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.

What Do "They" Mean by Rigor?

One of the latest buzz words being thrown around is rigor. Who is throwing this word around? Mostly I hear it from administrators; people who have the job of evaluating teachers. What is usually said is, "You need to increase the level of rigor in your classroom." I would imagine that most teachers stare back at their principal, or whoever just made the statement, like the puppy who turns his head to the side in that adorable gesture that means, "I'm listening but I don't understand." Unfortunately most of those same teachers will not ask the administrator for examples of what that means. Well, come on, would you go out on a public limb and admit you have no idea how to 'raise the level of rigor' in your classroom? Remember, this is the person writing your evaluation. Why would you put a seed in his brain about what might be perceived as a flaw in your ability to do your job? This is also a regrettable consequence of our current system but that's a topic for another day. (My suspicion is that the administrator wouldn't be able to give an example. You know what might be said..."I just know it when I see it." Yeah, thanks, that's helpful.)

What is rigor in the classroom? What does it look like in the classroom? Does it always look the same? 

One dictionary defines it as a noun meaning, 'the quality of being extremely thorough, exhaustive, or accurate; severity or strictness; demanding, difficult, or extreme. I can see how some of that can be translated into practice. The first part of the definition seems to be the responsibility of the teacher as he plans his lessons. In a previous posting about professionalism, I mentioned that teaching is built upon a large body of knowledge. It is mandatory that a teacher be very serious about the utterly exhaustive job of being thorough and accurate. But would an observer be able to see evidence of that thorough and accurate content knowledge? I would hope an administrator would bring in a content expert to ensure that no errors are being made. In the book, Teach Like a Champion by Doug Lemov, he describes a teaching technique he calls 'right is right'. The key idea of the technique is to set and defend a high standard of correctness in the classroom. When a teacher acknowledges as correct, answers that are barely formed, answers that use sloppy notation, or answers that do not use appropriate vocabulary, that teacher is robbing a student of a learning opportunity. That teacher has also just decreased the level of rigor in their classroom. Mr. Lemov believes that teachers must use the content and their expertise with the content to take all students outside their narrow band of experience. When a teacher is knowledgeable and enthusiastic about the beauty of their subject matter and can convey that to the students, the rigor increases.

I also think a rigorous classroom is one that gives students the opportunity to engage in the lesson in a way that helps them see the importance of mathematics. I was in a classroom recently where the objective of the lesson was to translate a word problem into an equation. Most of the students read the problem and started to solve it without writing an equation because an equation was not necessary to understand the problem. The teacher would not even validate any students who did not write an equation first. Once an equation was written and shared, the teacher proceeded to insist that every student solve the equation by going through the "proper" algebraic procedures. One young man sitting near to me was frustrated because he already had the answer to the problem. Does forcing every student to solve a problem in the exact same way make the classroom rigorous? Every problem that was presented during that period could be solved without writing an equation. Some of the equations were nothing more than arithmetic problems set equal to a variable. If the teacher would have thought more during the planning of the lesson, it could have been easily made more rigorous. How about carefully choosing or writing problems that are more difficult to solve without an equation? How about allowing the students to solve the problems in whatever creative ways they can and then comparing methods? Would that comparison of methods possibly lead to some amazing conversations?Might those conversations result in the equation writers convincing the non-equation writers to give it a try or vice versa?

I'm going to repeat myself and say that this generation of learners is not especially tolerant of being "talked at". They want to be in the conversation and they want to engage with learning. I've also watched them play video games. They are not afraid to try things, make mistakes, and revamp their strategies. Isn't that one of the habits of mind we want to promote in the classroom? One of the CCSS Mathematical Practices says that mathematically proficient students make sense of problems and persevere in solving them. Maybe if teachers plan appropriately to offer their students those opportunities they will never again be sitting with an administrator being told that they have to "increase the level of rigor" in their classroom.

I think I just wrote about professionalism again.