First of all, I must say I find this book very difficult to read. I don't really understand what it is getting at half the time and its examples seem far off to me... Therefore this blog contains my thoughts as I am reading this (my very honest thoughts :) )
Seeing What's in Front of You: This section made complete sense to me. Some times it is very difficult to know what's given, to think outside the box and to also not take the given for granted. It took me about a minute to see the cube in figure 2 on page 34, but once I did it was so obvious I don't know why I didn't realize it before! I also remember getting a problem like that discussed directly below that figure (1 + 2+ ... + 100 = ?) and thinking my only choice was to add them all up. After discovering a different way to look at that problem, I find myself using similar strategies with adding only a few numbers or even multiplying huge numbers.
Listing Attributes: When you are presented with an open ended problem and the givens are not given to you, it is very important to list attributes, to "put it all on the table" and take a look at what you have. It is important as the book says to not rule something out as useless until you can really know for sure. Then I kind of got lost. I understand that the authors were trying to take away some of the givens (change the attributes), but I don't really know what for. The problem I found interesting was trying to find a pattern/mathematical formula to determine the largest circular board we could cut from a square, but I know the majority of my students would not!
I then got thoroughly confused with the tree example. Again, I get that the authors are manipulating "givens" but still don't understand why. I don't understand the connection between the man and his diet pills? Is this so he doesn't eat the trees? Or so he can sit there forever without getting hungry? I just don't get it...
What have We Done?: This section cleared up a bit for me, but is still not explaining why we are doing this. Is it just to create more questions? What could I do with this in my classroom? From this section I got that there are three steps to this process (of what I don't know): List attributes, ask "what if that were not so?" and pose questions about our new assumptions. Is this just giving me ideas about how to have my students explore all possibilities? I can't imagine something like this going over well in one of my classrooms. So many of my students struggle with the givens! I can't imagine telling them to challenge those. I know what further explanation can help develop a stronger base of knowledge, but I find that next to impossible in our school systems. Are these problems we sit and create as educators or have our students create with us? Again, I don't really see this working realistically with students who are not interested in math, are lacking basic skills and who struggle to get by.
On page 48, the "What-If-not" strategy purpose becomes a little clearer to me. The authors discuss things like non-Euclidean geometry (gross) and the space time continuum and how they were examples of someone saying, "What if things are not the way we have always believed them to be? What would happen?" I am still struggling however to see how this can relate to my classroom. How can we cover curriculum by challenging all the mathematical norms? I feel that in proposing these ideas and then formulating new questions is getting so far away from the original problem. Is that the goal here? Thinking about this starts to freak me out. I barely have enough time to squeeze all the required curriculum in let alone developing a new set of questions and problems kind of related to what we start with.
I do really like the analyzing of the question "For what values is a^2 + b^2 <>
Cycling Questions: This is a great resource. Sometimes combining ideas can lead to even better ones. I completed a lot of the table on page 63 and tried to come up with some relationship. The fact that sometimes the number goes up and other times it doesn't is stumping me. I know something has to be effected by the numbers we get out when we square an integer, I just don't know how that would be related into a rule.
My Summary: I could see these strategies really useful from a teaching point of view. In creating curriculum (and hopefully a textbook some day) I am always looking for ways to challenge my students, to push their thinking and to analyze how much they truly understand. Thinking about What-If-Not and posing questions based on our new assumptions can get students thinking about more than the problem right in front of them and can develop their understanding of that topic. I do not think that this process is one that involves students, but is instead used by teachers to push and challenge their students.
I think while reading this chapter i shared several of your frustrations in thinking "how could the be useful and practical in teaching?" I think in the end i took comfort in that not every time i use 'what if not' strategy do I have to spend 40 minutes writing out questions... but rather it could be applied in some mini versions too.
ReplyDeleteI thought your summary point on this strategy being useful in creating curriculum was good too.
Hi Megan,
ReplyDeleteSorry for your frustration with the book. The point is, almost always, to think about mathematics in a different way and to learn something new by doing so. About the man, the ultimate diet means that he is functioning as a POINT, no depth, no breadth. For a real person, the answers to the questions would be incorrect, for we see with two eyes, thus in three dimensions, so we don't see along a single straight line. The important thing here is that, in thinking about a lattice made of up integral points, the lattice is so SPARSE in Cartesian space that the chance of seeing a line that goes up from one of those points, given a random line of sight is ZERO. This is kind of shocking. It has to do with the density of irrational points in space.
This is not a book about pedagogy per se - the purpose of the book is not to teach you specifics of how to teach with problems, but to provide you with new ways to approach problems and to have your students approach mathematics problems. This course is about teaching AND LEARNING through problem solving, and part of the reason for reading this book and for doing problem sets is for you to learn in (possibly) new ways through problem solving. Brown and Walter DO teach courses based on this book, but their courses would not by typical in middle or high school.
Hope this helps explain why we are using this book.
raven
More about why: many teachers have found that it is the least motivated students who are the most motivated by doing mathematics through problem solving. That doesn't mean going to the lengths that Brown and Walter go to, but it does mean posing interesting problems and letting students apply their intelligence and creativity to the problems. Doing interesting problems can motivate students to learn what they need to be able to work on the problems.
ReplyDeleteThe Brown and Walter book is what I am doing as a teacher to engage YOU in problem solving. The problems are hard enough to be challenging to most of the students in the class, and their way of thinking about problems is new to almost everyone in the class. Again, the point is not for you to do these exact strategies or use these exact problems in your class, but rather to experience yourself some different ways to think about and do mathematics. Then, you will need to decide what parts might apply to your class, how to apply them etc. The lesson plan part of the course is my effort to help you try some new things.
As I said in another posting (on my blog), I know that some of this is old hat for some students in the class who already teach with problems; and some of it is mathematics that is really hard, especially for some of our elementary teachers. But overall, I think the book hits some extremely important points for mathematics teachers using mathematical problem solving as a part of their teaching.