10 Things I Wish I Would Have Learned In Trigonometry Class

Okay, so I'm not going to write a list, sorry to disappoint. However, during my MTH 229 review I realized that I somehow missed a lot of really cool foundational ideas from trigonometry. It is a subject that I have always found fascinating, but also confusing. I remember asking questions, like "where does the sine graph come from," and getting really terrible answers. I remember being really overwhelmed with all of the identities and rules, but I think that is because I never really developed an understanding of why things work the way they do; I was just taught to memorize a few things and learned how to manipulate them.  So what I want to do here is spend a little time thinking about the things I wish I had a better understanding of early on, and perhaps some good ways to teach those things.

As I said, I memorized a few things, basically enough to help me solve the problems I was given. As an example of this I'm going to share something a little scary and embarrassing; until about a week ago I couldn't have properly explained why a circle is represented by the equation x ² + y ² = r ². I could use the equation to draw a graph and vice versa, but lacked the true understanding of why a circle was defined as such. The concept isn't hard, but I don't remember being taught, and I guess I was never aware that my knowledge was not understanding. One thing I do remember is using these 'special triangles,' the 30-60-90, and the 45-45-90 triangles pictured. For some reason I always remember these ratios and could use related trig functions to solve problems as necessary. We can see that they are related to the unit circle, which I vaguely remember being taught, but the triangles seem so much easier. The unit circle is so intimidating, especially to someone new to trigonometry, just look at that thing with all those square roots...yuck. I feel like the information is accessible if you know how to use it, but having to reconstruct or memorize it seems like a useless pain. I guess I tended toward the triangles because the made more sense to me, were easy to construct, and contained much of the same information as the unit circle.

So what I want to share now are a few ideas and resources that have helped me develop a much more thorough understanding of trigonometry over the last couple weeks. I'll begin with an article that discusses a way to introduce the topic of trigonometry without mentioning trigonometric functions. The teacher here assigns their students the task of drawing angles (every 10 degrees) within a circle of radius one, measuring the x and y values where the angle intersects the circle, and calculating the ratio of x/y. Students are encouraged to use shortcuts when they find them available, but also have the option of drawing each angle and calculating each value. Though it may seem a little tedious, I think this exercise would give students an opportunity to develop real understanding of what the trig functions are, and how they may be used. From here you would obviously introduce sine and cosine functions as well, you could compare ratios as the angle changes, and make connections amongst the functions. I recently used trig tables for the first time to do some basic calculations. I found them to be easy to use, and just looking at the tables gives you the opportunity to see a lot of patterns related to the functions.




Another resource I really like includes the diagram to the right; this particular diagram shows many of the trig identities built around the circle of radius one. The article that goes along with this visual actually explains it all really well using an analogy of a dome, a wall, and the ceiling. It does this in a way that allows us to construct, as opposed to just memorize, the diagrams and functions. The whole idea here is to build the understanding by making connections to the way these functions work and interact with each other. For example, it goes on to explain that Sine and Cosine are just percentages of the hypotenuse, which I think is a great strategy. It also includes a link to an interactive graph with a unit circle, graphs of the six common trig functions, as well as numerical representations of each of the functions. It is a really cool tool for discovery and understanding.



Finally, I want to talk about this interactive unit circle for sine and cosine. When I first played with it I though, oh yeah, this is pretty cool. Then I went back to it again and tried to really take in all the information that was being displayed; I was amazed. There is something really awesome about these interactive tools that allow you to see way more information than just plotting points on a graph; the ability to simultaneously see the circle, graph, and triangle created through all possible ranges is unbelievably helpful in building understanding. This is the graphic, that now famously, gave me understanding of the equation of a circle; I was just trying to figure out everything that was going on and it hit me. The screenshot below shows just some of the information available, but gives you an idea of the deep connections that can be made from the material. I think students could definitely benefit by doing some investigative work with this and making connections between the circle and triangles built within; I think they could either be given questions to answer, or even come up with questions of their own based on correlations they see. If nothing else, this is a great way to develop understanding of the sine and cosine graphs (another thing I historically struggled with). Again, as someone who has worked with the information in this graph quite a bit, spending a little time exploring helped deepen my understanding and build new connections between representations. Of all the learning tools and techniques I have discussed thus far, this is one that I will definitely use in teaching trigonometry.

