One thing I've been noticing this year as I find "challenge" problems online or give sets to my own students is that the first thing I acknowledge is the trust.
Often, we ask students to "Prove that..." or "Solve" in problems. All well and good in a classroom setting, but in a more "real-world" situation, the first thing that needs to be assessed is whether that problem CAN be solved. I think my own problem-solving strategies are sometimes directed by knowing that there is an answer before I even begin the work.
I trust that my teacher or problem-giver is giving me the opportunity to solve the problem when it is posed. That is, if I am not allowed a calculator, I trust that it CAN be solved without a calculator and that my answer will most likely be a whole number or at least a simple fraction. If I am in a 6th grade classroom then my problem will be "6th grade appropriate" (ie should be solved without calculus or other advanced methods).
Of course, this is not the case in higher mathematics where the main question can often be, "Is this solvable?" Certainly less so in every day life when we are faced with innumerable unsolvable problems. I'm not suggesting we give students unsolvable problems, but it might be interesting to mention this to students that we are in an artificial, safe environment in this classroom where these problems have already been solved and I (the teacher) am giving them to you knowing you should be able to complete them.
I appreciated this post.
ReplyDeleteI am currently a student in a mathematics education program and I will be student teaching in a month or so. I've been looking for blog posts for some different perspectives, and I found your blog. This post I found particularly interesting, since the classes I am in right now often do involve problems that may not always be solvable. I think it's definitely important to include problems like this in any curriculum, because it adds a higher level of thinking and problem solving. As you mentioned, this doesn't necessarily mean purposefully giving students problems that can't be solved (this would almost definitely just frustrate the students) but possibly having the students find out why some problems can't be solved, or having them come up with their own unsolvable problems. Great post!
ReplyDeleteI thought this was an interesting way of approaching problem solving. As a 6th grade student I never once considered that a problem I was given actually could not have a solution, I was assuming I had everything I needed to proceed from there. I think this is just early enough to start influencing the students to question most things they are told, it could lead to them to more critical thinking and original ideas. Often times students will go through the motions if they are given everything they need and more when solving problems. It would be a worthy endeavor to layer the information in such a way that the student uses the first piece of info to find the next piece, and so-on until he/she realizes they have all they need now to find a solution.
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