Saturday, May 2, 2015

Nuts & Bolts of Online Teaching

After reading a chapter from Kristin Kipp's book on online and blended learning, I couldn't help but think, "why didn't we read this weeks ago?". The chapter provided some very grounded suggestions in creating a syllabus, managing student information, and how to maintain teacher presence. All of which were exciting to read, but I felt much too late in the sequence of readings. This chapter, and maybe more chapters from her book, could have been used to ground some of the other more theoretical readings. Don't get me wrong, I love reading about learning theory. I really do find it an interesting subject! I just wish we had tried to create more of a bridge between theory and practice. 

That said, what I mostly got out of Kipp's chapter was that online instructors must put a lot of thought into their course design. But, face-to-face teachers should put a lot of effort into their course designs also. In an online or blended format, teachers also need to consider how the user interface could affect how students engage with the course content. Well, face-to-face teachers should probably consider those things too. How does the classroom environment influence what and how students will engage with the content. Teachers should think about the environments in which their students will learn whether online or blended or face-to-face? Yup.

Actually, that there isn't really that much difference between high quality face-to-face teaching and online teaching is pretty much the same take-away I had from DiPietro et al's piece on best practices in online teaching. (As an aside, I think it was on Twitter this week that someone criticized the term "best practices" because it encourages copying the technique without analyzing its affordances and constraints and thus effectively shuts down innovation.) Their extremely detailed and extensively descriptive table of findings was interesting. While reading through it, I just kept thinking, "shouldn't all teachers be doing these things?". The medium is obviously different in online teaching, but the actions and intentions of the teacher aren't that different. Good teaching looks like good teaching whether in an online or blended or face-to-face format? Yup.

This entire semester hasn't really convinced me that digital technologies can supplant traditional ones. Supplement? Yes! Replace? Not sure. There's some research that suggests that writing notes by hand is better than typing on a computer. However students who need help visualizing some of the more esoteric and/or abstract mathematical concepts can reap huge rewards because of what digital technologies can do. Form should follow function. First we need to figure out what students are supposed to learn then we can choose the appropriate technologies.

Monday, April 27, 2015

Technological Tools for Teaching

Mishra and Koehler have a really interesting take on technology and teaching. I think I read somewhere that if you want to get ahead, you have to get a theory. It seems like they've done just that. Rather, they've expanded an existing theory to explicitly include educational technologies.

I'm pretty familiar with Shulman's construct of Pedagogical Content Knowledge. (In the math ed world, Deborah Ball and a few others have proposed a special domain of teacher knowledge called Mathematical Knowledge for Teaching (MKT) which is located partly within PCK and partly with Content knowledge.) I really liked Misha & Koehler's Venn diagram for PK and CK and how they included a separate but overlapping domain for Technology Knowledge. I couldn't help but wonder if there are aspects of MKT that exist within the sphere of TK or any of the other categories they created. I'm sure there is. 

I also couldn't help but wonder if TK is more correctly placed within one or both of the other domains. Is it really a distinct domain of teacher knowledge? Maybe it is. I could go either way on it. Maybe the tipping point for me is that TK is not just about knowing about and how to use both traditional (black/white boards) and advanced technologies (computers/mobile devices) but is also about the ability to learn about existing and emerging technologies. When we started to introduce Interactive White Boards at my school, there were more than a few teachers who resisted them and complained loudly that the IWBs were placed in the center of the existing white boards essentially removing at least 1/3 of the board space. I liked the idea of the IWB but was also wary about its location in the room. Now, I am so glad that my principal had the foresight to put it front and center in the classrooms. I cannot imagine teaching without an IWB now. Many of the same teachers never really explored what an IWB can really do for your instruction. Students used to marvel at things that I did on the board; things that seemed routine to me but that other teachers hadn't tried to do.* I always figured that I couldn't do any permanent damage and that I should just try some stuff. We used to characterize that kind of attitude towards technology as being a "digital native". Mishra and Koehler conjecture that it's part of a special domain of teacher knowledge. Perhaps it is.

Perhaps TK is becoming so much a part of what we do that it really isn't separate domain of teacher knowledge. Last semester I had the opportunity to teach a course on technology in mathematics classes. I had anticipated that the students in the class would want to know about how to use hand-held graphing calculators to enhance instruction. I was very wrong. We did a few activities but they students seemed mostly disinterested. I have a few ideas why: 1) graphing calculators have permeated mathematics instruction so thoroughly that the students in the class, who were both preservice and inservice teachers, had already developed a familiarity with what the graphing calculators could do ; 2) the way I was using the calculators had not really occurred to them before and what I was doing was maybe a little too weird and different to really understand ; 3) the graphing calculators have been around for a while and the graphing capabilities are subpar when compared with newer technologies ; 4) the graphing calculator has a pretty steep learning curve and it often produces static results while newer technologies are considerably more intuitive and are much more adept at producing dynamic results which makes the graphing calculator a difficult tool to use. These are just a few of the reasons why I think the students were uninterested in graphing calculator but did take to other technologies.

