The definition of the spring constant is the applied force per unit length of stretch of the spring. So for a constant applied force the spring with the largest spring constant will stretch the least distance i.e. the spring will be “stiffer”.
This simple demonstration has two springs of the same material but different diameters and obviously different spring constants. At the beginning the demonstrator held up two springs of the same diameter but different materials (steel and plastic) and said they had different spring constants but the video does not show them being tested.
This video could be used as the impetus for students to test several springs to investigate what it is about their construction that determines the value of the spring constant.
For more advanced students you could get them to determine the relationship that governs springs i.e. F=-kx by first getting them to draw a free-body force diagram. Then ask “Why does the spring stop stretching?” “What is in balance when the spring stops / is at equilibrium?” Hmm.. so the force the spring applies to the mass opposes the gravitational force and is in the opposite direction to the displacement of the mass from the initial position. So while the spring constant is a constant of proportionally the formula showing the relationship has to have a negative in it to show that the restoring force is in the opposite direction to the displacement of the mass. Then you could ask them to graph the relationship and explain their graph and the free-body force diagram to their class mates. In this way you can see if they have unpacked the explanation you gave them. The explanation is very concise and precise. There are no jargon words but it has to be unpacked to get meaning out of it.
"the force the spring applies to the mass opposes the gravitational force" This is where the free-body force diagram can help with understanding.
"the force the spring applies to the mass... is in the opposite direction to the displacement of the mass from the initial position" Again the free-body force diagram is useful in unpacking this.
"the spring constant is a constant of proportionally" You may have to explain using several different examples what a constant of proportionality is.
Physicists have these patterns of speech so that they can communicate precisely with each other, as do many cultures, but this way of communication is exclusive for the novice who does not understand that the phrase contains more meaning than the individual words. As physics educators it is our responsibility to provide novices with opportunities to unpack the language of the discipline and with opportunities to practice using it themselves so that they too can become experts and engage in the conversation.