Is it possible to educate you differently?‘Educate’ in the real sense of the word; not to transmit from the teachers to the students some information about mathematics or history or geography, but in the very instruction of these subjects to bring about a change in your own mind.”
J. Krishnamurti, “ On Education”, 1974, Krishnamurti Foundation of America
My middle child was interested in math and logic from the very beginning. We noticed early on that she tends to make generalizations and logical conclusions about everything around her. As a toddler, she would say things like “Moms love their babies, otherwise they would not nurse them. Animals also nurse their babies. Animals love their babies too. Mom, are we animals, too?”
Noticing this inclination, we decided to introduce some math. She quickly learned to count, so we moved on to negative numbers, binary systems, arithmetic progression, and so on. She seemed to enjoy it all – until she started school. She was a well-behaved child who learned well and didn’t get into trouble. However, she would come home from school both bored and exhausted. I was worried that she would lose her interest both in math and in learning in general. I decided to start a “thinking” circle for kids of her age to provide them with both food for thought and social-intellectual stimulation they crave.
This book is not about mathematics the way it is introduced in school. Rather, I use mathematics to teach problem-solving skills to kids. In this book, I summarize my class observations, share my experience and suggestions, as well as provide sample lesson plans for parents. These problems were presented to six to nine year old children, though older kids and even adults may have a lot of fun solving them. I do not assume that the children have any specific mathematical background.
While a single child might be able to solve these problems, it is much more fun and stimulating for the kids to solve them together. I highly recommend that you find a few friends for your child who enjoy this type of activity. By making problem-solving a social activity, you will help your children learn to present their point of view to a group, to defend their ideas and to form a special type of friendship.
In Chapter 2 I describe the problem solving skills that this book is intended to teach and discuss the lesson topics. In Chapter 3, you will find the recommended lesson plans, my observations as to what makes the problems difficult for students, sample student responses, and possible solutions. Problems are labeled by topic and problem-solving approach. In Chapter 4, I put together some strategy games. Appendix A provides a list of helpful resources.
As I cannot foresee the future of my children, the best preparation I can give them is to teach them to think independently, to learn problem solving skills and to know themselves. Unfortunately, the schools are ill equipped to provide this kind of training. The teachers are forced to teach a rigid curriculum, without being able to adjust it according to the children’s’ needs and abilities. The child is controlled by fear (bad grades) often not taught to recognize their own mental process, their unstated biases and assumptions. The children are not taught to acknowledge mistakes as a natural part of the learning process – in contrast, they are taught to avoid them by any means, suppressing their curiosity and insight.
While choosing the problems for the class, I have tried my best to select problems that simultaneously challenge and engage children. I was especially keen on bringing in so-called “insight problems” that require the person to re-evaluate his vision of the problem, to be aware of his smallest misconceptions about the problem statement and problem approaches, to be able to step back from a working solution to look for a better one. My observations show that these are some of the problems most welcomed by kids.
Most people see insight problems as more difficult, and tend to introduce them much later down the road, in late middle or high school. I haven’t seen much rigorous material of this type targeted at elementary school kids. Many adults consider these problems too difficult even for themselves, and would not think of presenting them to their children. I argue that six to nine year-olds are at exactly the right age to encounter these types of problems. Moreover, insight problems boost their motivation to learn.
Young children have an innate ability to solve non-standard problems because every new problem they encounter is, to them, non-standard. If not nurtured from the very beginning, this ability disappears by the time the kids reach middle school and high school.
My purpose in presenting these problems to young students is not to teach them any specific approach or topic. It is rather to help them learn about the ways in which their brains work, to listen to themselves, to question themselves and others, and to develop a mind that is curious and alert.
My first full-time job after graduate school was working as an astronaut instructor at the Johnson Space Center. My job description included learning about the prototyped spacecraft, writing training manuals and developing training sessions for malfunction scenarios. Challenging as this was, I always had clear objectives in my training sessions and knew the way to accomplish them. This time, my objectives are foggier and the means to accomplishing them much more exploratory. Nevertheless, I hope that my work helps to make our children’s education more meaningful and provide new insights into children’s mind.
Companies across a broad spectrum of industries recognize the need to hire employees who are ready to solve non-standard problems and to approach familiar problems from a different perspective. I believe that these abilities will become more and more sought after in the coming decades.
The most important thing I want to urge the parents to do, however, is to observe your children constantly and develop a feedback loop to adjust as needed…
...for advice is a dangerous gift, even from the wise to the wise, and all courses may run ill. J. R. R. Tolkien