*from NGSS Appendix F: Science and Engineering Practices in the NGSS*

Although there are differences in how mathematics and computational thinking are applied in science and in engineering, mathematics often brings these two fields together by enabling engineers to apply the mathematical form of scientific theories and by enabling scientists to use powerful information technologies designed by engineers. Both kinds of professionals can thereby accomplish investigations and analyses and build complex models, which might otherwise be out of the question. (NRC Framework, 2012, p. 65)

Students are expected to use mathematics to represent physical variables and their relationships, and to make quantitative predictions. Other applications of mathematics in science and engineering include logic, geometry, and at the highest levels, calculus. Computers and digital tools can enhance the power of mathematics by automating calculations, approximating solutions to problems that cannot be calculated precisely, and analyzing large data sets available to identify meaningful patterns. Students are expected to use laboratory tools connected to computers for observing, measuring, recording, and processing data. Students are also expected to engage in computational thinking, which involves strategies for organizing and searching data, creating sequences of steps called algorithms, and using and developing new simulations of natural and designed systems. Mathematics is a tool that is key to understanding science. As such, classroom instruction must include critical skills of mathematics. The NGSS displays many of those skills through the performance expectations, but classroom instruction should enhance all of science through the use of quality mathematical and computational thinking.

*from A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (pages 51)*

In **science**, mathematics and computation are fundamental tools for representing physical variables and their relationships. They are used for a range of tasks, such as constructing simulations, statistically analyzing data, and recognizing, expressing, and applying quantitative relationships. Mathematical and computational approaches enable predictions of the behavior of physical systems, along with the testing of such predictions. Moreover, statistical techniques are invaluable for assessing the significance of patterns or correlations.

In **engineering**, mathematical and computational representations of established relationships and principles are an integral part of design. For example, structural engineers create mathematically based analyses of designs to calculate whether they can stand up to the expected stresses of use and if they can be completed within acceptable budgets. Moreover, simulations of designs provide an effective test bed for the development of designs and their improvement.

from NGSS Appendix F: Science and Engineering Practices in the NGSS

In both science and engineering, mathematics and computation are fundamental tools for representing physical variables and their relationships. They are used for a range of tasks such as constructing simulations; solving equations exactly or approximately; and recognizing, expressing, and applying quantitative relationships.

Mathematical and computational approaches enable scientists and engineers to predict the behavior of systems and test the validity of such predictions.

K-2 | 3-5 | MS | HS |
---|---|---|---|

Mathematical and computational thinking in K–2 builds on prior experience and progresses to recognizing that mathematics can be used to describe the natural and designed world(s). | Mathematical and computational thinking in 3–5 builds on K–2 experiences and progresses to extending quantitative measurements to a variety of physical properties and using computation and mathematics to analyze data and compare alternative design solutions. | Mathematical and computational thinking in 6–8 builds on K–5 experiences and progresses to identifying patterns in large data sets and using mathematical concepts to support explanations and arguments. | Mathematical and computational thinking in 9-12 builds on K-8 and experiences and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. |

Decide when to use qualitative vs. quantitative data. | Decide if qualitative or quantitative data are best to determine whether a proposed object or tool meets criteria for success. | ||

Use counting and numbers to identify and describe patterns in the natural and designed world(s). | Organize simple data sets to reveal patterns that suggest relationships. | Use digital tools (e.g., computers) to analyze very large data sets for patterns and trends. | Create and/or revise a computational model or simulation of a phenomenon, designed device, process, or system. |

Describe, measure, and/or compare quantitative attributes of different objects and display the data using simple graphs. | Describe, measure, estimate, and/or graph quantities such as area, volume, weight, and time to address scientific and engineering questions and problems. | Use mathematical representations to describe and/or support scientific conclusions and design solutions. | Use mathematical, computational, and/or algorithmic representations of phenomena or design solutions to describe and/or support claims and/or explanations. |

Use quantitative data to compare two alternative solutions to a problem. | Create and/or use graphs and/or charts generated from simple algorithms to compare alternative solutions to an engineering problem. | Create algorithms (a series of ordered steps) to solve a problem. Apply mathematical concepts and/or processes (such as ratio, rate, percent, basic operations, and simple algebra) to scientific and engineering questions and problems. Use digital tools and/or mathematical concepts and arguments to test and compare proposed solutions to an engineering design problem. |
Apply techniques of algebra and functions to represent and solve scientific and engineering problems. Use simple limit cases to test mathematical expressions, computer programs, algorithms, or simulations of a process or system to see if a model “makes sense” by comparing the outcomes with what is known about the real world. Apply ratios, rates, percentages, and unit conversions in the context of complicated measurement problems involving quantities with derived or compound units (such as mg/mL, kg/m ^{3}, acre-feet, etc.). |

By grade 12, students should be able to

- Recognize dimensional quantities and use appropriate units in scientific applications of mathematical formulas and graphs.
- Express relationships and quantities in appropriate mathematical or algorithmic forms for scientific modeling and investigations.
- Recognize that computer simulations are built on mathematical models that incorporate underlying assumptions about the phenomena or systems being studied.
- Use simple test cases of mathematical expressions, computer programs, or simulations—that is, compare their outcomes with what is known about the real world—to see if they “make sense.”
- Use grade-level-appropriate understanding of mathematics and statistics in analyzing data.

K-2 | 3-5 | 6-8 | 9-12 |
---|---|---|---|

n/a | 5-PS1-2 5-ESS2-2 |
MS-PS4-1 MS-LS4-6 |
HS-PS1-7 HS-PS2-2 HS-PS2-4 HS-PS3-1 HS-PS4-1 HS-LS2-1 HS-LS2-2 HS-LS2-4 HS-LS2-6 HS-ESS1-4 HS-ESS3-3 HS-ESS3-6 HS-ETS1-4 |

Science Practices Continuum - Students' Performance

This tool is a continuum for each practice that shows how students' performance can progress over time. A teacher can use the continuum to assess students' abilities to engage in the practices and to inform future instruction. From Instructional Leadership for Science Practices.

Science Practices Continuum - Supervision

This tool is a continuum for each practice that shows how instruction can progress over time. An instructional supervisor can use the continuum to identify the current level for a practice in a science lesson. Then the supervisor can provide feedback, such as offering instructional strategies to help move future instruction farther along the continuum. From Instructional Leadership for Science Practices.

Potential Instructional Strategies for Using Mathematics and Computational Thinking

This instructional strategies document provide examples of strategies that teachers can use to support the science practice. Supervisors might share these strategies with teachers as they work on improving instruction of the science practices. Teachers might find these helpful for lesson planning and implementing science practices in their classrooms. From Instructional Leadership for Science Practices.

Bozemanscience Video