How to Teach Function Monotonicity Effectively: Strategies and Classroom Reflections

At this moment, it is crucial to clearly explain the definition of monotonicity, with difficulty lying in the domain of values and the transformation after taking differences, involving factorization, completing the square, etc., which are topics bridging middle and high school. Additionally, when writing monotonic intervals, one should not simply merge them; generally intervals can be expressed as open or closed, except when endpoint values are unattainable.

When designing the lesson, I also considered how to judge a function’s “trend” (monotonicity) under three representations. The tabular method compares successive function values, assuming an ordered left‑to‑right list; the graphical method is the most intuitive, with a left‑to‑right rise indicating increase and a decline indicating decrease; the definition method works directly with the analytic expression and unifies the previous two. I want students to appreciate the “beauty” of the definition. Understanding the definition also requires practice; students need to solve problems using the definition, encounter difficulties, and feel the joy of success. To introduce the new content, I used examples: the planetary order and orbital radii (tabular), GDP growth rates from 1988‑2010 (graphical), and a shot‑put function’s analytic expression (definition), letting students summarize the methods.

In practice, I split the material over one and a half class periods. In the first half‑class I introduced the definition of monotonicity and had students use the definition to prove monotonicity of linear, reciprocal, and quadratic functions on specific intervals. At the end I left a cubic‑function problem for further thought.

In the second class I reviewed the definition and summarized steps such as “domain”, “difference transformation”, “sign”, and “conclusion”. I invited a student (who often asks questions and performs well) to present his proof, and I pointed out that some steps were not rigorous, prompting the class to notice. Then I asked the class to prove the monotonicity of a root function. Students responded in three ways: some thought proof unnecessary because the result was obvious; some had no idea; and some solved it smoothly. The first and third responses were rare; most lacked a strategy. When I explained that rationalizing the numerator could help, some corrected me that the denominator should be rationalized, so I clarified both concepts. Regarding whether intervals can be merged, I gave two examples: merging a reciprocal function’s intervals is unreasonable, while a piecewise function’s monotonic intervals can be merged; for non‑mergeable cases I use a comma to separate. I then guided students to discover patterns such as “increasing + increasing = increasing”, pausing ten seconds before offering explanations like “increasing + increasing = increasing”, “decreasing + decreasing = decreasing”, “the opposite of an increasing function is decreasing”, and I added the “same‑increase opposite‑decrease” rule without detailed discussion. The class ended with an exercise where students judged the monotonic intervals of a piecewise function, with one student working on the board while I explained the answer.

After class a student asked how to solve absolute‑value inequalities, which relate to the piecewise functions we covered; their confusion suggested I had not clearly presented the steps on the board.

I attended Professor Cheng Zhi’s lesson on monotonicity. Compared with my own, his was more detailed; he placed the topic within the broader high‑school exam perspective, introduced knowledge points naturally, and clarified issues such as endpoint notation (parentheses vs. brackets) at the moment students encountered them. He also emphasized the definition’s keywords, spending ample time on them, whereas I rushed through the definition.

My current teaching style is fast and does not consider the student’s perspective; I merely convey my summarized experience instead of addressing problems students encounter during learning. “Student learning is primary, teacher instruction is secondary” captures this principle.