Science & Engineering Practices:
5. Using Mathematics and Computational Thinking
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. Mathematical and computational thinking in grades 3–5 builds on previous 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. - Decide if qualitative or quantitative data are best to determine whether a proposed object or tool meets criteria for success. - Organize simple data sets to reveal patterns that suggest relationships. - Describe, measure, estimate, and/or graph quantities (e.g.,area, volume, weight,time)to address scientific and engineering questions and problems. - Create and/or use graphs and/or charts generated from simple algorithms to compare alternative solutions to an engineering problem Another way to understand this practice... Scientists and engineers often use math for statistics and computational techniques, as well as for deduction and spatial thinking. What are statistics? If you hear that tomorrow there's a 20% chance of rain, or we have 4% unemployment, those are statistics. Statistics is the science of collecting data, analyzing and interpreting that data, and then presenting it to others (often in a visual form, like using pie charts or bar charts). For example, one study showed that in 2018, about 50% (half) of video gamers earned less than $30,000 a year. Scientists use statistics to study data, make inferences (based on samples), and justify how things are related. For example, you might see how body height relates to being a professional basketball player. Mathematical deduction can be used in mathematical models to make predictions. Computational thinking is the human ability to formulate problems so that their solutions can be represented as computational steps or algorithms. Scientists and engineers use these algorithms when developing computer simulations to represent real-world phenomena. Computational thinking embodies one of the most useful skills in science and engineering: the ability to break a large problem down into smaller pieces. A computer program solves each piece independently . Scientists and engineers benefit from this aspect of computational thinking when they try to describe systems [CCC-4]. They identify individual components of the system (abstraction), determining how the components behave independently, and then designing an architecture that allows the objects to interact. These behaviors can then be encoded in executable computer code (automation of algorithms) and analyzed to determine if the abstractions made were valid and the encoding of algorithms was correct (analysis). |
Mathematics is a tool for communication that functions as one of the languages of science and engineering. Numerical representation of quantities is the basis of all measurement in science. Representing data numerically and statistically allows students to determine and communicate the level of confidence or uncertainty in a stated result. The symbolic representation of variables allows scientists and engineers to concisely communicate their systems models. Graphical representations are the most common forms of communicating the findings of investigations.
The level of mathematical thinking applied in the science classroom should parallel the learning of new mathematical skills and practices expected by the CA CCSSM. In the primary grades students can calculate the difference between two measurements to find out how much a plant has grown and to make a graph of measurements collected over time to represent that same idea. As students begin to measure various quantities they will need to discuss and use a variety of units of measure. Starting in upper elementary grades, students encounter and discuss quantities in their scientific investigations that involve more than one type of unit of measure, such as speed as distance traveled divided by time taken, or density as mass per unit of volume. Graphical representations of data, and the recognition of linear relationships in the graphs of distance traveled versus time elapsed for an object moving at a constant speed, or for mass versus volume for objects made from a given substance, can help students grasp the new concepts. With appropriate support and discussion, the mathematical representation becomes a tool for developing the scientific idea and the scientific idea serves as a motivation for learning the mathematical skill. By high school, students will use and interpret a greater variety of graphical representations, algebraic relationships and basic statistical representations of results. Computational thinking is likewise developed progressively across the grades as students develop algorithms for automating computation and for describing behaviors of components in computer models. The learning progressions in the CA NGSS (appendix 1 of this framework) do not specify any computational thinking benchmarks for grades K–5 . At the middle grades level, students can implement simple algorithms for repeated calculations. For example, students with a data table with columns for the mass and volume of many samples can calculate the density of each sample in a third column. Once they understand that this is a repeated operation, they can either continue to carry it out over and over again or code the calculation algorithm into a spreadsheet or other coding language (e .g ., C++, Python, etc .). Students in the middle grades should be able to use digital tools to analyze large data sets, which often include such repeated calculations. Understanding computational processes and how computers are programmed to carry out tasks is also essential in interpreting, using, creating, and modifying computer simulations at the upper secondary level. In high school, modeling and simulation tools (e.g., StarLogo, NetLogo, Agentsheets, etc.) can greatly facilitate the development of models of complex systems. These tools can be introduced using a developmentally appropriate sequence of “Use-Modify-Create.” Students first use pre-existing computer models to run experiments. Over time they begin to modify the models with increasing levels of sophistication . For example, a student may initially want to change the color of a data point in a model result. Later the student may want to change some small aspect of the model’s behavior that requires modifying an algorithm. As students gain skills and confidence, they develop new computational projects of their own design. Within this “create” stage, all three key aspects of computational thinking: abstraction, automation and analysis, come into play. Mathematics and computational thinking are also essential to engineers. For example, mathematical inequalities can specify design constraints more precisely than words (e .g ., “must weigh less than x” instead of “should not be too heavy”). Like systems in science, computers can represent individual objects or components that are pieces of a design solution. Computer tools such as simplified computer-assisted design programs (e.g., Tinkercad, SOLIDWORKS, etc.) or simplified simulation builders (e.g , NetLogo, PowerSim, Scratch, etc.) can greatly facilitate the iterative design process for students at the high school level. The ability to use and code such tools can extend students’ capability to develop design solutions. In earlier grades, students can use simple tools such as drop and drag drawing tools to create visual representations or simple robotics kits where computational thinking allows students to create instructions that allow their robot to achieve specific engineering goals. |