using mathematics and computational thinking

Strategies to promote Belonging with Math/Computational thinking:

With different science tracks and levels within a classroom, teachers should attempt to cultivate a safe environment for students to take risks with potentially unfamiliar math concepts. It is important to develop a sense of community in order for individual students to feel that sense of belonging. Some ways that teachers can do this is to give plenty of opportunities for students to delve into math in groups (with appropriate individual tasks). This can start with asking students to engage with questions regarding relatable items like phone apps, computer games and more. Algorithms are often at the core of this work so presenting students with day to day examples can help students feel belonging within their community (how often they need to clean up their space as an example). To learn more visit here.

Strategies to promote Confidence in Math/computational thinking:

Similar to belonging, students with varying math skills should experience different ways that math and computational thinking can be used in science. Teachers can start simple with showing students common algorithms in a variety of visual and graphical representations. Students should also practice creating these and demonstrating logic, patterns and generalizations. Students can be scaffolded using checklists for graphing, spreadsheet templates, practice using functions, and process charts. Lots of practice with all of this can slowly build confidence in students that may not feel they have a strong STEM identity. To learn more visit here.

Learning Orientation supports for using Math and Computational Thinking:

The learning orientation we want to create here is that using math to investigate a phenomenon is often an iterative process and that mistakes are key to moving forward. In order to develop this attitude teachers should make sure to emphasize that right answers are not what they are seeking and there should be no attributing mistakes to ability/intelligence. In addition, teachers should model how to create, explain and evaluate math thinking throughout the lessons and through to the assessment. Error analysis should always play a part in using math and having students share out multiple methods is another way for students to see that there is no one ‘right answer’.To learn more visit here.

Promoting Autonomy in using Math and Computational Thinking:

Whenever possible students should have a choice in how to address a problem or phenomenon using math. Along with choice, teachers and students should identify and acknowledge even the smallest differences in how students tackle a problem. Both of these can help emphasize to students that there are often multiple possible solutions to help understand a phenomenon. To learn more visit here.

Promoting Relevance in using Math and Computational Thinking:

“Framing the use of mathematics and application of computational thinking within a phenomenon or design problem that is of interest to students may help motivate them to work hard on tasks involving these skills. (M-Plans)” In the classroom, this can mean providing students with examples of the benefits of using math to solve science problems in daily life such as how a moving roller coaster works. In addition, it helps students to understand how they may already be using math and computational thinking in their daily lives such as if they track statistics in sports or patterns in sheet music. Lastly, teachers should provide plenty of opportunities to practice common software such as Excel or Google Sheets (with plenty of support and scaffolding!). To learn more visit here.

Standard Name: HS-ETS1-4 Engineering Design

Standard: Use a computer simulation to model the impact of proposed solutions to a complex real-world problem with numerous criteria and constraints on interactions within and between systems relevant to the problem.

Observable Features of Student Performance by the end of the Course: 

1. Representation

• Students identify the following components from a given computer simulation:

  • • The complex real-world problem with numerous criteria and constraints; 

    • The system that is being modeled by the computational simulation, including the boundaries of the systems;

    • What variables can be changed by the user to evaluate the proposed solutions, tradeoffs, or other decisions; and

    • The scientific principle(s) and/or relationship(s) being used by the model.

2. Computational Modeling 

• Students use the given computer simulation to model the proposed solutions by: 

  • • Selecting logical and realistic inputs; and 

    • Using the model to simulate the effects of different solutions, tradeoffs, or other decisions.

3. Analysis

• Students compare the simulated results to the expected results.

• Students interpret the results of the simulation and predict the effects of the proposed solutions within and between systems relevant to the problem based on the interpretation.

• Students identify the possible negative consequences of solutions that outweigh their benefits. d Students identify the simulation’s limitations.

Standard Name: HS-ESS3-6 Earth and Human Activity

Standard: Use a computational representation to illustrate the relationships among Earth systems and how those relationships are being modified due to human activity.

Observable Features of Student Performance by the end of the Course: 

1. Representation

• Students identify and describe* the relevant components of each of the Earth systems modeled in the given computational representation, including system boundaries, initial conditions, inputs and outputs, and relationships that determine the interaction (e.g., the relationship between atmospheric CO2 and production of photosynthetic biomass and ocean acidification). 

2. Computational modeling

• Students use the given computational representation of Earth systems to illustrate and describe* relationships among at least two of Earth’s systems, including how the relevant components in each individual Earth system can drive changes in another, interacting Earth system. 

3. Analysis 

• Students use evidence from the computational representation to describe* how human activity could affect the relationships between the Earth’s systems under consideration.

Standard Name: HS-ESS1-4 Earth's Place in the Universe

Standard: Use mathematical or computational representations to predict the motion of orbiting objects in the solar system.

Observable Features of Student Performance by the end of the Course: 

1. Representation

• Students identify and describe* the following relevant components in the given mathematical or computational representations of orbital motion: the trajectories of orbiting bodies, including planets, moons, or human-made spacecraft; each of which depicts a revolving body’s eccentricity e = f/d, where f is the distance between foci of an ellipse, and d is the ellipse’s major axis length (Kepler’s first law of planetary motion).

