Looking Back and Moving Forward

I probably couldn’t even list, let alone discuss, all the things that I have learned this semester. I have gotten so many opportunities to push myself, to fail, to succeed, and to grow as a teacher. I’ve had some of the best times as an educator and some of the worst and I walked out of this semester feeling more passionate and excited about teaching than ever. These are some of the most important things I’ve learned about myself as a teacher.

This semester, I taught two sections of Honors Geometry and two sections of Honors Algebra II. As I moved forward, I found myself teaching a different lesson to my second section than my first. I chalked it up to having different students. But, that wasn’t the case. I realized that I was evaluating myself in the first section and noting things that I should change for the next class. Before this semester, I had not really known the benefits of reflection. The next time I would be teaching a class would be next semester and I would have time to improve it. I realized that the best time to improve lessons and to make changes is right after you teach it. I made little changes, like the order of activities or the way I worded something, in order to increase the opportunities for student learning. That’s a piece of advice that I will be sharing with others and taking with me in the future. Reflect, change, improve, alter, and better your lessons each day. Ensure that every time you teach it, it gets better. We shouldn’t move forward as teachers teaching the same lessons day in, day out, year after year. We should be constantly improving. The best way to do this is reflection.

Teaching an honors class was hard. There’s no way to hide that. It is. The pace is faster, the material more difficult, and the expectations more steep. The freshman students I had were doing math that I, and most people, didn’t see until junior year or later. I had these assumptions about freshman students and what they could or couldn’t handle in terms of mathematical challenge. I was wrong. I taught lessons that, by the time I reached the middle, I realized I needed a bigger challenge for them. These students thrive through challenging tasks and I wasn’t doing them justice. I wasn’t challenging them. Over the course of the semester, I improved on this. I learned how to find where my students were at with material and what types of challenges to throw at them. By the end of the semester, every time that I wrote a lesson, I went back and made it a little more difficult. The benefits of this were amazing. My students were coming up with solutions, explanations, and mathematical conversations that blew me away. As freshman they were thinking more critically than I had until college. From this, I learned an important lesson. We’re teaching for the students, so we should be creating lessons for the students. I should be making lessons that challenge my students every single day, whether they are low-achieving students or honors students. Challenges are what create rich mathematical learning.

One of the best strategies I have used this semester is implementing technological demonstrations. During a unit on angle theorems for circles (chord-chord, inscribed, etc.), I used Geogebra Tube to illustrate to students the concepts. These technology applets allow me to spend less time constructing examples for students, give students the opportunity to discover ideas,  provide the opportunity for increased engagement, and give more accurate depictions of concepts. Before this semester, I was tired of hearing about implementing technology. I felt like it was just people talking just to talk, who didn’t have any evidence that implementing technology would actually benefit student learning. In my own classroom, I was proved so wrong. I felt like my students gained a deep understanding of the concepts and some even wanted to continue playing with the programs at home! New teachers, especially student teachers, should use technology in their classrooms. Try certain programs to see if they work in your classroom.

Finally, I have found that think-pair-share works great for me. I really want students to be able to convey mathematical ideas to one another, both through writing and discussion. When there were conversations that I wanted to have with students, I always asked them to do a continuous write for 2-3 minutes to get their ideas solidified. Then, they would get into table groups and each share. Finally, we would discuss the ideas as a class. I love this strategy for many reasons. First, students are able to write about math. Students don’t get to do that very often, and they should. Second, the solidifying of ideas before class discussion really helps students to have something to say. At times when I have just tried to start a discussion, very few students would contribute. After a think-pair-share, I usually had to stop conversation or it would last all class. Finally, I think that it allows for group collaboration. Once students shared their ideas, many groups would discuss different ideas and build on them. The think-pair-share just elicits so many characteristics that I want my students to have.

Overall, I can’t imagine trading this experience for the world. I’ve learned countless lessons, met some amazing mentors, made connections with students that will last forever, and found myself as a teacher. I love this career and I can’t wait to move forward.


