Two years ago, two students sat side by side working on review problems for the upcoming test. Let’s call them William and Mary. William and Mary worked together on the problems, talking about them and calling me over to ask questions when they didn’t understand something. The next day, Mary got an A on the test and William got a D. This happened several times throughout the year. Both students had practiced and felt prepared for the exam, but the results were completely different. William was one of many students I have had over the course of my years of teaching who said to me, frustrated, “I studied and thought I knew everything, but I still bombed the test. Why?” Why indeed?

# Students Like Order

Yesterday and today, I was compiling the results of student surveys used to evaluate their teachers in the math department. We have one first year teacher who is struggling with classroom management. Not surprisingly, on the question which read “suggest one thing your teacher could do to improve”, 90% of her students said that she needed to be “more strict”, “send bad kids to the office”, “keep better control of the class”, “don’t let kids talk so much”, etc. One young man even admitted that he and his friends were not respectful to her and that she shouldn’t let them get away with that.

Kids like order. They might act like they don’t, but they do. And they said so, loud and clear, in their survey comments.

I am hoping that reading the students’ comments will help our new teacher realize that managing the classroom well is not just something that she needs to do for herself, to keep her sanity, or for her boss, to keep her job, but for the students, who honestly want her to be more strict with them.

# What Makes A Math Teacher “Great”?

Thinking back on my own experiences as a student myself and also on conversations that I have had with students over the years, I have been reflecting on what it means to be a great teacher. What qualities does a teacher possess that makes students call that teacher “good” or even “great”? What kind of teacher do students point to as someone who changed their lives and their perspectives and their way of thinking for the better?

**Competence:**

The first and probably most essential quality of a good teacher is competence, both in the subject and in the craft of teaching. Great math teachers understand their material deeply and can explain it in multiple ways. They build on students’ previous knowledge and can point to future applications of the mathematics they are teaching. They plan lessons that are purposeful, well-organized, and that engage students in building a strong conceptual understanding. They can diagnose student misconceptions and address those misconceptions with clear examples and explanations.

Learning goals are clearly understood, feedback is timely, and assessments and grading policies are fair. The classroom is well-organized and provides an atmosphere conducive to learning. Classroom policies are clearly explained and consistently enforced. The teacher is in charge of the classroom and the students respect his or her expertise and authority.

**Caring:**

Students want a teacher who cares about them as individuals. In addition to that, a great math teacher must also show students that they will do everything in their power to help each student be successful in their math class. One way this caring is demonstrated is by purposefully building relationships, having conversations both in and outside of class, finding out about students’ interests, attending their extracurricular events, etc. Another way a teacher demonstrates caring is by working as hard for their students as they expect the students to work for them. Great teachers clearly show that they are putting forth a great deal of effort to make the mathematics they are teaching interesting, challenging, and yet accessible to any student willing to work for it. Caring teachers do not allow their students to settle for mediocrity. They challenge their advanced students and provide extra encouragement and support for the strugglers. Great teachers tell students why they are doing what they do in the classroom so that their students know that what the teacher is doing is for their benefit.

Students respect and learn from competent and caring teachers. A competent teacher earns their respect and a caring teacher earns their loyalty. But one more ingredient is needed to be a *great* teacher.

**Passion:**

Great teachers are passionate about their subject and about sharing their love for it with their students. Great math teachers are lifelong learners of math; they enjoy learning new mathematics or new ways of looking at things they already know. They think math is awesome and want their students to learn to love and appreciate math, too. To that end, a great teacher seeks to continually learn more about math and about how to teach it effectively. They connect with like-minded teachers and share ideas. They work to make their classroom a place where students can engage in mathematics in creative and thought-provoking ways. They make it their goal to help every one of their students find something intriguing and enjoyable in the world of mathematics.

Great teachers are also passionate about education. They are intrigued about how students think and learn and they seek out research and use it in their classrooms. Through practice, they become expert at helping students diagnose their strengths and weaknesses and how they can become more successful learners. They also continually seek out and try new classroom strategies that help students learn and retain what they learn. Great teachers keep getting better at what they do.

Great teachers change their students’ perspectives. After being in a great teacher’s math class, a student might say, “Your class was the first time that I saw how creative [interesting] [fun] [applicable] math could be.” Or “I never thought that I could do math before I took your class.” Or “Thanks for taking the time to really help me understand.” Or “I really like math. What would be a career that I should look into?” Or “I’m still not much of a math person, but I really enjoyed your class.”

