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.

Hard work makes awesome new tools worthwhile!

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clear, coherent, content, consolidated +

complete – what’s here is good stuff, but it needs a bit more mass. Can you think of a second example? Is there anything from your content classes? Can you compare with an example of being given the tool first?

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I thought this was a very interesting review of what we did. I looked at the results of our activity and discussion from a completely different perspective. I thought that our activity showed that we don’t always have all of the tools we need so we need to open to new tools, but you looked at it as if you persevere you can get close enough to get a useful answer, which is really the only thing necessary. Both very valid and interesting approaches so I’m glad I got your point of view! Good job!

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Kevin,

first I really enjoy your graphics in this (and your other) blog(s). They really add a certain flare to the overall flow/read of the piece. As for your actual content, I agree that encouraging students to persevere in the face of adversity in context of mathematical problems by having them develop their critical thinking/problem solving skills is a huge component of being a good teacher. Plus, as we all realized in this discussion, trying to do a problem by removing our prior mathematical experience and not using, in this case, our knowledge of fractions and ratios and such was extremely arduous.

That being said, I still believe that our approach to how we might use that problem, or similar ones, in our own classes will vary depending on the content of the each class. However, it will still be our job, as good instructors, to push our students to develop their cognitive skills to get them to be the best they can be.

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Your last comment stuck with me when I went through this. We, as educators, do need to allow our students to explore problems before diving into them. This will enable them to get a handle on the problem to search for the tools that they may need to answer the problem at hand. I do have one question though. Are we, as educators, looking for a specific tool for them to use for the problem that we give them or are we allowing them to look at it from any perspective? Example: If we are working with a problem that we believe we can solve using fractions, would they be able to get around not using fractions because it is not a “tool” that they have acquired just yet?

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I really enjoyed this blog post Kevin! Your final sentence really hit it right on the nail: “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”. This is a reminder that showing the students “short cuts” are not really teaching them the skill of problem solving. As teachers we need to explore what the students know and challenge them to go above and beyond what they think they know to what they really know!

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