Proofs

In my educational career I have had to write proofs in three classes.  Given how many classes I've taken, three classes isn't very many.  And since I don't plan on writing very many proofs in the rest of my life, what's the point?

 When we first started discussing the values of proofs in class a few days ago, my first thought was, "they're more trouble than they're worth."  Of course I was thinking about my high school geometry class in which everyone complained about them and how this reaction hasn't changed since I've come to college.  But I didn't truly understand the purpose and implications until a couple days ago.

An important implication that we discussed in class was that proofs establish critical and logical thinking. They are essentially arguments that, if written well enough, will never fail. That is a very powerful skill, to write effectively enough that what you've said can never be contradicted. So although proofs seem like they're very specific to whatever math conjecture you're trying to prove and require some foundation work to learn applicable theorems, they really establish writing skills and logical thinking that can be applied to many areas of life. Thus they're very important for students to learn.

One of the biggest problems I have with proofs is that the purpose of the proof is lost in the details of memorizing theorems and definitions. I find this particularly relevant in high school geometry where it's expected that the student will memorize all of the relationships between the angles and such. As teachers, I think it's important that we practice the names of the relationships and theorems and make sure students understand what they mean before they attempt to write a proof. I believe that if we have them try to piece things together too soon, the task will seem too daunting and students will become too frustrated, which will lead to them writing proofs off as “too hard.”

On his blog, Daniel Schneider emphasizes the maze-like structure of a proof.  That is, given a beginning and an end, find the path between those points. I think this is a very important connection to establish because it takes the the bewildering nature of a proof and simplifies it to a well understood maze. So with this simple concept in mind, and with full understanding of applicable theorems, I think students can easily tackle the daunting task of writing a proof and be able to realize their importance.

Simplifying Trig Functions

As a tutor at the university Math Center I see all kinds of students working of various levels of math with various levels of understanding. The other day a student asked me a question about his trigonometry homework.  The problem asked the student to verify that the following equation was true

After I went through the steps the student asked in regard to the fourth step (the one I've highlighted above), "Are we really supposed to know that you can factor out a sine squared?"  I was a little surprised and immediately blurted out a resounding "Yes."  To me that step was seemed like a simple thing I learned in algebra when I was factoring equations to find roots, long before I ever took trigonometry.  But it made me realize that that connection between algebra and trig doesn't happen for all students.  There's a tendency so see math classes with different names as completely different animals, when in reality they are all interconnected and require knowledge from the previous courses.  With this in mind I showed the student that this trig function is similar to x^2+x^2y and I asked him if he would recognize that he could factor out an x^2, so it would look like x^2(1+y).  Luckily, this change of perspective illuminated the reasoning behind factoring out a sin^2(x) and he had no trouble understanding how we solved the trig problem.

Students in the tutoring center also have the common misunderstanding that the trig function is being multiplied by x (or theta, etc.).  This is an unfortunate misconception because it misses what it means to be a function.  Sine, cosine, tangent, and their inverses mean nothing without their inputs--that's why they're called trig functions.  

Overall I find that talking to students who don't understand the material right away is extremely helpful because it makes me realize what holes I need to fill as a future teacher.  To me math is like breathing, easy and unconscious, but to others it's laborious and they need as many details as possible, especially in seemingly foreign topic like trigonometry.  But by understanding these common misconceptions I feel better about helping my future students simplify trig functions.  And I believe that these could be simple things to teach in the classroom.  As a teacher, I just need to remember to relate new concepts back to old ones and convince the students that there are many similar rules that apply to different functions, even though it may not be obvious at first.

Problem Solving

In class a few days ago, somebody said, "Mathematicians aren't afraid of failure.  If one method doesn't work out, we just try another."  This assumingly inconsequential comment stuck with me and as I thought about it more and more, it became clear to me that this is one of the reasons that I am good at math; I try and try again.  In fact, it's a defining feature of people who are good at problem solving.  First you try to understand the problem and find connections to things you already understand.  Then you try to solve the original problem by using methods that have solved or helped the related concepts.  If it works, great, you're done!  Usually though, it doesn't work on the first try.  This results in returning to the beginning to make sure you fully understand the problem or can find different connections that might work.  Then we try these new connections (repeating the first connections is by definition insane, since they didn't work the first time). Hopefully the second time works, but if not, no worries, we just return to the beginning and try again.

This is exactly how I go about solving a math problem.  (I actually think solving a math problem is easier than a real life problem because mathematical properties do not change, so I can completely rule out any connection that did not work.)  I've found from tutoring college math students that there are two main pitfalls of problem solving: (1) Not establishing correct connections, and (2) Becoming frustrated and failing to go back to the beginning when one attempt has failed.  The first problem could be corrected by a variety of things like studying and attempting similar problems.  The second pitfall is a result of lack of confidence and perseverance.  A mathematician's confidence is not docked by failing to solve one problem because they have solved tons and tons of problems in their lifetime, so not understanding one is not bad.  They also know that not every problem that they've solved has been accomplished on the first try.  For a student who has not had a lot of experience solving math problems, failing to solve one problem may seem overwhelming.  They will probably attribute it to not knowing anything, when really they might have just missed that one important connection that would solve the problem.  When student's don't believe that they can solve a problem just by reevaluating their method, they quickly give up and do not attempt the problem again.

