ABSTRACT
The following five Standards for Mathematical Practice, SMP #2, SMP #3, SMP #4, SMP #7, and SMP #8, serve as the door. Your students’ mathematics journey continues as they apply these practices, which continue to promote processes, proficiencies, and conceptual understanding in the Common Core mathematics classroom. The implementation of these five practices often occurs in tandem as they are integrated into daily content explorations.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits 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.
(Common Core State Standards Initiative, 2012b) In high school, students experience situations that allow them to connect mathematics not only to other contexts, but to other mathematics. By now, students are becoming mathematically mature enough to investigate complex tasks and to explain what strategy they employed and why. Students in high school have expanded their mathematical knowledge to include algebraic and geometric concepts as well as a deeper knowledge of statistical concepts. The symbols, operations, and properties that are associated with these various concepts should be as accessible to high-school students as basic arithmetic was to elementary students when developing strategies for their problem-solving tasks.3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
(Common Core State Standards Initiative, 2012b) By high school, students should be asking more sophisticated questions as well as answering and offering justifications for more sophisticated questions. Teachers continue to facilitate this development by leading mathematical discussions with probing questions and by providing rich, challenging, and meaningful tasks. Students in high school should be held to a high standard when discussing and writing about mathematical concepts. Informal arguments should begin to give way to more formal, structured justifications of various sorts.4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
(Common Core State Standards Initiative, 2012b) In high school, students should begin to formulate questions about their world that they can employ mathematics to investigate. Student models of real-world phenomena utilize a diverse assortment of tools, from charts, graphs, and dynamic software drawings to technology for calculations and simulations. High-school students should also be able to move seamlessly between various representations of mathematical concepts.7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 − 3(x − y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
(Common Core State Standards Initiative, 2012b) By high school, students should be discerning patterns and developing conjectures regarding the patterns they have observed. They see the structures and how they can be expanded from basic arithmetic operations and applied to more complex operations, such as those involving algebraic expressions, imaginary numbers, and vectors. They understand which properties hold for which systems and why. Their observations and resulting conjectures are more sophisticated and mature.8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y − 2)/(x − 1) = 3. Noticing the regularity in the way terms cancel when expanding (x − 1)(x + 1), (x − 1)(x 2 + x + 1), and (x − 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
(Common Core State Standards Initiative, 2012b) High-school students are able to take the culmination of their mathematical knowledge and begin applying it to new problem scenarios. They can adapt, adjust, and apply what they have observed to new situations. Teachers will continue to offer challenging new mathematical tasks throughout the high-school student’s mathematical journey.