One anecdote can't make a theory, but it might provide a little evidence that maybe TK is just part of what we do as teachers. Teachers with better / more developed TK or TPCK can use all sorts of technologies better or more flexibly. They can also adapt to new technologies faster because they have that kind of disposition. And maybe they even seek out or invent new technologies when existing ones can't do what he/she wants. It's certainly an interesting construct and I'd like to learn some more about it.

*Kids used to marvel at the IWBs. Nowadays there's a lot less awe. I think its because the crop of students I have now have had IWBs for pretty much their entire school career. It's hard to get excited about something that's been part of your routine for 10 years.

Sunday, April 26, 2015


I've been playing video games for almost as long as I can remember. We had a Magnavox Odessey 4000, then an Atari 2600, then a Nintendo Entertainment System. For as many fond memories I have of playing football outside with the kids in my neighborhood, I have even fonder ones of my brother and I sitting in our basement trying to beat Contra. My roommates sophomore and junior years in college had a Sega Genesis, which they used to play an NHL game incessantly. I've never liked playing sports games, so I sat out those sessions. A college professor introduced me to the Sony PlayStation and the revolutionary games that were being created for that system, like the Resident Evil and Final Fantasy series. I bought myself one when I was a senior and spent many Saturdays during my first year of teaching in front of the TV. I'm pretty sure that I should not try to accurately chronicle how much of my life I've poured into Final Fantasy VII, VIII, and IX. I used to visit friends and we would play Grand Theft Auto III or Final Fantasy X or Guitar Hero pretty much non-stop for an entire weekend. Even though a lot of the games we played were single-player, we used to watch and give suggestions to each other or cheer each other on. More eyes on the screen meant that few items went unnoticed. It also meant that we progressed through the games a little faster because if the person holding the controller couldn't figure out what do to, someone else in the room had an idea. After an extended break from gaming, and missing it quite a bit, my incredible partner bought me a PS3. I only play on weekend mornings now, because I know I could lose myself in the games.

There was a time when you almost had to read the instruction manual for a game because the controls for each game would be slightly different and often the designers would put tips and hints into the manual. Do games even have instruction manuals anymore? I haven't read one in years. Most of the games that I like to play build in a tutorial and then set you loose. You figure stuff out along the way by accomplishing the tasks that move the story forward. I guess I never really thought of it as learning, but I suppose that you do have to learn how to interact with the environment of the game world.

I can absolutely see the appeal of using video games and virtual environments in education. (I did love to play "Where in the World is Carmen Sandiego?") Games are designed to give you instant feedback on what you've done correctly or where you've made a mistake. You don't get a bad grade if you mess up. Even if you're playing a game where there is a score element, chances are you can replay that section of the game again to improve your score. (Some games are even designed in such a way that you can't maximize your score until you revisit early sections with the better abilities you acquire later in the game.) I wonder if a game that has an overt educational message or tone can be designed to do similar things. Maybe it can. I'm wary though. I wonder if what a player learns through a video game or through a virtual environment transfers to other domains. Perhaps it depends on what the player learns. 

But, assessing what students know through a virtual environment seems really promising. The idea that a virtual world can give educators a better sense of what students know and can do in novel situations is very intriguing. I would use something like that in my classroom, if it were available. We have some computer based assessments, but they are basically just a paper-and-pencil test on the computer. Science assessments seems to be a near-perfect fit. Sometimes math and science get lumped together, but I think there is a bigger divide between them than people realize. Science is pretty much based on empirical experiments while math relies heavily on thought experiments. Im sure someone can come up with a good math assessment that isn't just algorithm recall and implementation - which is often what we end up assessing in math class anyway. It's really hard to figure out if your students have developed any mathematical habits of mind. Hard, but not impossible. Maybe I should try my hand at that for a while...

Saturday, April 25, 2015

New Literacies

Thompson's book chapter on the "New Literacies" was enlightening. I did feel like it was really multiple chapters that had been unceremoniously lumped together. Each of the new literacies - data, video, photo - felt like it could have been expanded to fill a chapter a piece and form a section of his book. Maybe in the expanded second edition.