A particular student who I have worked with a few times in the university Math Center comes to mind whenever I think of perseverance.  Every time he has a new question, I approach him and he says, "I have this problem, but I don't know how to do it."  I try to probe a bit deeper to see if he actually does know, but won't admit it, but he's adamant that he has no idea how to start solving the problem.  One day we were working on log problems and he went through the usual routine of saying he doesn't know how to solve any of the problems and my probing led to a sequence of incorrect attempts at solving the problem.  So I took him through a few examples and pointed out where he can use regular algebra to get the log alone, and from there I helped him understand why we can switch from an equation with a log to an equation with an exponent.  So we did a few more problems and every time he started to say, "I don't know what to do next," I said, yes you do, look at the other problems, the methods are all related.  Finally after going through about four problems he started picking up the sequence of steps and recognizing when we can convert the log to an exponential function and vice versa.  By the end, he tried two slightly different problems in a row with no help from me and did them perfectly!  It really showed me that his need to understand the material (even if it was only to get a good grade) pushed him to persevere even though he initially did not believe in himself and truly did not know where to begin to solve a problem.  Once I made sure he understood the foundations of the problems and built his confidence by walking him through the problems while I slowly turned the decision making over to him, he truly had a transformation in his attitude and his skill with logs.

So there you have it, the big difference between mathematicians and non-mathematicians.  Mathematicians have a bigger repertoire of things they can attempt in order to solve a problem (reasonable, since they have more practice) and the knowledge that not everything can be solved on the first attempt.  In my opinion, these two things are not completely out of reach to non-mathematicians.  All you need is experience and confidence.  A good memory doesn't hurt either.

Formulas

Math is riddled with formulas that provide short cuts to solutions.  They are simplifications of long hours put in by great mathematicians to help us find a quick answer.  These mathematicians probably assume that we understand all of the "behind the scenes" work and intuition of the formulas, but that's rarely the case.  In my opinion, this is very unfortunate because by explaining how we get these formulas, we could deepen students' understanding and not only help them remember formulas, but know why and how they work.

I saw an example of misunderstanding in my classroom observations.  The algebra students were given two points, (3,4),(-5,2) and asked to find the slope.  Some students graphed the two points and found the slope by counting and dividing rise over run.  Others attempted to use the formula Y2 -Y1/X2-X1.  As I walked around the room, looking at students' work on their whiteboards, I noticed the numerators all seemed to be 4-4.  I was so confused as to how they were getting the second 4.  This point was clarified for me when the teacher asked one student to explain his work to the class.  The student started reading off the generic formula, but instead of saying Y2, as in the second Y, he said Y squared.  Ah, they were squaring 2 to get the second 4.  It was a simple problem of not reading the formula correctly.  However, this could have been avoided if the students' had understood intuitively what the slope formula was saying.  Clearly, in the graphs they understood rise over run, as the difference between the Y coordinates divided by the difference in the X coordinates, which is exactly what the slope formula says, just with some subscript to clarify the coordinates' order.  But the students missed the connection between that explanation and the creation of the slope formula.  If this point had been clarified, I believe that they would not have been confused about the subscript notation because they would have known Y squared has no use in rise over run.  

Quadratic Functions

When I received the "Deepening Quadratic Understanding" worksheet, I thought I already had a solid knowledge base of what it meant for a function to be quadratic.  Of course we all know it forms a parabola, but there are other important relationships that sometimes go unnoticed.

My group in class decided to find the significance between quadratics and their second differences in the output.  One of my peers suggested using a Excel spreadsheet to organize our findings, which was much faster than plugging in data points by hand and finding the differences between points (one great use of technology).  With lots of data quickly organized, it was easy to see a relationship between our functions and their second differences in the outputs.  We noticed that the second differences were always twice the leading coefficient.  The name "second differences" also made us think about second derivatives of the functions.  We found that the second derivative was also equivalent to the second differences and the value of 2a (when "a" is the value of the leading coefficient).  This makes sense because in a quadratic function, the degree is always 2.  So 2 is multiplied by the leading coefficient, a, as it is when taking the derivative.  Graphically, taking the first derivative will give us a linear equation, which is represented by a consistent change in slope, just like the first differences increase or decrease by a constant amount.  The second derivative names the slope of the line, and is represented graphically by a constant horizontal line and in a table, gives a constant second difference.

Here is a short example of what our excel sheet looked like,

As you can see, the value of the second difference was always 6, which is also equal to 2a because for this example a=3.

Working through these exercises is important for a few reasons.  First, it is important to work through exercises on your own time to expose yourself to important concepts that you might take for granted.  New discoveries about old concepts might seem insignificant if you already know how to work with the concepts, but these new things might be the link that students need in order to understand the concept fully for the first time.  Secondly, doing these exercises highlighted the importance of working with groups.  In groups, individuals can benefit from working with others who can contribute different ways of thinking (I, for one, would never have thought of organizing our data on an excel spreadsheet, but it was far more efficient than my method of doing it by hand.)  These exercises, enriching our ideas of quadratic functions, were not only helpful in understanding quadratic functions, but also understanding processes of thinking, and working towards enlightening solution.

Many students ask, "When are we going to use this?" whenever they are confronted with a difficult math problem, or whenever they feel like challenging the teacher.  Dan Meyer has created ways to involve students more in their math learning by connecting math problems to everyday items, such as a water tank, then asking students what they would like to know about the water tank.  This relates mathematical formula to an object that they are familiar with.  I believe that if teachers utilize this method in their classrooms, students might become more interested in and engaged in what they're learning.  Additionally we can expand this concept to accompanying math problems with student's drawings, instead of just relying on the numbers and formula.  If students make pictures to illustrate what the problem is asking it might help them realize that the problem is not just a math problem with numbers and symbols, but it connects to the wider world.  An example of this could involve the measurements of a picture frame.  When students draw out the frame, they might realize that it is simpler than they thought and it is relatable.  Yes, it still involves math, but now it is also relevant to students' everyday lives.