Evaluating Validity of Nonstandard Computational Strategies

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In math, there is no one right way to solve a problem. There are multiple paths we can take to reach a correct solution. However, some paths lead to incorrect solutions. Learn about evaluating these different types of paths in this lesson.

Nonstandard Computational Strategies

Different people problem solve differently. This is seen in all grade levels. For example, take this picture problem.


nonstandard computation


Now, some might look at this problem and answer 7 balls altogether. Others might look at this problem and give 3 as an answer. Which one is correct? Which one used a nonstandard computational strategy? In the textbook, the answer may say 7 as the problem seems to ask how many balls are there in the picture. But is 3 a wrong answer? To answer that, we have to consider if the computational strategy used to arrive at that answer is valid or not. We'll define a nonstandard computational strategy as a problem solving approach that is different than what is expected.

Importance of Evaluation

This is where evaluating the validity of such a nonstandard computational strategy is important. Without evaluating these nonstandard approaches to problem solving, teachers may inadvertently be dismissing a valid or correct way of thinking. Many times, gifted students will think in nontraditional ways. Many of the great mathematicians thought in nontraditional ways for their time in order to prove all the theorems and postulates that we now use on a daily basis. Also, evaluating a computational strategy allows the teacher to understand how a student is understanding the material. This allows the teacher to adjust the teaching to match the students' abilities and knowledge.

We can list these reasons as follows.

  • Nonstandard computational strategies may be a sign of giftedness.
  • Evaluating validity allows teachers to adapt teaching to match students' abilities and knowledge.

Examples

Going back to the example presented at the beginning of this lesson, which one is the correct answer? Is it the 7 or the 3?


nonstandard computation


It turns out that both are correct, depending on which computational strategy is used. If the question is viewed as asking how many balls there are in total, then 7 is the correct answer. However, if the question is viewed as asking how many pink balls there are, then the correct answer is 3.

Here's another example problem.

  • Subtract 2 - 8

To answer this problem, the student may use the traditional computational strategy of direct subtraction to get 2 - 8 = -6. A nontraditional computational strategy here might involve simply finding the distance between the numbers 2 and 8 for an answer of 6. Still, another nontraditional computational strategy might be to take 2 and add it to the negative 8 to get negative 6. Which computational strategy is correct here?

Evaluating Such a Strategy

To evaluate a computational strategy, more is involved than simply looking at the answer. If a student provides an incorrect answer, we many not know why the student gave an incorrect answer if we don't ask why. Evaluating a computational strategy involves asking the student how he or she solved the problem.

For the ball problem, the student that answered 3 may say that she saw a pink ball in the problem, so she thought the problem was only referring to pink balls. Because this was her understanding of the problem, the answer is correct. She herself did not make a mistake. The teacher can see that this student views such details as important. To help this student grow, the teacher can give the student other detailed problems to see which information the student may pick up on that other students have not picked up on.

For the subtraction problem, the second nonstandard computational strategy is valid because the reasoning behind it works: 2 - 8 = -8 + 2 = -6.

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