An epiphany after 17 years of teaching

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?

Then it came to me. Mary was not satisfied until she fully understand WHY her answer was correct and William was satisfied just to GET the correct answer, whether or not he understood why. William and I had a big “Ah-Hah” moment at the end of the year and I encouraged him to take the class again. He had made big strides in his effort; now he knew what else he needed to do to be successful. He became a different student, getting a B in the class, and grew confident in his ability to do math. (Precalculus, no less!)
Now I make a BIG DEAL out of telling my students that getting the right answer is not the goal; understanding the mathematics is the goal. I have found that students who work hard and study and then do badly on my tests, for the most part, are students who try to memorize procedures and formulas without understanding them. This year I have noticed a marked improvement in the overall quality and success of my students now that they are aware of what it means to  understand concepts deeply instead of just memorizing procedures. Students now diagnose their own poor test grades: “I didn’t really understand the concepts; I just memorized stuff” instead of wondering “Why did I do so badly on the test when I studied so hard?” Students are actually putting more thought into understanding their homework instead of just doing it
So spend time helping students with their study habits. Most students who give up on math don’t realize that they are sabotaging their own goals by not knowing how to learn math. We need to help them understand themselves as much as we need to help them understand the math.

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?


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.


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.


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:

Law of Cosines

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


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.

The 5 W’s and an H of grading — Part 2 What, When, and Where

What are the criteria we use to grade students?

  1. Mastery of content
  2. Good study habits and organizational skills
  3. Classroom behavior unrelated to academics

Starting with #3:  I used to have a behavior/extra credit/participation grade in which students were awarded 10 point per 9 weeks. They all started out with 10/10 points and then would lose points for being tardy or for excessive talking (off topic). With SBG, I dropped that grade because I wanted grades to reflect only mastery of content.  Unfortunately, now I have no incentive for discouraging tardies, because we are all expected to deal with tardies within our own classrooms.  So I understand why teachers include behavior as part of their classroom grade.  It is hard not to.

#2:  I think that when we teachers intend to measure mastery of content we are really grading how well students know how to “play the school game”.  Well-organized students with planners and neat notebooks full of resources, who turn their homework in on time and are prepared for tests, of course tend to get better grades.  Teachers often assume that those students also work harder and care more about their academic success than the students who lose their homework, don’t follow directions, and forget that the test is today.  But that is not necessarily true.  Disorganized students usually care very much and are very frustrated by their own disorganization.  Such students might work hard on an assignment and then not remember where they put it or study hard for a test, but study the wrong things.  Or maybe they want to study but lose their notes or leave their textbook at school.  Giving second chances and multiple ways to demonstrate knowledge and skills helps these students to truly demonstrate what they know. I agree that life skills such as responsibility, organizational skills, and punctuality are important, but if we claim to be grading mastery of content, then our grading system should not punish a kid just for being more disorganized or less mature than another student.

#3:  Accurately measuring mastery of content is an art, one that I have yet to master.  It is fairly easy to measure skills and even conceptual understanding, but I need to work on how to measure creative thinking and problem solving.

When should we grade?

 If we want to grade mastery of content, then when we grade is very important.  We should grade in the formative sense as often as possible, with immediate feedback, so students do not reinforce bad habits.  I need to do more of this.  Students who want to learn always are eager to know if their answers are correct.  Hopefully I can help all of my students to develop that eagerness.

We should grade in the summative sense when students clearly understand what the standards are and how we expect them to be able to demonstrate their knowledge, and after they have had adequate time to practice.   This is a nice ideal but hard to achieve with limited time and a diverse group of students who learn at different rates.  Once again, it is important to give specific feedback and to allow students a chance to improve and retest.

Also, we should grade and return all work and assessments as soon as possible.  Feedback is most helpful when the work is fresh in the students’ minds.

Where should we grade?

 I mentioned in my previous post that I like to grade in my comfy chair at home with a hot cup of tea.  Also, if you are like me, you have graded papers almost everywhere and probably have a red pen in your car or purse.  However, I did have a more insightful thought.  Grading in class in front of students can be really helpful.  This could be done with sample student work, with anonymous answers collected from the class, or one-on-one with students.  Being able to discuss what work is good, what needs to be improved, what I am looking for when I grade, what common mistakes are, etc., is really helpful to students.  Grading can also could be done with video or by email to give students more personalized feedback.

The 5 W’s and an H of grading Part 1- Who?

Having just completed my first semester of Standards Based Grading (SBG), I have been doing a great deal of reflecting about what worked and didn’t work and also just thinking about the whole process of grading in general.  I have come up with more questions than answers.  But that is how we learn, right?  So I am going to write down my questions and my reflections and invite feedback from my wonderful colleagues out there in the MBToS.  After a brain wave I got while lying in bed this morning (I love sleeping in!) I decided to frame my thoughts using the traditional Who, What, When, Where, Why, and How questions.

Who is most responsible for the grade that is earned, the student or the teacher? (You may be surprised. Or not.)

What criteria should be used to determine the student’s grade? (A big debate is always raging about this one.)

When should grades be given? (Another very interesting question.)

Where does grading take place?  (For me, preferably in my comfy chair at home with a cup of hot tea and maybe a familiar movie playing in the background to keep me awake. Smile. . . . Actually, there may be more to this question.  Stay tuned.)

