standards for mathematical practice make sense of problems and persevere in solving them. – first...

Standards for Mathematical Practice

• Make sense of problems and persevere in solving them.– First explain meaning of problem to themselves– Analyze, conjecture, plan– Consider analogous problems– Try simpler forms of the original problem– Can explain correspondence between graphs, charts/tables, verbal descriptions, equations– Check their answers – Understand approaches of others; see correspondences between the various approaches


• Reason abstractly and quantitatively.– Make sense of quantities and their relationships in problem situations– Have ability to both decontextualize (abstract a given situation and

represent it symbolically) and to contextualize (consider the actual meaning of the various parts of the situation)

– Ability to create a coherent representation of the problem – consider units involve, meaning of quantities as well as how to compute them

– Know and flexibly use different properties of operations and objects


• Construct viable arguments and critique the reasoning of others.– Understand and use state assumptions, definitions, previously

established results in constructing arguments– Make conjectures and build logical progression of statements to

explore the truth of those conjectures– Analyze situations by breaking them into cases– Recognize and use counter examples– Justify conclusions– Reason inductively about data– Compare effectiveness of two plausible arguments– Read/analyze/question the arguments of others


• Model with mathematics.– Apply known mathematics to solve real world problems– Can comfortably make assumptions and approximations to simplify

complicated situations– Realize such assumptions/approximations may require later

adjustment– Identify important quantities and map their relationships using a

variety of tools: diagrams, two-way tables, graphs, flowcharts, formulas

– Can analyze relationships mathematically to draw conclusions– Routinely interpret the results in the context of the situation– Reflect on whether the results make sense


• Use appropriate tools strategically.– Consider available tools when solving mathematical problems:

pencil/paper, concrete models, ruler, protractor, calculator, spread-sheet, computer algebra system, dynamic geometry software, etc.

– Sufficiently familiar with tools to recognize the insight that can be gained from their use and their limitations

– Strategically use estimation to detect possible errors– Identify relevant external mathematical resources (websites, etc.) and

use them effectively– Able to use technological tools to explore and deepen understanding

of concepts


• Attend to precision.– Communicate precisely to others– Use clear definitions in discussion with others and in their own

reasoning– State meaning of symbols they choose (including equal sign)

consistently and appropriately– Specify units of measures and label axes to clarify correspondence

with quantities in a problem– Calculate accurately and efficiently– Express numerical answers with appropriate degree of precision– Provide carefully formulated explanations– By high school – examine claims and explicitly use definitions.


• Look for and make use of structure.– Examine carefully to discern pattern or structure

• Early: 3 + 7 more is same as 7 + 3 more• Early: Sort shapes by number of sides• Later: 7 x 8 equals 7 x 5 + 7 x 3• Later: x2 (squared) + 9x + 14 – can see the 9 as 2 + 7 and the 14 as 2 x 7

– Recognize significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems

– Can step back for overview and shift perspective– View complicated items as single objects or as being composed of

several objects


• Look for and express regularity in repeated reasoning– Notice if calculations are repeated– Look for both general methods and shortcuts

• Example: recognize repeating decimal when dividing 25 by 11• Example: abstract equation (y-2)/(x-1)=3 by paying attention to the calculation of

slope when repeatedly checking whether points are on the line through (1,2) with slope 3

– Maintain oversight of the process, while attending to details– Continually evaluate reasonableness of their intermediate results

standards for mathematical practice make sense of problems and persevere in solving them. – first...

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