New Jersey STEM Innovation Fellowship

The 2024 NJ STEM Innovation Fellowship program will form a Professional Learning Community (PLC) that includes elementary and middle school teachers as well as university faculty members. The PLC will learn to implement a design innovation that develops foundational through advanced understanding of the mathematical concepts of equality and equivalence. The innovation uses a familiar and ancient technology – the balance scale – as a powerful learning tool that students can think with as they explore and come to understand increasingly complex ideas. Because the PLC includes math and science educators at all levels, this innovation will support learning from a child’s earliest mathematical explorations through college graduation and into the workforce.

What are equality and equivalence? At the elementary level, children are introduced to these concepts through the basic arithmetic operations of addition, subtraction, multiplication, and division. Understanding the concept of equality – as in 5 = 3 + 2 – prepares children for more complex mathematical concepts such as equations and equivalent fractions. These concepts are essential for building a solid mathematical foundation and preparing students for middle and high school mathematics.

In middle school, students encounter a broader range of mathematical concepts built upon equality and equivalence, from ratios and percents to linear equations and functions. Understanding equivalence – as in 2/3 = 6/9, 10 – 7 = 6 ÷ 2, and 4x + 12 = 4(x + 3) – is needed to solve problems involving rate, ratio, slope, and similarity. Without these concepts, students cannot make connections across the mathematical topics that would enable them to solve real-world problems.

In high school, students explore more advanced topics such as polynomial and exponential functions, rational expressions and equations, similar figures, and theorems and modeling involving area and volume. Mastery of equality and equivalence remains important for understanding these subjects, as students must be able to manipulate equations, solve complex problems, and make logical deductions based on mathematical principles.

Without a solid grounding in the foundational concepts, students may find themselves unready for college-level coursework in mathematics and natural science. Students who struggle with first-year math/science courses are less likely to persist in STEM majors or enter the STEM workforce (Chen, 2013; Cohen & Kelly, 2020; PCAST, 2012). The innovative use of a balance scale as a mathematical model for teaching the concepts of equality and equivalence provides students with an engaging, meaningful, hands-on approach to thinking and reasoning about abstract concepts and deepening their understandings.

References

• Chen, X. (2013). STEM Attrition: College Students’ Paths into and out of STEM Fields. Statistical Analysis Report. NCES 2014-001. National Center for Education Statistics.
• Cohen, R., & Kelly, A. M. (2020). Mathematics as a factor in community college STEM performance, persistence, and degree attainment. Journal of Research in Science Teaching, 57(2), 279-307.
• Kaput, J. J., & Roschelle, J. (2002). The mathematics of change and variation from a millennial perspective: New content, new context. In Rethinking the mathematics curriculum (pp. 155-170). Routledge.
• President’s Council of Advisors on Science and Technology (2012). Engage to Excel: Producing One Million Additional College Graduates with Degrees in Science, Technology, Engineering, and Mathematics. Retrieved from https://obamawhitehouse.archives.gov/sites/default/files/microsites/ostp/pcast-engage-to-excel-final_2-25-12.pdf
• Roschelle, J., Kaput, J. J., & Stroup, W. (2012). SimCalc: Accelerating students’ engagement with the mathematics of change. In Innovations in science and mathematics education (pp. 60-88). Routledge.

Application of Innovation to Natural Sciences

The concepts of equality and equivalence are used by students to understand and succeed in natural science courses at the secondary and college levels. In many cases, students use an actual balance scale as part of the science learning activity. Some examples of equality and equivalence in biological and physical sciences include:

Physics

Mechanics

Students encounter the concept of equilibrium in the context of equality in forces. For example, when studying objects in static equilibrium or calculating forces in a system, students must use the idea of equality to solve problems.

Electricity and Magnetism

In electromagnetism, Ohm’s Law (V = IR) and Kirchhoff’s Laws (for conservation of charge and energy) students must use equivalence and equality to understand and analyze electrical circuits and phenomena.

Thermodynamics

The concept of energy balance is necessary for understanding the equality between energy input and output in various processes. Equivalence is used to understand the relationship between different forms of energy (e.g. kinetic and potential) and their conversions.

Quantum Mechanics

In more advanced physics courses, equivalence is used to understand particle-wave duality and quantum superposition. The concept of equality is applied in equations for the conservation of energy and momentum in quantum interactions.

Chemistry

Chemical Reactions

Stoichiometry is based on the principle of equality, where chemical equations must be balanced by students to ensure that the number of atoms of each element remains the same before and after a reaction. This is used for determining reaction yields and understanding reaction mechanisms.

Equilibrium

In chemical equilibrium, the equilibrium constant expression involves the equality of forward and reverse reaction rates. Students use this concept to predict the direction of a reaction under different conditions and calculate equilibrium concentrations.

Quantitative Analysis

Students use equality and equivalence in quantitative analysis techniques such as titrations, where the equivalence point signifies the equality between the moles of reactants and products. This is needed to determine concentration of unknown substances.

Biology

Population Dynamics

Mathematical models for population growth and dynamics use the concept of equality to study the way birth rates are expected to equal death rates under certain conditions. Students also apply equivalence in models for carrying capacity and limiting factors.

Biochemical Pathways

When studying biochemical pathways and metabolic processes, students use equivalence to understand mass and energy conservation. Students use the concept of mathematical equality to study stoichiometry of chemical reactions within cells and the balance between substrate and product concentrations.

Earth Science

Geological Processes

Mathematical models for erosion, sedimentation, and tectonic movements are used to predict changes in the Earth over time. These require students to understand equations in which quantities before and after an event are equal.

Climate Modeling

Students use equivalence for climate modeling in equations for the balance between incoming and outgoing radiation to determine the planetary energy budget. This helps students understand climate change predictions and impacts.

Funders

We would like to acknowledge our funders for the fellowship. Their generosity & support has been invaluable to the development and sustainability of this program.