B/W Chapter 4- What if Not?

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.

Problem #7: Arithmagons

The idea of a secret number on the points and the sum of those numbers on the side stumped me for a second. For some reason I kept thinking of the sum as the length of the sides and the secret number as some sort of angle. I was trying to incorporate geometry where I only needed a little algebra. Once I wrapped my mind around what I was doing, I started making equations and manipulating them. I eventually could prove that the magic numbers were 1, 10 and 17.

I generalized this with more letters and got the secret numbers (A, B & C) in terms of the sides (x, y & z). My brain works much better with manipulation, variables and algebra. I could make these statements, prove it with some numbers and I would be satisfied with this.

My boyfriend actually walked by and took a look at it, when I explained it with numbers he was so confused. This led me to my written work, justifying the equations I had previously come up with.

I was even more satisfied with this answer. Again, my brain was ready to just accept the equations, simply because I believed in my own mathematical ability. Once I actually explained it, the equations were completely correct and I could justify them.

Next I tried to move on to one of the more complicated shapes given in the problem set. I made variables and equations and tried to go about manipulating like I did the first time.

I figured I would focus on one shape at a time. The first one I tried to tackle was the parallelogram shaped-thing on the top right. I couldn't do anything with the equations! Every time I subbed something in it ended in a roundabout where the magic number part would algebraically disappear. I plan on giving this some time and revisiting. Any one else have some thoughts on how to start?

Problem #6: Copper Plate Multiplication

First of all I was confused about the name of this method. I googled copper plates and found no relation between them and what we were doing in this problem and copper plate multiplication yielded no results.

I checked to make sure that the product of the two numbers given was in fact the answer you get from this method using a calculator. My intial response was to multiply these numbers like I was taught in elementary school. I realized in doing this I was breaking up the numbers found in this new method to follow through with the "carrying" process.

I stopped half way through and tried to figure out where the each of the numbers came from in the rombus shaped group. For some reason the middle number group was where I wanted to start. I could easily see that in that row the numbers were the products of each of the numbers in the problem that were on top of each other. I workd my way down from that row to find the pattern in the next 4 rows. They were all two digit products of the diagonals of the numbers.

After I recongnized some sort of a pattern, I tried recreating it in order

to draw more conclusions. I wrote the numbers out huge and tried to use different collors to show where I was getting the numbers from. I started with the biggest diagonal multiplying the bottom 6 by the top 5, then making my diagonals smaller and smaller until they were the numbers directly above each other. Once I got to that point, I switched the direction of the diagonals and ended with finally multiplying the bottom 9 by the top 7. I don't know how clear it will be, but it looks pretty :)

In creating this several questions/assumptions came to mind and I tried a few examples of my own to either prove or disprove them. My inital questions/assumptions are in green and my answers are written in black after I did a few examples:

Here are the examples I did:

Once I got used to this method, I kind of liked it. It took me awhile to be able to do it with out checking the orginial to make sure I was multiplying the right things at the right time. It was neat to see this method broken down and to even start thinking about the place values. It would also be interesting to look at the "standard" method and break that apart as well.

Polya: Part 1- In the Classroom

First of all, I must say I was not looking forward to reading this book. Just glancing through it, I was dreading how boring it was going to be. That being said, I really enjoyed this first part. I really feel that Polya has some great ideas and delivers it in a way that makes sense and isn't painful at all to read. I was actually a little bummed when I finished it!!