I am going to restrict my comments to data literacy. Not because I have nothing to say about the other ones but because I've been thinking about quantitative literacy for a long time. Data (quantitative) literacy is a vital skill. Since the economic collapse in 2008, I've been wondering about whom many people got themselves into trouble purchase houses or cars or whatever that they couldn't really afford but were lulled into thinking that they could. I know it's not really a mortgage broker's job to dissuade a potential homeowner from getting an inappropriate loan but maybe they should consult more. How many Wall Street stockbrokers and other financial retailers didn't really understand the mathematics behind the derivatives they were selling? Quantitative literacy is important because data is becoming more prevalent in society. (You can't even be a 1st grade teacher without being confronted with data on your students almost daily.) Because of the vaulted status that mathematics holds in our society and because so many people have anxiety linked to numbers, knowledgable people can use data and mathematics intimidate the less literate (or numerate, if you prefer). To become responsible citizens, people must learn that just because there is a number attached to a statement that does not automatically imply that the statement should be believed. The mathematics classroom seems like a natural place to expose students to data analysis. Some people might argue that practical applications of data should be the only thing we teach in school. I think that is misguided. We can't limit math class to data analysis. It would be like only teaching kids how to fill out job applications in English class. Sure, job applications have reading and writing but its just one kind. Mathematics has existed since the beginning of humanity because it can be used to answer questions about the world and it can be used to explore questions with no physical manifestation.  

Puntambekar and Hubscher present an informative argument about how the term "scaffolding", now so prominent in classrooms, had taken on a very different meaning than originally intended. Scaffolding was designed to be one-on-one, tutor and student. As the idea of scaffolds were applied to the classroom, the notions behind what constitutes a scaffold have been generalized. Puntambekar and Hubscher argue that maybe we've overgeneralized too much. I couldn't help but wonder if the term scaffold was really appropriate. Regardless, they present a few suggestions for how to implement scaffolds in classrooms to keep with the spirit, if not the letter, of the original idea of a scaffold. I found it refreshing that the researchers took into account that classrooms are (often) diverse and complex learning environments. I anticipated their suggestion that scaffolding tools be removed from environment over time as students need them less. That seemed perfectly reasonable to me.

I hadn't encountered Zygotsky's work until I started graduate school. I knew about Thorndike and Skinner and behaviorism, and Piaget and constructivism, but not Zygotsky and Brunner, and cognitive psychology. I wonder if other teacher preparation programs introduce the concept of Zone of Proximal Development. From the Puntambekar and Hubscher article, I got the sense that some teachers don't really understand ZPD. Without some basic understanding of ZPD, teachers would naturally struggle with how to structure supports for students and the importance of removing them when students no longer need them. I see this subtle tension whenever we use manipulatives in math classrooms. Some teachers and parents will argue against using base-10 blocks to model addition and subtraction because students need to know how to add and subtract without them. But that's not the right argument. Some students will need to use the models because the abstraction of adding and subtracting is beyond their *current* cognitive development. The base-10 blocks are one way for an adult to provide guidance to a student. Maybe addition and subtraction without a model are outside of the student's current ability but by providing the base-10 blocks, those concepts move into the child's ZPD. Maybe. I feel like I might have to read Mind in Society again over the summer. 

Is there a grand unified learning theory?

As I was reading Yasmin's Kafai's chapter on Constructionism from the Cambridge Handbook of the Learning Sciences, I couldn't help but think about James Greeno's chapter from the same text. They present two different, but compatible, theories of learning. (To be clear, I mean "theory" scientifically, as a way to explain data collected by experimentation or observation, not "theory" colloquially, as an idea or thought one has about something.) It felt to me that Greeno was writing about learning as participation in activity and Kafai was writing about learning as acquiring knowledge through construction. I remembered another piece I read a while ago by Anna Sfard where discussed learning through the participation and acquisition metaphors and cautioned against privileging one over the other. I don't think that Greeno or Kafai (at least in these pieces) was trying to say that their individual learning theory was superior to any other one. However, it is tempting to try to explain all learning through one lens... even though I think it's pointless.

I know I shouldn't pick on this, but Kafai's chapter seemed less like a survey of constructionism as it did a homage to the work of Seymour Papert. That said, it was an interesting read. I don't think I ever really considered the differences between constructivism and constructionism. It's easy to say that constructivism is a theory of knowledge while constructionism is a theory of learning, but that doesn't really explain the difference. At the core, constructivism is descriptive (what are the stages of knowledge development) where constructionism aims to be prescriptive (what teachers can do to induce learning). You can't teach a little kid object permanence. Either he has or he doesn't. But you can give a little kid Logo and help them figure out how to draw a square. 

