PCK and TPACK...Teaching Keeps Evolving

It is primary Election Day in Maryland which means a day off for teachers and students. I suppose I should be pondering my election choice but I made that determination months ago. So, instead, I've spent the morning catching up on my professional reading and thinking about an upcoming interview.

This morning I sat on my deck having a cup of coffee and reading the Spring 2016 NCSM Journal summarizing research on the technological knowledge of secondary mathematics teachers and its effect on student achievement. When I finished the article, I started to think about things that circle around my brain regularly; how difficult it is to be an effective teacher. 

There was a time in the not so distant past that HR people would accept career-changers with strong mathematics content knowledge to receive alternative pathways to teaching. When these retired engineers, veterinarians, and other math-related applicants were hired in August and had resigned by November, no one seemed to understand what had happened. But L.S. Shulman in the late 1980's and then Deborah Ball in the early 2000's started to realize and research the difference between knowing math and knowing math to teach it. Pedagogical Content Knowledge (PCK) is very different than being able to use math in an occupation. Teachers must understand the instructional techniques specific to the subject matter. Only when a teacher has the pedagogical knowledge to accompany the content knowledge will they be on a path to affect student learning. Without PCK, teachers don't have all the tools necessary for establishing an environment conducive to learning. When teachers (and most often career-changers with no real teaching experience) have no or low PCK, they rely on strategies they experienced as learners, utilize teacher-directed as their main instructional strategy, or use repetitive examples during instruction. 

Teachers now need to layer the use of technology onto the appropriate instructional strategies they use to develop content. Researchers have now defined a new kind of PCK (Neiss, 2014) that defines the knowledge necessary to teach mathematics with technology. From this perspective, teachers need to understand technological content knowledge, technological pedagogical knowledge, and how to incorporate that knowledge to anticipate the learning needs of students. Teachers must now promote the development of mathematical understanding through the use of technology. For this, the TPACK model was developed to show how decisions about curriculum, assessment, teaching, and learning are related. There are 5 teacher use levels within the themes. A teacher is categorized (these levels can shift depending on the theme or piece of technology) as recognizing, accepting, adapting, exploring, or advancing.

Of course there is a sweet spot where content knowledge, pedagogical knowledge, and technological knowledge overlap on the Venn diagram. Much as Charlotte Danielson describes her 'distinguished' category as a place that teachers visit periodically, I would imagine that teachers would find it hard to live in the TPACK sweet spot for every lesson every day. So, what are the barriers keeping teachers out of the sweet spot and what can resource teachers, coaches, and/or administrators do to help?

Researchers since 1999 including P.A. Ertmer and A. Ottenbreit-Leftwich have identified barriers to teachers' integration of technology in the math classroom. First-order barriers include lack of training opportunities to hardware and software problems, while second-order barriers are about teachers' attitudes, beliefs, and skills. Combining these barriers with low PCK result in low level learning experiences for students.

There is much talk around individualization and customization of student learning, but what is happening with personalization and customization of professional development. Maybe professional development designed by analyzing teachers' needs and developing their PCK will enable them to become more successful in identifying their students' needs, interpreting student error patterns, engaging their students in a true conceptual development of mathematics, thus being open to increasing their instructional capacity. This kind of professional development might also open the door to layering on the use of technology on a higher level of the TPACK model. A teacher with a low pedagogical content knowledge will, most likely, have difficulty progressing beyond the recognizing or accepting level of technology integration. A teacher with a low PCK with its inherent unproductive beliefs will only view technology as an efficient way to confirm computation. With the proper individualized professional development, a teacher could see technology as a way to engage students in high-level thinking activities using technology as the valuable, high-powered learning tool is is meant to be. Without changing teacher beliefs about how they define student learning and increasing their PCK, technology integration will not happen just because a teacher has calculators or computers available in the classroom.

So, as I finish my coffee and head out to vote, I will reflect on the article a bit more and try to develop some specific strategies for raising both PCK and TPACK in the teachers with whom I am lucky to work and have conversations. Please exercise your right to vote!

Mathematics is a Story

As I look forward to entering my 40th school year, 26 in the mathematics classroom and the rest as a secondary mathematics resource teacher, I spend most of my waking hours thinking about the art and science of teaching. I have finally hit upon a metaphor that seems to resonate with educators.

