To explain a difficult idea without oversimplifying it, make the route into the idea easier while keeping the conditions that make it true. You can use plain language, a small example, and a clear sequence without pretending the subject has no exceptions. The important distinction is between removing an obstacle to understanding and removing part of the meaning.
A listener doesn’t need every detail at once, but they do need an explanation they can safely build on. If the introductory version teaches something you must later undo, the simplification may have gone too far.
Start with what the learner should be able to do
“Understand percentages” is too broad to guide a useful explanation. “Explain why a 20 percent fall followed by a 20 percent rise doesn’t restore the starting value” gives you a specific task. It tells you which details are essential and which can wait.
The same approach works beyond mathematics. A learner might need to distinguish a cause from a correlation, explain why two instructions conflict, or choose the appropriate setting for a piece of equipment. Name the action before deciding how much background to include.
This keeps the explanation honest about its scope. You are helping someone understand one relationship well enough to use it. You aren’t claiming that a short lesson covers the entire subject.
Check the knowledge your explanation assumes
Carnegie Mellon’s Eberly Center explains that prior knowledge can support learning or get in its way. A learner may lack a prerequisite, but they may also hold a plausible idea that leads them toward the wrong interpretation. These problems require different explanations.
Before explaining percentage changes, ask what someone thinks “20 percent” means. If they answer “twenty out of a hundred,” you have a starting point. If they assume that twenty percent always means twenty units, address that distinction before discussing successive changes.
Ask a small question or use a quick example rather than beginning with a long diagnostic exercise. “What is ten percent of fifty?” can reveal whether the learner understands that a percentage depends on a base amount. The answer helps you choose where the explanation should begin.
Keep the condition inside the explanation
Many misleading simplifications remove a condition and leave behind an absolute statement. “A larger sample gives a better answer” sounds easy to remember, but it can hide problems with how the sample was collected. “More feedback improves learning” ignores whether the feedback is relevant, understandable, and usable.
A better introductory explanation keeps the main condition visible. You don’t need to list every exception immediately. You do need to include the qualification that would change what the listener does.
For percentage changes, the essential condition is that each percentage applies to a particular starting amount. Keep returning to that amount in the example. Without it, the learner may memorize a calculation without understanding why the result changes.
Work through one small example all the way
Suppose an amount starts at 100. A fall of 20 percent removes 20, leaving 80. If that remaining amount then rises by 20 percent, the increase is 16 because the new base is 80. The final amount is 96.
The two percentages look symmetrical, but they describe different changes because they apply to different amounts. Write the base beside each calculation rather than presenting a row of numbers without explanation. The learner should be able to point to what changed between the first step and the second.
Now ask what increase would restore 80 to 100. The missing amount is 20, and 20 is one quarter of 80, so the required increase is 25 percent. This extension checks whether the learner understands the base amount rather than merely remembering that the first example ended at 96.
Label hypothetical figures as examples when the setting could make them look like real data. In a workplace explanation, don’t let an invented budget, customer figure, or performance result acquire the authority of an actual measurement.
Use an analogy to show a relationship, then show its limit
An analogy is useful when it gives the learner a familiar relationship to work with. It becomes risky when the familiar situation carries extra assumptions that don’t belong to the new idea. The closer the analogy feels, the easier it can be to overlook the mismatch.
For instance, describing working memory as a small desk can help explain why managing several unfamiliar pieces of information at once is difficult. But a desk has a fixed physical area, whereas what a person can handle depends partly on knowledge, organization, and the task. If the analogy matters to the explanation, state that limitation.
You might say, “The desk comparison is useful for thinking about competing demands. It doesn’t mean everyone has the same fixed capacity for every activity.” That preserves the helpful image without turning it into a complete theory.
Return to the actual subject after the analogy has done its job. A learner who can describe the desk beautifully but cannot explain the original idea has learned the comparison instead of the concept.
Show a case that looks similar but works differently
A single example can accidentally teach the learner to recognize its surface features. A second example should test the boundary of the idea, not simply repeat the same calculation with different names.
After the percentage example, compare subtracting 20 units and adding 20 units. Starting at 100, those operations do restore the original amount because the changes are equal in absolute size. The contrast helps the learner see why “percent” cannot be treated as a decorative label on an ordinary number.
In other subjects, compare a genuine example with a near miss. A message can sound polite while leaving the requested action unclear. A chart can show two measures moving together without establishing why either changed. Ask the learner to explain which feature makes the difference.
Explain decisions that an expert makes silently
Experts often skip steps because those steps feel obvious. A learner may see the correct method and still have no idea how to choose it. Include the decision that comes before the operation: what are you looking for, and why does this approach fit?
In the percentage example, say why you use 80 as the base for the second change. In a writing lesson, explain why a sentence belongs in the introduction rather than the conclusion. In software instruction, explain what the selected option changes before asking someone to click it.
This doesn’t require narrating every small movement. Focus on the points where a beginner could reasonably choose a different path. Those are the places where your judgment contributes more than a polished demonstration.
Let the learner attempt a different case
After explaining, ask for a prediction or a short solution before showing the answer. A person who follows your reasoning while you provide every step may still struggle to produce it independently. The attempt helps you see which parts of the explanation are available to them without your prompts.
For example, ask whether a 10 percent decrease followed by a 10 percent increase returns an amount to its starting value. Have the learner explain the expected direction before calculating. If they understand the changing base, they can reason about the result without relying entirely on arithmetic.
Keep the exercise small enough that a mistake reveals something useful. Adding several new difficulties at once makes it hard to tell whether the original explanation worked. If the learner gets stuck, ask which amount the next percentage applies to rather than immediately completing the calculation.
Add detail in response to a real question
Once the central relationship is clear, you can introduce a formula, a more difficult example, or an exception. Each addition should answer a question the learner can now recognize. That gives the extra detail a place in their understanding.
A general formula for successive percentage changes becomes more meaningful after the learner sees why the base changes. Without that explanation, the formula may feel like a procedure to memorize. The sequence matters as much as the amount of information.
If the explanation is part of a presentation, plan these stages before building slides. Our guide to structuring a presentation can help you put examples and transitions in an order that serves the argument.
Check what your simplest version leaves behind
At the end, ask the learner to explain the idea to someone else using a new example. Listen for the essential condition, the reasoning, and the boundary. A short explanation that preserves those elements is more useful than a fluent repetition of your original words.
Also ask yourself what a reasonable person might misunderstand from your wording. Would they think the rule always applies? Would they mistake a model for a literal description? Would they know when they need more information before acting?
A warm delivery helps people stay with a lesson, but it doesn’t replace this check. Our article on why a friendly teaching video can still be hard to learn from explores that difference. A successful explanation leaves the learner able to make a distinction they couldn’t make before, while understanding where that distinction applies.
Source and further reading
Carnegie Mellon University Eberly Center: Learning Principles — the role of prior knowledge, practice, and feedback in learning. The percentage examples above are worked illustrations created for this guide.