My prayer is that every teacher at Covenant Christian High School seeks to be a great teacher. We owe it first to our Lord Jesus, whose name we bear as a Christian school. We owe it secondly to the families who pay a great deal of money for an “excellent, Christ-centered education” and who entrust their children to us with that expectation. And, of course, we owe it to our students. (And I wouldn’t feel good about myself if I was a mediocre teacher!)

This list of qualifications is daunting. But there is hope! Being a great math teacher, like being a great math student, can be learned. It is not something we __are__, it is something we __become__, with desire, dedication, and practice. Let’s all strive to have a growth mindset about our teaching!

# Epiphany about the Law of Cosines

I have been teaching Precalculus/Trigonometry for 18 years and I just figured out a much simpler way to derive the Law of Cosines that the students can actually do themselves, with a little guidance from me perhaps, instead of watching me do a more complicated version while their eyes glaze over and groans ensue.

Here it is:

Start with a right triangle and recall the Pythagorean Theorem. Then draw a second triangle where sides a and b are the same, but side c is shorter because angle C is less than 90 degrees. We don’t know how (yet) to find side C, so we divide triangle 2 into two right triangles and then we can use the Pythagorean Theorem to write a system of equations. Solving the system of equations gives us c in terms of a, b, and x. Then find x using cosine of the known angle C (let students know up front that the given information is the measure of angle C and the side lengths a and b (a SAS triangle).

I used the same diagram in previous years, but I didn’t use a system of equations to simplify the relationship first, so the substitutions used both sine and cosine and required multiplying out binomials with sine and cosine in them and simplifying with the Pythagorean trig property. The derivation used a lot of nice math, but was so long and cumbersome that it became meaningless to the kids and the first question out of their mouths was “Do we have to know how to do this for the test?” rather than, “Oh, that makes sense,” which is the response that I hope this newer method will elicit.

**ANOTHER EPIPHANY! **I have students who forget they can use the Law of Cosines and who would divide triangle 2 into two right triangles to find side C anyway. So I should let them find side C the long way and then ASK THEM IF THEY WOULD LIKE TO KNOW A QUICKER WAY TO DO THIS! The old Dan Meyer aspirin for a headache approach.

**After deriving the formula: **Then follow up by actually constructing 3 triangles with the same sides a and b but different angles for C and show how the term abcosC is the amount of change (+ or -) in the length of side C and that, when C = 90 degrees, abcosC = 0, so the Law of Cosines is just a special case of the Pythagorean Theorem.

# Three encouraging and freeing things I’ve realized this summer.

1. After trying lots of new ideas at school last year–some of which I liked and some of which I didn’t–I finally realized that I need to teach the way that suits me best, the way that best utilizes my strengths and my passions, and not worry about what works best for other people. In a sense, I threw away the guilt of not doing all the cool things I read about in everyone else’s blogs and gave myself the permission to just be me. I still intend to try new things–that is something that I like to do–but I don’t need to feel guilty for not embracing the educational trend du jour if it doesn’t suit me. That is very freeing! 2. Like every teacher out there, I have continually fought the “cover every required topic” versus “teach the concepts well and enjoy math” battle. I have always leaned towards the “teach the concepts and enjoy math” side, but struggled with the fact that I always ended up leaving out topics which might potentially leave students inadequately prepared for future classes. But last week I listened to the MathOut podcast interview of Fawn Nguyen (check out all of Adrian Pumphrey’s great podcasts at MathedOut). When Adrian asked Fawn how she managed to teach rich problem-solving lessons and still cover the standards, she actually said that she doesn’t worry too much about covering all of the standards–she considers it more important to teach kids (hopefully) to love math and be problem solvers*. Wow! That was also very freeing to hear! Now I will enjoy planning my year in Precalculus with a goal of teaching well, rather than worrying so much about how I can possibly cover everything (especially since I already know that I can’t). *For those of you who think that Fawn sounds like an irresponsible teacher, she does not completely ignore standards–she is just not enslaved to them. Listen to the podcast and hear what she has to say. 3. I checked out *Why Students Don’t Like School* by Daniel Willingham from the library on my Kindle and immediately realized that I wanted to highlight things and I needed my own copy to keep. When I purchased it, I also purchased one of those “people who bought this book also bought this book” books called *How Children Succeed*:* Grit, Curiosity, and the Hidden Power of Character* by Paul Tough (“grit” by Tough *smile*). So I put down the first book to read the second and found myself following my husband around the house reading passages to him. The essence of the book is that students need more than smarts to be successful in school and in life, they need moral fiber–grit, determination, self-control, etc.–the ability to deal with challenge and failure and grow stronger as a result. This is not earth-shaking to those of us who teach, but how to teach it is the challenge (and I haven’t finished that part of the book yet). However, in reading about a teacher who has been successful in developing these traits in her students, I realized that I have done the same things with my students and that spending time doing those things is important. It was one of those moments when reading someone else say things that I already believe and do validated my beliefs and made me that much more passionate about acting on them. (And I also look forward to getting back to the first book, which I think will also be enlightening and encouraging.) To my fellow teachers, I hope your summer is equally uplifting.