Building an Effective Interactive Math Activity

Throughout the semester we have used technology like Desmos and Geogebra to facilitate learning of different topics. Though different methods are used, it usually involves some correlation between equations and graphs, which I have found helpful in developing understanding. Specifically, sliders are something new to me; I have found them to be very helpful as I can manipulate different terms in an equation and immediately see how it affects the graphical representation. From there I can evaluate trends and build understating based on multiple sources, equation and graph.

I decided that it would fun and practical to learn how to make an activity; so, with students in mind, I built one in Desmos. One of my favorite activities up to this point has been the Marbleslide activities that we have done using a couple different types of equations, so I decided to make one for exponentials and logarithmics. I wanted to include both types with the idea that it would help students make connections between the two related forms. 

When developing this activity I realized that a lot more planning has to go into it than I would have originally thought. The first thing I had to think about was, "what do I want students to get out of this?" I decide that I wanted to design an activity that would make students think about how the different terms in exponential and logarithmic equations relate to a graph, and how those graphs relate to each other. As such, I started with two activities that allowed the use of sliders to change the graph. The first challenge was uses an exponential equation and the second uses a logarithmic, though the ball drop and stars were in the same location. I wanted them to see that, even though the functions are inverses you can still make the graphs look very close to, or identical to each other. Moving forward I wanted to make sure that the game offered an opportunity to use a variety of different graphical shapes, and in a few cases I actually stipulated that the student must use one of each graph type (exponential and logarithmic) in collecting all of the stars. I also tried vary the difficulty of the tasks, creating some that were more simple, and one that I considered quite challenging.

This was a fun learning experience for me and something that I will definitely use in the future with my students.

Here is my finished product:

https://student.desmos.com/?prepopulateCode=cw6gs

Thinking About What Makes a Good Teacher

MTH 229, "Mathematical Activities for Secondary Teachers," has definitely got me thinking about activities, but perhaps more importantly how to prepare and teach the activities. The material is great, and every day I find myself saying "I have to remember that." I appreciate all the different ways we are learning to do things and the connections we are making between mathematical concepts, but what I really want to make sure I remember is the strategies and the ideas behind the lessons. We have not only been talking about math, but what strategies work well in teaching math, what keeps student's engaged, and the basic mentality behind planning a lesson. So what I want to do here is start gathering those ideas in one place so that I can think about how they all work together, reflect on what I was thinking when I underlined that note and put '!!!' next to it, and have a more permanent place to reference my thoughts. 

Something we have discussed many times, and I want to be sure never to forget, is to always keep the goal of the lesson in mind when planning. And hopefully that is not just something like, 'learn polynomial division.' I want to make sure that they understand the material, but maybe more importantly how it can be useful to them. Not only in their daily lives or careers but how it relates to other mathematical topics we have discussed as well. We need to plan those connections in lessons, not just mention it and hope that a few kids pick it up. In other words, I want to make sure that I am planning ahead an appropriate lesson, to teach specific objectives, that promotes overall understanding. In addition, I think it is important to think about how we are going to measure the success of the lesson. Have the students gained what we had hoped they would from it? 

I suppose this seems like a good point to transition to the idea of homework. I can see it as a way to assess where the students are and if they have understood a lesson. It can also serve as extra practice with the material. However, we must consider individual student needs and the fact that certain students have life situations that don't permit time outside of school for extra work. Something that stuck with me from our class discussion on homework is the question, 'is it supporting the students that need it most?' Given all that, if there is homework, the question of whether or not to grade it, and how to grade it comes to mind as well. My first inclination is to say that I would not want to attach a grade to homework. 

Grades in general offer even further debate. It seems to be a growing trend, that I agree with, that grades should reflect an overall understanding of the material, not necessarily how well you do on a single test. It wasn't until late in my college career that I had a math professor who allowed us to re-assess on learning objectives that we had not yet shown understanding for. I will never forget how brilliant I thought that was. Not only does it reduce stress for students in so many ways, but it hits much closer to the point of assessments. If by the end of the class, you can show thorough understanding of the material, your grade should reflect that. A grade should not suffer because you don't test well, were sick, got buried in other classes for a couple weeks, or just life.