Road Trip Across the USA

Last week, my students worked on mastering the distance and midpoint formulas using a really amazing lesson, linked here:


My goal in this lesson was for students to practice using the distance and midpoint formulas while staying engaged, working in collaborative groups, and applying math to real-life scenarios. Students worked in groups of four to plan a trip across the United States. They were given a map with a coordinate plane and were required to travel through ten states over a 5 day period. Students were to find the distance traveled each day using the formula. Then, they had to stop each day exactly halfway between their start and endpoints for gas. They were to find this pit stop point using the midpoint formula.

Students in both of my classes were incredibly engaged throughout the lesson. They were discussing mathematics and were actually excited to do the project. Students were on task, focused, explaining mathematics to each other, working together to solve problems, and getting a lot of practice with the concepts. My favorite lessons involve controlled chaos and this one had a lot of it. As it became one of my favorite lessons to date, I wanted to know what my kids thought. Were they as excited about this lesson as I was? Here’s a sample of their feedback:

“I really enjoyed this project because it wasn’t just boring problems.”

“I prefer packets over projects like this because I can work independently and I can solve things problem by problem without having to pull many things together.”

“I enjoy the bookwork because everyone gets the same answer, but I really like a break once in awhile with a fun project that applies the math with the real world”

“I enjoy doing bookwork because it is very orderly. But I enjoy projects because they allow me to do math differently and be away from the norm. I like both equally.”

“I like this activity because it puts math into real life and it is easier for me and less boring.”

“Well I like the project itself because it’s a more interesting way to practice the concepts. I really don’t like how we had to work in groups.”

“I liked this better than packet work because there were far less problems, but I feel I learned the same info.”

“I enjoyed doing the map because it was a lot more hands on and the book is not. I feel like I learned more and in the book I would not understand things as well.”

“I love working on projects like this. I understand what we are doing better and I love to work in groups.”

“I prefer doing activities like these because they help me use real things which help me understand the info better.”

“I find doing an activity like this was fun. But I also learn well by writing notes and trying examples in class. So, I think I would like doing both (like doing these activities every now and then).”

My favorite response is the one that said that they had far less problems to do, but learned the same info. Isn’t that the point? Students in math are so used to getting drowned by repetitious practice problems. Instead, we should be giving students problems like these. Ones that give students an opportunity to learn the material, practice the material, and master the material without feeling like they just did 100 problems. My CT even mentioned to me that I should ensure that students had actually learned the formulas, as opposed to just skating through the activity. The next day, part of my warm-up asked them to write down the formulas without looking in their notes. Almost universally, every single student had the correct formulas written down and knew how to use them.

I couldn’t imagine how much my students’ responses would impact my teaching. So many teachers just do their jobs without realizing that we really work for our students. We are there simply to ensure that they succeed. How are we supposed to do that without asking them what they need and what they prefer? I could spend every single day of an entire school year doing activities like this USA activity and be perfectly happy. Some of my students probably could too. But not all of them. We need to be listening to students and seeking out their input. In this case, I was thankful that my students loved this lesson as much as I did.


#MichEd in the Classroom

#MichEd in the Classroom

A couple weeks ago, I had the opportunity to work with Brad Wilson and #MichEd, an effort to connect Michigan teachers and students and highlight the amazing things going on around Michigan schools. If you want to know more about them, head here: http://miched.net/. You can also check out more pictures from his visit by clicking on the picture above.

This experience has, by far, been one of the best I’ve had in the world of education. Brad visited the classroom that I am currently teacher assisting in to highlight what new teachers are bringing into the classroom. The lesson that I taught was a hands-on activity in which students were to create their own hubcaps, focusing on maintaining rotational symmetry. This served as an introduction to the concept, forcing students to use manipulatives in order to understand the ideas behind rotational symmetry. Students used paper plates, angle rulers, and wooden shapes to construct artistic hubcaps that were rotationally symmetric. They learned about what it means to have rotational symmetry, what an angle of rotation was, and the order of rotational symmetry. My students gained so much from this activity. They were having fun, talking about mathematical concepts, and working collaboratively to achieve their goal. (Plus, we got some pretty awesome new decorations for the classroom out of it too.)