# Will we ever get state standards for high school math in Indiana?

I probably should be figuring out what in the heck I’ll be teaching in Precalc tomorrow, but I’m just so frustrated I could scream. So I decided to write and vent a bit…

Indiana has finally released its draft of new state standards for math and language arts for public comment. Laudably, the Indiana Dept of Education is inviting teachers and the public to comment freely on the proposed standards before they are finalized. After three years of having standards and then not having them finished yet and then having different ones and then changing those again and then definitely going with Common Core and then definitely not going with Common Core, I was hoping that what the IDOE was promising–standards in place by April 1st–was finally going to happen. However, here are the standards that they came up with–a laundry list of 139 algebra standards, 96 geometry standards, 80 data analysis, probability, and statistics standards, 70 discrete math standards, 51 calculus standards, 24 “process, problem-solving, practice” standards, and a few misc others. The standards are not divided in any way by subject or year, grouped by major content strands, or even organized sequentially in the order in which they would normally be taught. There is not indication of how many courses would be required to teach all of these standards and which of these standards would be required for all students and which would be considered non-required or elective standards. Basically, they just took our 2000 Indiana standards, our 2009 Indiana standards, the Common Core standards, NCTM standards, and possibly some other lists of standards and put them all together into one really lo-o-o-ong list with no organization whatsoever. Talk about a mile wide and an inch deep!! There is absolutely no way that they are going to have the high school standards even close to done by April 1st. So, wearily we soldier on, never knowing if what we are teaching now is what they will want us to be teaching next year, and what they will be testing our students on the year after that.

# I am terrible at leading classroom discussions.

I think that I do fairly well at engaging my students when I lecture. And my assessments require students to demonstrate in various ways that they truly understand the concepts. I have implemented some successful projects. But I am JUST PLAIN HORRIBLE at facilitating meaningful and rich mathematical discourse in my classroom. Inspired by wonderful blogs and websites, I have been trying to lecture less and get kids to think for themselves and engage in rich tasks and have meaningful discussion . . .with almost zero success so far. On Monday I decided to begin my unit on Sequences and Series by using some patterns from visualpatterns.org

These patterns are SO cool! All of the students were engaged some of the time and some of the students were engaged all of the time (thanks, Abe Lincoln), but the conversation was never lively and I felt like we never really accomplished anything . So often, when I try something new, I have a interesting activity and I know what I want the kids to do or to learn, but I have no idea how to get there. I don’t know how to plan a classroom discussion. How do I get the conversation started? How do I facilitate a student-driven activity and yet accomplish specific learning objectives? (Looking for some feedback here!)

Some things I know I did wrong:

- I didn’t start with a question that everyone could answer. I probably should have said, “Tell me something you notice about this pattern,” and then written down all of the student ideas. (any other ideas for good starter questions?)
- I revealed too much information at the beginning. The website gives the answer to the 43rd pattern and asks for the equation. I should have just shown them the pattern first without the additional information. I think students were intimidated by the question about the equation.
- I should have taken the time to anticipate possible student answers and plan out my questioning strategy ahead of time instead of winging it!

Thanks to two wonderful bloggers for their inspiration and help: Michael Fenton’s recent posts about his One Minute Makeovers of his old assessments helped me to realize that trying something new and failing is normal and is better than not trying something new at all. I can learn and grow better at this! And Dan Meyer’s post today on Teaching the “Boring” Bits was especially timely and full of helpful advice as I navigate the new and uncharted waters of facilitating meaningful classroom dialogue.