3 - Teaching

Though I hate being forced to say it, I believe I am an excellent teacher. Continued improvement requires healthy humility, but when I consider my teaching path I am happy, though not content, with where I am. When I was happy saying I was an excellent teacher I wasn't half the teacher I am now. My improvement has come as a direct result of working with excellent colleagues here, having opportunities to plan or teach together, and sharing frequent, immersive discussions of teaching. Though lately our departmental needs have strongly dictated what I teach, I have a broad variety of teaching interests and have taught Math 096, 097, 110, 202, 210, 227, 327 and 341 in addition to the more math ed flavored Math 221, 222, 322, 229, 329, 603, 629, 641 and Ed 331. I've led or shared independent studies in 408 and 495 also.

I write the vast majority of the activities that I use in class, and my classes are designed to be places of active learning. Most of the other activities are modifications of colleagues' work. Students work in groups, usually with options and choices to involve them in choosing their direction. I structure classes using a workshop model, that begins with students making connections, followed by direction or focus from the teacher, the bulk of the class in groups working cooperatively on a problem or problems and ending in a reflection period that may be individual, group or whole class.

Two structures that greatl

The structure I often use to evaluate students is inspired by Vygotsky's model of the zone of proximal development. Information on what the students can do is important because their new learning has to build on what they can do, and research shows that knowledge that connects to previous knowledge is longer lasting. Information on what students are ready for now is crucial to planning for instruction and knowing when to move on. Information about what the students can not do yet is helpful in terms of planning demonstrations and plotting a course towards the material that is currently a struggle.

I also evaluate students' work using a communication rubric. I'm looking for work that is:

• Clear

• Coherent - moves towards the objective in a sensible way

• Complete - meets all the requirements of the workshop or problem

• Consolidated

  • spontaneous work: includes reflection that ties work together

  • planned or revised writing: exhibits structure suitable to the assignment

• Content - demonstrates understanding of mathematics or pedagogy from the objective

In Standards Based Grading I use a rubric that evaluates the understanding demonstrated, which encourages explanation.

The area of evaluation I'm working on the most right now is how to write constructive feedback that is focused and valuable to the student. This is the crucial condition of learning of response. One of the reasons I am moving away from grades as feedback is that it is of little use to the student in making improvements. This also includes being sure to comment on areas of strength. Specific positive feedback is something many students have not experienced in a K-12 math classroom. Timeliness of feedback is important also. I have tried to address students' past concerns about slow turnaround by better structuring assignments, managing due dates amongst all my courses and making assignments more interesting for me to read. I am greatly curious about student approaches to problems and portfolio assignments, and they are often a pleasure into which to delve. More frequent SBG assessments are also quicker to turn around than longer tests.

Planning

I plan for courses using backwards design. First think about the goals and objectives, then think about how I can assess whether they have met these objectives. Then I look for a logical sequence of activities to equip the students to meet the objectives. For math ed classes this includes thinking about which pedagogical topics will pair best with which content topics, but even in a pure math class I like to consider the five mathematical processes from the National Council of Teachers of Mathematics (NCTM) and when would be best to focus on which process. No one would accuse me of over-planning a semester. The positive side of this is that I can respond to students' understanding by moving on sooner than I would have thought possible or dwelling longer than I would have guessed necessary. The negative side of this has sometimes been perception of a lack of organization from students, though not much recently. One of the benefits of math education courses is that it is germane to share my teacher thinking about scheduling and planning, so I do this and also clearly give students an idea of the upcoming schedule. The use of daily agendas in class has also helped organize me and the students.

I am prone to taking leaps that I don't always know how they will work out. In fall 2010 one class came in while they were filming the Grand Valley lipdub. This led to a class discussion, showing a few lipdubs, and analyzing why students would be willing to put forth so much effort for a project like that, when the conventional wisdom is that students are lazy and unwilling to work. A great discussion that led to a request to make our own class video. We had strong support of about 90% of the class with the rest quite against it. We worked as a class towards making it acceptable to all, and the preliminary work was done in choice workshops. But not enough was done ahead of time, and as the day of the class drew near I was quite depressed about how much effort and course time went towards something that was going to be a flop. But when the filming day came (and one of the two cameras failed) it all came together. The video was fun, but not the most significant part to me. It became a project that was the students', and changed their perception of our course and, for several of them, what they envisioned doing with their students.

