Good Teachers Work Really Hard

Are we conscious of how hard good teachers work on a daily basis? Bad teachers seem to get all the press. Bad teachers don't work very hard. They do as little as possible to get through a day and go home.

Recently, a good teacher asked me to spend the day in her classroom. She was teaching a lesson on geometric transformations and she said that experience has shown that an extra set of hands would be beneficial. The class is a group of challenging seventh graders who have a tendency to get off task if anything out of the ordinary happens. The teacher knew that, although my presence would be considered out of routine for them, the amount of new materials (namely compasses) needed for the lesson would cause a disruption. I was excited to be able to spend a whole day in one classroom. My full-time position as a secondary mathematics resource teacher requires that I travel to 26 middle schools, sometimes 2 in a day. That doesn't allow for much one-on-one time with a teacher unless scheduled ahead of time.

I arrived in the teacher's classroom at 8:05 am. The teacher was already there and she told me that she woke up with a splitting headache. Although she had taken some ibuprofen, she said that they didn't seem to be helping. She had arrived at school at 7:30 and had prepared her classroom, boardwork and materials, for the day. The students started to arrive at 8:15. This is when a teacher has to be ready to "hit the ground running". Most every student needs a piece of the teacher's time. The reasons are endless and usually unique; ranging from personal to academic. Many of them just want to touch base with a caring adult before they start their day. This teacher genuinely cares for her students and wants to have this time available for them. This is one of the reasons why she needs to have all her preparations finished. She will teach three 90-minute periods in the course of the day.

The first class was basically receptive to the lesson although an outside observer would have thought they had arrived in the room for the first time ever. As the teacher carefully asked questions to access the students' prior knowledge and make connections among that previous learning and today's lesson, the students mostly stared blankly at her. Every once in a while someone would raise a hand and blather some inane response that had no relationship to the question that had been asked. The teacher never lost her cool. Mr. Lemov, who wrote the book Teach Like a Champion, calls this emotional constancy. This teacher is the master of emotional constancy. She knows that middle school students spend their days basking in drama but she doesn't allow it to take her from the objective. She smiles, rephrases the questions and moves on until the students give up the nonsense and get back on task. I think some of the behavior was for my benefit. A 13 year old always thinks getting attention from the new person in the room is a good idea even if the attention is garnered for all the wrong reasons. So, although the lesson moves forward in the allotted 90 minutes, it is obvious to me that the headache has not dissipated.

With hardly 4 minutes between classes, the next group of seventh graders arrives. The difference is that these students are more awake than the previous group. That means more interaction, more minutes since arrival at 8:20, and more issues with which to deal. For this class, the teacher sits down at the ELMO (a newer version of the overhead projector that displays documents) and tries to begin the process of accessing prior knowledge to begin the lesson. The students are not responding and I ask her if I can say something to the class. I ask them how they feel about their teacher and they overwhelmingly call out that she is their favorite teacher. (I suspected as much or I wouldn't have asked the question.) I then tell them that she has a bad headache and ask them if they would be willing to stop the disruptive behavior such as calling out and talking to their neighbors. They say 'yes' and stop for maybe 3 minutes. The rest of the lesson was about half as productive as the first class.

When the class ends, the teacher has about 5 minutes to report to lunch duty. She tells me that her headache is no better but it's too soon to take any more ibuprofen. If you don't teach, you have no idea what a middle school cafeteria sounds like. Try to imagine sitting in a bustling crowd with no noise buffers. Now try to imagine having a headache in the same situation. By the way, a teacher on cafeteria duty does not just sit on the sidelines. It is an active 30-minute duty from beginning to end. After cafeteria duty, the teacher has time to eat her lunch and grade some papers. I was able to have a quiet lunch with the math department chair. We had planned to have a conversation while we ate. This also gave the teacher some time to herself after cafeteria duty.

There is a third class before the day ends. The third class is an average between first and second. There are only about 3 students that attempt to run the show. By this time, the skin under the teacher's eyes has visibly darkened. She looks tired and I can tell the headache is still there. When the 90 minutes is up, I look at her and say that I bet she is ready to call it a day and go home. Before she can answer, about 8 girls arrive in her room. It just so happens that she is the girls' basketball coach and they have a practice today. After the rest of the giggling, excited girls arrive, they all head down to the gym. As I pack up my things to go home, I can only think, "OMG...a headache combined with how many bouncing balls for the next hour"?

Sometimes professionalism and dedication need a second look. And that day in a school made me realize how hard good teachers work everyday!

Know What You Teach

 I recently attended a meeting in my state with the supervisors of mathematics from all the jurisdictions in the state. Under the leadership of the state mathematics supervisors, we are all grappling with the transition from our current mathematics standards to the Common Core State Standards. This transition is made more difficult because our state is in the PARCC consortia, which has released very little information regarding the assessments that will steer the new curriculum. The other stumbling block is the existence of the current testing program which is expected to last until 2013/2014. The current testing program assesses content on a lower, skill-based level than what is being hinted at on the new assessments. Principals are very reluctant to transition away from current standards if schools will still receive sanctions due to poor performance on the current tests. I guess every state is wrestling with the same dilemmas.