Chapter 6 from How People Learn was also an insightful read. At the beginning, I was fearful that it was going to advocate for a Learner-Centered environment over a Knowledge-Centered environment. Then the authors introduce Assessment-Centered and Community-Centered environments and I got all confused. I found it satisfactory to learn that the authors were saying that all four of these centers should be present in any kind of learning environment. I've always found that a good theory should helpfully explain what I already know. These four things seem so obvious now that they've been named. A good teacher will try to figure out what students bring to the classroom and leverage that information into the design of the environment (learner-centered). She will also make sure that norms and ways of knowing and core concepts of the discipline are a major component of the course work (knowledge-centered). Of course, the teacher needs to figure out what students have learned and must give students feedback about what they still need to learn (assessment-centered). And the classroom is a community located within and alongside other communities. The teacher has to take that into account when designing activities (community-centered). I think I try to do these things but I have to be more intentional about them. 

Personal aside: Since I've read a few different pieces about computer programming (Kafai's Constructionism chapter and Resnick et al's short piece on Scratch) I've become even more convinced that including some programming in my math classes could enhance student learning. I just have to figure out how to incorporate it. I was the kid who learned to program in Basic in elementary school and I tinkered with Logo in an after school program. I taught programming early in my career and I really wish I had kept up with it. My skills are rusty, but I think I should pick it up again. 

Thursday, April 23, 2015


I like to think that I have a pretty good working knowledge of the main learning theories - behaviorism, cognitivism, situativity - but it's always good to get a little refresher on them. James Greeno's excellent synthesis of the situative research perspective in the Cambridge Handbook of Learning Sciences made me think about my classroom environment and how students might be interacting within it.

My big take away from Greeno's piece was that "[s]ituativity is a general scientific perspective and as such does not say what educational practices should be adopted". I thought that Greeno was implying throughout the piece that you cannot create a situative classroom or situative learning environment. Rather, situativity is an analytical framework that researchers in the learning sciences can use to try to understand how people interact and learn within some kind of system. That system could be a classroom or a school, but it does not have to be either of those.* 

However, since I work in a school and teach classes in that school, I started to wonder about how my classroom environment, or any classroom environment, could be analyzed or examined through a situative lens. It seems to me that the classroom is a kind of activity system and that what students learn about content of the class is intricately linked to the learning environment. How students interact with mathematical concepts, with each other, with the teacher in my classroom might be is almost certainly different than how students are expected to to behave in other classrooms. Can differences in the classroom system explain or account for differences in student achievement? I think Greeno might say yes. 

Furthermore, classroom activity systems exist within the larger school activity systems. What happens when a single classroom system rejects the prevailing norms of the larger school system? Is that a sustainable situation for the teacher and the students? What will students learn about the content of the class? Will students be able to participate within the systems authentically? I guess I'm thinking about how many teachers design their instruction in such a way that students are positioned as passive recipients of knowledge where other teachers expect students to actively make sense of the subject matter. Do students find it difficult to switch between these environments?

A situative analysis could also helpful in framing the short article on the Scratch programming language. The activity system has been constructed in such a way that participants can interact with the language individually or they can make their creations public to garner feedback from a wider audience. The threefold goals of tinkerability, meaningfulness, and social interaction can be viewed as providing learners opportunities to engage in authentic problem solving and inquiry in much the same way that professionals do. Even if programming is not a career goal, making sense of problems and persevering in solving them is a goal most educators can support.

As an aside, I believe strongly that we teach much too much computation in high school mathematics classes. We expect students to be able to do algebra that a computer can do much better. As such, we never get to the analysis and problem solving that (it seems) everyone values more than mathematical facts. Programming has felt to me (for a long time) like a nearly perfect way to incorporate critical thinking and problem solving into math class. The real mathematics is in developing the algorithm to solve a family of problem not from executing the algorithm flawlessly on a contrived assessment. I'm not sure that I can get enough student buy-in but I want to try. 

*Greeno passingly referred to Lave and Wenger's work. They studied some apprentice-like situations (butchering, midwifery) and posited the existence of a construct they called "legitimate peripheral participation". This means that novices to a system perform roles that are outside the core practices of the system but are necessary for the system to run effectively. The novices gradually take on more responsibility as they become full members of the activity. 

Wednesday, April 22, 2015

Can schools meet the challenges of the digital revolution?

After reading a few things about emerging technologies and what the future world might look like, I'm not sure. As a public school teacher for over 15 years, I know that schools in their current state are not the answer. They often do not serve their populations well. But, I'm not ready to abandon schools just yet.

