“Show your work”, “Show your work”, “Show your work”, etc.! The all too often phrase said by math teachers to students. And the students reply is often like, “But, I got the right answer!”.
So, why isn’t the answer good enough for most math teachers? The answer to this question is that good mathematics is not about the answer, it is about reasoning, which will eventually lead to the correct answer.
Good mathematics is about reasoning. Solutions in mathematics should never involve “leaps of faith”, guessing, or the idea that it just seemed right. The beauty of mathematics is there is always a clear, logical argument as to why the solution is correct. The ideal should be that all students can learn how to present these arguments in such a way that anyone, anywhere can follow the solution. This is what I am calling a universal solution.
What constitutes a universal solution?
Well, first of all, it should be factually correct. All information contained in a solution should be true and accurate. Mathematics is built upon definitions and axioms. It is from there that we can launch forward to proving new ideas. In classrooms, most students are not focusing on the proof of new ideas, but learning to explain clearly the ones that already exist. Second, the solution should address the problem in a clear and coherent manner and not contain any irrelevant information. Finally, there should be a logical flow, from step to step, with sound reasoning as to why each claim can be made. In summary, always ask yourself, could my solution be understood by anyone, anywhere, anytime?
Example of a Universal Solution
Explain which value is greater, 0.003 or 0.0006691?
When comparing numbers, the digit to the left of any other digit will always represent a higher place value. For example, a 9 in the ones place will always be less than any digit that is in a higher place value, like even a 1 in the tens place. For 0.003, the first non-zero digit, 3, occupies the thousandths place while in 0.0006691, the digit in the thousandths place is a 0. Although there is a 6 in the ten-thousandths position and 6 is greater than 3, thousandths are greater than ten-thousandths, so any digit in the thousandths position will always be more than a higher digit in a lesser position. Therefore, 0.003 > 0.0006691.