Tag Archives: independent learning

Flipping, Week 1: Stop the Blabbing.

One of my major goals this year is to stop talking so much. Even in my tenth year, I still spend far too much time explaining, questioning, and presenting in front of the class.

The nature of this talking has changed a lot though. When I first started teaching, it was almost all explaining. That's what I thought good teaching was all about - if you could just explain it the right way, then everyone would get the concept you are teaching, right? A perfect lesson consisting of a perfect development, a perfect explanation of all concepts, perfect example problems, and perfect students. This is how I looked at it during my summer training, and before I got into my classroom.

That changed pretty quickly once I actually got started. Explanations were important, but more important was getting students to be somehow involved. My coaching from administration was focused on good questioning over talking and explaining as a way to do this, so I put a lot of energy into this during my first couple years, and it has since stuck.

The problem is that I am often addicted to asking questions when it's really time for students to get working and thinking on their own. I can ask questions like crazy, which might have really impressed administrators in my room at one time, but it probably infuriated (and still infuriates) my students to no end. As good of a question I could have asked, they were still just sitting there thinking and not doing any active learning on their own. Furthermore, when the one or two students do answer a question I ask, it isn't necessarily a real indication of what thinking is going on in the heads of the other students in the room. Students who self select to participate make for a bad sample for the level of understanding in the rest of the room.

The technique to address poor participation (as pushed by my administration in my first couple years) was to cold call students. This is a bad sample in the other direction - pushing a student to go from full listening mode to full participation mode with the rest of students is not an effective way to make dialogue an important part of what goes on in your classroom every day. This is especially the case for students that have poor self esteem about math in the first place. Good conversation is rarely one or two to many. When was the last time you saw twenty people actively involved in a discussion? Why would you really try to get that going in a classroom when it doesn't work for a room full of adults at a faculty meeting?

In reality, real learning doesn't look like a kid staring into space pondering a good question. It involves experimenting, testing a theory, writing down an idea and trying it out. It involves taking what you have produced out of your own thinking and getting active and reliable feedback.

Back to my main point - I am attempting this year to put any direct instruction for a particular day's lesson in two minute video chunks, and limiting the number of these to no more than four per day. A few really nice things have happened since doing this:

  • I'm putting a lot of thought into exactly which ideas are best left to video or direct instruction, and which ideas will come out through conversation and the activities. Some things are better taught in a big group, I'm not going to deny this to be the case. Some things are better learned one-on-one, and thinking about the difference has really changed the way I organize the activities for the day.
  • My students are spending a lot less time listening to me, and more time engaged in the videos and what I ask them to do. The videos sometimes include straight example problems, but I try to include a couple things for students to actually do, write down, or talk about with the person next to them as they watch. The conversations that students have during the videos are really rich (and remarkably on-topic), and are so much more useful than having me tell them things while they stare at me.
  • I can do other things while they are working on the activities I give them. I can see how they are watching the video and make suggestions on things they should be writing down. I can test their knowledge by asking one-on-one questions and get a really good sense for the level of understanding for each student. I can look at quizzes I gave at the beginning of the period, make comments on them, and have a conversation with the students about their work before the end of the period! The quality of my interaction with students has been much higher than before, which has resulted in larger amounts of quality feedback. That is really the goal here.
  • My ESOL students are loving it. They can take time with vocabulary, which is the hard part for them, and make note of the mathematics concepts at the same time. Some students are using the videos to create their own glossaries in other languages. I've always suggested that students do this, but until now, I haven't seen them do it so well, let alone of their own volition.
  • The learning in my room is messier than ever before - everyone is at different points and is having different conversations. There are papers all over the place. Students are crowded around tables working and are facing all different directions. Seeing this sort of thing happen my first year would have meant that this was a spent lesson, that I had lost them. Here, it just looks like (and is so far proving to me) good learning experiences for students.
  • This post is partly in response to one of the new blogger prompts about what I want my students to remember ten years down the road. I really don't care if they remember how to factor quadratics. Moving to a more student-centered learning model though has made the students in charge of making sure they understand what they are learning. I would love if students tell me in ten years that they learned how to learn something new in my class. Real learning is messy, and actually doing math (not watching), making mistakes and growing from feedback is part of the game. There's not as much room for that in the more traditional math structure of "I-Do,We-do,You-do" model because the last part is where the real learning happens. Maximizing that part (and simultaneously providing ways for that feedback to happen) is the real meat of teaching, and it's where I am focusing my energy this year.

