The hard part is not only to encourage the child to try something new once, but to stick to it and give the new skill a really good try. It may require trying something a number of times.
Encourage the growth mindset in your child. Praise the process, such as effort, not the personality, such as the fact that the child is "smart" or is "good" at something. This goes for both the old skills already mastered by the child, and the new skills. For example, instead of saying: "You are good at karate", say "I like it that you put so much effort in mastering these moves". See more examples below, as well as in Dweck, 2017.
Lead by example. At this age, children still try to be like their parents. If you practice something yourself routinely, and the child observes it, they are more likely to try this skill.
Use peer pressure (sparingly). Use this only in moderation, and emphasize the positive aspects of peer pressure. Peer pressure should be a spice, not the main dish. For example, if the entire family is going swimming or biking, the child who does not want to try these skills may decide to overcome the inertia or fear, and try to learn the new skills. The desire to be in the in-group is strong in children.
How to praise:
Note that the examples below refer mostly to academic subjects, but they can be generalized to other areas, such as hobbies, sports, etc.
Praise what they do, not what they are. Praise should emphasize behaviors that are under the recipient’s control. Highlight hard work,
good strategy use, determination, and persistence rather than praising
abilities that are seen as fixed or innate.
Avoid comparing to others. Avoid praise that explicitly compares your students or employees to their peers. Instead, compare their
current performance to their own past performance, in order to
emphasize the value of improvement and keep the focus on getting
(Halvorson, 2012, Chapter 13, p. 162)
Does this mean we can’t praise our children enthusiastically when they
do something great? Should we try to restrain our admiration for their
successes? Not at all. It just means that we should keep away from a
certain kind of praise—praise that judges their intelligence or
talent. Or praise that implies that we’re proud of them for their
intelligence or talent rather than for the work they put in.
We can appreciate them as much as we want for the growth-oriented
process—what they accomplished through practice, study, persistence,
and good strategies. And we can ask them about their work in a way
that recognizes and shows interest in their efforts and choices.
“You really studied for your test and your improvement shows it. You
read the material over several times, you outlined it, and you tested
yourself on it. It really worked!”
“I like the way you tried all kinds of strategies on that math problem
until you finally got it. You thought of a lot of different ways to do
it and found the one that worked!”
“I like that you took on that challenging project for your science
class. It will take a lot of work—doing the research, designing the
apparatus, buying the parts, and building it. Boy, you’re going to
learn a lot of great things.”
“I know school used to be easy for you and you used to feel like the
smart kid all the time. But the truth is that you weren’t using your
brain to the fullest. I’m really excited about how you’re stretching
yourself now and working to learn hard things.”
“That homework was so long and involved. I really admire the way you
concentrated and finished it.”
“That picture has so many beautiful colors. Tell me about them.”
“You put so much thought into this essay. It really makes me
understand Shakespeare in a new way.”
“The passion you put into that piano piece gives me a real feeling of
joy. How do you feel when you play it?”
What about a student who worked hard and didn’t do well?
“I liked the effort you put in, but let’s work together some more and
figure out what it is you don’t understand.”
“We all have different learning curves. It may take more time for you
to catch on to this and be comfortable with this material, but if you
keep at it like this you will.”
“Everyone learns in a different way. Let’s keep trying to find the way
that works for you.”
(Dweck, 2017, pp 250-251)
Halvorson HG. Succeed: How We Can Reach Our Goals. New York: Penguin Group; 2012: https://www.amazon.com/Succeed-How-Can-Reach-Goals/dp/0452297710
Dweck CS. Mindset: Changing the Way You Think to Fulfil Your Potential. London: Robinson; 2017: https://www.amazon.com/gp/product/0345472322
(Note: Amazon has only an older, 2007, edition)
Carol Dweck: The power of believing that you can improve. TED Talk: https://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve?
Carol Dweck: The Growth Mindset. Talks at Google: https://www.youtube.com/watch?v=-71zdXCMU6A
Mindset Kit: Do's and don'ts of praise: https://www.mindsetkit.org/topics/praise-process-not-person/dos-donts-of-praise
Dweck, C. (2015). Carol Dweck Revisits the 'Growth Mindset'. Education Week. 35(5): https://www.edweek.org/ew/articles/2015/09/23/carol-dweck-revisits-the-growth-mindset.html
NOTE: Carol Dweck's research has been criticized with respect to replicability: https://en.wikipedia.org/wiki/Carol_Dweck#Criticism
How to help my son improve without being discouraging?: https://parenting.stackexchange.com/a/39008/33055