In an interesting piece by Collins and Halverson, they lay out an argument for why schools reject technology and the digital revolution. There were a few parts of the argument that struck me as very true and some that struck me as not particularly current. The piece is relatively old by digital standards, being published in 2009. I'm sure that in 2009 many facets of the piece were more true than they are now. (In 2009, I didn't have a SmartBoard in my classroom. Now, I can't imagine teaching without one.)  Perhaps Collins & Halverson need to write a follow-up documenting what schools have and have not done to incorporate digital tools into classrooms.

Here are my major disagreements with Collins & Halverson's argument: (1) They put a lot of faith in the ability of computers to respond to the interests of students. I'm not sure that their faith has borne itself out yet. Computers are very good at some things. Interacting authentically with a human is not one of them. No computers have yet been able to pass the Turing test, which is widely accepted as the bar for human-like behavior. Simulations are realistic, but always programmed not authentic. Games can have open-world features but are governed by algorithms; they are not truly interactive. Computers can deliver content but they are not very good at providing personalized feedback. (2) One part of their argument entails giving learners, primarily children, more significant control over what they learn. Obviously children should be encouraged to follow their passions. However, over a century ago, John Dewey warned against the dangers of a curriculum that was too child-centered. Children do not know enough of the world to make the most informed choices. Schools, teachers, parents have a responsibility to expose children to different facets of how we make sense of the world. I know that anecdotes should be taken lightly, but I know that I would never have pursued mathematics without the guidance or the excellent high school math teachers who inspired me with their passion for the subject. (3) It was only a small portion, but the idea that in adult life one rarely needs to have readily available "knowledge in the head" rang hollow for me. When you go to buy a car, do you want the salesperson to know the facts about different models or should they read them to you from the brochure? I can read the specs online myself! I came to the dealership to interact with a person. Would you feel good about an English teacher who needed to check the text of Romeo & Juliet to figure out which one was a Montague and which was a Capulet? Certainly not. I want my physician to know some things about what might be causing my fever and cough even if he/she has to check his/her computer for a backup diagnosis. Internalize knowledge is still important, but so is knowing how to use resources to support your work.

I agree with a lot of what they said. Here goes: (1) Schooling needs to change from knowledge acquisition to learning to learn. This is absolutely true. However, I think they failed to take into account that this kind of schooling is different from what parents experienced. That means that parents will be expected to support their children in a type of education with which they are unfamiliar. The issues with learning to learn is probably most prevalent in mathematics. Many new elementary school mathematics programs focus on ways of thinking instead of fast fact recall. You only have to scroll through Facebook to see the objections to the "new math". Yes, parents object to teachers who want children to think mathematically and construct viable arguments and use multiple representations instead of memorizing facts. Crazy, but true. (2) Math teachers need to stop teaching computation skills and need to teach thinking skills. There are some substantial objections to the Common Core State Standards, but they did put out the Standards for Mathematical Practice. These are ways of thinking and acting that are prevalent in the mathematical sciences but also cut across disciplines. It is far more valuable to have students analyze the conclusions drawn from quantitative data than it is to have them graph a context-free parabola. Mathematics can be co-opted by people who understand it, or who think they understand it, to intimidate people who don't. That's a dangerous situation. 

Can schools meet the challenges of the digital revolution? I don't know. But I hope that schools can make some changes to content and format and pedagogy so that they remain relevant places of learning far into the future. 

Thursday, March 26, 2015

A Silver Bullet for Math Teaching?

This school year, I had the opportunity to step away from the classroom and move into a district level teacher leadership position. I teach one class in the morning, and then I visit K-12 math teachers around the district for the rest of the day. Or I attend meetings and provide input about the needs and desires of the math department. Or I plan professional development sessions for the middle school and high school math teachers. And I'm supposed to do some teacher coaching too.

It sounds kind of great, doesn't it?

I do not like it. There are a lot of reasons why but I'm not writing about that right now. Maybe another time. What I do want to reflect on now is the challenges of how to respond when teachers ask me questions about teaching. 

A few days ago, I was at a school and stopped in to see a teacher I had visited before. We have a pretty good rapport and she said hello when I walked in. Mind, I hadn't been to the school since November. I saw that all of her students were on computers working on various math tutorials. I've been in the room for about 30 seconds and then this conversation takes place:

Teacher: So, do you have any advice?

Me: About what?

Teacher: About what we can do to teach math better?

Me: Um...