    Here's to keeping it going as long as possible!

A tale of two classrooms - which is yours?

Consider the two scenarios below in the context of your own classroom, or if you are an administrator, in the context of how you might react to the following situations occurring in your teacher's classrooms. Assume the class skill level is normally distributed from weak to strong.

Situation one:

You are teaching a lesson in a mathematics class on a skills-heavy topic - perhaps solving a quadratic equation with rational roots. You have a lesson completely planned, a great intuitive hook problem at the beginning, and plenty of pivotal questions to shape student understanding around the process. Perhaps you have a carefully crafted exploration that guides students to figure out for themselves exactly how the procedure works. You have students work in groups to create a set of procedures to follow, and then students individually solve practice problems and compare to each other to check their work and help each other.

Situation two:

You are again teaching students to solve a quadratic equation with rational roots. You give them the set of practice problems at the beginning of the class and briefly review what it means to solve an equation - what should your final answer look like on your paper? You then give them textbooks, laptops with internet access, Geogebra, graphing calculators, whiteboards - all sorts of materials and tell the students your expectation is that they learn using whatever method works for them how to solve the equation. Some look on Youtube for hints. Some students might already know how to solve the equation - those students quickly tell their friends how to do so. Some decide to graph the quadratic function, get the solutions to the equation first, and then try to get those answers algebraically. You find that some students are struggling, so you are able to give additional help to those students, and they do seem to understand the general procedure after getting some help from online videos and their peers in the class. By the end, everyone has solved at least a couple of these types of problems on their own.

Suppose also that the next day you give the students a quiz with two of these problems, the second with an additional layer of difficulty. The strongest students get both questions correct, and the rest get at least the simplest question correct, with some fundamental flaw in reasoning or procedure for the second. In other words, I want the measured outcome of both situations to be roughly the same.

Before I go on, let me be clear about my own background here. When I was first trained to teach in New York City public schools, I was expected to teach lessons fitting the mould of the first scenario. The "I-do, we-do, you-do" model or the developmental lesson were the names often given to this type of classroom. The principal expected teachers to stick to a well defined structure for each lesson, and he was in and out of classrooms frequently to ensure that this was the case. The idea was that the structure helped with classroom management, made learning objectives clear to students, and made it easy for students to take notes and keep track of what they learned. Another part of doing things this way was that there was some level of control over how students were guided to an answer. If the activities or examples are shrewdly selected, a lesson doesn't devolve into situations in which it is necessary to say "Yes, [generic shortcut that students will find if it exist] works in this case, but it won't always do so."

Since leaving that school, I've taught in environments in which I've been able to experiment a bit more and try new instructional methods. In my current school, I am supported to use whichever methods I choose to help my students learn. I find, however, that since my mind is not really made up, I go back and forth. I am more likely to use the first situation in Calculus and geometry, and the second in physics and algebra two, but there are exceptions.

Which of these classrooms is yours? What are the advantages and disadvantages of each? Since I'm the one writing, I get to share first.

Situation one has always been my go-to model for helping students that are weak in arithmetic, algebraic skills, or overall organization. These students benefit from seeing clear examples of what to do, and then from getting opportunities to practice either with guidance through whole class, pair, or independent work. In many cases these students are not sure how best they learn, so they follow steps they are given and trust that the path their teacher has selected will be one that will eventually lead them to success. In addition, my presentation and activities can be carefully chosen to make it so that students are not just memorizing a procedure, but are required to go through thinking to understand the mathematical thinking involved.