The main reason Brad was there, both in my classroom and later in a discussion with other Grand Valley teacher assistants, was to understand what new ideas future teachers were bringing to the table and how we were implementing them in the classroom. One of the main ideas in the lesson that I taught was working with manipulatives. Were they necessary in building an understanding for my students? Why did I choose to use them? I truly believe that students need to construct their own understanding. Too much of mathematics education of the past has focused on teachers spewing out information and hoping students would absorb it. This isn’t how people learn math. That’s not how people learn anything. Instead, we need to be guides in students’ education. Throughout my lesson, students were constructing their own meaning on what rotational symmetry meant. They had guidelines to follow, but were forced to make decisions on how to place shapes in order to maintain the symmetry they desired. Some students split their hubcap into 8 slices and made each slice rotationally symmetric. Some students made 8 slices but made every other slice rotationally symmetric. And they could argue why it worked. The student who made all 8 slices identical only had to rotate their hubcap to the next slice to reach a point of symmetry. The student who skipped slices had to rotate two slices. But, still, they both had rotational symmetry. It’s these ideas that students develop THEMSELVES that are important. I could tell them what rotational symmetry was. I could show them examples. But, that doesn’t teach them how to DO mathematics. They aren’t DOING anything.

During our discussion with Brad, all of us GVSU students discussed how we have seen math change. We’ve been lucky enough to have the opportunity to be a part of the movement toward better math teaching, and better teaching in general. When we were in school, we were part of the “old” ways. We were set there to absorb information that our teachers spit out. We were sponges. Which also meant that we could wring ourselves out after a test and forget everything we learned. Until college. That’s when it changed for all of us. We had professors and classes that changed almost everything we knew about math education, what DOING math really was. We were using manipulatives for the first time since elementary school. We were constructing pictorial representations. We were explaining our thinking. FOR THE FIRST TIME. As 21-year-old college mathematics majors, we were doing math for the first time. Why? Why didn’t our teachers teach us in the way we were learning now? So much has changed since we learned math that we had to learn it all over again. And, it’s for the better. We’re taking this new idea of doing math into classrooms with us. We’re teaching our students what it means to do math and not just absorb information. We’re teaching them how to be problem-solvers, how to work together. We’re teaching them how to construct their own learning and discover ideas. We’re teaching them how to explain their thinking, develop conceptual understandings, and really DO math.

Brad is creating a podcast with all of the things we talked about over at #MichEd in the near future, so I won’t give everything away. But, we’re focusing on the future and we’re focusing on the students. We’re doing what’s best for them, not what’s easy, and not what our teachers of the past have done. There are so many great teachers in Michigan flooding our schools with great ideas, innovative strategies, and a passion for teaching students. This next generation of Michigan teachers feels the same.


Student voices should be the loudest ones in your head.

Student voices should be the loudest ones in your head.

Too often, teachers are making choices about instruction and content delivery without truly thinking of their students. But, why is that happening? Shouldn’t all teachers be basing their lessons almost entirely on the needs of their students? Student voices should be the loudest ones in your head as you’re planning.

This podcast, by #MichEd host Brad Wilson, highlights what students want, what they’re looking for in school, and what makes them learn best. The majority of students discussed hands-on learning. This type of learning keeps students engaged, interested, and willing to do the work. They need the stimulation in order to retain and understand content. Students also talked about making choices. They want to make choices about topics they work with, pacing of content, peers they are in a group with, and how they demonstrate their knowledge. Isn’t this what teachers have been recently trying to push in the field of education? Why haven’t we just been listening to students?

One of the main points that I found so interesting is that students know how they learn best. As teachers, we shouldn’t take this away from them. We want our students to be skilled in metacognition and self-awareness. If our students understand themselves enough to know how they learn, then we need to let them learn in that way. Now, I know its near impossible in a classroom to let each student learn exactly how they want everyday, but the alternative is not to teach how we want to everyday instead. Teachers should be taking into account the learning styles of students and giving them the opportunity to learn in the way they learn best as often as possible. If the majority of your class consists of hands-on, visual learners, the majority of your lessons should be hands-on and visual. Give students options. They are capable of making decisions and should be encouraged to do so. The podcast highlights many students who know exactly how they learn, and who even noted that other students learn differently from them. If students have choices of how to receive content, we can engage so many more students than by forcing them to learn how we think they should learn.

After all, our whole job is for them. Teaching is all about the student. Their voices should be the loudest we hear.