But, this wasn't the point of this posting. The point I'd like to address is the issue that concerns all of the supervisors; teacher content knowledge. Every time this group gets together the conversation always turns to the topic of professional development and how much content development needs to be done. If you're reading this and are not a math teacher, this is the point where you might get a bit scared, especially if you have children in school. Many teachers have a comfort level with the content they are teaching. They aren't necessarily fluent in the content that comes before or after their course. Is this a problem? It's a huge problem. When a teacher doesn't know how their content fits in the big picture of mathematics, it's difficult to make connections for the students. Many teachers that I've worked with have never taken the time to look back at the curriculum guide that preceded theirs or the one that comes after. Unfortunately, that means that many teachers are teaching every concept as if it's brand new without giving the students any credit for bringing prior knowledge to the table. The scenario that sends me up the wall goes like this: "Ms. Smith, last year my teacher showed me how to do it this way." "I don't care. That was last year and this year you have to do it this way." Heaven forbid that Ms. Smith should look at the methodology, determine it's mathematical significance, and if nothing else, acknowledge how that previous method connects to 'her way'.

With this in mind, I'll describe a recent school visit and a conversation I had with a 17-year veteran teacher. I was observing the teacher's class as they were practicing division of fractions, a subject that very few humans understand. (That's not a smart--- comment! It's true.) The teacher presented a problem and asked a student to show the solution. The problem was 13/12 divided by 10/12. The student went to the board and beautifully showed the algorithm of multiplying by the reciprocal and coming up with the correct answer of 1 and 3/10. (I'm realizing that it's difficult to blog when you need mathematical symbols.) 

All was fine until a student raised his hand and said, "Weren't we working with twelfths? How can the answer have tenths?" The teacher responded, "That's just the way it works and that's how it simplifies." WHAT??? It was obvious that the teacher was not any more knowledgeable than knowing the algorithm. I guess that in 17 years she has never been asked that question before or else she's been answering it the same way for 17 years and never bothered to delve deeper into the "why". I won't veer off onto my typical rant about professionalism at this moment.

I had a chance to talk to the teacher after the lesson and asked her about the student's question. She admitted that she had no idea why the algorithm works. I first talked to her about the idea that both fractions in her example had common denominators. The problem was basically asking how many groups of 10/12 could be formed from 13/12. She could see that one group could be formed and she realized that there were 3 "of something" left over. The sticking point is what are those things that are left over. It seems that they are twelfths. But if you had 3/12 as a remainder, the answer would have been 1 and 1/4. Even when I drew a picture for her, it still seems that 3/12 remain. That's the misconception. The remainder to a division problem is always how many items are left based on how a group is defined. When working with whole numbers, most people can define the remainder. Many even know how to change the remainder to a fraction by placing it over the divisor. In this fractional problem, how is a group defined? A group is considered to be ten items. It just so happens that these items are called 'twelfths'. So, there are three items  out of ten, making a remainder of 3/10. Students can see this when the fractions have common denominators. They also quickly determine their own algorithm for dividing fractions. They'll find common denominators, then the problem is nothing more than dividing the numerators and dividing the denominators. Because you have common denominators, when you divide them, the quotient is always one. That allows the students to realize that all they have to do is divide the numerators WHEN THE DENOMINATORS ARE THE SAME. 

I've used a scenario when developing this concept with students that talks about a manufacturer and a packaging company. The first fraction defines what the manufacturer produces (in the original problem...twelfths). It also tell how many the manufacturer sends to the packaging company. The second fraction defines how many the manufacturer wants in one package (in the original problem...ten). The packager tells the manufacturer that he can make one package but only has three of the next ten to make another package. Students usually have no problem understanding how a problem that started with twelfths can end with tenths.

If students are more sophisticated, the standard algorithm can be developed using complex fractions; meaning a fraction in the numerator and the second fraction in the denominator. In order to simplify the complex denominator, change it to one by multiplying by the reciprocal. That forces the numerator to be multiplied by that same value to maintain equality. At least that explains why the standard algorithm works...not just making students memorize rules with no comprehension.

I'd like to be able to end this posting by saying the 17-year veteran had a clear understanding of division of fractions after our conversation but "the deer was still staring into the headlights" when our time was up. I did leave her by saying that I was always available to talk whenever she comes upon something in her curriculum that she only has a surface understanding of. I also encouraged her to look at the previous course curriculum guide. It just so happens that the previous guide develops the concept of dividing fractions using the common denominator algorithm (because I wrote it :-)). Fancy that...

What Did You Get on Your Report Card?

