7. Look for and make use of Structure.
The mathematically proficient student is capable of making close observations to discern patterns or “structures”. These structures form the basics of mathematics. The ability to discern these structures is vital to problem solving. The structures may be as simplistic as a number line; 2 follows 1 and 3 follows 2, or they may be incredibly complex. Mathematical growth requires building an ever greater awareness of structures and interconnectedness.
Specific instances could include recognizing multiplication is a specific number of sets, or that fractions represent the present parts of a whole. Students start by learning to count in order; 1, 2, 3,…. Later students learn more complex structures such as properties of operation, which can be used to add, subtract, multiply and divide. Recognition of even more complex structures allow for solving difficult algebraic equations, creating geometric proofs and statistical analysis.
Finally, students must be able to describe structures in writing, orally and through visual representations. Critical thinking skills and a true understanding of how the structures are formed is required to process such translations.
5. Use Appropriate Tools Strategically.
Math proficiency standard 5 requires students to use appropriate math tools and use them strategically. Tools used for mathematics are varied and include pencil and paper, calculators, computer software, Ipads with programmed instruction lessons, rulers, compasses, protractors, manipulatives and countless other items. Students should show proficiency using all applicable tools. Showing proficiency requires realizing what tools can be used, and how they should be used dependent upon the problem being solved. Students should recognize the tool’s value and limitations.
Tools help students better visualize and understand problem solving by providing a deeper level of learning. Students are able to look at problems in different ways and compare solutions by using the data provided by utilizing mathematical tools. This supports learning at higher levels, and facilitates a more comprehensive approach to solving both mathematical and everyday problems. Carpenters, tool-makers, scientists and countless other professions require the proficient use of mathematical tools to complete projects.
While many bemoan the fact today’s students utilize computers and calculators, like the slide-rule and trigonometric tables of old, these tools allow students to solve problems using extremely large numbers or great amounts of data. The graphing calculator and internet allow students to explore mathematical content that in the past would not have been possible until much later in the educational process.