I believe that mathematics is a complete and coherent (beautiful) story that begins at birth and ends, I guess, at death. For most, it formally begins in school and continues until a child is out of school. Each of a child's teachers is the sacred keeper of a chapter of this story. It is the teacher's job to make their chapter the most engaging piece of the story possible. Characters, plot, storyline must all develop in a logical sequence. These story aspects must fit coherently into place based on what came before. Connections must be made to previous characters and students need to understand how the characters will fit into the subsequent chapters. The story needs to be exciting, must capture attention, must motivate the students to want to hear and learn more. Think about what has happened to each and every one of us when we started to read a book and it was boring or, for whatever reason, didn't grab our attention. It doesn't matter if someone else tells you what a great book it is, most of us will not pick it back up. Especially when there are so many other options available. Once interest is lost in a story, it's difficult to gain that interest back. Now, think about a story where you were interested but all of a sudden, new characters are introduced or the plot line seems to diverge. Well, now the story loses me for another reason and, most likely, I've lost interest.

I'm thinking about teachers who use slang instead of correct mathematics terminology. As a student, I'm losing the storyline and the characters. Think about the teacher who teaches tricks instead of conceptual understanding. As a student, I can't follow the plot line. Think of the teacher who never lets students engage in a discussion about the story. As a student, I'm tuning out because I have all these thoughts in my head about the story but no one lets me share. Wow! What is the actual probability of any student getting 12+ amazing storytellers in their school life? Unfortunately, the probability must be near zero. This has to change! Every student is entitled to teachers who believe that their chapter of the story is sacred and worth conveying as if it is the beautiful wisdom, the flame of life. This means that teachers need to understand the content and they need to understand the pedagogy that underpins that content.

As I begin this new school year, I am committed to having conversations with math teachers about how their chapter of the mathematics story can be developed and shared with students to keep them clambering for more. I want every math teacher to know the joy when their class period is over and the students say, "WHAT, class is over already?!"

Is It Really New?

Went to a mandatory two-day training last week to be introduced to a digital platform for housing our curriculum. The platform is a component of a system our county has been using for about 8 years. Throughout the training, the facilitators touted the benefits for teachers of having the curriculum available digitally. We were shown how the teachers would be able to take the core documents and develop and change them to fit the needs of their students. They would be able to add resources to make the curriculum fit their needs. WAIT! HOLD UP! Haven't our teachers been doing that forever? Maybe not storing their changes digitally but certainly they have been supplementing the county curriculum however and whenever they chose. Many of them have supplemented and changed programs so much that the core purpose of the course is barely recognizable. As our office has pushed to make courses student-centered; writing instructional guides for the curriculum that suggest interactive activities, connections among content, and ways to improve the pedagogy, many teachers have consistently supplemented or ignored the suggestions giving students worksheets and tricks; offering the same old same old and wondering why their students can't retain information or problem solve. How will a digital platform change instruction?

Regardless of how the curriculum is housed, the teacher is the strongest indicator of whether students will learn the content. Great teachers need nothing more than a good, solid curriculum and their expertise, creativity, and relationship-building skills. It doesn't matter whether the course is built on paper, carried around on a thumb-drive, or accessible online. Showing a video that peels away the layers of a cell is no more effective than when a teacher had a pile of transparencies overlaid in such a way to peel away those same layers. If we think students are more receptive to watching and hearing a stranger on a video present content then we better look inward to find out why they aren't tuned into us as their teachers.

The biggest disappointment at the end of the last day of the training was when the facilitator showed a video of the digital platform being used by a teacher in another state. This commercial for the program did more to turn off the audience in the room than anything that had happened previously. It was the end of the second day and we were starting to brainstorm the capabilities of the program. This video showed the worse case scenario. The teacher in the video had built a 'playlist' of assets that became her 'lesson'. She then turned on the playlist, which played on a Smartboard, and the students sat taking notes. She basically was unnecessary after she hit 'play'. REALLY?! How is this different than death by PowerPoint or death by teacher droning on for 45 minutes? This was new? No, this was kids staring at a screen taking notes while someone spewed information.

Thank goodness our curriculum office personnel were appalled at this scenario. The video gave us a chance to share ideas for using the platform that would make our classrooms look different; a chance to talk about how to use the platform to individualize learning, and a chance to realize that without the principals on our side this will not work. We have been trying to change what happens in the classroom for many years. But, what is evaluated at the schoolhouse level will always trump any pedagogy shifts initiated in the curriculum offices. If our principals interpret what that teacher did as 'using technology', no change will occur.

Nothing is really new. Great teachers will use whatever means necessary to help students achieve. The means may look different but great teachers will never become obsolete because they will continue to grow and reflect. It will be up to the rest of us, principals and curriculum offices, to make sure the other teachers step up and attempt to achieve greatness or at least effectiveness.

Posted with Blogsy

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.....

 

Posted with Blogsy

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!

Posted with Blogsy

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!

Posted with Blogsy

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.

Posted with Blogsy

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.

 

Posted with Blogsy

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.

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.

Professionalism

The tag line at the beginning of this blog mentions the word professional. That is a loaded word in teaching. Many teachers bemoan the fact that the public does not treat teachers as professionals. If asked to elaborate, those same teachers will include the fact that teachers are not paid on a scale with other professionals. Teachers spend an equivalent amount of time in college, especially if it is taken into consideration that teachers must continue to accrue credits to renew their credentials. So, where is the break-down in perception?