2. Mathematical or computational modeling

• Students use the given mathematical or computational representations of orbital motion to depict that the square of a revolving body’s period of revolution is proportional to the cube of its distance to a gravitational center (𝑇 2 ∝ 𝑅 3 , where T is the orbital period and R is the semimajor axis of the orbit — Kepler’s third law of planetary motion). 

3. Analysis

• Students use the given mathematical or computational representation of Kepler’s second law of planetary motion (an orbiting body sweeps out equal areas in equal time) to predict the relationship between the distance between an orbiting body and its star, and the object’s orbital velocity (i.e., that the closer an orbiting body is to a star, the larger its orbital velocity will be).

• Students use the given mathematical or computational representation of Kepler’s third law of planetary motion (𝑇 2 ∝ 𝑅 3 , where T is the orbital period and R is the semi-major axis of the orbit) to predict how either the orbital distance or orbital period changes given a change in the other variable.

• Students use Newton’s law of gravitation plus his third law of motion to predict how the acceleration of a planet towards the sun varies with its distance from the sun, and to argue qualitatively about how this relates to the observed orbits

Standard Name: HS-LS4-6 Biological Evolution: Unity and Diversity

Standard: Create or revise a simulation to test a solution to mitigate adverse impacts of human activity on biodiversity.*

Observable Features of Student Performance by the end of the Course: 

1. Representation

• • Students create or revise a simulation that: 

    • • Models effects of human activity (e.g., overpopulation, overexploitation, adverse habitat alterations, pollution, invasive species, changes in climate) on a threatened or endangered species or to the genetic variation within a species; and 

      • Provides quantitative information about the effect of the solutions on threatened or endangered species. 

  • Students describe* the components that are modeled by the computational simulation, including human activity (e.g., overpopulation, overexploitation, adverse habitat alterations, pollution, invasive species, changes in climate) and the factors that affect biodiversity.

  • Students describe* the variables that can be changed by the user to evaluate the proposed solutions, tradeoffs, or other decisions.

2. Computational modeling

• • Students use logical and realistic inputs for the simulation that show an understanding of the reliance of ecosystem function and productivity on biodiversity, and that take into account the constraints of cost, safety, and reliability as well as cultural, and environmental impacts.

  • Students use the simulation to identify possible negative consequences of solutions that would outweigh their benefits.

3. Analysis 

• • Students compare the simulation results to expected results.

  • Students analyze the simulation results to determine whether the simulation provides sufficient information to evaluate the solution.

  • Students identify the simulation’s limitations.

  • Students interpret the simulation results, and predict the effects of the specific design solutions on biodiversity based on the interpretation.

4. Revision

• • Students revise the simulation as needed to provide sufficient information to evaluate the solution.

Standard Name: HS-LS2-4 Ecosystems: Interactions, Energy, and Dynamics

Standard: Use mathematical representations to support claims for the cycling of matter and flow of energy among organisms in an ecosystem.

Observable Features of Student Performance by the end of the Course: 

1. Representation

• Students identify and describe* the components in the mathematical representations that are relevant to supporting the claims. The components could include relative quantities related to organisms, matter, energy, and the food web in an ecosystem.

• Students identify the claims about the cycling of matter and energy flow among organisms in an ecosystem.

2. Mathematical modeling

• Students describe* how the claims can be expressed as a mathematical relationship in the mathematical representations of the components of an ecosystem

• Students use the mathematical representation(s) of the food web to: 

  • • Describe* the transfer of matter (as atoms and molecules) and flow of energy upward between organisms and their environment; 

    • Identify the transfer of energy and matter between tropic levels; and 

    • Identify the relative proportion of organisms at each trophic level by correctly identifying producers as the lowest trophic level having the greatest biomass and energy and consumers decreasing in numbers at higher trophic levels. 

3. Analysis

• Students use the mathematical representation(s) to support the claims that include the idea that matter flows between organisms and their environment.

• Students use the mathematical representation(s) to support the claims that include the idea that energy flows from one trophic level to another as well as through the environment.

• Students analyze and use the mathematical representation(s) to account for the energy not transferred to higher trophic levels but which is instead used for growth, maintenance, or repair, and/or transferred to the environment, and the inefficiencies in transfer of matter and energy.

Standard Name: HS-PS4-1 Waves and their Applications in Technologies for Information Transfer

Standard: Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media.

Observable Features of Student Performance by the end of the Course: 

1. Representation

• Students identify and describe* the relevant components in the mathematical representations: 

  • Mathematical values for frequency, wavelength, and speed of waves traveling in various specified media; and 

  • The relationships between frequency, wavelength, and speed of waves traveling in various specified media. 

2. Mathematical modeling

• Students show that the product of the frequency and the wavelength of a particular type of wave in a given medium is constant, and identify this relationship as the wave speed according to the mathematical relationship 𝑣 = 𝑓𝜆.