Another thing I have been thinking about as a result of this class is group work and whole class discussion. I think that group work can be greatly beneficial in different ways. Working on your own and then sharing with a group gives you time to think on your own before collaborating and seeing other ideas. Working with a group on a white board (VNPS) gives everyone the opportunity to participate in real time and see other thought processes immediately. I think that group work can create a safe space for students to share ideas and build relationships. Whole class work is a slightly different dynamic. This provides opportunities for small groups to present to each other, methods of standing with thumbs up (down, side, etc..), or clickers to poll the group for answers. I think that small groups presenting to the larger group helps create safe space and community, while thumbs and clickers provide anonymity and offer the instructor a good baseline of class understanding. I think we have to consider the right method of group work for each situation, and remember that perceived randomization of groups is important for students.

Finally, I want to talk a little bit about methods of discovery and ways to make things more fun. I think a discovery approach to mathematics can be really fun and engaging. By that I mean lessons where students are guided to find the answers for themselves but never specifically told how to do the problems. I experienced this a little bit in class just the other day with the Rational Function Puzzle. I slowly pieced this back together as we discussed it in class and other students brought things up. It is always really rewarding when you feel like you are figuring math out on your own without someone telling you how to do it, and I think this generates better understanding as well. Technological tools are great for this as well. I had worked with geogebra some in the past and it is cool, but Desmos is my new favorite. Not only is it easy to use and intuitive, but the sliders make understanding what happens when you change the value of a variable so much easier to see for me. Additionally, the games and programs people have made for both geogebra and Desmos are fun and functional learning tools ( Quadratics game, marbleslide, Rational Function Puzzle).

I realize that I basically brought up a bunch of ideas but didn't really settle on anything; at this point I am unable to say, 'this is how I am going to run my classroom.' I recognize that I have a lot more questions than answers, but I think that's okay. I don't know what the lesson plans will look like, or how I am going to assess the success or failure of a lesson plan. It is a little intimidating and sometimes overwhelming to think about having my own classroom and being somewhat responsible for at least the mathematical ideas of these kids. But on the other hand, I think it's good that I'm a little scared, and I know it is only because I care. I just want to make sure that I put as much effort into learning how to be a good teacher as I do into learning the math. Spending time thinking about the things listed here is a good start to being where I want to be. I am going to make mistakes, but I am going to keep trying to figure out the best way to solve the problem, it's what mathematicians do. 

Mathematical Autobiography

The idea of a writing a blog is not only completely foreign to me, but also invokes a little anxiety. As such, I suppose the best way to get started is to share a little bit about myself and how I ended up where I am. 

My earliest memory of a math class is memorizing multiplication tables in third grade. I remember not wanting to just memorize the tables, but to try make sense of them. Specifically, I remember noticing patterns with multiplying by nine. Something like, 'oh, just multiply by ten and then subtract, that's way easier.' I thought that was the coolest thing. A few years on down the road I remember doing algebra and geometry and always enjoyed solving a tough problem and finally getting the right answer. Geometry was always one of my favorites, and still is; I like being able to see what I'm working with, it seems more tangible perhaps. I was never the really smart kid who could just look at things and get it. I had to study, but when I did I always enjoyed math and it came easier to me that it did for others. I think this actually help me explain things to others. 

Unfortunately, I severely underachieved in high school. Though with little effort I could have taken all the math classes my school offered and surely done well, I only made it through trigonometry... barely. There are a number of reasons for my lack of effort there, but that's a different story all together. I guess the point is, math is something I always liked but high school was almost the end of my mathematics career. 

Fast forward a few years down the road and I was in a job that I didn't hate, but most definitely didn't love. I could have made a decent living, but the one thing I knew for sure was that I didn't want to get stuck there for the rest of my life. To me, that meant I needed an education. I quit my job, moved to Grand Rapids, and started college fresh at the age of 29. Wow, was that a change of pace. 

