From Solving Worked Examples to Independent Problem-Solving: Why Learn Proof in Senior Secondary Mathematics?
Many parents have told me something similar: “My child gets almost all their homework correct, but as soon as they face a test and the questions look slightly different, they do not know where to begin.”
Does getting every homework question right mean that a child has truly learned the material? In my view, successfully completing worked examples and independently solving an unfamiliar problem are two different abilities. The former can be achieved through imitation; the latter requires understanding.
To distinguish between the two, I often look at whether a student can explain why a method must work. In senior secondary mathematics, the most direct way to train this ability is through proof. Proof is included in both the IB Mathematics AA HL and A-Level curricula; yet even where it is not formally required, the thinking behind proof benefits every student.
A seemingly simple question
Prove that the equation
x3+x+1=0x3+x+1=0
has no integer solution.
There is no complicated calculation involved, and parents may wish to try it too. In class, I usually see three types of response.
First: trying values one by one
Some students substitute x=0,1,−1,2,−2x=0,1,−1,2,−2, find that none gives zero, and then conclude: “Therefore, there is no integer solution.”
This is a natural approach and shows that the student understands what a solution is. The problem, however, is that there are infinitely many integers. Testing five values can only show that those five values are not solutions; it cannot prove that none of them is. If a child believes this completes the task, it often suggests that they are used to “getting an answer” without yet recognising that mathematics requires a statement to hold in every possible case.
Second: observing the graph
Other students may use a calculator to sketch the graph of y=x3+x+1y=x3+x+1. They see that the curve crosses the xx-axis only once, between x=−1x=−1 and x=0x=0: when x=−1x=−1, the function value is −1−1, while when x=0x=0, it is 11. Therefore, the root lies between two integers.
This goes a step further than testing values, because the student is beginning to use the overall properties of a function. However, a graph can only show what appears within the displayed window. To make the argument complete, one must also explain why the curve can cross the xx-axis only once—for example, by showing that the function is always increasing. A student who realises that “what the graph shows still needs justification” is already thinking in the manner of proof.
Third: examining the structure of the numbers
The third method is the most concise. Rewrite the expression as
x3+x+1=x(x2+1)+1.x3+x+1=x(x2+1)+1.
Now consider two cases. If xx is even, then x(x2+1)x(x2+1) is even. If xx is odd, then x2x2 is odd, so x2+1x2+1 is even; hence the product is again even.
In other words, for every integer xx, x(x2+1)x(x2+1) is even. Adding 11 therefore always gives an odd number. An odd number cannot equal zero, so the equation has no integer solution.
This argument uses only the ideas of odd and even numbers, yet deals with infinitely many integers at once. Its value lies not in a clever trick, but in the key question the student has asked: “What property is shared by all integers?”
Proof develops independent problem-solving
Placed side by side, these three approaches reveal an important difference: the first looks for an answer; the second observes a pattern; the third explains the reason. When facing an unfamiliar problem, students who have developed the habit of asking “why?” will usually progress further, because they do not depend on whether a new question looks like one they have seen before.
This is also why I do not believe that mathematics learning should be judged by marks alone. Marks tell us how many questions a child got right this time. Whether they can explain their reasoning tells us more about whether they will be able to think independently when they encounter a new problem next time.
Three questions parents can ask
Parents do not need to know senior secondary mathematics to understand how their children are thinking. They can simply ask:
“Why did you choose this as your first step?”
Being able to give a reason is better than simply saying, “That is how the teacher taught us.”
“If the numbers in the question changed, would this method still work?”
This helps reveal whether the child understands the conditions under which a method applies.
“How do you know you have not missed any other cases?”
This goes to the very heart of proof.
Children may not be able to answer fully every time—and that is perfectly normal. What matters is helping them develop the habit of recognising that mathematics is not only about reaching an answer; it is also about being able to convince others why that answer is correct.
The graph of the function is strictly increasing and crosses the xx-axis only once. Its only real solution is approximately −0.682−0.682, which lies between −1−1 and 00, and is therefore not an integer.
Written by: Ken Sir
Founder of Mathetactics Education and an IB Mathematics tutor, specialising in IB and A-Level Mathematics. He completed his studies at the University of Hong Kong on a three-year full-tuition scholarship, has authored several senior secondary mathematics reference books, and was invited by Longman to deliver public examination analysis seminars.
Original Article: Hong Kong Economic Journal Company Limited https://edu.hkej.com/php/article.detail.php?aid=64857
English translation provided by AI.
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