ABSTRACT
The following three Standards for Mathematical Practice, SMP #1, SMP #5, and SMP #6, serve as the frame for the door. Students who venture through this door are on a journey that promotes processes, proficiencies, and practices in the Common Core mathematics classroom. These three practices are pivotal not only because they develop conceptual understanding of the content, but also because they play an integral role in the implementation of the other five practices. These practices should permeate the mathematics classroom environment and become part of the daily fabric of both mathematics instruction and the students’ mathematics experience.
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
(Common Core State Standards Initiative, 2012b) As students move from middle school into high school, they begin to develop new mathematical knowledge by investigating problems that arise from mathematics as well as other contexts. Tasks need to be carefully chosen to give students at all levels a point of entry and yet be both cognitively demanding and interesting enough to motivate the student to keep at it until a reasonable solution is obtained. Talking, writing, and reflecting about mathematics are just as important for high-school students as they were for students in earlier grades. Through mathematical discourse and writing, students explain and justify their strategies and approaches to problem solving. Students have many more strategies at their command now due to the ever-increasing complexity of the mathematical concepts that they are investigating. Situations or problems that are interesting to the student can serve as the springboard from which students can investigate and develop understanding of many mathematical concepts. Students can begin to see the important aspects of the topic being investigated as well as the usefulness of the mathematics.5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
(Common Core State Standards Initiative, 2012b) A tool is some type of device that allows a person to carry out a particular function. There are many tools that can be used in a high-school mathematics classroom. Everything from paper and pencil to manipulatives to various forms of technology, including calculators and dynamic computer software, can be thought of as a tool. These become tools of investigation for students as they are doing mathematics. Students need to experience mathematics, and the use of tools of investigation allows them that experience. High-school students come with a diverse toolbox containing the various tools that were available to them in K–8. Many students in high school still benefit from the use of concrete models while also using sophisticated computer software, graphing calculators, and handheld computing devices. Students understand how to use these tools effectively and know when and why it is more efficient and strategic to use each tool. Students need to know and understand the benefits as well as the limitations of the various tools. They need to recognize that tools are a way to explore and deepen their understanding of various concepts.6.Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school, they have learned to examine claims and make explicit use of definitions.
(Common Core State Standards Initiative, 2012b) NCTM considers communicationa way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public. When students are challenged to think and reason about mathematics and to communicate the results of their thinking to others orally or in writing, they learn to be clear and convincing.
(NCTM, 2005, p. 60) Communication can be oral or written; it can employ a visual model or graphic. Students should not memorize a definition just to spit it out again. The purpose of learning vocabulary is to use it later to facilitate building the structures of mathematics. Vocabulary in any discipline usually has connotations specific for that discipline. That is one of the challenges in a technical subject such as mathematics. However, the teacher is the person responsible for monitoring students’ precise use of vocabulary, units of measure, symbols, and other mathematical language.