Most teachers, if you catch them with their guard down in the teacher's lounge, will admit that they have a hard time getting their students to think. By that, they will mean that it is challenging to get students to think for themselves, without prompting or hints (or downright explicit instructions) on what to do next. I think the problem has to do with the fact that we rarely ASK students to think for themselves.
Instead, we lead them by the hand and attempt to minimize the struggle, the uncomfortable silence, and the fear of unknowing that always accompany real thinking and problem solving. Perhaps, the answer to the thinking problem is the equivalent of throwing students into the deep end and allowing them to learn, individually, the very important non-cognitive skills of persistence, focused effort, ambition, & resourcefulness along with the required content skills. Learning occurs in the struggle. We must think about options, conditions, and possible solutions, wrestling with pro's & con's of each, then apply what we know and can figure out in order to solve our problems. We must each be capable of doing that on our own in order to have confidence in our ability to navigate this world and overcome the challenges we will face.
So how do we teach for thinking? I believe there are several ways but one of my favorites is via problem posing and problem solving activities. Please understand that I am using the term problem defined as "something you don't know how to do." By that I mean NOT an exercise, NOT a question that's just like the 10 questions you've already worked on the board for them, and NOT something that they did last year & the year before that. I mean PROBLEM: something you don't know how to do (as, obviously, if you know how to do it, then it's not a problem). Again, please note my words and don't misunderstand: "something you don't know how to do" NOT something that's impossible to do. I am not advocating giving unsolvable riddles to our students and simply standing by and watching them squirm (no matter how much we may want to do so at various times in the school year! *grin*). What I'm saying is that we give our students problems: questions that they are capable of answering, using or building upon skills they have already learned, but that they have not actually ever done before.
Of course you immediately realize then that problems are rather unique to the solver. By that I mean that something that may be a problem for me might not be a problem for you, or something that would certainly be a problem for a 4th grader would not at all be a problem for a student in high school.
Teaching via problem posing and solving goes hand in hand with teaching for thinking because the focus shifts from the answer to the method, from the solution to the strategy, and from the notion of one right way to the question of "how else could we have solved this?" I encourage my students to employ Polya's problem solving process and various heuristics (i.e. strategies) which I both teach directly and model throughout the class. My students learn what questions to ask themselves as they try to solve a problem (a critical thinking skill!) and how to communicate their struggles and progress along the way.
I have my collection of favorite problems for various topics as I'm sure many of you do. Here's one of them: (the "story" aspects of the problem vary depending upon my audience; in this one I'm assuming a group of young adults in the US and purposefully being rather stereotypical in my descriptions)
Let's say you're at the mall shopping when suddenly you see someone running between the kiosks. It's a man, you realize, and he doesn't notice the "Wet Floor" sign off to one side. He loses his balance, falls, and slides headlong into one of those giant metal trash cans with a huge crash. You, being the good Samaritan that you are, rush over to see if he's ok. Brushing his hair away from his eyes, you notice he's bleeding and end up getting blood on your hands as he struggles to get to his feet. Trying not to be judgmental, you can't help but notice he's wearing a prominent marijuana leaf necklace, sporting several tattoos, and has a pack of cigarettes folded into one sleeve of his T-shirt of rather questionable taste. Undeterred, though a bit more cautious, you turn him over to the paramedics who've now arrived to help as the crash caused quite a commotion. As you walk away you again notice the blood and go into the restroom to wash your hands. The soap stings and you suddenly realize that you must have cut your finger last night when you were making dinner. You hadn't noticed the wound before. That night you can't stop thinking about the incident and you begin worrying about blood-borne diseases like HIV. The next day you go to your doctor and tell her what happened and your fears. She agrees that testing is wise so she draws some blood and, as she does so, explains the medical test by telling you this, "The test I'm using is 98% accurate in detecting HIV. That means that if 100 people have the HIV virus and we administer this test, it will be right 98% of the time (i.e. 98 people) and indicate a positive result (that the person has the virus). It will be wrong 2% of the time (i.e. 2 people) and tell them they don't have the virus when in fact they do. Likewise, if we were to test 100 people who don't have the HIV virus, the test would be correct 98% of the time and report a negative result (that the person does not have the virus). It would be wrong 2% of the time and tell those 2 people that they have the virus when in fact they don't." She then asks you whether or not this all makes sense to you and you respond that it does and you understand. As you leave your doctor's office, she lets you know that the wait time for the results is approximately 3 days. So, you go home and attempt to wait patiently. Questions from Part 1 are: "How would you be feeling during this wait time? What thoughts would be going through your head? What questions, if any, would you have during this time?" The wait is over and you finally get a call from your doctor who tells you the results are in and asks you to come in to talk with her. You arrive at the office and sit down. She says, "I'm sorry I have bad news. The test result came back positive indicating that you have the HIV virus." The question from Part 2 is: "What is the probability that you really have the HIV virus given the fact that the test says you do?"
I invite you to answer the question before you proceed. Email me if you want to discuss your solution or have questions in your thinking process.
The specific mathematics topic that is primary in this problem is conditional statistics and this one problem, explored in depth, could easily allow you and your students to accomplish most of these Common Core Mathematics Standards: (plus other important topics as well)
It goes without saying that I'm not going to simply give you the answer here in this blog -- what would I be teaching you about thinking if I did that?? :-) Instead, I will offer to virtually meet with you via Skype if you'd like to discuss your thinking or share your solutions, just contact me to arrange it! Also, I will leave you with another educator's thoughts on thinking, statistics, and conditional probability (Note: "spoilers" are included in his presentation so don't watch before you've tackled the problem on your own first!):
Join me in teaching for thinking, striving for struggle, and encouraging persistence in our students!
The Solver Blog
Author: Dr. Diana S. Perdue