Through out this section, many things stood out to me: things that happen constantly in a math class that could be changed to help our students better solve "problems" and things that I never really thought of before. This blog is going to be a list of those things:

• Students should have a reasonable share of the work (pg. 1). I think often times as teachers we give students work, if it is too hard, we tend to completely dumb it down giving students not necessarily the answers, but pretty darn close. If the work is too easy to begin with, the students might complete it without help, but what's the point? they we not challenged and chances are had very little to think about.
• Purpose #4: Common Sense (pg. 3) Some times I feel like my students have none when it comes to math!! The suggestion of "Look at the unknown! And try to think of a familiar problem..." is so foreign to my students. They cannot take one problem and see connections between it and others and I think part of that is our fault as teachers (will revisit later in this post)
• Solving problems is a practical skill (pg. 4) To develop a skill takes practice and imitation. Our students learn to solve problems by doing many of them. I think a lot of times in math class we beat "kill and drill" type problems to death and every once and a while give them a story problem or application to work on just expecting them to know how to approach them. We need to condition our students and constantly give them practice in order to succeed.
• #6. Four Phases of Problem solving (pg. 5-6) Understand the problem, find connections, carry out our plan, review and discuss the solution. So important!! I am making this a list and hanging it up in my classroom. When we put a problem in front of our students and say solve this, most struggle to begin. Having a list of what to do can be so helpful to them.
• #7. Understanding the Problem (pg. 6-7) "It is foolish to answer a question you do not understand" yet another thing that will go up in my classroom this week! I know that I always try to get my students to answer questions like "What is the end goal?", "What does 'solve' mean?" or "How do we know we are done?", but I feel that they do not ever ask these questions of themselves. I feel that they get into robot-mode in math class: see a problem, attempt it with not goal or agenda in mind (basically regurgitate memorized information trying to copy what I just did).
• Good Ideas (pg. 9) "Mere remembering is not enough for a good idea, but we cannot have any good idea without recollecting some pertinent facts..." My students honestly think that math problems can only be solved by remembering steps about how to do each and every single one. They have a very hard time connecting bits and pieces to individual problems and I know I am guilty of perpetuating this. If students have a question about a problem, I tend to help them solve that problem and not really give them any tools to solve something similar.
• Forgetting "The Plan" (pg. 13) Like the last paragraph stated my students try to remember the plan for each specific problem and often times it is forgotten. Polya states that this tends to happen "if the student received his plan from outside, and accepted it on the authority of the teacher". I think this happens even to my students who perform well in classes. They are simply better at remembering the plan! We need to develop ways for students to find this plan on their own, and to make it theirs so that it can be applied to different problems and later on down the road.
• Accidental Questions (pg. 14) "...there is some danger that the answer to an incidental questions may become the main difficulty for the majority of the students". As teachers we need to be careful of the questions we ask. I have run into a situation where in trying to clarify or get my students thinking I have ended up thoroughly confusing my students.
• Good vs. Bad Questions (pg. 21) "The suggestions (we give our students) must be simple and natural because otherwise they cannot be unobtrusive". I think that 'obtrusive' questions are the norm in most math classes. We tend to led our students into answering a specific questions about a specific problem to get that specific answer. The students finish that problem, move on to the next and have the same difficulty with that one. Our questions should led our students into thinking through a problem on their own, guiding them into the right path.

B/W Chapter 3- Accepting

Example 2: Isosceles Triangle

Observations

1. Are all isosceles related in some way?
2. Why are these triangles so important?
3. Is there a way for the sides to be equal lengths and not the angles?
4. How is the thrid side related to the first two? Is this always the case?

I would have to agree with the authors about people familiar with Geometry only coming up with questions like 1 & 2. It has bee a very long time since I have taken geometry and I've never taught it, so I tended to just hink about the shape (not rerally the math). But for problems like x² + y² = z² I only approached it from a mathematical sense because I am so familiar with algebra.

Example 3: Geoboards

What would I do with it?

1. Make a fun design with different colored rubber bands
2. See how far I could stretch a rubber band around the nails
3. Determine how many shapes I can make with only using a certain number of pegs

Example 4: Pythagorean Triples

Questions, Patterns, Observations

1. There is a much bigger gap between corresponding x's and y's than y's and z's
2. x is mostly odd, y is mostly even z is all odd
3. How many triples are there if we say x is 1-100?

"How much is lost by searching for a less exact analysis or a less precise strategy?"

