There have been a number of studies that show that students have difficulty with mathematical reasoning. I’ve seen it in my own classroom (both math and science). Students can take an equation or formula, plug in the numbers, and get the correct answer, but they can’t explain why what they did was correct or have difficulty solving a slightly different type of problem using that same equation or formula. The article points out that that “teaching approaches often tend to concentrate on verification and devalue or omit exploration and explanation.” I try to incorporate different avenues to encourage students to explore and explain their work, but they are hesitant or resistant to doing anything but show you that they got the right answer. They don’t think beyond “I’ve used the correct formula and gotten the right answer, I’m done.” This was particularly prevalent with a unit in Chemistry about Gas Laws (a topic that is very heavy in algebraic manipulation). As a class, we would walk through how to solve the problem and I would ask “How do I know my answer makes sense?” The smart-aleck comment I got was “well you’re the teacher so of course it’s right”. The point I was trying to make was to go back and look at the particular law and what the concept says. The math should follow along from that and if they have done the work correctly, knowledge of the larger concept will allow them to check their own work (without having me there to say “yes” or “no”).

Interactive geometry software are computer programs that allow students to explore geometry concepts. It allows them to see how the relationships in geometry work and how they all interrelate. It allows students to see that we aren’t lying when we say that it’s all connected because the evidence to support that statement is right in front of them. Students look at why the geometrical shapes and constructions are the way that the are, and what elements need to be there for the whole thing to work. The main issue is with student interaction with the software. Students have a hard time ensuring that the shape they use is correct for the program to recognize and for them to move on and actually work with the geometry. I think if the interface was less finicky, along with clear instruction on how the program worked and its limitations, I think this could be a powerful tool in the classroom.

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I agree that students are resistant to explaining their thinking. However, if it becomes an expectation of the class, teachers can usually overcome that initial resistance. As you note, that is a very important aspect of mathematics learning–to be able to really understand enough to explain it. DGE’s can be neat tools to facilitate that explanation and also to develop conceptual understanding.

I completely agree! Even when I go about solving problems to make sure students are working it out correctly. Or to practice to make sure I’m solid at a concept before I try to teach it. I go directly to solving it and seeing if I got the right answer. That is how many of us learned, especially with math. There is a right answer, and when we find it, we often stop and move on to the next. It never was addressed that we continue on with the problem and dig deeper into the meaning behind the steps or the general ideas of it. I think when students are able to see it, hear it, have pictures, ECT it reaches out to more of our senses and allows us to recall/remember and hopefully start to generate questions or meaningful solutions besides just the “right answer”