I plan for individual lessons by looking for problematic situations that offer a good opportunity to address the objectives. This is about engagement and immersion to me. Taking into consideration the information from the students' informal assessments, I try to make the activity in the sweet spot of Vygotsky's center zone. I usually err to the side of too challenging, but that can be rectified on the spot with more instructional support. I do try to build in extensions and open-ended problems for groups that need more challenge, but I'm willing to try that on the spot also. For the most part I use heterogenous groups so that there is usually a need for table discussion for everyone to understand. I will use homogenous groups when doing centers or when there is a particular group of students with whom I want to work directly myself. Most often homogenous groups come about when students have a choice of activity and rearrange themselves by their choices. I typically have more planned for a class period than could possibly fit in, which can get me into trouble if I try to fit it all in.

Instruction

I love being in the classroom, with students of any age. From preschool through teachers with decades of experience. The problem of creating the conditions of learning for a diverse group of students is endlessly fascinating to me and deeply rewarding.

Both of the classroom observations capture pretty typical classroom periods for me - though we're not always that successful in hitting the learning objectives. The reports capture the cooperative nature of class, the emphasis on active learning and how much I value reflection and discussion. What they don't capture, being in a late-semester class and a graduate class of pretty independent learners, is how much I strive for that independence and cooperation as the semester goes on. It is definitely not always successful with every class, but every class does make progress. I struggle responsibility as a condition of learning, because students have so little experience being in charge of their own learning. The workshop structure has been invaluable for this, as it is a flexible structure that allows me to loosen the reigns while maintaining a continuity of form. Students recognize the value of reflection for consolidation of learning and the vast majority get better at it throughout the semester.

Two common features of my classroom that can be a bit unusual are games and technology use and games.

Technology

Of course, technology in and of itself is not unusual, especially in our department at our university. But I do probably involve it more than most, because I am interested in it as learner support, barrier removal and activity sponsor. As learner support, the ability to automate formerly time-consuming tasks has a dramatic impact on class time use and student ability to make up for deficiencies. For example, graphing calculators allowing us to talk about what we learn from graphs instead of using all our time just making them. Or students being able integrate a function to solve a problem by computer even if they do not recall the by hand method. Barrier removal is related - technology like GeoGebra or Wolfram|Alpha (links to the programs) makes so much more mathematics available to the student. I find technology often to be an invitation to activity because of this. Whereas a student might feel intimidated by the mathematics the technology is appealing and enabling. Or it enables them to make contributions that are more lasting; for example, my 229 students are building a wiki for high school math teachers.

Teaching future teachers raises other issues because technology should impact what we teach also. Do we need to teach taking square roots by hand or using logarithm tables? No. Should we teach how to graph functions or integration techniques? Less clear. And that's just the technology we have now. Five years from now is difficult to even imagine. Twenty five years? Impossible. So I feel a responsibility to teach students how to learn how to use new technology as well as the technology itself.

There is a glog of my technology use (a kind of internet poster). I regularly use (meaning have students use) Twitter, blogging, wikis, Google Apps, Wolfram|Alpha, GeoGebra and more. Of course, this means I do actually use these things also

Games

I value games for engagement, skill practice and context. Even in classrooms where educational games are more common, student interest jumps when a game is introduced. In classrooms where they are uncommon, it can be shocking to the teacher just how much student interest jumps. At all levels there are skills that are necessary for mathematics, and games will keep learners practicing longer than they would without. As these move from clunky paper games to neat electronic ones, this will be an even better effect. Games also make an interesting context for mathematics, and I will use them to get students to explore a mathematical structure or to gather data to analyze later.

Professional Development

I feel like I am constantly improving as a teacher; as my assessment improves it reveals that more of my students are achieving more of what I care about. The main sources of this are my colleagues and the internet. Work with my colleagues has been crucial, and having a department with such amazing teachers has been a real gift. Some of those I've learned the most from include (but are definitely not limited to) Char Beckmann, Esther Billings, Steve Blair, Dave Coffey, Georgi Klein, Gary Klingler, Jan Shroyer, Rebecca Walker, Clark Wells, Pam Wells, Matt Wyneken, Paul Yu and more; these are the people with whom I can remember specific conversations that impacted my teaching. I do have to single out Dave for being a particularly challenging and frequent source of growth. In addition, connection to other teachers through blogging and Twitter has been amazing. I started blogging as a way to record and link to resources I found valuable, and that drew me into a fascinating collection of writing by teachers and mathematicians who shared their thinking generously and authentically. The reading recommendations on my blog give a sense of this material. I began using Twitter because one of those writers, Maria Anderson, described how it impacted one of her online classes. I thought that what she described would be very helpful with some of the challenges we have with student teacher assistants, whom we see infrequently. I always try to pilot technology before using with students, so I signed up for Twitter, feeling a little silly. However, it has been great. The only problem is really too many connections, conversations and resources are available and it's easy to fall down the rabbit hole. (Some of my most recent favorited tweets.)