I really liked this question. I think in many math classrooms approximations and educated guesses are regarded as wrong. It is very important for students to look at a problem like 214 + 497 and realize their answer should be around 700 (and if they get something far away form that they need to go back and check it). So many times my students are so focused on getting the answer that they aren't checking to see if it makes sense in problem or if they could estimate the answer. Making guesses and predictions is a very important life strategy that math class could help foster if we change our focus.

B/W Chapter 2- Accepting v. Challenging

FIRST LOOK: x² + y² = z² “What are some answers?

Looking at this I think of the answers I gave to problem set 1 (lengths of the sides of a triangle, circle equation, numbers that work) and am still thinking about why x, y and z are used. Is it on purpose? Is it simply because x is the most commonly used variable and y and z go with it? Could it be for 3 dimentional space with an x, y and z-axis?

"Students and teachers do not usualy ask questions... rather, they are interested in making sure that their students understand and excecute what is expected of them" (pg. 14) This is math class summed up in one sentance. There are so many things that would be interesting or even fun to explore in a math class but we don't. We are focused on our students being able to add two digit numbers, not caring if they know why or how it works. Obviously time is a huge reason for this, but how much different would students view math if this was the case?? If it wasn't just a list of rules and procedures created to get "right answers"?

SECOND LOOK: x² + y² = z² “What are some questions?

I already thought of some questions above, but now I'm thinking more about what questions I could ask my students about an equation like this: What shape comes to mind when you see this equation? (I could argue for circle or right triangle) How could we solve for one of the variables? What is a story or issue that could be described by this equation? How many solutions are there? Can x, y and z be negatives?

A NEW PERSPECTIVE: I know I'm thinking like my students right now, but all of those examples took the equation and manipulated it. I thought we were just asking about this equation, not about any new ones we could could come up with... Observation #5: Accepting the given! Most of my students wouldn't think twice about accepting what was given to them and not strying away from that for fear of getting the wrong answer. "...the given is a starting point for investigations that modify it." (pg 18)

B/W Chapter 1- Introduction

The very first page discusses a topic that I have been thinking about a lot lately. The model given implies that students look at problems gven to them by textbooks or teachers and their only option is to solve them. At first I thought that the table (on page 1) was a very strange way to get this point across. The more I studied it, I came to really like it. It allows for a path where the student both presents a problem and solves it or where the authority can solve a problem a students has generated. Often times in math I think my students really think that we are pulling problems out of thin air. The are not seeing the connections we do and don't see the point of solving many of the problems they are given.

This goes right along with problem solving being viewed as a "spectator sport" (pg. 2). My students take no interest or pride in solving problems, they simply want the answer so they can be done with it. To many of my students, they are not trying to "know" or understand, only how to get the answer. On page 5, "right" answers are discussed. Especially in math class, a fear or getting the wrong answer hinders our students ability to think and reseason through problems they are given.

At the very end of this chapter, the authors discuss reasons for main topics to be reiterrated. Did anyone else notice that reason 6 came after reason 4 and reason 5 is the same as reason 1?? I did find it a little funny that they repeated the fact that it might take a second time or two reading through "novel" ideas before they sink in :)

Problem #5: Palindromes

A number like 12321 is called a palindrome because it reads the same backwards as forwards. A friend of mine claims that all palindromes with four digits are exactly divisible by 11. Are they?

To start this problem, I began with smaller palindromes. Obviously the digits 1-9 are read the same forwards and backwards, but as you got into two digit (11, 22, 33..., 99) and three digit (111, 121, 131, ... , 999) they all seemed to be multiples of 11.

To look at 4 digit palindromes, I wanted to make a list of them (or at least see how many there are of them). I started my list and realized that there was a pattern:

Since there are only 90 of them, it wouldn't be completely absurd to try all of them, but I knew there had to be a more efficient way :)

I decided to just look at the first set (from 1001 to 1991) and see if I could find some sort of pattern. I actually tested all 10 of these and they were all divisible by 11. In writing these out I noticed that they were smaller numbers flipped around: 10 gave me 10 01, 11 gave me 11 11, and so on. Glancing up at my table, that would be the case for all of the numbers 10- 99 repeated and flipped. In writing out the first set I also noticed that the difference between all these numbers was 110 (a number also divisible by 11!)

