Thursday, September 29, 2016

Order of Operations Misconceptions

Recently, I had the opportunity to interview a student going through a math problem and I was able to discover several misconceptions about how students think about a common topic in mathematics: order of operations.

I chose this problem and this topic because order of operations is found in mathematics courses from middle school through college. However, some students still have many misconceptions about it and how it is solved.

Here's what I learned about student understanding of order of operations:


  • Other grouping symbols are often not recognized as forms of parentheses in PEMDAS.
    • When students are just taught to memorize the acronym PEMDAS, they may fail to recognize the connection between other grouping symbols and parentheses. In this example, the brackets caused some confusion because of lack of immediate recognition. 
  • Students taught to memorize PEMDAS may always simplify parentheses first, even when they don't need to be simplified.
    • In the example above, -5 and -2 are in parentheses, but they cannot be simplified further because they are only a number in parentheses rather than an expression. However, if students just memorize an acronym and do not have a deeper understanding, they will immediately jump to anywhere that they see parentheses rather than knowing that it really involves simplifying within the parentheses.
  • Students can have difficulty associating the exponent with the entire grouping symbol, rather than the last number of the group. 
    • Students in the example above may want to square 5 or -5 rather than the entire group because it is in closest proximity. Students who are able to immediately recognize other grouping symbols as parentheses in PEMDAS in Misconception #1 are better able to avoid this misconception.


Other things that I learned from this interview:
1. I noticed that I frequently interchanged the words SOLVE and SIMPLIFY when asking the student about the problem. This can be so confusing for students because they are not synonymous! I need to check my use of mathematical terms if I want my students to understand the true meaning of the vocabulary.

2. I need to ask questions about the right answers as much as I ask questions about the wrong ones. Students can become conditioned when they hear questions to know that they are wrong and immediately correct their response, rather than develop deep thinking about the material. They will search for what they think you want to hear instead of thinking critically about whether their answer was actually correct or incorrect. If I question their correct and incorrect thinking, they become accountable all the time, not just when they are mistaken.

This was a huge learning opportunity for me to check my own difficulties with questioning and learn order of operations misconceptions from a student perspective.

Thursday, September 15, 2016

Quizzes and Tests and Warm-ups oh my!

For my assessment class, we had the opportunity to interview two teachers at our school about what assessment was and how they assess in their classes. In our EDU 315 assessment class we have read a lot of articles about formative assessments, summative assessments, and the importance of checking for understanding in our classrooms. However, there are only so many articles I can read before my head starts spinning. Hearing from other teachers first-hand about how they institute assessment practices in their classrooms helps me to visualize how I can implement my own assessment practices someday.

Collaboration among teachers is incredibly crucial and I learned so much with talking with these two math teachers about what they do in their classrooms. This practice, although not through formal interviews, is definitely something I want to bring to whatever school I will be teaching at next year. I want to hear the opinions of others in order to learn and develop my own assessment philosophy.

A few things important I learned:

  • Modifications aren't only for students with IEPs.
    • This can look different for varying students but can be anything from circling only certain problems to complete, not counting it against a student if they don't finish, etc. For example, we had a student who worked tirelessly on a quiz today but didn't have the opportunity to finish the last two problems. My CT said that instead of just counting those questions as wrong she would simply take the number correct out of the number that he completed. Obviously I knew that I should/must accommodate students with IEPs, but I didn't know how to accommodate others who struggled in the classroom. If a student works slower than the other students and automatically gets the questions wrong that they don't get completed, as a teacher I am not setting them up for success. Often times we are asked the question of being "fair" in the classroom, but through my assessment interviews I learned that fair sometimes has to look different and you have to meet every kid's learning needs. 
  • Assessment can look different in different classes.
    • Prior to my interviews, I had the misconception that the same teacher would assess similarly in all of their classes. However, I learned that although teacher preferences certainly determine how students are assessed in the classroom, the course structure is also a factor. The teachers at Williams assess frequently in Math I and Foundations of Math I because it is an EOC course, the students are mostly freshmen, and schools often look at Math I EOC data. However, in a Math II course, the assessments are much less frequent. The structure of Math II and other math courses doesn't lend itself as much to the daily quizzes and check-ups that you see in Math I. 
  • Assessment should be data-driven, generate data, and have a purpose. 
    • One surprising thing that I heard is that we shouldn't just assess for the sake of assessing. In my assessment course, we have talked about how great assessment is and why we use it, but we haven't really talked about when we don't use it. When should we not assess our students? I think that this point will help me think about why I am using a particular assessment, how it ties back to the standard I am trying to address, what data led me to that decision, and what data I hope to gain from that assessment.
  • Self-assess! Total class failure is a reflection of you, not them.  
    • It's okay to admit that you failed. Reteach! That is what assessment is supposed to teach you. The goal is that the students obtain mastery of the material, not that they make good grades on the assessment. If this means that you have to take a day and reteach a lesson, then do it. Learn from your assessments so that your students can learn from you.
I could certainly write more about all that I learned about assessment through these interviews, but these were some of my greatest take-aways. I can't wait to learn more about assessment and how it will drive instruction in my classroom!

Thursday, September 8, 2016

Let's do this!

I've been browsing through Twitter and reading blog posts by math teachers that are rockstars, plain and simple. I figured I might as well give it a shot so I can pretend to have my life together as well. This blog will hopefully serve as a way to highlight the highs, lows, and everywhere in between of my classes, methods, and student teaching experience at Williams this year - as well as into my future as a teacher after that!

As for the title of my blog - exponential growth. I chose this title for three main reasons. One, it's just a great function. Two, I want my students to grow exponentially in my classroom through engagement and learning opportunities. And finally, I want to continue to grow exponentially throughout my time as a teacher whether I am a student teacher or have been teaching for thirty years. I am a firm believer that I can always learn how to become a better teacher for my students from the people around me through reaching out and asking questions. We as teachers are always encouraging our students to ask questions, but how often do we take the time to ask questions for ourselves to improve our own learning? I believe this is the only way for exponential growth to happen.

As for how I'm going to achieve this growth, I am so fortunate to have some of the best professors, CT, Elon alums, and classmates. I have already learned so much in both my methods and assessment classes about asking the right questions and how to implement formative assessment effectively. My CT is awesome! I just started at Williams this past week, but she has such a positive demeanor with her students. Every class period she starts off with "good news" and encourages her students to share something good in their lives with the class. The relationships that she forms with the students encourages me to build the relationships with my students as I remind myself that students don't care what I know until they know that I care. Next: ELON ALUMS. The most helpful people on the planet. I have reached out to several in the past few weeks and have received lengthy emails with wonderful advice, tons of resources via Twitter, and I'm actually interviewing one alum for my assessment class who currently teaches at Williams. It is so incredibly encouraging how sweet these people have been to me when they have their own lives and worries as teachers, but instead have taken the time to care and invest in me as a teacher as well. I cannot put into words my gratefulness and cannot wait to "repay" (if it is even possible) to all of the younger Elon teachers. Finally, the people going through this with me. We're gonna make it. Even if some people tell us teaching will give us the worst days of our lives, it will also give us the best days of our lives. I'm thankful for classmates who push me, who let me tell math jokes, and who most importantly ask questions with me about what it means to be a teacher.

2 weeks of classes down, 1 week in the school down, a lifetime to go.
LET'S DO THIS.