A story
Once, I gave my daughter and her friend the famous three cottages problem. Drawn on a piece of paper are three cottages and three wells. The task is to draw the roads that connect each cottage to each well, that is nine roads in total, such that no two roads cross each other.
The problem appeared to be very enticing to them; after all, drawing was their favorite subject. They eagerly started working on it. One drawing, two drawings, then three, then more…, but to no avail. They knew they needed to draw nine lines but somehow they could only get up to eight; the last road always had to cross one of the existing roads. The solution seemed to be very close but always escaped them at the last moment.
During lunch, I asked them about their progress. Knowing what would motivate kids at this age (7-8 years old): an instant reward, I promised them that I would let them watch the short cartoon that they had mentioned earlier. Then one of them had an idea: “Can I split a road from one cottage to two different wells?”. “Sure, you can”, I answered much to their delight. So after lunch they set out to work right away with this newfound idea.
Below is one such innovative solution. Note that the road that the person from house C takes to well 2 and well 3 are split in the middle; that is within the rule. Unfortunately, the road from house B to well 2 and the road from house C to well 3 still cross each other.
Undeterred, they continued working. First separately, then together. There was such lively discussion and great enthusiasm beyond what I had expected.
I have to admit that I gave them too hard a problem. The problem sounds simple but the solution is not, as often the case in mathematics. I didn’t mean for them to find the solution; I just wanted to see they how far they were willing to carry on. So after almost an hour, I told them it was time to stop.
They looked very unsatisfied. My daughter declared “We’ve wasted the whole playdate and still couldn’t solve the problem!”. Her friend echoed “That’s right. PLAYdate is time to play, not to do math problems”. How fast they forgot that just a few minutes away they were so absorbed in the problem! I explained to them that it was not a waste of time, for thinking would make them smarter, just like doing exercises would make them stronger. When they insisted that I tell them the solution, I again told them that they wouldn’t become smarter if I told them the solution, just like if their daddies lifted them up high they wouldn’t become stronger, but if they learned to jump or climb up high they would certainly be. They seemed to be content with my answers. With that, we finished our math education for the day and moved on to real play. And of course I offered them to watch the cartoon which they happily accepted.
The next day, on the way back from school, my daughter told me that she had again tried to find the solution during recess time. At this point, there was no more reward; apparently she kept thinking about it because it was interesting enough for her. I told her that no matter what she did, the last road would always have to cross another road. She immediately asked me “Why? How are you certain of that?”. I told her to she would find it out some day. Deep inside I was glad that we were moving in the right direction because instead of accepting the answer she was asking for a proof.
Should we give kids answers?
In learning math, remembering a definition or a fact is a very small part; it’s all about problem solving. One can make up an entirely new problem by twisting an old one, rendering the old solution unusable for the new problem. That is why learners should find solutions themselves, instead of being told the solutions. That may seem too much to ask of a young kid, as the process of solving a new problem often involves some sort of struggling and entails frustration. However, that is the nature of learning and when the kids get used to this learning habit, they will be better not only at math but at learning other subjects as well. Here are a few tips on how to help kids become independent problem solvers.
Give them the right problems
- The problems should be interesting enough for the kids to work on their own for some time. This is especially important in the beginning when we are trying to get them interested in math.
- Avoid problems that have a single impasse. When kids get stuck they tend to think that there is a special trick that is beyond their power. Choose problems whose solutions consist of several steps that the kids can take on their own or with little help. They need to see that hard work payoffs, that by working diligently and systematically most of the time they will find the solutions.
The problem I described earlier is a special one. I chose to tell the story because of the level of excitement and unusual tenacity that the kids displayed when attempting to solve it. Usually I only give them problems that they can solve given what they have already known.
Ask them to focus 20 minutes at a time
If they get stuck, it is okay and actually beneficial to let them have a break and return to the problem at a later time or on another day, that is switching between focused and diffuse thinking modes. More details on this will be discussed in the article about learning techniques.
Only give hints after the kids tell you what they have tried
It is a familiar scene but one that often upsets parents and dreads teachers: you give the kids a problem, they read it and become silent, and remain silent for a long time even after you ask them to speak up. Kids usually do not write things down. They keep thinking in their head hoping for a solution to pop out. When pressed, they may tell you that are still thinking or they don’t know how to do it.
It is not always easy and may require some tactics anywhere from encouragement to cold call, from jokes to promises of rewards of some kinds, but by all means try to get them talk and show you, either in words or on paper what they were thinking. Comment on their progress and give them hints to move forward. Part of the learning is for them to explain to others their thought process. And a lot of the time they come up with the solution while explaining their approach to the instructor or among themselves.
Discuss the solution at a delayed time
Depending on the problem and your objective, the discussion may take place within the same session, or the next day, or it could be delayed for a week, a month, or even years in our case with the three cottages problem. The important point is that this delay gives the kids time to think through, to dive deeper and understand the material better whether or not they will eventually find the solution themselves.
It is not easy to let young children struggle and it takes time to guide them through the process but often the joy they show when they finally solve a challenging problem is worth all the effort.