I profoundly enjoyed my math teaching class this past Wednesday January 28th. Why though?
A critical teaching concept finally made sense, how does solving problems with only our previous tools and skills allow us to understand a new technique better? I learned that through persevering in problem solving a student can get close to the correct answer or get the correct answer using what they know. Then, the teacher can show them the new tool that makes the problem much more understandable! The understanding of this seemingly fundamental idea amazed me!
We used a chemistry mixture problem in class to try to understand the importance of fractions, the common denominator, and ratios.
The beakers are filled with blue or water. The most difficult problem that we encountered had 3 beakers of blue and 2 beakers of clear liquid, while the other had 4 beakers of blue and 3 beakers of clear. We found that we could not solve this problem in any other way but with fractions after struggling with it for over a half hour. We tried canceling white beakers, we tried canceling blue beakers , and we tried cancelling white and blue beakers, but none of these techniques worked because mixture ratios and fractions are not equally affected by adding and subtracting the beakers. These techniques cause the denominator to change often times making the fraction seem closer to 1/2. When working with ratios we were able to see the importance of taking care to preserve the ratios between blue beakers and clear beakers that were originally presented in the problem. In the end, we found that the best way to truly report the equal ratio of blue to clear beakers as the original problem presented for each mixture, was to utilize fractions and a common denominator. The importance of common denominators and fractions was displayed for us as future teachers and learners because we struggled and persevered in problem solving without these tools.
I look forward to helping out my students by having them spend time using the tools they already know to solve problems, before they transition to new methods to add to their tool box. I think that as educators we need to be very careful to let students explore problems before aiding them with a new tool that makes the problem significantly easier.