As I was hiking the other day, I couldn't resist snapping this photo (below) for the exact purpose of asking others how they would solve the conversion question:
How many pounds are in 17 1/5 tons?
This is an example a question that can be answered in many ways. As a math tutor, I always encourage my students to solve a problem using the approach that makes the most sense to them. The truth is that there are multiple ways to add, subtract, multiply, and divide. If a student is taught to only use one way to do any of the four operations, it is a sure way to discourage mathematical thinking and reasoning. Instead, it only encourages the student to mimic a strategy and not think on his/her own.
When I was learning math in elementary school, I was only taught the standard algorithms for addition, subtraction, multiplication, and division. Only when I was older did I start to feel confident enough in my math skills to try to solve a problem using something other than the standard algorithm. Straying from the standard algorithm does not mean you are "breaking math." Instead, you're applying what you know and reasoning your way through the problem. I will write future posts on this topic.
number sense and numeracy
In an article published by the National Council of Teachers of Mathematics, Hilde Howden describes number sense as a
“...good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms. Since textbooks are limited to paper-and-pencil orientation, they can only suggest ideas to be investigated, they cannot replace the "doing of mathematics" that is essential for the development of number sense. No substitute exists for a skillful teacher and an environment that fosters curiosity and exploration at all grade levels."
I like to include estimation in my tutoring sessions for this reason, especially when looking at answers to make sure they are correct. It is helpful to recognize when an answer simply doesn't make (good number) sense. If a student gets an answer incorrect during a session, I ask the student to take a step back, look at the answer, and see if it "makes sense." It's even a good thing to do when an answer is correct. Estimation can help identify possible correct answers and eliminate incorrect answers on a multiple-choice exam.
Solving the same problem different ways and learning how to communicate strategies with others develops numeracy. It is like building a muscle that takes time to develop. By communicating and sharing answers, an awareness slowly builds that allows us to choose a more efficient strategy for future problems. As Cathy Smith states in her article in Mathematics in School, "It is mathematically empowering - and interesting - to use an appropriate method for a problem." I have been working on my numeracy skills as an adult and can really see how approaching problems in different ways makes me appreciate math in a whole new way.
How many pounds are in 17 1/5 tons?
I posted this question to my relatively new tutoring Instagram account and asked for the reasoning that people used. I also provided three solutions that I could think of that day. The truth is that I may have used three different solutions the following day . The fact that I may use one strategy one day and a different one the next day shows that I can be flexible with my thinking. This is part of what makes math beautiful to me. Math is a language and we may say something one way one day and slightly differently another day. Children should feel that they can also be flexible with their mathematical thinking. When we allow them to do so, they feel more confident and able to understand the math they are doing. Being able to experiment with numbers, have different relationships with them, and communicate strategies to others builds good number sense and numeracy. These are skills that can be carried throughout future math classes and life.
There are six strategies presented below to answer the question "How many pounds are in 17 1/5 tons?" Some were shared from my Instagram post (thank you for your comments!). There are many additional ways that it could be solved. If you have an approach that isn't shown here, I'd love to hear it! Leave a comment here or send me a message at melissa@learnwithme123.com including your strategy to the problem.
References:
Howden, H. (1989). Teaching Number Sense. The Arithmetic Teacher, 36(6), 6–11. http://www.jstor.org/stable/41194455
Smith, C. (1999). Pencil and Paper Numeracy. Mathematics in School, 28(5), 10–13. http://www.jstor.org/stable/30215425
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Melissa Agocs
melissa@learnwithme123.com
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