At this point, I think I mumbled something about not knowing what they were doing or how the year was going or that I couldn't possibly give any feedback since I had only just stepped foot in the classroom. The weird thing is, she asked me the same question during lunch, "Do you have any suggestions for what we can do to teach better?" Again, I had to mumble something and try to avoid the conversation. What could I possibly say?

It's taken a few days of unpacking and reflecting, but I think I know why the question has stuck with me. I should also put out there that I think the question was an earnest attempt to engage me in a conversation about pedagogy but nonetheless, two things about it has bothered me a bit. First, I think the question betrayed her belief about her own teaching. She believes that her mathematics pedagogy is not good and is in need of repair. Perhaps she was being self-deprecating; perhaps she was telling me what she thought I wanted to hear; perhaps she was trying to learn something. It is entirely possible that she was really asking for help to improve her practice but didn't really know how to ask. I'd be happy to do some coaching with her, if that's what she really wanted but I suspect that is not what she wanted.

The second thing about the question is more important to me. It felt to me like the teacher expected some kind of silver bullet. She was hoping for a quick fix. She expected me to have some new technique that she had never heard before that would turn her instruction on its head, be easy to implement, and raise standardized test scores in weeks. I have no such tricks. I doubt those tricks exist. My suggestions for how to improve your math teaching isn't popular or easy. Here goes:

1) Make sure you know the math - Seems obvious, right? It's not. Read the textbook. The student edition AND the teacher edition. Work the problems. Work all of the problems. Don't skim the notes to the teacher. Most modern textbooks for elementary teachers (and middle and high school too) have pretty substantial notes to the teacher that alert the teacher to student misconceptions and tips for teaching. The best books also explain why a particular topic is important and how that topic fits into the overall curriculum plan. It's also worthwhile to plan with your colleagues and collaborate on the problems and explore the mathematics. Learn different approaches and algorithms and models. Be sure that you can attack a problem from multiple angles. Be able to respond to students' questions about "why" with something more than, "that's the rule". Make sure you know the math.*

2) Make connections with kids - Seems obvious, right? It isn't. I suspect that we all knew or had a teacher that seemed to hate kids. You know the ones: the teacher who snaps at kids when they have a question, the teacher who talks down to children who are just trying to learn, the teacher who blames a child for his inability to keep up with the rest of the class, the teacher who thinks that some kids just aren't capable of learning math. The list could go on and on. I have a lot of colleagues who spend their day talking at kids, a.k.a. lecturing, and them complain that kids don't pay attention and they haven't learned any math. I say, of course kids haven't learned. You need to engage kids where they are and nudge them towards understanding mathematical concepts. If you leave work bothered because your students have not complied with your requests, you need to ask yourself why the students didn't comply. Maybe they didn't understand the instructions. On the other hand, if you leave work every day wondering what you could do differently to help them learn more and better math, that's a sign that you're trying to improve. Keep at it. Talk to colleagues about it. But not the people who hate kids.  

3) Only scratch when you have an itch - Every single mathematical concept or algorithm was developed as the answer to a question. A lot of times, we teach a concept or procedure before kids consider why anyone would want to know that. Think about these different ways to start a unit on fractions: "Today we're going to start talking about fractions. This can get tricky so make sure you pay attention." versus "I was wondering, are there numbers between numbers?" Which one creates an itch? One of these openers will almost certainly deviate from the district sanctioned pacing guide but might result in deep and meaningful understanding of fractions. When we don't ask any questions, we essentially tell kids what to think. I sometimes like to call this, "opening everyone's presents for them". Even worse, we give students the false impression that mathematics has no human aspect, that the rules of mathematics are irrefutable because they have been gifted to us from the Creator, that mathematics is done by memorizing and acting with certainty and not by understanding and exploring. I'm not saying that we need to give kids a practical, real-world example of how mathematics is useful. Sometimes there are physical situations that lend themselves to mathematics. Sometimes we try too hard to manufacture a scenario where impose mathematics that no one would actually do. A question about an interesting phenomenon can trump a physical situation. Think about which one will create a better itch to scratch.

4) Be patient - This one seems especially obvious but it is probably the least apparent suggestion. You have to understand learning theories. You have to remember that kids learn at different paces and in different ways. Kids will learn math. But, it might not be on the timetable that the textbook or the school district give you. It will be stressful for you and for them. But you have to embrace the stress and not let it consume you. Most of all, you have to rid yourself of the noxious belief that some kids can do math and other kids cannot. I firmly believe that all kids can learn math. (Excepting children with severe developmental disabilities.) If a strategy you find appealing isn't working then you need to try another one. Use a different kind of representation. Get a concrete model. Draw a picture. Use base-10 blocks. Don't resort to giving the child an algorithm before he or she is ready for it. You're not helping when you prematurely move him to the finish. She needs to get there on her own with your support and guidance. He will never get there if you don't give him adequate time to play, experiment, and figure things out.

