Having hammered away at the importance of student self-confidence and positive attitude as a condition for success in math (part of the larger discussion of applying Pólya's teaching principles), it's important to step back and point out that a lot of people have made horrible, costly mistakes thanks to positive thinking and the influence of motivational speakers (for example).
With fantastic successes on one side and horror stories on the other, it is tempting to call this a wash, but if you think like a statistician (and you should always think like a statistician) and start breaking things down, you'll find that a few common sense rules can tell you when to assume the best and when to prepare for the worst.
Being pragmatic about being positive
For the purposes of this discussion, let's decide on a fairly precise definition of what we mean by positive thinking:
To apply positive thinking to a task, you act under the assumption that, given reasonable and intelligently applied effort, the probability of success is close to one;
Furthermore, this assumption will not be reassessed unless there is a major change in the situation.
The advantages to this approach are: we can waste a great deal of time and energy worrying; overestimating risk can cause us to prematurely abandon projects; thoughts of failure can cause us to "flinch," to hold back and not give the task our best effort. Avoiding these things can allow positive thinking to create self-fulfilling prophecies.
The disadvantages are that underestimating the probability of failure can cause us to waste resources on projects with negative expected value and, more importantly, failing to pay attention to warning signs can leave us vulnerable to otherwise avoidable disasters.
We could have a general discussion at this point about the relative weight of these advantages and disadvantages but it wouldn't be very productive because neither risk nor reward are evenly distributed. In many if not most situations, a fairly clear case can be made for either positive or cautious thinking. To determine which approach is best for a given situation, think about these rules of thumb:
Before you commit yourself, try to think realistically about the expected value in terms of other people's success rates;
Never bet more than you're willing to lose;
Consider collateral damage (are you putting your spouse and children at risk of hardship?);
Is there incremental payoff? This last one is extremely important. If you decide to start a restaurant or move to NYC to make it on Broadway, and you fail, then you will probably have very little to show for the effort. If, on the other hand, you decide to lose forty pounds through diet and exercise or to go from being a C student to an A student, then there is incremental pay off for your hard work even if you fail to achieve your goal.
All of this leads us back to the original point. We often associate positive thinking with business and entrepreneurship where it is, more often than not, a bad idea, while in education, where we have every reason to encourage positive thinking, we are constantly hearing people like Michele Rhee complain that we spend too much time building up kids' self-esteem.
Don't let the posturing and tough talk fool you. Self-esteem is good for kids and you should do everything you can to convince them that they are capable of doing every problem their teacher gives them, as long as they put in the effort.
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