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.