Don’t Leave Analytical Thinking Behind!
From Alexander S. Belenky
Every American wants her/his children to be well educated, and today, almost everyone agrees that studying mathematics successfully is the key to achieving this goal. Indeed, mathematical studies develop analytical thinking, the most critical ingredient to successfully studying any other subject.

Yet, the existing system of teaching school mathematics mostly prepares students for passing tests, which doesn't necessarily develop analytical thinking in students, and may even cripple their ability to ever develop it. The existing system follows a traditional approach to solving new problems. That is, first, people try to find whether they've faced similar problems. Second, they try to solve new problems based on the past experience. The stronger the ability to recognize problems as already known, and the more the past experience in solving known problems, the better are the person's chances of handling new problems successfully.

Currently, mathematics teachers train school students to solve equations, to make geometric calculations, etc., using particular techniques. Then the students learn to find which of the studied techniques are suitable for solving problems required by the curriculum. As long as test scores remain the major priority in public education, the memorizing of rote techniques, often helping students pass the tests, will remain a preferred style of learning mathematics in schools.

However, not all students (and not even all adults) are capable of memorizing and reasoning such techniques without necessarily comprehending their logic. Moreover, such a memorizing often doesn't help in solving unfamiliar problems, even within the school curriculum. What students learn in mathematics doesn't help them in studying other school subjects, while memorizing mathematical facts consumes more and more time at each new grade. Difficulties in learning school mathematics currently experienced by many suggest that the existing system of teaching school mathematics is not effective. But teaching left-handed children to write with their right hand is also rarely successful, whereas these children can perfectly well learn to write with their left hand.

Teaching the logic of finding solutions to mathematical problems based upon a few basic conceptions and facts produces remarkably different results. Developing the ability to establish logical connections between any problem and these basic facts and conceptions—underlying analytical thinking—gradually makes a student capable of solving unfamiliar problems, particularly within the school curriculum. To help develop this ability, first, the teachers should explain basic mathematical concepts and facts and make sure that the students comprehend them. Only after that should the students study techniques for solving particular problems while focusing on establishing logical links between the basic facts and every problem that they are trained to solve.

Not everyone can learn in the same manner, and my experience in tutoring "hard" or even "hopeless" (from their teachers' viewpoint) school students suggests that many children experiencing difficulties in studying mathematics in a traditional manner often do better under more abstract and formal teaching, accompanied by daily, one- to two hours of home exercises. Currently, I teach systems analysis for future applied mathematicians at the National Research University Higher School of Economics in Moscow, Russia. Though all my students are talented people, who have gone through a set of challenging tests and exams to acquire the right to study at this most prestigious Universities in Russia, not all of them have the same level of analytical thinking. However, all of them are ready and willing to improve it, and “proving every step” in the reasoning, along with providing illustrative geometric images of mathematical facts and properties under study, helps a great deal.

One important principle of developing analytical thinking is to remember to challenge every statement, no matter who makes it, and to prove (or to request to prove) every step of every reasoning. This should become a “pleasant routine” for an interested student, and, to a certain extent, even part of her\his lifestyle. Another important principle is to constantly ask yourself questions and to try to figure out what exactly you do not understand and cannot prove, and what hinders you to do it.

The major challenges in developing analytical thinking in a student are a) to convince a student that she\he can do it, b) to explain that this activity requires her\his everyday work, and c) to develop a personal curriculum for developing this skill for every particular student in the class while teaching the subject for the whole audience.

Developing a system for teaching mathematics effectively and encouraging every student to learn requires meticulously selected individuals. They must be interested in and capable of teaching the logic of solving mathematical problems and creating effective methodological materials using contemporary visual techniques. I consider the MIT BLOSSOMS collection to be a unique tool for both school teachers and school students. All the problems discussed there within each particular topic keep the students challenged and captured, which is undoubtedly a necessary condition of successfully developing analytical thinking.

Moreover, the best way to motivate a school student to work on developing analytical thinking everyday is to demonstrate how it works for her\him personally. Only a challenging problem solved by this student for the first time in her\his life as a result of applying analytical thinking will make the student sure that a) she\he is capable of developing this type of thinking, and b) this may be worthy of doing. Wisely incorporated into the school curricula, all the well-thought-out video presentations from MIT’s BOLSSOMS collection look excellently positioned to create this assurance in school students, making an invaluable contribution to dramatically changing for the better the whole system of teaching math and science in schools.

Alexander S. Belenky is a professor of Applications of Mathematics in the Department of Mathematics for Economics and a leading scientist at the Decision Choice and Analysis Laboratory at the National Research University Higher School of Economics in Moscow. He is also a visiting scholar at the MIT Center for Engineering Systems Fundamentals.

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