Standards for Mathematical Practice

As a Self-Directed Learner know that:

The Standards for Mathematical Practice (California Common Core)

describe varieties of expertise that mathematics educators (INCLUDING YOURSELF!) at all levels should seek to ____ in their students.  ????????


These practices rest on important “____ ____ ____ ” with longstanding importance in mathematics education.  ????????

processes and proficiencies

The first of these are the National Council of Mathematics Teachers (NCTM) ____ ____ of

  • problem solving,
  • reasoning and proof,
  • communication,
  • representation, and
  • connections.    ????????

process standards

The second are the strands of ____ ____ specified in the National Research Council’s report “Adding It Up”.  ????????

mathematical proficiency

The strands of mathematical proficiency are:

  • adaptive ____,
  • strategic ____,
  • conceptual ____ (comprehension of mathematical concepts, operations and relations),
  • procedural ____ (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and
  • productive ____
    • (habitual inclination to see mathematics as sensible, useful, and worthwhile,
    • coupled with a belief in diligence and one’s own efficacy).   ????????

reasoning    competence     understanding     fluency     disposition

1)  Make sense of problems and persevere in solving them.

Mathematically proficient students start by ____ to themselves the meaning of a ____ and looking for entry points to its solution. ????????

explaining    problem

They ____  givens, constraints, relationships, and goals. ????????


They make ____ about the form and meaning of the solution and plan a ____  pathway rather than simply jumping into a solution attempt. ????????

conjectures   solution

They consider ____ problems, and try special cases and simpler forms of the original problem in order to gain ____  into its solution. ????????

analogous   insight

They monitor and evaluate their____ and change ____ if necessary. ????????

progress   course

Older students might, depending on the context of the problem, ____ algebraic expressions or change the viewing window on their graphing calculator to get the information they need. ????????


Mathematically proficient students can explain ____ between equations, verbal descriptions, tables, and graphs or draw ____  of important features and relationships, graph data, and search for regularity or trends. ????????

correspondences   diagrams

Younger students might rely on using ____  objects or pictures to help conceptualize and solve a problem. ????????


Mathematically proficient students ____ their answers to problems using a different method, and they continually ask themselves, “Does this make ____ ?” ????????

check   sense

They can understand the ____  of others to solving complex problems and ____ correspondences between different approaches. ????????

approaches   identify

2)  Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their ____ in problem situations. ????????


They bring two complementary abilities to bear on problems involving ____ ____. ????????

quantitative relationships

The first is the ability to ____ — ????????


to ____ a given situation and represent it symbolically and manipulate the representing ____ as if they have a life of their own, without necessarily attending to their referents— ????????

abstract   symbols

and the second is the ability to ____, ????????


to pause as needed during the manipulation process in order to probe into the ____ for the symbols involved. ????????


Quantitative reasoning entails habits of creating a ____ ____ of the problem at hand; ????????

coherent representation

considering the ____ involved; ????????


attending to the____ of quantities, not just how to ____ them; and ????????

meaning   compute

knowing and flexibly using different ____ of operations and objects. ????????


3) Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated ____, definitions, and previously established results in constructing ____. ????????

assumptions    arguments

They make conjectures and build a logical ____ of statements to explore the truth of their conjectures. ????????


They are able to analyze situations by breaking them into ____, and can recognize and use ____. ????????

cases     counterexamples

They ____ their conclusions, communicate them to others, and respond to the ____ of others. ????????

justify    arguments

They reason ____ about data, making plausible arguments that take into account the ____ from which the data arose. ????????

inductively    context

Mathematically proficient students are also able to compare the effectiveness of two ____ arguments, distinguish ____ logic or reasoning from that which is flawed, and—if there is a____ in an argument—explain what it is. ????????

plausible    correct   flaw

Elementary students can construct arguments using concrete____ such as objects, drawings, diagrams, and actions. ????????


Such arguments can make sense and be correct, even though they are not ____ or made formal until later grades. ????????


Later, students learn to determine domains to which an ____ applies.????????


Students at all grades can listen to or read the ____  of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. ????????