As you can see this led me to creating an equation (the relationship had to be linear). For set number 1 it was 1001 + 110x where x was any number 1-9. Since 1001 and 110 are both divisible by 11, any multiple of 110 added to 1001 must be as well. I wrote out what set #2's equation would have been and found a way to relate it to set number one. Since 2002 is really 2*1001, I created a more generalized argument. Set #n could be figured out using 1001*n + 110*x where n and x are both any number between 1-9. Using my same logic that any multiple of a number divisible by 11 is allso divisible by 11 and the sum of any two numbers divisible by 11 is also divisble, we can conclude that this proof works.

As I was typing this I just thought of another way!! If we factor 1001*n + 110*x, we can pull out an 11 to get 11(91*n + 10*x) which obviously makes it divisible by 11!

Problem #4: Weights

Given a two-pan-balance and weights of 1, 3, 9, 27, 81, 243, and 729 grams, what weights can you measure? If you want to upgrade your set of weights, what is the next weight you might buy?

After rewriting the different

weights I had it was easy to see that they were all powers of 3 (0-6) so I answered the second question very quickly. The next weight in the pattern woul

d have to be 3^7 which is 2187 grams.

I then went back to the first question. From first glance I was only thinking about addition- putting a weight or set of weights on one side and using an unknown weight to balance that out. I was also thinking about only using two weights for some reason. So I wrote out all the possible combinations of two weights on one side. I came up with 21 different possiblities and also figured out the smallest

I could measure was 1 gram and the largest unknown would be 1093 grams.

All this talk about combinations started me thinking about put

ting more than two weights on the one side. So using the combination formula, I figured out the number possibilities with 1 to 7 weights on the scale. Using only one side of the scale, there are 127 different unknown weights I could measure. SInce I didn't take the time to figure out what each of these possiblities were, this didn't really help me that much :/

I stepped away from the problem for a bit and returned with a new take on this problem.

For some reason when I first sta

rted this problem, I only thought about putting the known weights on one side and trying to balance them out with an unknown (addition). But my new idea led me to subtraction (weights on both sides!). To come to this conclusion, I wrote out all the weights I knew up to 36 and tried thinking about ways I could "fill in the blanks". I drew myself some pictures to help out.

Since this wasn't very orgainized in my head I decided to make a table. I listed numbers 1-20 and came up with combinations of how to measure them, some with addition, some with subtraction. There was no pattern I could see in how I could come up with the numbers and also no pattern on when they were added or subtracted.

From this table I want to say that there is a way to measure all the possible number combinations (from 1 gram to 1093 grams), but I know I can't just do that without trying all of those numbers.

I know this isn't a conclusive proof, I'm just a bit at a loss on where to go next... I'll be sure to revisit this later this week.

In thinking about this a little later, I thought about maybe proving that this works for all prime numbers 1-1093. Since all other numbers are divisible by something else, I figured I could start to make a case around that. In looking up how many numbers are prime, I realized that was a horrible idea (there's around 100). So I don't really know where else to go with this. I know I can't just say because I've proved it for the first 20 numbers, I must work for the rest, but I am completely stumped...

Problem #3: Rectangular Parallelpiped

Problem 3. Parallelepiped

Find the diagonal of a rectangular parallelepiped of which the length, the width, and the height are known.

I got excited when I read this problem, I felt that it was one I could really tackle. I drew myself a picture of a parallelepiped (once I looked it up, I honestly I no idea what it was) and drew the diagonal. I realized that I could make a right triangle using the parallelepiped faces and sides where the diagonal is the hypotenuse. Since one of the sides of the triangle is the top face of the shape, I redrew that and found the diagonal of the rectangle.

The other side of the main triangle is the height, so plugging that into the Pythagorean Theorem, I got that the diagonal is the square root of the sum of the width squared, height squared and length squared.