• Students use the data to show that the wave speed for a particular type of wave changes as the medium through which the wave travels changes.

• Students predict the relative change in the wavelength of a wave when it moves from one medium to another (thus different wave speeds using the mathematical relationship 𝑣 = 𝑓𝜆). Students express the relative change in terms of cause (different media) and effect (different wavelengths but same frequency).

3. Analysis

• Using the mathematical relationship 𝑣 = 𝑓𝜆, students assess claims about any of the three quantities when the other two quantities are known for waves traveling in various specified media.

• Students use the mathematical relationships to distinguish between cause and correlation with respect to the supported claims.

Standard Name: HS-PS3-1 Energy

Standard: Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.

Observable Features of Student Performance by the end of the Course: 

1. Representation

• Students identify and describe* the components to be computationally modeled, including: 

  • • The boundaries of the system and that the reference level for potential energy = 0 (the potential energy of the initial or final state does not have to be zero); 

    • The initial energies of the system’s components (e.g., energy in fields, thermal energy, kinetic energy, energy stored in springs — all expressed as a total amount of Joules in each component), including a quantification in an algebraic description to calculate the total initial energy of the system;

    • The energy flows in or out of the system, including a quantification in an algebraic description with flow into the system defined as positive; and

    • The final energies of the system components, including a quantification in an algebraic description to calculate the total final energy of the system. 

2. Computational Modeling

• Students use the algebraic descriptions of the initial and final energy state of the system, along with the energy flows to create a computational model (e.g., simple computer program, spreadsheet, simulation software package application) that is based on the principle of the conservation of energy.

• Students use the computational model to calculate changes in the energy of one component of the system when changes in the energy of the other components and the energy flows are known. 

3. Analysis

• Students use the computational model to predict the maximum possible change in the energy of one component of the system for a given set of energy flows.

• Students identify and describe* the limitations of the computational model, based on the assumptions that were made in creating the algebraic descriptions of energy changes and flows in the system.

Standard Name: HS-PS2-4 Motion and Stability: Forces and Interactions

Standard: Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.

Observable Features of Student Performance by the end of the Course: 

1. Representation

• Students clearly define the system of the interacting objects that is mathematically represented.

• Using the given mathematical representations, students identify and describe* the gravitational attraction between two objects as the product of their masses divided by the separation distance squared (Fg = −G m1m2 d2 ), where a negative force is understood to be attractive.

• Using the given mathematical representations, students identify and describe* the electrostatic force between two objects as the product of their individual charges divided by the separation distance squared (Fe = k q1q2 d2 ), where a negative force is understood to be attractive.

2. Mathematical modeling

• Students correctly use the given mathematical formulas to predict the gravitational force between objects or predict the electrostatic force between charged objects.

3. Analysis 

• Based on the given mathematical models, students describe* that the ratio between gravitational and electric forces between objects with a given charge and mass is a pattern that is independent of distance.

• Students describe* that the mathematical representation of the gravitational field (Fg = −G m1m2 d2 ) only predicts an attractive force because mass is always positive.

• Students describe* that the mathematical representation of the electric field (Fe = k q1q2 d2 ) predicts both attraction and repulsion because electric charge can be either positive or negative. 

• Students use the given formulas for the forces as evidence to describe* that the change in the energy of objects interacting through electric or gravitational forces depends on the distance between the objects.

Standard Name: HS-PS1-7 Matter and its Interactions

Standard: Use mathematical representations to support the claim that atoms, and therefore mass, are conserved during a chemical reaction.

Observable Features of Student Performance by the end of the Course: 

1. Representation

• Students identify and describe* the relevant components in the mathematical representations: 

  • • Quantities of reactants and products of a chemical reaction in terms of atoms, moles, and mass; 

    • Molar mass of all components of the reaction; 

    • Use of balanced chemical equation(s); and 

    • Identification of the claim that atoms, and therefore mass, are conserved during a chemical reaction.

• The mathematical representations may include numerical calculations, graphs, or other pictorial depictions of quantitative information. 

• Students identify the claim to be supported: that atoms, and therefore mass, are conserved during a chemical reaction.

2. Mathematical modeling

• Students use the mole to convert between the atomic and macroscopic scale in the analysis.

• Given a chemical reaction, students use the mathematical representations to 

  • • Predict the relative number of atoms in the reactants versus the products at the atomic molecular scale; and 

    • Calculate the mass of any component of a reaction, given any other component.

3. Analysis

• Students describe* how the mathematical representations (e.g., stoichiometric calculations to show that the number of atoms or number of moles is unchanged after a chemical reaction where a specific mass of reactant is converted to product) support the claim that atoms, and therefore mass, are conserved during a chemical reaction.

• Students describe* how the mass of a substance can be used to determine the number of atoms, molecules, or ions using moles and mole relationships (e.g., macroscopic to atomic molecular scale conversion using the number of moles and Avogadro’s number).