In the larger context of teaching mathematical thinking, however, this method can lead to students expecting or relying on the teacher to provide the frame work for learning. It can (though does not necessarily, depending on the group) lead to a mindset on the part of students that it's the teacher's job to explain everything and make it easy to learn. I do believe in my responsibility to know how to explain or present material in many different ways to help students, but there are some concepts that just aren't easy. They may take work, practice, and interaction with me and the other students to understand and apply.

Situation two offers a bit more in terms of empowering the students to take control of their learning. It lets the students choose how they learn a concept best, whether by direct instruction, watching a video, reading example problems, or working with peers. If students learn the material on their own, have seen it before, or grasp the concept quickly, this offers many opportunities for using that knowledge to help other students or challenge them with more difficult questions. It does not require that material be presented in a linear fashion, from simple to complex, because it offers opportunities to jump back and forth, working backwards and from different representations to eventually come to an understanding.

In many cases, this offers the opportunity for the teacher to show what it looks like when figuring something out or learning something for the first time. I have read many people that refer to this position as the 'learner-in-chief', a concept I really like because I think students need to see that learning is non-linear, filled with mistakes and the testing of theories. Getting it right the first time, while nice when it happens, is not the norm. Sharing this fact with students can be a valuable learning experience. While it is nice to see a concept presented perfectly, it contrasts with the real learning process that is a lot more messy.

I have seen a couple negative factors that need to be considered in implementation, the first concerning the weak students. These are often the students that perhaps lack the background knowledge to figure out a mathematical procedure, or the self control to sit and figure something out on their own. What is nice in the second situation, assuming the other students know they must complete the assigned lesson and work for the day, is that the group of these students is a smaller one than the entire class. It is an example of differentiation in action - the students that need direct instruction to learn, get it. Those that do not, are able to reinforce and apply their learning habits by learning on their own. This situation also presumes the students are motivated to learn the concepts, though being able to do so in their own way and being held accountable for their learning may improve how some students react to the prospect of coming to your classroom each day.

Another downside that I've seen in practice is also a downside of students teaching each other mathematical processes. Students will often teach 'just the steps' and none of the understanding. While this is not the end of the world, it is something that teachers must reinforce with their students. The idea that mathematics is not just a list of problems, but a way of thinking, is strengthened by the arrangement in situation two. If the arrangement of resources available to students is sufficiently broad, the students will be able to piece together the overall concepts as a group. This entire process needs to be modeled, however, early on in the year to teach students both how to do it and what the expectations are.

For administrators, I imagine that walking into a classroom like this can result in an initial feeling of chaos or disorder, and might therefore lead to the feeling that this is less ideal than that presented in situation one. To be clear - it is possible to run a classroom poorly in both situations, and classroom management is essential to maximize the student learning occurring in both. Ultimately, a classroom filled with students that are all learning in their own way to reach a given set of learning standards, is the holy grail. It is important to be given the opportunity, training, time to interact with colleagues, and the necessary resources to make this feasible in every classroom. The important part, chaos or no chaos, is to determine whether (or not) learning is happening in the classroom. My main point is that there is fundamental difference in the philosophy of learning between the two classrooms.

Which is better? I'm not sure. I go back and forth between the two, depending on the concepts we are exploring on a particular day, or he problems we are looking at. Some of the most fulfilling lessons I have taught have involved giving the students a challenging problem and letting them figure it out in their own way. Yesterday in Calculus we did a number of activities that led to the Fundamental Theorem, but I was guiding the way. I think keeping it balanced is the way to go, but that's partly because I haven't structured my courses to be taught completely one way or the other. Maybe, in moving to Standards Based Grading, it might make it more natural to move toward more of situation two.

What do you think?