There's Never Enough Time

I've been in a number of classrooms lately when the displayed objective has been something to the general effect of "SWBAT (teachers will know that stupid combination of letters) review content in order to be successful on tomorrow's test." The teacher then proceeds to hand out a packet of worksheets, play a game, or have the students skim their textbook to determine what topics they are shaky on.

What is the purpose of taking a whole class period to review? Is there a good way to do it? Most math teachers seem to feel that this is a valuable use of time. What I've determined from observing and listening to the students is that this becomes the time to confirm that they might not do well on tomorrow's test. Does it motivate them to go home and study for the test? Not, if the students I spoke to are being honest. One young man didn't have a single problem correct on his worksheet. When I tried to help him, he said, "I know how to do this stuff." I guess because I was a visitor in the classroom, he didn't quite trust my opinion that he needed some help. In another class, the teacher had the class broken into 2 teams to play a game that used problems from the chapter. As I watched I realized that only the team member who was 'on' was doing the problem. Her teammates were encouraging her but the teacher said they couldn't help her. Everyone was supposed to be doing all the problems and confirming their answers. the 'good' kids were doing that. When all was said and done, not every student on either team had a chance to solve a problem. Plus, what does the teacher really know about what the students understand? The teacher might know that a student knows or doesn't know how to solve one specific example. There has to be a better way!

My opinion...if you're going to spend a whole class period to review the day before a test, find and use a rich task that utilizes all the skills of the unit. Allow the students to work in collaborative teams to work on an engaging task together. Make sure the problem requires the students to use the content from the chapter in connection with prior learning. If the teacher circulates while the students work, he may have a chance to find out where the students are being successful and where they are still struggling. Individual intervention can be done or the teacher can pull the class together briefly to clarify something with which everyone is struggling.

I just think that if one class period is used for every test review...hmmmmmm, about 10 to 12 units times 2 (1 for review, 1 for test), the teacher and students have used 20 to 24 days. Add in a possible day after the test for error analysis and all the days lost for mandatory standardized testing AND there goes more than a month of valuable instruction. No wonder there is never enough time.

I think that teachers need to learn to be more efficient at continually building review into their lessons. If mathematics is truly a 'building block' curriculum, then each day should have very natural connections to the day before and the day after. We need to look at the big picture, the story of the entire course, and find ways to tell that story without chopping it up. Maybe there's never enough time because we are blaming all the wrong reasons? Again, hmmmmmmmmmmm.....

 

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Hurricane Sandy...Exciting Times

The east coast just survived a major storm. It certainly wasn't the first and won't be the last. Our schools have been closed for 2 days, so kids and many teachers are very happy. It is a wonderful feeling to not have to venture out of the house in dangerous conditions. Not everyone has that luxury; witnessed by the cars and trucks on the highways whenever the news would flash to a live picture of the interstates. But, must the closing of schools be a disruption to student learning?

As we, as a society, become more and more connected by the Internet, texting, etc., is there a way to put some sort of virtual learning in place so that our students don't really lose two days of school? My brain is coming up with many reasons why this isn't possible at this moment in time, but I'd rather think about how amazing this could be. The naysayers who read this can comment on the negative and list them, if they'd like to waste energy on that list.

Last night, even as each of my co-workers reported that they had lost power, they still had their smartphones and were able to continue to stay connected. Maybe our school systems need to begin to brainstorm ways to put some of these learning opportunities in place. This is such an exciting time. Putting virtual learning opportunities into place could be one more way that our school system could capitalize on days when the schoolhouse can't be open. Witness how quickly your own children become bored when the power goes out. But, as mentioned, my own son never lost his cellphone service. He was texting, etc throughout the storm. Could your students be Tweeting with you when school is closed? Could teachers have Facebook groups? Could you all come up with even better ideas?

Very exciting times!

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The Avengers

Watching the BluRay of the Avengers...less than 5 minutes in and Loki says, " I'm burdened with glorious purpose."

As a reflective teacher, I think this is my new mantra. In my mind, this says it all!

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Equity

The word equity has been used in education for many years. Mostly it seems to refer to whether all student sub-groups are being given the same opportunities to learn. It has been argued that girls are not pressed to justify answers or even answer higher-level questions. It has also been argued that the same is true for certain minority students. It is my premise that the majority of teachers do not do this consciously. I think that most teachers have a nurturing streak that sometimes interferes with equal opportunity for all students. By this I mean...most teachers don't want to embarrass their students or put them on the spot. So I've watched as teachers ask a student a question and if the student can't answer immediately, they move to another student. The initial student is off the hook but what did they learn? I'll let that question hang for my readers to ponder.

But...the more I read and think about the art and science of teaching mathematics, the more I believe that equity is a far greater issue than looking at the opportunities given to sub-groups. I'm starting to believe that most every decision a teacher makes while planning an executing their lessons can be a gained or lost opportunity to teach all students equally.

