A Concrete Approach to Probability Education

10 October 2026

Probability is ubiquitous in everyday life, and yet, students often struggle to grasp how chance, risk, and formal probability relate to one another. A new approach from Cambridge researchers offers a way forward by starting with real-world risk communication and teaching students to think in terms of expected frequencies.

The Discrepancy Between Risk and Uncertainty

Confusion can arise because students encounter probabilistic language in both literary and mathematical forms. Some risks can be quantified precisely—the likelihood of flipping heads on a fair coin is 50%, and that of drawing a red card from a deck is 50%. Additionally, a doctor may be able to say, “out of 100 patients like you, we would expect 16 to have a heart attack or stroke in the next ten years”. This is an interpretation of probabilistic language that is easier to reason with.

However, there is another category of chance for which numbers are much less certain—uncertainty, as defined by Frank Knight: unmeasurable events where the odds of an outcome are indeterminable. For example, a technician may show their confidence in a repair by saying, "I think it's likely". Likely? The student needs to know when to treat probability quantitatively and when to treat it qualitatively.

The advantages of expected frequencies are manifold. They are easy to reason with, directly refer to the real world, and are built on everyday language. And as Cambridge researchers Spiegelhalter and Gage argue, frequency trees may become fundamental to representing probabilistic reasoning—coming before probability trees in the learning sequence.

The Cambridge System, Step by Step

These researchers advocate for a probative pedagogy that includes:

1. Starting the lesson with a simplified problem, such as what colors might result when a die is rolled

2. Physically modeling the problem, such as using plastic cups to represent blind draws

3. Carrying out classroom experiments, using randomizing devices such as coins, dice, or roulette wheels

4. Pooling empirical data from the experiments into 2x2 tables

5. Constructing frequency trees to sketch out the relationship between outcomes

6. Drawing Venn diagrams, where students can see categories and joint memberships

7. Spotting patterns, such as how a 3/8 and 5/8 frequency must fall together

8. Moving to expected frequency trees, expressions for the *expectation*

9. Introducing probability trees once the student has numerical confidence

This teaching approach relies on experiential learning. Cambridge's module begins with news headlines to trigger recognition of probabilistic language, and then introduces the connections to topics such as weather forecasting, political polling, and risk assessment. Students use the language of probability in practical contexts before any mathematical notation is introduced.

Beyond a Model—A Cognitive Asset

The strength of this approach is that it aligns with research in both education and risk communication. For one, students in grades 5 to 11 were surveyed by researchers Slattery and others, comprising 2,726 students in total. Higher-grade students were more likely to show higher levels of evaluating and interpreting chance language, a core part of statistical literacy. These findings suggest that understanding probabilistic language is a complex pathway that improves with education.

But the issue also goes to the heart of how we as humans intuitively think about uncertainty. As a separate Cambridge source notes, the difference between measurable risk, when we assign numbers to chances, and unmeasurable uncertainty, can be a subtle one. By teaching through concrete scenarios, classrooms can better reflect the everyday thinking of their students.

As a closing note, remember that this is not just about better math teaching. It's about students becoming more fluent in the formal language of probability as it relates to real-life situations. If our teaching could bridge that gap, imagine the impact: a citizenry that is more competent at drawing meaning from probabilistic statements, and more alert to what is measurable and what is not in the face of uncertainty. In the age of risk and adversity, don't these seem like crucial gains?