The Checking Manifesto

Checking your answer is part of doing the problem. You can never be sure you are right, but you can gain more likelihood of being correct, by checking.

You need to develop a feeling of self-confidence that comes after proper checking. It's a good feeling grounded in worry, because constructive worrying entails thinking carefully about what could go wrong, and making sure that it did not. Do you worry about not getting an A? Do you worry constructively?

The self-confidence you should look for is knowledge that "I have a realistic chance of getting an A." It comes from having known how to check every problem on the test, and from worrying enough to have done the checking.

Don't be fooled by the confidence that comes from just knowing how to solve the problem. Proper checking requires that you solve the problem again, by a different technique.

Since checking is part of doing the problem, checking by calculator constitutes doing the problem with a calculator. If it's a non-calculator problem, you should only check with a calculator while first learning the concepts.

Since checking is part of doing the problem, checking the answer in the back of the book constitutes getting the answer from the back of the book. You should only look at the answer in the back of the book while first learning the concepts. The answer in the back of the book is a poor substitute for a teacher. When your answer is wrong, how often do I tell you the right answer?

Checking Techniques

• Look at the problem in various ways. (This has more benefits than just for checking. Some day you'll encounter a similar problem which does not yield to your first attack, but when you look at the problem a new way, you will find a solution.)
• Don't stop with one check. If there are other effective ways to check, use them. Often different ways of checking will be good at detecting different kinds of errors.
• Your math lessons often focus on one technique, but don't let that make you forget that other techniques apply. Think of 2 techniques for each of the following problems.
  • How much is 11 times 12?
  • What is the area of a trapezoid?
  • How much is 2/3 plus 3/5?
  The new techniques may be nifty, but the old ones are true, and are often more efficient when they do apply, or can be used to find an approximation to the answer.
• Make an accurate picture, and measure the answer. This is excellent for lengths and areas. It gives an approximate result (see below).
• Approximate -- avoid messy details that distract from the important stuff, to get an answer which you know should be near the right one, while at the same time avoiding several opportunites for error.
  This is a wonderful approach which you should use almost all of the time, because it will catch all of the errors that have the most devastating effect on your answer.
  • Approximate the statement of the problem. Simplify shapes. Simplify numbers. Make a mental note of how much impact you expect your simplification to have on the answer. If you change the shape a lot, it could make the approximation half of or double the right answer, but that will still catch lots of kinds of errors.
  • If the problem has units, think of the problem concretely. For example, maybe you've got a parallelogram that's 200 cm on one side and 400 cm on the other. That's comparable to the living room. If your answer is that the carpet will cost $2.79 then common sense will help you know that you've made an error.
  • Approximate as you calculate. Keep only 1 significant figure. But if you do this, just watch out for subtraction, because 5 - 5 yields 0 and if both of the 5's are approximations then you don't know whether you've got -1, 0, or +1 and there is a very significant difference between those numbers. Addition, mulitplication and division are usually OK, and subtraction is often OK (as in 10 - 1).

• If it's a "puzzle" problem, check your answer directly to see whether it satisfies the requirements of the problem.

  Consider the problem
  x² - 2x = 2(1 - x)
  A 10th grader adds x to both sides and gets
  x² - x = 2(1)
  x² - x - 2 = 0
  (x - 2)(x + 1) = 0
  concluding that x = -1 or x = 2. A second tenth grader does
  x² - 2x = 2 - 2x
  x² = 2
  and says that x is the square root of 2. Neither one checks his or her answer. A sixth grader listens in and does some figuring, and says that the first student is wrong. Which student has done the best mathematics, the one who factored, the one who got the right answer, or the one who checked?

  There are lots more puzzle problems than just in algebra.

• If you have trouble with - (minus) signs, and one of those tricky things gets into the problem you're working on, then think through how much impact that sign has on the result. If it could have a big effect on the result then you may feel depressed but actually you are lucky! You'll be able to catch an error by redoing your problem with a diagram or with another means of approximating. Otherwise try to check a smaller segment of the problem, in which the sign is tricky but where its impact is significant.

Exercises

• Go through your returned homework for the past years, and find all of the X-marks and sad faces. Check those problems in writing so that Dad can check your check. Do the same thing with some of the problems you marked ^ck because you looked in the back of the book.
• Get SATs and solve the math problems, not by the usual techniques, but by checking all of the multiple choice answers and eliminating the wrong ones.
• Do you worry about tests? I hope so, because your record of checking is not great. Worry constructively. Check all of your problems. Evaluate how you checked each problem:
  • How many different ways did you solve the problem?
  • Did your checking give you confidence that your answer is in the ballpark (which is good) or did it also give you confidence that it's exactly right (which is better)?
  • Did your checking verify everything from the beginning to the end? Did you reread the problem?
  • Did you verify that your final answer is an answer to the question asked?
  • Do you wish you had a better way to check this problem? Make Dad tell you one. He may have a hard time coming up with a really nice way to do it, but he'll give a $10 reward to the first kid who presents a checking problem that he doesn't have some answer for.
  • Do you still wish you had a better way to check this problem? Now's your chance, while you're taking tests at home. It's perfectly fair as long as you don't discuss the particular problem. In the future you won't often have this luxury. Don't be chicken, ask!

Checking Rules

When practicing checking, avoid solving the problem if possible. Just indicate your method briefly in one sentence. Then check 2 multiple choice answers, or estimate the answer. Show your work when checking or estimating. Finally, do not look at the answer in the back of the book.

The point of not looking in the back of the book is this:
You can know for yourself whether you’ve got a good answer, but it takes practice. If your custom is to finish a problem quickly, by comparing your answer with another, then you lose the chance to develop self-confidence. You also lose the chance to reflect on the tricky aspects of the problems later on. There’s usually one problem that you can come back to with a fresh insight at the end of the problem set, but not if you already know whether your answer was “right or wrong”.

Emphasis

Post this on your wall until you know it by heart.