Observe
Students begin each conceptual unit by observing simple, carefully selected phenomena or observational experiments.
How PUM Works
Learning physics through the practices of science.
Physics Union Mathematics (PUM) is grounded in the Investigative Science Learning Environment (ISLE), a comprehensive learning system that helps students learn physics by engaging in the kinds of reasoning, evidence-based thinking, and model building used in science itself. PUM adapts that logic for a 6–12 continuum with a strong emphasis on mathematical reasoning.
Learning physics through practice
The logic of PUM is based on the elements of scientific reasoning—inductive, analogical, and hypothetico-deductive—and on what is known about how students learn. Rather than beginning with final formulas or finished explanations, students begin with concrete experiences and observations and use those experiences to build and test ideas.
Find patterns
They analyze those observations, search for patterns, and look for regularities in what they have seen.
Build explanations
Students devise qualitative explanations based on evidence and discussion rather than memorized statements.
Test and revise
They use their explanations to make predictions about new experiments, then revise or strengthen those explanations based on results.
From qualitative ideas to quantitative models
Once students have built an initial qualitative understanding of a phenomenon, the process is repeated using quantitative mathematical representations. This progression helps students move from intuitive sense-making to more precise forms of reasoning without losing the underlying meaning of the physics.
Progressive differentiation
PUM helps students move from broad qualitative ideas to more detailed quantitative understanding. This progression focuses simultaneously on the structure of the knowledge being developed and the learning processes of students.
Model building
Students are not simply handed equations. They are guided toward identifying quantities, collecting data, finding relationships, and constructing mathematical models that help explain and predict physical behavior.
Multiple representations
A central feature of PUM is its purposeful use of multiple representations. Students learn to move between abstract and concrete ways of understanding so that equations remain connected to physical meaning.
Connecting ideas
Students work with words, sketches, process diagrams, graphs, mathematical relationships, and experimental evidence. Each representation helps illuminate a different aspect of the same physical situation.
Supporting reasoning
Different representations make it easier for students to identify patterns, compare ideas, and check for consistency. For example, a motion diagram, force diagram, and symbolic statement should all describe the same process coherently.
Expanding access
Multiple representations provide repeated exposure to ideas and engage learners in different ways. This makes the approach especially helpful for students who may struggle if physics is presented only in abstract symbolic form.
An emphasis on mathematics
PUM specifically focuses on developing mathematical reasoning through physics. The curriculum builds on students’ intrinsic sense-making and helps them make sense of quantities, relationships, graphs, and equations rather than treating mathematics as a separate or purely procedural task.
Pattern recognition
Students collect and analyze data to find relationships among variables and to identify meaningful regularities in physical processes.
Proportional reasoning
The modules help students reason about ratios, rates, scaling, units, and the meaning of physical quantities in context.
Graphs and symbolic reasoning
Students learn to interpret graphs, connect them to physical situations, and use symbols and equations as meaningful representations rather than detached procedures.
Invention of quantities
Students are guided in devising and conceptualizing new physical quantities, helping them understand where formulas come from and what they represent.
Scientific abilities and experimentation
PUM is designed not only to help students learn content, but also to help them acquire abilities used in science, mathematics, and engineering. Students are asked to reason from evidence, to evaluate explanations, and to design and analyze experiments.
Testing ideas
Students use their explanations to make predictions about what should happen in new situations, then compare those predictions to actual results.
Dealing with anomalous data
When results do not fit an expectation, students are asked to consider what needs revision—either in the explanation, the assumptions, or the interpretation of the evidence.
Communicating reasoning
Students learn to represent their thinking clearly, justify claims with evidence, and discuss ideas collaboratively as part of a learning community.
In the classroom
A PUM lesson often begins with a simple phenomenon that students can reason about directly. The goal is not to deliver an answer immediately, but to create the conditions for students to notice, question, test, and refine their understanding.
Students might begin by holding a tennis ball and a medicine ball, one in each hand, and discussing the interactions involved. They may reason that the air pushes differently on the two objects. PUM then invites them to test that hypothesis with evidence, such as an experiment involving an object suspended by a spring under a vacuum jar.
If the observation does not match the initial prediction, students are not told simply that they are wrong. Instead, they are asked to revise their explanation in light of what they observed. This is a key part of how PUM helps students experience physics as a process of knowledge construction.
That same logic carries forward when students collect numerical data, build relationships among quantities, and develop equations that describe physical behavior with increasing precision.
Conceptual understanding
Students build ideas from evidence rather than relying only on memorized statements.
Students build ideas from evidence rather than relying only on memorized statements.
Mathematical sense-making
Mathematics becomes a tool for understanding and modeling physical processes.
Mathematics becomes a tool for understanding and modeling physical processes.
Scientific habits of mind
Students learn to test ideas, weigh evidence, and revise explanations.
Students learn to test ideas, weigh evidence, and revise explanations.
Confidence and belonging
The approach supports students in seeing themselves as capable participants in science.
The approach supports students in seeing themselves as capable participants in science.
Why this matters
PUM is designed to help students do more than cover physics topics. It helps them understand how scientific ideas are built, tested, and applied; how mathematical reasoning supports physical understanding; and how careful evidence-based thinking can lead to deeper learning.