These are not glamorous suggestions, but I think they are true. More than my own belief about teaching, I think we can find pretty solid research for each of these suggestions. It's not an easy path to take, but it can work. There is no silver bullet for teaching math. Teaching math well entails knowing the subject and how to represent it in a variety of ways, connecting with students and their individual learning progressions, and asking questions instead of giving all the answers. It's hard work... and I love working at it. 

*By the way, it's still pretty much an open question about how much math you need to know to teach it well. There is a very robust area of research devoted to this very topic. Many scholars have worked for many years investigating a construct called "Mathematical Knowledge for Teaching". Still, we don't know exactly what comprises MKT. 

Monday, March 9, 2015

Why all the buzz around "flipping" the classroom?

I have a problem with the flipped classroom movement.

OK, maybe not the movement itself, but I do have a problem with the usual justifications used by the flipped classroom movement. Every single thing I have ever seen about the flipped classroom movement has stated that the main reason to flip is so that students can watch videos of the content at their own pace and come to class prepared to work problems or investigate further. This approach leaves the teacher free to work individually with students.

I don't even know where to begin, but I'll try.

First, the entire movement seems to be predicated on the notion that lecture is the primary instructional method. It's 2015, people! If you're lecturing more than you're engaging kids in conversation about mathematical ideas or discovering interesting properties of mathematical structures or undertaking problem solving, then you're probably not a very good math teacher. There, I said it.

Second, is the ridiculous idea that lecturing is the reason why you can't work individually with students. If you can't find time during class to address a particular student's need, then you're probably not a very good math teacher. There, I said it again.

Third, the videos that are often peddled to math teachers are nothing more than how-to's. These videos tell kids how to get answers. They do not help kids learn how to think mathematically. If all you're doing in math class is telling kids how to perform procedures, not getting them involved in the solutions and not motivating the algorithms with the interesting and vital historical questions that accompany the problems that the algorithms were invented to solve, then you're probably not a very good math teacher.  Yup, I said it a third time.

OK, I'll soften a bit. I can see where a flipping aspects of the classroom might be beneficial. You might want to show students a short video to get them interested in a topic. It could be worth having students view the content ahead of time and then take all of class time to explore. That I get. But, is the flipped classroom really all that different from what we expect in other disciplines? In literature, students read a section of a novel and then come to class prepared to discuss it. That's not flipping the classroom, that's just good teaching. In history, the same thing should happen. Maybe it doesn't, but it should.

I don't have any problem with students preparing in advance; that's a very good thing. I don't have any problem with not talking at kids; that's a good thing too. You have to get kids involved in learning! It's not about flipping the classroom; it's about good teaching.

Sunday, March 1, 2015

The future classroom

I've been trying to envision what my classroom might look like in 10 years. It occurred to me today that I have absolutely no idea what might be on the horizon for me. But there are some things that I'd like to happen. 

First, I think there's a temptation to project physical attributes into the classroom of the future. My ideal classroom has looked pretty much the same in my mind for many years. I would rather have tables and chairs instead of desks and incandescent lights to replace the fluorescent ones. Maybe a wall color that isn't institutional beige and a floor that doesn't look like in belongs in a hospital. Books, games, puzzles, video game consoles, computers, outlets for charging hand-held devices, and some kind of interactive projection technology. Basically, I want a Muggle version of the Gryffindor common room (or, a hipster coffee shop).

The physicality of the space is a pipe-dream; it's never going to happen and I know that. The changes to the environment are simply the window dressing for the kind of education with which I'd like to be involved. I want students to collaborate more. They should work on projects that are compelling and authentic. Not necessarily practical or real-world, but interesting.*  Students can consult outside sources, in real time or on video, to work on their questions. They can make a record of what they've accomplished, either digitally or manually, so that they know where they've come from and where they might go next. The engage with different kinds of tools to help them make progress in their task. Maybe the tool is a dynamic geometry suite or maybe it is hand-held graphing technology or maybe it is pencil and paper. The tool chosen assists in the task completion. They present their work to their peers, have it constructively and good-naturedly critiqued and then they revise it to make it better.  

In the best kind of environment my students will near the completion of one task and realize that they have developed more questions to pursue. Wouldn't that be amazing?

*An awful lot of mathematics was done for purposes of curiosity and only later took on some usefulness. The computer was a thought experiment before someone built one and realized its utility.