I have heard the argument that everybody goes to school and therefore, everybody has an idea of what a teacher's job entails. Many will argue that this entitles the public to an opinion on how best to do a teacher's job. Certainly that argument negates the public from second guessing doctors or lawyers. In fact, I can think of no other professional career in which the majority of the public has participated. But, I'd like to explore another side to the break-down of why teachers may not always be considered professionals.

The best definition of professional that I've come across comes from some of the literature I've read on professional learning communities. A professional is someone who understands that they are building a career on an ever-changing, ever-expanding body of knowledge. A professional realizes that it is imperative to remain current on the trends and research of their profession. If a doctor was not reading their professional journals and keeping up with the latest developments in their field, I doubt their practice would remain vibrant or viable. Patients have a tendency to want the best and most recent treatments available. Those patients (clients) change doctors when the care they are receiving is perceived to be lacking.

I recently asked a secondary mathematics teacher what he was reading. He started to describe a NY Times best seller. I stopped him to clarify that I was asking what he was reading professionally. He said, "I don't read that crap". WOW...really!? There's a professional. I told him that he may as well have slapped me in the face. It seemed like a real affront to all the teachers who try to keep abreast of the changes in their field of expertise. I gave him my analogy about a patient being able to change doctors when becoming aware that your doctor is about to bring out the leaches (yes, I know that some doctors are taking a new look at using leaches). I asked him what recourse his students have when their teacher is still teaching as if it's 1954. There are very few students who can change their schedule to change teachers. And then there's no guarantee that when you get the better math teacher you won't get the antiquated science teacher.

If teachers are ever to be considered professionals on par with doctors or lawyers, they are going to have to seriously consider the rights and responsibilities of the designation. They are going to have to become a community of learners who understand that the profession as a whole is only as strong as the weakest link. There are so many professional organizations for mathematics teachers. A great beginning would be NCTM. I strongly recommend that mathematics professional educators start to embrace the changes in mathematics education. Engage in a conversation based around an article in a journal. Start to talk about what your classroom will look like as the United States transitions to the Common Core State Standards. How will the Standards for Mathematical Practice and the ideas of what it means to be a 21st century learner change your practice?

Honestly, I see how my 20 year old son learns. He has little patience for sitting without opportunities to engage in the discussion. He wants to obtain information on his own and use it to discuss and solve problems. If teachers continue to "stand and deliver" without giving students opportunities to fully participate, I think there will be two outcomes; the United States will stand firmly in 25th place in the world and teachers will make themselves obsolete. In the 21st century, information is easily obtained. Teachers need to be the ones who help students take the information overload and put it together in meaningful ways. That's what I think a professional understands.

Questioning 101

Effective teachers understand that learning is about exploring the unknown and that such exploration begins with questions. Not questions that are simply lectures in disguise; not yes-or-no questions that don't spark lively discussion. But, questions that open a door to deeper understanding. A teacher's questions should be planned as carefully as the lesson. Those questions should become the basis of the plot line of the story that the lesson tells. If the students were to listen to and answer, to themselves, every question that the teacher asks, their understanding of the lesson would be fairly complete by the end of the lesson. This would also occur because the teacher would be carefully calling on students to assist that understanding.

There are basically three types of questions that teachers use as they teach. The recall question is intended to elicit stored data from prior knowledge. It's close to a stimulus response mechanism. These are used effectively when a teacher wants to quickly review and bring to mind information that will be needed for the lesson at hand. (i.e. What is the title of the chapter?, Name the steps needed to solve this equation.) The teacher should be able to ask these questions of anyone in the class but if you want the review to be efficient, ask students who know the answers. Analytical questions require processing and are usually associated with cause and effect. They are used when you want students to distinguish, group, and explain what they know. (i.e. What does this topic have to do with yesterday's topic?, In what ways is this method different than the one we used previously?) These questions get to the heart of whether the students understand. Ask them slowly, with emphasis, and give all students a chance to ponder the question. Keep class attention as students answer because the answers are necessary for continued discussion and informal assessment. Application questions ask the student to move beyond the immediate information to arrive at their own constructed knowledge. Students theorize, state examples, judge, and extrapolate. (i.e. What would happen if every student knew what you now know?, What information would you like to ignore if you could?) These questions are asked sparingly or given in written form. That way all students have a chance to answer. These questions give students a chance to embed learning on a deeper, emotional level.

High-powered questions allow the students an opportunity to think. High-powered questions often time create quiet, often awkward moments. These awkward, quiet moments allow the most productive thinking to occur.

[read Teaching with the Brain in Mind by Eric Jensen]