Ask Questions!


One of the most important things I’ve found in teaching so far is the importance of asking meaningful questions to deepen mathematical understanding. I want my students to be mathematical thinkers and to be able to discuss their thinking with me and with their peers. I think that so much of mathematics education has focused on answers and procedures. Instead of that, I want to instill in my students the idea that the journey to get to answers, the thinking, is what doing math is all about. That’s what is important. 

At my latest observation, I asked my coordinator, Jon, to focus on the questions I was asking students and how they can influence the thinking of my students. I wanted to analyze how well my questions were actually inspiring thinking within my students and if I could be more effective at this. 

One of the main realizations we had was how often I was starting questions with “why?” That was even the whole question a lot of times. Though this is a good start to the way I want students to be thinking, it comes off as a bit aggressive. Jon talked about how there are genuine questions we can ask as teachers and how these are often more effective. For example, we ask a lot of questions like “what is the distance formula?” for repetition purposes, but these aren’t genuine questions. We know the answer. Instead, a genuine question would be “how did you think about that?” We don’t know how our students’ thought processes work and having them explain it is what I would want my ideal student to be able to do– to explain, in detail, how they thought about the math involved. 

Another idea we came across was the issue of how to reword questions when students are not responding well. When my students struggled with the answer, it was hard for me to reword it in a different enough way to get them to understand. Then, for time management purposes, I would often give them or lead them into answers that I wanted them to discover or illustrate to me themselves. Having ideas for multiple questions before the lesson will help me to get students to understand what I’m asking. This will take a lot of planning, but will definitely be worth it. 

My final “grow” moment from this lesson was about addressing student misconceptions. When students come up with ideas that are incorrect, but common (and understandable), I have a hard time knowing how to respond. I know that saying “no” in any part of my response is something I don’t want to do. This has a lot to do with my opinion that students who hear the word “no” will shut down and not listen to any of the rest of my response. For example, in this particular lesson, I had asked students which side was the hypotenuse and how they knew. One of my students responded and said “I know that side is the hypotenuse because it’s the one that’s diagonal.” My response in the moment was “yeah, but, actually, it doesn’t always have to be. We know it’s the hypotenuse because it’s across from the right angle.” I immediately regretted saying the “yeah, but…” part because I don’t want my students to be confused. Jon and I talked about saying something like “that’s really interesting! Let’s look at this,” showing them an example of something that disproves their idea. In this way, we are not shooting students down, not giving them incorrect ideas, and clearing up misconceptions (that much of the class shared.) 

Now what? 

In the future, I want to be conscious of the ideas that we discussed. Firstly, I want to change my question starters and begin asking more genuine questions. I think my students will really respond better to questions about their thinking, rather than “why?” Why questions seem to be looking for the mathematically correct answer of why, which isn’t my goal at all. So, I want to focus more on the goal I have and which questions seem to be getting there. Second, I want to plan, plan, plan. I want to plan multiple questions that address the big ideas of the lesson and how to get students to be able to answer them. This will take practice and experience, but I’m enthusiastic to start trying. Last, I want to spend a minute or two actually addressing misconceptions. I don’t want to just tell them why something is wrong, I want to show them. I want my students to see exactly why the hypotenuse is not always the diagonal line in order to better understand and retain that information. I think all of these things will help me to increase the mathematical thinking of my students. ‘Cause, after all, it’s the journey that’s important, right? 


Middle School Mathematics

I have officially begun my Teacher Assisting semester. In just three days in the classroom, I can honestly say that I have never felt more at home. I’ve wanted to be a teacher my entire life, and it’s amazing to see how confident I am that this is what I want to do for the rest of my life. One of the most important things I’ve taken away from this past week is that my reluctance to teach middle school has diminished. I had always thought that I wanted to teach high school and swore I would never teach middle schoolers. Despite my beliefs, I am honestly starting to love the middle school atmosphere. The students are honest, intelligent, and just coming into their own as self-regulating learners. I think it is absolutely inspiring how influential middle school mathematics teachers are. They bridge the gap between elementary and high school math content and have the opportunity to produce students who love mathematics. This past week in my placement has definitely opened my eyes to what I want to pursue in the future as a teacher.