Back to math. I started out at Grand Rapids Community College and had to get all of my basics out of the way. That means starting with college algebra, I had forgotten everything in the ten years I was out of school. I was pretty sure that I wanted to be in the sciences somewhere, probably physics, maybe engineering. Teaching had kind of always been in the back of my mind, by I wasn't set on it. Through all my work experience I had always ended up in training roles, and I just have a knack for helping people learn new things. 

As I made my way through the classes at GRCC I rediscovered what had drawn me to mathematics as a child. People look at me like I'm crazy when I say it, but math is fun. I love solving puzzles and figuring out how and why things work. That's math. The 'refresher' courses were a breeze for me, but I really did need to see all that again. I finally got to calculus, new material for me, and it blew my mind. I thought calculus was pretty much the coolest thing ever. It was interesting because I was using this tool that I had such a limited understanding of, to gain a deeper understanding of things that I had been using for ever, but had taken for granted. I'm thinking of how we derived a bunch of formulas for area and volume. Later on I got a deeper understanding of how calculus worked and my mind was blown again.

It was somewhere in that first year of calculus that two things happened which would both have a big impact on the direction I chose to take in education. First, I decided that math was too cool all by itself and I didn't feel the need to focus on a different place to use it, like physics. Though applied mathematics is great, I just wanted to focus on the math itself. Second, I started tutoring math at GRCC. I remember someone came to our class to put the word out about tutoring and kind of recruit people to do it. I thought it sounded like something I would be good at, and fun. I was hired in the tutoring lab part-time and loved it. 

I could probably write a whole paper just on my tutoring experiences. It never felt like work and the time always flew by. Helping empower people to solve problem and seeing them have those 'ah-ha' moments when they realize they figured something out are priceless. I'm sure there were times when it was frustrating, but honestly all I remember is feeling really satisfied every day I left work there. I felt like I was doing something important, and that made me feel good. And not only was I able to help others, but my knowledge in the material was growing every day as I tried to find new ways to explain things that someone different would understand. It was through that first tutoring experience that I knew I wanted to be in the education field in some capacity. 

After I finished up at GRCC and transferred to Grand Valley I didn't have the time to tutor anymore, but somewhere in there I decided that I wanted to get my masters degree in mathematics and teach at a two year college like GRCC. I think there was a lot that went into that decision, but mostly I just had a great experience there and saw that I could make a difference in a role like that. I finished up my B.S. in Mathematics at GVSU and overall really enjoyed the process. Advanced Calculus made me feel a little dumb, but I hear that's common. 

At this point something else unexpected happened. I started doing some more tutoring work with GRCC and Grand Rapids Public Schools, which was all great, and looking into masters programs. Unfortunately, the masters programs just didn't feel right. It wasn't that I didn't think that I could do the work, it was that the work just didn't inspire the same passion as the undergraduate math did. I applied for a few jobs with student services at GRCC but nothing came of that, and once again I felt lost. I knew I wanted to be in education, and I knew I liked math, I just didn't know what to do with it. Teaching high school seems like an obvious choice but for some reason I had always had an aversion to that. I think I had this idea in my head that I wanted to teach 'the good stuff,' to people who cared about it. Well, they would either like math in the same way I did, or would be diligent enough to at least try because they needed it for their degree. 

I figured I should give the high school thing a fair shake and started doing some substitute teaching. Though it can be a little hard to pick your preferred assignments I was able to get into a few math classes, and even met a teacher that started calling me when she needed someone to fill in. Through this process I changed my mind about high school and had some realizations about what I wanted to accomplish. I realized that the education piece is just as important to me as the math. It isn't enough to be good at math and like it, I want to know how to best help others learn it. In high school I have to opportunity to help students learn to appreciate math, or at least not fear it, before the get to college. Or if they don't continue education, at least teach them enough to have a basic understanding of the math that surrounds us every day. I know this a career path that will challenge and stimulate me, and at the end of the day leave me feeling like I am doing something worth while. Plus, in high school I can coach a sport and have summers off, so there's that. 

So here I am, working on a 'teachable minor,' headed toward a teaching certificate and a Masters in Education. I am excited to teach math, and to gain deeper understanding in order to teach it well.