Finding Errors

  As I mentioned yesterday, its hard for students to find the error they made if they do not get the correct answer.  I've been wondering about techniques I can include to help students learn to find their errors.

It seems that once the student has completed a problem, their mind shuts the door on it and moves on because they are finished with it and don't need to check it.

One suggestion I ran across is to have a poster in the classroom for the top 11 errors made in math calculations hung somewhere in the room so they can check it before they move on.

1.  Did not distribute the outside term to both terms inside the parenthesis. This includes not distributing the negative sign with the number.

2. Multiplying by 2 instead of squaring.  In other words they multiply by the exponent, instead of applying the power.

3. Adding instead of subtracting or vice versa.

4. Adding instead of multiplying or vice versa.

5. Misplacing or loosing a decimal.

6. Making a rounding error.

7. Forgetting to carry a number or to borrow.

8. Forgetting to change the inequality sign when dividing or multiplying by a negative.

9. Making a mistake when cross multiplying ratios.

10.  Making a mistake when adding/subtracting/multiplying/dividing a fraction.

11. Omitting units or incorrectly converting units. 

I think I'm going to run this list of common mistakes off and give each student a copy so they can use it to double check their steps.  Of course, I'll have to model its use but if I use it regularly, perhaps they will choose to use their list.

It is also suggested that the teacher change the way they identify mistakes for students.  Rather than saying  "You made a mistake", say "I'm glad you made the mistake, it means you are thinking about the problem and you can learn from it."  I tend to let the student know they missed a step when solving it, so go back and check to see if they can tell where they missed the step. 

In addition it is good for the teacher to make a mistake, correct it, and let the students know what the mistake was and why they did it.  It shows that teachers are not infallible. Teachers are human.  Too often students are under the mistaken impression that math teachers are extremely smart, like Einstein.  Its important to show them we are human.  Make it normal to look at mistakes so they are no longer something to be feared but celebrated.

When a student makes a mistake, it is important to correct it but also to understand why the mistake was made.  By correcting the error and knowing why it was made, it gives the student a personal sense of success. Furthermore, the type of the mistake provides an assessment for the teacher.  The mistakes let the teacher know, what has not been mastered yet.

In a sense, this is something that should be started in elementary but it isn't always so it is necessary to work with students in high school.

Let me know what you think.  I love to hear from my readers.  Have a good day.

Changing Perceptions of Mistakes Part 2.

   Yesterday, I discussed societal perceptions of math being only right or wrong so if you make a mistake you are wrong and a failure but as we've seen that is not always true.  Making a mistake is just a signal letting the individual know it may be a misunderstanding of the math process, it might be a sign that was lost, or numbers switched.

Today, I'm looking at ways to help students learn to identify the type of mistake they are making so they can work on becoming more proficient.  It is not easy to work something else into your day but I figured out how I can use some of these suggestions with my students.

1. Write a problem on the board with several solutions.  Ask students to rank the solutions from best to worst. Next have the student discuss the ranking with their neighbor and create the criteria for ranking the solutions.  Finally, create a list of the most common mistakes and suggest ways to catch them or prevent them.

2.  Assign a select number of problems for students to complete.  Ask them why they think their answer is correct or incorrect.  Ask students questions to help them see different ways of reflecting on their thinking. 

3.  The teacher needs  to monitor student work to see areas of misconceptions so the teacher can help clarify those areas.

4.  Take time in class to create a list of the top ten mistakes they do such as multiply the number by two (the exponent) rather than squaring it.

5.  Help individual students create a list of their own common mistakes to use as they do their work such as switching digits in a subtraction problem so they don't have to borrow.

6.  Have students mark the spot where they run into problems as they work the problem so if their answer is incorrect, they can return to see if that is where the mistake occurred.  This also identifies something they can ask for additional help in learning because it lets them know what they still don't know.

My final thought on this is that I have not explicitly taught my students to learn to analyze their errors.  I have just started including one math problem in their warm-ups that will have an incorrect answer.  Their job will be to determine where the error occurs and what the exact error is.  How can I require them to find their own errors if I have not taught them the process?

Tomorrow, check out teaching students to self-correct.  The first step is to change people's mindset from getting a problem wrong is failure to an incorrect answer is actually just telling us where we need a bit more work.  Teaching them to self-correct is the next step.