One of the unique features of the Common Core Standards for Mathematics is the focus on Content standards and Process standards. These process standards (there are 8 of them) are not a checklist of teacher to-dos, but rather they are proficiencies for students to experience and demonstrate as they master the content standards.
Each of the eight Standards for Mathematical Practice begins with three words—Mathematically proficient students. This language establishes an expectation for evidence of student growth toward proficiency in each of these eight practices as part of the K–12 mathematics learning experience.
The Standards for Mathematical Practice describe what students are doing as they engage in learning the CCSS for mathematics content standards. How should students engage with mathematics tasks and interact with their fellow students? How well do teachers develop students’ engagement in mathematics reflecting the CCSS Mathematical Practices?
One of the fundamental shifts of the Common Core for Mathematics is that how students learn the mathematics, is now as important as what students learn. And, as a school leader, you need to both support (celebrate) and hold accountable teacher lesson planning design and implementation that intentionally plans for these student proficiencies.
Over the next few blogs, I will outline ideas for each of the eight mathematical practices around two critical questions:
1. What is the intent of each Mathematical practice?
2. How will you ensure each collaborative team addresses the CCSS Mathematical practice on a unit-by-unit basis?
It is up to you and your teachers to shift instruction and provide evidence that students are actually developing each practice.
Mathematical Practice 1: Make Sense of Problems and Persevere in Solving Them
Mathematical Practice 1, “Make sense of problems and persevere in solving them,” refers to the ability of students to explain to themselves (and others) the meaning of a mathematical task or problem and look for entry points to its solution (NGA & CCSSO, 2010, p. 5).
What Is the Intent of Mathematical Practice 1?
Problem solving is one of the hallmarks of mathematics and is the essence of doing mathematics (NCTM, 1989). When students are engaged in problem solving, it means they are drawing on their understanding of mathematical concepts and procedures with the goal to reach a successful response to the problem.
As you study the expectations for Mathematical Practice 1, you will notice several areas for student proficiency including:
1. Students make conjectures about the meaning of a solution and plan a solution pathway.
2. Students try special cases or simpler forms to gain insight (they hypothesize and test conjectures).
3. Students monitor and evaluate their progress and discuss with others.
4. Students understand multiple approaches and ask the question, “Does this solution make sense?”
5. Students explain correspondences between equations, tables, graphs, verbal descriptions, and data and search for regularity, patterns, or trends.
Successful problem solving does not mean that students will always conclude with the correct response to a problem, but rather that students will undertake a genuine effort to engage in the problem-solving process, drawing on learning resources described in the other practices such as appropriate tools, using their prior knowledge, engaging in mathematical discourse with other students, and asking questions to make progress in the problem solving process. Successful problem solvers also recognize that powerful learning can be experienced even when an appropriate answer to a problem ultimately evades the student.
How Can Collaborative Teams Address Mathematical Practice 1?
Teachers play the important role in supporting students’ ability to make sense of problems and persevere in solving them. The first of these roles is the presentation of appropriate problems or tasks for students to solve. While it seems that appropriate is subjective, there are six questions you can present to teachers for discussion within their collaborative teams when planning lessons to assess the quality of problem solving within a common or shared mathematical task.
As we develop common tasks and problems to be used during the unit, we should consider:
1. Is the problem interesting to students?
2. Does the problem involve meaningful mathematics?
3. Does the problem provide an opportunity for students to apply and extend mathematics?
4. Is the problem challenging for students? Does it apply a complexity of reasoning at the DOK level 3 or 4?
5. Does the problem support the use of multiple strategies or solution pathways?
6. Will students’ interactions with the problem and peers reveal information about their mathematics understanding?
Observing students’ interactions with a mathematical task (for example, students’ work, discourse, tools, and representations) will provide information about how their thinking is hindered or evolving by interaction with the problem or task selected. This list of questions is not exhaustive, but it is a beginning step toward examining problems for the potential benefit they can provide for advancing students’ mathematical problem solving and learning.
Your leadership role is to ensure teachers work in collaborative teams to discuss how to help students understand that the answer is not the final step in the problem-solving process. A great deal of mathematical learning can happen when students are guided to explain and justify processes and check the reasonableness of the solution. After teaching lessons within the unit, teachers on the team should ask:
“Is there evidence that students are learning other ways of solving the problem? Is there evidence that students are making and learning mathematical connections to other problems and mathematical connections as they persevere in solving the problem?”
As a school leader, you must focus deliberate attention on implementing the CCSS Mathematical Practices, part of your challenge will be to envision and teach what the practices "look and sound" like in the classroom as part of instruction.
The student tasks teachers design, the questions they ask in the classroom, and the discourse in which students participate will all combine to advance students’ abilities to engage with peers in the Mathematical Practices.