Sunday, February 22, 2015

Mathematical Knowledge Space

I've been thinking a lot about Linear Knowledge Space and Random Knowledge Space. (Incidentally, the term "Random" seems to connote haphazard and unconnected, and I don't think that's the point. If I come up with a better term, I'll put it out there.)

Linear Knowledge Space seems pretty self-explanatory. Start at the beginning and proceed to the end. This is certainly the most common kind of presentation of mathematics seen in schools. Orderly and hierarchical. Step 1, Step 2, etc. And don't dare miss any steps!

This view of mathematical knowledge is the one that permeates the culture. It is popularly believed that "math people"* are rigid and linear and excessively rational in their thoughts and actions. School mathematics often gets framed, or positioned, in these linear knowledge spaces. The conventional wisdom says that in order to learn section 12.5 of the textbook, you must have first mastered chapter 1-11 and sections 12.1-12.4. But, where is the evidence that this is necessary? Check any two textbooks for the same course (at any level) and you will see that there are differences. These differences could be minor, like the order of the topics, or they could be major, such as the philosophical stance on mathematics that the authors take. Regardless, differences exist. If mathematics was linear, there would be only one line!

History could be our guide here. Mathematics, like all other human endeavors (and mathematics is as susceptible to humans foibles as any other discipline) is written by the victor. If history of mathematics can teach us anything, it should be that mathematics was developed in fits and starts, through argument, experimentation, and collaboration. A question was posed and people tried to answer it. Newton and Leibniz knew an awful lot about Calculus and they didn't have any idea what a "limit" was. Weierstrass comes along (about 100 years later) and defines "limit" to everyone's satisfaction, despite a century of doing calculus without that definition. What is the first chapter of pretty much every major calculus textbook? Limits. There is absolutely no historical reason for it. We force students to learn linearly that which does not need to be linear. What we need is better questions to answer.

"But", you say, "But, certainly there are topics that have prerequisite knowledge." Yes, of course there are. It's pretty hard to learn how to multiply if you are sketchy on what it means to add. However, the analogy that learning mathematics is like building a tower is fundamentally flawed. Learning mathematics is more like building a city. Yes, there are towers. But there are expansions to the towers and there are other buildings, and some of those buildings have additions. And there's an infrastructure of roads, and maybe even tunnels, that connect the various parts of the city. Put another way, mathematics is less like a novel and more like short stories that take place in the same universe. Do you have to have seen episodes 1-99 of "Friends" before watching the 100th one? Certainly not. Could it help you understand something different about the characters if you have seen episodes 1-99? Quite possibly.

Bob and Ellen Kaplan, whom I consider mentors, have said that when teaching math, if you want the children to learn topic A then you should ask them about topic B where A is necessary. I think this is the "sweet spot" that instruction needs to hit. There are prerequisites, but many fewer than we think there are. Mathematical Knowledge Space isn't necessarily Linear, but it's not completely Random either. It doesn't matter if the instruction is face-to-face or online. What matters is that there is a compelling question, some strategies to answer it, and collaboration. What turns it from the answer to a specific problem into mathematical knowledge is connecting the specific answer to answers of other problems.

*Frankly, that there are "math people" and "non-math people" is a ridiculous notion.

Tuesday, February 10, 2015

The most important reason to understand online teaching and learning is...

... because they exist.

As our lives become more and more technologically integrated it becomes vital to understand how people consume, use, and generate information. That is, what do we learn in this increasingly digital age, and how do we learn it.

How many times has this happened: you are out with friends and you will be involved in a conversation where some details are missing. For example, when a particular event started, or what years a TV show aired, or the name of an actor in a movie, or the title of a sing or book, or how to convert from cups to gallons, or whatever. After a few minutes of banter and deciding that you do not know the answer to the question, someone says, "if only there was a way that we could figure that out". At which point several people take out their phones, retreat into the internet for a few minutes, and emerge with some facts that address the question. 

This is the most obvious (and maybe most prevalent) kind of online learning. The sort where facts can be found relatively quickly. I'm not sure how much teaching is happening in the above scenario. The question I wonder about is whether this fact acquisition is the only type of learning that can happen in online environment. A search engine is much better than a human at coming up with facts. But, a computer is not very good at synthesizing information or asking good questions. It is not even very good at so-called adaptive instruction. (Technically, the program/website/app is not reacting at all. It is only following an algorithm designed by a human.)

My questions are: How does a human teacher use technology to enhance learning? How does a teacher use digital technologies to move beyond fact acquisition and procedural knowledge?