Why do we give students grades?   (O0-00, this should be a good one.)

How do I implement what I consider to be best practices in grading?  (Ay, there’s the rub.)

This post (and hopefully I will have time to go through all of these) will concentrate on Who.  Who is responsible for the grade that is earned?

All teachers with any sense realize that both the student and the teacher contribute to the grade, but we like to think, especially as math teachers, that our grading system is a reasonably objective reflection of what students actually know–their mastery of the standards of our class.  Ideally, if a student is willing to put in the time and effort, he should be able to master the standards and earn a good grade.  I switched to SBG this year in part because my previous system of grading was flawed in this area and I wanted something better.  However, who sets the standards? The teacher (OK, the state and the district and the school, too, but certainly not the student).  Who is responsible for creating or procuring lessons, resources, and practice activities to help students master the standards?  The teacher.  Who decides how to measure the mastery of those standards? The teacher.  What I choose to measure and how and when I choose to measure it, along with how well I teach, has a huge effect on student grades.

And how objective am I in my assessments?  I have trouble being totally consistent from student to student on a single quiz, so how consistent am I from unit to unit or from year to year?  I know that I cover fewer topics and different topics than I did 10 years ago or even 5 years ago and I wonder if I am grading more strictly or more leniently from year to year.  Does an A in my class vary from year to year?  Do I sometimes give the benefit of the doubt to my good students but not to my poor students when deciding between two grades?  I will not answer those questions because I know the answer.  Guilty as charged.

We all know those extremists in the teaching profession:

  • The teachers who never give A’s and who delight in failing everybody, nobly clinging to their “high standards” while their colleagues “slide down the slippery slopes of academic mediocrity”.  Any failure on the student’s part is solely blamed on the student, of course.
  • The teachers who ensure the “success” of their students by giving generous test curves and copious amounts of extra credit, giving high grades for learning almost nothing.  (There are various reasons for this behavior:  Feeling sorry for the students, lack of time or effort to fix the real problem, compensating or covering up for inept teaching or grading practices, and/or bowing to administrative or parental pressure to bring up grades are the ones that occur to me.)

Although I am not either of these, I have sometimes leaned one way and sometimes the other in my grading mentality, even compensating for one by doing the other.  (Am I the only one?  I think not.)

The point I am trying to make in all of this rambling reflection is that not matter how hard we try to be objective, assigning grades is very subjective.  I wish–oh, do I wish–that I could just teach without giving grades.  (Am I the only one?  I think not.)

What brought on all of this agonizing self-reflection, you ask?  My final exam.  After a semester of SBG quizzes with multiple chances to retake, I gave a cumulative, mostly multiple-choice final exam with no retake, worth 20% of the semester grade.  My reasons for giving a one shot cumulative final:  College-bound juniors and seniors need to learn how to take these types of tests, students review and relearn material they would otherwise completely forget while studying for the final, part of mastery is retaining material and being able to recall more than a few isolated skills at a time.  I stand by these reasons for giving the test.  I even had several students finally figure out some concepts for the first time while reviewing for the final.  My reason for the 20%?  I wanted it to affect their grade.  An A student who couldn’t get at least a B- on the final was not an A student in my mind.  Yet 20% allowed a student to get even a low F without lowering their grade more than one letter grade, so it wouldn’t destroy all of their hard work over the semester.  I have always had a wide spread of grades on the final, from perfect scores all the way down to the 30% range.  But I didn’t expect the grades I got this year.  After a semester of SBG, with multiple retakes, I anticipated a higher level of mastery than previous years.  Instead, I got 6 A’s, 19 B’s, 3 C’s, 4 D’s, and 22 F’s.  I believe that my test and my grading of it was comparable to previous years.  I have always had more low grades on the final than on other tests, but the number of F’s and the huge gap between the A’s and B’s (about half my students) and the C’s , D’s, and F’s was more pronounced.  I know that some of the F’s were due lack of study and effort.  I also anticipated F’s from some students who had struggled all year long.  But it doesn’t seem that allowing multiple retakes on concepts all year really helped my lower students retain concepts any better than they did before SBG.  However, the almost total absence of C’s indicates that my average students did seem to benefit and achieve higher mastery then previously (I think).

I know that grading policies are only part of teaching, but how did my grading policies affect student learning?  I’m not sure.  I was definitely hoping for better results.  Everybody ended up passing the class (without any tweaking on my part) and I only had a few D’s, but 14 of my 54 students had lower grades (1/2 to 1 letter grade lower) as a result of failing the exam.  It seems a fair assessment of their mastery (retaining concepts), but perhaps is also an indictment of my teaching and grading methods.  I’d love to know what you think.

MTBoS Mission #5

Did my first Twitter Chat last Thursday.  It was a good experience, but not my favorite way to spend an hour and a half on a school night.  I felt like it took a lot of time to say very little, although I did glean a few useful ideas on which to follow up.  The best part was discovering some more like-minded Precalc teachers to follow on Twitter and possibly exchange ideas with later.  I’m not sure if I’ll do it again, but I am definitely finding Twitter to be a wealth of wonderful people and enough ideas and resources to make my head spin.  I think I tend to be more of a blog person.