I had several interesting conversations lately about grading; a subject that I've been thinking deeply about for several years. I started thinking about it on a deeper level when my son was in tenth grade. My husband and I went for a parent-teacher conference with his geometry teacher. My son was "failing" the class and we wanted to understand what he was not doing. The teacher explained that our son sat in the front row, always did his homework, participated in class discussions in a meaningful way, and understood the main topics in the subject. After hearing the teacher describe our son thusly, we were extremely confused about his apparent failings. This is when we heard the comment that was to set me on a path to force teachers to talk about their grading policies; a subject that most teachers do not want to discuss. What did the teacher say, you ask? She said, " Most of the students in my class fail". WHAT??? My response to her was two-fold. I told her that from the perspective of a parent, I was baffled and confused because I didn't know how to help my son be successful in her class. (I guess I need to tell you that his test average was the aspect in his grade that was the dominating factor...but more about that later, maybe.) But, then I told her that as a teacher, I was embarrassed that she was wearing that statement as a badge of honor. Couldn't she see that in some cases her students' grades were part of her responsibility? Believe me, I know about the students who do not prepare for class and whose attendance greatly affects their grade, but she admitted that my son did not fit in that category. I asked her if she truly believed that an 'F' would describe what my son knew and could do in geometry. She said, "no, but that's how the numbers work out". Again, WHAT???? 

In some ways, I think math teachers might be the worst because they work with straight numbers. They devise formulas, weights, point systems, and the like to rationalize the grades the students "earn". Of course they do...because no one teaches how to grade in college. Even those fancy formulas are subjective and is one teacher's fancy formula equivalent to the teacher teaching the exact same course across the hall? An example...Billy is in an algebra class and getting good grades. For some reason (probably to accommodate his trumpet lessons :-)), Billy has to have a schedule change. He gets moved into the algebra class of another teacher. All of a sudden, his grades are much lower. His mother is on his case asking him what has happened and what is he going to do to bring up his grade. Billy is baffled and desperately trying to explain to his mother that he hasn't changed anything. He tries to explain that he has no idea what is happening but of course, his mother doesn't think he's telling the whole truth. What most people outside the teaching profession don't understand is that Billy hasn't changed a thing. His new teacher has a completely different fancy formula for determining grades.  So, what is the answer? This is what teachers have to start discussing.

Back to the conversation I had recently. The math department chair from one of the schools at which I'm assigned as a resource relayed that the principal said that homework should be given a particular weight when determining grades. He said that because their school is located in a lower socio-economic area and many students don't do their homework, the homework portion of the grade should be almost insignificant when deciphering final grades. The math teachers in the department were "all over the place" in how they felt about this mandate. I think most were just upset that the principal used the description of their population as a rationale for any policy. He is in his first year as principal and had previously been at a more middle-class school (where I presume students do their homework). 

Where do I stand on this issue? I want teachers to think carefully about all their grading policies and I think they should be coming to some consistency within a school and possibly within their district. As far as homework, I think teachers have to wrestle with the question of whether their homework policy is academic or behavioral. If a homework policy is formulated to be punitive, it has lost all value for students. By that I mean, is a teacher's homework policy designed to be a "gotcha" or an "I'll show you who's in charge". Let's go back to Billy. Billy is passionate about playing his trumpet and he is a good student. Sometimes he practices his trumpet in lieu of doing his math homework, mostly on nights when he understands what has been taught in class and he feels he doesn't need the extra practice. He gets A's and B's on all his tests and quizzes and always does his classwork. He has demonstrated that he understands the content. BUT, he doesn't get an A or B on his report card because his teacher penalizes him for missing homework assignments, thinking silently, 'he doesn't deserve an A or B because he doesn't do everything I ask'. Academic homework policy or punitive? Will his report card grade really describe who Billy is as a math student? 

My opinion...I think homework should sit off to the side as a teacher determines grades. Determine grades based on what a child knows and can demonstrate using all manners of assessment. If the child is on some sort of cusp, look at the homework completion and then use it to make decisions. Does the child's homework show diligence and understanding? Does a lack of homework completion mean that the homework might have been unnecessary for this student? If a child is failing the class, there are probably other reasons besides not doing homework. Is their a valid reason for penalizing a student for a lack of homework completion?

Let's start talking with our colleagues about our grading policies instead of talking 'about' students. When we get together, let's have professional conversations about our craft. Let's start with a conversation about grading. I found that every teacher I talked to had a very strong opinion about their personal method of grading.

BTW...I asked my son's geometry teacher to consider the fact that my son would be "wearing" her geometry grade like a ball-and-chain for the rest of his academic career, long after she could even remember who he was. That grade would be averaged into his GPA and become part of his high school transcript when he applied to college. I don't think she had any understanding of the point I was making. She had her grading policy and she was sticking to it. The end of the story was that my son ended up with a C in the course. We never heard from the teacher again and I think she just wanted to be left alone. My husband and I always wondered whether the C truly indicated how our son was performing in geometry or was it the "I'll just give a C so they don't bother me anymore".

If you're interested, read Tranforming Classroom Grading by Marzano. There are also other excellent, thought provoking books and articles published on the subject of grading.