2, 3, 4″ in Guess and Check is my favorite for helping students attend to issues of precision, accuracy, and reasonableness. In addition, some of the partner questioning structures in the Get Unstuck, Plan and Reflect, and Wonder activities help students develop habits of mind around asking for and reflecting on precision in language and calculation.
7. Look for and make use of structure.
Looking at the activity series as a tool to help students solve word problems makes it seem disconnected from this sort of looking for structure and connections in specific mathematical content areas. But stepping back and thinking about applying the activities in the context of teaching content might help illuminate how it is that we get students to become people who “look for and make use of structure.” Three activities really stand out for me.
The first is “I Notice/I Wonder.” Proficient noticers see the structure that’s there; proficient wonders seek out even hidden structures. As students practice noticing and wondering in the context of solving problems and learning content, they become better able to see and more disposed to look for structure.
The second is Solve a Simpler Problem. The activities there, while focused on standards methods of simplifying problems or looking for analogous problems, exploit the structure of problems or of the objects within problems. When students look to see how they could make a problem simpler and whether they’ve changed the problem, they are grappling with its underlying structure. When they change a problem to an analogous problem or relate it to a problem they’ve solved before, they’re again using the ability to see structure. And when they re-represent the numbers in the problem, they are realizing that numbers are flexible and they have the power to decompose and recompose them as needed.
Finally, the Play activities help students dig deeper into noticing and wondering by carefully analyzing the “clues” or structure given in a problem or context and allowing themselves to tweak, change, restate, or re-represent them. Here is where students learn to add auxiliary lines (or experiment with taking away given lines) in geometry problems, to recognize the significance of the constraints in the problem (i.e. to see that a problem with two unknowns and one constraint has many possible solutions), to try multiple strategies and representations before getting bogged down in one, and to generally approach problems and math content with an openness to exploring them to find deeper and deeper structures.
8. Look for and express regularity in repeated reasoning.
In the Make a Table strategy (which should really be called Make a Table and Look for Patterns) we offer several ways for students to explore and talk through patterns they see in repeated calculations. We encourage students to look for and describe patterns both horizontally and vertically, as well as to describe what’s happening “over and over again.” Even the simple activity provided in an extension of the Guess and Check activities (courtesy of Dr. Math), in which students do calculations without ever simplifying any steps, help them see and exploit repeated reasoning.