PLANNING: How is the teacher utilizing the resources available for planning and executing their lessons? In our county, every course comes with a curriculum guide. These guides are created using the combined talents of the master teachers who teach that course. Their ideas, instructional strategies, and best practices are written into the guide, giving all teachers of that course proven ways to help students be successful. The writers also purposefully write connections among lessons and units. If a teacher, not aware of the big picture of the course, chooses to skip the mathematical task in one unit, they've lost the opportunity to continually refer to it as the course progresses. When a teacher makes the decision not to use the curriculum guide and the prescribed strategies and activities, they have made a decision to rob their students of proven opportunities. (an aside...I know there are some readers who are starting to fume at this point. I know that there are other ways to present content.) The hard fact is that our school system is the 25th largest school system in the United States. We have around 11,000 algebra I students. We also have movement within our county. By that I mean that many families move from an existing area of the county to another part of the county. When a student changes schools, there needs to be consistency. So, Teacher A decides not to follow the curriculum guide and skips any task that requires gathering materials or manipulatives or engages the students in discussion. That teacher has just robbed their students of an opportunity. The students who never leave the school have no idea that other students in the county are being given richer opportunities to learn the mathematics on a deeper level. To me, this becomes an equity issue. What gives a teacher who has signed a contract to teach the curriculum the right to make decisions like that. (Now for the readers who are really fuming...I'm not talking about the teachers who are thoroughly planned and provide their students with alternative lessons that maintain the integrity and fidelity of the written curriculum. I am talking about the teachers who disregard a rich task and change the lesson into a direct instruction, "I talk, you listen" experience). Unfortunately students can not easily get a new teacher if they don't change schools.

LANGUAGE: I hear mathematics teachers using slang terms all the time as they develop content. I think this is a huge equity issue. I have teachers argue with me and rationalize their use of incorrect terminology, like saying plug instead of substitute. One 6th grade teacher said that he and his peers in high school in calc III used those slang terms all the time and so did their teacher (who he says was the best teacher he ever had). My rebuttal to him was that by the time he was a senior in high school, he was 'in the club' so to speak. Experts in various fields have a tendency to create short-cuts in their communication when speaking to peers. What I tried to explain to him is that he had been given opportunities to hear and use the correct vocabulary and symbolism as he progressed, giving him the chance to get into the 'club' and speak in the shortened language. I told him that he was teaching 11-year olds who should be given every opportunity to hear correct language and see correct use of the mathematical symbolism so that maybe someday they, too, could get in the club. Every time a teacher takes short-cuts, whether with language or by teaching some trick for a mathematical process before developing the concept, they are robbing students of opportunities from which many of them may never recover. That is an equity issue.

I believe that teachers need to remember that they are teachers every single second that students are present, whether in the classroom or the cafeteria, or talking to them individually. Every interaction becomes an opportunity to show students the importance and beauty of mathematics. If we consciously think about our roles of assisting our students to see and believe in the importance of mathematics and model this love of subject with the execution of well-planned lessons, I don't think we'd have to worry about equity.

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Flipped Classroom...New Idea?

I have been reading about the notion of the 'flipped classroom' for about a year. The new young teachers who are getting much credit for this "new" idea should be very proud. Honestly, my thought is that they have taken a tried and true technique and added some bells and whistles to it. Jonathan Bergman and Aaron Sarns, both from a school in Colorado, early on, discovered a piece of software that would allow them to post PowerPoints online. In their earliest iteration of the technique, the teachers designed their PowerPoint presentations to be viewed by students who had been absent from class. The online lectures started to spread beyond just their students and the two teachers were soon asked to speak to other teachers about this idea of a flipped classroom. The idea evolved to mean that students were expected to watch podcasts and online videos to prepare for class the next day. This frees up valuable class time for the students to engage in cooperative endeavors such as projects or other activities that use and extend the ideas from the homework. The teacher can facilitate the learning of the connections of skills instead of constantly teaching those skills that underlie the real world applications.

What a powerful idea...but...not new.

My school system has been using the UCSMP Transition Mathematics program since about 1985 and I taught this program from that time until I left the classroom in 2000. Since coming to the Secondary Mathematics Office, I have been deemed the 'keeper of the flame' for the program. Most recently, I was the lead writer and editor of the curriculum rewriting when the third edition of the textbook was released. (But, I'm slightly off the topic.)

The cornerstone of this program (and all of the prior and subsequent programs in the series) require the students to read ahead. The students read the lesson from their textbook that will be taught in the next class. They read, take notes, and complete some exercises that cover the ideas from the reading. The exercises are nothing more than questions that clarify whether the students know what the main ideas of the lesson are. The students are instructed to use their notes to answer the questions. If they can't answer a question using just their notes, they should go back to the text and enhance their notes. Never are the students required to read for mastery of content. Their prereading is nothing more than reading a movie review or the summary on a book jacket. The students bring some knowledge to class which allows the teacher to form the lesson in a way that does more than regurgitate the reading. If the teacher plans the lesson in such a way that reiterates the text, the majority of students will stop prereading. Well heck, wouldn't you? Think of your college classes that required a textbook that was only used to build the muscles it took to carry it. If the professor spewed back the content verbatim, why take the time to read?

The students come to class with their notes, each having a personal understanding of what was read. The teacher's job is to design a lesson that uses the content in such a way that all students have a more thorough understanding of what they preread. They are also encouraged to enhance and add to their notes during the lesson. The prereading also gives the students a chance to formulate questions to ask the teacher or their classmates. So, isn't this the original version of a flipped classroom? Hasn't prereading always been a technique used to give students a chance to prepare for the content that will be presented in class?

As I mentioned previously, this idea of giving this technique a new name, the flipped classroom, is wonderful if it encourages students and parents to buy into the idea of preparing or taking some responsibility for bringing something to each and every lesson. But, let's not act like this is a bold new frontier. Anytime students are asked to be prepared for class and held responsible for being prepared allows the teacher to take all students farther and achieve more in the little time time available in class.

 

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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...