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The Common Core Standards of Mathematical Practice
The Standards of Mathematical Practice describe the “processes and proficiencies” that maximize students’ deep and lasting conceptual understanding in mathematics.
The first listed Standard of Mathematical Practice is “Make sense of problems and persevere in solving them.” The process of making sense of anything involves (a) evaluating it yourself first; (b) sharing your thinking with others; and (c) reflecting on your interactions with others to refine your thinking. Encouraging students to habitually think, share, and reflect serves them as both budding mathematicians and as problem solvers. The most effective strategy for successful implementation of this standard is to present ample opportunities for students to talk, talk, talk in your classroom. Discourse, particularly in small groups or with a partner, builds confidence and provides different mathematical views.
For example:
1. After presenting a problem and having students briefly think about it themselves, they discuss their solution pathway and accompanying reasoning with a partner. Their ideas may be validated or tweaked, but are always recognized.
2. As students solve the problem, allow them to seek advice and help from their partner. This builds a sense of confidence and teamwork.
3. After solving the problem, invite students to share their results and reasoning in small groups. This reflective practice allows students to revisit and justify their thinking, learn the approaches of others, and identify relationships between different solution pathways.
The mathematical practices are not standards taught in isolation. Think of them as the sauces that blend into every content “dish,” adding flavor, interest, and relevance.
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Posted by Meg Susi on March 29, 2011
Your analogy to sauces, blended into the content, is great. Just like you may have to acquire a taste for a new food, students will have to acquire the talent for having this discourse.
My first year teaching was at the beginning of a district movement to have students explain their thinking and justify their reasoning. At the beginning, I watched in horror as some of my most talented math students would immediately change their answer when asked to justify or explain it. They did this because they assumed that if you were questioning them, their answer was wrong. At the beginning, asking students to explain their thinking was like pulling teeth. A normally chatty bunch of students would prefer to sit silently rather than explain. They saw value only in the correct answer, not the discourse.
I found modeling was key to changing the expectations of the math classroom! This type of dialogue had never been required of our students before and they didn't know what it sounded like. I would regularly think aloud, modeling the process for the students. We would have discussions about the explanations, how could we improve them, make them more clear, and use more mathematical vocabulary. The more they heard the easier it became. They knew what this discourse should sound like. They learned that the "right" answer for a math problem could no longer be presented simply in the form of numbers. After initial resistance, students did learn to enjoy the process. No longer were students sharing an answer and being told that they were wrong, they were being asked to explain how they came to that result. Even if their initial answer was incorrect, as they explained, I could validate some part of their thinking and guide them in the correct direction without making them feel as if their efforts were wasted. It would also help other students who may have been thinking in the same way.
The ability to communicate their reasoning is itself a skill that students need to develop through practice and application, and just like any other skill in mathematics becomes easier with practice. Therefore, I saw a dramatic difference in the abilities of each successive year of students. The ability to conceptualize and verbalize how you solve a problem is not only a tool for math, but a skill onto itself that is valuable well beyond the classroom and well worth the instructional time.