Students build proofs by ____  and proofs by ____. CA 3.1 (for higher mathematics only). ????????

induction      contradiction

4) Model with mathematics.

Mathematically proficient students can ____ the mathematics they know to solve problems arising in everyday life, society, and the workplace. ????????


In early grades, this might be as simple as ____  an addition equation to describe a situation. ????????


In middle grades, a student might apply ____  reasoning to plan a school event or analyze a problem in the community. ????????


By high school, a student might use geometry to solve a ____  problem or use a function to describe how one quantity of interest ____ on another. ????????

design     depends

Mathematically proficient students who can apply what they know are comfortable making  ____ and approximations to simplify a complicated situation, realizing that these may need ____  later. ????????

assumptions    revision

They are able to identify important quantities in a ____  situation and map their relationships using such ____ as diagrams, two-way tables, graphs, flowcharts and formulas. ????????

practical    tools

They can analyze those relationships mathematically to draw ____. ????????


They routinely interpret their mathematical results in the ____ of the situation and reflect on whether the results make sense, possibly improving the ____ if it has not served its purpose. ????????

context      model

5) Use appropriate tools strategically.

Mathematically proficient students consider the available tools when ____  a mathematical problem.????????


These ____ 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 ____  to be gained and their ____. ????????

insight      limitations

For example, mathematically proficient high school students analyze graphs of ____ and ____  generated using a graphing calculator. ????????

functions       solutions

They detect possible errors by strategically using ____  and other mathematical knowledge. ????????


When making mathematical models, they know that technology can enable them to ____ the results of  varying assumptions, explore ____, and compare predictions with ____. ????????

visualize       consequences        data

Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a ____, and use them to ____ or solve problems. ????????

website     pose

They are able to use technological tools to ____  and ____  their understanding of concepts. ????????

explore      deepen

6) Attend to precision.

Mathematically proficient students try to communicate ____ to others. ????????


They try to use clear definitions in ____ with others and in their own reasoning. ????????


They state the meaning of the ____ they choose, including using the equal sign consistently and appropriately. ????????


They are careful about specifying units of ____, and labeling axes to clarify the correspondence with quantities in a problem. ????????


They calculate accurately and ____, express numerical answers with a degree of precision appropriate for the problem context. ????????


In the elementary grades, students give carefully formulated ____  to each other. ????????


By the time they reach high school they have learned to examine claims and make ____ use of definitions. ????????


7) Look for and make use of structure.

Mathematically proficient students look closely to discern a ____ or structure. ????????


Young students, for example, might notice that three and seven more is the ____ ____ as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. ????????

same amount

Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the ____ ____  . In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 +7. ????????

distributive property

They recognize the significance of an existing line in a ____ ____ and can use the strategy of drawing an auxiliary line for solving problems.  ????????

geometric figure

They also can step back for an overview and ____  perspective. ????????


They can see complicated things, such as some algebraic expressions, as single objects or as being ____  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 ____ cannot be more than 5 for any real numbers x and y. ????????


8) Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are ____, 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 ____ ____ whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3.  ????????

repeatedly check

Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the ____ ____  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. ????????

general formula

They continually evaluate the reasonableness of their____  results. ????????


Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the ________ as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. ????????

subject matter

Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in ____ ___. ????????

subject matter

The Standards for Mathematical Content are a balanced combination of  and understanding. ????????


Expectations that begin with the word “understand” are often especially good opportunities to ____ the practices to the content. ????????


Students who lack understanding of a topic may rely on ____ too heavily. ????????


Without a flexible base from which to work, they may be less likely to

consider ____ problems, ????????


represent problems ____, ????????


justify ____, ????????


apply the mathematics to ____ situations, ????????


use technology mindfully to ____ with the mathematics, ????????


explain the mathematics  to other students, ????????


step back for an , or ????????


deviate from a known ____ to find a shortcut. ????????


In short, a lack of understanding effectively prevents a student from ____ in the mathematical practices. ????????


In this respect, those content standards which set an  of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice.????????


These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to ____ ____ the curriculum, instruction, assessment, professional development, and student achievement in mathematics. ????????

qualitatively improve