Breaststroke Kick is a tough technique for some young kids, and likewise it's a tough skill for instructors to teach. Much of the trouble with teaching breaststroke kick has to do with the verbage we use as instructors. I hear a lot of "point your feet" coming from instructors -- when "pointing" the feet is the opposite position we'd like to see from a novice breaststroker. In fact, we'd like to see the feet flexed from the onset of the kick (heels up), to the finish of the kick (feet together). Sure, it's great at the end of the kick to point the feet before getting into the next kick cycle, but the pointed foot in this case is for the advanced breaststroker -- not the beginner.
This is the "Flex Position" in Breaststroke kick -- we want kids to learn this position.
This is the "Point Position". When we tell swimmers to "Point their toes" this is what the athlete does. We tell them to "Point the toes" because we figure if the athlete can directionally situate their toes toward something, the problem will take care of itself. I've spent hours telling kids to "Point the toes toward the wall" before I figured out what I was doing wrong.
Here is a good drill you can do with young kids to teach effective breaststroke kick. I've seen plenty of kids who otherwise couldn't get it -- simply "get it" with this drill.
Step 1: the athlete situates on the ground. Place a kickboard on their feet like seen here. Teach the athlete that their feet are in the "flexed" position. Let them hold the kickboard for a full minute.
Step 2: Toss a few more kickboards on the top. The extra weight will create added kinesthetic awareness of the "Flex Position". Continue to talk about the flex position with the athlete.
Step 3: Press lightly on the kickboards, applying more pressure toward the athlete's toe side. Explain each time you press that the pressing down on the kickboard helps the athlete into the "Flex Position". They will feel it in their feet and ankles.
Getting into the Flex Position, and holding it throughout the kick cycle, determines in many cases if an athlete is performing a legal or illegal kick. I like to go through the above drill on land for 5 minutes, then ask the athlete to get in the water -- without a kickboard -- and kick a few "flex position" kicks on the wall. The whole time the athlete is on the wall, the instuctor should verbalize the goal: "Flex, Flex, Flex...."
Once the athlete can flex their feet on the wall (hands on the wall, feet toward the middle of the pool), then the athlete is ready to attempt a few strokes of breaststroke, or a few kicks on a kickboard. When I find someone who has gotten it, I usually walk up and down the pool with them a few times, saying "Flex, Flex, Flex" over and again.
Once you can get a young athlete to feel the "Flex Position" and perform it for a practice, they will have the basics of that skill for their entire swimming carreer.